id	sid	tid	token	lemma	pos
ejpam-6940	1	1	european	european	PROPN
ejpam-6940	1	2	journal	journal	PROPN
ejpam-6940	1	3	of	of	ADP
ejpam-6940	1	4	pure	pure	ADJ
ejpam-6940	1	5	and	and	CCONJ
ejpam-6940	1	6	applied	applied	ADJ
ejpam-6940	1	7	mathematics	mathematic	NOUN
ejpam-6940	1	8	2025	2025	NUM
ejpam-6940	1	9	,	,	PUNCT
ejpam-6940	1	10	vol	vol	NOUN
ejpam-6940	1	11	.	.	PROPN
ejpam-6940	1	12	18	18	NUM
ejpam-6940	1	13	,	,	PUNCT
ejpam-6940	1	14	issue	issue	NOUN
ejpam-6940	1	15	4	4	NUM
ejpam-6940	1	16	,	,	PUNCT
ejpam-6940	1	17	article	article	NOUN
ejpam-6940	1	18	number	number	NOUN
ejpam-6940	1	19	6940	6940	NUM
ejpam-6940	1	20	issn	issn	VERB
ejpam-6940	1	21	1307	1307	NUM
ejpam-6940	1	22	-	-	SYM
ejpam-6940	1	23	5543	5543	NUM
ejpam-6940	1	24	–	–	PUNCT
ejpam-6940	1	25	ejpam.com	ejpam.com	X
ejpam-6940	1	26	published	publish	VERB
ejpam-6940	1	27	by	by	ADP
ejpam-6940	1	28	new	new	PROPN
ejpam-6940	1	29	york	york	PROPN
ejpam-6940	1	30	business	business	PROPN
ejpam-6940	1	31	global	global	PROPN
ejpam-6940	1	32	on	on	ADP
ejpam-6940	1	33	edge	edge	NOUN
ejpam-6940	1	34	q	q	NOUN
ejpam-6940	1	35	-	-	PUNCT
ejpam-6940	1	36	algebras	algebras	ADJ
ejpam-6940	1	37	ananya	ananya	PROPN
ejpam-6940	1	38	anantayasethi1,∗	anantayasethi1,∗	PROPN
ejpam-6940	1	39	,	,	PUNCT
ejpam-6940	1	40	kittisak	kittisak	PROPN
ejpam-6940	1	41	saengsura1	saengsura1	PROPN
ejpam-6940	1	42	,	,	PUNCT
ejpam-6940	1	43	yeni	yeni	PROPN
ejpam-6940	1	44	susanti2	susanti2	PROPN
ejpam-6940	1	45	,	,	PUNCT
ejpam-6940	1	46	napaporn	napaporn	ADJ
ejpam-6940	1	47	sarasit3,∗	sarasit3,∗	NOUN
ejpam-6940	1	48	1	1	NUM
ejpam-6940	1	49	department	department	NOUN
ejpam-6940	1	50	of	of	ADP
ejpam-6940	1	51	mathematics	mathematic	NOUN
ejpam-6940	1	52	,	,	PUNCT
ejpam-6940	1	53	faculty	faculty	NOUN
ejpam-6940	1	54	of	of	ADP
ejpam-6940	1	55	science	science	NOUN
ejpam-6940	1	56	,	,	PUNCT
ejpam-6940	1	57	mahasarakham	mahasarakham	PROPN
ejpam-6940	1	58	university	university	PROPN
ejpam-6940	1	59	,	,	PUNCT
ejpam-6940	1	60	mahasarakham	mahasarakham	PROPN
ejpam-6940	1	61	,	,	PUNCT
ejpam-6940	1	62	44150	44150	NUM
ejpam-6940	1	63	,	,	PUNCT
ejpam-6940	1	64	thailand	thailand	PROPN
ejpam-6940	1	65	2	2	NUM
ejpam-6940	1	66	department	department	NOUN
ejpam-6940	1	67	of	of	ADP
ejpam-6940	1	68	mathematics	mathematic	NOUN
ejpam-6940	1	69	,	,	PUNCT
ejpam-6940	1	70	faculty	faculty	NOUN
ejpam-6940	1	71	of	of	ADP
ejpam-6940	1	72	mathematics	mathematic	NOUN
ejpam-6940	1	73	and	and	CCONJ
ejpam-6940	1	74	natural	natural	ADJ
ejpam-6940	1	75	sciences	science	NOUN
ejpam-6940	1	76	,	,	PUNCT
ejpam-6940	1	77	universitas	universita	NOUN
ejpam-6940	1	78	gadjah	gadjah	PROPN
ejpam-6940	1	79	mada	mada	PROPN
ejpam-6940	1	80	,	,	PUNCT
ejpam-6940	1	81	yogyakarta	yogyakarta	PROPN
ejpam-6940	1	82	55281	55281	NUM
ejpam-6940	1	83	,	,	PUNCT
ejpam-6940	1	84	indonesia	indonesia	PROPN
ejpam-6940	1	85	3	3	NUM
ejpam-6940	1	86	division	division	NOUN
ejpam-6940	1	87	of	of	ADP
ejpam-6940	1	88	mathematics	mathematic	NOUN
ejpam-6940	1	89	,	,	PUNCT
ejpam-6940	1	90	faculty	faculty	NOUN
ejpam-6940	1	91	of	of	ADP
ejpam-6940	1	92	engineering	engineering	NOUN
ejpam-6940	1	93	,	,	PUNCT
ejpam-6940	1	94	rajamangala	rajamangala	PROPN
ejpam-6940	1	95	university	university	PROPN
ejpam-6940	1	96	of	of	ADP
ejpam-6940	1	97	technology	technology	PROPN
ejpam-6940	1	98	isan	isan	PROPN
ejpam-6940	1	99	,	,	PUNCT
ejpam-6940	1	100	khon	khon	PROPN
ejpam-6940	1	101	kaen	kaen	PROPN
ejpam-6940	1	102	40000	40000	NUM
ejpam-6940	1	103	,	,	PUNCT
ejpam-6940	1	104	thailand	thailand	PROPN
ejpam-6940	1	105	abstract	abstract	NOUN
ejpam-6940	1	106	.	.	PUNCT
ejpam-6940	2	1	the	the	DET
ejpam-6940	2	2	concept	concept	NOUN
ejpam-6940	2	3	of	of	ADP
ejpam-6940	2	4	an	an	DET
ejpam-6940	2	5	edge	edge	NOUN
ejpam-6940	2	6	in	in	ADP
ejpam-6940	2	7	q	q	NOUN
ejpam-6940	2	8	-	-	PUNCT
ejpam-6940	2	9	algebra	algebra	NOUN
ejpam-6940	2	10	is	be	AUX
ejpam-6940	2	11	introduced	introduce	VERB
ejpam-6940	2	12	in	in	ADP
ejpam-6940	2	13	this	this	DET
ejpam-6940	2	14	work	work	NOUN
ejpam-6940	2	15	.	.	PUNCT
ejpam-6940	3	1	we	we	PRON
ejpam-6940	3	2	explore	explore	VERB
ejpam-6940	3	3	some	some	DET
ejpam-6940	3	4	properties	property	NOUN
ejpam-6940	3	5	of	of	ADP
ejpam-6940	3	6	edge	edge	NOUN
ejpam-6940	3	7	q	q	NOUN
ejpam-6940	3	8	-	-	PUNCT
ejpam-6940	3	9	algebras	algebras	X
ejpam-6940	3	10	.	.	PUNCT
ejpam-6940	4	1	the	the	DET
ejpam-6940	4	2	characterization	characterization	NOUN
ejpam-6940	4	3	of	of	ADP
ejpam-6940	4	4	subsets	subset	NOUN
ejpam-6940	4	5	of	of	ADP
ejpam-6940	4	6	an	an	DET
ejpam-6940	4	7	edge	edge	NOUN
ejpam-6940	4	8	q	q	NOUN
ejpam-6940	4	9	-	-	NOUN
ejpam-6940	4	10	algebra	algebra	NOUN
ejpam-6940	4	11	to	to	PART
ejpam-6940	4	12	be	be	AUX
ejpam-6940	4	13	subalgebras	subalgebras	PROPN
ejpam-6940	4	14	is	be	AUX
ejpam-6940	4	15	provided	provide	VERB
ejpam-6940	4	16	.	.	PUNCT
ejpam-6940	5	1	we	we	PRON
ejpam-6940	5	2	show	show	VERB
ejpam-6940	5	3	that	that	SCONJ
ejpam-6940	5	4	for	for	ADP
ejpam-6940	5	5	an	an	DET
ejpam-6940	5	6	edge	edge	NOUN
ejpam-6940	5	7	q	q	NOUN
ejpam-6940	5	8	-	-	NOUN
ejpam-6940	5	9	algebra	algebra	NOUN
ejpam-6940	5	10	x	x	PUNCT
ejpam-6940	5	11	of	of	ADP
ejpam-6940	5	12	order	order	NOUN
ejpam-6940	5	13	n	n	CCONJ
ejpam-6940	5	14	,	,	PUNCT
ejpam-6940	5	15	there	there	PRON
ejpam-6940	5	16	are	be	VERB
ejpam-6940	5	17	2n−1	2n−1	NUM
ejpam-6940	5	18	subalgebras	subalgebra	NOUN
ejpam-6940	5	19	of	of	ADP
ejpam-6940	5	20	x.	x.	PROPN
ejpam-6940	5	21	moreover	moreover	ADV
ejpam-6940	5	22	,	,	PUNCT
ejpam-6940	5	23	the	the	DET
ejpam-6940	5	24	set	set	NOUN
ejpam-6940	5	25	of	of	ADP
ejpam-6940	5	26	all	all	DET
ejpam-6940	5	27	subalgebras	subalgebras	PROPN
ejpam-6940	5	28	forms	form	VERB
ejpam-6940	5	29	a	a	DET
ejpam-6940	5	30	semigroup	semigroup	NOUN
ejpam-6940	5	31	with	with	ADP
ejpam-6940	5	32	a	a	DET
ejpam-6940	5	33	right	right	ADJ
ejpam-6940	5	34	identity	identity	NOUN
ejpam-6940	5	35	.	.	PUNCT
ejpam-6940	6	1	beside	beside	ADP
ejpam-6940	6	2	this	this	PRON
ejpam-6940	6	3	,	,	PUNCT
ejpam-6940	6	4	the	the	DET
ejpam-6940	6	5	concept	concept	NOUN
ejpam-6940	6	6	of	of	ADP
ejpam-6940	6	7	ideal	ideal	NOUN
ejpam-6940	6	8	is	be	AUX
ejpam-6940	6	9	discussed	discuss	VERB
ejpam-6940	6	10	.	.	PUNCT
ejpam-6940	7	1	we	we	PRON
ejpam-6940	7	2	obtain	obtain	VERB
ejpam-6940	7	3	that	that	SCONJ
ejpam-6940	7	4	the	the	DET
ejpam-6940	7	5	set	set	NOUN
ejpam-6940	7	6	of	of	ADP
ejpam-6940	7	7	all	all	DET
ejpam-6940	7	8	ideals	ideal	NOUN
ejpam-6940	7	9	forms	form	VERB
ejpam-6940	7	10	a	a	DET
ejpam-6940	7	11	left	left	ADJ
ejpam-6940	7	12	zero	zero	NUM
ejpam-6940	7	13	semigroup	semigroup	NOUN
ejpam-6940	7	14	and	and	CCONJ
ejpam-6940	7	15	a	a	DET
ejpam-6940	7	16	simple	simple	ADJ
ejpam-6940	7	17	semigroup	semigroup	NOUN
ejpam-6940	7	18	.	.	PUNCT
ejpam-6940	8	1	finally	finally	ADV
ejpam-6940	8	2	,	,	PUNCT
ejpam-6940	8	3	we	we	PRON
ejpam-6940	8	4	describe	describe	VERB
ejpam-6940	8	5	all	all	DET
ejpam-6940	8	6	possible	possible	ADJ
ejpam-6940	8	7	structures	structure	NOUN
ejpam-6940	8	8	of	of	ADP
ejpam-6940	8	9	edge	edge	NOUN
ejpam-6940	8	10	q	q	NOUN
ejpam-6940	8	11	-	-	PUNCT
ejpam-6940	8	12	algebras	algebra	VERB
ejpam-6940	8	13	and	and	CCONJ
ejpam-6940	8	14	enumerate	enumerate	VERB
ejpam-6940	8	15	all	all	DET
ejpam-6940	8	16	members	member	NOUN
ejpam-6940	8	17	of	of	ADP
ejpam-6940	8	18	a	a	DET
ejpam-6940	8	19	class	class	NOUN
ejpam-6940	8	20	of	of	ADP
ejpam-6940	8	21	all	all	DET
ejpam-6940	8	22	edge	edge	NOUN
ejpam-6940	8	23	q	q	NOUN
ejpam-6940	8	24	-	-	PUNCT
ejpam-6940	8	25	algebras	algebras	X
ejpam-6940	8	26	.	.	PUNCT
ejpam-6940	9	1	we	we	PRON
ejpam-6940	9	2	prove	prove	VERB
ejpam-6940	9	3	that	that	SCONJ
ejpam-6940	9	4	there	there	PRON
ejpam-6940	9	5	are	be	VERB
ejpam-6940	9	6	exactly	exactly	ADV
ejpam-6940	9	7	2n	2n	NUM
ejpam-6940	9	8	2−3n+2	2−3n+2	NUM
ejpam-6940	9	9	edge	edge	NOUN
ejpam-6940	9	10	qalgebras	qalgebra	NOUN
ejpam-6940	9	11	of	of	ADP
ejpam-6940	9	12	order	order	NOUN
ejpam-6940	9	13	n.	n.	PROPN
ejpam-6940	9	14	finally	finally	ADV
ejpam-6940	9	15	,	,	PUNCT
ejpam-6940	9	16	we	we	PRON
ejpam-6940	9	17	show	show	VERB
ejpam-6940	9	18	a	a	DET
ejpam-6940	9	19	connection	connection	NOUN
ejpam-6940	9	20	between	between	ADP
ejpam-6940	9	21	q	q	NOUN
ejpam-6940	9	22	-	-	PUNCT
ejpam-6940	9	23	algebras	algebra	NOUN
ejpam-6940	9	24	and	and	CCONJ
ejpam-6940	9	25	d	d	NOUN
ejpam-6940	9	26	-	-	PUNCT
ejpam-6940	9	27	algebras	algebras	X
ejpam-6940	9	28	.	.	PUNCT
ejpam-6940	10	1	we	we	PRON
ejpam-6940	10	2	obtain	obtain	VERB
ejpam-6940	10	3	that	that	SCONJ
ejpam-6940	10	4	every	every	DET
ejpam-6940	10	5	edge	edge	NOUN
ejpam-6940	10	6	d	d	X
ejpam-6940	10	7	-	-	PUNCT
ejpam-6940	10	8	algebra	algebra	NOUN
ejpam-6940	10	9	is	be	AUX
ejpam-6940	10	10	a	a	DET
ejpam-6940	10	11	q	q	NOUN
ejpam-6940	10	12	-	-	PUNCT
ejpam-6940	10	13	algebra	algebra	NOUN
ejpam-6940	10	14	.	.	PUNCT
ejpam-6940	11	1	precisely	precisely	ADV
ejpam-6940	11	2	,	,	PUNCT
ejpam-6940	11	3	every	every	DET
ejpam-6940	11	4	edge	edge	NOUN
ejpam-6940	11	5	d	d	X
ejpam-6940	11	6	-	-	PUNCT
ejpam-6940	11	7	algebra	algebra	NOUN
ejpam-6940	11	8	is	be	AUX
ejpam-6940	11	9	an	an	DET
ejpam-6940	11	10	edge	edge	NOUN
ejpam-6940	11	11	q	q	NOUN
ejpam-6940	11	12	-	-	NOUN
ejpam-6940	11	13	algebra	algebra	NOUN
ejpam-6940	11	14	.	.	PUNCT
ejpam-6940	12	1	2020	2020	NUM
ejpam-6940	12	2	mathematics	mathematic	NOUN
ejpam-6940	12	3	subject	subject	NOUN
ejpam-6940	12	4	classifications	classification	NOUN
ejpam-6940	12	5	:	:	PUNCT
ejpam-6940	12	6	03g25	03g25	NUM
ejpam-6940	12	7	,	,	PUNCT
ejpam-6940	12	8	03g27	03g27	NOUN
ejpam-6940	12	9	,	,	PUNCT
ejpam-6940	12	10	06f35	06f35	NUM
ejpam-6940	12	11	,	,	PUNCT
ejpam-6940	12	12	20k01	20k01	NUM
ejpam-6940	12	13	key	key	ADJ
ejpam-6940	12	14	words	word	NOUN
ejpam-6940	12	15	and	and	CCONJ
ejpam-6940	12	16	phrases	phrase	NOUN
ejpam-6940	12	17	:	:	PUNCT
ejpam-6940	12	18	q	q	X
ejpam-6940	12	19	-	-	PUNCT
ejpam-6940	12	20	algebra	algebra	NOUN
ejpam-6940	12	21	,	,	PUNCT
ejpam-6940	12	22	subalgebra	subalgebra	NOUN
ejpam-6940	12	23	,	,	PUNCT
ejpam-6940	12	24	ideal	ideal	ADJ
ejpam-6940	12	25	,	,	PUNCT
ejpam-6940	12	26	edge	edge	NOUN
ejpam-6940	12	27	,	,	PUNCT
ejpam-6940	12	28	edge	edge	NOUN
ejpam-6940	12	29	q	q	NOUN
ejpam-6940	12	30	-	-	PUNCT
ejpam-6940	12	31	algebra	algebra	NOUN
ejpam-6940	12	32	,	,	PUNCT
ejpam-6940	12	33	semigroup	semigroup	PROPN
ejpam-6940	12	34	1	1	NUM
ejpam-6940	12	35	.	.	PUNCT
ejpam-6940	13	1	introduction	introduction	NOUN
ejpam-6940	13	2	and	and	CCONJ
ejpam-6940	13	3	preliminaries	preliminary	NOUN
ejpam-6940	13	4	back	back	ADV
ejpam-6940	13	5	in	in	ADP
ejpam-6940	13	6	the	the	DET
ejpam-6940	13	7	period	period	NOUN
ejpam-6940	13	8	of	of	ADP
ejpam-6940	13	9	the	the	DET
ejpam-6940	13	10	late	late	ADJ
ejpam-6940	13	11	20th	20th	ADJ
ejpam-6940	13	12	century	century	NOUN
ejpam-6940	14	1	,	,	PUNCT
ejpam-6940	14	2	k.	k.	PROPN
ejpam-6940	14	3	iseki	iseki	PROPN
ejpam-6940	14	4	and	and	CCONJ
ejpam-6940	14	5	y.	y.	PROPN
ejpam-6940	14	6	imai	imai	PROPN
ejpam-6940	14	7	introduced	introduce	VERB
ejpam-6940	14	8	two	two	NUM
ejpam-6940	14	9	classes	class	NOUN
ejpam-6940	14	10	of	of	ADP
ejpam-6940	14	11	logical	logical	ADJ
ejpam-6940	14	12	algebras	algebra	NOUN
ejpam-6940	14	13	which	which	PRON
ejpam-6940	14	14	are	be	AUX
ejpam-6940	14	15	called	call	VERB
ejpam-6940	14	16	bck	bck	NOUN
ejpam-6940	14	17	-	-	PUNCT
ejpam-6940	14	18	algebra	algebra	PROPN
ejpam-6940	14	19	and	and	CCONJ
ejpam-6940	14	20	bci	bci	NOUN
ejpam-6940	14	21	-	-	NOUN
ejpam-6940	14	22	algebra	algebra	NOUN
ejpam-6940	14	23	in	in	ADP
ejpam-6940	14	24	1966	1966	NUM
ejpam-6940	14	25	[	[	X
ejpam-6940	14	26	1	1	NUM
ejpam-6940	14	27	,	,	PUNCT
ejpam-6940	14	28	2	2	NUM
ejpam-6940	14	29	]	]	PUNCT
ejpam-6940	14	30	.	.	PUNCT
ejpam-6940	15	1	the	the	DET
ejpam-6940	15	2	class	class	NOUN
ejpam-6940	15	3	of	of	ADP
ejpam-6940	15	4	bck	bck	PROPN
ejpam-6940	15	5	-	-	PUNCT
ejpam-6940	15	6	algebras	algebras	PROPN
ejpam-6940	15	7	is	be	AUX
ejpam-6940	15	8	a	a	DET
ejpam-6940	15	9	proper	proper	ADJ
ejpam-6940	15	10	subclass	subclass	NOUN
ejpam-6940	15	11	of	of	ADP
ejpam-6940	15	12	the	the	DET
ejpam-6940	15	13	class	class	NOUN
ejpam-6940	15	14	of	of	ADP
ejpam-6940	15	15	bci	bci	PROPN
ejpam-6940	15	16	-	-	PUNCT
ejpam-6940	15	17	algebras	algebra	NOUN
ejpam-6940	15	18	.	.	PUNCT
ejpam-6940	16	1	later	later	ADV
ejpam-6940	16	2	,	,	PUNCT
ejpam-6940	16	3	in	in	ADP
ejpam-6940	16	4	1978	1978	NUM
ejpam-6940	16	5	k.	k.	PROPN
ejpam-6940	16	6	iseki	iseki	PROPN
ejpam-6940	16	7	and	and	CCONJ
ejpam-6940	16	8	s.	s.	PROPN
ejpam-6940	16	9	tanaka	tanaka	PROPN
ejpam-6940	16	10	discussed	discuss	VERB
ejpam-6940	16	11	the	the	DET
ejpam-6940	16	12	theory	theory	NOUN
ejpam-6940	16	13	of	of	ADP
ejpam-6940	16	14	bck	bck	NOUN
ejpam-6940	16	15	-	-	PUNCT
ejpam-6940	16	16	algebras	algebras	NOUN
ejpam-6940	16	17	in	in	ADP
ejpam-6940	16	18	[	[	X
ejpam-6940	16	19	3	3	NUM
ejpam-6940	16	20	]	]	PUNCT
ejpam-6940	16	21	.	.	PUNCT
ejpam-6940	17	1	since	since	SCONJ
ejpam-6940	17	2	then	then	ADV
ejpam-6940	17	3	,	,	PUNCT
ejpam-6940	17	4	many	many	ADJ
ejpam-6940	17	5	new	new	ADJ
ejpam-6940	17	6	kinds	kind	NOUN
ejpam-6940	17	7	of	of	ADP
ejpam-6940	17	8	algebras	algebra	NOUN
ejpam-6940	17	9	which	which	PRON
ejpam-6940	17	10	are	be	AUX
ejpam-6940	17	11	related	relate	VERB
ejpam-6940	17	12	to	to	PART
ejpam-6940	17	13	bck	bck	VERB
ejpam-6940	17	14	/	/	SYM
ejpam-6940	17	15	bcialgebras	bcialgebra	NOUN
ejpam-6940	17	16	are	be	AUX
ejpam-6940	17	17	introduced	introduce	VERB
ejpam-6940	17	18	.	.	PUNCT
ejpam-6940	18	1	in	in	ADP
ejpam-6940	18	2	1983	1983	NUM
ejpam-6940	18	3	,	,	PUNCT
ejpam-6940	18	4	q.	q.	PROPN
ejpam-6940	18	5	p.	p.	PROPN
ejpam-6940	18	6	hu	hu	PROPN
ejpam-6940	19	1	and	and	CCONJ
ejpam-6940	19	2	x.	x.	PROPN
ejpam-6940	19	3	li	li	PROPN
ejpam-6940	19	4	introduced	introduce	VERB
ejpam-6940	19	5	the	the	DET
ejpam-6940	19	6	notion	notion	NOUN
ejpam-6940	19	7	of	of	ADP
ejpam-6940	19	8	bch	bch	PROPN
ejpam-6940	19	9	-	-	PUNCT
ejpam-6940	19	10	algebra	algebra	NOUN
ejpam-6940	19	11	which	which	PRON
ejpam-6940	19	12	is	be	AUX
ejpam-6940	19	13	a	a	DET
ejpam-6940	19	14	generalization	generalization	NOUN
ejpam-6940	19	15	of	of	ADP
ejpam-6940	19	16	bck	bck	PROPN
ejpam-6940	19	17	/	/	SYM
ejpam-6940	19	18	bci	bci	NOUN
ejpam-6940	19	19	-	-	PUNCT
ejpam-6940	19	20	algebras	algebras	X
ejpam-6940	20	1	[	[	X
ejpam-6940	20	2	4	4	NUM
ejpam-6940	20	3	,	,	PUNCT
ejpam-6940	20	4	5	5	NUM
ejpam-6940	20	5	]	]	PUNCT
ejpam-6940	20	6	.	.	PUNCT
ejpam-6940	21	1	in	in	ADP
ejpam-6940	21	2	1984	1984	NUM
ejpam-6940	21	3	,	,	PUNCT
ejpam-6940	21	4	a	a	DET
ejpam-6940	21	5	class	class	NOUN
ejpam-6940	21	6	of	of	ADP
ejpam-6940	21	7	bcc	bcc	PROPN
ejpam-6940	21	8	-	-	PUNCT
ejpam-6940	21	9	algebras	algebras	PROPN
ejpam-6940	21	10	was	be	AUX
ejpam-6940	21	11	presented	present	VERB
ejpam-6940	21	12	by	by	ADP
ejpam-6940	21	13	y.	y.	PROPN
ejpam-6940	21	14	komori	komori	PROPN
ejpam-6940	22	1	[	[	X
ejpam-6940	22	2	6	6	NUM
ejpam-6940	22	3	]	]	PUNCT
ejpam-6940	22	4	,	,	PUNCT
ejpam-6940	22	5	and	and	CCONJ
ejpam-6940	22	6	then	then	ADV
ejpam-6940	22	7	in	in	ADP
ejpam-6940	22	8	1992	1992	NUM
ejpam-6940	22	9	w.a	w.a	PROPN
ejpam-6940	22	10	.	.	PUNCT
ejpam-6940	22	11	dudek	dudek	PROPN
ejpam-6940	22	12	discussed	discuss	VERB
ejpam-6940	22	13	some	some	DET
ejpam-6940	22	14	properties	property	NOUN
ejpam-6940	22	15	of	of	ADP
ejpam-6940	22	16	this	this	DET
ejpam-6940	22	17	algebra	algebra	NOUN
ejpam-6940	22	18	in	in	ADP
ejpam-6940	22	19	[	[	X
ejpam-6940	22	20	7	7	NUM
ejpam-6940	22	21	]	]	PUNCT
ejpam-6940	22	22	.	.	PUNCT
ejpam-6940	23	1	∗corresponding	∗corresponde	VERB
ejpam-6940	23	2	author	author	NOUN
ejpam-6940	23	3	.	.	PUNCT
ejpam-6940	24	1	∗corresponding	∗corresponde	VERB
ejpam-6940	24	2	author	author	NOUN
ejpam-6940	24	3	.	.	PUNCT
ejpam-6940	25	1	doi	doi	NOUN
ejpam-6940	25	2	:	:	PUNCT
ejpam-6940	25	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6940	https://doi.org/10.29020/nybg.ejpam.v18i4.6940	ADJ
ejpam-6940	25	4	email	email	NOUN
ejpam-6940	25	5	addresses	address	VERB
ejpam-6940	25	6	:	:	PUNCT
ejpam-6940	25	7	ananya.a@msu.ac.th	ananya.a@msu.ac.th	ADP
ejpam-6940	25	8	(	(	PUNCT
ejpam-6940	25	9	a.	a.	NOUN
ejpam-6940	25	10	anantayasethi	anantayasethi	PROPN
ejpam-6940	25	11	)	)	PUNCT
ejpam-6940	25	12	,	,	PUNCT
ejpam-6940	26	1	kittisak.s@msu.ac.th	kittisak.s@msu.ac.th	PROPN
ejpam-6940	26	2	(	(	PUNCT
ejpam-6940	26	3	k.	k.	PROPN
ejpam-6940	26	4	saengsura	saengsura	PROPN
ejpam-6940	26	5	)	)	PUNCT
ejpam-6940	26	6	,	,	PUNCT
ejpam-6940	26	7	yeni-math@ugm.ac.id	yeni-math@ugm.ac.id	PROPN
ejpam-6940	26	8	(	(	PUNCT
ejpam-6940	26	9	y.	y.	PROPN
ejpam-6940	26	10	susanti	susanti	PROPN
ejpam-6940	26	11	)	)	PUNCT
ejpam-6940	26	12	,	,	PUNCT
ejpam-6940	26	13	napaporn.sr@rmuti.ac.th	napaporn.sr@rmuti.ac.th	PROPN
ejpam-6940	26	14	(	(	PUNCT
ejpam-6940	26	15	n.	n.	NOUN
ejpam-6940	26	16	sarasit	sarasit	PROPN
ejpam-6940	26	17	)	)	PUNCT
ejpam-6940	26	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6940	26	19	1	1	NUM
ejpam-6940	26	20	copyright	copyright	NOUN
ejpam-6940	26	21	:	:	PUNCT
ejpam-6940	26	22	©	©	PROPN
ejpam-6940	26	23	2025	2025	NUM
ejpam-6940	26	24	the	the	DET
ejpam-6940	26	25	author(s	author(s	NOUN
ejpam-6940	26	26	)	)	PUNCT
ejpam-6940	26	27	.	.	PUNCT
ejpam-6940	27	1	(	(	PUNCT
ejpam-6940	27	2	cc	cc	NOUN
ejpam-6940	27	3	by	by	ADP
ejpam-6940	27	4	-	-	PUNCT
ejpam-6940	27	5	nc	nc	PROPN
ejpam-6940	27	6	4.0	4.0	NUM
ejpam-6940	27	7	)	)	PUNCT
ejpam-6940	27	8	a.	a.	NOUN
ejpam-6940	27	9	anantayasethi	anantayasethi	PROPN
ejpam-6940	27	10	et	et	PROPN
ejpam-6940	27	11	al	al	PROPN
ejpam-6940	27	12	.	.	PUNCT
ejpam-6940	27	13	/	/	SYM
ejpam-6940	27	14	eur	eur	PROPN
ejpam-6940	27	15	.	.	PUNCT
ejpam-6940	28	1	j.	j.	PROPN
ejpam-6940	28	2	pure	pure	PROPN
ejpam-6940	28	3	appl	appl	PROPN
ejpam-6940	28	4	.	.	PROPN
ejpam-6940	28	5	math	math	PROPN
ejpam-6940	28	6	,	,	PUNCT
ejpam-6940	28	7	18	18	NUM
ejpam-6940	28	8	(	(	PUNCT
ejpam-6940	28	9	4	4	NUM
ejpam-6940	28	10	)	)	PUNCT
ejpam-6940	28	11	(	(	PUNCT
ejpam-6940	28	12	2025	2025	NUM
ejpam-6940	28	13	)	)	PUNCT
ejpam-6940	28	14	,	,	PUNCT
ejpam-6940	28	15	6940	6940	NUM
ejpam-6940	28	16	2	2	NUM
ejpam-6940	28	17	of	of	ADP
ejpam-6940	28	18	14	14	NUM
ejpam-6940	28	19	a	a	DET
ejpam-6940	28	20	d	d	NOUN
ejpam-6940	28	21	-	-	PUNCT
ejpam-6940	28	22	algebra	algebra	NOUN
ejpam-6940	28	23	,	,	PUNCT
ejpam-6940	28	24	another	another	DET
ejpam-6940	28	25	generalization	generalization	NOUN
ejpam-6940	28	26	of	of	ADP
ejpam-6940	28	27	bck	bck	PROPN
ejpam-6940	28	28	/	/	SYM
ejpam-6940	28	29	bci	bci	NOUN
ejpam-6940	28	30	-	-	PUNCT
ejpam-6940	28	31	algebras	algebras	X
ejpam-6940	28	32	,	,	PUNCT
ejpam-6940	28	33	was	be	AUX
ejpam-6940	28	34	appeared	appear	VERB
ejpam-6940	28	35	in	in	ADP
ejpam-6940	28	36	1999	1999	NUM
ejpam-6940	28	37	by	by	ADP
ejpam-6940	28	38	j.	j.	PROPN
ejpam-6940	28	39	neggers	neggers	PROPN
ejpam-6940	28	40	and	and	CCONJ
ejpam-6940	28	41	h.s	h.s	PROPN
ejpam-6940	28	42	.	.	PROPN
ejpam-6940	28	43	kim	kim	PROPN
ejpam-6940	29	1	[	[	X
ejpam-6940	29	2	8	8	NUM
ejpam-6940	29	3	]	]	PUNCT
ejpam-6940	29	4	.	.	PUNCT
ejpam-6940	30	1	the	the	DET
ejpam-6940	30	2	authors	author	NOUN
ejpam-6940	30	3	explored	explore	VERB
ejpam-6940	30	4	relations	relation	NOUN
ejpam-6940	30	5	between	between	ADP
ejpam-6940	30	6	d	d	NOUN
ejpam-6940	30	7	-	-	PUNCT
ejpam-6940	30	8	algebras	algebras	PROPN
ejpam-6940	30	9	and	and	CCONJ
ejpam-6940	30	10	bckalgebras	bckalgebras	PROPN
ejpam-6940	30	11	.	.	PUNCT
ejpam-6940	31	1	the	the	DET
ejpam-6940	31	2	concept	concept	NOUN
ejpam-6940	31	3	of	of	ADP
ejpam-6940	31	4	edge	edge	NOUN
ejpam-6940	31	5	in	in	ADP
ejpam-6940	31	6	d	d	NOUN
ejpam-6940	31	7	-	-	PUNCT
ejpam-6940	31	8	algebras	algebras	PROPN
ejpam-6940	31	9	is	be	AUX
ejpam-6940	31	10	also	also	ADV
ejpam-6940	31	11	provided	provide	VERB
ejpam-6940	31	12	.	.	PUNCT
ejpam-6940	32	1	they	they	PRON
ejpam-6940	32	2	showed	show	VERB
ejpam-6940	32	3	that	that	SCONJ
ejpam-6940	32	4	every	every	DET
ejpam-6940	32	5	d	d	ADJ
ejpam-6940	32	6	-	-	ADJ
ejpam-6940	32	7	transitive	transitive	ADJ
ejpam-6940	32	8	edge	edge	NOUN
ejpam-6940	32	9	d	d	NOUN
ejpam-6940	32	10	-	-	PUNCT
ejpam-6940	32	11	algebra	algebra	NOUN
ejpam-6940	32	12	is	be	AUX
ejpam-6940	32	13	a	a	DET
ejpam-6940	32	14	bck	bck	NOUN
ejpam-6940	32	15	-	-	PUNCT
ejpam-6940	32	16	algebra	algebra	NOUN
ejpam-6940	32	17	.	.	PUNCT
ejpam-6940	33	1	moreover	moreover	ADV
ejpam-6940	33	2	,	,	PUNCT
ejpam-6940	33	3	they	they	PRON
ejpam-6940	33	4	obtained	obtain	VERB
ejpam-6940	33	5	a	a	DET
ejpam-6940	33	6	correspondence	correspondence	NOUN
ejpam-6940	33	7	between	between	ADP
ejpam-6940	33	8	oriented	orient	VERB
ejpam-6940	33	9	digraphs	digraphs	NOUN
ejpam-6940	33	10	and	and	CCONJ
ejpam-6940	33	11	edge	edge	NOUN
ejpam-6940	33	12	d	d	NOUN
ejpam-6940	33	13	-	-	PUNCT
ejpam-6940	33	14	algebras	algebra	NOUN
ejpam-6940	33	15	such	such	ADJ
ejpam-6940	33	16	that	that	SCONJ
ejpam-6940	33	17	every	every	DET
ejpam-6940	33	18	edge	edge	NOUN
ejpam-6940	33	19	d	d	NOUN
ejpam-6940	33	20	-	-	PUNCT
ejpam-6940	33	21	algebra	algebra	NOUN
ejpam-6940	33	22	produces	produce	VERB
ejpam-6940	33	23	an	an	DET
ejpam-6940	33	24	oriented	orient	VERB
ejpam-6940	33	25	digraph	digraph	NOUN
ejpam-6940	33	26	.	.	PUNCT
ejpam-6940	34	1	some	some	DET
ejpam-6940	34	2	years	year	NOUN
ejpam-6940	34	3	later	later	ADV
ejpam-6940	34	4	,	,	PUNCT
ejpam-6940	34	5	neggers	negger	NOUN
ejpam-6940	34	6	and	and	CCONJ
ejpam-6940	34	7	kim	kim	PROPN
ejpam-6940	34	8	also	also	ADV
ejpam-6940	34	9	introduced	introduce	VERB
ejpam-6940	34	10	the	the	DET
ejpam-6940	34	11	notion	notion	NOUN
ejpam-6940	34	12	of	of	ADP
ejpam-6940	34	13	q	q	NOUN
ejpam-6940	34	14	-	-	PUNCT
ejpam-6940	34	15	algebras	algebras	ADJ
ejpam-6940	34	16	and	and	CCONJ
ejpam-6940	34	17	b	b	NOUN
ejpam-6940	34	18	-	-	PUNCT
ejpam-6940	34	19	algebras	algebras	X
ejpam-6940	34	20	.	.	PUNCT
ejpam-6940	35	1	in	in	ADP
ejpam-6940	35	2	2001	2001	NUM
ejpam-6940	35	3	,	,	PUNCT
ejpam-6940	35	4	an	an	DET
ejpam-6940	35	5	algebra	algebra	NOUN
ejpam-6940	35	6	which	which	PRON
ejpam-6940	35	7	related	relate	VERB
ejpam-6940	35	8	to	to	PART
ejpam-6940	35	9	bck	bck	VERB
ejpam-6940	35	10	/	/	SYM
ejpam-6940	35	11	bci	bci	NOUN
ejpam-6940	35	12	-	-	PUNCT
ejpam-6940	35	13	algebras	algebras	PROPN
ejpam-6940	35	14	was	be	AUX
ejpam-6940	35	15	emerged	emerge	VERB
ejpam-6940	35	16	,	,	PUNCT
ejpam-6940	35	17	so	so	ADV
ejpam-6940	35	18	called	call	VERB
ejpam-6940	35	19	q	q	NOUN
ejpam-6940	35	20	-	-	PUNCT
ejpam-6940	35	21	algebra	algebra	NOUN
ejpam-6940	35	22	,	,	PUNCT
ejpam-6940	35	23	by	by	ADP
ejpam-6940	35	24	j.	j.	PROPN
ejpam-6940	35	25	neggers	neggers	PROPN
ejpam-6940	35	26	,	,	PUNCT
ejpam-6940	35	27	s.	s.	PROPN
ejpam-6940	35	28	ahn	ahn	PROPN
ejpam-6940	35	29	and	and	CCONJ
ejpam-6940	35	30	h.	h.	PROPN
ejpam-6940	35	31	s.	s.	PROPN
ejpam-6940	35	32	kim	kim	PROPN
ejpam-6940	36	1	[	[	X
ejpam-6940	36	2	9	9	NUM
ejpam-6940	36	3	]	]	PUNCT
ejpam-6940	36	4	.	.	PUNCT
ejpam-6940	37	1	a	a	DET
ejpam-6940	37	2	q	q	NOUN
ejpam-6940	37	3	-	-	PUNCT
ejpam-6940	37	4	algebra	algebra	NOUN
ejpam-6940	37	5	is	be	AUX
ejpam-6940	37	6	an	an	DET
ejpam-6940	37	7	algebraic	algebraic	ADJ
ejpam-6940	37	8	system	system	NOUN
ejpam-6940	37	9	(	(	PUNCT
ejpam-6940	37	10	x	x	NOUN
ejpam-6940	37	11	;	;	PUNCT
ejpam-6940	37	12	∗	∗	NOUN
ejpam-6940	37	13	,	,	PUNCT
ejpam-6940	37	14	0	0	NUM
ejpam-6940	37	15	)	)	PUNCT
ejpam-6940	37	16	consists	consist	VERB
ejpam-6940	37	17	of	of	ADP
ejpam-6940	37	18	a	a	DET
ejpam-6940	37	19	non	non	ADJ
ejpam-6940	37	20	-	-	ADJ
ejpam-6940	37	21	empty	empty	ADJ
ejpam-6940	37	22	set	set	NOUN
ejpam-6940	37	23	x	x	NOUN
ejpam-6940	37	24	,	,	PUNCT
ejpam-6940	38	1	a	a	DET
ejpam-6940	38	2	constant	constant	ADJ
ejpam-6940	38	3	0	0	NUM
ejpam-6940	38	4	∈	∈	NOUN
ejpam-6940	38	5	x	x	X
ejpam-6940	38	6	and	and	CCONJ
ejpam-6940	38	7	a	a	DET
ejpam-6940	38	8	binary	binary	ADJ
ejpam-6940	38	9	operation	operation	NOUN
ejpam-6940	38	10	∗	∗	NOUN
ejpam-6940	38	11	defined	define	VERB
ejpam-6940	38	12	on	on	ADP
ejpam-6940	38	13	x	x	PUNCT
ejpam-6940	38	14	that	that	PRON
ejpam-6940	38	15	yields	yield	VERB
ejpam-6940	38	16	the	the	DET
ejpam-6940	38	17	following	follow	VERB
ejpam-6940	38	18	three	three	NUM
ejpam-6940	38	19	conditions	condition	NOUN
ejpam-6940	38	20	(	(	PUNCT
ejpam-6940	38	21	q1	q1	PROPN
ejpam-6940	38	22	)	)	PUNCT
ejpam-6940	38	23	,	,	PUNCT
ejpam-6940	38	24	(	(	PUNCT
ejpam-6940	38	25	q2	q2	NOUN
ejpam-6940	38	26	)	)	PUNCT
ejpam-6940	38	27	and	and	CCONJ
ejpam-6940	38	28	(	(	PUNCT
ejpam-6940	38	29	q3	q3	PROPN
ejpam-6940	38	30	)	)	PUNCT
ejpam-6940	38	31	as	as	ADP
ejpam-6940	38	32	the	the	DET
ejpam-6940	38	33	following	following	NOUN
ejpam-6940	38	34	:	:	PUNCT
ejpam-6940	38	35	for	for	ADP
ejpam-6940	38	36	any	any	DET
ejpam-6940	38	37	x	x	NOUN
ejpam-6940	38	38	,	,	PUNCT
ejpam-6940	38	39	y	y	PROPN
ejpam-6940	38	40	,	,	PUNCT
ejpam-6940	38	41	z	z	PROPN
ejpam-6940	38	42	∈	∈	PROPN
ejpam-6940	38	43	x	x	X
ejpam-6940	38	44	,	,	PUNCT
ejpam-6940	38	45	(	(	PUNCT
ejpam-6940	38	46	q1	q1	PROPN
ejpam-6940	38	47	)	)	PUNCT
ejpam-6940	38	48	x	x	SYM
ejpam-6940	38	49	∗	∗	NOUN
ejpam-6940	38	50	x	x	SYM
ejpam-6940	38	51	=	=	SYM
ejpam-6940	38	52	0	0	NUM
ejpam-6940	38	53	,	,	PUNCT
ejpam-6940	38	54	(	(	PUNCT
ejpam-6940	38	55	q2	q2	NOUN
ejpam-6940	38	56	)	)	PUNCT
ejpam-6940	38	57	x	x	SYM
ejpam-6940	38	58	∗	∗	NOUN
ejpam-6940	38	59	0	0	NUM
ejpam-6940	39	1	=	=	SYM
ejpam-6940	39	2	x	x	NOUN
ejpam-6940	39	3	,	,	PUNCT
ejpam-6940	39	4	(	(	PUNCT
ejpam-6940	39	5	q3	q3	PROPN
ejpam-6940	39	6	)	)	PUNCT
ejpam-6940	39	7	(	(	PUNCT
ejpam-6940	39	8	x	x	SYM
ejpam-6940	39	9	∗	∗	PROPN
ejpam-6940	39	10	y	y	NOUN
ejpam-6940	39	11	)	)	PUNCT
ejpam-6940	39	12	∗	∗	NOUN
ejpam-6940	39	13	z	z	NOUN
ejpam-6940	39	14	=	=	SYM
ejpam-6940	39	15	(	(	PUNCT
ejpam-6940	39	16	x	x	X
ejpam-6940	39	17	∗	∗	PROPN
ejpam-6940	39	18	z	z	NOUN
ejpam-6940	39	19	)	)	PUNCT
ejpam-6940	39	20	∗	∗	NOUN
ejpam-6940	39	21	y.	y.	NOUN
ejpam-6940	40	1	we	we	PRON
ejpam-6940	40	2	omit	omit	VERB
ejpam-6940	40	3	the	the	DET
ejpam-6940	40	4	symbol	symbol	NOUN
ejpam-6940	40	5	∗	∗	NOUN
ejpam-6940	40	6	for	for	ADP
ejpam-6940	40	7	a	a	DET
ejpam-6940	40	8	convenient	convenient	ADJ
ejpam-6940	40	9	reason	reason	NOUN
ejpam-6940	40	10	.	.	PUNCT
ejpam-6940	41	1	let	let	VERB
ejpam-6940	41	2	us	we	PRON
ejpam-6940	41	3	mention	mention	VERB
ejpam-6940	41	4	here	here	ADV
ejpam-6940	41	5	,	,	PUNCT
ejpam-6940	41	6	later	later	ADV
ejpam-6940	41	7	on	on	ADV
ejpam-6940	41	8	we	we	PRON
ejpam-6940	41	9	will	will	AUX
ejpam-6940	41	10	denote	denote	VERB
ejpam-6940	41	11	the	the	DET
ejpam-6940	41	12	symbol	symbol	NOUN
ejpam-6940	41	13	x	x	PUNCT
ejpam-6940	41	14	as	as	ADP
ejpam-6940	41	15	a	a	DET
ejpam-6940	41	16	q	q	NOUN
ejpam-6940	41	17	-	-	NOUN
ejpam-6940	41	18	algebra	algebra	NOUN
ejpam-6940	41	19	(	(	PUNCT
ejpam-6940	41	20	x	x	NOUN
ejpam-6940	41	21	;	;	PUNCT
ejpam-6940	41	22	∗	∗	NOUN
ejpam-6940	41	23	,	,	PUNCT
ejpam-6940	41	24	0	0	NUM
ejpam-6940	41	25	)	)	PUNCT
ejpam-6940	41	26	unless	unless	SCONJ
ejpam-6940	41	27	otherwise	otherwise	ADV
ejpam-6940	41	28	specified	specify	VERB
ejpam-6940	41	29	.	.	PUNCT
ejpam-6940	42	1	in	in	ADP
ejpam-6940	42	2	[	[	X
ejpam-6940	42	3	9	9	NUM
ejpam-6940	42	4	]	]	PUNCT
ejpam-6940	42	5	,	,	PUNCT
ejpam-6940	42	6	the	the	DET
ejpam-6940	42	7	authors	author	NOUN
ejpam-6940	42	8	presented	present	VERB
ejpam-6940	42	9	some	some	DET
ejpam-6940	42	10	connections	connection	NOUN
ejpam-6940	42	11	of	of	ADP
ejpam-6940	42	12	bck	bck	PROPN
ejpam-6940	42	13	/	/	SYM
ejpam-6940	42	14	bci	bci	PROPN
ejpam-6940	42	15	/	/	SYM
ejpam-6940	42	16	bch	bch	PROPN
ejpam-6940	42	17	-	-	PUNCT
ejpam-6940	42	18	algebras	algebras	PROPN
ejpam-6940	42	19	and	and	CCONJ
ejpam-6940	42	20	q	q	NOUN
ejpam-6940	42	21	-	-	PUNCT
ejpam-6940	42	22	algebras	algebras	X
ejpam-6940	42	23	.	.	PUNCT
ejpam-6940	43	1	they	they	PRON
ejpam-6940	43	2	showed	show	VERB
ejpam-6940	43	3	that	that	SCONJ
ejpam-6940	43	4	a	a	DET
ejpam-6940	43	5	q	q	NOUN
ejpam-6940	43	6	-	-	NOUN
ejpam-6940	43	7	algebra	algebra	NOUN
ejpam-6940	43	8	x	x	PUNCT
ejpam-6940	43	9	satisfying	satisfy	VERB
ejpam-6940	43	10	the	the	DET
ejpam-6940	43	11	condition	condition	NOUN
ejpam-6940	43	12	(	(	PUNCT
ejpam-6940	43	13	a	a	X
ejpam-6940	43	14	):	):	PUNCT
ejpam-6940	43	15	for	for	ADP
ejpam-6940	43	16	all	all	DET
ejpam-6940	43	17	x	x	NOUN
ejpam-6940	43	18	,	,	PUNCT
ejpam-6940	43	19	y	y	PROPN
ejpam-6940	43	20	∈	∈	PROPN
ejpam-6940	44	1	x	x	X
ejpam-6940	44	2	,	,	PUNCT
ejpam-6940	44	3	xy	xy	PROPN
ejpam-6940	44	4	=	=	SYM
ejpam-6940	44	5	0	0	PROPN
ejpam-6940	44	6	and	and	CCONJ
ejpam-6940	44	7	yx	yx	X
ejpam-6940	44	8	=	=	SYM
ejpam-6940	44	9	0	0	NUM
ejpam-6940	44	10	implies	imply	VERB
ejpam-6940	44	11	x	x	PUNCT
ejpam-6940	44	12	=	=	SYM
ejpam-6940	44	13	y	y	PROPN
ejpam-6940	44	14	,	,	PUNCT
ejpam-6940	44	15	is	be	AUX
ejpam-6940	44	16	a	a	DET
ejpam-6940	44	17	bch	bch	NOUN
ejpam-6940	44	18	-	-	PUNCT
ejpam-6940	44	19	algebra	algebra	NOUN
ejpam-6940	44	20	.	.	PUNCT
ejpam-6940	45	1	a	a	DET
ejpam-6940	45	2	q	q	NOUN
ejpam-6940	45	3	-	-	NOUN
ejpam-6940	45	4	algebra	algebra	NOUN
ejpam-6940	45	5	x	x	PUNCT
ejpam-6940	45	6	satisfying	satisfy	VERB
ejpam-6940	45	7	the	the	DET
ejpam-6940	45	8	conditions	condition	NOUN
ejpam-6940	45	9	(	(	PUNCT
ejpam-6940	45	10	a	a	X
ejpam-6940	45	11	)	)	PUNCT
ejpam-6940	45	12	and	and	CCONJ
ejpam-6940	45	13	(	(	PUNCT
ejpam-6940	45	14	b	b	NOUN
ejpam-6940	45	15	):	):	PUNCT
ejpam-6940	45	16	(	(	PUNCT
ejpam-6940	45	17	xy)(xz	xy)(xz	NOUN
ejpam-6940	45	18	)	)	PUNCT
ejpam-6940	45	19	=	=	SYM
ejpam-6940	45	20	zy	zy	PROPN
ejpam-6940	45	21	for	for	ADP
ejpam-6940	45	22	all	all	DET
ejpam-6940	45	23	x	x	PROPN
ejpam-6940	45	24	,	,	PUNCT
ejpam-6940	45	25	y	y	PROPN
ejpam-6940	45	26	,	,	PUNCT
ejpam-6940	45	27	z	z	PROPN
ejpam-6940	45	28	∈	∈	PROPN
ejpam-6940	45	29	x	x	X
ejpam-6940	45	30	,	,	PUNCT
ejpam-6940	45	31	is	be	AUX
ejpam-6940	45	32	a	a	DET
ejpam-6940	45	33	bci	bci	NOUN
ejpam-6940	45	34	-	-	NOUN
ejpam-6940	45	35	algebra	algebra	NOUN
ejpam-6940	45	36	.	.	PUNCT
ejpam-6940	46	1	a	a	DET
ejpam-6940	46	2	q	q	NOUN
ejpam-6940	46	3	-	-	NOUN
ejpam-6940	46	4	algebra	algebra	NOUN
ejpam-6940	46	5	x	x	PUNCT
ejpam-6940	46	6	is	be	AUX
ejpam-6940	46	7	a	a	DET
ejpam-6940	46	8	bck	bck	NOUN
ejpam-6940	46	9	-	-	PUNCT
ejpam-6940	46	10	algebra	algebra	NOUN
ejpam-6940	46	11	if	if	SCONJ
ejpam-6940	46	12	the	the	DET
ejpam-6940	46	13	conditions	condition	NOUN
ejpam-6940	46	14	(	(	PUNCT
ejpam-6940	46	15	a	a	X
ejpam-6940	46	16	)	)	PUNCT
ejpam-6940	46	17	,	,	PUNCT
ejpam-6940	46	18	(	(	PUNCT
ejpam-6940	46	19	b	b	X
ejpam-6940	46	20	)	)	PUNCT
ejpam-6940	46	21	and	and	CCONJ
ejpam-6940	46	22	(	(	PUNCT
ejpam-6940	46	23	c	c	NOUN
ejpam-6940	46	24	):	):	PUNCT
ejpam-6940	46	25	for	for	ADP
ejpam-6940	46	26	all	all	DET
ejpam-6940	46	27	x	x	NOUN
ejpam-6940	46	28	,	,	PUNCT
ejpam-6940	46	29	y	y	PROPN
ejpam-6940	46	30	,	,	PUNCT
ejpam-6940	46	31	z	z	PROPN
ejpam-6940	46	32	∈	∈	PROPN
ejpam-6940	46	33	x	x	X
ejpam-6940	46	34	,	,	PUNCT
ejpam-6940	46	35	(	(	PUNCT
ejpam-6940	46	36	xy)x	xy)x	X
ejpam-6940	46	37	=	=	SYM
ejpam-6940	46	38	0	0	NUM
ejpam-6940	46	39	,	,	PUNCT
ejpam-6940	46	40	are	be	AUX
ejpam-6940	46	41	hold	hold	NOUN
ejpam-6940	46	42	.	.	PUNCT
ejpam-6940	47	1	the	the	DET
ejpam-6940	47	2	concepts	concept	NOUN
ejpam-6940	47	3	of	of	ADP
ejpam-6940	47	4	subalgebra	subalgebra	NOUN
ejpam-6940	47	5	,	,	PUNCT
ejpam-6940	47	6	g	g	NOUN
ejpam-6940	47	7	-	-	PUNCT
ejpam-6940	47	8	part	part	NOUN
ejpam-6940	47	9	and	and	CCONJ
ejpam-6940	47	10	ideal	ideal	NOUN
ejpam-6940	47	11	were	be	AUX
ejpam-6940	47	12	also	also	ADV
ejpam-6940	47	13	offered	offer	VERB
ejpam-6940	47	14	in	in	ADP
ejpam-6940	47	15	[	[	X
ejpam-6940	47	16	9	9	NUM
ejpam-6940	47	17	]	]	PUNCT
ejpam-6940	47	18	.	.	PUNCT
ejpam-6940	48	1	a	a	DET
ejpam-6940	48	2	non	non	ADJ
ejpam-6940	48	3	-	-	ADJ
ejpam-6940	48	4	empty	empty	ADJ
ejpam-6940	48	5	subset	subset	NOUN
ejpam-6940	48	6	s	s	NOUN
ejpam-6940	48	7	of	of	ADP
ejpam-6940	48	8	x	x	PRON
ejpam-6940	48	9	is	be	AUX
ejpam-6940	48	10	a	a	DET
ejpam-6940	48	11	subalgebra	subalgebra	NOUN
ejpam-6940	48	12	if	if	SCONJ
ejpam-6940	48	13	ab	ab	PROPN
ejpam-6940	48	14	∈	∈	PROPN
ejpam-6940	48	15	s	s	X
ejpam-6940	48	16	for	for	ADP
ejpam-6940	48	17	any	any	DET
ejpam-6940	48	18	a	a	PRON
ejpam-6940	48	19	and	and	CCONJ
ejpam-6940	48	20	b	b	NOUN
ejpam-6940	48	21	in	in	ADP
ejpam-6940	48	22	s.	s.	PROPN
ejpam-6940	48	23	it	it	PRON
ejpam-6940	48	24	is	be	AUX
ejpam-6940	48	25	easy	easy	ADJ
ejpam-6940	48	26	to	to	PART
ejpam-6940	48	27	see	see	VERB
ejpam-6940	48	28	that	that	SCONJ
ejpam-6940	48	29	a	a	DET
ejpam-6940	48	30	subset	subset	NOUN
ejpam-6940	48	31	{	{	PUNCT
ejpam-6940	48	32	0	0	NUM
ejpam-6940	48	33	}	}	PUNCT
ejpam-6940	48	34	is	be	AUX
ejpam-6940	48	35	a	a	DET
ejpam-6940	48	36	subalgebra	subalgebra	NOUN
ejpam-6940	48	37	of	of	ADP
ejpam-6940	48	38	a	a	DET
ejpam-6940	48	39	q	q	NOUN
ejpam-6940	48	40	-	-	PUNCT
ejpam-6940	48	41	algebra	algebra	NOUN
ejpam-6940	48	42	x	x	PUNCT
ejpam-6940	48	43	since	since	SCONJ
ejpam-6940	48	44	00	00	NUM
ejpam-6940	49	1	=	=	SYM
ejpam-6940	49	2	0	0	NUM
ejpam-6940	49	3	by	by	ADP
ejpam-6940	49	4	(	(	PUNCT
ejpam-6940	49	5	q1	q1	PROPN
ejpam-6940	49	6	)	)	PUNCT
ejpam-6940	49	7	.	.	PUNCT
ejpam-6940	50	1	a	a	DET
ejpam-6940	50	2	non	non	ADJ
ejpam-6940	50	3	-	-	ADJ
ejpam-6940	50	4	empty	empty	ADJ
ejpam-6940	50	5	subset	subset	NOUN
ejpam-6940	50	6	i	i	PRON
ejpam-6940	50	7	of	of	ADP
ejpam-6940	50	8	x	x	PUNCT
ejpam-6940	50	9	is	be	AUX
ejpam-6940	50	10	an	an	DET
ejpam-6940	50	11	ideal	ideal	NOUN
ejpam-6940	50	12	of	of	ADP
ejpam-6940	50	13	x	x	PRON
ejpam-6940	50	14	if	if	SCONJ
ejpam-6940	50	15	the	the	DET
ejpam-6940	50	16	following	follow	VERB
ejpam-6940	50	17	conditions	condition	NOUN
ejpam-6940	50	18	(	(	PUNCT
ejpam-6940	50	19	i1	i1	PROPN
ejpam-6940	50	20	)	)	PUNCT
ejpam-6940	50	21	and	and	CCONJ
ejpam-6940	50	22	(	(	PUNCT
ejpam-6940	50	23	i2	i2	PROPN
ejpam-6940	50	24	)	)	PUNCT
ejpam-6940	50	25	are	be	AUX
ejpam-6940	50	26	hold	hold	ADJ
ejpam-6940	50	27	:	:	PUNCT
ejpam-6940	50	28	(	(	PUNCT
ejpam-6940	50	29	i1	i1	PROPN
ejpam-6940	50	30	)	)	PUNCT
ejpam-6940	50	31	0	0	PUNCT
ejpam-6940	51	1	∈	∈	PROPN
ejpam-6940	51	2	i	i	PRON
ejpam-6940	51	3	;	;	PUNCT
ejpam-6940	51	4	(	(	PUNCT
ejpam-6940	51	5	i2	i2	PROPN
ejpam-6940	51	6	)	)	PUNCT
ejpam-6940	51	7	for	for	ADP
ejpam-6940	51	8	x	x	X
ejpam-6940	51	9	,	,	PUNCT
ejpam-6940	51	10	y	y	PROPN
ejpam-6940	51	11	∈	∈	PROPN
ejpam-6940	51	12	x	x	PRON
ejpam-6940	51	13	,	,	PUNCT
ejpam-6940	51	14	if	if	SCONJ
ejpam-6940	51	15	xy	xy	PROPN
ejpam-6940	51	16	∈	∈	PROPN
ejpam-6940	52	1	i	i	PRON
ejpam-6940	52	2	,	,	PUNCT
ejpam-6940	52	3	y	y	PROPN
ejpam-6940	52	4	∈	∈	PROPN
ejpam-6940	53	1	i	i	PRON
ejpam-6940	53	2	,	,	PUNCT
ejpam-6940	53	3	then	then	ADV
ejpam-6940	53	4	x	x	PART
ejpam-6940	53	5	∈	∈	PROPN
ejpam-6940	53	6	i.	i.	NOUN
ejpam-6940	53	7	the	the	DET
ejpam-6940	53	8	subsets	subset	NOUN
ejpam-6940	53	9	{	{	PUNCT
ejpam-6940	53	10	0	0	NUM
ejpam-6940	53	11	}	}	PUNCT
ejpam-6940	53	12	and	and	CCONJ
ejpam-6940	53	13	x	x	PRON
ejpam-6940	53	14	are	be	AUX
ejpam-6940	53	15	obviously	obviously	ADV
ejpam-6940	53	16	ideals	ideal	NOUN
ejpam-6940	53	17	of	of	ADP
ejpam-6940	53	18	x.	x.	NOUN
ejpam-6940	53	19	an	an	DET
ejpam-6940	53	20	ideal	ideal	ADJ
ejpam-6940	53	21	i	i	PRON
ejpam-6940	53	22	of	of	ADP
ejpam-6940	53	23	x	x	PRON
ejpam-6940	53	24	is	be	AUX
ejpam-6940	53	25	called	call	VERB
ejpam-6940	53	26	a	a	DET
ejpam-6940	53	27	zero	zero	NUM
ejpam-6940	53	28	ideal	ideal	NOUN
ejpam-6940	53	29	if	if	SCONJ
ejpam-6940	53	30	i	i	PRON
ejpam-6940	53	31	=	=	PUNCT
ejpam-6940	53	32	{	{	PUNCT
ejpam-6940	53	33	0	0	NUM
ejpam-6940	53	34	}	}	PUNCT
ejpam-6940	53	35	,	,	PUNCT
ejpam-6940	53	36	otherwise	otherwise	ADV
ejpam-6940	53	37	i	i	PRON
ejpam-6940	53	38	is	be	AUX
ejpam-6940	53	39	a	a	DET
ejpam-6940	53	40	non	non	ADJ
ejpam-6940	53	41	-	-	ADJ
ejpam-6940	53	42	zero	zero	NUM
ejpam-6940	53	43	ideal	ideal	NOUN
ejpam-6940	53	44	of	of	ADP
ejpam-6940	53	45	x	x	X
ejpam-6940	53	46	.	.	PUNCT
ejpam-6940	54	1	the	the	DET
ejpam-6940	54	2	g	g	NOUN
ejpam-6940	54	3	-	-	PUNCT
ejpam-6940	54	4	part	part	NOUN
ejpam-6940	54	5	of	of	ADP
ejpam-6940	54	6	x	x	PRON
ejpam-6940	54	7	,	,	PUNCT
ejpam-6940	54	8	denoted	denote	VERB
ejpam-6940	54	9	by	by	ADP
ejpam-6940	54	10	g(x	g(x	PROPN
ejpam-6940	54	11	)	)	PUNCT
ejpam-6940	54	12	,	,	PUNCT
ejpam-6940	54	13	is	be	AUX
ejpam-6940	54	14	defined	define	VERB
ejpam-6940	54	15	by	by	ADP
ejpam-6940	54	16	g(x	g(x	NOUN
ejpam-6940	54	17	)	)	PUNCT
ejpam-6940	55	1	=	=	PRON
ejpam-6940	55	2	{	{	PUNCT
ejpam-6940	55	3	a	a	DET
ejpam-6940	55	4	∈	∈	NOUN
ejpam-6940	55	5	x	x	PUNCT
ejpam-6940	55	6	|	|	ADV
ejpam-6940	55	7	0a	0a	VERB
ejpam-6940	55	8	=	=	SYM
ejpam-6940	55	9	a	a	PRON
ejpam-6940	55	10	}	}	PUNCT
ejpam-6940	55	11	.	.	PUNCT
ejpam-6940	56	1	it	it	PRON
ejpam-6940	56	2	is	be	AUX
ejpam-6940	56	3	easy	easy	ADJ
ejpam-6940	56	4	to	to	PART
ejpam-6940	56	5	see	see	VERB
ejpam-6940	56	6	that	that	DET
ejpam-6940	56	7	0	0	NUM
ejpam-6940	56	8	∈	∈	PROPN
ejpam-6940	56	9	g(x	g(x	NOUN
ejpam-6940	56	10	)	)	PUNCT
ejpam-6940	56	11	since	since	SCONJ
ejpam-6940	56	12	00	00	NUM
ejpam-6940	56	13	=	=	SYM
ejpam-6940	56	14	0	0	PROPN
ejpam-6940	56	15	.	.	PUNCT
ejpam-6940	57	1	the	the	DET
ejpam-6940	57	2	authors	author	NOUN
ejpam-6940	57	3	in	in	ADP
ejpam-6940	57	4	[	[	X
ejpam-6940	57	5	9	9	NUM
ejpam-6940	57	6	]	]	PUNCT
ejpam-6940	57	7	obtained	obtain	VERB
ejpam-6940	57	8	the	the	DET
ejpam-6940	57	9	characterization	characterization	NOUN
ejpam-6940	57	10	of	of	ADP
ejpam-6940	57	11	g(x	g(x	NOUN
ejpam-6940	57	12	)	)	PUNCT
ejpam-6940	57	13	which	which	PRON
ejpam-6940	57	14	is	be	AUX
ejpam-6940	57	15	an	an	DET
ejpam-6940	57	16	ideal	ideal	NOUN
ejpam-6940	57	17	of	of	ADP
ejpam-6940	57	18	x	x	SYM
ejpam-6940	57	19	when	when	SCONJ
ejpam-6940	57	20	|x|	|x|	PROPN
ejpam-6940	57	21	≤	≤	NOUN
ejpam-6940	57	22	3	3	NUM
ejpam-6940	57	23	.	.	PUNCT
ejpam-6940	58	1	they	they	PRON
ejpam-6940	58	2	also	also	ADV
ejpam-6940	58	3	provided	provide	VERB
ejpam-6940	58	4	that	that	SCONJ
ejpam-6940	58	5	every	every	DET
ejpam-6940	58	6	subalgebra	subalgebra	NOUN
ejpam-6940	58	7	s	s	VERB
ejpam-6940	58	8	of	of	ADP
ejpam-6940	58	9	x	x	NOUN
ejpam-6940	58	10	,	,	PUNCT
ejpam-6940	58	11	g(x	g(x	NOUN
ejpam-6940	58	12	)	)	PUNCT
ejpam-6940	58	13	∩	∩	NOUN
ejpam-6940	58	14	s	s	PART
ejpam-6940	58	15	=	=	SYM
ejpam-6940	58	16	g(s	g(s	PROPN
ejpam-6940	58	17	)	)	PUNCT
ejpam-6940	58	18	where	where	SCONJ
ejpam-6940	58	19	g(s	g(s	NOUN
ejpam-6940	58	20	)	)	PUNCT
ejpam-6940	58	21	=	=	PRON
ejpam-6940	59	1	{	{	PUNCT
ejpam-6940	59	2	x	x	PUNCT
ejpam-6940	59	3	∈	∈	PROPN
ejpam-6940	59	4	s	s	VERB
ejpam-6940	59	5	|	|	ADV
ejpam-6940	59	6	0x	0x	NOUN
ejpam-6940	59	7	=	=	SYM
ejpam-6940	59	8	x	x	SYM
ejpam-6940	59	9	}	}	PUNCT
ejpam-6940	59	10	.	.	PUNCT
ejpam-6940	60	1	example	example	NOUN
ejpam-6940	61	1	1	1	NUM
ejpam-6940	61	2	.	.	PUNCT
ejpam-6940	61	3	let	let	VERB
ejpam-6940	61	4	x	x	PUNCT
ejpam-6940	61	5	=	=	PUNCT
ejpam-6940	61	6	{	{	PUNCT
ejpam-6940	61	7	0	0	NUM
ejpam-6940	61	8	,	,	PUNCT
ejpam-6940	61	9	a	a	DET
ejpam-6940	61	10	,	,	PUNCT
ejpam-6940	61	11	b	b	NOUN
ejpam-6940	61	12	,	,	PUNCT
ejpam-6940	61	13	c	c	NOUN
ejpam-6940	61	14	,	,	PUNCT
ejpam-6940	61	15	d	d	NOUN
ejpam-6940	61	16	,	,	PUNCT
ejpam-6940	61	17	f	f	NOUN
ejpam-6940	61	18	}	}	PUNCT
ejpam-6940	61	19	.	.	PUNCT
ejpam-6940	62	1	define	define	VERB
ejpam-6940	62	2	a	a	DET
ejpam-6940	62	3	binary	binary	ADJ
ejpam-6940	62	4	operations	operation	NOUN
ejpam-6940	62	5	∗	∗	NOUN
ejpam-6940	62	6	on	on	ADP
ejpam-6940	62	7	x	x	PUNCT
ejpam-6940	62	8	as	as	ADP
ejpam-6940	62	9	the	the	DET
ejpam-6940	62	10	following	follow	VERB
ejpam-6940	62	11	table	table	NOUN
ejpam-6940	62	12	:	:	PUNCT
ejpam-6940	62	13	a.	a.	NOUN
ejpam-6940	62	14	anantayasethi	anantayasethi	PROPN
ejpam-6940	62	15	et	et	PROPN
ejpam-6940	62	16	al	al	PROPN
ejpam-6940	62	17	.	.	PUNCT
ejpam-6940	62	18	/	/	SYM
ejpam-6940	62	19	eur	eur	PROPN
ejpam-6940	62	20	.	.	PUNCT
ejpam-6940	63	1	j.	j.	PROPN
ejpam-6940	63	2	pure	pure	PROPN
ejpam-6940	63	3	appl	appl	PROPN
ejpam-6940	63	4	.	.	PROPN
ejpam-6940	63	5	math	math	PROPN
ejpam-6940	63	6	,	,	PUNCT
ejpam-6940	63	7	18	18	NUM
ejpam-6940	63	8	(	(	PUNCT
ejpam-6940	63	9	4	4	NUM
ejpam-6940	63	10	)	)	PUNCT
ejpam-6940	63	11	(	(	PUNCT
ejpam-6940	63	12	2025	2025	NUM
ejpam-6940	63	13	)	)	PUNCT
ejpam-6940	63	14	,	,	PUNCT
ejpam-6940	63	15	6940	6940	NUM
ejpam-6940	63	16	3	3	NUM
ejpam-6940	63	17	of	of	ADP
ejpam-6940	63	18	14	14	NUM
ejpam-6940	63	19	∗	∗	NOUN
ejpam-6940	63	20	0	0	NUM
ejpam-6940	64	1	a	a	DET
ejpam-6940	64	2	b	b	NOUN
ejpam-6940	64	3	c	c	NOUN
ejpam-6940	64	4	d	d	X
ejpam-6940	64	5	f	f	PROPN
ejpam-6940	64	6	0	0	NUM
ejpam-6940	64	7	0	0	NUM
ejpam-6940	64	8	a	a	DET
ejpam-6940	64	9	c	c	NOUN
ejpam-6940	64	10	b	b	PROPN
ejpam-6940	64	11	c	c	PROPN
ejpam-6940	64	12	b	b	PROPN
ejpam-6940	64	13	a	a	DET
ejpam-6940	64	14	a	a	DET
ejpam-6940	64	15	0	0	NUM
ejpam-6940	64	16	b	b	NOUN
ejpam-6940	64	17	c	c	PROPN
ejpam-6940	64	18	b	b	PROPN
ejpam-6940	64	19	c	c	PROPN
ejpam-6940	64	20	b	b	PROPN
ejpam-6940	64	21	b	b	PROPN
ejpam-6940	64	22	c	c	NOUN
ejpam-6940	64	23	0	0	NUM
ejpam-6940	65	1	a	a	DET
ejpam-6940	65	2	0	0	NUM
ejpam-6940	65	3	a	a	DET
ejpam-6940	65	4	c	c	NOUN
ejpam-6940	65	5	c	c	NOUN
ejpam-6940	65	6	d	d	NOUN
ejpam-6940	65	7	a	a	DET
ejpam-6940	65	8	0	0	NUM
ejpam-6940	65	9	a	a	DET
ejpam-6940	65	10	0	0	NUM
ejpam-6940	66	1	d	d	NOUN
ejpam-6940	67	1	d	d	PROPN
ejpam-6940	68	1	c	c	NOUN
ejpam-6940	68	2	0	0	NUM
ejpam-6940	69	1	a	a	DET
ejpam-6940	69	2	0	0	NUM
ejpam-6940	69	3	a	a	DET
ejpam-6940	69	4	f	f	X
ejpam-6940	70	1	f	f	PROPN
ejpam-6940	70	2	d	d	NOUN
ejpam-6940	70	3	a	a	DET
ejpam-6940	70	4	0	0	NUM
ejpam-6940	70	5	a	a	DET
ejpam-6940	70	6	0	0	NUM
ejpam-6940	71	1	it	it	PRON
ejpam-6940	71	2	is	be	AUX
ejpam-6940	71	3	a	a	DET
ejpam-6940	71	4	routine	routine	NOUN
ejpam-6940	71	5	to	to	PART
ejpam-6940	71	6	check	check	VERB
ejpam-6940	71	7	that	that	PRON
ejpam-6940	71	8	x	x	PRON
ejpam-6940	71	9	is	be	AUX
ejpam-6940	71	10	a	a	DET
ejpam-6940	71	11	q	q	NOUN
ejpam-6940	71	12	-	-	NOUN
ejpam-6940	71	13	algebra	algebra	NOUN
ejpam-6940	71	14	.	.	PUNCT
ejpam-6940	72	1	a	a	DET
ejpam-6940	72	2	subset	subset	NOUN
ejpam-6940	72	3	s	s	X
ejpam-6940	72	4	=	=	X
ejpam-6940	72	5	{	{	PUNCT
ejpam-6940	72	6	0	0	NUM
ejpam-6940	72	7	,	,	PUNCT
ejpam-6940	72	8	a	a	DET
ejpam-6940	72	9	,	,	PUNCT
ejpam-6940	72	10	b	b	NOUN
ejpam-6940	72	11	,	,	PUNCT
ejpam-6940	72	12	c	c	NOUN
ejpam-6940	72	13	,	,	PUNCT
ejpam-6940	72	14	d	d	X
ejpam-6940	72	15	}	}	PUNCT
ejpam-6940	72	16	is	be	AUX
ejpam-6940	72	17	a	a	DET
ejpam-6940	72	18	subalgebra	subalgebra	NOUN
ejpam-6940	72	19	of	of	ADP
ejpam-6940	72	20	x	x	X
ejpam-6940	72	21	but	but	CCONJ
ejpam-6940	72	22	a	a	DET
ejpam-6940	72	23	subset	subset	NOUN
ejpam-6940	72	24	t	t	NOUN
ejpam-6940	72	25	=	=	PUNCT
ejpam-6940	72	26	{	{	PUNCT
ejpam-6940	72	27	0	0	NUM
ejpam-6940	72	28	,	,	PUNCT
ejpam-6940	72	29	f	f	X
ejpam-6940	72	30	}	}	PUNCT
ejpam-6940	72	31	is	be	AUX
ejpam-6940	72	32	not	not	PART
ejpam-6940	72	33	a	a	DET
ejpam-6940	72	34	subalgebra	subalgebra	NOUN
ejpam-6940	72	35	since	since	SCONJ
ejpam-6940	72	36	0	0	NUM
ejpam-6940	72	37	∗	∗	NOUN
ejpam-6940	72	38	f	f	NOUN
ejpam-6940	72	39	=	=	SYM
ejpam-6940	72	40	b	b	PROPN
ejpam-6940	72	41	/∈	/∈	PROPN
ejpam-6940	72	42	t	t	PROPN
ejpam-6940	72	43	.	.	PUNCT
ejpam-6940	73	1	moreover	moreover	ADV
ejpam-6940	73	2	,	,	PUNCT
ejpam-6940	73	3	a	a	DET
ejpam-6940	73	4	subset	subset	NOUN
ejpam-6940	73	5	t	t	NOUN
ejpam-6940	73	6	is	be	AUX
ejpam-6940	73	7	not	not	PART
ejpam-6940	73	8	an	an	DET
ejpam-6940	73	9	ideal	ideal	NOUN
ejpam-6940	73	10	since	since	SCONJ
ejpam-6940	73	11	c	c	NOUN
ejpam-6940	73	12	∗	∗	X
ejpam-6940	73	13	f	f	NOUN
ejpam-6940	73	14	=	=	SYM
ejpam-6940	73	15	0	0	NUM
ejpam-6940	73	16	∈	∈	PROPN
ejpam-6940	73	17	t	t	PROPN
ejpam-6940	73	18	and	and	CCONJ
ejpam-6940	73	19	f	f	PROPN
ejpam-6940	73	20	∈	∈	PROPN
ejpam-6940	73	21	t	t	PROPN
ejpam-6940	73	22	but	but	CCONJ
ejpam-6940	73	23	c	c	PROPN
ejpam-6940	73	24	/∈	/∈	PROPN
ejpam-6940	74	1	t	t	PROPN
ejpam-6940	74	2	.	.	PUNCT
ejpam-6940	75	1	let	let	AUX
ejpam-6940	75	2	consider	consider	VERB
ejpam-6940	75	3	a	a	DET
ejpam-6940	75	4	subset	subset	NOUN
ejpam-6940	76	1	i	i	PRON
ejpam-6940	76	2	=	=	PUNCT
ejpam-6940	76	3	{	{	PUNCT
ejpam-6940	76	4	0	0	NUM
ejpam-6940	76	5	,	,	PUNCT
ejpam-6940	76	6	a	a	PRON
ejpam-6940	76	7	}	}	PUNCT
ejpam-6940	76	8	.	.	PUNCT
ejpam-6940	77	1	it	it	PRON
ejpam-6940	77	2	is	be	AUX
ejpam-6940	77	3	not	not	PART
ejpam-6940	77	4	difficult	difficult	ADJ
ejpam-6940	77	5	to	to	PART
ejpam-6940	77	6	check	check	VERB
ejpam-6940	77	7	that	that	SCONJ
ejpam-6940	77	8	i	i	PRON
ejpam-6940	77	9	is	be	AUX
ejpam-6940	77	10	an	an	DET
ejpam-6940	77	11	ideal	ideal	NOUN
ejpam-6940	77	12	and	and	CCONJ
ejpam-6940	77	13	i	i	NOUN
ejpam-6940	77	14	=	=	SYM
ejpam-6940	77	15	g(x	g(x	NOUN
ejpam-6940	77	16	)	)	PUNCT
ejpam-6940	77	17	.	.	PUNCT
ejpam-6940	78	1	in	in	ADP
ejpam-6940	78	2	2004	2004	NUM
ejpam-6940	78	3	,	,	PUNCT
ejpam-6940	78	4	s.	s.	PROPN
ejpam-6940	78	5	s.	s.	PROPN
ejpam-6940	78	6	ahn	ahn	PROPN
ejpam-6940	78	7	,	,	PUNCT
ejpam-6940	78	8	h.	h.	PROPN
ejpam-6940	78	9	kim	kim	PROPN
ejpam-6940	78	10	and	and	CCONJ
ejpam-6940	78	11	h.	h.	PROPN
ejpam-6940	78	12	d.	d.	PROPN
ejpam-6940	78	13	lee	lee	PROPN
ejpam-6940	78	14	discussed	discuss	VERB
ejpam-6940	78	15	the	the	DET
ejpam-6940	78	16	homomorphisms	homomorphism	NOUN
ejpam-6940	78	17	of	of	ADP
ejpam-6940	78	18	q	q	NOUN
ejpam-6940	78	19	-	-	NOUN
ejpam-6940	78	20	algebra	algebra	NOUN
ejpam-6940	78	21	in	in	ADP
ejpam-6940	78	22	[	[	X
ejpam-6940	78	23	10	10	NUM
ejpam-6940	78	24	]	]	PUNCT
ejpam-6940	78	25	.	.	PUNCT
ejpam-6940	79	1	they	they	PRON
ejpam-6940	79	2	introduced	introduce	VERB
ejpam-6940	79	3	the	the	DET
ejpam-6940	79	4	self	self	NOUN
ejpam-6940	79	5	-	-	PUNCT
ejpam-6940	79	6	maps	map	NOUN
ejpam-6940	79	7	of	of	ADP
ejpam-6940	79	8	q	q	NOUN
ejpam-6940	79	9	-	-	PUNCT
ejpam-6940	79	10	algebras	algebra	NOUN
ejpam-6940	79	11	which	which	PRON
ejpam-6940	79	12	are	be	AUX
ejpam-6940	79	13	called	call	VERB
ejpam-6940	79	14	a	a	DET
ejpam-6940	79	15	right	right	ADJ
ejpam-6940	79	16	map	map	NOUN
ejpam-6940	79	17	and	and	CCONJ
ejpam-6940	79	18	a	a	DET
ejpam-6940	79	19	left	left	ADJ
ejpam-6940	79	20	map	map	NOUN
ejpam-6940	79	21	.	.	PUNCT
ejpam-6940	80	1	they	they	PRON
ejpam-6940	80	2	obtained	obtain	VERB
ejpam-6940	80	3	that	that	SCONJ
ejpam-6940	80	4	a	a	DET
ejpam-6940	80	5	right	right	ADJ
ejpam-6940	80	6	map	map	NOUN
ejpam-6940	80	7	is	be	AUX
ejpam-6940	80	8	an	an	DET
ejpam-6940	80	9	endomorphism	endomorphism	NOUN
ejpam-6940	80	10	whenever	whenever	SCONJ
ejpam-6940	80	11	x	x	PRON
ejpam-6940	80	12	is	be	AUX
ejpam-6940	80	13	a	a	DET
ejpam-6940	80	14	positive	positive	ADJ
ejpam-6940	80	15	implicative	implicative	ADJ
ejpam-6940	80	16	q	q	NOUN
ejpam-6940	80	17	-	-	PUNCT
ejpam-6940	80	18	algebra	algebra	NOUN
ejpam-6940	80	19	,	,	PUNCT
ejpam-6940	80	20	i.e.	i.e.	X
ejpam-6940	80	21	(	(	PUNCT
ejpam-6940	80	22	xy)(xz	xy)(xz	NOUN
ejpam-6940	80	23	)	)	PUNCT
ejpam-6940	80	24	=	=	SYM
ejpam-6940	80	25	xyz	xyz	NOUN
ejpam-6940	80	26	for	for	ADP
ejpam-6940	80	27	all	all	DET
ejpam-6940	80	28	x	x	PROPN
ejpam-6940	80	29	,	,	PUNCT
ejpam-6940	80	30	y	y	PROPN
ejpam-6940	80	31	,	,	PUNCT
ejpam-6940	80	32	z	z	NOUN
ejpam-6940	80	33	∈	∈	PROPN
ejpam-6940	80	34	x.	x.	NOUN
ejpam-6940	80	35	in	in	ADP
ejpam-6940	80	36	2011	2011	NUM
ejpam-6940	80	37	,	,	PUNCT
ejpam-6940	80	38	another	another	DET
ejpam-6940	80	39	self	self	NOUN
ejpam-6940	80	40	-	-	PUNCT
ejpam-6940	80	41	map	map	NOUN
ejpam-6940	80	42	of	of	ADP
ejpam-6940	80	43	a	a	DET
ejpam-6940	80	44	q	q	NOUN
ejpam-6940	80	45	-	-	PUNCT
ejpam-6940	80	46	algebra	algebra	NOUN
ejpam-6940	80	47	x	x	NOUN
ejpam-6940	80	48	,	,	PUNCT
ejpam-6940	80	49	called	call	VERB
ejpam-6940	80	50	a	a	DET
ejpam-6940	80	51	right	right	ADJ
ejpam-6940	80	52	fixed	fix	VERB
ejpam-6940	80	53	map	map	NOUN
ejpam-6940	80	54	,	,	PUNCT
ejpam-6940	80	55	is	be	AUX
ejpam-6940	80	56	discussed	discuss	VERB
ejpam-6940	80	57	and	and	CCONJ
ejpam-6940	80	58	examined	examine	VERB
ejpam-6940	80	59	their	their	PRON
ejpam-6940	80	60	properties	property	NOUN
ejpam-6940	80	61	in	in	ADP
ejpam-6940	80	62	[	[	X
ejpam-6940	80	63	11	11	NUM
ejpam-6940	80	64	]	]	PUNCT
ejpam-6940	80	65	by	by	ADP
ejpam-6940	80	66	s.	s.	PROPN
ejpam-6940	80	67	m.	m.	PROPN
ejpam-6940	80	68	lee	lee	PROPN
ejpam-6940	80	69	.	.	PUNCT
ejpam-6940	81	1	the	the	DET
ejpam-6940	81	2	author	author	NOUN
ejpam-6940	81	3	showed	show	VERB
ejpam-6940	81	4	that	that	SCONJ
ejpam-6940	81	5	the	the	DET
ejpam-6940	81	6	set	set	NOUN
ejpam-6940	81	7	of	of	ADP
ejpam-6940	81	8	all	all	DET
ejpam-6940	81	9	right	right	ADV
ejpam-6940	81	10	fixed	fix	VERB
ejpam-6940	81	11	maps	map	NOUN
ejpam-6940	81	12	of	of	ADP
ejpam-6940	81	13	x	x	X
ejpam-6940	81	14	is	be	AUX
ejpam-6940	81	15	a	a	DET
ejpam-6940	81	16	q	q	NOUN
ejpam-6940	81	17	-	-	NOUN
ejpam-6940	81	18	algebra	algebra	NOUN
ejpam-6940	81	19	under	under	ADP
ejpam-6940	81	20	a	a	DET
ejpam-6940	81	21	binary	binary	ADJ
ejpam-6940	81	22	operation	operation	NOUN
ejpam-6940	81	23	which	which	PRON
ejpam-6940	81	24	is	be	AUX
ejpam-6940	81	25	induced	induce	VERB
ejpam-6940	81	26	from	from	ADP
ejpam-6940	81	27	a	a	DET
ejpam-6940	81	28	binary	binary	ADJ
ejpam-6940	81	29	operation	operation	NOUN
ejpam-6940	81	30	on	on	ADP
ejpam-6940	81	31	x.	x.	NOUN
ejpam-6940	81	32	in	in	ADP
ejpam-6940	81	33	2010	2010	NUM
ejpam-6940	81	34	,	,	PUNCT
ejpam-6940	81	35	s.	s.	PROPN
ejpam-6940	81	36	s.	s.	PROPN
ejpam-6940	81	37	ahn	ahn	PROPN
ejpam-6940	81	38	and	and	CCONJ
ejpam-6940	81	39	s.	s.	PROPN
ejpam-6940	81	40	e.	e.	PROPN
ejpam-6940	81	41	kang	kang	PROPN
ejpam-6940	81	42	proposed	propose	VERB
ejpam-6940	81	43	the	the	DET
ejpam-6940	81	44	concept	concept	NOUN
ejpam-6940	81	45	of	of	ADP
ejpam-6940	81	46	atom	atom	NOUN
ejpam-6940	81	47	in	in	ADP
ejpam-6940	81	48	q	q	NOUN
ejpam-6940	81	49	-	-	PUNCT
ejpam-6940	81	50	algebras	algebras	X
ejpam-6940	81	51	.	.	PUNCT
ejpam-6940	82	1	an	an	DET
ejpam-6940	82	2	element	element	NOUN
ejpam-6940	82	3	w	w	PROPN
ejpam-6940	82	4	of	of	ADP
ejpam-6940	82	5	x	x	PUNCT
ejpam-6940	82	6	is	be	AUX
ejpam-6940	82	7	an	an	DET
ejpam-6940	82	8	atom	atom	NOUN
ejpam-6940	82	9	if	if	SCONJ
ejpam-6940	82	10	xw	xw	PROPN
ejpam-6940	82	11	=	=	SYM
ejpam-6940	82	12	0	0	PROPN
ejpam-6940	82	13	implies	imply	VERB
ejpam-6940	82	14	x	x	PUNCT
ejpam-6940	82	15	=	=	SYM
ejpam-6940	82	16	w	w	PROPN
ejpam-6940	82	17	for	for	ADP
ejpam-6940	82	18	all	all	PRON
ejpam-6940	82	19	x	x	SYM
ejpam-6940	82	20	∈	∈	ADJ
ejpam-6940	82	21	x.	x.	NOUN
ejpam-6940	83	1	they	they	PRON
ejpam-6940	83	2	showed	show	VERB
ejpam-6940	83	3	that	that	SCONJ
ejpam-6940	83	4	any	any	DET
ejpam-6940	83	5	subalgebra	subalgebra	NOUN
ejpam-6940	83	6	of	of	ADP
ejpam-6940	83	7	x	x	PUNCT
ejpam-6940	83	8	is	be	AUX
ejpam-6940	83	9	an	an	DET
ejpam-6940	83	10	ideal	ideal	NOUN
ejpam-6940	83	11	of	of	ADP
ejpam-6940	83	12	x	x	SYM
ejpam-6940	83	13	whenever	whenever	SCONJ
ejpam-6940	83	14	every	every	DET
ejpam-6940	83	15	non	non	ADJ
ejpam-6940	83	16	-	-	ADJ
ejpam-6940	83	17	zero	zero	NUM
ejpam-6940	83	18	element	element	NOUN
ejpam-6940	83	19	of	of	ADP
ejpam-6940	83	20	x	x	PUNCT
ejpam-6940	83	21	is	be	AUX
ejpam-6940	83	22	an	an	DET
ejpam-6940	83	23	atom	atom	NOUN
ejpam-6940	83	24	[	[	X
ejpam-6940	83	25	12	12	NUM
ejpam-6940	83	26	]	]	PUNCT
ejpam-6940	83	27	.	.	PUNCT
ejpam-6940	84	1	in	in	ADP
ejpam-6940	84	2	2024	2024	NUM
ejpam-6940	84	3	the	the	DET
ejpam-6940	84	4	authors	author	NOUN
ejpam-6940	84	5	in	in	ADP
ejpam-6940	84	6	[	[	X
ejpam-6940	84	7	13	13	NUM
ejpam-6940	84	8	]	]	PUNCT
ejpam-6940	84	9	examined	examine	VERB
ejpam-6940	84	10	some	some	DET
ejpam-6940	84	11	properties	property	NOUN
ejpam-6940	84	12	of	of	ADP
ejpam-6940	84	13	atoms	atom	NOUN
ejpam-6940	84	14	in	in	ADP
ejpam-6940	84	15	q	q	NOUN
ejpam-6940	84	16	-	-	PUNCT
ejpam-6940	84	17	algebras	algebras	X
ejpam-6940	84	18	.	.	PUNCT
ejpam-6940	85	1	they	they	PRON
ejpam-6940	85	2	showed	show	VERB
ejpam-6940	85	3	some	some	DET
ejpam-6940	85	4	relations	relation	NOUN
ejpam-6940	85	5	between	between	ADP
ejpam-6940	85	6	atoms	atom	NOUN
ejpam-6940	85	7	and	and	CCONJ
ejpam-6940	85	8	the	the	DET
ejpam-6940	85	9	set	set	NOUN
ejpam-6940	85	10	g	g	NOUN
ejpam-6940	85	11	-	-	PUNCT
ejpam-6940	85	12	part	part	NOUN
ejpam-6940	85	13	which	which	PRON
ejpam-6940	85	14	is	be	AUX
ejpam-6940	85	15	related	relate	VERB
ejpam-6940	85	16	to	to	ADP
ejpam-6940	85	17	the	the	DET
ejpam-6940	85	18	concept	concept	NOUN
ejpam-6940	85	19	of	of	ADP
ejpam-6940	85	20	ideal	ideal	NOUN
ejpam-6940	85	21	.	.	PUNCT
ejpam-6940	86	1	they	they	PRON
ejpam-6940	86	2	proved	prove	VERB
ejpam-6940	86	3	that	that	SCONJ
ejpam-6940	86	4	every	every	DET
ejpam-6940	86	5	element	element	NOUN
ejpam-6940	86	6	of	of	ADP
ejpam-6940	86	7	g(x	g(x	NOUN
ejpam-6940	86	8	)	)	PUNCT
ejpam-6940	86	9	is	be	AUX
ejpam-6940	86	10	an	an	DET
ejpam-6940	86	11	atom	atom	NOUN
ejpam-6940	86	12	whenever	whenever	SCONJ
ejpam-6940	86	13	g(x	g(x	NOUN
ejpam-6940	86	14	)	)	PUNCT
ejpam-6940	86	15	is	be	AUX
ejpam-6940	86	16	an	an	DET
ejpam-6940	86	17	ideal	ideal	NOUN
ejpam-6940	86	18	.	.	PUNCT
ejpam-6940	87	1	the	the	DET
ejpam-6940	87	2	notion	notion	NOUN
ejpam-6940	87	3	of	of	ADP
ejpam-6940	87	4	strong	strong	ADJ
ejpam-6940	87	5	atoms	atom	NOUN
ejpam-6940	87	6	was	be	AUX
ejpam-6940	87	7	also	also	ADV
ejpam-6940	87	8	offered	offer	VERB
ejpam-6940	87	9	in	in	ADP
ejpam-6940	87	10	[	[	X
ejpam-6940	87	11	13	13	NUM
ejpam-6940	87	12	]	]	PUNCT
ejpam-6940	87	13	,	,	PUNCT
ejpam-6940	87	14	which	which	PRON
ejpam-6940	87	15	was	be	AUX
ejpam-6940	87	16	inspired	inspire	VERB
ejpam-6940	87	17	from	from	ADP
ejpam-6940	87	18	the	the	DET
ejpam-6940	87	19	concept	concept	NOUN
ejpam-6940	87	20	of	of	ADP
ejpam-6940	87	21	strong	strong	ADJ
ejpam-6940	87	22	atom	atom	NOUN
ejpam-6940	87	23	in	in	ADP
ejpam-6940	87	24	bck	bck	NOUN
ejpam-6940	87	25	-	-	PUNCT
ejpam-6940	87	26	algebra	algebra	PROPN
ejpam-6940	87	27	(	(	PUNCT
ejpam-6940	87	28	see	see	VERB
ejpam-6940	87	29	[	[	X
ejpam-6940	87	30	14	14	NUM
ejpam-6940	87	31	]	]	NUM
ejpam-6940	87	32	)	)	PUNCT
ejpam-6940	87	33	.	.	PUNCT
ejpam-6940	88	1	they	they	PRON
ejpam-6940	88	2	proved	prove	VERB
ejpam-6940	88	3	that	that	SCONJ
ejpam-6940	88	4	x	x	PRON
ejpam-6940	88	5	does	do	AUX
ejpam-6940	88	6	not	not	PART
ejpam-6940	88	7	contain	contain	VERB
ejpam-6940	88	8	a	a	DET
ejpam-6940	88	9	strong	strong	ADJ
ejpam-6940	88	10	atom	atom	NOUN
ejpam-6940	88	11	whenever	whenever	SCONJ
ejpam-6940	88	12	x	x	PRON
ejpam-6940	88	13	contains	contain	VERB
ejpam-6940	88	14	a	a	DET
ejpam-6940	88	15	non	non	ADJ
ejpam-6940	88	16	-	-	ADJ
ejpam-6940	88	17	zero	zero	NUM
ejpam-6940	88	18	ideal	ideal	NOUN
ejpam-6940	88	19	g(x	g(x	NOUN
ejpam-6940	88	20	)	)	PUNCT
ejpam-6940	88	21	.	.	PUNCT
ejpam-6940	89	1	the	the	DET
ejpam-6940	89	2	concept	concept	NOUN
ejpam-6940	89	3	of	of	ADP
ejpam-6940	89	4	fuzzy	fuzzy	ADJ
ejpam-6940	89	5	set	set	NOUN
ejpam-6940	89	6	on	on	ADP
ejpam-6940	89	7	q	q	NOUN
ejpam-6940	89	8	-	-	PUNCT
ejpam-6940	89	9	algebra	algebra	NOUN
ejpam-6940	89	10	can	can	AUX
ejpam-6940	89	11	be	be	AUX
ejpam-6940	89	12	found	find	VERB
ejpam-6940	89	13	in	in	ADP
ejpam-6940	89	14	[	[	X
ejpam-6940	89	15	15	15	NUM
ejpam-6940	89	16	]	]	PUNCT
ejpam-6940	89	17	and	and	CCONJ
ejpam-6940	89	18	[	[	X
ejpam-6940	89	19	16	16	NUM
ejpam-6940	89	20	]	]	PUNCT
ejpam-6940	89	21	.	.	PUNCT
ejpam-6940	90	1	the	the	DET
ejpam-6940	90	2	authors	author	NOUN
ejpam-6940	90	3	provided	provide	VERB
ejpam-6940	90	4	some	some	DET
ejpam-6940	90	5	properties	property	NOUN
ejpam-6940	90	6	of	of	ADP
ejpam-6940	90	7	fuzzy	fuzzy	ADJ
ejpam-6940	90	8	q	q	NOUN
ejpam-6940	90	9	-	-	PUNCT
ejpam-6940	90	10	ideals	ideal	NOUN
ejpam-6940	90	11	,	,	PUNCT
ejpam-6940	90	12	fuzzy	fuzzy	ADJ
ejpam-6940	90	13	prime	prime	ADJ
ejpam-6940	90	14	ideals	ideal	NOUN
ejpam-6940	90	15	and	and	CCONJ
ejpam-6940	90	16	fuzzy	fuzzy	ADJ
ejpam-6940	90	17	relations	relation	NOUN
ejpam-6940	90	18	of	of	ADP
ejpam-6940	90	19	q	q	NOUN
ejpam-6940	90	20	-	-	PUNCT
ejpam-6940	90	21	algebras	algebras	X
ejpam-6940	90	22	.	.	PUNCT
ejpam-6940	91	1	recently	recently	ADV
ejpam-6940	91	2	,	,	PUNCT
ejpam-6940	91	3	in	in	ADP
ejpam-6940	91	4	2025	2025	NUM
ejpam-6940	91	5	the	the	DET
ejpam-6940	91	6	authors	author	NOUN
ejpam-6940	91	7	in	in	ADP
ejpam-6940	91	8	[	[	X
ejpam-6940	91	9	17	17	NUM
ejpam-6940	91	10	]	]	PUNCT
ejpam-6940	91	11	discussed	discuss	VERB
ejpam-6940	91	12	the	the	DET
ejpam-6940	91	13	concept	concept	NOUN
ejpam-6940	91	14	of	of	ADP
ejpam-6940	91	15	ideal	ideal	NOUN
ejpam-6940	91	16	in	in	ADP
ejpam-6940	91	17	q	q	NOUN
ejpam-6940	91	18	-	-	NOUN
ejpam-6940	91	19	algebra	algebra	NOUN
ejpam-6940	91	20	.	.	PUNCT
ejpam-6940	92	1	they	they	PRON
ejpam-6940	92	2	provided	provide	VERB
ejpam-6940	92	3	a	a	DET
ejpam-6940	92	4	characterization	characterization	NOUN
ejpam-6940	92	5	of	of	ADP
ejpam-6940	92	6	ideals	ideal	NOUN
ejpam-6940	92	7	which	which	PRON
ejpam-6940	92	8	is	be	AUX
ejpam-6940	92	9	related	relate	VERB
ejpam-6940	92	10	to	to	ADP
ejpam-6940	92	11	the	the	DET
ejpam-6940	92	12	g	g	NOUN
ejpam-6940	92	13	-	-	PUNCT
ejpam-6940	92	14	part	part	NOUN
ejpam-6940	92	15	.	.	PUNCT
ejpam-6940	93	1	they	they	PRON
ejpam-6940	93	2	showed	show	VERB
ejpam-6940	93	3	that	that	SCONJ
ejpam-6940	93	4	every	every	DET
ejpam-6940	93	5	g	g	NOUN
ejpam-6940	93	6	-	-	PUNCT
ejpam-6940	93	7	part	part	NOUN
ejpam-6940	93	8	that	that	PRON
ejpam-6940	93	9	is	be	AUX
ejpam-6940	93	10	an	an	DET
ejpam-6940	93	11	ideal	ideal	NOUN
ejpam-6940	93	12	,	,	PUNCT
ejpam-6940	93	13	is	be	AUX
ejpam-6940	93	14	an	an	DET
ejpam-6940	93	15	abelian	abelian	ADJ
ejpam-6940	93	16	group	group	NOUN
ejpam-6940	93	17	.	.	PUNCT
ejpam-6940	94	1	in	in	ADP
ejpam-6940	94	2	this	this	DET
ejpam-6940	94	3	work	work	NOUN
ejpam-6940	94	4	,	,	PUNCT
ejpam-6940	94	5	we	we	PRON
ejpam-6940	94	6	introduce	introduce	VERB
ejpam-6940	94	7	the	the	DET
ejpam-6940	94	8	notion	notion	NOUN
ejpam-6940	94	9	of	of	ADP
ejpam-6940	94	10	edge	edge	NOUN
ejpam-6940	94	11	q	q	NOUN
ejpam-6940	94	12	-	-	PUNCT
ejpam-6940	94	13	algebras	algebras	X
ejpam-6940	94	14	.	.	PUNCT
ejpam-6940	95	1	we	we	PRON
ejpam-6940	95	2	describe	describe	VERB
ejpam-6940	95	3	all	all	DET
ejpam-6940	95	4	possible	possible	ADJ
ejpam-6940	95	5	edge	edge	NOUN
ejpam-6940	95	6	q	q	NOUN
ejpam-6940	95	7	-	-	PUNCT
ejpam-6940	95	8	algebras	algebra	NOUN
ejpam-6940	95	9	of	of	ADP
ejpam-6940	95	10	order	order	NOUN
ejpam-6940	95	11	n.	n.	NOUN
ejpam-6940	95	12	we	we	PRON
ejpam-6940	95	13	examine	examine	VERB
ejpam-6940	95	14	some	some	DET
ejpam-6940	95	15	properties	property	NOUN
ejpam-6940	95	16	of	of	ADP
ejpam-6940	95	17	edge	edge	NOUN
ejpam-6940	95	18	q	q	NOUN
ejpam-6940	95	19	-	-	PUNCT
ejpam-6940	95	20	algebras	algebras	X
ejpam-6940	95	21	.	.	PUNCT
ejpam-6940	96	1	the	the	DET
ejpam-6940	96	2	connection	connection	NOUN
ejpam-6940	96	3	between	between	ADP
ejpam-6940	96	4	d	d	NOUN
ejpam-6940	96	5	-	-	PUNCT
ejpam-6940	96	6	algebras	algebra	NOUN
ejpam-6940	96	7	and	and	CCONJ
ejpam-6940	96	8	edge	edge	NOUN
ejpam-6940	96	9	q	q	NOUN
ejpam-6940	96	10	-	-	PUNCT
ejpam-6940	96	11	algebras	algebras	PROPN
ejpam-6940	96	12	is	be	AUX
ejpam-6940	96	13	provided	provide	VERB
ejpam-6940	96	14	.	.	PUNCT
ejpam-6940	97	1	the	the	DET
ejpam-6940	97	2	concepts	concept	NOUN
ejpam-6940	97	3	of	of	ADP
ejpam-6940	97	4	subalgebras	subalgebras	PROPN
ejpam-6940	97	5	and	and	CCONJ
ejpam-6940	97	6	ideals	ideal	NOUN
ejpam-6940	97	7	are	be	AUX
ejpam-6940	97	8	discussed	discuss	VERB
ejpam-6940	97	9	in	in	ADP
ejpam-6940	97	10	edge	edge	NOUN
ejpam-6940	97	11	q	q	NOUN
ejpam-6940	97	12	-	-	PUNCT
ejpam-6940	97	13	algebras	algebras	X
ejpam-6940	97	14	.	.	PUNCT
ejpam-6940	98	1	we	we	PRON
ejpam-6940	98	2	give	give	VERB
ejpam-6940	98	3	necessary	necessary	ADJ
ejpam-6940	98	4	and	and	CCONJ
ejpam-6940	98	5	sufficient	sufficient	ADJ
ejpam-6940	98	6	conditions	condition	NOUN
ejpam-6940	98	7	for	for	ADP
ejpam-6940	98	8	subsets	subset	NOUN
ejpam-6940	98	9	of	of	ADP
ejpam-6940	98	10	an	an	DET
ejpam-6940	98	11	edge	edge	NOUN
ejpam-6940	98	12	q	q	NOUN
ejpam-6940	98	13	-	-	NOUN
ejpam-6940	98	14	algebra	algebra	NOUN
ejpam-6940	98	15	to	to	PART
ejpam-6940	98	16	be	be	AUX
ejpam-6940	98	17	subalgebras	subalgebras	PROPN
ejpam-6940	98	18	.	.	PUNCT
ejpam-6940	99	1	we	we	PRON
ejpam-6940	99	2	also	also	ADV
ejpam-6940	99	3	provide	provide	VERB
ejpam-6940	99	4	some	some	DET
ejpam-6940	99	5	properties	property	NOUN
ejpam-6940	99	6	in	in	ADP
ejpam-6940	99	7	edge	edge	NOUN
ejpam-6940	99	8	q	q	NOUN
ejpam-6940	99	9	-	-	PUNCT
ejpam-6940	99	10	algebras	algebra	NOUN
ejpam-6940	99	11	which	which	PRON
ejpam-6940	99	12	are	be	AUX
ejpam-6940	99	13	related	relate	VERB
ejpam-6940	99	14	to	to	ADP
ejpam-6940	99	15	the	the	DET
ejpam-6940	99	16	concept	concept	NOUN
ejpam-6940	99	17	of	of	ADP
ejpam-6940	99	18	ideals	ideal	NOUN
ejpam-6940	99	19	.	.	PUNCT
ejpam-6940	100	1	we	we	PRON
ejpam-6940	100	2	obtain	obtain	VERB
ejpam-6940	100	3	that	that	SCONJ
ejpam-6940	100	4	the	the	DET
ejpam-6940	100	5	set	set	NOUN
ejpam-6940	100	6	of	of	ADP
ejpam-6940	100	7	all	all	DET
ejpam-6940	100	8	ideals	ideal	NOUN
ejpam-6940	100	9	forms	form	VERB
ejpam-6940	100	10	a	a	DET
ejpam-6940	100	11	right	right	ADJ
ejpam-6940	100	12	zero	zero	NUM
ejpam-6940	100	13	semigroup	semigroup	NOUN
ejpam-6940	100	14	and	and	CCONJ
ejpam-6940	100	15	a	a	DET
ejpam-6940	100	16	simple	simple	ADJ
ejpam-6940	100	17	semigroup	semigroup	NOUN
ejpam-6940	100	18	.	.	PUNCT
ejpam-6940	101	1	finally	finally	ADV
ejpam-6940	101	2	,	,	PUNCT
ejpam-6940	101	3	we	we	PRON
ejpam-6940	101	4	describe	describe	VERB
ejpam-6940	101	5	all	all	DET
ejpam-6940	101	6	possible	possible	ADJ
ejpam-6940	101	7	structures	structure	NOUN
ejpam-6940	101	8	of	of	ADP
ejpam-6940	101	9	edge	edge	NOUN
ejpam-6940	101	10	q	q	NOUN
ejpam-6940	101	11	-	-	PUNCT
ejpam-6940	101	12	algebras	algebra	VERB
ejpam-6940	101	13	and	and	CCONJ
ejpam-6940	101	14	enumerate	enumerate	VERB
ejpam-6940	101	15	all	all	DET
ejpam-6940	101	16	members	member	NOUN
ejpam-6940	101	17	of	of	ADP
ejpam-6940	101	18	a	a	DET
ejpam-6940	101	19	class	class	NOUN
ejpam-6940	101	20	of	of	ADP
ejpam-6940	101	21	all	all	DET
ejpam-6940	101	22	edge	edge	NOUN
ejpam-6940	101	23	q	q	NOUN
ejpam-6940	101	24	-	-	PUNCT
ejpam-6940	101	25	algebras	algebras	X
ejpam-6940	101	26	.	.	PUNCT
ejpam-6940	101	27	a.	a.	PROPN
ejpam-6940	101	28	anantayasethi	anantayasethi	PROPN
ejpam-6940	101	29	et	et	PROPN
ejpam-6940	101	30	al	al	PROPN
ejpam-6940	101	31	.	.	PUNCT
ejpam-6940	101	32	/	/	SYM
ejpam-6940	101	33	eur	eur	PROPN
ejpam-6940	101	34	.	.	PUNCT
ejpam-6940	102	1	j.	j.	PROPN
ejpam-6940	102	2	pure	pure	PROPN
ejpam-6940	102	3	appl	appl	PROPN
ejpam-6940	102	4	.	.	PROPN
ejpam-6940	102	5	math	math	PROPN
ejpam-6940	102	6	,	,	PUNCT
ejpam-6940	102	7	18	18	NUM
ejpam-6940	102	8	(	(	PUNCT
ejpam-6940	102	9	4	4	NUM
ejpam-6940	102	10	)	)	PUNCT
ejpam-6940	102	11	(	(	PUNCT
ejpam-6940	102	12	2025	2025	NUM
ejpam-6940	102	13	)	)	PUNCT
ejpam-6940	102	14	,	,	PUNCT
ejpam-6940	102	15	6940	6940	NUM
ejpam-6940	102	16	4	4	NUM
ejpam-6940	102	17	of	of	ADP
ejpam-6940	102	18	14	14	NUM
ejpam-6940	102	19	2	2	NUM
ejpam-6940	102	20	.	.	PUNCT
ejpam-6940	102	21	edge	edge	PROPN
ejpam-6940	102	22	q	q	NOUN
ejpam-6940	102	23	-	-	PUNCT
ejpam-6940	102	24	algebras	algebras	ADV
ejpam-6940	102	25	let	let	VERB
ejpam-6940	102	26	a	a	PRON
ejpam-6940	102	27	and	and	CCONJ
ejpam-6940	102	28	b	b	NOUN
ejpam-6940	102	29	be	be	AUX
ejpam-6940	102	30	non	non	ADJ
ejpam-6940	102	31	-	-	ADJ
ejpam-6940	102	32	empty	empty	ADJ
ejpam-6940	102	33	subsets	subset	NOUN
ejpam-6940	102	34	of	of	ADP
ejpam-6940	102	35	a	a	DET
ejpam-6940	102	36	q	q	NOUN
ejpam-6940	102	37	-	-	PUNCT
ejpam-6940	102	38	algebra	algebra	NOUN
ejpam-6940	102	39	x.	x.	NOUN
ejpam-6940	102	40	we	we	PRON
ejpam-6940	102	41	define	define	VERB
ejpam-6940	102	42	a	a	DET
ejpam-6940	102	43	product	product	NOUN
ejpam-6940	102	44	ab	ab	NOUN
ejpam-6940	102	45	in	in	ADP
ejpam-6940	102	46	a	a	DET
ejpam-6940	102	47	usual	usual	ADJ
ejpam-6940	102	48	way	way	NOUN
ejpam-6940	102	49	as	as	SCONJ
ejpam-6940	102	50	follow	follow	VERB
ejpam-6940	102	51	:	:	PUNCT
ejpam-6940	102	52	ab	ab	PROPN
ejpam-6940	102	53	=	=	PUNCT
ejpam-6940	102	54	{	{	PUNCT
ejpam-6940	102	55	ab	ab	PROPN
ejpam-6940	102	56	|	|	ADV
ejpam-6940	102	57	a	a	DET
ejpam-6940	102	58	∈	∈	PROPN
ejpam-6940	102	59	a	a	PRON
ejpam-6940	102	60	,	,	PUNCT
ejpam-6940	102	61	b	b	PROPN
ejpam-6940	102	62	∈	∈	PROPN
ejpam-6940	102	63	b	b	NOUN
ejpam-6940	102	64	}	}	PUNCT
ejpam-6940	102	65	.	.	PUNCT
ejpam-6940	103	1	if	if	SCONJ
ejpam-6940	103	2	b	b	X
ejpam-6940	103	3	=	=	PRON
ejpam-6940	103	4	{	{	PUNCT
ejpam-6940	103	5	x	x	NOUN
ejpam-6940	103	6	}	}	PUNCT
ejpam-6940	103	7	,	,	PUNCT
ejpam-6940	103	8	we	we	PRON
ejpam-6940	103	9	denote	denote	VERB
ejpam-6940	103	10	ab	ab	PROPN
ejpam-6940	103	11	and	and	CCONJ
ejpam-6940	103	12	ba	ba	PROPN
ejpam-6940	103	13	by	by	ADP
ejpam-6940	103	14	ax	ax	NOUN
ejpam-6940	103	15	and	and	CCONJ
ejpam-6940	103	16	xa	xa	PROPN
ejpam-6940	103	17	,	,	PUNCT
ejpam-6940	103	18	respectively	respectively	ADV
ejpam-6940	103	19	.	.	PUNCT
ejpam-6940	104	1	proposition	proposition	NOUN
ejpam-6940	104	2	1	1	NUM
ejpam-6940	104	3	.	.	PUNCT
ejpam-6940	105	1	let	let	VERB
ejpam-6940	105	2	a	a	DET
ejpam-6940	105	3	,	,	PUNCT
ejpam-6940	105	4	b	b	NOUN
ejpam-6940	105	5	and	and	CCONJ
ejpam-6940	105	6	c	c	AUX
ejpam-6940	105	7	be	be	AUX
ejpam-6940	105	8	non	non	ADJ
ejpam-6940	105	9	-	-	ADJ
ejpam-6940	105	10	empty	empty	ADJ
ejpam-6940	105	11	subsets	subset	NOUN
ejpam-6940	105	12	of	of	ADP
ejpam-6940	105	13	a	a	DET
ejpam-6940	105	14	q	q	NOUN
ejpam-6940	105	15	-	-	PUNCT
ejpam-6940	105	16	algebra	algebra	NOUN
ejpam-6940	105	17	x.	x.	NOUN
ejpam-6940	105	18	then	then	ADV
ejpam-6940	105	19	the	the	DET
ejpam-6940	105	20	following	follow	VERB
ejpam-6940	105	21	properties	property	NOUN
ejpam-6940	105	22	are	be	AUX
ejpam-6940	105	23	valid	valid	ADJ
ejpam-6940	105	24	:	:	PUNCT
ejpam-6940	105	25	(	(	PUNCT
ejpam-6940	105	26	i	i	NOUN
ejpam-6940	105	27	)	)	PUNCT
ejpam-6940	105	28	(	(	PUNCT
ejpam-6940	106	1	ab)c	ab)c	PROPN
ejpam-6940	106	2	=	=	SYM
ejpam-6940	106	3	(	(	PUNCT
ejpam-6940	106	4	ac)b	ac)b	PROPN
ejpam-6940	106	5	.	.	PUNCT
ejpam-6940	106	6	(	(	PUNCT
ejpam-6940	106	7	ii	ii	NOUN
ejpam-6940	106	8	)	)	PUNCT
ejpam-6940	106	9	a{0	a{0	PROPN
ejpam-6940	106	10	}	}	PUNCT
ejpam-6940	106	11	=	=	SYM
ejpam-6940	106	12	a0	a0	PROPN
ejpam-6940	106	13	=	=	PUNCT
ejpam-6940	106	14	a.	a.	NOUN
ejpam-6940	106	15	(	(	PUNCT
ejpam-6940	106	16	iii	iii	NOUN
ejpam-6940	106	17	)	)	PUNCT
ejpam-6940	106	18	if	if	SCONJ
ejpam-6940	106	19	a	a	DET
ejpam-6940	106	20	⊆	⊆	NUM
ejpam-6940	106	21	b	b	NOUN
ejpam-6940	106	22	,	,	PUNCT
ejpam-6940	106	23	then	then	ADV
ejpam-6940	106	24	ac	ac	PROPN
ejpam-6940	106	25	⊆	⊆	NUM
ejpam-6940	106	26	bc	bc	PROPN
ejpam-6940	106	27	and	and	CCONJ
ejpam-6940	106	28	ca	can	AUX
ejpam-6940	106	29	⊆	⊆	NUM
ejpam-6940	106	30	cb	cb	PROPN
ejpam-6940	106	31	.	.	PROPN
ejpam-6940	106	32	(	(	PUNCT
ejpam-6940	106	33	iv	iv	X
ejpam-6940	106	34	)	)	PUNCT
ejpam-6940	106	35	if	if	SCONJ
ejpam-6940	106	36	0	0	NUM
ejpam-6940	106	37	∈	∈	PROPN
ejpam-6940	106	38	b	b	NOUN
ejpam-6940	106	39	,	,	PUNCT
ejpam-6940	106	40	then	then	ADV
ejpam-6940	106	41	a	a	DET
ejpam-6940	106	42	⊆	⊆	NUM
ejpam-6940	106	43	ab	ab	X
ejpam-6940	106	44	.	.	PUNCT
ejpam-6940	107	1	(	(	PUNCT
ejpam-6940	107	2	v	v	NOUN
ejpam-6940	107	3	)	)	PUNCT
ejpam-6940	107	4	if	if	SCONJ
ejpam-6940	107	5	a	a	DET
ejpam-6940	107	6	∩b	∩b	NOUN
ejpam-6940	107	7	̸=	̸=	NOUN
ejpam-6940	107	8	∅	∅	NOUN
ejpam-6940	107	9	,	,	PUNCT
ejpam-6940	107	10	then	then	ADV
ejpam-6940	107	11	0	0	NUM
ejpam-6940	107	12	∈	∈	PROPN
ejpam-6940	107	13	ab	ab	PROPN
ejpam-6940	107	14	.	.	PUNCT
ejpam-6940	107	15	proof	proof	NOUN
ejpam-6940	107	16	.	.	PUNCT
ejpam-6940	108	1	(	(	PUNCT
ejpam-6940	108	2	i	i	NOUN
ejpam-6940	108	3	)	)	PUNCT
ejpam-6940	108	4	follows	follow	VERB
ejpam-6940	108	5	directly	directly	ADV
ejpam-6940	108	6	from	from	ADP
ejpam-6940	108	7	the	the	DET
ejpam-6940	108	8	condition	condition	NOUN
ejpam-6940	108	9	(	(	PUNCT
ejpam-6940	108	10	q3	q3	PROPN
ejpam-6940	108	11	)	)	PUNCT
ejpam-6940	108	12	.	.	PUNCT
ejpam-6940	109	1	(	(	PUNCT
ejpam-6940	109	2	ii	ii	NOUN
ejpam-6940	109	3	)	)	PUNCT
ejpam-6940	109	4	by	by	ADP
ejpam-6940	109	5	the	the	DET
ejpam-6940	109	6	condition	condition	NOUN
ejpam-6940	109	7	(	(	PUNCT
ejpam-6940	109	8	q2	q2	NOUN
ejpam-6940	109	9	)	)	PUNCT
ejpam-6940	109	10	,	,	PUNCT
ejpam-6940	109	11	we	we	PRON
ejpam-6940	109	12	get	get	VERB
ejpam-6940	109	13	a{0	a{0	ADJ
ejpam-6940	109	14	}	}	PUNCT
ejpam-6940	109	15	=	=	SYM
ejpam-6940	109	16	{	{	PUNCT
ejpam-6940	109	17	a0	a0	NOUN
ejpam-6940	109	18	|	|	CCONJ
ejpam-6940	109	19	a	a	DET
ejpam-6940	109	20	∈	∈	PROPN
ejpam-6940	109	21	a	a	DET
ejpam-6940	109	22	}	}	PUNCT
ejpam-6940	109	23	=	=	SYM
ejpam-6940	109	24	{	{	PUNCT
ejpam-6940	109	25	a	a	DET
ejpam-6940	109	26	|	|	NOUN
ejpam-6940	109	27	a	a	DET
ejpam-6940	109	28	∈	∈	NOUN
ejpam-6940	109	29	a	a	DET
ejpam-6940	109	30	}	}	PUNCT
ejpam-6940	109	31	=	=	PUNCT
ejpam-6940	109	32	a.	a.	NOUN
ejpam-6940	109	33	(	(	PUNCT
ejpam-6940	109	34	iii	iii	X
ejpam-6940	109	35	)	)	PUNCT
ejpam-6940	109	36	it	it	PRON
ejpam-6940	109	37	is	be	AUX
ejpam-6940	109	38	obvious	obvious	ADJ
ejpam-6940	109	39	.	.	PUNCT
ejpam-6940	110	1	(	(	PUNCT
ejpam-6940	110	2	iv	iv	X
ejpam-6940	110	3	)	)	PUNCT
ejpam-6940	110	4	by	by	ADP
ejpam-6940	110	5	(	(	PUNCT
ejpam-6940	110	6	ii	ii	NOUN
ejpam-6940	110	7	)	)	PUNCT
ejpam-6940	110	8	,	,	PUNCT
ejpam-6940	110	9	(	(	PUNCT
ejpam-6940	110	10	iii	iii	NOUN
ejpam-6940	110	11	)	)	PUNCT
ejpam-6940	110	12	and	and	CCONJ
ejpam-6940	110	13	since	since	SCONJ
ejpam-6940	110	14	0	0	NUM
ejpam-6940	110	15	∈	∈	PROPN
ejpam-6940	110	16	b	b	PROPN
ejpam-6940	110	17	,	,	PUNCT
ejpam-6940	110	18	then	then	ADV
ejpam-6940	110	19	a	a	DET
ejpam-6940	110	20	=	=	PUNCT
ejpam-6940	110	21	a{0	a{0	ADJ
ejpam-6940	110	22	}	}	PUNCT
ejpam-6940	110	23	⊆	⊆	NUM
ejpam-6940	110	24	ab	ab	PROPN
ejpam-6940	110	25	.	.	PUNCT
ejpam-6940	111	1	(	(	PUNCT
ejpam-6940	111	2	v	v	NOUN
ejpam-6940	111	3	)	)	PUNCT
ejpam-6940	111	4	let	let	VERB
ejpam-6940	111	5	x	x	PART
ejpam-6940	111	6	∈	∈	VERB
ejpam-6940	111	7	a	a	DET
ejpam-6940	111	8	∩b	∩b	NOUN
ejpam-6940	111	9	.	.	PUNCT
ejpam-6940	112	1	then	then	ADV
ejpam-6940	112	2	by	by	ADP
ejpam-6940	112	3	the	the	DET
ejpam-6940	112	4	condition	condition	NOUN
ejpam-6940	112	5	(	(	PUNCT
ejpam-6940	112	6	q1	q1	PROPN
ejpam-6940	112	7	)	)	PUNCT
ejpam-6940	112	8	,	,	PUNCT
ejpam-6940	112	9	0	0	NUM
ejpam-6940	112	10	=	=	SYM
ejpam-6940	112	11	xx	xx	NUM
ejpam-6940	112	12	∈	∈	PROPN
ejpam-6940	112	13	ab	ab	PROPN
ejpam-6940	112	14	.	.	PUNCT
ejpam-6940	113	1	concerning	concern	VERB
ejpam-6940	113	2	to	to	ADP
ejpam-6940	113	3	the	the	DET
ejpam-6940	113	4	concept	concept	NOUN
ejpam-6940	113	5	of	of	ADP
ejpam-6940	113	6	subalgebra	subalgebra	NOUN
ejpam-6940	113	7	in	in	ADP
ejpam-6940	113	8	algebras	algebra	NOUN
ejpam-6940	113	9	,	,	PUNCT
ejpam-6940	113	10	in	in	ADP
ejpam-6940	113	11	general	general	ADJ
ejpam-6940	113	12	a	a	DET
ejpam-6940	113	13	subalgebra	subalgebra	NOUN
ejpam-6940	113	14	is	be	AUX
ejpam-6940	113	15	a	a	DET
ejpam-6940	113	16	non	non	ADJ
ejpam-6940	113	17	-	-	ADJ
ejpam-6940	113	18	empty	empty	ADJ
ejpam-6940	113	19	subset	subset	NOUN
ejpam-6940	113	20	which	which	PRON
ejpam-6940	113	21	is	be	AUX
ejpam-6940	113	22	closed	close	VERB
ejpam-6940	113	23	.	.	PUNCT
ejpam-6940	114	1	in	in	ADP
ejpam-6940	114	2	q	q	NOUN
ejpam-6940	114	3	-	-	PUNCT
ejpam-6940	114	4	algebras	algebras	X
ejpam-6940	114	5	,	,	PUNCT
ejpam-6940	114	6	we	we	PRON
ejpam-6940	114	7	get	get	VERB
ejpam-6940	114	8	a	a	DET
ejpam-6940	114	9	sharp	sharp	ADJ
ejpam-6940	114	10	condition	condition	NOUN
ejpam-6940	114	11	as	as	SCONJ
ejpam-6940	114	12	seen	see	VERB
ejpam-6940	114	13	in	in	ADP
ejpam-6940	114	14	the	the	DET
ejpam-6940	114	15	following	follow	VERB
ejpam-6940	114	16	proposition	proposition	NOUN
ejpam-6940	114	17	:	:	PUNCT
ejpam-6940	114	18	proposition	proposition	NOUN
ejpam-6940	114	19	2	2	NUM
ejpam-6940	114	20	.	.	PUNCT
ejpam-6940	115	1	let	let	VERB
ejpam-6940	115	2	a	a	PRON
ejpam-6940	115	3	and	and	CCONJ
ejpam-6940	115	4	b	b	NOUN
ejpam-6940	115	5	be	be	AUX
ejpam-6940	115	6	non	non	ADJ
ejpam-6940	115	7	-	-	ADJ
ejpam-6940	115	8	empty	empty	ADJ
ejpam-6940	115	9	subsets	subset	NOUN
ejpam-6940	115	10	of	of	ADP
ejpam-6940	115	11	a	a	DET
ejpam-6940	115	12	q	q	NOUN
ejpam-6940	115	13	-	-	PUNCT
ejpam-6940	115	14	algebra	algebra	NOUN
ejpam-6940	115	15	x.	x.	NOUN
ejpam-6940	115	16	then	then	ADV
ejpam-6940	115	17	(	(	PUNCT
ejpam-6940	115	18	i	i	NOUN
ejpam-6940	115	19	)	)	PUNCT
ejpam-6940	115	20	a	a	PRON
ejpam-6940	115	21	is	be	AUX
ejpam-6940	115	22	a	a	DET
ejpam-6940	115	23	subalgebra	subalgebra	NOUN
ejpam-6940	115	24	of	of	ADP
ejpam-6940	115	25	x	x	PUNCT
ejpam-6940	115	26	if	if	SCONJ
ejpam-6940	115	27	and	and	CCONJ
ejpam-6940	116	1	only	only	ADV
ejpam-6940	116	2	if	if	SCONJ
ejpam-6940	116	3	aa	aa	NOUN
ejpam-6940	116	4	=	=	PUNCT
ejpam-6940	116	5	a.	a.	NOUN
ejpam-6940	116	6	(	(	PUNCT
ejpam-6940	116	7	ii	ii	NOUN
ejpam-6940	116	8	)	)	PUNCT
ejpam-6940	116	9	if	if	SCONJ
ejpam-6940	116	10	a	a	DET
ejpam-6940	116	11	⊆	⊆	NUM
ejpam-6940	116	12	b	b	NOUN
ejpam-6940	116	13	and	and	CCONJ
ejpam-6940	116	14	b	b	PROPN
ejpam-6940	116	15	is	be	AUX
ejpam-6940	116	16	a	a	DET
ejpam-6940	116	17	subalgebra	subalgebra	NOUN
ejpam-6940	116	18	,	,	PUNCT
ejpam-6940	116	19	then	then	ADV
ejpam-6940	116	20	ab	ab	PROPN
ejpam-6940	116	21	⊆	⊆	NUM
ejpam-6940	116	22	b.	b.	NOUN
ejpam-6940	116	23	proof	proof	NOUN
ejpam-6940	116	24	.	.	PUNCT
ejpam-6940	117	1	(	(	PUNCT
ejpam-6940	117	2	i	i	NOUN
ejpam-6940	117	3	)	)	PUNCT
ejpam-6940	117	4	assume	assume	VERB
ejpam-6940	117	5	that	that	SCONJ
ejpam-6940	117	6	a	a	PRON
ejpam-6940	117	7	is	be	AUX
ejpam-6940	117	8	a	a	DET
ejpam-6940	117	9	subalgebra	subalgebra	NOUN
ejpam-6940	117	10	of	of	ADP
ejpam-6940	117	11	x	x	PRON
ejpam-6940	117	12	,	,	PUNCT
ejpam-6940	117	13	then	then	ADV
ejpam-6940	117	14	aa	aa	NOUN
ejpam-6940	117	15	⊆	⊆	NUM
ejpam-6940	117	16	a.	a.	NOUN
ejpam-6940	117	17	since	since	SCONJ
ejpam-6940	117	18	0	0	NUM
ejpam-6940	117	19	∈	∈	PROPN
ejpam-6940	117	20	a	a	PRON
ejpam-6940	117	21	,	,	PUNCT
ejpam-6940	117	22	then	then	ADV
ejpam-6940	117	23	by	by	ADP
ejpam-6940	117	24	proposition	proposition	NOUN
ejpam-6940	117	25	1(iv	1(iv	NUM
ejpam-6940	117	26	)	)	PUNCT
ejpam-6940	117	27	,	,	PUNCT
ejpam-6940	117	28	a	a	DET
ejpam-6940	117	29	⊆	⊆	NUM
ejpam-6940	117	30	aa	aa	NOUN
ejpam-6940	117	31	.	.	PUNCT
ejpam-6940	118	1	thus	thus	ADV
ejpam-6940	118	2	,	,	PUNCT
ejpam-6940	118	3	aa	aa	NOUN
ejpam-6940	118	4	=	=	PUNCT
ejpam-6940	118	5	a.	a.	NOUN
ejpam-6940	118	6	the	the	DET
ejpam-6940	118	7	converse	converse	NOUN
ejpam-6940	118	8	direction	direction	NOUN
ejpam-6940	118	9	is	be	AUX
ejpam-6940	118	10	clear	clear	ADJ
ejpam-6940	118	11	.	.	PUNCT
ejpam-6940	119	1	(	(	PUNCT
ejpam-6940	119	2	ii	ii	NOUN
ejpam-6940	119	3	)	)	PUNCT
ejpam-6940	119	4	assume	assume	VERB
ejpam-6940	119	5	that	that	SCONJ
ejpam-6940	119	6	b	b	PROPN
ejpam-6940	119	7	is	be	AUX
ejpam-6940	119	8	a	a	DET
ejpam-6940	119	9	subalgebra	subalgebra	NOUN
ejpam-6940	119	10	of	of	ADP
ejpam-6940	119	11	x	x	PUNCT
ejpam-6940	119	12	and	and	CCONJ
ejpam-6940	119	13	∅	∅	NOUN
ejpam-6940	119	14	̸=	̸=	PROPN
ejpam-6940	119	15	a	a	DET
ejpam-6940	119	16	⊆	⊆	NUM
ejpam-6940	119	17	b.	b.	NOUN
ejpam-6940	119	18	then	then	ADV
ejpam-6940	119	19	by	by	ADP
ejpam-6940	119	20	(	(	PUNCT
ejpam-6940	119	21	i	i	NOUN
ejpam-6940	119	22	)	)	PUNCT
ejpam-6940	119	23	and	and	CCONJ
ejpam-6940	119	24	proposition	proposition	NOUN
ejpam-6940	119	25	1(iii	1(iii	NUM
ejpam-6940	119	26	)	)	PUNCT
ejpam-6940	119	27	,	,	PUNCT
ejpam-6940	119	28	we	we	PRON
ejpam-6940	119	29	get	get	VERB
ejpam-6940	119	30	ab	ab	PRON
ejpam-6940	119	31	⊆	⊆	NUM
ejpam-6940	119	32	bb	bb	NUM
ejpam-6940	119	33	⊆	⊆	NUM
ejpam-6940	119	34	b.	b.	NOUN
ejpam-6940	119	35	next	next	ADV
ejpam-6940	119	36	,	,	PUNCT
ejpam-6940	119	37	we	we	PRON
ejpam-6940	119	38	will	will	AUX
ejpam-6940	119	39	introduce	introduce	VERB
ejpam-6940	119	40	the	the	DET
ejpam-6940	119	41	notion	notion	NOUN
ejpam-6940	119	42	of	of	ADP
ejpam-6940	119	43	an	an	DET
ejpam-6940	119	44	edge	edge	NOUN
ejpam-6940	119	45	q	q	NOUN
ejpam-6940	119	46	-	-	NOUN
ejpam-6940	119	47	algebra	algebra	NOUN
ejpam-6940	119	48	.	.	PUNCT
ejpam-6940	120	1	in	in	ADP
ejpam-6940	120	2	d	d	NOUN
ejpam-6940	120	3	-	-	PUNCT
ejpam-6940	120	4	algebra	algebra	PROPN
ejpam-6940	120	5	,	,	PUNCT
ejpam-6940	120	6	j.	j.	PROPN
ejpam-6940	120	7	neggers	neggers	PROPN
ejpam-6940	120	8	and	and	CCONJ
ejpam-6940	120	9	h.	h.	PROPN
ejpam-6940	120	10	s.	s.	PROPN
ejpam-6940	120	11	kim	kim	PROPN
ejpam-6940	120	12	introduced	introduce	VERB
ejpam-6940	120	13	the	the	DET
ejpam-6940	120	14	concept	concept	NOUN
ejpam-6940	120	15	of	of	ADP
ejpam-6940	120	16	edge	edge	NOUN
ejpam-6940	120	17	d	d	NOUN
ejpam-6940	120	18	-	-	PUNCT
ejpam-6940	120	19	algebras	algebras	PROPN
ejpam-6940	120	20	in	in	ADP
ejpam-6940	120	21	1999	1999	NUM
ejpam-6940	120	22	[	[	X
ejpam-6940	120	23	8	8	NUM
ejpam-6940	120	24	]	]	PUNCT
ejpam-6940	120	25	.	.	PUNCT
ejpam-6940	121	1	a	a	DET
ejpam-6940	121	2	d	d	X
ejpam-6940	121	3	-	-	PUNCT
ejpam-6940	121	4	algebra	algebra	NOUN
ejpam-6940	121	5	consists	consist	VERB
ejpam-6940	121	6	of	of	ADP
ejpam-6940	121	7	a	a	DET
ejpam-6940	121	8	non	non	ADJ
ejpam-6940	121	9	-	-	ADJ
ejpam-6940	121	10	empty	empty	ADJ
ejpam-6940	121	11	set	set	NOUN
ejpam-6940	121	12	x	x	PUNCT
ejpam-6940	121	13	with	with	ADP
ejpam-6940	121	14	a	a	DET
ejpam-6940	121	15	constant	constant	ADJ
ejpam-6940	121	16	0	0	NUM
ejpam-6940	121	17	∈	∈	NOUN
ejpam-6940	121	18	x	x	PUNCT
ejpam-6940	121	19	together	together	ADV
ejpam-6940	121	20	with	with	ADP
ejpam-6940	121	21	a	a	DET
ejpam-6940	121	22	binary	binary	ADJ
ejpam-6940	121	23	operation	operation	NOUN
ejpam-6940	121	24	on	on	ADP
ejpam-6940	121	25	x	x	PUNCT
ejpam-6940	121	26	satisfying	satisfy	VERB
ejpam-6940	121	27	the	the	DET
ejpam-6940	121	28	following	follow	VERB
ejpam-6940	121	29	axioms	axiom	NOUN
ejpam-6940	121	30	:	:	PUNCT
ejpam-6940	121	31	for	for	ADP
ejpam-6940	121	32	all	all	DET
ejpam-6940	121	33	x	x	NOUN
ejpam-6940	121	34	,	,	PUNCT
ejpam-6940	121	35	y	y	PROPN
ejpam-6940	121	36	,	,	PUNCT
ejpam-6940	121	37	z	z	PROPN
ejpam-6940	121	38	∈	∈	PROPN
ejpam-6940	121	39	x	x	X
ejpam-6940	121	40	,	,	PUNCT
ejpam-6940	121	41	(	(	PUNCT
ejpam-6940	121	42	d1	d1	NOUN
ejpam-6940	121	43	)	)	PUNCT
ejpam-6940	121	44	xx	xx	NUM
ejpam-6940	122	1	=	=	SYM
ejpam-6940	122	2	0	0	NUM
ejpam-6940	122	3	,	,	PUNCT
ejpam-6940	122	4	(	(	PUNCT
ejpam-6940	122	5	d2	d2	PROPN
ejpam-6940	122	6	)	)	PUNCT
ejpam-6940	122	7	x0	x0	PROPN
ejpam-6940	123	1	=	=	PUNCT
ejpam-6940	123	2	0	0	NUM
ejpam-6940	123	3	,	,	PUNCT
ejpam-6940	123	4	(	(	PUNCT
ejpam-6940	123	5	d3	d3	PROPN
ejpam-6940	123	6	)	)	PUNCT
ejpam-6940	123	7	xy	xy	PROPN
ejpam-6940	124	1	=	=	PUNCT
ejpam-6940	124	2	0	0	NUM
ejpam-6940	124	3	and	and	CCONJ
ejpam-6940	124	4	yx	yx	X
ejpam-6940	124	5	=	=	SYM
ejpam-6940	124	6	0	0	NUM
ejpam-6940	124	7	imply	imply	VERB
ejpam-6940	124	8	x	x	X
ejpam-6940	124	9	=	=	PUNCT
ejpam-6940	124	10	y.	y.	PROPN
ejpam-6940	124	11	a.	a.	NOUN
ejpam-6940	124	12	anantayasethi	anantayasethi	PROPN
ejpam-6940	124	13	et	et	PROPN
ejpam-6940	124	14	al	al	PROPN
ejpam-6940	124	15	.	.	PUNCT
ejpam-6940	124	16	/	/	SYM
ejpam-6940	124	17	eur	eur	PROPN
ejpam-6940	124	18	.	.	PUNCT
ejpam-6940	125	1	j.	j.	PROPN
ejpam-6940	125	2	pure	pure	PROPN
ejpam-6940	125	3	appl	appl	PROPN
ejpam-6940	125	4	.	.	PROPN
ejpam-6940	125	5	math	math	PROPN
ejpam-6940	125	6	,	,	PUNCT
ejpam-6940	125	7	18	18	NUM
ejpam-6940	125	8	(	(	PUNCT
ejpam-6940	125	9	4	4	NUM
ejpam-6940	125	10	)	)	PUNCT
ejpam-6940	125	11	(	(	PUNCT
ejpam-6940	125	12	2025	2025	NUM
ejpam-6940	125	13	)	)	PUNCT
ejpam-6940	125	14	,	,	PUNCT
ejpam-6940	125	15	6940	6940	NUM
ejpam-6940	125	16	5	5	NUM
ejpam-6940	125	17	of	of	ADP
ejpam-6940	125	18	14	14	NUM
ejpam-6940	125	19	both	both	CCONJ
ejpam-6940	126	1	d	d	NOUN
ejpam-6940	126	2	-	-	PUNCT
ejpam-6940	126	3	algebras	algebra	NOUN
ejpam-6940	126	4	and	and	CCONJ
ejpam-6940	126	5	q	q	NOUN
ejpam-6940	126	6	-	-	PUNCT
ejpam-6940	126	7	algebras	algebra	NOUN
ejpam-6940	126	8	are	be	AUX
ejpam-6940	126	9	generalizations	generalization	NOUN
ejpam-6940	126	10	of	of	ADP
ejpam-6940	126	11	bci	bci	PROPN
ejpam-6940	126	12	/	/	SYM
ejpam-6940	126	13	bck	bck	PROPN
ejpam-6940	126	14	/	/	SYM
ejpam-6940	126	15	bch	bch	PROPN
ejpam-6940	126	16	-	-	PUNCT
ejpam-6940	126	17	algebras	algebras	PROPN
ejpam-6940	126	18	.	.	PUNCT
ejpam-6940	127	1	but	but	CCONJ
ejpam-6940	127	2	both	both	PRON
ejpam-6940	127	3	of	of	ADP
ejpam-6940	127	4	them	they	PRON
ejpam-6940	127	5	are	be	AUX
ejpam-6940	127	6	independent	independent	ADJ
ejpam-6940	127	7	,	,	PUNCT
ejpam-6940	127	8	i.e.	i.e.	X
ejpam-6940	127	9	a	a	DET
ejpam-6940	127	10	q	q	NOUN
ejpam-6940	127	11	-	-	PUNCT
ejpam-6940	127	12	algebra	algebra	NOUN
ejpam-6940	127	13	need	need	AUX
ejpam-6940	127	14	not	not	PART
ejpam-6940	127	15	be	be	AUX
ejpam-6940	127	16	a	a	DET
ejpam-6940	127	17	d	d	NOUN
ejpam-6940	127	18	-	-	PUNCT
ejpam-6940	127	19	algebra	algebra	NOUN
ejpam-6940	127	20	and	and	CCONJ
ejpam-6940	127	21	vice	vice	NOUN
ejpam-6940	127	22	versa	versa	ADV
ejpam-6940	127	23	.	.	PUNCT
ejpam-6940	128	1	the	the	DET
ejpam-6940	128	2	following	follow	VERB
ejpam-6940	128	3	example	example	NOUN
ejpam-6940	128	4	shows	show	VERB
ejpam-6940	128	5	this	this	DET
ejpam-6940	128	6	fact	fact	NOUN
ejpam-6940	128	7	:	:	PUNCT
ejpam-6940	128	8	example	example	NOUN
ejpam-6940	128	9	2	2	X
ejpam-6940	128	10	.	.	PUNCT
ejpam-6940	128	11	let	let	VERB
ejpam-6940	128	12	x	x	PUNCT
ejpam-6940	128	13	=	=	PUNCT
ejpam-6940	128	14	{	{	PUNCT
ejpam-6940	128	15	0	0	NUM
ejpam-6940	128	16	,	,	PUNCT
ejpam-6940	128	17	a	a	DET
ejpam-6940	128	18	,	,	PUNCT
ejpam-6940	128	19	b	b	NOUN
ejpam-6940	128	20	,	,	PUNCT
ejpam-6940	128	21	c	c	NOUN
ejpam-6940	128	22	}	}	PUNCT
ejpam-6940	128	23	,	,	PUNCT
ejpam-6940	128	24	y	y	PROPN
ejpam-6940	128	25	=	=	PUNCT
ejpam-6940	128	26	{	{	PUNCT
ejpam-6940	128	27	0	0	NUM
ejpam-6940	128	28	,	,	PUNCT
ejpam-6940	128	29	x	x	NOUN
ejpam-6940	128	30	,	,	PUNCT
ejpam-6940	128	31	y	y	PROPN
ejpam-6940	128	32	,	,	PUNCT
ejpam-6940	128	33	z	z	NOUN
ejpam-6940	128	34	}	}	PUNCT
ejpam-6940	128	35	and	and	CCONJ
ejpam-6940	128	36	t	t	NOUN
ejpam-6940	128	37	=	=	SYM
ejpam-6940	128	38	{	{	PUNCT
ejpam-6940	128	39	0	0	NUM
ejpam-6940	128	40	,	,	PUNCT
ejpam-6940	128	41	γ	γ	X
ejpam-6940	128	42	,	,	PUNCT
ejpam-6940	128	43	β	β	X
ejpam-6940	128	44	,	,	PUNCT
ejpam-6940	128	45	ν	ν	NOUN
ejpam-6940	128	46	}	}	PUNCT
ejpam-6940	128	47	.	.	PUNCT
ejpam-6940	129	1	define	define	VERB
ejpam-6940	129	2	binary	binary	ADJ
ejpam-6940	129	3	operations	operation	NOUN
ejpam-6940	129	4	∗	∗	NOUN
ejpam-6940	129	5	on	on	ADP
ejpam-6940	129	6	x	x	NOUN
ejpam-6940	129	7	,	,	PUNCT
ejpam-6940	129	8	•	•	NOUN
ejpam-6940	129	9	on	on	ADP
ejpam-6940	129	10	y	y	PROPN
ejpam-6940	129	11	and	and	CCONJ
ejpam-6940	129	12	◦	◦	VERB
ejpam-6940	129	13	on	on	ADP
ejpam-6940	129	14	t	t	PROPN
ejpam-6940	129	15	as	as	ADP
ejpam-6940	129	16	the	the	DET
ejpam-6940	129	17	following	following	ADJ
ejpam-6940	129	18	tables	table	NOUN
ejpam-6940	129	19	:	:	PUNCT
ejpam-6940	129	20	∗	∗	NOUN
ejpam-6940	129	21	0	0	NUM
ejpam-6940	130	1	a	a	DET
ejpam-6940	130	2	b	b	NOUN
ejpam-6940	130	3	c	c	NOUN
ejpam-6940	130	4	0	0	NUM
ejpam-6940	130	5	0	0	NUM
ejpam-6940	130	6	0	0	NUM
ejpam-6940	130	7	0	0	NUM
ejpam-6940	130	8	0	0	NUM
ejpam-6940	130	9	a	a	DET
ejpam-6940	130	10	a	a	DET
ejpam-6940	130	11	0	0	NUM
ejpam-6940	130	12	c	c	NOUN
ejpam-6940	130	13	b	b	PROPN
ejpam-6940	130	14	b	b	PROPN
ejpam-6940	130	15	c	c	NOUN
ejpam-6940	130	16	0	0	NUM
ejpam-6940	130	17	0	0	NUM
ejpam-6940	131	1	a	a	DET
ejpam-6940	131	2	c	c	NOUN
ejpam-6940	131	3	c	c	PROPN
ejpam-6940	131	4	a	a	DET
ejpam-6940	131	5	c	c	NOUN
ejpam-6940	131	6	0	0	NUM
ejpam-6940	131	7	•	•	NOUN
ejpam-6940	131	8	0	0	NUM
ejpam-6940	131	9	x	x	SYM
ejpam-6940	131	10	y	y	PROPN
ejpam-6940	131	11	z	z	NOUN
ejpam-6940	131	12	0	0	NUM
ejpam-6940	131	13	0	0	NUM
ejpam-6940	131	14	x	x	SYM
ejpam-6940	132	1	z	z	NOUN
ejpam-6940	132	2	y	y	NOUN
ejpam-6940	132	3	x	x	PUNCT
ejpam-6940	132	4	x	x	X
ejpam-6940	132	5	0	0	PUNCT
ejpam-6940	133	1	y	y	PROPN
ejpam-6940	133	2	z	z	PROPN
ejpam-6940	133	3	y	y	PROPN
ejpam-6940	133	4	y	y	PROPN
ejpam-6940	133	5	z	z	NOUN
ejpam-6940	133	6	0	0	PUNCT
ejpam-6940	134	1	x	x	SYM
ejpam-6940	135	1	z	z	NOUN
ejpam-6940	135	2	z	z	NOUN
ejpam-6940	135	3	y	y	NOUN
ejpam-6940	135	4	x	x	SYM
ejpam-6940	135	5	0	0	NUM
ejpam-6940	136	1	◦	◦	NOUN
ejpam-6940	136	2	0	0	NUM
ejpam-6940	136	3	γ	γ	X
ejpam-6940	136	4	β	β	X
ejpam-6940	136	5	ν	ν	X
ejpam-6940	136	6	0	0	NUM
ejpam-6940	136	7	0	0	NUM
ejpam-6940	136	8	0	0	NUM
ejpam-6940	136	9	0	0	NUM
ejpam-6940	136	10	0	0	NUM
ejpam-6940	136	11	γ	γ	PROPN
ejpam-6940	136	12	γ	γ	X
ejpam-6940	136	13	0	0	NUM
ejpam-6940	136	14	γ	γ	X
ejpam-6940	136	15	0	0	PUNCT
ejpam-6940	136	16	β	β	X
ejpam-6940	136	17	β	β	X
ejpam-6940	136	18	0	0	NUM
ejpam-6940	136	19	0	0	NUM
ejpam-6940	136	20	β	β	SYM
ejpam-6940	136	21	ν	ν	X
ejpam-6940	136	22	ν	ν	X
ejpam-6940	136	23	ν	ν	NOUN
ejpam-6940	136	24	ν	ν	X
ejpam-6940	136	25	0	0	PUNCT
ejpam-6940	136	26	then	then	ADV
ejpam-6940	136	27	(	(	PUNCT
ejpam-6940	136	28	x	x	NOUN
ejpam-6940	136	29	;	;	PUNCT
ejpam-6940	136	30	∗	∗	NOUN
ejpam-6940	136	31	,	,	PUNCT
ejpam-6940	136	32	0	0	NUM
ejpam-6940	136	33	)	)	PUNCT
ejpam-6940	136	34	is	be	AUX
ejpam-6940	136	35	a	a	DET
ejpam-6940	136	36	d	d	NOUN
ejpam-6940	136	37	-	-	PUNCT
ejpam-6940	136	38	algebra	algebra	NOUN
ejpam-6940	136	39	and	and	CCONJ
ejpam-6940	136	40	(	(	PUNCT
ejpam-6940	136	41	y	y	PROPN
ejpam-6940	136	42	;	;	PUNCT
ejpam-6940	136	43	•	•	X
ejpam-6940	136	44	,	,	PUNCT
ejpam-6940	136	45	0	0	NUM
ejpam-6940	136	46	)	)	PUNCT
ejpam-6940	136	47	is	be	AUX
ejpam-6940	136	48	a	a	DET
ejpam-6940	136	49	q	q	NOUN
ejpam-6940	136	50	-	-	PUNCT
ejpam-6940	136	51	algebra	algebra	NOUN
ejpam-6940	136	52	.	.	PUNCT
ejpam-6940	137	1	since	since	SCONJ
ejpam-6940	137	2	0	0	NUM
ejpam-6940	137	3	∗	∗	NOUN
ejpam-6940	137	4	b	b	NOUN
ejpam-6940	137	5	=	=	SYM
ejpam-6940	137	6	c	c	PROPN
ejpam-6940	137	7	̸=	̸=	PROPN
ejpam-6940	137	8	0	0	NUM
ejpam-6940	137	9	,	,	PUNCT
ejpam-6940	137	10	then	then	ADV
ejpam-6940	137	11	the	the	DET
ejpam-6940	137	12	condition	condition	NOUN
ejpam-6940	137	13	(	(	PUNCT
ejpam-6940	137	14	q2	q2	NOUN
ejpam-6940	137	15	)	)	PUNCT
ejpam-6940	137	16	is	be	AUX
ejpam-6940	137	17	not	not	PART
ejpam-6940	137	18	satisfied	satisfied	ADJ
ejpam-6940	137	19	in	in	ADP
ejpam-6940	137	20	x.	x.	NOUN
ejpam-6940	137	21	then	then	ADV
ejpam-6940	137	22	(	(	PUNCT
ejpam-6940	137	23	x	x	X
ejpam-6940	137	24	;	;	PUNCT
ejpam-6940	137	25	∗	∗	NOUN
ejpam-6940	137	26	,	,	PUNCT
ejpam-6940	137	27	0	0	NUM
ejpam-6940	137	28	)	)	PUNCT
ejpam-6940	137	29	is	be	AUX
ejpam-6940	137	30	not	not	PART
ejpam-6940	137	31	a	a	DET
ejpam-6940	137	32	q	q	NOUN
ejpam-6940	137	33	-	-	NOUN
ejpam-6940	137	34	algebra	algebra	NOUN
ejpam-6940	137	35	.	.	PUNCT
ejpam-6940	138	1	moreover	moreover	ADV
ejpam-6940	138	2	,	,	PUNCT
ejpam-6940	138	3	(	(	PUNCT
ejpam-6940	138	4	y	y	NOUN
ejpam-6940	138	5	;	;	PUNCT
ejpam-6940	138	6	•	•	X
ejpam-6940	138	7	,	,	PUNCT
ejpam-6940	138	8	0	0	NUM
ejpam-6940	138	9	)	)	PUNCT
ejpam-6940	138	10	is	be	AUX
ejpam-6940	138	11	not	not	PART
ejpam-6940	138	12	a	a	DET
ejpam-6940	138	13	d	d	NOUN
ejpam-6940	138	14	-	-	NOUN
ejpam-6940	138	15	algebra	algebra	NOUN
ejpam-6940	138	16	since	since	SCONJ
ejpam-6940	138	17	0	0	NUM
ejpam-6940	138	18	•	•	NOUN
ejpam-6940	138	19	y	y	NOUN
ejpam-6940	138	20	=	=	PUNCT
ejpam-6940	138	21	z	z	PROPN
ejpam-6940	138	22	̸=	̸=	PROPN
ejpam-6940	138	23	0	0	NUM
ejpam-6940	138	24	,	,	PUNCT
ejpam-6940	138	25	i.e.	i.e.	X
ejpam-6940	138	26	the	the	DET
ejpam-6940	138	27	condition	condition	NOUN
ejpam-6940	138	28	(	(	PUNCT
ejpam-6940	138	29	d2	d2	PROPN
ejpam-6940	138	30	)	)	PUNCT
ejpam-6940	138	31	is	be	AUX
ejpam-6940	138	32	not	not	PART
ejpam-6940	138	33	valid	valid	ADJ
ejpam-6940	138	34	.	.	PUNCT
ejpam-6940	139	1	the	the	DET
ejpam-6940	139	2	algebra	algebra	PROPN
ejpam-6940	139	3	(	(	PUNCT
ejpam-6940	139	4	t	t	NOUN
ejpam-6940	139	5	;	;	PUNCT
ejpam-6940	139	6	◦	◦	NOUN
ejpam-6940	139	7	,	,	PUNCT
ejpam-6940	139	8	0	0	NUM
ejpam-6940	139	9	)	)	PUNCT
ejpam-6940	139	10	is	be	AUX
ejpam-6940	139	11	a	a	DET
ejpam-6940	139	12	d	d	NOUN
ejpam-6940	139	13	-	-	PUNCT
ejpam-6940	139	14	algebra	algebra	NOUN
ejpam-6940	139	15	and	and	CCONJ
ejpam-6940	139	16	a	a	DET
ejpam-6940	139	17	q	q	NOUN
ejpam-6940	139	18	-	-	NOUN
ejpam-6940	139	19	algebra	algebra	NOUN
ejpam-6940	139	20	.	.	PUNCT
ejpam-6940	140	1	a	a	DET
ejpam-6940	140	2	d	d	X
ejpam-6940	140	3	-	-	PUNCT
ejpam-6940	140	4	algebra	algebra	NOUN
ejpam-6940	140	5	x	x	PUNCT
ejpam-6940	140	6	is	be	AUX
ejpam-6940	140	7	said	say	VERB
ejpam-6940	140	8	to	to	PART
ejpam-6940	140	9	be	be	AUX
ejpam-6940	140	10	edge	edge	NOUN
ejpam-6940	140	11	if	if	SCONJ
ejpam-6940	140	12	xx	xx	NUM
ejpam-6940	140	13	=	=	SYM
ejpam-6940	140	14	{	{	PUNCT
ejpam-6940	140	15	0	0	NUM
ejpam-6940	140	16	,	,	PUNCT
ejpam-6940	140	17	x	x	NOUN
ejpam-6940	140	18	}	}	PUNCT
ejpam-6940	140	19	for	for	ADP
ejpam-6940	140	20	all	all	DET
ejpam-6940	140	21	x	x	SYM
ejpam-6940	140	22	∈	∈	ADJ
ejpam-6940	140	23	x.	x.	NOUN
ejpam-6940	140	24	from	from	ADP
ejpam-6940	140	25	example	example	NOUN
ejpam-6940	140	26	2	2	NUM
ejpam-6940	140	27	,	,	PUNCT
ejpam-6940	140	28	a	a	DET
ejpam-6940	140	29	d	d	NOUN
ejpam-6940	140	30	-	-	PUNCT
ejpam-6940	140	31	algebra	algebra	NOUN
ejpam-6940	140	32	(	(	PUNCT
ejpam-6940	140	33	t	t	NOUN
ejpam-6940	140	34	;	;	PUNCT
ejpam-6940	140	35	◦	◦	NOUN
ejpam-6940	140	36	,	,	PUNCT
ejpam-6940	140	37	0	0	NUM
ejpam-6940	140	38	)	)	PUNCT
ejpam-6940	140	39	is	be	AUX
ejpam-6940	140	40	an	an	DET
ejpam-6940	140	41	edge	edge	NOUN
ejpam-6940	140	42	d	d	NOUN
ejpam-6940	140	43	-	-	NOUN
ejpam-6940	140	44	algebra	algebra	NOUN
ejpam-6940	140	45	.	.	PUNCT
ejpam-6940	141	1	in	in	ADP
ejpam-6940	141	2	[	[	X
ejpam-6940	141	3	8	8	NUM
ejpam-6940	141	4	]	]	PUNCT
ejpam-6940	141	5	the	the	DET
ejpam-6940	141	6	authors	author	NOUN
ejpam-6940	141	7	obtained	obtain	VERB
ejpam-6940	141	8	some	some	DET
ejpam-6940	141	9	properties	property	NOUN
ejpam-6940	141	10	of	of	ADP
ejpam-6940	141	11	edge	edge	NOUN
ejpam-6940	141	12	d	d	NOUN
ejpam-6940	141	13	-	-	PUNCT
ejpam-6940	141	14	algebras	algebras	PROPN
ejpam-6940	141	15	as	as	ADP
ejpam-6940	141	16	the	the	DET
ejpam-6940	141	17	following	following	NOUN
ejpam-6940	141	18	:	:	PUNCT
ejpam-6940	141	19	proposition	proposition	NOUN
ejpam-6940	141	20	3	3	NUM
ejpam-6940	141	21	.	.	PUNCT
ejpam-6940	142	1	[	[	X
ejpam-6940	142	2	8	8	NUM
ejpam-6940	142	3	]	]	PUNCT
ejpam-6940	142	4	let	let	VERB
ejpam-6940	142	5	x	x	PRON
ejpam-6940	142	6	be	be	AUX
ejpam-6940	142	7	an	an	DET
ejpam-6940	142	8	edge	edge	NOUN
ejpam-6940	142	9	d	d	NOUN
ejpam-6940	142	10	-	-	NOUN
ejpam-6940	142	11	algebra	algebra	NOUN
ejpam-6940	142	12	.	.	PUNCT
ejpam-6940	143	1	then	then	ADV
ejpam-6940	143	2	x0	x0	PROPN
ejpam-6940	143	3	=	=	PUNCT
ejpam-6940	144	1	x	x	PUNCT
ejpam-6940	144	2	for	for	ADP
ejpam-6940	144	3	all	all	DET
ejpam-6940	144	4	x	x	SYM
ejpam-6940	144	5	∈	∈	NOUN
ejpam-6940	144	6	x.	x.	NOUN
ejpam-6940	144	7	now	now	ADV
ejpam-6940	144	8	we	we	PRON
ejpam-6940	144	9	describe	describe	VERB
ejpam-6940	144	10	the	the	DET
ejpam-6940	144	11	concept	concept	NOUN
ejpam-6940	144	12	of	of	ADP
ejpam-6940	144	13	edge	edge	NOUN
ejpam-6940	144	14	in	in	ADP
ejpam-6940	144	15	q	q	NOUN
ejpam-6940	144	16	-	-	PUNCT
ejpam-6940	144	17	algebra	algebra	NOUN
ejpam-6940	144	18	motivated	motivate	VERB
ejpam-6940	144	19	by	by	ADP
ejpam-6940	144	20	[	[	X
ejpam-6940	144	21	8	8	NUM
ejpam-6940	144	22	]	]	PUNCT
ejpam-6940	144	23	.	.	PUNCT
ejpam-6940	145	1	definition	definition	NOUN
ejpam-6940	145	2	1	1	NUM
ejpam-6940	145	3	.	.	PUNCT
ejpam-6940	146	1	a	a	DET
ejpam-6940	146	2	q	q	NOUN
ejpam-6940	146	3	-	-	NOUN
ejpam-6940	146	4	algebra	algebra	NOUN
ejpam-6940	146	5	x	x	PUNCT
ejpam-6940	146	6	is	be	AUX
ejpam-6940	146	7	said	say	VERB
ejpam-6940	146	8	to	to	PART
ejpam-6940	146	9	be	be	AUX
ejpam-6940	146	10	edge	edge	NOUN
ejpam-6940	146	11	if	if	SCONJ
ejpam-6940	146	12	ax	ax	NOUN
ejpam-6940	146	13	=	=	SYM
ejpam-6940	146	14	{	{	PUNCT
ejpam-6940	146	15	0	0	NUM
ejpam-6940	146	16	,	,	PUNCT
ejpam-6940	146	17	a	a	PRON
ejpam-6940	146	18	}	}	PUNCT
ejpam-6940	146	19	for	for	ADP
ejpam-6940	146	20	all	all	DET
ejpam-6940	146	21	a	a	DET
ejpam-6940	146	22	∈	∈	PROPN
ejpam-6940	146	23	x.	x.	NOUN
ejpam-6940	146	24	example	example	NOUN
ejpam-6940	147	1	3	3	X
ejpam-6940	147	2	.	.	PUNCT
ejpam-6940	147	3	let	let	VERB
ejpam-6940	147	4	x	x	PUNCT
ejpam-6940	147	5	=	=	PUNCT
ejpam-6940	147	6	{	{	PUNCT
ejpam-6940	147	7	0	0	NUM
ejpam-6940	147	8	,	,	PUNCT
ejpam-6940	147	9	x	x	NOUN
ejpam-6940	147	10	,	,	PUNCT
ejpam-6940	147	11	y	y	PROPN
ejpam-6940	147	12	,	,	PUNCT
ejpam-6940	147	13	z	z	NOUN
ejpam-6940	147	14	}	}	PUNCT
ejpam-6940	147	15	and	and	CCONJ
ejpam-6940	147	16	y	y	PROPN
ejpam-6940	147	17	=	=	PUNCT
ejpam-6940	147	18	{	{	PUNCT
ejpam-6940	147	19	0	0	NUM
ejpam-6940	147	20	,	,	PUNCT
ejpam-6940	147	21	a	a	DET
ejpam-6940	147	22	,	,	PUNCT
ejpam-6940	147	23	b	b	NOUN
ejpam-6940	147	24	,	,	PUNCT
ejpam-6940	147	25	c	c	NOUN
ejpam-6940	147	26	,	,	PUNCT
ejpam-6940	147	27	d	d	NOUN
ejpam-6940	147	28	,	,	PUNCT
ejpam-6940	147	29	f	f	NOUN
ejpam-6940	147	30	}	}	PUNCT
ejpam-6940	147	31	.	.	PUNCT
ejpam-6940	148	1	define	define	VERB
ejpam-6940	148	2	binary	binary	ADJ
ejpam-6940	148	3	operations	operation	NOUN
ejpam-6940	148	4	∗	∗	NOUN
ejpam-6940	148	5	on	on	ADP
ejpam-6940	148	6	x	x	PUNCT
ejpam-6940	148	7	and	and	CCONJ
ejpam-6940	148	8	•	•	NOUN
ejpam-6940	148	9	on	on	ADP
ejpam-6940	148	10	y	y	PROPN
ejpam-6940	148	11	as	as	ADP
ejpam-6940	148	12	the	the	DET
ejpam-6940	148	13	following	following	ADJ
ejpam-6940	148	14	tables	table	NOUN
ejpam-6940	148	15	:	:	PUNCT
ejpam-6940	148	16	∗	∗	NOUN
ejpam-6940	148	17	0	0	NUM
ejpam-6940	149	1	x	x	SYM
ejpam-6940	149	2	y	y	PROPN
ejpam-6940	149	3	z	z	NOUN
ejpam-6940	149	4	0	0	NUM
ejpam-6940	149	5	0	0	NUM
ejpam-6940	149	6	0	0	NUM
ejpam-6940	149	7	0	0	NUM
ejpam-6940	149	8	0	0	NUM
ejpam-6940	150	1	x	x	SYM
ejpam-6940	150	2	x	x	SYM
ejpam-6940	150	3	0	0	NUM
ejpam-6940	150	4	0	0	NUM
ejpam-6940	150	5	0	0	NUM
ejpam-6940	151	1	y	y	PROPN
ejpam-6940	151	2	y	y	PROPN
ejpam-6940	151	3	y	y	PROPN
ejpam-6940	151	4	0	0	PUNCT
ejpam-6940	152	1	y	y	PROPN
ejpam-6940	152	2	z	z	PROPN
ejpam-6940	152	3	z	z	NOUN
ejpam-6940	152	4	0	0	NUM
ejpam-6940	152	5	0	0	NUM
ejpam-6940	152	6	0	0	NUM
ejpam-6940	152	7	•	•	NOUN
ejpam-6940	152	8	0	0	NUM
ejpam-6940	153	1	a	a	DET
ejpam-6940	153	2	b	b	NOUN
ejpam-6940	153	3	c	c	NOUN
ejpam-6940	153	4	d	d	X
ejpam-6940	153	5	f	f	PROPN
ejpam-6940	153	6	0	0	NUM
ejpam-6940	153	7	0	0	NUM
ejpam-6940	154	1	a	a	DET
ejpam-6940	154	2	f	f	PROPN
ejpam-6940	154	3	b	b	PROPN
ejpam-6940	154	4	b	b	PROPN
ejpam-6940	154	5	b	b	PROPN
ejpam-6940	154	6	a	a	PRON
ejpam-6940	154	7	a	a	PRON
ejpam-6940	154	8	0	0	NUM
ejpam-6940	154	9	b	b	NOUN
ejpam-6940	154	10	f	f	X
ejpam-6940	154	11	f	f	PROPN
ejpam-6940	155	1	f	f	PROPN
ejpam-6940	155	2	b	b	PROPN
ejpam-6940	155	3	b	b	PROPN
ejpam-6940	155	4	c	c	PROPN
ejpam-6940	155	5	0	0	NUM
ejpam-6940	155	6	a	a	DET
ejpam-6940	155	7	a	a	DET
ejpam-6940	155	8	a	a	DET
ejpam-6940	155	9	c	c	NOUN
ejpam-6940	155	10	c	c	NOUN
ejpam-6940	155	11	b	b	PROPN
ejpam-6940	155	12	a	a	PRON
ejpam-6940	155	13	0	0	NUM
ejpam-6940	155	14	0	0	NUM
ejpam-6940	155	15	0	0	NUM
ejpam-6940	156	1	d	d	PROPN
ejpam-6940	156	2	d	d	PROPN
ejpam-6940	156	3	b	b	PROPN
ejpam-6940	156	4	a	a	PRON
ejpam-6940	156	5	0	0	NUM
ejpam-6940	156	6	0	0	NUM
ejpam-6940	156	7	0	0	NUM
ejpam-6940	157	1	f	f	PROPN
ejpam-6940	157	2	f	f	PROPN
ejpam-6940	157	3	b	b	PROPN
ejpam-6940	157	4	a	a	DET
ejpam-6940	157	5	0	0	NUM
ejpam-6940	157	6	0	0	NUM
ejpam-6940	157	7	0	0	NUM
ejpam-6940	158	1	then	then	ADV
ejpam-6940	158	2	(	(	PUNCT
ejpam-6940	158	3	x	x	NOUN
ejpam-6940	158	4	;	;	PUNCT
ejpam-6940	158	5	∗	∗	NOUN
ejpam-6940	158	6	,	,	PUNCT
ejpam-6940	158	7	0	0	NUM
ejpam-6940	158	8	)	)	PUNCT
ejpam-6940	158	9	and	and	CCONJ
ejpam-6940	158	10	(	(	PUNCT
ejpam-6940	158	11	y	y	PROPN
ejpam-6940	158	12	;	;	PUNCT
ejpam-6940	158	13	•	•	X
ejpam-6940	158	14	,	,	PUNCT
ejpam-6940	158	15	0	0	NUM
ejpam-6940	158	16	)	)	PUNCT
ejpam-6940	158	17	are	be	AUX
ejpam-6940	158	18	q	q	NOUN
ejpam-6940	158	19	-	-	PUNCT
ejpam-6940	158	20	algebras	algebras	X
ejpam-6940	158	21	.	.	PUNCT
ejpam-6940	159	1	since	since	SCONJ
ejpam-6940	159	2	0x	0x	NUM
ejpam-6940	159	3	=	=	SYM
ejpam-6940	159	4	{	{	PUNCT
ejpam-6940	159	5	0	0	NUM
ejpam-6940	159	6	}	}	PUNCT
ejpam-6940	159	7	,	,	PUNCT
ejpam-6940	159	8	xx	xx	X
ejpam-6940	159	9	=	=	SYM
ejpam-6940	159	10	{	{	PUNCT
ejpam-6940	159	11	0	0	NUM
ejpam-6940	159	12	,	,	PUNCT
ejpam-6940	159	13	x	x	NOUN
ejpam-6940	159	14	}	}	PUNCT
ejpam-6940	159	15	,	,	PUNCT
ejpam-6940	159	16	yx	yx	X
ejpam-6940	159	17	=	=	PUNCT
ejpam-6940	159	18	{	{	PUNCT
ejpam-6940	159	19	0	0	NUM
ejpam-6940	159	20	,	,	PUNCT
ejpam-6940	159	21	y	y	NOUN
ejpam-6940	159	22	}	}	PUNCT
ejpam-6940	159	23	and	and	CCONJ
ejpam-6940	159	24	zx	zx	NUM
ejpam-6940	159	25	=	=	SYM
ejpam-6940	159	26	{	{	PUNCT
ejpam-6940	159	27	0	0	NUM
ejpam-6940	159	28	,	,	PUNCT
ejpam-6940	159	29	z	z	NOUN
ejpam-6940	159	30	}	}	PUNCT
ejpam-6940	159	31	,	,	PUNCT
ejpam-6940	159	32	then	then	ADV
ejpam-6940	159	33	x	x	PUNCT
ejpam-6940	159	34	is	be	AUX
ejpam-6940	159	35	an	an	DET
ejpam-6940	159	36	edge	edge	NOUN
ejpam-6940	159	37	q	q	NOUN
ejpam-6940	159	38	-	-	NOUN
ejpam-6940	159	39	algebra	algebra	NOUN
ejpam-6940	159	40	.	.	PUNCT
ejpam-6940	160	1	since	since	SCONJ
ejpam-6940	160	2	ay	ay	PROPN
ejpam-6940	160	3	=	=	PUNCT
ejpam-6940	160	4	{	{	PUNCT
ejpam-6940	160	5	0	0	NUM
ejpam-6940	160	6	,	,	PUNCT
ejpam-6940	160	7	a	a	PRON
ejpam-6940	160	8	,	,	PUNCT
ejpam-6940	160	9	b	b	NOUN
ejpam-6940	160	10	,	,	PUNCT
ejpam-6940	160	11	f	f	PROPN
ejpam-6940	160	12	}	}	PUNCT
ejpam-6940	160	13	̸=	̸=	PROPN
ejpam-6940	160	14	{	{	PUNCT
ejpam-6940	160	15	0	0	NUM
ejpam-6940	160	16	,	,	PUNCT
ejpam-6940	160	17	a	a	PRON
ejpam-6940	160	18	}	}	PUNCT
ejpam-6940	160	19	,	,	PUNCT
ejpam-6940	160	20	then	then	ADV
ejpam-6940	160	21	y	y	PROPN
ejpam-6940	160	22	is	be	AUX
ejpam-6940	160	23	not	not	PART
ejpam-6940	160	24	edge	edge	NOUN
ejpam-6940	160	25	.	.	PUNCT
ejpam-6940	161	1	a.	a.	NOUN
ejpam-6940	161	2	anantayasethi	anantayasethi	PROPN
ejpam-6940	161	3	et	et	PROPN
ejpam-6940	161	4	al	al	PROPN
ejpam-6940	161	5	.	.	PUNCT
ejpam-6940	161	6	/	/	SYM
ejpam-6940	161	7	eur	eur	PROPN
ejpam-6940	161	8	.	.	PUNCT
ejpam-6940	162	1	j.	j.	PROPN
ejpam-6940	162	2	pure	pure	PROPN
ejpam-6940	162	3	appl	appl	PROPN
ejpam-6940	162	4	.	.	PROPN
ejpam-6940	162	5	math	math	PROPN
ejpam-6940	162	6	,	,	PUNCT
ejpam-6940	162	7	18	18	NUM
ejpam-6940	162	8	(	(	PUNCT
ejpam-6940	162	9	4	4	NUM
ejpam-6940	162	10	)	)	PUNCT
ejpam-6940	162	11	(	(	PUNCT
ejpam-6940	162	12	2025	2025	NUM
ejpam-6940	162	13	)	)	PUNCT
ejpam-6940	162	14	,	,	PUNCT
ejpam-6940	162	15	6940	6940	NUM
ejpam-6940	162	16	6	6	NUM
ejpam-6940	162	17	of	of	ADP
ejpam-6940	162	18	14	14	NUM
ejpam-6940	162	19	next	next	ADJ
ejpam-6940	162	20	proposition	proposition	NOUN
ejpam-6940	162	21	provides	provide	VERB
ejpam-6940	162	22	a	a	DET
ejpam-6940	162	23	connection	connection	NOUN
ejpam-6940	162	24	between	between	ADP
ejpam-6940	162	25	d	d	NOUN
ejpam-6940	162	26	-	-	PUNCT
ejpam-6940	162	27	algebras	algebra	NOUN
ejpam-6940	162	28	and	and	CCONJ
ejpam-6940	162	29	q	q	NOUN
ejpam-6940	162	30	-	-	PUNCT
ejpam-6940	162	31	algebras	algebras	X
ejpam-6940	162	32	.	.	PUNCT
ejpam-6940	163	1	we	we	PRON
ejpam-6940	163	2	show	show	VERB
ejpam-6940	163	3	a	a	DET
ejpam-6940	163	4	sufficient	sufficient	ADJ
ejpam-6940	163	5	condition	condition	NOUN
ejpam-6940	163	6	for	for	ADP
ejpam-6940	163	7	a	a	DET
ejpam-6940	163	8	d	d	NOUN
ejpam-6940	163	9	-	-	NOUN
ejpam-6940	163	10	algebra	algebra	NOUN
ejpam-6940	163	11	to	to	PART
ejpam-6940	163	12	be	be	AUX
ejpam-6940	163	13	a	a	DET
ejpam-6940	163	14	q	q	NOUN
ejpam-6940	163	15	-	-	PUNCT
ejpam-6940	163	16	algebra	algebra	NOUN
ejpam-6940	163	17	.	.	PUNCT
ejpam-6940	164	1	proposition	proposition	NOUN
ejpam-6940	164	2	4	4	NUM
ejpam-6940	164	3	.	.	PUNCT
ejpam-6940	165	1	every	every	DET
ejpam-6940	165	2	edge	edge	NOUN
ejpam-6940	165	3	d	d	X
ejpam-6940	165	4	-	-	PUNCT
ejpam-6940	165	5	algebra	algebra	NOUN
ejpam-6940	165	6	is	be	AUX
ejpam-6940	165	7	a	a	DET
ejpam-6940	165	8	q	q	NOUN
ejpam-6940	165	9	-	-	PUNCT
ejpam-6940	165	10	algebra	algebra	NOUN
ejpam-6940	165	11	,	,	PUNCT
ejpam-6940	165	12	more	more	ADV
ejpam-6940	165	13	specific	specific	ADJ
ejpam-6940	165	14	it	it	PRON
ejpam-6940	165	15	is	be	AUX
ejpam-6940	165	16	an	an	DET
ejpam-6940	165	17	edge	edge	NOUN
ejpam-6940	165	18	q	q	NOUN
ejpam-6940	165	19	-	-	NOUN
ejpam-6940	165	20	algebra	algebra	NOUN
ejpam-6940	165	21	.	.	PUNCT
ejpam-6940	166	1	proof	proof	NOUN
ejpam-6940	166	2	.	.	PUNCT
ejpam-6940	167	1	let	let	VERB
ejpam-6940	167	2	x	x	PRON
ejpam-6940	167	3	be	be	AUX
ejpam-6940	167	4	an	an	DET
ejpam-6940	167	5	edge	edge	NOUN
ejpam-6940	167	6	d	d	NOUN
ejpam-6940	167	7	-	-	PUNCT
ejpam-6940	167	8	algebra	algebra	NOUN
ejpam-6940	167	9	.	.	PUNCT
ejpam-6940	168	1	we	we	PRON
ejpam-6940	168	2	want	want	VERB
ejpam-6940	168	3	to	to	PART
ejpam-6940	168	4	show	show	VERB
ejpam-6940	168	5	that	that	SCONJ
ejpam-6940	168	6	x	x	PRON
ejpam-6940	168	7	is	be	AUX
ejpam-6940	168	8	a	a	DET
ejpam-6940	168	9	q	q	NOUN
ejpam-6940	168	10	-	-	PUNCT
ejpam-6940	168	11	algebra	algebra	NOUN
ejpam-6940	168	12	.	.	PUNCT
ejpam-6940	169	1	the	the	DET
ejpam-6940	169	2	condition	condition	NOUN
ejpam-6940	169	3	(	(	PUNCT
ejpam-6940	169	4	q1	q1	PROPN
ejpam-6940	169	5	)	)	PUNCT
ejpam-6940	169	6	and	and	CCONJ
ejpam-6940	169	7	(	(	PUNCT
ejpam-6940	169	8	q2	q2	NOUN
ejpam-6940	169	9	)	)	PUNCT
ejpam-6940	169	10	follow	follow	VERB
ejpam-6940	169	11	from	from	ADP
ejpam-6940	169	12	the	the	DET
ejpam-6940	169	13	condition	condition	NOUN
ejpam-6940	169	14	(	(	PUNCT
ejpam-6940	169	15	d1	d1	NOUN
ejpam-6940	169	16	)	)	PUNCT
ejpam-6940	169	17	and	and	CCONJ
ejpam-6940	169	18	proposition	proposition	NOUN
ejpam-6940	169	19	3	3	NUM
ejpam-6940	169	20	,	,	PUNCT
ejpam-6940	169	21	respectively	respectively	ADV
ejpam-6940	169	22	.	.	PUNCT
ejpam-6940	170	1	let	let	VERB
ejpam-6940	170	2	x	x	PRON
ejpam-6940	170	3	,	,	PUNCT
ejpam-6940	170	4	y	y	PROPN
ejpam-6940	170	5	,	,	PUNCT
ejpam-6940	170	6	z	z	NOUN
ejpam-6940	170	7	∈	∈	PROPN
ejpam-6940	170	8	x.	x.	NOUN
ejpam-6940	171	1	we	we	PRON
ejpam-6940	171	2	calculate	calculate	VERB
ejpam-6940	171	3	(	(	PUNCT
ejpam-6940	171	4	xy)z	xy)z	PROPN
ejpam-6940	171	5	and	and	CCONJ
ejpam-6940	171	6	(	(	PUNCT
ejpam-6940	171	7	xz)y	xz)y	PROPN
ejpam-6940	171	8	.	.	PUNCT
ejpam-6940	172	1	since	since	SCONJ
ejpam-6940	172	2	x	x	PRON
ejpam-6940	172	3	is	be	AUX
ejpam-6940	172	4	an	an	DET
ejpam-6940	172	5	edge	edge	NOUN
ejpam-6940	172	6	d	d	NOUN
ejpam-6940	172	7	-	-	PUNCT
ejpam-6940	172	8	algebra	algebra	NOUN
ejpam-6940	172	9	,	,	PUNCT
ejpam-6940	172	10	then	then	ADV
ejpam-6940	172	11	xy	xy	PROPN
ejpam-6940	172	12	∈	∈	PROPN
ejpam-6940	172	13	xx	xx	NUM
ejpam-6940	173	1	=	=	SYM
ejpam-6940	173	2	{	{	PUNCT
ejpam-6940	173	3	0	0	NUM
ejpam-6940	173	4	,	,	PUNCT
ejpam-6940	173	5	x	x	NOUN
ejpam-6940	173	6	}	}	PUNCT
ejpam-6940	173	7	,	,	PUNCT
ejpam-6940	173	8	i.e.	i.e.	X
ejpam-6940	173	9	xy	xy	X
ejpam-6940	173	10	=	=	SYM
ejpam-6940	173	11	0	0	NUM
ejpam-6940	173	12	or	or	CCONJ
ejpam-6940	173	13	xy	xy	NOUN
ejpam-6940	173	14	=	=	PUNCT
ejpam-6940	174	1	x.	x.	NOUN
ejpam-6940	174	2	if	if	SCONJ
ejpam-6940	174	3	xy	xy	PROPN
ejpam-6940	174	4	=	=	SYM
ejpam-6940	174	5	0	0	PROPN
ejpam-6940	174	6	,	,	PUNCT
ejpam-6940	174	7	then	then	ADV
ejpam-6940	174	8	by	by	ADP
ejpam-6940	174	9	(	(	PUNCT
ejpam-6940	174	10	d2	d2	PROPN
ejpam-6940	174	11	)	)	PUNCT
ejpam-6940	174	12	(	(	PUNCT
ejpam-6940	174	13	xy)z	xy)z	NOUN
ejpam-6940	174	14	=	=	SYM
ejpam-6940	174	15	0z	0z	NUM
ejpam-6940	175	1	=	=	SYM
ejpam-6940	175	2	0	0	X
ejpam-6940	175	3	.	.	PUNCT
ejpam-6940	176	1	since	since	SCONJ
ejpam-6940	176	2	xz	xz	PROPN
ejpam-6940	176	3	∈	∈	PROPN
ejpam-6940	176	4	xx	xx	NUM
ejpam-6940	176	5	=	=	SYM
ejpam-6940	176	6	{	{	PUNCT
ejpam-6940	176	7	0	0	NUM
ejpam-6940	176	8	,	,	PUNCT
ejpam-6940	176	9	x	x	NOUN
ejpam-6940	176	10	}	}	PUNCT
ejpam-6940	176	11	,	,	PUNCT
ejpam-6940	176	12	there	there	PRON
ejpam-6940	176	13	follows	follow	VERB
ejpam-6940	176	14	(	(	PUNCT
ejpam-6940	176	15	xz)y	xz)y	PROPN
ejpam-6940	176	16	∈	∈	PROPN
ejpam-6940	176	17	{	{	PUNCT
ejpam-6940	176	18	0y	0y	NUM
ejpam-6940	176	19	,	,	PUNCT
ejpam-6940	176	20	xy	xy	INTJ
ejpam-6940	176	21	}	}	PUNCT
ejpam-6940	176	22	=	=	PUNCT
ejpam-6940	176	23	{	{	PUNCT
ejpam-6940	176	24	0	0	NUM
ejpam-6940	176	25	}	}	PUNCT
ejpam-6940	176	26	.	.	PUNCT
ejpam-6940	177	1	thus	thus	ADV
ejpam-6940	177	2	,	,	PUNCT
ejpam-6940	177	3	(	(	PUNCT
ejpam-6940	177	4	xy)z	xy)z	PUNCT
ejpam-6940	177	5	=	=	SYM
ejpam-6940	177	6	(	(	PUNCT
ejpam-6940	177	7	xz)y	xz)y	PROPN
ejpam-6940	177	8	.	.	PUNCT
ejpam-6940	177	9	now	now	ADV
ejpam-6940	177	10	we	we	PRON
ejpam-6940	177	11	assume	assume	VERB
ejpam-6940	177	12	xy	xy	X
ejpam-6940	178	1	=	=	PUNCT
ejpam-6940	179	1	x.	x.	NOUN
ejpam-6940	180	1	then	then	ADV
ejpam-6940	180	2	(	(	PUNCT
ejpam-6940	180	3	xy)z	xy)z	PROPN
ejpam-6940	180	4	=	=	SYM
ejpam-6940	180	5	xz	xz	PROPN
ejpam-6940	180	6	∈	∈	PROPN
ejpam-6940	180	7	xx	xx	NUM
ejpam-6940	181	1	=	=	SYM
ejpam-6940	182	1	{	{	PUNCT
ejpam-6940	182	2	0	0	NUM
ejpam-6940	182	3	,	,	PUNCT
ejpam-6940	182	4	x	x	NOUN
ejpam-6940	182	5	}	}	PUNCT
ejpam-6940	182	6	.	.	PUNCT
ejpam-6940	183	1	if	if	SCONJ
ejpam-6940	183	2	xz	xz	PROPN
ejpam-6940	183	3	=	=	SYM
ejpam-6940	183	4	0	0	PROPN
ejpam-6940	183	5	,	,	PUNCT
ejpam-6940	183	6	then	then	ADV
ejpam-6940	183	7	by	by	ADP
ejpam-6940	183	8	(	(	PUNCT
ejpam-6940	183	9	d2	d2	PROPN
ejpam-6940	183	10	)	)	PUNCT
ejpam-6940	183	11	(	(	PUNCT
ejpam-6940	183	12	xy)z	xy)z	PUNCT
ejpam-6940	183	13	=	=	SYM
ejpam-6940	183	14	xz	xz	PROPN
ejpam-6940	183	15	=	=	SYM
ejpam-6940	183	16	0	0	PUNCT
ejpam-6940	183	17	=	=	SYM
ejpam-6940	183	18	0y	0y	NOUN
ejpam-6940	183	19	=	=	SYM
ejpam-6940	183	20	(	(	PUNCT
ejpam-6940	183	21	xz)y	xz)y	PROPN
ejpam-6940	183	22	.	.	PUNCT
ejpam-6940	184	1	if	if	SCONJ
ejpam-6940	184	2	xz	xz	PROPN
ejpam-6940	184	3	=	=	SYM
ejpam-6940	184	4	x	x	PROPN
ejpam-6940	184	5	,	,	PUNCT
ejpam-6940	184	6	then	then	ADV
ejpam-6940	184	7	(	(	PUNCT
ejpam-6940	184	8	xy)z	xy)z	PROPN
ejpam-6940	184	9	=	=	SYM
ejpam-6940	184	10	xz	xz	PROPN
ejpam-6940	185	1	=	=	PUNCT
ejpam-6940	185	2	x	x	PUNCT
ejpam-6940	186	1	=	=	PUNCT
ejpam-6940	186	2	xy	xy	PROPN
ejpam-6940	186	3	=	=	PUNCT
ejpam-6940	186	4	(	(	PUNCT
ejpam-6940	186	5	xz)y	xz)y	PROPN
ejpam-6940	186	6	.	.	PUNCT
ejpam-6940	186	7	therefore	therefore	ADV
ejpam-6940	186	8	,	,	PUNCT
ejpam-6940	186	9	(	(	PUNCT
ejpam-6940	186	10	xy)z	xy)z	PUNCT
ejpam-6940	186	11	=	=	SYM
ejpam-6940	186	12	(	(	PUNCT
ejpam-6940	186	13	xz)y	xz)y	PROPN
ejpam-6940	186	14	for	for	ADP
ejpam-6940	186	15	all	all	DET
ejpam-6940	186	16	x	x	NOUN
ejpam-6940	186	17	,	,	PUNCT
ejpam-6940	186	18	y	y	PROPN
ejpam-6940	186	19	,	,	PUNCT
ejpam-6940	186	20	z	z	PROPN
ejpam-6940	186	21	∈	∈	PROPN
ejpam-6940	186	22	x.	x.	NOUN
ejpam-6940	186	23	hence	hence	ADV
ejpam-6940	186	24	,	,	PUNCT
ejpam-6940	186	25	the	the	DET
ejpam-6940	186	26	condition	condition	NOUN
ejpam-6940	186	27	(	(	PUNCT
ejpam-6940	186	28	q3	q3	PROPN
ejpam-6940	186	29	)	)	PUNCT
ejpam-6940	186	30	is	be	AUX
ejpam-6940	186	31	fulfilled	fulfil	VERB
ejpam-6940	186	32	.	.	PUNCT
ejpam-6940	187	1	thus	thus	ADV
ejpam-6940	187	2	,	,	PUNCT
ejpam-6940	187	3	x	x	PRON
ejpam-6940	187	4	is	be	AUX
ejpam-6940	187	5	a	a	DET
ejpam-6940	187	6	q	q	NOUN
ejpam-6940	187	7	-	-	PUNCT
ejpam-6940	187	8	algebra	algebra	NOUN
ejpam-6940	187	9	.	.	PUNCT
ejpam-6940	188	1	since	since	SCONJ
ejpam-6940	188	2	x	x	PROPN
ejpam-6940	188	3	is	be	AUX
ejpam-6940	188	4	edge	edge	NOUN
ejpam-6940	188	5	,	,	PUNCT
ejpam-6940	188	6	then	then	ADV
ejpam-6940	188	7	x	x	PUNCT
ejpam-6940	188	8	is	be	AUX
ejpam-6940	188	9	an	an	DET
ejpam-6940	188	10	edge	edge	NOUN
ejpam-6940	188	11	q	q	NOUN
ejpam-6940	188	12	-	-	NOUN
ejpam-6940	188	13	algebra	algebra	NOUN
ejpam-6940	188	14	.	.	PUNCT
ejpam-6940	189	1	the	the	DET
ejpam-6940	189	2	converse	converse	NOUN
ejpam-6940	189	3	of	of	ADP
ejpam-6940	189	4	proposition	proposition	NOUN
ejpam-6940	189	5	4	4	NUM
ejpam-6940	189	6	is	be	AUX
ejpam-6940	189	7	not	not	PART
ejpam-6940	189	8	true	true	ADJ
ejpam-6940	189	9	,	,	PUNCT
ejpam-6940	189	10	i.e.	i.e.	X
ejpam-6940	189	11	there	there	PRON
ejpam-6940	189	12	is	be	VERB
ejpam-6940	189	13	an	an	DET
ejpam-6940	189	14	edge	edge	NOUN
ejpam-6940	189	15	q	q	NOUN
ejpam-6940	189	16	-	-	PUNCT
ejpam-6940	189	17	algebra	algebra	NOUN
ejpam-6940	189	18	which	which	PRON
ejpam-6940	189	19	is	be	AUX
ejpam-6940	189	20	not	not	PART
ejpam-6940	189	21	an	an	DET
ejpam-6940	189	22	edge	edge	NOUN
ejpam-6940	189	23	d	d	NOUN
ejpam-6940	189	24	-	-	NOUN
ejpam-6940	189	25	algebra	algebra	NOUN
ejpam-6940	189	26	as	as	SCONJ
ejpam-6940	189	27	shown	show	VERB
ejpam-6940	189	28	in	in	ADP
ejpam-6940	189	29	the	the	DET
ejpam-6940	189	30	following	follow	VERB
ejpam-6940	189	31	example	example	NOUN
ejpam-6940	189	32	.	.	PUNCT
ejpam-6940	190	1	example	example	NOUN
ejpam-6940	191	1	4	4	NUM
ejpam-6940	191	2	.	.	PUNCT
ejpam-6940	191	3	let	let	VERB
ejpam-6940	191	4	x	x	PUNCT
ejpam-6940	191	5	=	=	PUNCT
ejpam-6940	191	6	{	{	PUNCT
ejpam-6940	191	7	0	0	NUM
ejpam-6940	191	8	,	,	PUNCT
ejpam-6940	191	9	α	α	PROPN
ejpam-6940	191	10	,	,	PUNCT
ejpam-6940	191	11	η	η	PROPN
ejpam-6940	191	12	,	,	PUNCT
ejpam-6940	191	13	µ	µ	NOUN
ejpam-6940	191	14	}	}	PUNCT
ejpam-6940	191	15	and	and	CCONJ
ejpam-6940	191	16	a	a	DET
ejpam-6940	191	17	binary	binary	ADJ
ejpam-6940	191	18	operation	operation	NOUN
ejpam-6940	191	19	∗	∗	NOUN
ejpam-6940	191	20	be	be	AUX
ejpam-6940	191	21	defined	define	VERB
ejpam-6940	191	22	on	on	ADP
ejpam-6940	191	23	x	x	PUNCT
ejpam-6940	191	24	as	as	ADP
ejpam-6940	191	25	the	the	DET
ejpam-6940	191	26	following	follow	VERB
ejpam-6940	191	27	table	table	NOUN
ejpam-6940	191	28	.	.	PUNCT
ejpam-6940	192	1	∗	∗	NOUN
ejpam-6940	192	2	0	0	NUM
ejpam-6940	192	3	α	α	PROPN
ejpam-6940	192	4	η	η	PROPN
ejpam-6940	192	5	µ	µ	X
ejpam-6940	192	6	0	0	NUM
ejpam-6940	192	7	0	0	NUM
ejpam-6940	192	8	0	0	NUM
ejpam-6940	192	9	0	0	NUM
ejpam-6940	192	10	0	0	NUM
ejpam-6940	193	1	α	α	NOUN
ejpam-6940	193	2	α	α	NOUN
ejpam-6940	193	3	0	0	PUNCT
ejpam-6940	193	4	α	α	NOUN
ejpam-6940	193	5	0	0	NUM
ejpam-6940	193	6	η	η	PROPN
ejpam-6940	193	7	η	η	PROPN
ejpam-6940	193	8	0	0	PROPN
ejpam-6940	193	9	0	0	NUM
ejpam-6940	193	10	η	η	PROPN
ejpam-6940	193	11	µ	µ	X
ejpam-6940	193	12	µ	µ	X
ejpam-6940	193	13	0	0	NUM
ejpam-6940	193	14	µ	µ	NOUN
ejpam-6940	193	15	0	0	NUM
ejpam-6940	193	16	then	then	ADV
ejpam-6940	193	17	(	(	PUNCT
ejpam-6940	193	18	x	x	NOUN
ejpam-6940	193	19	;	;	PUNCT
ejpam-6940	193	20	∗	∗	NOUN
ejpam-6940	193	21	,	,	PUNCT
ejpam-6940	193	22	0	0	NUM
ejpam-6940	193	23	)	)	PUNCT
ejpam-6940	193	24	is	be	AUX
ejpam-6940	193	25	an	an	DET
ejpam-6940	193	26	edge	edge	NOUN
ejpam-6940	193	27	q	q	NOUN
ejpam-6940	193	28	-	-	NOUN
ejpam-6940	193	29	algebra	algebra	NOUN
ejpam-6940	193	30	.	.	PUNCT
ejpam-6940	194	1	since	since	SCONJ
ejpam-6940	194	2	α	α	PROPN
ejpam-6940	194	3	∗	∗	X
ejpam-6940	194	4	µ	µ	X
ejpam-6940	194	5	=	=	SYM
ejpam-6940	194	6	0	0	NUM
ejpam-6940	194	7	and	and	CCONJ
ejpam-6940	194	8	µ	µ	NOUN
ejpam-6940	194	9	∗	∗	X
ejpam-6940	194	10	α	α	NOUN
ejpam-6940	194	11	=	=	SYM
ejpam-6940	194	12	0	0	PROPN
ejpam-6940	195	1	but	but	CCONJ
ejpam-6940	195	2	α	α	PROPN
ejpam-6940	195	3	̸=	̸=	PROPN
ejpam-6940	195	4	µ	µ	NUM
ejpam-6940	195	5	,	,	PUNCT
ejpam-6940	195	6	then	then	ADV
ejpam-6940	195	7	the	the	DET
ejpam-6940	195	8	condition	condition	NOUN
ejpam-6940	195	9	(	(	PUNCT
ejpam-6940	195	10	d3	d3	PROPN
ejpam-6940	195	11	)	)	PUNCT
ejpam-6940	195	12	is	be	AUX
ejpam-6940	195	13	not	not	PART
ejpam-6940	195	14	satisfied	satisfied	ADJ
ejpam-6940	195	15	.	.	PUNCT
ejpam-6940	196	1	thus	thus	ADV
ejpam-6940	196	2	,	,	PUNCT
ejpam-6940	196	3	(	(	PUNCT
ejpam-6940	196	4	x	x	X
ejpam-6940	196	5	;	;	PUNCT
ejpam-6940	196	6	∗	∗	NOUN
ejpam-6940	196	7	,	,	PUNCT
ejpam-6940	196	8	0	0	NUM
ejpam-6940	196	9	)	)	PUNCT
ejpam-6940	196	10	is	be	AUX
ejpam-6940	196	11	not	not	PART
ejpam-6940	196	12	a	a	DET
ejpam-6940	196	13	d	d	NOUN
ejpam-6940	196	14	-	-	NOUN
ejpam-6940	196	15	algebra	algebra	NOUN
ejpam-6940	196	16	.	.	PUNCT
ejpam-6940	197	1	next	next	ADV
ejpam-6940	197	2	,	,	PUNCT
ejpam-6940	197	3	we	we	PRON
ejpam-6940	197	4	examine	examine	VERB
ejpam-6940	197	5	some	some	DET
ejpam-6940	197	6	properties	property	NOUN
ejpam-6940	197	7	of	of	ADP
ejpam-6940	197	8	edge	edge	NOUN
ejpam-6940	197	9	q	q	NOUN
ejpam-6940	197	10	-	-	PUNCT
ejpam-6940	197	11	algebras	algebra	NOUN
ejpam-6940	197	12	.	.	PUNCT
ejpam-6940	198	1	proposition	proposition	NOUN
ejpam-6940	198	2	5	5	NUM
ejpam-6940	198	3	.	.	PUNCT
ejpam-6940	199	1	let	let	VERB
ejpam-6940	199	2	x	x	PRON
ejpam-6940	199	3	be	be	AUX
ejpam-6940	199	4	an	an	DET
ejpam-6940	199	5	edge	edge	NOUN
ejpam-6940	199	6	q	q	NOUN
ejpam-6940	199	7	-	-	NOUN
ejpam-6940	199	8	algebra	algebra	NOUN
ejpam-6940	199	9	and	and	CCONJ
ejpam-6940	199	10	let	let	VERB
ejpam-6940	199	11	a	a	PRON
ejpam-6940	199	12	and	and	CCONJ
ejpam-6940	199	13	b	b	NOUN
ejpam-6940	199	14	be	be	AUX
ejpam-6940	199	15	non	non	ADJ
ejpam-6940	199	16	-	-	ADJ
ejpam-6940	199	17	empty	empty	ADJ
ejpam-6940	199	18	subsets	subset	NOUN
ejpam-6940	199	19	of	of	ADP
ejpam-6940	199	20	x.	x.	NOUN
ejpam-6940	199	21	then	then	ADV
ejpam-6940	199	22	the	the	DET
ejpam-6940	199	23	following	follow	VERB
ejpam-6940	199	24	properties	property	NOUN
ejpam-6940	199	25	are	be	AUX
ejpam-6940	199	26	valid	valid	ADJ
ejpam-6940	199	27	.	.	PUNCT
ejpam-6940	200	1	(	(	PUNCT
ejpam-6940	200	2	i	i	NOUN
ejpam-6940	200	3	)	)	PUNCT
ejpam-6940	200	4	0x	0x	NOUN
ejpam-6940	200	5	=	=	SYM
ejpam-6940	200	6	{	{	PUNCT
ejpam-6940	200	7	0	0	NUM
ejpam-6940	200	8	}	}	PUNCT
ejpam-6940	200	9	.	.	PUNCT
ejpam-6940	201	1	(	(	PUNCT
ejpam-6940	201	2	ii	ii	NOUN
ejpam-6940	201	3	)	)	PUNCT
ejpam-6940	201	4	if	if	SCONJ
ejpam-6940	201	5	0	0	NUM
ejpam-6940	201	6	∈	∈	PROPN
ejpam-6940	201	7	a	a	NOUN
ejpam-6940	201	8	,	,	PUNCT
ejpam-6940	201	9	then	then	ADV
ejpam-6940	201	10	0	0	NUM
ejpam-6940	201	11	∈	∈	PROPN
ejpam-6940	201	12	ab	ab	PROPN
ejpam-6940	201	13	.	.	PUNCT
ejpam-6940	201	14	proof	proof	NOUN
ejpam-6940	201	15	.	.	PUNCT
ejpam-6940	202	1	(	(	PUNCT
ejpam-6940	202	2	i	i	NOUN
ejpam-6940	202	3	)	)	PUNCT
ejpam-6940	202	4	it	it	PRON
ejpam-6940	202	5	is	be	AUX
ejpam-6940	202	6	clear	clear	ADJ
ejpam-6940	202	7	.	.	PUNCT
ejpam-6940	203	1	(	(	PUNCT
ejpam-6940	203	2	ii	ii	NOUN
ejpam-6940	203	3	)	)	PUNCT
ejpam-6940	203	4	let	let	VERB
ejpam-6940	203	5	0	0	NUM
ejpam-6940	203	6	∈	∈	PROPN
ejpam-6940	203	7	a	a	PRON
ejpam-6940	203	8	and	and	CCONJ
ejpam-6940	203	9	b	b	PROPN
ejpam-6940	203	10	∈	∈	PROPN
ejpam-6940	203	11	b	b	PROPN
ejpam-6940	203	12	,	,	PUNCT
ejpam-6940	203	13	then	then	ADV
ejpam-6940	203	14	by	by	ADP
ejpam-6940	203	15	(	(	PUNCT
ejpam-6940	203	16	i	i	NOUN
ejpam-6940	203	17	)	)	PUNCT
ejpam-6940	203	18	there	there	PRON
ejpam-6940	203	19	follows	follow	VERB
ejpam-6940	203	20	that	that	PRON
ejpam-6940	203	21	0	0	NUM
ejpam-6940	204	1	=	=	SYM
ejpam-6940	204	2	0b	0b	NOUN
ejpam-6940	204	3	∈	∈	PROPN
ejpam-6940	204	4	0b	0b	NOUN
ejpam-6940	204	5	⊆	⊆	NUM
ejpam-6940	204	6	ab	ab	PROPN
ejpam-6940	204	7	.	.	PROPN
ejpam-6940	205	1	for	for	ADP
ejpam-6940	205	2	any	any	DET
ejpam-6940	205	3	a	a	DET
ejpam-6940	205	4	∈	∈	PROPN
ejpam-6940	205	5	x	x	NOUN
ejpam-6940	205	6	,	,	PUNCT
ejpam-6940	205	7	a	a	DET
ejpam-6940	205	8	subset	subset	NOUN
ejpam-6940	205	9	ax	ax	NOUN
ejpam-6940	205	10	normally	normally	ADV
ejpam-6940	205	11	is	be	AUX
ejpam-6940	205	12	not	not	PART
ejpam-6940	205	13	necessarily	necessarily	ADV
ejpam-6940	205	14	a	a	DET
ejpam-6940	205	15	subalgebra	subalgebra	NOUN
ejpam-6940	205	16	of	of	ADP
ejpam-6940	205	17	x.	x.	NOUN
ejpam-6940	205	18	for	for	ADP
ejpam-6940	205	19	example	example	NOUN
ejpam-6940	205	20	,	,	PUNCT
ejpam-6940	205	21	a	a	DET
ejpam-6940	205	22	set	set	NOUN
ejpam-6940	205	23	ay	ay	NOUN
ejpam-6940	205	24	=	=	PUNCT
ejpam-6940	205	25	{	{	PUNCT
ejpam-6940	205	26	0	0	NUM
ejpam-6940	205	27	,	,	PUNCT
ejpam-6940	205	28	a	a	DET
ejpam-6940	205	29	,	,	PUNCT
ejpam-6940	205	30	b	b	NOUN
ejpam-6940	205	31	,	,	PUNCT
ejpam-6940	205	32	f	f	NOUN
ejpam-6940	205	33	}	}	PUNCT
ejpam-6940	205	34	in	in	ADP
ejpam-6940	205	35	example	example	NOUN
ejpam-6940	205	36	3	3	NUM
ejpam-6940	205	37	is	be	AUX
ejpam-6940	205	38	not	not	PART
ejpam-6940	205	39	a	a	DET
ejpam-6940	205	40	subalgebra	subalgebra	NOUN
ejpam-6940	205	41	of	of	ADP
ejpam-6940	205	42	(	(	PUNCT
ejpam-6940	205	43	y	y	PROPN
ejpam-6940	205	44	;	;	PUNCT
ejpam-6940	205	45	•	•	X
ejpam-6940	205	46	,	,	PUNCT
ejpam-6940	205	47	0	0	NUM
ejpam-6940	205	48	)	)	PUNCT
ejpam-6940	205	49	since	since	SCONJ
ejpam-6940	205	50	b	b	NUM
ejpam-6940	205	51	•	•	NOUN
ejpam-6940	205	52	a	a	DET
ejpam-6940	205	53	=	=	SYM
ejpam-6940	205	54	c	c	X
ejpam-6940	205	55	̸∈	̸∈	PROPN
ejpam-6940	205	56	ay	ay	PROPN
ejpam-6940	205	57	.	.	PUNCT
ejpam-6940	206	1	but	but	CCONJ
ejpam-6940	206	2	for	for	ADP
ejpam-6940	206	3	edge	edge	NOUN
ejpam-6940	206	4	q	q	NOUN
ejpam-6940	206	5	-	-	PUNCT
ejpam-6940	206	6	algebras	algebras	ADV
ejpam-6940	206	7	we	we	PRON
ejpam-6940	206	8	get	get	VERB
ejpam-6940	206	9	the	the	DET
ejpam-6940	206	10	following	follow	VERB
ejpam-6940	206	11	positive	positive	ADJ
ejpam-6940	206	12	result	result	NOUN
ejpam-6940	206	13	.	.	PUNCT
ejpam-6940	207	1	proposition	proposition	NOUN
ejpam-6940	207	2	6	6	NUM
ejpam-6940	207	3	.	.	PUNCT
ejpam-6940	208	1	if	if	SCONJ
ejpam-6940	208	2	x	x	PRON
ejpam-6940	208	3	is	be	AUX
ejpam-6940	208	4	an	an	DET
ejpam-6940	208	5	edge	edge	NOUN
ejpam-6940	208	6	q	q	NOUN
ejpam-6940	208	7	-	-	NOUN
ejpam-6940	208	8	algebra	algebra	NOUN
ejpam-6940	208	9	,	,	PUNCT
ejpam-6940	208	10	then	then	ADV
ejpam-6940	208	11	ax	ax	NOUN
ejpam-6940	208	12	is	be	AUX
ejpam-6940	208	13	a	a	DET
ejpam-6940	208	14	subalgebra	subalgebra	NOUN
ejpam-6940	208	15	of	of	ADP
ejpam-6940	208	16	x	x	PUNCT
ejpam-6940	208	17	for	for	ADP
ejpam-6940	208	18	all	all	DET
ejpam-6940	208	19	a	a	DET
ejpam-6940	208	20	∈	∈	NOUN
ejpam-6940	208	21	x.	x.	NOUN
ejpam-6940	208	22	proof	proof	NOUN
ejpam-6940	208	23	.	.	PUNCT
ejpam-6940	209	1	let	let	VERB
ejpam-6940	209	2	a	a	DET
ejpam-6940	209	3	∈	∈	NOUN
ejpam-6940	209	4	x.	x.	NOUN
ejpam-6940	209	5	then	then	ADV
ejpam-6940	209	6	ax	ax	NOUN
ejpam-6940	209	7	=	=	PUNCT
ejpam-6940	209	8	{	{	PUNCT
ejpam-6940	209	9	0	0	NUM
ejpam-6940	209	10	,	,	PUNCT
ejpam-6940	209	11	a	a	PRON
ejpam-6940	209	12	}	}	PUNCT
ejpam-6940	209	13	.	.	PUNCT
ejpam-6940	210	1	it	it	PRON
ejpam-6940	210	2	is	be	AUX
ejpam-6940	210	3	easy	easy	ADJ
ejpam-6940	210	4	to	to	PART
ejpam-6940	210	5	verify	verify	VERB
ejpam-6940	210	6	that	that	PRON
ejpam-6940	210	7	00	00	PUNCT
ejpam-6940	210	8	,	,	PUNCT
ejpam-6940	210	9	a0	a0	PROPN
ejpam-6940	210	10	and	and	CCONJ
ejpam-6940	210	11	aa	aa	NOUN
ejpam-6940	210	12	are	be	AUX
ejpam-6940	210	13	elements	element	NOUN
ejpam-6940	210	14	in	in	ADP
ejpam-6940	210	15	ax	ax	NOUN
ejpam-6940	210	16	.	.	PUNCT
ejpam-6940	211	1	now	now	ADV
ejpam-6940	211	2	we	we	PRON
ejpam-6940	211	3	calculate	calculate	VERB
ejpam-6940	211	4	0a	0a	PROPN
ejpam-6940	211	5	.	.	PUNCT
ejpam-6940	212	1	since	since	SCONJ
ejpam-6940	212	2	0a	0a	PROPN
ejpam-6940	212	3	∈	∈	PROPN
ejpam-6940	212	4	0x	0x	NOUN
ejpam-6940	212	5	,	,	PUNCT
ejpam-6940	212	6	by	by	ADP
ejpam-6940	212	7	proposition	proposition	NOUN
ejpam-6940	212	8	5(i	5(i	NOUN
ejpam-6940	212	9	)	)	PUNCT
ejpam-6940	212	10	we	we	PRON
ejpam-6940	212	11	get	get	VERB
ejpam-6940	212	12	0a	0a	NOUN
ejpam-6940	212	13	=	=	SYM
ejpam-6940	212	14	0	0	NUM
ejpam-6940	212	15	∈	∈	PROPN
ejpam-6940	212	16	ax	ax	NOUN
ejpam-6940	212	17	.	.	PUNCT
ejpam-6940	213	1	therefore	therefore	ADV
ejpam-6940	213	2	,	,	PUNCT
ejpam-6940	213	3	ax	ax	NOUN
ejpam-6940	213	4	is	be	AUX
ejpam-6940	213	5	closed	close	VERB
ejpam-6940	213	6	and	and	CCONJ
ejpam-6940	213	7	then	then	ADV
ejpam-6940	213	8	ax	ax	NOUN
ejpam-6940	213	9	is	be	AUX
ejpam-6940	213	10	a	a	DET
ejpam-6940	213	11	subalgebra	subalgebra	NOUN
ejpam-6940	213	12	of	of	ADP
ejpam-6940	213	13	x.	x.	PROPN
ejpam-6940	213	14	a.	a.	PROPN
ejpam-6940	213	15	anantayasethi	anantayasethi	PROPN
ejpam-6940	213	16	et	et	PROPN
ejpam-6940	213	17	al	al	PROPN
ejpam-6940	213	18	.	.	PUNCT
ejpam-6940	213	19	/	/	SYM
ejpam-6940	213	20	eur	eur	PROPN
ejpam-6940	213	21	.	.	PUNCT
ejpam-6940	214	1	j.	j.	PROPN
ejpam-6940	214	2	pure	pure	PROPN
ejpam-6940	214	3	appl	appl	PROPN
ejpam-6940	214	4	.	.	PROPN
ejpam-6940	214	5	math	math	PROPN
ejpam-6940	214	6	,	,	PUNCT
ejpam-6940	214	7	18	18	NUM
ejpam-6940	214	8	(	(	PUNCT
ejpam-6940	214	9	4	4	NUM
ejpam-6940	214	10	)	)	PUNCT
ejpam-6940	214	11	(	(	PUNCT
ejpam-6940	214	12	2025	2025	NUM
ejpam-6940	214	13	)	)	PUNCT
ejpam-6940	214	14	,	,	PUNCT
ejpam-6940	214	15	6940	6940	NUM
ejpam-6940	214	16	7	7	NUM
ejpam-6940	214	17	of	of	ADP
ejpam-6940	214	18	14	14	NUM
ejpam-6940	214	19	next	next	ADV
ejpam-6940	215	1	,	,	PUNCT
ejpam-6940	215	2	we	we	PRON
ejpam-6940	215	3	will	will	AUX
ejpam-6940	215	4	examine	examine	VERB
ejpam-6940	215	5	some	some	DET
ejpam-6940	215	6	conditions	condition	NOUN
ejpam-6940	215	7	that	that	PRON
ejpam-6940	215	8	lead	lead	VERB
ejpam-6940	215	9	the	the	DET
ejpam-6940	215	10	product	product	NOUN
ejpam-6940	215	11	of	of	ADP
ejpam-6940	215	12	subsets	subset	NOUN
ejpam-6940	215	13	of	of	ADP
ejpam-6940	215	14	a	a	DET
ejpam-6940	215	15	q	q	NOUN
ejpam-6940	215	16	-	-	NOUN
ejpam-6940	215	17	algebra	algebra	NOUN
ejpam-6940	215	18	x	x	PUNCT
ejpam-6940	215	19	to	to	PART
ejpam-6940	215	20	be	be	AUX
ejpam-6940	215	21	a	a	DET
ejpam-6940	215	22	subalgebra	subalgebra	NOUN
ejpam-6940	215	23	of	of	ADP
ejpam-6940	215	24	x.	x.	NOUN
ejpam-6940	215	25	proposition	proposition	NOUN
ejpam-6940	215	26	7	7	NUM
ejpam-6940	215	27	.	.	PUNCT
ejpam-6940	216	1	let	let	VERB
ejpam-6940	216	2	x	x	PRON
ejpam-6940	216	3	be	be	AUX
ejpam-6940	216	4	an	an	DET
ejpam-6940	216	5	edge	edge	NOUN
ejpam-6940	216	6	q	q	NOUN
ejpam-6940	216	7	-	-	NOUN
ejpam-6940	216	8	algebra	algebra	NOUN
ejpam-6940	216	9	and	and	CCONJ
ejpam-6940	216	10	let	let	VERB
ejpam-6940	216	11	∅	∅	NOUN
ejpam-6940	216	12	̸=	̸=	PROPN
ejpam-6940	216	13	a	a	PRON
ejpam-6940	216	14	,	,	PUNCT
ejpam-6940	216	15	b	b	PROPN
ejpam-6940	216	16	⊆	⊆	NUM
ejpam-6940	216	17	x.	x.	NOUN
ejpam-6940	216	18	if	if	SCONJ
ejpam-6940	216	19	0	0	NUM
ejpam-6940	216	20	∈	∈	PROPN
ejpam-6940	216	21	a	a	DET
ejpam-6940	216	22	or	or	CCONJ
ejpam-6940	216	23	a∩b	a∩b	PROPN
ejpam-6940	216	24	̸=	̸=	PROPN
ejpam-6940	216	25	∅	∅	NOUN
ejpam-6940	216	26	then	then	ADV
ejpam-6940	216	27	ab	ab	PROPN
ejpam-6940	216	28	is	be	AUX
ejpam-6940	216	29	a	a	DET
ejpam-6940	216	30	subalgebra	subalgebra	NOUN
ejpam-6940	216	31	of	of	ADP
ejpam-6940	216	32	x.	x.	NOUN
ejpam-6940	216	33	proof	proof	PROPN
ejpam-6940	216	34	.	.	PUNCT
ejpam-6940	217	1	assume	assume	VERB
ejpam-6940	217	2	that	that	SCONJ
ejpam-6940	217	3	0	0	NUM
ejpam-6940	217	4	∈	∈	NOUN
ejpam-6940	217	5	a.	a.	NOUN
ejpam-6940	217	6	then	then	ADV
ejpam-6940	217	7	by	by	ADP
ejpam-6940	217	8	proposition	proposition	NOUN
ejpam-6940	217	9	5(ii	5(ii	NUM
ejpam-6940	217	10	)	)	PUNCT
ejpam-6940	217	11	,	,	PUNCT
ejpam-6940	217	12	0	0	NUM
ejpam-6940	217	13	∈	∈	PROPN
ejpam-6940	218	1	ab	ab	PROPN
ejpam-6940	218	2	.	.	PUNCT
ejpam-6940	218	3	let	let	VERB
ejpam-6940	218	4	x	x	PRON
ejpam-6940	218	5	,	,	PUNCT
ejpam-6940	218	6	y	y	PROPN
ejpam-6940	218	7	∈	∈	PROPN
ejpam-6940	218	8	ab	ab	PROPN
ejpam-6940	218	9	.	.	PUNCT
ejpam-6940	219	1	since	since	SCONJ
ejpam-6940	219	2	x	x	PRON
ejpam-6940	219	3	is	be	AUX
ejpam-6940	219	4	an	an	DET
ejpam-6940	219	5	edge	edge	NOUN
ejpam-6940	219	6	q	q	NOUN
ejpam-6940	219	7	-	-	NOUN
ejpam-6940	219	8	algebra	algebra	NOUN
ejpam-6940	219	9	,	,	PUNCT
ejpam-6940	219	10	then	then	ADV
ejpam-6940	219	11	xy	xy	PROPN
ejpam-6940	219	12	∈	∈	PROPN
ejpam-6940	219	13	xx	xx	NUM
ejpam-6940	220	1	=	=	SYM
ejpam-6940	220	2	{	{	PUNCT
ejpam-6940	220	3	0	0	NUM
ejpam-6940	220	4	,	,	PUNCT
ejpam-6940	220	5	x	x	NOUN
ejpam-6940	220	6	}	}	PUNCT
ejpam-6940	220	7	.	.	PUNCT
ejpam-6940	221	1	there	there	PRON
ejpam-6940	221	2	follows	follow	VERB
ejpam-6940	221	3	that	that	PRON
ejpam-6940	221	4	xy	xy	PROPN
ejpam-6940	222	1	=	=	SYM
ejpam-6940	222	2	0	0	NUM
ejpam-6940	222	3	or	or	CCONJ
ejpam-6940	222	4	xy	xy	NOUN
ejpam-6940	222	5	=	=	PUNCT
ejpam-6940	223	1	x.	x.	NOUN
ejpam-6940	223	2	thus	thus	ADV
ejpam-6940	223	3	,	,	PUNCT
ejpam-6940	223	4	xy	xy	PROPN
ejpam-6940	223	5	∈	∈	PROPN
ejpam-6940	223	6	ab	ab	PROPN
ejpam-6940	223	7	.	.	PUNCT
ejpam-6940	223	8	therefore	therefore	ADV
ejpam-6940	223	9	,	,	PUNCT
ejpam-6940	223	10	ab	ab	PROPN
ejpam-6940	223	11	is	be	AUX
ejpam-6940	223	12	a	a	DET
ejpam-6940	223	13	subalgebra	subalgebra	NOUN
ejpam-6940	223	14	.	.	PUNCT
ejpam-6940	224	1	assume	assume	VERB
ejpam-6940	224	2	now	now	ADV
ejpam-6940	224	3	that	that	SCONJ
ejpam-6940	224	4	a	a	DET
ejpam-6940	224	5	∩	∩	ADJ
ejpam-6940	224	6	b	b	NOUN
ejpam-6940	224	7	̸=	̸=	PROPN
ejpam-6940	224	8	∅.	∅.	X
ejpam-6940	224	9	by	by	ADP
ejpam-6940	224	10	proposition	proposition	NOUN
ejpam-6940	224	11	1(v	1(v	NUM
ejpam-6940	224	12	)	)	PUNCT
ejpam-6940	224	13	,	,	PUNCT
ejpam-6940	224	14	0	0	NUM
ejpam-6940	224	15	∈	∈	PROPN
ejpam-6940	224	16	ab	ab	PROPN
ejpam-6940	224	17	.	.	PUNCT
ejpam-6940	225	1	using	use	VERB
ejpam-6940	225	2	the	the	DET
ejpam-6940	225	3	same	same	ADJ
ejpam-6940	225	4	argument	argument	NOUN
ejpam-6940	225	5	,	,	PUNCT
ejpam-6940	225	6	we	we	PRON
ejpam-6940	225	7	get	get	VERB
ejpam-6940	225	8	ab	ab	PROPN
ejpam-6940	225	9	is	be	AUX
ejpam-6940	225	10	closed	closed	ADJ
ejpam-6940	225	11	.	.	PUNCT
ejpam-6940	226	1	hence	hence	ADV
ejpam-6940	226	2	,	,	PUNCT
ejpam-6940	226	3	ab	ab	PROPN
ejpam-6940	226	4	is	be	AUX
ejpam-6940	226	5	a	a	DET
ejpam-6940	226	6	subalgebra	subalgebra	NOUN
ejpam-6940	226	7	of	of	ADP
ejpam-6940	226	8	x.	x.	NOUN
ejpam-6940	226	9	the	the	DET
ejpam-6940	226	10	converse	converse	NOUN
ejpam-6940	226	11	of	of	ADP
ejpam-6940	226	12	proposition	proposition	NOUN
ejpam-6940	226	13	7	7	NUM
ejpam-6940	226	14	is	be	AUX
ejpam-6940	226	15	not	not	PART
ejpam-6940	226	16	true	true	ADJ
ejpam-6940	226	17	as	as	SCONJ
ejpam-6940	226	18	shown	show	VERB
ejpam-6940	226	19	in	in	ADP
ejpam-6940	226	20	the	the	DET
ejpam-6940	226	21	following	follow	VERB
ejpam-6940	226	22	example	example	NOUN
ejpam-6940	226	23	.	.	PUNCT
ejpam-6940	227	1	example	example	NOUN
ejpam-6940	228	1	5	5	NUM
ejpam-6940	228	2	.	.	PUNCT
ejpam-6940	228	3	let	let	VERB
ejpam-6940	228	4	x	x	PUNCT
ejpam-6940	228	5	=	=	PUNCT
ejpam-6940	228	6	{	{	PUNCT
ejpam-6940	228	7	0	0	NUM
ejpam-6940	228	8	,	,	PUNCT
ejpam-6940	228	9	a	a	DET
ejpam-6940	228	10	,	,	PUNCT
ejpam-6940	228	11	b	b	NOUN
ejpam-6940	228	12	,	,	PUNCT
ejpam-6940	228	13	c	c	NOUN
ejpam-6940	228	14	}	}	PUNCT
ejpam-6940	228	15	and	and	CCONJ
ejpam-6940	228	16	let	let	VERB
ejpam-6940	228	17	a	a	DET
ejpam-6940	228	18	binary	binary	ADJ
ejpam-6940	228	19	operation	operation	NOUN
ejpam-6940	228	20	∗	∗	NOUN
ejpam-6940	228	21	be	be	AUX
ejpam-6940	228	22	defined	define	VERB
ejpam-6940	228	23	on	on	ADP
ejpam-6940	228	24	x	x	PART
ejpam-6940	228	25	as	as	SCONJ
ejpam-6940	228	26	follow	follow	NOUN
ejpam-6940	228	27	:	:	PUNCT
ejpam-6940	228	28	∗	∗	NOUN
ejpam-6940	228	29	0	0	PUNCT
ejpam-6940	229	1	a	a	DET
ejpam-6940	229	2	b	b	NOUN
ejpam-6940	229	3	c	c	NOUN
ejpam-6940	229	4	0	0	NUM
ejpam-6940	229	5	0	0	NUM
ejpam-6940	229	6	0	0	NUM
ejpam-6940	229	7	0	0	NUM
ejpam-6940	229	8	0	0	NUM
ejpam-6940	229	9	a	a	DET
ejpam-6940	229	10	a	a	DET
ejpam-6940	229	11	0	0	NUM
ejpam-6940	229	12	a	a	DET
ejpam-6940	229	13	0	0	NUM
ejpam-6940	229	14	b	b	PROPN
ejpam-6940	229	15	b	b	PROPN
ejpam-6940	229	16	0	0	NUM
ejpam-6940	229	17	0	0	NUM
ejpam-6940	229	18	0	0	NUM
ejpam-6940	230	1	c	c	NOUN
ejpam-6940	230	2	c	c	NOUN
ejpam-6940	230	3	c	c	NOUN
ejpam-6940	230	4	c	c	NOUN
ejpam-6940	230	5	0	0	NUM
ejpam-6940	231	1	it	it	PRON
ejpam-6940	231	2	is	be	AUX
ejpam-6940	231	3	a	a	DET
ejpam-6940	231	4	routine	routine	NOUN
ejpam-6940	231	5	to	to	PART
ejpam-6940	231	6	check	check	VERB
ejpam-6940	231	7	that	that	PRON
ejpam-6940	231	8	(	(	PUNCT
ejpam-6940	231	9	x	x	X
ejpam-6940	231	10	;	;	PUNCT
ejpam-6940	231	11	∗	∗	NOUN
ejpam-6940	231	12	,	,	PUNCT
ejpam-6940	231	13	0	0	NUM
ejpam-6940	231	14	)	)	PUNCT
ejpam-6940	231	15	is	be	AUX
ejpam-6940	231	16	a	a	DET
ejpam-6940	231	17	q	q	NOUN
ejpam-6940	231	18	-	-	PUNCT
ejpam-6940	231	19	algebra	algebra	NOUN
ejpam-6940	231	20	.	.	PUNCT
ejpam-6940	232	1	since	since	SCONJ
ejpam-6940	232	2	0x	0x	NUM
ejpam-6940	232	3	=	=	SYM
ejpam-6940	232	4	{	{	PUNCT
ejpam-6940	232	5	0	0	NUM
ejpam-6940	232	6	}	}	PUNCT
ejpam-6940	232	7	,	,	PUNCT
ejpam-6940	232	8	ax	ax	NOUN
ejpam-6940	232	9	=	=	SYM
ejpam-6940	232	10	{	{	PUNCT
ejpam-6940	232	11	0	0	NUM
ejpam-6940	232	12	,	,	PUNCT
ejpam-6940	232	13	a	a	PRON
ejpam-6940	232	14	}	}	PUNCT
ejpam-6940	232	15	,	,	PUNCT
ejpam-6940	232	16	bx	bx	X
ejpam-6940	232	17	=	=	PUNCT
ejpam-6940	232	18	{	{	PUNCT
ejpam-6940	232	19	0	0	NUM
ejpam-6940	232	20	,	,	PUNCT
ejpam-6940	232	21	b	b	NOUN
ejpam-6940	232	22	}	}	PUNCT
ejpam-6940	232	23	and	and	CCONJ
ejpam-6940	232	24	cx	cx	NOUN
ejpam-6940	232	25	=	=	PUNCT
ejpam-6940	232	26	{	{	PUNCT
ejpam-6940	232	27	0	0	NUM
ejpam-6940	232	28	,	,	PUNCT
ejpam-6940	232	29	c	c	NOUN
ejpam-6940	232	30	}	}	PUNCT
ejpam-6940	232	31	,	,	PUNCT
ejpam-6940	232	32	then	then	ADV
ejpam-6940	232	33	x	x	PUNCT
ejpam-6940	232	34	is	be	AUX
ejpam-6940	232	35	an	an	DET
ejpam-6940	232	36	edge	edge	NOUN
ejpam-6940	232	37	.	.	PUNCT
ejpam-6940	233	1	let	let	VERB
ejpam-6940	233	2	a	a	PRON
ejpam-6940	233	3	=	=	SYM
ejpam-6940	233	4	{	{	PUNCT
ejpam-6940	233	5	b	b	NOUN
ejpam-6940	233	6	}	}	PUNCT
ejpam-6940	233	7	and	and	CCONJ
ejpam-6940	233	8	b	b	X
ejpam-6940	233	9	=	=	PUNCT
ejpam-6940	233	10	{	{	PUNCT
ejpam-6940	233	11	c	c	NOUN
ejpam-6940	233	12	}	}	PUNCT
ejpam-6940	233	13	.	.	PUNCT
ejpam-6940	234	1	then	then	ADV
ejpam-6940	234	2	ab	ab	PROPN
ejpam-6940	234	3	=	=	PUNCT
ejpam-6940	234	4	{	{	PUNCT
ejpam-6940	234	5	b	b	NOUN
ejpam-6940	234	6	∗	∗	NOUN
ejpam-6940	234	7	c	c	NOUN
ejpam-6940	234	8	}	}	PUNCT
ejpam-6940	234	9	=	=	SYM
ejpam-6940	234	10	{	{	PUNCT
ejpam-6940	234	11	0	0	NUM
ejpam-6940	234	12	}	}	PUNCT
ejpam-6940	234	13	is	be	AUX
ejpam-6940	234	14	a	a	DET
ejpam-6940	234	15	subalgebra	subalgebra	NOUN
ejpam-6940	234	16	.	.	PUNCT
ejpam-6940	235	1	but	but	CCONJ
ejpam-6940	235	2	neither	neither	PRON
ejpam-6940	235	3	0	0	NUM
ejpam-6940	235	4	∈	∈	PROPN
ejpam-6940	235	5	a	a	PRON
ejpam-6940	235	6	nor	nor	CCONJ
ejpam-6940	235	7	a	a	DET
ejpam-6940	235	8	∩	∩	ADJ
ejpam-6940	235	9	b	b	NOUN
ejpam-6940	235	10	̸=	̸=	PROPN
ejpam-6940	235	11	∅.	∅.	ADV
ejpam-6940	235	12	thus	thus	ADV
ejpam-6940	235	13	,	,	PUNCT
ejpam-6940	235	14	the	the	DET
ejpam-6940	235	15	converse	converse	NOUN
ejpam-6940	235	16	of	of	ADP
ejpam-6940	235	17	proposition	proposition	NOUN
ejpam-6940	235	18	7	7	NUM
ejpam-6940	235	19	is	be	AUX
ejpam-6940	235	20	not	not	PART
ejpam-6940	235	21	true	true	ADJ
ejpam-6940	235	22	.	.	PUNCT
ejpam-6940	236	1	corollary	corollary	ADJ
ejpam-6940	236	2	1	1	NUM
ejpam-6940	236	3	.	.	PUNCT
ejpam-6940	237	1	let	let	VERB
ejpam-6940	237	2	x	x	PRON
ejpam-6940	237	3	be	be	AUX
ejpam-6940	237	4	an	an	DET
ejpam-6940	237	5	edge	edge	NOUN
ejpam-6940	237	6	q	q	NOUN
ejpam-6940	237	7	-	-	NOUN
ejpam-6940	237	8	algebra	algebra	NOUN
ejpam-6940	237	9	and	and	CCONJ
ejpam-6940	237	10	let	let	VERB
ejpam-6940	237	11	a	a	DET
ejpam-6940	237	12	,	,	PUNCT
ejpam-6940	237	13	b	b	NOUN
ejpam-6940	237	14	be	be	AUX
ejpam-6940	237	15	non	non	ADJ
ejpam-6940	237	16	-	-	ADJ
ejpam-6940	237	17	empty	empty	ADJ
ejpam-6940	237	18	subsets	subset	NOUN
ejpam-6940	237	19	of	of	ADP
ejpam-6940	237	20	x.	x.	NOUN
ejpam-6940	237	21	then	then	ADV
ejpam-6940	237	22	the	the	DET
ejpam-6940	237	23	following	follow	VERB
ejpam-6940	237	24	properties	property	NOUN
ejpam-6940	237	25	are	be	AUX
ejpam-6940	237	26	true	true	ADJ
ejpam-6940	237	27	.	.	PUNCT
ejpam-6940	238	1	(	(	PUNCT
ejpam-6940	238	2	i	i	NOUN
ejpam-6940	238	3	)	)	PUNCT
ejpam-6940	238	4	if	if	SCONJ
ejpam-6940	238	5	a	a	PRON
ejpam-6940	238	6	is	be	AUX
ejpam-6940	238	7	a	a	DET
ejpam-6940	238	8	subalgebra	subalgebra	NOUN
ejpam-6940	238	9	of	of	ADP
ejpam-6940	238	10	x	x	PRON
ejpam-6940	238	11	,	,	PUNCT
ejpam-6940	238	12	then	then	ADV
ejpam-6940	238	13	ab	ab	PROPN
ejpam-6940	238	14	is	be	AUX
ejpam-6940	238	15	a	a	DET
ejpam-6940	238	16	subalgebra	subalgebra	NOUN
ejpam-6940	238	17	of	of	ADP
ejpam-6940	238	18	x.	x.	PROPN
ejpam-6940	238	19	(	(	PUNCT
ejpam-6940	238	20	ii	ii	PROPN
ejpam-6940	238	21	)	)	PUNCT
ejpam-6940	238	22	if	if	SCONJ
ejpam-6940	238	23	a	a	PRON
ejpam-6940	238	24	is	be	AUX
ejpam-6940	238	25	an	an	DET
ejpam-6940	238	26	ideal	ideal	NOUN
ejpam-6940	238	27	of	of	ADP
ejpam-6940	238	28	x	x	PRON
ejpam-6940	238	29	,	,	PUNCT
ejpam-6940	238	30	then	then	ADV
ejpam-6940	238	31	ab	ab	PROPN
ejpam-6940	238	32	is	be	AUX
ejpam-6940	238	33	a	a	DET
ejpam-6940	238	34	subalgebra	subalgebra	NOUN
ejpam-6940	238	35	of	of	ADP
ejpam-6940	238	36	x.	x.	NOUN
ejpam-6940	238	37	proof	proof	NOUN
ejpam-6940	238	38	.	.	PUNCT
ejpam-6940	239	1	(	(	PUNCT
ejpam-6940	239	2	i	i	NOUN
ejpam-6940	239	3	)	)	PUNCT
ejpam-6940	239	4	follows	follow	VERB
ejpam-6940	239	5	directly	directly	ADV
ejpam-6940	239	6	from	from	ADP
ejpam-6940	239	7	proposition	proposition	NOUN
ejpam-6940	239	8	7	7	NUM
ejpam-6940	239	9	since	since	SCONJ
ejpam-6940	239	10	0	0	NUM
ejpam-6940	239	11	is	be	AUX
ejpam-6940	239	12	a	a	DET
ejpam-6940	239	13	member	member	NOUN
ejpam-6940	239	14	of	of	ADP
ejpam-6940	239	15	a.	a.	PROPN
ejpam-6940	239	16	(	(	PUNCT
ejpam-6940	239	17	ii	ii	NOUN
ejpam-6940	239	18	)	)	PUNCT
ejpam-6940	239	19	assume	assume	VERB
ejpam-6940	239	20	that	that	SCONJ
ejpam-6940	239	21	a	a	PRON
ejpam-6940	239	22	is	be	AUX
ejpam-6940	239	23	an	an	DET
ejpam-6940	239	24	ideal	ideal	NOUN
ejpam-6940	239	25	of	of	ADP
ejpam-6940	239	26	x.	x.	NOUN
ejpam-6940	239	27	by	by	ADP
ejpam-6940	239	28	(	(	PUNCT
ejpam-6940	239	29	i1	i1	PROPN
ejpam-6940	239	30	)	)	PUNCT
ejpam-6940	239	31	,	,	PUNCT
ejpam-6940	239	32	0	0	NUM
ejpam-6940	239	33	is	be	AUX
ejpam-6940	239	34	a	a	DET
ejpam-6940	239	35	member	member	NOUN
ejpam-6940	239	36	of	of	ADP
ejpam-6940	239	37	i	i	PRON
ejpam-6940	239	38	there	there	PRON
ejpam-6940	239	39	follows	follow	VERB
ejpam-6940	239	40	by	by	ADP
ejpam-6940	239	41	proposition	proposition	NOUN
ejpam-6940	239	42	7	7	NUM
ejpam-6940	239	43	that	that	SCONJ
ejpam-6940	239	44	ab	ab	PROPN
ejpam-6940	239	45	is	be	AUX
ejpam-6940	239	46	a	a	DET
ejpam-6940	239	47	subalgebra	subalgebra	NOUN
ejpam-6940	239	48	of	of	ADP
ejpam-6940	239	49	x.	x.	PROPN
ejpam-6940	239	50	corollary	corollary	PROPN
ejpam-6940	240	1	2	2	X
ejpam-6940	240	2	.	.	PUNCT
ejpam-6940	241	1	let	let	VERB
ejpam-6940	241	2	x	x	PRON
ejpam-6940	241	3	be	be	AUX
ejpam-6940	241	4	an	an	DET
ejpam-6940	241	5	edge	edge	NOUN
ejpam-6940	241	6	q	q	NOUN
ejpam-6940	241	7	-	-	NOUN
ejpam-6940	241	8	algebra	algebra	NOUN
ejpam-6940	241	9	.	.	PUNCT
ejpam-6940	242	1	then	then	ADV
ejpam-6940	242	2	the	the	DET
ejpam-6940	242	3	following	follow	VERB
ejpam-6940	242	4	properties	property	NOUN
ejpam-6940	242	5	are	be	AUX
ejpam-6940	242	6	true	true	ADJ
ejpam-6940	242	7	.	.	PUNCT
ejpam-6940	243	1	(	(	PUNCT
ejpam-6940	243	2	i	i	NOUN
ejpam-6940	243	3	)	)	PUNCT
ejpam-6940	243	4	for	for	ADP
ejpam-6940	243	5	any	any	DET
ejpam-6940	243	6	a	a	PRON
ejpam-6940	243	7	,	,	PUNCT
ejpam-6940	243	8	b	b	PROPN
ejpam-6940	243	9	∈	∈	PROPN
ejpam-6940	243	10	x	x	NOUN
ejpam-6940	243	11	,	,	PUNCT
ejpam-6940	243	12	ax	ax	NOUN
ejpam-6940	243	13	∪	∪	X
ejpam-6940	243	14	bx	bx	PROPN
ejpam-6940	243	15	is	be	AUX
ejpam-6940	243	16	a	a	DET
ejpam-6940	243	17	subalgebra	subalgebra	NOUN
ejpam-6940	243	18	of	of	ADP
ejpam-6940	243	19	x.	x.	PROPN
ejpam-6940	243	20	(	(	PUNCT
ejpam-6940	243	21	ii	ii	NOUN
ejpam-6940	243	22	)	)	PUNCT
ejpam-6940	243	23	let	let	VERB
ejpam-6940	243	24	∅	∅	NOUN
ejpam-6940	243	25	̸=	̸=	PROPN
ejpam-6940	243	26	λ	λ	PROPN
ejpam-6940	243	27	⊆	⊆	NUM
ejpam-6940	243	28	x.	x.	NOUN
ejpam-6940	243	29	then	then	ADV
ejpam-6940	243	30	∪	∪	ADJ
ejpam-6940	243	31	a∈λ	a∈λ	NOUN
ejpam-6940	243	32	ax	ax	NOUN
ejpam-6940	243	33	is	be	AUX
ejpam-6940	243	34	a	a	DET
ejpam-6940	243	35	subalgebra	subalgebra	NOUN
ejpam-6940	243	36	of	of	ADP
ejpam-6940	243	37	x.	x.	PROPN
ejpam-6940	243	38	a.	a.	PROPN
ejpam-6940	243	39	anantayasethi	anantayasethi	PROPN
ejpam-6940	243	40	et	et	PROPN
ejpam-6940	243	41	al	al	PROPN
ejpam-6940	243	42	.	.	PUNCT
ejpam-6940	243	43	/	/	SYM
ejpam-6940	243	44	eur	eur	PROPN
ejpam-6940	243	45	.	.	PUNCT
ejpam-6940	244	1	j.	j.	PROPN
ejpam-6940	244	2	pure	pure	PROPN
ejpam-6940	244	3	appl	appl	PROPN
ejpam-6940	244	4	.	.	PROPN
ejpam-6940	244	5	math	math	PROPN
ejpam-6940	244	6	,	,	PUNCT
ejpam-6940	244	7	18	18	NUM
ejpam-6940	244	8	(	(	PUNCT
ejpam-6940	244	9	4	4	NUM
ejpam-6940	244	10	)	)	PUNCT
ejpam-6940	244	11	(	(	PUNCT
ejpam-6940	244	12	2025	2025	NUM
ejpam-6940	244	13	)	)	PUNCT
ejpam-6940	244	14	,	,	PUNCT
ejpam-6940	244	15	6940	6940	NUM
ejpam-6940	244	16	8	8	NUM
ejpam-6940	244	17	of	of	ADP
ejpam-6940	244	18	14	14	NUM
ejpam-6940	244	19	proof	proof	NOUN
ejpam-6940	244	20	.	.	PUNCT
ejpam-6940	245	1	(	(	PUNCT
ejpam-6940	245	2	i	i	NOUN
ejpam-6940	245	3	)	)	PUNCT
ejpam-6940	245	4	let	let	VERB
ejpam-6940	245	5	a	a	DET
ejpam-6940	245	6	,	,	PUNCT
ejpam-6940	245	7	b	b	X
ejpam-6940	245	8	∈	∈	PROPN
ejpam-6940	245	9	x.	x.	NOUN
ejpam-6940	246	1	since	since	SCONJ
ejpam-6940	246	2	x	x	PRON
ejpam-6940	246	3	is	be	AUX
ejpam-6940	246	4	an	an	DET
ejpam-6940	246	5	edge	edge	NOUN
ejpam-6940	246	6	q	q	NOUN
ejpam-6940	246	7	-	-	NOUN
ejpam-6940	246	8	algebra	algebra	NOUN
ejpam-6940	246	9	,	,	PUNCT
ejpam-6940	246	10	then	then	ADV
ejpam-6940	246	11	ax∪bx	ax∪bx	X
ejpam-6940	246	12	=	=	PUNCT
ejpam-6940	246	13	{	{	PUNCT
ejpam-6940	246	14	0	0	NUM
ejpam-6940	246	15	,	,	PUNCT
ejpam-6940	246	16	a}∪{0	a}∪{0	NOUN
ejpam-6940	246	17	,	,	PUNCT
ejpam-6940	246	18	b	b	NOUN
ejpam-6940	246	19	}	}	PUNCT
ejpam-6940	246	20	=	=	SYM
ejpam-6940	246	21	{	{	PUNCT
ejpam-6940	246	22	0	0	NUM
ejpam-6940	246	23	,	,	PUNCT
ejpam-6940	246	24	a	a	DET
ejpam-6940	246	25	,	,	PUNCT
ejpam-6940	246	26	b	b	NOUN
ejpam-6940	246	27	}	}	PUNCT
ejpam-6940	246	28	.	.	PUNCT
ejpam-6940	247	1	set	set	VERB
ejpam-6940	247	2	a	a	PRON
ejpam-6940	247	3	=	=	PUNCT
ejpam-6940	247	4	{	{	PUNCT
ejpam-6940	247	5	0	0	NUM
ejpam-6940	247	6	,	,	PUNCT
ejpam-6940	247	7	a	a	DET
ejpam-6940	247	8	,	,	PUNCT
ejpam-6940	247	9	b	b	NOUN
ejpam-6940	247	10	}	}	PUNCT
ejpam-6940	247	11	and	and	CCONJ
ejpam-6940	247	12	b	b	X
ejpam-6940	247	13	=	=	SYM
ejpam-6940	247	14	{	{	PUNCT
ejpam-6940	247	15	0	0	NUM
ejpam-6940	247	16	}	}	PUNCT
ejpam-6940	247	17	,	,	PUNCT
ejpam-6940	247	18	then	then	ADV
ejpam-6940	247	19	by	by	ADP
ejpam-6940	247	20	proposition	proposition	NOUN
ejpam-6940	247	21	1(ii	1(ii	NUM
ejpam-6940	247	22	)	)	PUNCT
ejpam-6940	247	23	,	,	PUNCT
ejpam-6940	247	24	ab	ab	PROPN
ejpam-6940	247	25	=	=	PUNCT
ejpam-6940	247	26	a	a	PROPN
ejpam-6940	247	27	=	=	X
ejpam-6940	247	28	{	{	PUNCT
ejpam-6940	247	29	0	0	NUM
ejpam-6940	247	30	,	,	PUNCT
ejpam-6940	247	31	a	a	PRON
ejpam-6940	247	32	,	,	PUNCT
ejpam-6940	247	33	b	b	NOUN
ejpam-6940	247	34	}	}	PUNCT
ejpam-6940	247	35	=	=	NOUN
ejpam-6940	247	36	ax	ax	NOUN
ejpam-6940	247	37	∪	∪	X
ejpam-6940	247	38	bx	bx	PROPN
ejpam-6940	247	39	.	.	PUNCT
ejpam-6940	248	1	since	since	SCONJ
ejpam-6940	248	2	0	0	NUM
ejpam-6940	248	3	∈	∈	PROPN
ejpam-6940	248	4	a	a	PRON
ejpam-6940	248	5	,	,	PUNCT
ejpam-6940	248	6	then	then	ADV
ejpam-6940	248	7	by	by	ADP
ejpam-6940	248	8	proposition	proposition	NOUN
ejpam-6940	248	9	7	7	NUM
ejpam-6940	248	10	,	,	PUNCT
ejpam-6940	248	11	ab	ab	NOUN
ejpam-6940	248	12	=	=	NOUN
ejpam-6940	248	13	ax	ax	NOUN
ejpam-6940	248	14	∪	∪	X
ejpam-6940	248	15	bx	bx	PROPN
ejpam-6940	248	16	is	be	AUX
ejpam-6940	248	17	a	a	DET
ejpam-6940	248	18	subalgebra	subalgebra	NOUN
ejpam-6940	248	19	of	of	ADP
ejpam-6940	248	20	x.	x.	PROPN
ejpam-6940	248	21	(	(	PUNCT
ejpam-6940	248	22	ii	ii	NOUN
ejpam-6940	248	23	)	)	PUNCT
ejpam-6940	248	24	let	let	VERB
ejpam-6940	248	25	a	a	DET
ejpam-6940	248	26	∈	∈	ADJ
ejpam-6940	248	27	λ	λ	NOUN
ejpam-6940	248	28	⊆	⊆	NUM
ejpam-6940	248	29	x.	x.	NOUN
ejpam-6940	248	30	since	since	SCONJ
ejpam-6940	248	31	x	x	PRON
ejpam-6940	248	32	is	be	AUX
ejpam-6940	248	33	an	an	DET
ejpam-6940	248	34	edge	edge	NOUN
ejpam-6940	248	35	q	q	NOUN
ejpam-6940	248	36	-	-	NOUN
ejpam-6940	248	37	algebra	algebra	NOUN
ejpam-6940	248	38	,	,	PUNCT
ejpam-6940	248	39	then	then	ADV
ejpam-6940	248	40	ax	ax	NOUN
ejpam-6940	248	41	=	=	PUNCT
ejpam-6940	248	42	{	{	PUNCT
ejpam-6940	248	43	0	0	NUM
ejpam-6940	248	44	,	,	PUNCT
ejpam-6940	248	45	a	a	PRON
ejpam-6940	248	46	}	}	PUNCT
ejpam-6940	248	47	.	.	PUNCT
ejpam-6940	249	1	there	there	PRON
ejpam-6940	249	2	follows	follow	VERB
ejpam-6940	249	3	∪	∪	ADJ
ejpam-6940	249	4	a∈λ	a∈λ	NOUN
ejpam-6940	249	5	ax	ax	NOUN
ejpam-6940	249	6	=	=	PUNCT
ejpam-6940	249	7	λ	λ	X
ejpam-6940	249	8	∪	∪	X
ejpam-6940	249	9	{	{	PUNCT
ejpam-6940	249	10	0	0	NUM
ejpam-6940	249	11	}	}	PUNCT
ejpam-6940	249	12	.	.	PUNCT
ejpam-6940	250	1	set	set	VERB
ejpam-6940	250	2	a	a	DET
ejpam-6940	250	3	=	=	X
ejpam-6940	250	4	λ	λ	X
ejpam-6940	250	5	∪	∪	X
ejpam-6940	250	6	{	{	PUNCT
ejpam-6940	250	7	0	0	NUM
ejpam-6940	250	8	}	}	PUNCT
ejpam-6940	250	9	and	and	CCONJ
ejpam-6940	250	10	b	b	X
ejpam-6940	250	11	=	=	PUNCT
ejpam-6940	250	12	{	{	PUNCT
ejpam-6940	250	13	0	0	NUM
ejpam-6940	250	14	}	}	PUNCT
ejpam-6940	250	15	.	.	PUNCT
ejpam-6940	251	1	then	then	ADV
ejpam-6940	251	2	ab	ab	PROPN
ejpam-6940	251	3	=	=	PUNCT
ejpam-6940	251	4	λ	λ	X
ejpam-6940	251	5	∪	∪	X
ejpam-6940	251	6	{	{	PUNCT
ejpam-6940	251	7	0	0	NUM
ejpam-6940	251	8	}	}	PUNCT
ejpam-6940	251	9	=	=	SYM
ejpam-6940	251	10	∪	∪	ADP
ejpam-6940	251	11	a∈λ	a∈λ	NOUN
ejpam-6940	251	12	ax	ax	NOUN
ejpam-6940	251	13	.	.	PUNCT
ejpam-6940	252	1	since	since	SCONJ
ejpam-6940	252	2	0	0	NUM
ejpam-6940	252	3	∈	∈	PROPN
ejpam-6940	252	4	a	a	PRON
ejpam-6940	252	5	,	,	PUNCT
ejpam-6940	252	6	by	by	ADP
ejpam-6940	252	7	proposition	proposition	NOUN
ejpam-6940	252	8	7	7	NUM
ejpam-6940	252	9	ab	ab	NOUN
ejpam-6940	252	10	=	=	PUNCT
ejpam-6940	252	11	∪	∪	ADP
ejpam-6940	252	12	a∈λ	a∈λ	NOUN
ejpam-6940	252	13	ax	ax	NOUN
ejpam-6940	252	14	is	be	AUX
ejpam-6940	252	15	a	a	DET
ejpam-6940	252	16	subalgebra	subalgebra	NOUN
ejpam-6940	252	17	of	of	ADP
ejpam-6940	252	18	x.	x.	NOUN
ejpam-6940	252	19	for	for	ADP
ejpam-6940	252	20	a	a	DET
ejpam-6940	252	21	q	q	NOUN
ejpam-6940	252	22	-	-	PUNCT
ejpam-6940	252	23	algebra	algebra	NOUN
ejpam-6940	252	24	x	x	NOUN
ejpam-6940	252	25	,	,	PUNCT
ejpam-6940	252	26	we	we	PRON
ejpam-6940	252	27	denote	denote	VERB
ejpam-6940	252	28	the	the	DET
ejpam-6940	252	29	set	set	NOUN
ejpam-6940	252	30	of	of	ADP
ejpam-6940	252	31	all	all	DET
ejpam-6940	252	32	subalgebras	subalgebra	NOUN
ejpam-6940	252	33	of	of	ADP
ejpam-6940	252	34	x	x	PUNCT
ejpam-6940	252	35	by	by	ADP
ejpam-6940	252	36	sub(x	sub(x	PROPN
ejpam-6940	252	37	)	)	PUNCT
ejpam-6940	252	38	.	.	PUNCT
ejpam-6940	253	1	next	next	ADJ
ejpam-6940	253	2	proposition	proposition	NOUN
ejpam-6940	253	3	provides	provide	VERB
ejpam-6940	253	4	the	the	DET
ejpam-6940	253	5	characterization	characterization	NOUN
ejpam-6940	253	6	of	of	ADP
ejpam-6940	253	7	subalgebras	subalgebra	NOUN
ejpam-6940	253	8	of	of	ADP
ejpam-6940	253	9	an	an	DET
ejpam-6940	253	10	edge	edge	NOUN
ejpam-6940	253	11	q	q	NOUN
ejpam-6940	253	12	-	-	PUNCT
ejpam-6940	253	13	algebra	algebra	NOUN
ejpam-6940	253	14	and	and	CCONJ
ejpam-6940	253	15	enumerate	enumerate	VERB
ejpam-6940	253	16	all	all	PRON
ejpam-6940	253	17	of	of	ADP
ejpam-6940	253	18	subalgebras	subalgebra	NOUN
ejpam-6940	253	19	of	of	ADP
ejpam-6940	253	20	any	any	DET
ejpam-6940	253	21	edge	edge	NOUN
ejpam-6940	253	22	q	q	NOUN
ejpam-6940	253	23	-	-	PUNCT
ejpam-6940	253	24	algebra	algebra	ADJ
ejpam-6940	253	25	x.	x.	NOUN
ejpam-6940	253	26	proposition	proposition	NOUN
ejpam-6940	253	27	8	8	NUM
ejpam-6940	253	28	.	.	PUNCT
ejpam-6940	254	1	let	let	VERB
ejpam-6940	254	2	x	x	PRON
ejpam-6940	254	3	be	be	AUX
ejpam-6940	254	4	an	an	DET
ejpam-6940	254	5	edge	edge	NOUN
ejpam-6940	254	6	q	q	NOUN
ejpam-6940	254	7	-	-	NOUN
ejpam-6940	254	8	algebra	algebra	NOUN
ejpam-6940	254	9	and	and	CCONJ
ejpam-6940	254	10	let	let	VERB
ejpam-6940	254	11	|x|	|x|	PROPN
ejpam-6940	254	12	=	=	SYM
ejpam-6940	254	13	n	n	PROPN
ejpam-6940	254	14	for	for	ADP
ejpam-6940	254	15	some	some	DET
ejpam-6940	254	16	positive	positive	ADJ
ejpam-6940	254	17	integer	integer	NOUN
ejpam-6940	254	18	n.	n.	NOUN
ejpam-6940	254	19	then	then	ADV
ejpam-6940	254	20	the	the	DET
ejpam-6940	254	21	following	follow	VERB
ejpam-6940	254	22	conditions	condition	NOUN
ejpam-6940	254	23	are	be	AUX
ejpam-6940	254	24	hold	hold	ADJ
ejpam-6940	254	25	:	:	PUNCT
ejpam-6940	254	26	(	(	PUNCT
ejpam-6940	254	27	i	i	NOUN
ejpam-6940	254	28	)	)	PUNCT
ejpam-6940	254	29	for	for	ADP
ejpam-6940	254	30	∅	∅	NOUN
ejpam-6940	254	31	̸=	̸=	PROPN
ejpam-6940	254	32	s	s	PART
ejpam-6940	254	33	⊆	⊆	NUM
ejpam-6940	254	34	x	x	NOUN
ejpam-6940	254	35	,	,	PUNCT
ejpam-6940	254	36	s	s	X
ejpam-6940	254	37	is	be	AUX
ejpam-6940	254	38	a	a	DET
ejpam-6940	254	39	subalgebra	subalgebra	NOUN
ejpam-6940	254	40	of	of	ADP
ejpam-6940	254	41	x	x	PUNCT
ejpam-6940	254	42	if	if	SCONJ
ejpam-6940	254	43	and	and	CCONJ
ejpam-6940	255	1	only	only	ADV
ejpam-6940	255	2	if	if	SCONJ
ejpam-6940	255	3	0	0	NUM
ejpam-6940	255	4	∈	∈	PROPN
ejpam-6940	255	5	s.	s.	PROPN
ejpam-6940	255	6	(	(	PUNCT
ejpam-6940	255	7	ii	ii	NOUN
ejpam-6940	255	8	)	)	PUNCT
ejpam-6940	255	9	|sub(x)|	|sub(x)|	NOUN
ejpam-6940	255	10	=	=	SYM
ejpam-6940	255	11	2n−1	2n−1	NUM
ejpam-6940	255	12	.	.	PUNCT
ejpam-6940	255	13	proof	proof	NOUN
ejpam-6940	255	14	.	.	PUNCT
ejpam-6940	256	1	(	(	PUNCT
ejpam-6940	256	2	i	i	NOUN
ejpam-6940	256	3	)	)	PUNCT
ejpam-6940	256	4	(	(	PUNCT
ejpam-6940	256	5	⇒	⇒	PROPN
ejpam-6940	256	6	)	)	PUNCT
ejpam-6940	256	7	it	it	PRON
ejpam-6940	256	8	is	be	AUX
ejpam-6940	256	9	clear	clear	ADJ
ejpam-6940	256	10	.	.	PUNCT
ejpam-6940	257	1	(	(	PUNCT
ejpam-6940	257	2	⇐	⇐	NOUN
ejpam-6940	257	3	)	)	PUNCT
ejpam-6940	257	4	let	let	VERB
ejpam-6940	257	5	s	s	PRON
ejpam-6940	257	6	be	be	AUX
ejpam-6940	257	7	a	a	DET
ejpam-6940	257	8	non	non	ADJ
ejpam-6940	257	9	-	-	ADJ
ejpam-6940	257	10	empty	empty	ADJ
ejpam-6940	257	11	subset	subset	NOUN
ejpam-6940	257	12	of	of	ADP
ejpam-6940	257	13	x	x	PUNCT
ejpam-6940	257	14	and	and	CCONJ
ejpam-6940	257	15	assume	assume	VERB
ejpam-6940	257	16	0	0	NUM
ejpam-6940	257	17	∈	∈	PROPN
ejpam-6940	257	18	s.	s.	PROPN
ejpam-6940	257	19	by	by	ADP
ejpam-6940	257	20	proposition	proposition	NOUN
ejpam-6940	257	21	1(ii	1(ii	NUM
ejpam-6940	257	22	)	)	PUNCT
ejpam-6940	257	23	,	,	PUNCT
ejpam-6940	257	24	s{0	s{0	PROPN
ejpam-6940	257	25	}	}	PUNCT
ejpam-6940	257	26	=	=	SYM
ejpam-6940	257	27	s.	s.	PROPN
ejpam-6940	257	28	set	set	VERB
ejpam-6940	257	29	a	a	PRON
ejpam-6940	257	30	=	=	SYM
ejpam-6940	257	31	s	s	PROPN
ejpam-6940	257	32	,	,	PUNCT
ejpam-6940	257	33	b	b	X
ejpam-6940	257	34	=	=	SYM
ejpam-6940	257	35	{	{	PUNCT
ejpam-6940	257	36	0	0	NUM
ejpam-6940	257	37	}	}	PUNCT
ejpam-6940	257	38	and	and	CCONJ
ejpam-6940	257	39	then	then	ADV
ejpam-6940	257	40	by	by	ADP
ejpam-6940	257	41	proposition	proposition	NOUN
ejpam-6940	257	42	7	7	NUM
ejpam-6940	257	43	there	there	ADV
ejpam-6940	257	44	follows	follow	VERB
ejpam-6940	257	45	ab	ab	PROPN
ejpam-6940	257	46	=	=	PUNCT
ejpam-6940	257	47	s{0	s{0	PROPN
ejpam-6940	257	48	}	}	PUNCT
ejpam-6940	257	49	=	=	SYM
ejpam-6940	257	50	s	s	VERB
ejpam-6940	257	51	is	be	AUX
ejpam-6940	257	52	a	a	DET
ejpam-6940	257	53	subalgebra	subalgebra	NOUN
ejpam-6940	257	54	of	of	ADP
ejpam-6940	257	55	x.	x.	PROPN
ejpam-6940	257	56	(	(	PUNCT
ejpam-6940	257	57	ii	ii	PROPN
ejpam-6940	257	58	)	)	PUNCT
ejpam-6940	257	59	from	from	ADP
ejpam-6940	257	60	(	(	PUNCT
ejpam-6940	257	61	i	i	NOUN
ejpam-6940	257	62	)	)	PUNCT
ejpam-6940	257	63	we	we	PRON
ejpam-6940	257	64	conclude	conclude	VERB
ejpam-6940	257	65	that	that	SCONJ
ejpam-6940	257	66	any	any	DET
ejpam-6940	257	67	subset	subset	NOUN
ejpam-6940	257	68	of	of	ADP
ejpam-6940	257	69	x	x	PUNCT
ejpam-6940	257	70	containing	contain	VERB
ejpam-6940	257	71	a	a	DET
ejpam-6940	257	72	constant	constant	ADJ
ejpam-6940	257	73	0	0	NUM
ejpam-6940	257	74	is	be	AUX
ejpam-6940	257	75	a	a	DET
ejpam-6940	257	76	subalgebra	subalgebra	NOUN
ejpam-6940	257	77	of	of	ADP
ejpam-6940	257	78	x.	x.	NOUN
ejpam-6940	257	79	there	there	PRON
ejpam-6940	257	80	follows	follow	VERB
ejpam-6940	257	81	that	that	SCONJ
ejpam-6940	257	82	|sub(x)|	|sub(x)|	NOUN
ejpam-6940	257	83	=	=	PUNCT
ejpam-6940	257	84	n−1∑	n−1∑	PROPN
ejpam-6940	257	85	i=0	i=0	PROPN
ejpam-6940	257	86	(	(	PUNCT
ejpam-6940	257	87	n−	n−	NOUN
ejpam-6940	257	88	1	1	NUM
ejpam-6940	257	89	i	i	NOUN
ejpam-6940	257	90	)	)	PUNCT
ejpam-6940	258	1	=	=	SYM
ejpam-6940	258	2	2n−1	2n−1	NUM
ejpam-6940	258	3	.	.	PUNCT
ejpam-6940	258	4	proposition	proposition	NOUN
ejpam-6940	258	5	9	9	NUM
ejpam-6940	258	6	.	.	PUNCT
ejpam-6940	259	1	let	let	VERB
ejpam-6940	259	2	x	x	PRON
ejpam-6940	259	3	be	be	AUX
ejpam-6940	259	4	an	an	DET
ejpam-6940	259	5	edge	edge	NOUN
ejpam-6940	259	6	q	q	NOUN
ejpam-6940	259	7	-	-	NOUN
ejpam-6940	259	8	algebra	algebra	NOUN
ejpam-6940	259	9	.	.	PUNCT
ejpam-6940	260	1	then	then	ADV
ejpam-6940	260	2	sub(x	sub(x	PROPN
ejpam-6940	260	3	)	)	PUNCT
ejpam-6940	260	4	is	be	AUX
ejpam-6940	260	5	a	a	DET
ejpam-6940	260	6	semigroup	semigroup	NOUN
ejpam-6940	260	7	with	with	ADP
ejpam-6940	260	8	a	a	DET
ejpam-6940	260	9	right	right	ADJ
ejpam-6940	260	10	identity	identity	NOUN
ejpam-6940	260	11	{	{	PUNCT
ejpam-6940	260	12	0	0	NUM
ejpam-6940	260	13	}	}	PUNCT
ejpam-6940	260	14	.	.	PUNCT
ejpam-6940	261	1	proof	proof	NOUN
ejpam-6940	261	2	.	.	PUNCT
ejpam-6940	262	1	let	let	VERB
ejpam-6940	262	2	a	a	DET
ejpam-6940	262	3	,	,	PUNCT
ejpam-6940	262	4	b	b	NOUN
ejpam-6940	262	5	,	,	PUNCT
ejpam-6940	262	6	c	c	PROPN
ejpam-6940	262	7	∈	∈	PROPN
ejpam-6940	262	8	sub(x	sub(x	PROPN
ejpam-6940	262	9	)	)	PUNCT
ejpam-6940	262	10	.	.	PUNCT
ejpam-6940	263	1	by	by	ADP
ejpam-6940	263	2	corollary	corollary	ADJ
ejpam-6940	263	3	1	1	NUM
ejpam-6940	263	4	,	,	PUNCT
ejpam-6940	263	5	ab	ab	PROPN
ejpam-6940	263	6	is	be	AUX
ejpam-6940	263	7	a	a	DET
ejpam-6940	263	8	subalgebra	subalgebra	NOUN
ejpam-6940	263	9	and	and	CCONJ
ejpam-6940	263	10	then	then	ADV
ejpam-6940	263	11	sub(x	sub(x	PROPN
ejpam-6940	263	12	)	)	PUNCT
ejpam-6940	263	13	is	be	AUX
ejpam-6940	263	14	closed	close	VERB
ejpam-6940	263	15	.	.	PUNCT
ejpam-6940	264	1	since	since	SCONJ
ejpam-6940	264	2	0	0	NUM
ejpam-6940	264	3	∈	∈	PROPN
ejpam-6940	264	4	b	b	NOUN
ejpam-6940	264	5	,	,	PUNCT
ejpam-6940	264	6	by	by	ADP
ejpam-6940	264	7	proposition	proposition	NOUN
ejpam-6940	264	8	1(iv	1(iv	NUM
ejpam-6940	264	9	)	)	PUNCT
ejpam-6940	264	10	we	we	PRON
ejpam-6940	264	11	get	get	VERB
ejpam-6940	264	12	a	a	DET
ejpam-6940	264	13	⊆	⊆	NUM
ejpam-6940	264	14	ab	ab	NOUN
ejpam-6940	264	15	.	.	PUNCT
ejpam-6940	265	1	since	since	SCONJ
ejpam-6940	265	2	x	x	PRON
ejpam-6940	265	3	is	be	AUX
ejpam-6940	265	4	edge	edge	NOUN
ejpam-6940	265	5	and	and	CCONJ
ejpam-6940	265	6	0	0	NUM
ejpam-6940	265	7	∈	∈	PROPN
ejpam-6940	265	8	a	a	PRON
ejpam-6940	265	9	,	,	PUNCT
ejpam-6940	265	10	then	then	ADV
ejpam-6940	265	11	ab	ab	PROPN
ejpam-6940	265	12	⊆	⊆	NUM
ejpam-6940	265	13	ax	ax	NOUN
ejpam-6940	265	14	=	=	PUNCT
ejpam-6940	265	15	a	a	DET
ejpam-6940	265	16	∪	∪	X
ejpam-6940	265	17	{	{	PUNCT
ejpam-6940	265	18	0	0	NUM
ejpam-6940	265	19	}	}	PUNCT
ejpam-6940	265	20	=	=	NOUN
ejpam-6940	265	21	a.	a.	NOUN
ejpam-6940	265	22	therefore	therefore	ADV
ejpam-6940	265	23	,	,	PUNCT
ejpam-6940	265	24	a	a	DET
ejpam-6940	265	25	⊆	⊆	NUM
ejpam-6940	265	26	ab	ab	ADP
ejpam-6940	265	27	⊆	⊆	NUM
ejpam-6940	265	28	a.	a.	NOUN
ejpam-6940	265	29	thus	thus	ADV
ejpam-6940	265	30	,	,	PUNCT
ejpam-6940	265	31	ab	ab	PROPN
ejpam-6940	265	32	=	=	NOUN
ejpam-6940	265	33	a.	a.	NOUN
ejpam-6940	265	34	there	there	ADV
ejpam-6940	265	35	follows	follow	VERB
ejpam-6940	265	36	(	(	PUNCT
ejpam-6940	265	37	ab)c	ab)c	PROPN
ejpam-6940	265	38	=	=	SYM
ejpam-6940	265	39	ac	ac	PROPN
ejpam-6940	265	40	=	=	PUNCT
ejpam-6940	265	41	a	a	PROPN
ejpam-6940	265	42	=	=	X
ejpam-6940	265	43	ab	ab	NOUN
ejpam-6940	265	44	=	=	PUNCT
ejpam-6940	265	45	a(bc	a(bc	PROPN
ejpam-6940	265	46	)	)	PUNCT
ejpam-6940	265	47	.	.	PUNCT
ejpam-6940	266	1	this	this	PRON
ejpam-6940	266	2	gives	give	VERB
ejpam-6940	266	3	an	an	DET
ejpam-6940	266	4	associative	associative	ADJ
ejpam-6940	266	5	law	law	NOUN
ejpam-6940	266	6	.	.	PUNCT
ejpam-6940	267	1	hence	hence	ADV
ejpam-6940	267	2	,	,	PUNCT
ejpam-6940	267	3	sub(x	sub(x	PROPN
ejpam-6940	267	4	)	)	PUNCT
ejpam-6940	267	5	is	be	AUX
ejpam-6940	267	6	a	a	DET
ejpam-6940	267	7	semigroup	semigroup	NOUN
ejpam-6940	267	8	.	.	PUNCT
ejpam-6940	268	1	moreover	moreover	ADV
ejpam-6940	268	2	,	,	PUNCT
ejpam-6940	268	3	by	by	ADP
ejpam-6940	268	4	proposition	proposition	NOUN
ejpam-6940	268	5	1(ii	1(ii	NUM
ejpam-6940	268	6	)	)	PUNCT
ejpam-6940	268	7	a{0	a{0	PROPN
ejpam-6940	268	8	}	}	PUNCT
ejpam-6940	268	9	=	=	PUNCT
ejpam-6940	268	10	a	a	PRON
ejpam-6940	268	11	this	this	PRON
ejpam-6940	268	12	shows	show	VERB
ejpam-6940	268	13	that	that	SCONJ
ejpam-6940	268	14	{	{	PUNCT
ejpam-6940	268	15	0	0	X
ejpam-6940	268	16	}	}	PUNCT
ejpam-6940	268	17	is	be	AUX
ejpam-6940	268	18	a	a	DET
ejpam-6940	268	19	right	right	ADJ
ejpam-6940	268	20	identity	identity	NOUN
ejpam-6940	268	21	.	.	PUNCT
ejpam-6940	269	1	from	from	ADP
ejpam-6940	269	2	proposition	proposition	NOUN
ejpam-6940	269	3	6	6	NUM
ejpam-6940	269	4	,	,	PUNCT
ejpam-6940	269	5	we	we	PRON
ejpam-6940	269	6	get	get	VERB
ejpam-6940	269	7	an	an	DET
ejpam-6940	269	8	information	information	NOUN
ejpam-6940	269	9	that	that	SCONJ
ejpam-6940	269	10	a	a	DET
ejpam-6940	269	11	set	set	NOUN
ejpam-6940	269	12	ax	ax	NOUN
ejpam-6940	269	13	is	be	AUX
ejpam-6940	269	14	a	a	DET
ejpam-6940	269	15	subalgebra	subalgebra	NOUN
ejpam-6940	269	16	for	for	ADP
ejpam-6940	269	17	any	any	DET
ejpam-6940	269	18	element	element	NOUN
ejpam-6940	269	19	a	a	PRON
ejpam-6940	269	20	in	in	ADP
ejpam-6940	269	21	an	an	DET
ejpam-6940	269	22	edge	edge	NOUN
ejpam-6940	269	23	q	q	NOUN
ejpam-6940	269	24	-	-	NOUN
ejpam-6940	269	25	algebra	algebra	NOUN
ejpam-6940	269	26	x.	x.	NOUN
ejpam-6940	269	27	but	but	CCONJ
ejpam-6940	269	28	this	this	DET
ejpam-6940	269	29	kind	kind	NOUN
ejpam-6940	269	30	of	of	ADP
ejpam-6940	269	31	subset	subset	NOUN
ejpam-6940	269	32	need	need	AUX
ejpam-6940	269	33	not	not	PART
ejpam-6940	269	34	be	be	AUX
ejpam-6940	269	35	an	an	DET
ejpam-6940	269	36	ideal	ideal	NOUN
ejpam-6940	269	37	.	.	PUNCT
ejpam-6940	270	1	for	for	ADP
ejpam-6940	270	2	example	example	NOUN
ejpam-6940	270	3	,	,	PUNCT
ejpam-6940	270	4	a	a	DET
ejpam-6940	270	5	subset	subset	ADJ
ejpam-6940	270	6	ax	ax	NOUN
ejpam-6940	270	7	of	of	ADP
ejpam-6940	270	8	x	x	PUNCT
ejpam-6940	270	9	in	in	ADP
ejpam-6940	270	10	example	example	NOUN
ejpam-6940	270	11	5	5	NUM
ejpam-6940	270	12	is	be	AUX
ejpam-6940	270	13	not	not	PART
ejpam-6940	270	14	an	an	DET
ejpam-6940	270	15	ideal	ideal	NOUN
ejpam-6940	270	16	since	since	SCONJ
ejpam-6940	270	17	b	b	NOUN
ejpam-6940	270	18	∗	∗	NOUN
ejpam-6940	270	19	a	a	DET
ejpam-6940	270	20	=	=	SYM
ejpam-6940	270	21	0	0	NUM
ejpam-6940	270	22	∈	∈	PROPN
ejpam-6940	270	23	ax	ax	NOUN
ejpam-6940	270	24	,	,	PUNCT
ejpam-6940	270	25	a	a	DET
ejpam-6940	270	26	∈	∈	NOUN
ejpam-6940	270	27	ax	ax	NOUN
ejpam-6940	270	28	but	but	CCONJ
ejpam-6940	270	29	b	b	X
ejpam-6940	270	30	̸∈	̸∈	PROPN
ejpam-6940	270	31	ax	ax	NOUN
ejpam-6940	270	32	.	.	PUNCT
ejpam-6940	271	1	next	next	ADV
ejpam-6940	271	2	,	,	PUNCT
ejpam-6940	271	3	we	we	PRON
ejpam-6940	271	4	will	will	AUX
ejpam-6940	271	5	show	show	VERB
ejpam-6940	271	6	a	a	DET
ejpam-6940	271	7	condition	condition	NOUN
ejpam-6940	271	8	for	for	ADP
ejpam-6940	271	9	a	a	DET
ejpam-6940	271	10	subset	subset	NOUN
ejpam-6940	271	11	of	of	ADP
ejpam-6940	271	12	an	an	DET
ejpam-6940	271	13	edge	edge	NOUN
ejpam-6940	271	14	q	q	NOUN
ejpam-6940	271	15	-	-	NOUN
ejpam-6940	271	16	algebra	algebra	NOUN
ejpam-6940	271	17	x	x	PUNCT
ejpam-6940	271	18	in	in	ADP
ejpam-6940	271	19	the	the	DET
ejpam-6940	271	20	form	form	NOUN
ejpam-6940	271	21	ax	ax	NOUN
ejpam-6940	271	22	,	,	PUNCT
ejpam-6940	271	23	a	a	DET
ejpam-6940	271	24	∈	∈	NOUN
ejpam-6940	271	25	x	x	X
ejpam-6940	271	26	to	to	PART
ejpam-6940	271	27	be	be	AUX
ejpam-6940	271	28	an	an	DET
ejpam-6940	271	29	ideal	ideal	NOUN
ejpam-6940	271	30	of	of	ADP
ejpam-6940	271	31	x.	x.	PROPN
ejpam-6940	271	32	a.	a.	PROPN
ejpam-6940	271	33	anantayasethi	anantayasethi	PROPN
ejpam-6940	271	34	et	et	PROPN
ejpam-6940	271	35	al	al	PROPN
ejpam-6940	271	36	.	.	PUNCT
ejpam-6940	271	37	/	/	SYM
ejpam-6940	271	38	eur	eur	PROPN
ejpam-6940	271	39	.	.	PUNCT
ejpam-6940	272	1	j.	j.	PROPN
ejpam-6940	272	2	pure	pure	PROPN
ejpam-6940	272	3	appl	appl	PROPN
ejpam-6940	272	4	.	.	PROPN
ejpam-6940	272	5	math	math	PROPN
ejpam-6940	272	6	,	,	PUNCT
ejpam-6940	272	7	18	18	NUM
ejpam-6940	272	8	(	(	PUNCT
ejpam-6940	272	9	4	4	NUM
ejpam-6940	272	10	)	)	PUNCT
ejpam-6940	272	11	(	(	PUNCT
ejpam-6940	272	12	2025	2025	NUM
ejpam-6940	272	13	)	)	PUNCT
ejpam-6940	272	14	,	,	PUNCT
ejpam-6940	272	15	6940	6940	NUM
ejpam-6940	272	16	9	9	NUM
ejpam-6940	272	17	of	of	ADP
ejpam-6940	272	18	14	14	NUM
ejpam-6940	272	19	proposition	proposition	NOUN
ejpam-6940	272	20	10	10	NUM
ejpam-6940	272	21	.	.	PUNCT
ejpam-6940	273	1	let	let	VERB
ejpam-6940	273	2	x	x	PRON
ejpam-6940	273	3	be	be	AUX
ejpam-6940	273	4	an	an	DET
ejpam-6940	273	5	edge	edge	NOUN
ejpam-6940	273	6	q	q	NOUN
ejpam-6940	273	7	-	-	NOUN
ejpam-6940	273	8	algebra	algebra	NOUN
ejpam-6940	273	9	and	and	CCONJ
ejpam-6940	273	10	let	let	VERB
ejpam-6940	273	11	a	a	DET
ejpam-6940	273	12	∈	∈	NOUN
ejpam-6940	273	13	x.	x.	NOUN
ejpam-6940	273	14	then	then	ADV
ejpam-6940	273	15	ax	ax	NOUN
ejpam-6940	273	16	is	be	AUX
ejpam-6940	273	17	an	an	DET
ejpam-6940	273	18	ideal	ideal	NOUN
ejpam-6940	273	19	if	if	SCONJ
ejpam-6940	273	20	and	and	CCONJ
ejpam-6940	273	21	only	only	ADV
ejpam-6940	273	22	if	if	SCONJ
ejpam-6940	273	23	wa	wa	PROPN
ejpam-6940	273	24	=	=	PROPN
ejpam-6940	273	25	w	w	PROPN
ejpam-6940	273	26	for	for	ADP
ejpam-6940	273	27	all	all	DET
ejpam-6940	273	28	w	w	NOUN
ejpam-6940	273	29	∈	∈	NOUN
ejpam-6940	273	30	x\{a	x\{a	PRON
ejpam-6940	273	31	}	}	PUNCT
ejpam-6940	273	32	.	.	PUNCT
ejpam-6940	274	1	proof	proof	NOUN
ejpam-6940	274	2	.	.	PUNCT
ejpam-6940	275	1	let	let	VERB
ejpam-6940	275	2	a	a	DET
ejpam-6940	275	3	∈	∈	ADJ
ejpam-6940	275	4	x	x	NOUN
ejpam-6940	275	5	,	,	PUNCT
ejpam-6940	275	6	and	and	CCONJ
ejpam-6940	275	7	assume	assume	VERB
ejpam-6940	275	8	that	that	SCONJ
ejpam-6940	275	9	ax	ax	NOUN
ejpam-6940	275	10	is	be	AUX
ejpam-6940	275	11	an	an	DET
ejpam-6940	275	12	ideal	ideal	NOUN
ejpam-6940	275	13	of	of	ADP
ejpam-6940	275	14	x.	x.	NOUN
ejpam-6940	275	15	suppose	suppose	VERB
ejpam-6940	275	16	that	that	SCONJ
ejpam-6940	275	17	there	there	PRON
ejpam-6940	275	18	is	be	VERB
ejpam-6940	275	19	an	an	DET
ejpam-6940	275	20	element	element	NOUN
ejpam-6940	275	21	w	w	PROPN
ejpam-6940	275	22	∈	∈	PROPN
ejpam-6940	275	23	x	x	X
ejpam-6940	275	24	,	,	PUNCT
ejpam-6940	275	25	w	w	PROPN
ejpam-6940	275	26	̸=	̸=	PROPN
ejpam-6940	275	27	a	a	DET
ejpam-6940	275	28	such	such	ADJ
ejpam-6940	275	29	that	that	DET
ejpam-6940	275	30	wa	wa	PROPN
ejpam-6940	275	31	̸=	̸=	PROPN
ejpam-6940	275	32	w.	w.	PROPN
ejpam-6940	275	33	since	since	SCONJ
ejpam-6940	275	34	wa	wa	PROPN
ejpam-6940	275	35	∈	∈	PROPN
ejpam-6940	275	36	wx	wx	PROPN
ejpam-6940	275	37	=	=	SYM
ejpam-6940	275	38	{	{	PUNCT
ejpam-6940	275	39	0	0	NUM
ejpam-6940	275	40	,	,	PUNCT
ejpam-6940	275	41	w	w	NOUN
ejpam-6940	275	42	}	}	PUNCT
ejpam-6940	275	43	and	and	CCONJ
ejpam-6940	275	44	wa	wa	X
ejpam-6940	275	45	̸=	̸=	PROPN
ejpam-6940	275	46	w	w	PROPN
ejpam-6940	275	47	,	,	PUNCT
ejpam-6940	275	48	then	then	ADV
ejpam-6940	275	49	wa	wa	ADV
ejpam-6940	275	50	=	=	NOUN
ejpam-6940	275	51	0	0	PROPN
ejpam-6940	275	52	.	.	PUNCT
ejpam-6940	276	1	there	there	PRON
ejpam-6940	276	2	follows	follow	VERB
ejpam-6940	276	3	wa	wa	PROPN
ejpam-6940	276	4	∈	∈	PROPN
ejpam-6940	276	5	ax	ax	NOUN
ejpam-6940	276	6	.	.	PUNCT
ejpam-6940	277	1	since	since	SCONJ
ejpam-6940	277	2	wa	wa	PROPN
ejpam-6940	277	3	∈	∈	PROPN
ejpam-6940	277	4	ax	ax	NOUN
ejpam-6940	277	5	,	,	PUNCT
ejpam-6940	277	6	a	a	DET
ejpam-6940	277	7	∈	∈	NOUN
ejpam-6940	277	8	ax	ax	NOUN
ejpam-6940	277	9	and	and	CCONJ
ejpam-6940	277	10	ax	ax	NOUN
ejpam-6940	277	11	is	be	AUX
ejpam-6940	277	12	an	an	DET
ejpam-6940	277	13	ideal	ideal	NOUN
ejpam-6940	277	14	,	,	PUNCT
ejpam-6940	277	15	then	then	ADV
ejpam-6940	277	16	w	w	PROPN
ejpam-6940	277	17	∈	∈	PROPN
ejpam-6940	277	18	ax	ax	NOUN
ejpam-6940	277	19	.	.	PUNCT
ejpam-6940	278	1	this	this	PRON
ejpam-6940	278	2	gives	give	VERB
ejpam-6940	278	3	a	a	DET
ejpam-6940	278	4	contradiction	contradiction	NOUN
ejpam-6940	278	5	.	.	PUNCT
ejpam-6940	279	1	hence	hence	ADV
ejpam-6940	279	2	,	,	PUNCT
ejpam-6940	279	3	for	for	ADP
ejpam-6940	279	4	all	all	DET
ejpam-6940	279	5	w	w	NOUN
ejpam-6940	279	6	∈	∈	NOUN
ejpam-6940	279	7	x\{a	x\{a	PRON
ejpam-6940	279	8	}	}	PUNCT
ejpam-6940	279	9	,	,	PUNCT
ejpam-6940	279	10	wa	wa	NOUN
ejpam-6940	279	11	=	=	PROPN
ejpam-6940	279	12	w.	w.	PROPN
ejpam-6940	279	13	for	for	ADP
ejpam-6940	279	14	the	the	DET
ejpam-6940	279	15	converse	converse	NOUN
ejpam-6940	279	16	direction	direction	NOUN
ejpam-6940	279	17	,	,	PUNCT
ejpam-6940	279	18	assume	assume	VERB
ejpam-6940	279	19	wa	wa	PROPN
ejpam-6940	279	20	=	=	PROPN
ejpam-6940	279	21	w	w	PROPN
ejpam-6940	279	22	for	for	ADP
ejpam-6940	279	23	all	all	DET
ejpam-6940	279	24	w	w	NOUN
ejpam-6940	279	25	∈	∈	NOUN
ejpam-6940	279	26	x\{a	x\{a	PRON
ejpam-6940	279	27	}	}	PUNCT
ejpam-6940	279	28	.	.	PUNCT
ejpam-6940	280	1	we	we	PRON
ejpam-6940	280	2	want	want	VERB
ejpam-6940	280	3	to	to	PART
ejpam-6940	280	4	show	show	VERB
ejpam-6940	280	5	that	that	DET
ejpam-6940	280	6	ax	ax	NOUN
ejpam-6940	280	7	=	=	PUNCT
ejpam-6940	280	8	{	{	PUNCT
ejpam-6940	280	9	0	0	NUM
ejpam-6940	280	10	,	,	PUNCT
ejpam-6940	280	11	a	a	PRON
ejpam-6940	280	12	}	}	PUNCT
ejpam-6940	280	13	is	be	AUX
ejpam-6940	280	14	an	an	DET
ejpam-6940	280	15	ideal	ideal	NOUN
ejpam-6940	280	16	.	.	PUNCT
ejpam-6940	281	1	if	if	SCONJ
ejpam-6940	281	2	a	a	DET
ejpam-6940	281	3	=	=	NOUN
ejpam-6940	281	4	0	0	NUM
ejpam-6940	281	5	,	,	PUNCT
ejpam-6940	281	6	then	then	ADV
ejpam-6940	281	7	it	it	PRON
ejpam-6940	281	8	is	be	AUX
ejpam-6940	281	9	obvious	obvious	ADJ
ejpam-6940	281	10	that	that	SCONJ
ejpam-6940	281	11	ax	ax	NOUN
ejpam-6940	281	12	is	be	AUX
ejpam-6940	281	13	an	an	DET
ejpam-6940	281	14	ideal	ideal	NOUN
ejpam-6940	281	15	of	of	ADP
ejpam-6940	281	16	x.	x.	NOUN
ejpam-6940	281	17	assume	assume	VERB
ejpam-6940	281	18	now	now	ADV
ejpam-6940	281	19	a	a	DET
ejpam-6940	281	20	̸=	̸=	PROPN
ejpam-6940	281	21	0	0	NUM
ejpam-6940	281	22	.	.	PUNCT
ejpam-6940	282	1	let	let	VERB
ejpam-6940	282	2	xy	xy	PRON
ejpam-6940	282	3	∈	∈	PROPN
ejpam-6940	282	4	ax	ax	NOUN
ejpam-6940	282	5	and	and	CCONJ
ejpam-6940	282	6	y	y	PROPN
ejpam-6940	282	7	∈	∈	PROPN
ejpam-6940	282	8	ax	ax	NOUN
ejpam-6940	282	9	.	.	PUNCT
ejpam-6940	283	1	if	if	SCONJ
ejpam-6940	283	2	y	y	PROPN
ejpam-6940	283	3	=	=	SYM
ejpam-6940	283	4	0	0	PROPN
ejpam-6940	283	5	,	,	PUNCT
ejpam-6940	283	6	then	then	ADV
ejpam-6940	283	7	x	x	X
ejpam-6940	283	8	=	=	PUNCT
ejpam-6940	284	1	x0	x0	PROPN
ejpam-6940	284	2	=	=	PUNCT
ejpam-6940	284	3	xy	xy	PROPN
ejpam-6940	284	4	∈	∈	PROPN
ejpam-6940	284	5	ax	ax	NOUN
ejpam-6940	284	6	.	.	PUNCT
ejpam-6940	285	1	if	if	SCONJ
ejpam-6940	285	2	y	y	PROPN
ejpam-6940	285	3	=	=	SYM
ejpam-6940	285	4	a	a	PROPN
ejpam-6940	285	5	,	,	PUNCT
ejpam-6940	285	6	then	then	ADV
ejpam-6940	285	7	xa	xa	PROPN
ejpam-6940	285	8	∈	∈	PROPN
ejpam-6940	285	9	ax	ax	NOUN
ejpam-6940	285	10	.	.	PUNCT
ejpam-6940	286	1	there	there	PRON
ejpam-6940	286	2	follows	follow	VERB
ejpam-6940	286	3	xa	xa	PROPN
ejpam-6940	286	4	=	=	SYM
ejpam-6940	286	5	0	0	PROPN
ejpam-6940	286	6	or	or	CCONJ
ejpam-6940	286	7	xa	xa	PROPN
ejpam-6940	286	8	=	=	NOUN
ejpam-6940	286	9	a.	a.	NOUN
ejpam-6940	286	10	suppose	suppose	VERB
ejpam-6940	286	11	xa	xa	PROPN
ejpam-6940	286	12	=	=	SYM
ejpam-6940	286	13	a.	a.	PROPN
ejpam-6940	286	14	then	then	ADV
ejpam-6940	286	15	x	x	X
ejpam-6940	286	16	̸=	̸=	PROPN
ejpam-6940	286	17	a	a	DET
ejpam-6940	286	18	otherwise	otherwise	ADV
ejpam-6940	286	19	a	a	PRON
ejpam-6940	286	20	=	=	X
ejpam-6940	286	21	xa	xa	PROPN
ejpam-6940	286	22	=	=	SYM
ejpam-6940	286	23	aa	aa	PROPN
ejpam-6940	286	24	=	=	SYM
ejpam-6940	286	25	0	0	PROPN
ejpam-6940	286	26	,	,	PUNCT
ejpam-6940	286	27	a	a	DET
ejpam-6940	286	28	contradiction	contradiction	NOUN
ejpam-6940	286	29	.	.	PUNCT
ejpam-6940	287	1	hence	hence	ADV
ejpam-6940	287	2	,	,	PUNCT
ejpam-6940	287	3	xa	xa	PROPN
ejpam-6940	287	4	=	=	PROPN
ejpam-6940	288	1	0	0	PROPN
ejpam-6940	288	2	.	.	PUNCT
ejpam-6940	288	3	then	then	ADV
ejpam-6940	288	4	by	by	ADP
ejpam-6940	288	5	assumption	assumption	NOUN
ejpam-6940	288	6	x	x	X
ejpam-6940	288	7	̸∈	̸∈	PROPN
ejpam-6940	288	8	x\{a	x\{a	X
ejpam-6940	288	9	}	}	PUNCT
ejpam-6940	288	10	.	.	PUNCT
ejpam-6940	289	1	therefore	therefore	ADV
ejpam-6940	289	2	,	,	PUNCT
ejpam-6940	289	3	x	x	X
ejpam-6940	289	4	=	=	PUNCT
ejpam-6940	289	5	a	a	DET
ejpam-6940	289	6	∈	∈	PROPN
ejpam-6940	289	7	ax	ax	NOUN
ejpam-6940	289	8	.	.	PUNCT
ejpam-6940	290	1	altogether	altogether	ADV
ejpam-6940	290	2	,	,	PUNCT
ejpam-6940	290	3	ax	ax	NOUN
ejpam-6940	290	4	is	be	AUX
ejpam-6940	290	5	an	an	DET
ejpam-6940	290	6	ideal	ideal	NOUN
ejpam-6940	290	7	of	of	ADP
ejpam-6940	290	8	x.	x.	NOUN
ejpam-6940	290	9	in	in	ADP
ejpam-6940	290	10	a	a	DET
ejpam-6940	290	11	q	q	NOUN
ejpam-6940	290	12	-	-	PUNCT
ejpam-6940	290	13	algebra	algebra	NOUN
ejpam-6940	290	14	x	x	NOUN
ejpam-6940	290	15	,	,	PUNCT
ejpam-6940	290	16	a	a	DET
ejpam-6940	290	17	product	product	NOUN
ejpam-6940	290	18	i1i2	i1i2	X
ejpam-6940	290	19	of	of	ADP
ejpam-6940	290	20	ideals	ideals	PROPN
ejpam-6940	290	21	i1	i1	PROPN
ejpam-6940	290	22	and	and	CCONJ
ejpam-6940	290	23	i2	i2	PROPN
ejpam-6940	290	24	of	of	ADP
ejpam-6940	290	25	x	x	PROPN
ejpam-6940	290	26	is	be	AUX
ejpam-6940	290	27	not	not	PART
ejpam-6940	290	28	necessarily	necessarily	ADV
ejpam-6940	290	29	an	an	DET
ejpam-6940	290	30	ideal	ideal	NOUN
ejpam-6940	290	31	as	as	SCONJ
ejpam-6940	290	32	can	can	AUX
ejpam-6940	290	33	be	be	AUX
ejpam-6940	290	34	seen	see	VERB
ejpam-6940	290	35	in	in	ADP
ejpam-6940	290	36	the	the	DET
ejpam-6940	290	37	following	follow	VERB
ejpam-6940	290	38	example	example	NOUN
ejpam-6940	290	39	.	.	PUNCT
ejpam-6940	291	1	example	example	NOUN
ejpam-6940	292	1	6	6	NUM
ejpam-6940	292	2	.	.	PUNCT
ejpam-6940	293	1	let	let	VERB
ejpam-6940	293	2	x	x	PUNCT
ejpam-6940	293	3	=	=	PUNCT
ejpam-6940	293	4	{	{	PUNCT
ejpam-6940	293	5	0	0	NUM
ejpam-6940	293	6	,	,	PUNCT
ejpam-6940	293	7	a	a	DET
ejpam-6940	293	8	,	,	PUNCT
ejpam-6940	293	9	b	b	NOUN
ejpam-6940	293	10	,	,	PUNCT
ejpam-6940	293	11	c	c	NOUN
ejpam-6940	293	12	,	,	PUNCT
ejpam-6940	293	13	d	d	NOUN
ejpam-6940	293	14	,	,	PUNCT
ejpam-6940	293	15	f	f	NOUN
ejpam-6940	293	16	}	}	PUNCT
ejpam-6940	293	17	.	.	PUNCT
ejpam-6940	294	1	define	define	VERB
ejpam-6940	294	2	a	a	DET
ejpam-6940	294	3	binary	binary	ADJ
ejpam-6940	294	4	operation	operation	NOUN
ejpam-6940	294	5	∗	∗	NOUN
ejpam-6940	294	6	on	on	ADP
ejpam-6940	294	7	x	x	PUNCT
ejpam-6940	294	8	as	as	ADP
ejpam-6940	294	9	the	the	DET
ejpam-6940	294	10	following	follow	VERB
ejpam-6940	294	11	table	table	NOUN
ejpam-6940	294	12	:	:	PUNCT
ejpam-6940	294	13	∗	∗	NOUN
ejpam-6940	294	14	0	0	PUNCT
ejpam-6940	295	1	a	a	DET
ejpam-6940	295	2	b	b	NOUN
ejpam-6940	295	3	c	c	NOUN
ejpam-6940	295	4	d	d	X
ejpam-6940	295	5	f	f	PROPN
ejpam-6940	295	6	0	0	NUM
ejpam-6940	295	7	0	0	NUM
ejpam-6940	296	1	a	a	DET
ejpam-6940	296	2	c	c	NOUN
ejpam-6940	296	3	d	d	NOUN
ejpam-6940	296	4	c	c	PROPN
ejpam-6940	296	5	c	c	NOUN
ejpam-6940	296	6	a	a	DET
ejpam-6940	296	7	a	a	DET
ejpam-6940	296	8	0	0	NUM
ejpam-6940	296	9	d	d	NOUN
ejpam-6940	296	10	c	c	NOUN
ejpam-6940	297	1	d	d	PROPN
ejpam-6940	297	2	d	d	PROPN
ejpam-6940	297	3	b	b	PROPN
ejpam-6940	297	4	b	b	PROPN
ejpam-6940	297	5	c	c	NOUN
ejpam-6940	297	6	0	0	NUM
ejpam-6940	297	7	a	a	DET
ejpam-6940	297	8	0	0	NUM
ejpam-6940	297	9	0	0	NUM
ejpam-6940	298	1	c	c	NOUN
ejpam-6940	298	2	c	c	PROPN
ejpam-6940	298	3	b	b	PROPN
ejpam-6940	298	4	a	a	DET
ejpam-6940	298	5	0	0	NUM
ejpam-6940	298	6	a	a	DET
ejpam-6940	298	7	a	a	PROPN
ejpam-6940	298	8	d	d	X
ejpam-6940	298	9	d	d	PROPN
ejpam-6940	298	10	c	c	NOUN
ejpam-6940	298	11	0	0	NUM
ejpam-6940	299	1	a	a	DET
ejpam-6940	299	2	0	0	NUM
ejpam-6940	299	3	0	0	NUM
ejpam-6940	299	4	f	f	NOUN
ejpam-6940	299	5	f	f	PROPN
ejpam-6940	299	6	c	c	NOUN
ejpam-6940	299	7	0	0	NUM
ejpam-6940	299	8	a	a	DET
ejpam-6940	299	9	0	0	NUM
ejpam-6940	299	10	0	0	NUM
ejpam-6940	300	1	then	then	ADV
ejpam-6940	300	2	(	(	PUNCT
ejpam-6940	300	3	x	x	NOUN
ejpam-6940	300	4	;	;	PUNCT
ejpam-6940	300	5	∗	∗	NOUN
ejpam-6940	300	6	,	,	PUNCT
ejpam-6940	300	7	0	0	NUM
ejpam-6940	300	8	)	)	PUNCT
ejpam-6940	300	9	is	be	AUX
ejpam-6940	300	10	a	a	DET
ejpam-6940	300	11	q	q	NOUN
ejpam-6940	300	12	-	-	PUNCT
ejpam-6940	300	13	algebra	algebra	NOUN
ejpam-6940	300	14	.	.	PUNCT
ejpam-6940	301	1	the	the	DET
ejpam-6940	301	2	subsets	subsets	PROPN
ejpam-6940	301	3	i1	i1	PROPN
ejpam-6940	301	4	=	=	PUNCT
ejpam-6940	301	5	{	{	PUNCT
ejpam-6940	301	6	0	0	NUM
ejpam-6940	301	7	,	,	PUNCT
ejpam-6940	301	8	a	a	PRON
ejpam-6940	301	9	}	}	PUNCT
ejpam-6940	301	10	and	and	CCONJ
ejpam-6940	301	11	i2	i2	PROPN
ejpam-6940	301	12	=	=	PUNCT
ejpam-6940	301	13	{	{	PUNCT
ejpam-6940	301	14	0	0	NUM
ejpam-6940	301	15	,	,	PUNCT
ejpam-6940	301	16	b	b	NOUN
ejpam-6940	301	17	,	,	PUNCT
ejpam-6940	301	18	d	d	PROPN
ejpam-6940	301	19	,	,	PUNCT
ejpam-6940	301	20	f	f	X
ejpam-6940	301	21	}	}	PUNCT
ejpam-6940	301	22	are	be	AUX
ejpam-6940	301	23	ideals	ideal	NOUN
ejpam-6940	301	24	of	of	ADP
ejpam-6940	301	25	(	(	PUNCT
ejpam-6940	301	26	x	x	NOUN
ejpam-6940	301	27	;	;	PUNCT
ejpam-6940	301	28	∗	∗	NOUN
ejpam-6940	301	29	,	,	PUNCT
ejpam-6940	301	30	0	0	NUM
ejpam-6940	301	31	)	)	PUNCT
ejpam-6940	301	32	.	.	PUNCT
ejpam-6940	302	1	let	let	AUX
ejpam-6940	302	2	consider	consider	VERB
ejpam-6940	302	3	a	a	DET
ejpam-6940	302	4	product	product	NOUN
ejpam-6940	302	5	i1i2	i1i2	ADP
ejpam-6940	302	6	=	=	PRON
ejpam-6940	302	7	{	{	PUNCT
ejpam-6940	302	8	0	0	NUM
ejpam-6940	302	9	,	,	PUNCT
ejpam-6940	302	10	c	c	NOUN
ejpam-6940	302	11	,	,	PUNCT
ejpam-6940	302	12	d	d	NOUN
ejpam-6940	302	13	}	}	PUNCT
ejpam-6940	302	14	.	.	PUNCT
ejpam-6940	303	1	since	since	SCONJ
ejpam-6940	303	2	a	a	DET
ejpam-6940	303	3	∗	∗	NOUN
ejpam-6940	303	4	c	c	NOUN
ejpam-6940	303	5	=	=	SYM
ejpam-6940	303	6	c	c	PROPN
ejpam-6940	303	7	∈	∈	PROPN
ejpam-6940	303	8	i1i2	i1i2	NOUN
ejpam-6940	303	9	and	and	CCONJ
ejpam-6940	303	10	c	c	PROPN
ejpam-6940	303	11	∈	∈	PROPN
ejpam-6940	304	1	i1i2	i1i2	X
ejpam-6940	304	2	but	but	CCONJ
ejpam-6940	304	3	a	a	DET
ejpam-6940	304	4	/∈	/∈	INTJ
ejpam-6940	304	5	i1i2	i1i2	NOUN
ejpam-6940	304	6	,	,	PUNCT
ejpam-6940	304	7	then	then	ADV
ejpam-6940	304	8	i1i2	i1i2	PROPN
ejpam-6940	304	9	is	be	AUX
ejpam-6940	304	10	not	not	PART
ejpam-6940	304	11	an	an	DET
ejpam-6940	304	12	ideal	ideal	NOUN
ejpam-6940	304	13	of	of	ADP
ejpam-6940	304	14	(	(	PUNCT
ejpam-6940	304	15	x	x	NOUN
ejpam-6940	304	16	;	;	PUNCT
ejpam-6940	304	17	∗	∗	NOUN
ejpam-6940	304	18	,	,	PUNCT
ejpam-6940	304	19	0	0	NUM
ejpam-6940	304	20	)	)	PUNCT
ejpam-6940	304	21	.	.	PUNCT
ejpam-6940	305	1	therefore	therefore	ADV
ejpam-6940	305	2	,	,	PUNCT
ejpam-6940	305	3	the	the	DET
ejpam-6940	305	4	set	set	NOUN
ejpam-6940	305	5	of	of	ADP
ejpam-6940	305	6	all	all	DET
ejpam-6940	305	7	ideals	ideal	NOUN
ejpam-6940	305	8	of	of	ADP
ejpam-6940	305	9	a	a	DET
ejpam-6940	305	10	q	q	NOUN
ejpam-6940	305	11	-	-	PUNCT
ejpam-6940	305	12	algebra	algebra	NOUN
ejpam-6940	305	13	is	be	AUX
ejpam-6940	305	14	not	not	PART
ejpam-6940	305	15	necessarily	necessarily	ADV
ejpam-6940	305	16	closed	close	VERB
ejpam-6940	305	17	.	.	PUNCT
ejpam-6940	306	1	in	in	ADP
ejpam-6940	306	2	an	an	DET
ejpam-6940	306	3	edge	edge	NOUN
ejpam-6940	306	4	q	q	NOUN
ejpam-6940	306	5	-	-	NOUN
ejpam-6940	306	6	algebra	algebra	NOUN
ejpam-6940	306	7	,	,	PUNCT
ejpam-6940	306	8	we	we	PRON
ejpam-6940	306	9	get	get	VERB
ejpam-6940	306	10	a	a	DET
ejpam-6940	306	11	good	good	ADJ
ejpam-6940	306	12	outcome	outcome	NOUN
ejpam-6940	306	13	,	,	PUNCT
ejpam-6940	306	14	i.e.	i.e.	X
ejpam-6940	306	15	the	the	DET
ejpam-6940	306	16	product	product	NOUN
ejpam-6940	306	17	of	of	ADP
ejpam-6940	306	18	ideals	ideal	NOUN
ejpam-6940	306	19	is	be	AUX
ejpam-6940	306	20	again	again	ADV
ejpam-6940	306	21	an	an	DET
ejpam-6940	306	22	ideal	ideal	NOUN
ejpam-6940	306	23	as	as	SCONJ
ejpam-6940	306	24	obtained	obtain	VERB
ejpam-6940	306	25	in	in	ADP
ejpam-6940	306	26	the	the	DET
ejpam-6940	306	27	following	follow	VERB
ejpam-6940	306	28	proposition	proposition	NOUN
ejpam-6940	306	29	.	.	PUNCT
ejpam-6940	307	1	proposition	proposition	NOUN
ejpam-6940	307	2	11	11	NUM
ejpam-6940	307	3	.	.	PUNCT
ejpam-6940	308	1	let	let	VERB
ejpam-6940	308	2	x	x	PRON
ejpam-6940	308	3	be	be	AUX
ejpam-6940	308	4	an	an	DET
ejpam-6940	308	5	edge	edge	NOUN
ejpam-6940	308	6	q	q	NOUN
ejpam-6940	308	7	-	-	NOUN
ejpam-6940	308	8	algebra	algebra	NOUN
ejpam-6940	308	9	.	.	PUNCT
ejpam-6940	309	1	if	if	SCONJ
ejpam-6940	309	2	a	a	PRON
ejpam-6940	309	3	and	and	CCONJ
ejpam-6940	309	4	b	b	NOUN
ejpam-6940	309	5	are	be	AUX
ejpam-6940	309	6	ideals	ideal	NOUN
ejpam-6940	309	7	of	of	ADP
ejpam-6940	309	8	x	x	NOUN
ejpam-6940	309	9	,	,	PUNCT
ejpam-6940	309	10	then	then	ADV
ejpam-6940	309	11	ab	ab	PROPN
ejpam-6940	309	12	is	be	AUX
ejpam-6940	309	13	an	an	DET
ejpam-6940	309	14	ideal	ideal	NOUN
ejpam-6940	309	15	.	.	PUNCT
ejpam-6940	310	1	proof	proof	NOUN
ejpam-6940	310	2	.	.	PUNCT
ejpam-6940	311	1	let	let	VERB
ejpam-6940	311	2	a	a	PRON
ejpam-6940	311	3	and	and	CCONJ
ejpam-6940	311	4	b	b	NOUN
ejpam-6940	311	5	be	be	AUX
ejpam-6940	311	6	ideals	ideal	NOUN
ejpam-6940	311	7	of	of	ADP
ejpam-6940	311	8	x.	x.	NOUN
ejpam-6940	311	9	let	let	VERB
ejpam-6940	311	10	a	a	DET
ejpam-6940	311	11	∈	∈	NOUN
ejpam-6940	311	12	a.	a.	NOUN
ejpam-6940	311	13	since	since	SCONJ
ejpam-6940	311	14	0	0	NUM
ejpam-6940	311	15	∈	∈	PROPN
ejpam-6940	311	16	b	b	NOUN
ejpam-6940	311	17	,	,	PUNCT
ejpam-6940	311	18	by	by	ADP
ejpam-6940	311	19	proposition	proposition	NOUN
ejpam-6940	311	20	1(iv	1(iv	NUM
ejpam-6940	311	21	)	)	PUNCT
ejpam-6940	311	22	,	,	PUNCT
ejpam-6940	311	23	a	a	DET
ejpam-6940	311	24	⊆	⊆	NUM
ejpam-6940	311	25	ab	ab	NOUN
ejpam-6940	311	26	.	.	PUNCT
ejpam-6940	312	1	since	since	SCONJ
ejpam-6940	312	2	x	x	PRON
ejpam-6940	312	3	is	be	AUX
ejpam-6940	312	4	an	an	DET
ejpam-6940	312	5	edge	edge	NOUN
ejpam-6940	312	6	,	,	PUNCT
ejpam-6940	312	7	ab	ab	PROPN
ejpam-6940	312	8	⊆	⊆	NUM
ejpam-6940	312	9	ax	ax	NOUN
ejpam-6940	312	10	=	=	PUNCT
ejpam-6940	312	11	{	{	PUNCT
ejpam-6940	312	12	0	0	NUM
ejpam-6940	312	13	,	,	PUNCT
ejpam-6940	312	14	a	a	PRON
ejpam-6940	312	15	}	}	PUNCT
ejpam-6940	312	16	.	.	PUNCT
ejpam-6940	313	1	thus	thus	ADV
ejpam-6940	313	2	,	,	PUNCT
ejpam-6940	313	3	we	we	PRON
ejpam-6940	313	4	can	can	AUX
ejpam-6940	313	5	conclude	conclude	VERB
ejpam-6940	313	6	that	that	SCONJ
ejpam-6940	313	7	ab	ab	PROPN
ejpam-6940	313	8	⊆	⊆	NUM
ejpam-6940	313	9	a	a	DET
ejpam-6940	313	10	∪	∪	ADJ
ejpam-6940	313	11	{	{	PUNCT
ejpam-6940	313	12	0	0	NUM
ejpam-6940	313	13	}	}	PUNCT
ejpam-6940	313	14	=	=	NOUN
ejpam-6940	313	15	a.	a.	NOUN
ejpam-6940	313	16	hence	hence	ADV
ejpam-6940	313	17	,	,	PUNCT
ejpam-6940	313	18	ab	ab	PROPN
ejpam-6940	313	19	=	=	PUNCT
ejpam-6940	313	20	a	a	PRON
ejpam-6940	313	21	so	so	ADV
ejpam-6940	313	22	that	that	SCONJ
ejpam-6940	313	23	ab	ab	PROPN
ejpam-6940	313	24	is	be	AUX
ejpam-6940	313	25	an	an	DET
ejpam-6940	313	26	ideal	ideal	NOUN
ejpam-6940	313	27	of	of	ADP
ejpam-6940	313	28	x.	x.	NOUN
ejpam-6940	313	29	as	as	ADP
ejpam-6940	313	30	a	a	DET
ejpam-6940	313	31	direct	direct	ADJ
ejpam-6940	313	32	consequence	consequence	NOUN
ejpam-6940	313	33	of	of	ADP
ejpam-6940	313	34	proposition	proposition	NOUN
ejpam-6940	313	35	11	11	NUM
ejpam-6940	313	36	we	we	PRON
ejpam-6940	313	37	have	have	VERB
ejpam-6940	313	38	the	the	DET
ejpam-6940	313	39	subsequence	subsequence	NOUN
ejpam-6940	313	40	corollary	corollary	NOUN
ejpam-6940	313	41	.	.	PUNCT
ejpam-6940	314	1	corollary	corollary	ADJ
ejpam-6940	314	2	3	3	NUM
ejpam-6940	314	3	.	.	PUNCT
ejpam-6940	315	1	if	if	SCONJ
ejpam-6940	315	2	a	a	PRON
ejpam-6940	315	3	and	and	CCONJ
ejpam-6940	315	4	b	b	NOUN
ejpam-6940	315	5	are	be	AUX
ejpam-6940	315	6	ideals	ideal	NOUN
ejpam-6940	315	7	of	of	ADP
ejpam-6940	315	8	an	an	DET
ejpam-6940	315	9	edge	edge	NOUN
ejpam-6940	315	10	q	q	NOUN
ejpam-6940	315	11	-	-	NOUN
ejpam-6940	315	12	algebra	algebra	NOUN
ejpam-6940	315	13	x	x	NOUN
ejpam-6940	315	14	,	,	PUNCT
ejpam-6940	315	15	then	then	ADV
ejpam-6940	315	16	ab	ab	PROPN
ejpam-6940	315	17	=	=	PUNCT
ejpam-6940	315	18	a.	a.	NOUN
ejpam-6940	315	19	for	for	ADP
ejpam-6940	315	20	a	a	DET
ejpam-6940	315	21	q	q	NOUN
ejpam-6940	315	22	-	-	PUNCT
ejpam-6940	315	23	algebra	algebra	NOUN
ejpam-6940	315	24	x	x	NOUN
ejpam-6940	315	25	,	,	PUNCT
ejpam-6940	315	26	we	we	PRON
ejpam-6940	315	27	denote	denote	VERB
ejpam-6940	315	28	i(x	i(x	PROPN
ejpam-6940	315	29	)	)	PUNCT
ejpam-6940	315	30	as	as	ADP
ejpam-6940	315	31	the	the	DET
ejpam-6940	315	32	set	set	NOUN
ejpam-6940	315	33	of	of	ADP
ejpam-6940	315	34	all	all	DET
ejpam-6940	315	35	ideals	ideal	NOUN
ejpam-6940	315	36	of	of	ADP
ejpam-6940	315	37	x.	x.	NOUN
ejpam-6940	315	38	by	by	ADP
ejpam-6940	315	39	proposition	proposition	NOUN
ejpam-6940	315	40	11	11	NUM
ejpam-6940	315	41	and	and	CCONJ
ejpam-6940	315	42	corollary	corollary	ADJ
ejpam-6940	315	43	3	3	NUM
ejpam-6940	315	44	,	,	PUNCT
ejpam-6940	315	45	we	we	PRON
ejpam-6940	315	46	obtain	obtain	VERB
ejpam-6940	315	47	that	that	SCONJ
ejpam-6940	315	48	the	the	DET
ejpam-6940	315	49	set	set	NOUN
ejpam-6940	315	50	of	of	ADP
ejpam-6940	315	51	all	all	DET
ejpam-6940	315	52	ideals	ideal	NOUN
ejpam-6940	315	53	of	of	ADP
ejpam-6940	315	54	an	an	DET
ejpam-6940	315	55	edge	edge	NOUN
ejpam-6940	315	56	q	q	NOUN
ejpam-6940	315	57	-	-	PUNCT
ejpam-6940	315	58	algebra	algebra	NOUN
ejpam-6940	315	59	forms	form	VERB
ejpam-6940	315	60	a	a	DET
ejpam-6940	315	61	semigroup	semigroup	NOUN
ejpam-6940	315	62	.	.	PUNCT
ejpam-6940	316	1	a.	a.	PROPN
ejpam-6940	316	2	anantayasethi	anantayasethi	PROPN
ejpam-6940	316	3	et	et	PROPN
ejpam-6940	316	4	al	al	PROPN
ejpam-6940	316	5	.	.	PUNCT
ejpam-6940	316	6	/	/	SYM
ejpam-6940	316	7	eur	eur	PROPN
ejpam-6940	316	8	.	.	PUNCT
ejpam-6940	317	1	j.	j.	PROPN
ejpam-6940	317	2	pure	pure	PROPN
ejpam-6940	317	3	appl	appl	PROPN
ejpam-6940	317	4	.	.	PROPN
ejpam-6940	317	5	math	math	PROPN
ejpam-6940	317	6	,	,	PUNCT
ejpam-6940	317	7	18	18	NUM
ejpam-6940	317	8	(	(	PUNCT
ejpam-6940	317	9	4	4	NUM
ejpam-6940	317	10	)	)	PUNCT
ejpam-6940	317	11	(	(	PUNCT
ejpam-6940	317	12	2025	2025	NUM
ejpam-6940	317	13	)	)	PUNCT
ejpam-6940	317	14	,	,	PUNCT
ejpam-6940	317	15	6940	6940	NUM
ejpam-6940	317	16	10	10	NUM
ejpam-6940	317	17	of	of	ADP
ejpam-6940	317	18	14	14	NUM
ejpam-6940	317	19	proposition	proposition	NOUN
ejpam-6940	317	20	12	12	NUM
ejpam-6940	317	21	.	.	PUNCT
ejpam-6940	318	1	let	let	VERB
ejpam-6940	318	2	x	x	PRON
ejpam-6940	318	3	be	be	AUX
ejpam-6940	318	4	an	an	DET
ejpam-6940	318	5	edge	edge	NOUN
ejpam-6940	318	6	q	q	NOUN
ejpam-6940	318	7	-	-	NOUN
ejpam-6940	318	8	algebra	algebra	NOUN
ejpam-6940	318	9	.	.	PUNCT
ejpam-6940	319	1	then	then	ADV
ejpam-6940	319	2	i(x	i(x	PROPN
ejpam-6940	319	3	)	)	PUNCT
ejpam-6940	319	4	is	be	AUX
ejpam-6940	319	5	a	a	DET
ejpam-6940	319	6	semigroup	semigroup	NOUN
ejpam-6940	319	7	.	.	PUNCT
ejpam-6940	320	1	proposition	proposition	NOUN
ejpam-6940	320	2	13	13	NUM
ejpam-6940	320	3	.	.	PUNCT
ejpam-6940	321	1	let	let	VERB
ejpam-6940	321	2	x	x	PRON
ejpam-6940	321	3	be	be	AUX
ejpam-6940	321	4	an	an	DET
ejpam-6940	321	5	edge	edge	NOUN
ejpam-6940	321	6	q	q	NOUN
ejpam-6940	321	7	-	-	NOUN
ejpam-6940	321	8	algebra	algebra	NOUN
ejpam-6940	321	9	.	.	PUNCT
ejpam-6940	322	1	then	then	ADV
ejpam-6940	322	2	i(x	i(x	PROPN
ejpam-6940	322	3	)	)	PUNCT
ejpam-6940	322	4	is	be	AUX
ejpam-6940	322	5	a	a	DET
ejpam-6940	322	6	subsemigroup	subsemigroup	NOUN
ejpam-6940	322	7	of	of	ADP
ejpam-6940	322	8	sub(x	sub(x	PROPN
ejpam-6940	322	9	)	)	PUNCT
ejpam-6940	322	10	.	.	PUNCT
ejpam-6940	323	1	proof	proof	NOUN
ejpam-6940	323	2	.	.	PUNCT
ejpam-6940	324	1	let	let	VERB
ejpam-6940	324	2	i	i	PRON
ejpam-6940	324	3	∈	∈	PROPN
ejpam-6940	324	4	i(x	i(x	PROPN
ejpam-6940	324	5	)	)	PUNCT
ejpam-6940	324	6	.	.	PUNCT
ejpam-6940	325	1	by	by	ADP
ejpam-6940	325	2	(	(	PUNCT
ejpam-6940	325	3	i1	i1	PROPN
ejpam-6940	325	4	)	)	PUNCT
ejpam-6940	325	5	,	,	PUNCT
ejpam-6940	325	6	0	0	NUM
ejpam-6940	325	7	∈	∈	PROPN
ejpam-6940	325	8	i	i	PRON
ejpam-6940	325	9	,	,	PUNCT
ejpam-6940	325	10	so	so	SCONJ
ejpam-6940	325	11	that	that	SCONJ
ejpam-6940	325	12	by	by	ADP
ejpam-6940	325	13	proposition	proposition	NOUN
ejpam-6940	325	14	8(i	8(i	NUM
ejpam-6940	325	15	)	)	PUNCT
ejpam-6940	325	16	,	,	PUNCT
ejpam-6940	325	17	i	i	PRON
ejpam-6940	325	18	is	be	AUX
ejpam-6940	325	19	a	a	DET
ejpam-6940	325	20	subalgebra	subalgebra	NOUN
ejpam-6940	325	21	.	.	PUNCT
ejpam-6940	326	1	thus	thus	ADV
ejpam-6940	326	2	,	,	PUNCT
ejpam-6940	326	3	i(x	i(x	PROPN
ejpam-6940	326	4	)	)	PUNCT
ejpam-6940	326	5	⊆	⊆	NUM
ejpam-6940	326	6	sub(x	sub(x	PROPN
ejpam-6940	326	7	)	)	PUNCT
ejpam-6940	326	8	.	.	PUNCT
ejpam-6940	327	1	since	since	SCONJ
ejpam-6940	327	2	i(x	i(x	PROPN
ejpam-6940	327	3	)	)	PUNCT
ejpam-6940	327	4	is	be	AUX
ejpam-6940	327	5	a	a	DET
ejpam-6940	327	6	semigroup	semigroup	NOUN
ejpam-6940	327	7	and	and	CCONJ
ejpam-6940	327	8	i(x	i(x	NOUN
ejpam-6940	327	9	)	)	PUNCT
ejpam-6940	327	10	⊆	⊆	NUM
ejpam-6940	327	11	sub(x	sub(x	PROPN
ejpam-6940	327	12	)	)	PUNCT
ejpam-6940	327	13	,	,	PUNCT
ejpam-6940	327	14	then	then	ADV
ejpam-6940	327	15	i(x	i(x	PROPN
ejpam-6940	327	16	)	)	PUNCT
ejpam-6940	327	17	is	be	AUX
ejpam-6940	327	18	a	a	DET
ejpam-6940	327	19	subsemigroup	subsemigroup	NOUN
ejpam-6940	327	20	of	of	ADP
ejpam-6940	327	21	sub(x	sub(x	PROPN
ejpam-6940	327	22	)	)	PUNCT
ejpam-6940	327	23	.	.	PUNCT
ejpam-6940	328	1	we	we	PRON
ejpam-6940	328	2	will	will	AUX
ejpam-6940	328	3	recall	recall	VERB
ejpam-6940	328	4	some	some	DET
ejpam-6940	328	5	concepts	concept	NOUN
ejpam-6940	328	6	of	of	ADP
ejpam-6940	328	7	semigroup	semigroup	PROPN
ejpam-6940	328	8	theory	theory	NOUN
ejpam-6940	328	9	.	.	PUNCT
ejpam-6940	329	1	a	a	DET
ejpam-6940	329	2	semigroup	semigroup	NOUN
ejpam-6940	329	3	s	s	VERB
ejpam-6940	329	4	is	be	AUX
ejpam-6940	329	5	a	a	DET
ejpam-6940	329	6	left	left	ADJ
ejpam-6940	329	7	zero	zero	NUM
ejpam-6940	329	8	semigroup	semigroup	NOUN
ejpam-6940	329	9	if	if	SCONJ
ejpam-6940	329	10	za	za	PROPN
ejpam-6940	329	11	=	=	PROPN
ejpam-6940	329	12	z	z	PROPN
ejpam-6940	329	13	for	for	ADP
ejpam-6940	329	14	all	all	DET
ejpam-6940	329	15	z	z	PROPN
ejpam-6940	329	16	,	,	PUNCT
ejpam-6940	329	17	a	a	DET
ejpam-6940	329	18	∈	∈	PROPN
ejpam-6940	329	19	s.	s.	PROPN
ejpam-6940	329	20	a	a	DET
ejpam-6940	329	21	non	non	ADJ
ejpam-6940	329	22	-	-	ADJ
ejpam-6940	329	23	empty	empty	ADJ
ejpam-6940	329	24	subset	subset	NOUN
ejpam-6940	330	1	i	i	PRON
ejpam-6940	330	2	of	of	ADP
ejpam-6940	330	3	s	s	PROPN
ejpam-6940	330	4	is	be	AUX
ejpam-6940	330	5	a	a	DET
ejpam-6940	330	6	left	leave	VERB
ejpam-6940	330	7	semigroup	semigroup	NOUN
ejpam-6940	330	8	ideal	ideal	NOUN
ejpam-6940	330	9	(	(	PUNCT
ejpam-6940	330	10	right	right	PROPN
ejpam-6940	330	11	semigroup	semigroup	PROPN
ejpam-6940	330	12	ideal	ideal	PROPN
ejpam-6940	330	13	)	)	PUNCT
ejpam-6940	330	14	if	if	SCONJ
ejpam-6940	330	15	si	si	PROPN
ejpam-6940	330	16	⊆	⊆	NUM
ejpam-6940	330	17	i	i	PRON
ejpam-6940	330	18	(	(	PUNCT
ejpam-6940	330	19	is	be	AUX
ejpam-6940	330	20	⊆	⊆	NUM
ejpam-6940	330	21	i	i	PROPN
ejpam-6940	330	22	,	,	PUNCT
ejpam-6940	330	23	respectively	respectively	ADV
ejpam-6940	330	24	)	)	PUNCT
ejpam-6940	330	25	.	.	PUNCT
ejpam-6940	331	1	if	if	SCONJ
ejpam-6940	331	2	i	i	PRON
ejpam-6940	331	3	is	be	AUX
ejpam-6940	331	4	both	both	CCONJ
ejpam-6940	331	5	a	a	DET
ejpam-6940	331	6	left	left	ADJ
ejpam-6940	331	7	semigroup	semigroup	NOUN
ejpam-6940	331	8	ideal	ideal	NOUN
ejpam-6940	331	9	and	and	CCONJ
ejpam-6940	331	10	a	a	DET
ejpam-6940	331	11	right	right	ADJ
ejpam-6940	331	12	semigroup	semigroup	PROPN
ejpam-6940	331	13	ideal	ideal	NOUN
ejpam-6940	331	14	,	,	PUNCT
ejpam-6940	331	15	then	then	ADV
ejpam-6940	331	16	i	i	PRON
ejpam-6940	331	17	is	be	AUX
ejpam-6940	331	18	a	a	DET
ejpam-6940	331	19	semigroup	semigroup	ADJ
ejpam-6940	331	20	ideal	ideal	NOUN
ejpam-6940	331	21	.	.	PUNCT
ejpam-6940	332	1	a	a	DET
ejpam-6940	332	2	semigroup	semigroup	ADJ
ejpam-6940	332	3	ideal	ideal	NOUN
ejpam-6940	332	4	(	(	PUNCT
ejpam-6940	332	5	left	leave	VERB
ejpam-6940	332	6	semigroup	semigroup	PROPN
ejpam-6940	332	7	ideal	ideal	NOUN
ejpam-6940	332	8	,	,	PUNCT
ejpam-6940	332	9	right	right	PROPN
ejpam-6940	332	10	semigroup	semigroup	PROPN
ejpam-6940	332	11	ideal	ideal	PROPN
ejpam-6940	332	12	)	)	PUNCT
ejpam-6940	332	13	i	i	PRON
ejpam-6940	332	14	such	such	ADJ
ejpam-6940	332	15	that	that	SCONJ
ejpam-6940	332	16	i	i	PRON
ejpam-6940	332	17	̸=	̸=	PROPN
ejpam-6940	332	18	s	s	VERB
ejpam-6940	332	19	is	be	AUX
ejpam-6940	332	20	called	call	VERB
ejpam-6940	332	21	a	a	DET
ejpam-6940	332	22	proper	proper	ADJ
ejpam-6940	332	23	semigroup	semigroup	ADJ
ejpam-6940	332	24	ideal	ideal	NOUN
ejpam-6940	332	25	(	(	PUNCT
ejpam-6940	332	26	left	leave	VERB
ejpam-6940	332	27	semigroup	semigroup	PROPN
ejpam-6940	332	28	ideal	ideal	NOUN
ejpam-6940	332	29	,	,	PUNCT
ejpam-6940	332	30	right	right	PROPN
ejpam-6940	332	31	semigroup	semigroup	PROPN
ejpam-6940	332	32	ideal	ideal	PROPN
ejpam-6940	332	33	)	)	PUNCT
ejpam-6940	332	34	.	.	PUNCT
ejpam-6940	333	1	a	a	DET
ejpam-6940	333	2	semigroup	semigroup	NOUN
ejpam-6940	333	3	s	s	VERB
ejpam-6940	333	4	is	be	AUX
ejpam-6940	333	5	a	a	DET
ejpam-6940	333	6	left	left	ADJ
ejpam-6940	333	7	simple	simple	ADJ
ejpam-6940	333	8	semigroup	semigroup	NOUN
ejpam-6940	333	9	if	if	SCONJ
ejpam-6940	333	10	s	s	PROPN
ejpam-6940	333	11	has	have	VERB
ejpam-6940	333	12	no	no	DET
ejpam-6940	333	13	proper	proper	ADJ
ejpam-6940	333	14	left	leave	VERB
ejpam-6940	333	15	semigroup	semigroup	NOUN
ejpam-6940	333	16	ideals	ideal	NOUN
ejpam-6940	333	17	.	.	PUNCT
ejpam-6940	334	1	a	a	DET
ejpam-6940	334	2	right	right	ADJ
ejpam-6940	334	3	simple	simple	ADJ
ejpam-6940	334	4	semigroup	semigroup	NOUN
ejpam-6940	334	5	and	and	CCONJ
ejpam-6940	334	6	a	a	DET
ejpam-6940	334	7	simple	simple	ADJ
ejpam-6940	334	8	semigroup	semigroup	NOUN
ejpam-6940	334	9	are	be	AUX
ejpam-6940	334	10	defined	define	VERB
ejpam-6940	334	11	in	in	ADP
ejpam-6940	334	12	an	an	DET
ejpam-6940	334	13	analogous	analogous	ADJ
ejpam-6940	334	14	way	way	NOUN
ejpam-6940	334	15	.	.	PUNCT
ejpam-6940	335	1	for	for	ADP
ejpam-6940	335	2	more	more	ADV
ejpam-6940	335	3	intensive	intensive	ADJ
ejpam-6940	335	4	details	detail	NOUN
ejpam-6940	335	5	in	in	ADP
ejpam-6940	335	6	semigroup	semigroup	PROPN
ejpam-6940	335	7	theory	theory	NOUN
ejpam-6940	335	8	we	we	PRON
ejpam-6940	335	9	refer	refer	VERB
ejpam-6940	335	10	to	to	ADP
ejpam-6940	335	11	[	[	X
ejpam-6940	335	12	18	18	NUM
ejpam-6940	335	13	]	]	PUNCT
ejpam-6940	335	14	.	.	PUNCT
ejpam-6940	336	1	proposition	proposition	NOUN
ejpam-6940	336	2	14	14	NUM
ejpam-6940	336	3	.	.	PUNCT
ejpam-6940	337	1	let	let	VERB
ejpam-6940	337	2	x	x	PRON
ejpam-6940	337	3	be	be	AUX
ejpam-6940	337	4	an	an	DET
ejpam-6940	337	5	edge	edge	NOUN
ejpam-6940	337	6	q	q	NOUN
ejpam-6940	337	7	-	-	NOUN
ejpam-6940	337	8	algebra	algebra	NOUN
ejpam-6940	337	9	.	.	PUNCT
ejpam-6940	338	1	then	then	ADV
ejpam-6940	338	2	i(x	i(x	PROPN
ejpam-6940	338	3	)	)	PUNCT
ejpam-6940	338	4	is	be	AUX
ejpam-6940	338	5	a	a	DET
ejpam-6940	338	6	left	left	ADJ
ejpam-6940	338	7	zero	zero	NUM
ejpam-6940	338	8	semigroup	semigroup	NOUN
ejpam-6940	338	9	.	.	PUNCT
ejpam-6940	339	1	proof	proof	NOUN
ejpam-6940	339	2	.	.	PUNCT
ejpam-6940	340	1	by	by	ADP
ejpam-6940	340	2	proposition	proposition	NOUN
ejpam-6940	340	3	12	12	NUM
ejpam-6940	340	4	,	,	PUNCT
ejpam-6940	340	5	i(x	i(x	PROPN
ejpam-6940	340	6	)	)	PUNCT
ejpam-6940	340	7	is	be	AUX
ejpam-6940	340	8	a	a	DET
ejpam-6940	340	9	semigroup	semigroup	NOUN
ejpam-6940	340	10	.	.	PUNCT
ejpam-6940	341	1	let	let	VERB
ejpam-6940	341	2	a	a	DET
ejpam-6940	341	3	,	,	PUNCT
ejpam-6940	341	4	b	b	PROPN
ejpam-6940	341	5	∈	∈	PROPN
ejpam-6940	341	6	i(x	i(x	NOUN
ejpam-6940	341	7	)	)	PUNCT
ejpam-6940	341	8	.	.	PUNCT
ejpam-6940	342	1	then	then	ADV
ejpam-6940	342	2	by	by	ADP
ejpam-6940	342	3	corollary	corollary	ADJ
ejpam-6940	342	4	3	3	NUM
ejpam-6940	342	5	,	,	PUNCT
ejpam-6940	342	6	aw	aw	INTJ
ejpam-6940	342	7	=	=	NOUN
ejpam-6940	342	8	a.	a.	NOUN
ejpam-6940	342	9	therefore	therefore	ADV
ejpam-6940	342	10	,	,	PUNCT
ejpam-6940	342	11	i(x	i(x	PROPN
ejpam-6940	342	12	)	)	PUNCT
ejpam-6940	342	13	is	be	AUX
ejpam-6940	342	14	a	a	DET
ejpam-6940	342	15	left	left	ADJ
ejpam-6940	342	16	zero	zero	NUM
ejpam-6940	342	17	semigroup	semigroup	NOUN
ejpam-6940	342	18	.	.	PUNCT
ejpam-6940	343	1	moreover	moreover	ADV
ejpam-6940	343	2	,	,	PUNCT
ejpam-6940	343	3	i(x	i(x	PROPN
ejpam-6940	343	4	)	)	PUNCT
ejpam-6940	343	5	is	be	AUX
ejpam-6940	343	6	a	a	DET
ejpam-6940	343	7	simple	simple	ADJ
ejpam-6940	343	8	semigroup	semigroup	NOUN
ejpam-6940	343	9	.	.	PUNCT
ejpam-6940	344	1	proposition	proposition	NOUN
ejpam-6940	344	2	15	15	NUM
ejpam-6940	344	3	.	.	PUNCT
ejpam-6940	345	1	let	let	VERB
ejpam-6940	345	2	x	x	PRON
ejpam-6940	345	3	be	be	AUX
ejpam-6940	345	4	an	an	DET
ejpam-6940	345	5	edge	edge	NOUN
ejpam-6940	345	6	q	q	NOUN
ejpam-6940	345	7	-	-	NOUN
ejpam-6940	345	8	algebra	algebra	NOUN
ejpam-6940	345	9	.	.	PUNCT
ejpam-6940	346	1	then	then	ADV
ejpam-6940	346	2	i(x	i(x	PROPN
ejpam-6940	346	3	)	)	PUNCT
ejpam-6940	346	4	is	be	AUX
ejpam-6940	346	5	a	a	DET
ejpam-6940	346	6	left	left	ADJ
ejpam-6940	346	7	simple	simple	ADJ
ejpam-6940	346	8	semigroup	semigroup	NOUN
ejpam-6940	346	9	.	.	PUNCT
ejpam-6940	347	1	proof	proof	NOUN
ejpam-6940	347	2	.	.	PUNCT
ejpam-6940	348	1	let	let	VERB
ejpam-6940	348	2	p	p	PRON
ejpam-6940	348	3	be	be	AUX
ejpam-6940	348	4	a	a	DET
ejpam-6940	348	5	left	left	ADJ
ejpam-6940	348	6	semigroup	semigroup	NOUN
ejpam-6940	348	7	ideal	ideal	NOUN
ejpam-6940	348	8	of	of	ADP
ejpam-6940	348	9	a	a	DET
ejpam-6940	348	10	semigroup	semigroup	ADJ
ejpam-6940	348	11	i(x	i(x	NOUN
ejpam-6940	348	12	)	)	PUNCT
ejpam-6940	348	13	.	.	PUNCT
ejpam-6940	349	1	then	then	ADV
ejpam-6940	349	2	i(x)p	i(x)p	VERB
ejpam-6940	349	3	⊆	⊆	NUM
ejpam-6940	349	4	p	p	NOUN
ejpam-6940	349	5	.	.	PUNCT
ejpam-6940	350	1	let	let	VERB
ejpam-6940	350	2	a	a	DET
ejpam-6940	350	3	∈	∈	PROPN
ejpam-6940	350	4	i(x	i(x	NOUN
ejpam-6940	350	5	)	)	PUNCT
ejpam-6940	350	6	and	and	CCONJ
ejpam-6940	350	7	b	b	X
ejpam-6940	350	8	∈	∈	PROPN
ejpam-6940	350	9	p	p	NOUN
ejpam-6940	350	10	.	.	PUNCT
ejpam-6940	351	1	then	then	ADV
ejpam-6940	351	2	ab	ab	PROPN
ejpam-6940	351	3	=	=	PUNCT
ejpam-6940	351	4	a	a	PRON
ejpam-6940	351	5	by	by	ADP
ejpam-6940	351	6	corollary	corollary	ADJ
ejpam-6940	351	7	3	3	NUM
ejpam-6940	351	8	.	.	PUNCT
ejpam-6940	352	1	there	there	PRON
ejpam-6940	352	2	follows	follow	VERB
ejpam-6940	352	3	that	that	SCONJ
ejpam-6940	352	4	i(x	i(x	PROPN
ejpam-6940	352	5	)	)	PUNCT
ejpam-6940	352	6	⊆	⊆	NUM
ejpam-6940	352	7	i(x)p	i(x)p	NOUN
ejpam-6940	352	8	⊆	⊆	NUM
ejpam-6940	352	9	p	p	NOUN
ejpam-6940	352	10	.	.	PUNCT
ejpam-6940	353	1	hence	hence	ADV
ejpam-6940	353	2	,	,	PUNCT
ejpam-6940	353	3	i(x	i(x	PROPN
ejpam-6940	353	4	)	)	PUNCT
ejpam-6940	354	1	=	=	SYM
ejpam-6940	355	1	p	p	NOUN
ejpam-6940	355	2	.	.	PUNCT
ejpam-6940	356	1	therefore	therefore	ADV
ejpam-6940	356	2	,	,	PUNCT
ejpam-6940	356	3	i(x	i(x	PROPN
ejpam-6940	356	4	)	)	PUNCT
ejpam-6940	356	5	does	do	AUX
ejpam-6940	356	6	not	not	PART
ejpam-6940	356	7	contain	contain	VERB
ejpam-6940	356	8	a	a	DET
ejpam-6940	356	9	proper	proper	ADJ
ejpam-6940	356	10	semigroup	semigroup	NOUN
ejpam-6940	356	11	left	leave	VERB
ejpam-6940	356	12	ideal	ideal	NOUN
ejpam-6940	356	13	.	.	PUNCT
ejpam-6940	357	1	thus	thus	ADV
ejpam-6940	357	2	,	,	PUNCT
ejpam-6940	357	3	i(x	i(x	PROPN
ejpam-6940	357	4	)	)	PUNCT
ejpam-6940	357	5	is	be	AUX
ejpam-6940	357	6	a	a	DET
ejpam-6940	357	7	left	left	ADJ
ejpam-6940	357	8	simple	simple	ADJ
ejpam-6940	357	9	semigroup	semigroup	NOUN
ejpam-6940	357	10	.	.	PUNCT
ejpam-6940	358	1	corollary	corollary	ADJ
ejpam-6940	358	2	4	4	NUM
ejpam-6940	358	3	.	.	PUNCT
ejpam-6940	359	1	let	let	VERB
ejpam-6940	359	2	x	x	PRON
ejpam-6940	359	3	be	be	AUX
ejpam-6940	359	4	an	an	DET
ejpam-6940	359	5	edge	edge	NOUN
ejpam-6940	359	6	q	q	NOUN
ejpam-6940	359	7	-	-	NOUN
ejpam-6940	359	8	algebra	algebra	NOUN
ejpam-6940	359	9	.	.	PUNCT
ejpam-6940	360	1	then	then	ADV
ejpam-6940	360	2	i(x	i(x	PROPN
ejpam-6940	360	3	)	)	PUNCT
ejpam-6940	360	4	is	be	AUX
ejpam-6940	360	5	a	a	DET
ejpam-6940	360	6	simple	simple	ADJ
ejpam-6940	360	7	semigroup	semigroup	NOUN
ejpam-6940	360	8	.	.	PUNCT
ejpam-6940	361	1	proof	proof	NOUN
ejpam-6940	361	2	.	.	PUNCT
ejpam-6940	362	1	it	it	PRON
ejpam-6940	362	2	follows	follow	VERB
ejpam-6940	362	3	from	from	ADP
ejpam-6940	362	4	proposition	proposition	NOUN
ejpam-6940	362	5	15	15	NUM
ejpam-6940	362	6	.	.	NOUN
ejpam-6940	363	1	3	3	X
ejpam-6940	363	2	.	.	X
ejpam-6940	363	3	enumeration	enumeration	NOUN
ejpam-6940	363	4	of	of	ADP
ejpam-6940	363	5	edge	edge	NOUN
ejpam-6940	363	6	q	q	NOUN
ejpam-6940	363	7	-	-	PUNCT
ejpam-6940	363	8	algebras	algebras	ADJ
ejpam-6940	363	9	in	in	ADP
ejpam-6940	363	10	this	this	DET
ejpam-6940	363	11	section	section	NOUN
ejpam-6940	363	12	,	,	PUNCT
ejpam-6940	363	13	we	we	PRON
ejpam-6940	363	14	describe	describe	VERB
ejpam-6940	363	15	all	all	DET
ejpam-6940	363	16	possible	possible	ADJ
ejpam-6940	363	17	structures	structure	NOUN
ejpam-6940	363	18	of	of	ADP
ejpam-6940	363	19	edge	edge	NOUN
ejpam-6940	363	20	q	q	NOUN
ejpam-6940	363	21	-	-	PUNCT
ejpam-6940	363	22	algebras	algebra	NOUN
ejpam-6940	363	23	of	of	ADP
ejpam-6940	363	24	order	order	NOUN
ejpam-6940	363	25	n	n	CCONJ
ejpam-6940	363	26	,	,	PUNCT
ejpam-6940	363	27	for	for	ADP
ejpam-6940	363	28	any	any	DET
ejpam-6940	363	29	positive	positive	ADJ
ejpam-6940	363	30	integer	integer	NOUN
ejpam-6940	363	31	n.	n.	NOUN
ejpam-6940	363	32	to	to	PART
ejpam-6940	363	33	do	do	VERB
ejpam-6940	363	34	this	this	PRON
ejpam-6940	363	35	we	we	PRON
ejpam-6940	363	36	need	need	VERB
ejpam-6940	363	37	to	to	PART
ejpam-6940	363	38	construct	construct	VERB
ejpam-6940	363	39	a	a	DET
ejpam-6940	363	40	q	q	NOUN
ejpam-6940	363	41	-	-	NOUN
ejpam-6940	363	42	algebra	algebra	NOUN
ejpam-6940	363	43	as	as	SCONJ
ejpam-6940	363	44	follows	follow	VERB
ejpam-6940	363	45	:	:	PUNCT
ejpam-6940	364	1	construction(∗	construction(∗	ADV
ejpam-6940	364	2	):	):	PUNCT
ejpam-6940	364	3	let	let	VERB
ejpam-6940	364	4	xn	xn	PUNCT
ejpam-6940	364	5	=	=	PUNCT
ejpam-6940	364	6	{	{	PUNCT
ejpam-6940	364	7	x1	x1	PROPN
ejpam-6940	364	8	,	,	PUNCT
ejpam-6940	364	9	x2	x2	PROPN
ejpam-6940	364	10	,	,	PUNCT
ejpam-6940	364	11	x3	x3	ADJ
ejpam-6940	364	12	,	,	PUNCT
ejpam-6940	364	13	.	.	PUNCT
ejpam-6940	364	14	.	.	PUNCT
ejpam-6940	364	15	.	.	PUNCT
ejpam-6940	365	1	,	,	PUNCT
ejpam-6940	365	2	xn	xn	X
ejpam-6940	365	3	}	}	PUNCT
ejpam-6940	365	4	be	be	AUX
ejpam-6940	365	5	a	a	DET
ejpam-6940	365	6	set	set	NOUN
ejpam-6940	365	7	of	of	ADP
ejpam-6940	365	8	order	order	NOUN
ejpam-6940	365	9	n.	n.	NOUN
ejpam-6940	365	10	we	we	PRON
ejpam-6940	365	11	define	define	VERB
ejpam-6940	365	12	a	a	DET
ejpam-6940	365	13	binary	binary	ADJ
ejpam-6940	365	14	operation	operation	NOUN
ejpam-6940	365	15	on	on	ADP
ejpam-6940	365	16	xn	xn	PROPN
ejpam-6940	365	17	as	as	SCONJ
ejpam-6940	365	18	follow	follow	VERB
ejpam-6940	365	19	:	:	PUNCT
ejpam-6940	365	20	for	for	ADP
ejpam-6940	365	21	xi	xi	PROPN
ejpam-6940	365	22	,	,	PUNCT
ejpam-6940	365	23	xj	xj	PROPN
ejpam-6940	365	24	∈	∈	PROPN
ejpam-6940	365	25	xn	xn	PROPN
ejpam-6940	365	26	,	,	PUNCT
ejpam-6940	365	27	xixj	xixj	PROPN
ejpam-6940	366	1	=	=	PUNCT
ejpam-6940	366	2			X
ejpam-6940	366	3	x1	x1	INTJ
ejpam-6940	367	1	if	if	SCONJ
ejpam-6940	367	2	i	i	PRON
ejpam-6940	367	3	=	=	SYM
ejpam-6940	367	4	j	j	PROPN
ejpam-6940	367	5	,	,	PUNCT
ejpam-6940	367	6	x1	x1	PROPN
ejpam-6940	367	7	if	if	SCONJ
ejpam-6940	367	8	i	i	PRON
ejpam-6940	367	9	=	=	NOUN
ejpam-6940	367	10	1	1	NUM
ejpam-6940	367	11	,	,	PUNCT
ejpam-6940	367	12	xi	xi	X
ejpam-6940	368	1	if	if	SCONJ
ejpam-6940	368	2	j	j	PROPN
ejpam-6940	368	3	=	=	SYM
ejpam-6940	368	4	1	1	NUM
ejpam-6940	368	5	,	,	PUNCT
ejpam-6940	368	6	a	a	DET
ejpam-6940	368	7	∈	∈	NOUN
ejpam-6940	368	8	{	{	PUNCT
ejpam-6940	368	9	x1	x1	PROPN
ejpam-6940	368	10	,	,	PUNCT
ejpam-6940	368	11	xi	xi	ADP
ejpam-6940	368	12	}	}	PUNCT
ejpam-6940	368	13	if	if	SCONJ
ejpam-6940	368	14	otherwise	otherwise	ADV
ejpam-6940	368	15	.	.	PUNCT
ejpam-6940	369	1	a.	a.	PROPN
ejpam-6940	369	2	anantayasethi	anantayasethi	PROPN
ejpam-6940	369	3	et	et	PROPN
ejpam-6940	369	4	al	al	PROPN
ejpam-6940	369	5	.	.	PUNCT
ejpam-6940	369	6	/	/	SYM
ejpam-6940	369	7	eur	eur	PROPN
ejpam-6940	369	8	.	.	PUNCT
ejpam-6940	370	1	j.	j.	PROPN
ejpam-6940	370	2	pure	pure	PROPN
ejpam-6940	370	3	appl	appl	PROPN
ejpam-6940	370	4	.	.	PROPN
ejpam-6940	370	5	math	math	PROPN
ejpam-6940	370	6	,	,	PUNCT
ejpam-6940	370	7	18	18	NUM
ejpam-6940	370	8	(	(	PUNCT
ejpam-6940	370	9	4	4	NUM
ejpam-6940	370	10	)	)	PUNCT
ejpam-6940	370	11	(	(	PUNCT
ejpam-6940	370	12	2025	2025	NUM
ejpam-6940	370	13	)	)	PUNCT
ejpam-6940	370	14	,	,	PUNCT
ejpam-6940	370	15	6940	6940	NUM
ejpam-6940	370	16	11	11	NUM
ejpam-6940	370	17	of	of	ADP
ejpam-6940	370	18	14	14	NUM
ejpam-6940	370	19	from	from	ADP
ejpam-6940	370	20	the	the	DET
ejpam-6940	370	21	construction	construction	NOUN
ejpam-6940	370	22	(	(	PUNCT
ejpam-6940	370	23	∗	∗	NOUN
ejpam-6940	370	24	)	)	PUNCT
ejpam-6940	370	25	,	,	PUNCT
ejpam-6940	370	26	in	in	ADP
ejpam-6940	370	27	the	the	DET
ejpam-6940	370	28	case	case	NOUN
ejpam-6940	370	29	i	i	PRON
ejpam-6940	370	30	̸=	̸=	PROPN
ejpam-6940	370	31	j	j	PROPN
ejpam-6940	370	32	and	and	CCONJ
ejpam-6940	370	33	i	i	PROPN
ejpam-6940	370	34	,	,	PUNCT
ejpam-6940	370	35	j	j	PROPN
ejpam-6940	370	36	̸=	̸=	PROPN
ejpam-6940	370	37	1	1	NUM
ejpam-6940	370	38	,	,	PUNCT
ejpam-6940	370	39	the	the	DET
ejpam-6940	370	40	product	product	NOUN
ejpam-6940	370	41	xixj	xixj	PROPN
ejpam-6940	370	42	∈	∈	PROPN
ejpam-6940	370	43	{	{	PUNCT
ejpam-6940	370	44	x1	x1	PROPN
ejpam-6940	370	45	,	,	PUNCT
ejpam-6940	370	46	xi	xi	ADJ
ejpam-6940	370	47	}	}	PUNCT
ejpam-6940	370	48	,	,	PUNCT
ejpam-6940	370	49	i.e.	i.e.	X
ejpam-6940	370	50	the	the	DET
ejpam-6940	370	51	product	product	NOUN
ejpam-6940	370	52	xixj	xixj	PROPN
ejpam-6940	370	53	is	be	AUX
ejpam-6940	370	54	either	either	CCONJ
ejpam-6940	370	55	x1	x1	PROPN
ejpam-6940	370	56	or	or	CCONJ
ejpam-6940	370	57	xi	xi	PROPN
ejpam-6940	370	58	.	.	PUNCT
ejpam-6940	371	1	we	we	PRON
ejpam-6940	371	2	denote	denote	VERB
ejpam-6940	371	3	here	here	ADV
ejpam-6940	371	4	x	x	X
ejpam-6940	371	5	∨	∨	NUM
ejpam-6940	371	6	y	y	PROPN
ejpam-6940	371	7	by	by	ADP
ejpam-6940	371	8	”	"	PUNCT
ejpam-6940	371	9	either	either	CCONJ
ejpam-6940	371	10	x	x	SYM
ejpam-6940	371	11	or	or	CCONJ
ejpam-6940	371	12	y	y	PROPN
ejpam-6940	371	13	”	"	PUNCT
ejpam-6940	371	14	.	.	PUNCT
ejpam-6940	372	1	then	then	ADV
ejpam-6940	372	2	we	we	PRON
ejpam-6940	372	3	obtain	obtain	VERB
ejpam-6940	372	4	the	the	DET
ejpam-6940	372	5	following	follow	VERB
ejpam-6940	372	6	cayley	cayley	ADJ
ejpam-6940	372	7	table	table	NOUN
ejpam-6940	372	8	:	:	PUNCT
ejpam-6940	373	1	x1	x1	PROPN
ejpam-6940	373	2	x2	x2	NOUN
ejpam-6940	373	3	x3	x3	ADJ
ejpam-6940	373	4	…	…	PUNCT
ejpam-6940	374	1	xi−1	xi−1	PROPN
ejpam-6940	374	2	xi	xi	ADP
ejpam-6940	374	3	xi+1	xi+1	PROPN
ejpam-6940	374	4	…	…	PUNCT
ejpam-6940	374	5	xn	xn	PROPN
ejpam-6940	375	1	x1	x1	NUM
ejpam-6940	375	2	x1	x1	NUM
ejpam-6940	376	1	x1	x1	NUM
ejpam-6940	376	2	x1	x1	PROPN
ejpam-6940	376	3	…	…	PUNCT
ejpam-6940	377	1	x1	x1	NUM
ejpam-6940	378	1	x1	x1	NUM
ejpam-6940	379	1	x1	x1	PROPN
ejpam-6940	379	2	…	…	PUNCT
ejpam-6940	380	1	x1	x1	NUM
ejpam-6940	381	1	x2	x2	NOUN
ejpam-6940	382	1	x2	x2	INTJ
ejpam-6940	382	2	x1	x1	PROPN
ejpam-6940	383	1	x1	x1	PROPN
ejpam-6940	383	2	∨	∨	NUM
ejpam-6940	383	3	x2	x2	PROPN
ejpam-6940	383	4	…	…	PUNCT
ejpam-6940	383	5	x1	x1	NUM
ejpam-6940	384	1	∨	∨	NUM
ejpam-6940	384	2	x2	x2	PROPN
ejpam-6940	384	3	x1	x1	PROPN
ejpam-6940	384	4	∨	∨	NUM
ejpam-6940	384	5	x2	x2	PROPN
ejpam-6940	384	6	x1	x1	PROPN
ejpam-6940	384	7	∨	∨	NUM
ejpam-6940	384	8	x2	x2	PROPN
ejpam-6940	384	9	…	…	PUNCT
ejpam-6940	384	10	x1	x1	NUM
ejpam-6940	384	11	∨	∨	NUM
ejpam-6940	384	12	x2	x2	NOUN
ejpam-6940	384	13	x3	x3	NOUN
ejpam-6940	385	1	x3	x3	PROPN
ejpam-6940	386	1	x1	x1	PROPN
ejpam-6940	387	1	∨	∨	NUM
ejpam-6940	387	2	x3	x3	PROPN
ejpam-6940	387	3	x1	x1	PROPN
ejpam-6940	387	4	…	…	PUNCT
ejpam-6940	387	5	x1	x1	NUM
ejpam-6940	387	6	∨	∨	NUM
ejpam-6940	387	7	x3	x3	PROPN
ejpam-6940	387	8	x1	x1	PROPN
ejpam-6940	387	9	∨	∨	NUM
ejpam-6940	387	10	x3	x3	PROPN
ejpam-6940	387	11	x1	x1	PROPN
ejpam-6940	387	12	∨	∨	NUM
ejpam-6940	387	13	x3	x3	PROPN
ejpam-6940	387	14	…	…	PUNCT
ejpam-6940	387	15	x1	x1	PRON
ejpam-6940	387	16	∨	∨	NUM
ejpam-6940	387	17	x3	x3	PROPN
ejpam-6940	387	18	...	...	PUNCT
ejpam-6940	387	19	...	...	PUNCT
ejpam-6940	387	20	...	...	PUNCT
ejpam-6940	387	21	...	...	PUNCT
ejpam-6940	387	22	...	...	PUNCT
ejpam-6940	387	23	...	...	PUNCT
ejpam-6940	387	24	...	...	PUNCT
ejpam-6940	387	25	...	...	PUNCT
ejpam-6940	387	26	...	...	PUNCT
ejpam-6940	387	27	...	...	PUNCT
ejpam-6940	388	1	xi	xi	X
ejpam-6940	388	2	xi	xi	NUM
ejpam-6940	388	3	x1	x1	PROPN
ejpam-6940	388	4	∨	∨	NUM
ejpam-6940	388	5	xi	xi	PROPN
ejpam-6940	388	6	x1	x1	PROPN
ejpam-6940	388	7	∨	∨	NOUN
ejpam-6940	388	8	xi	xi	PROPN
ejpam-6940	388	9	…	…	PUNCT
ejpam-6940	388	10	x1	x1	PROPN
ejpam-6940	389	1	∨	∨	NOUN
ejpam-6940	389	2	xi	xi	PROPN
ejpam-6940	390	1	x1	x1	NUM
ejpam-6940	390	2	x1	x1	PROPN
ejpam-6940	390	3	∨	∨	NUM
ejpam-6940	390	4	xi	xi	PROPN
ejpam-6940	390	5	…	…	PUNCT
ejpam-6940	390	6	x1	x1	PROPN
ejpam-6940	390	7	∨	∨	NOUN
ejpam-6940	390	8	xi	xi	X
ejpam-6940	390	9	...	...	PUNCT
ejpam-6940	390	10	...	...	PUNCT
ejpam-6940	390	11	...	...	PUNCT
ejpam-6940	390	12	...	...	PUNCT
ejpam-6940	390	13	...	...	PUNCT
ejpam-6940	390	14	...	...	PUNCT
ejpam-6940	390	15	...	...	PUNCT
ejpam-6940	390	16	...	...	PUNCT
ejpam-6940	390	17	...	...	PUNCT
ejpam-6940	390	18	...	...	PUNCT
ejpam-6940	391	1	xn	xn	PUNCT
ejpam-6940	391	2	xn	xn	PROPN
ejpam-6940	392	1	x1	x1	NUM
ejpam-6940	393	1	∨	∨	NUM
ejpam-6940	393	2	xn	xn	PROPN
ejpam-6940	393	3	x1	x1	NUM
ejpam-6940	394	1	∨	∨	PROPN
ejpam-6940	394	2	xn	xn	PROPN
ejpam-6940	395	1	…	…	PUNCT
ejpam-6940	395	2	x1	x1	NUM
ejpam-6940	395	3	∨	∨	NUM
ejpam-6940	395	4	xn	xn	PROPN
ejpam-6940	395	5	x1	x1	NUM
ejpam-6940	396	1	∨	∨	NUM
ejpam-6940	396	2	xn	xn	PROPN
ejpam-6940	396	3	x1	x1	NUM
ejpam-6940	397	1	∨	∨	PROPN
ejpam-6940	397	2	xn	xn	PROPN
ejpam-6940	398	1	…	…	PUNCT
ejpam-6940	398	2	x1	x1	PROPN
ejpam-6940	398	3	for	for	ADP
ejpam-6940	398	4	the	the	DET
ejpam-6940	398	5	set	set	NOUN
ejpam-6940	398	6	xn	xn	PROPN
ejpam-6940	398	7	,	,	PUNCT
ejpam-6940	398	8	the	the	DET
ejpam-6940	398	9	construction	construction	NOUN
ejpam-6940	398	10	(	(	PUNCT
ejpam-6940	398	11	∗	∗	NOUN
ejpam-6940	398	12	)	)	PUNCT
ejpam-6940	398	13	allows	allow	VERB
ejpam-6940	398	14	us	we	PRON
ejpam-6940	398	15	to	to	PART
ejpam-6940	398	16	get	get	VERB
ejpam-6940	398	17	2n	2n	NUM
ejpam-6940	398	18	2−3n+2	2−3n+2	NUM
ejpam-6940	398	19	algebraic	algebraic	ADJ
ejpam-6940	398	20	structures	structure	NOUN
ejpam-6940	398	21	.	.	PUNCT
ejpam-6940	399	1	let	let	VERB
ejpam-6940	399	2	eq(xn	eq(xn	PROPN
ejpam-6940	399	3	)	)	PUNCT
ejpam-6940	399	4	be	be	AUX
ejpam-6940	399	5	the	the	DET
ejpam-6940	399	6	set	set	NOUN
ejpam-6940	399	7	of	of	ADP
ejpam-6940	399	8	all	all	DET
ejpam-6940	399	9	algebras	algebra	NOUN
ejpam-6940	399	10	which	which	PRON
ejpam-6940	399	11	obtained	obtain	VERB
ejpam-6940	399	12	from	from	ADP
ejpam-6940	399	13	the	the	DET
ejpam-6940	399	14	construction	construction	NOUN
ejpam-6940	399	15	(	(	PUNCT
ejpam-6940	399	16	∗	∗	NOUN
ejpam-6940	399	17	)	)	PUNCT
ejpam-6940	399	18	.	.	PUNCT
ejpam-6940	400	1	example	example	NOUN
ejpam-6940	401	1	7	7	X
ejpam-6940	401	2	.	.	PUNCT
ejpam-6940	402	1	let	let	VERB
ejpam-6940	402	2	x3	x3	VERB
ejpam-6940	402	3	=	=	PUNCT
ejpam-6940	402	4	{	{	PUNCT
ejpam-6940	402	5	x1	x1	PROPN
ejpam-6940	402	6	,	,	PUNCT
ejpam-6940	402	7	x2	x2	PROPN
ejpam-6940	402	8	,	,	PUNCT
ejpam-6940	402	9	x3	x3	ADJ
ejpam-6940	402	10	}	}	PUNCT
ejpam-6940	402	11	.	.	PUNCT
ejpam-6940	403	1	the	the	DET
ejpam-6940	403	2	following	follow	VERB
ejpam-6940	403	3	algebras	algebras	PROPN
ejpam-6940	403	4	a	a	DET
ejpam-6940	403	5	,	,	PUNCT
ejpam-6940	403	6	b	b	NOUN
ejpam-6940	403	7	,	,	PUNCT
ejpam-6940	403	8	c	c	PROPN
ejpam-6940	403	9	and	and	CCONJ
ejpam-6940	403	10	d	d	PROPN
ejpam-6940	403	11	are	be	AUX
ejpam-6940	403	12	obtained	obtain	VERB
ejpam-6940	403	13	from	from	ADP
ejpam-6940	403	14	the	the	DET
ejpam-6940	403	15	construction	construction	NOUN
ejpam-6940	403	16	(	(	PUNCT
ejpam-6940	403	17	∗	∗	NOUN
ejpam-6940	403	18	):	):	PUNCT
ejpam-6940	404	1	x1	x1	PROPN
ejpam-6940	405	1	x2	x2	NOUN
ejpam-6940	405	2	x3	x3	VERB
ejpam-6940	405	3	x1	x1	NOUN
ejpam-6940	406	1	x1	x1	NUM
ejpam-6940	407	1	x1	x1	NUM
ejpam-6940	408	1	x1	x1	NUM
ejpam-6940	409	1	x2	x2	NOUN
ejpam-6940	410	1	x2	x2	INTJ
ejpam-6940	410	2	x1	x1	PROPN
ejpam-6940	411	1	x1	x1	NUM
ejpam-6940	412	1	x3	x3	NOUN
ejpam-6940	412	2	x3	x3	PROPN
ejpam-6940	412	3	x1	x1	NOUN
ejpam-6940	413	1	x1	x1	NUM
ejpam-6940	413	2	x1	x1	NUM
ejpam-6940	414	1	x2	x2	NOUN
ejpam-6940	414	2	x3	x3	VERB
ejpam-6940	414	3	x1	x1	NOUN
ejpam-6940	415	1	x1	x1	NUM
ejpam-6940	416	1	x1	x1	NUM
ejpam-6940	417	1	x1	x1	NUM
ejpam-6940	418	1	x2	x2	NOUN
ejpam-6940	419	1	x2	x2	INTJ
ejpam-6940	419	2	x1	x1	PROPN
ejpam-6940	420	1	x1	x1	NUM
ejpam-6940	421	1	x3	x3	ADJ
ejpam-6940	421	2	x3	x3	PROPN
ejpam-6940	422	1	x3	x3	VERB
ejpam-6940	423	1	x1	x1	PROPN
ejpam-6940	424	1	a	a	DET
ejpam-6940	424	2	b	b	NOUN
ejpam-6940	424	3	x1	x1	PROPN
ejpam-6940	425	1	x2	x2	NOUN
ejpam-6940	425	2	x3	x3	VERB
ejpam-6940	425	3	x1	x1	NOUN
ejpam-6940	426	1	x1	x1	NUM
ejpam-6940	427	1	x1	x1	NUM
ejpam-6940	428	1	x1	x1	NUM
ejpam-6940	429	1	x2	x2	NOUN
ejpam-6940	430	1	x2	x2	INTJ
ejpam-6940	430	2	x1	x1	NUM
ejpam-6940	431	1	x2	x2	NOUN
ejpam-6940	431	2	x3	x3	NOUN
ejpam-6940	432	1	x3	x3	VERB
ejpam-6940	432	2	x1	x1	NOUN
ejpam-6940	433	1	x1	x1	NUM
ejpam-6940	433	2	x1	x1	NUM
ejpam-6940	434	1	x2	x2	NOUN
ejpam-6940	434	2	x3	x3	VERB
ejpam-6940	434	3	x1	x1	NOUN
ejpam-6940	435	1	x1	x1	NUM
ejpam-6940	436	1	x1	x1	NUM
ejpam-6940	437	1	x1	x1	NUM
ejpam-6940	438	1	x2	x2	NOUN
ejpam-6940	439	1	x2	x2	INTJ
ejpam-6940	439	2	x1	x1	NUM
ejpam-6940	440	1	x2	x2	NOUN
ejpam-6940	440	2	x3	x3	ADJ
ejpam-6940	440	3	x3	x3	INTJ
ejpam-6940	441	1	x3	x3	VERB
ejpam-6940	442	1	x1	x1	PROPN
ejpam-6940	442	2	c	c	PROPN
ejpam-6940	443	1	d	d	X
ejpam-6940	443	2	it	it	PRON
ejpam-6940	443	3	is	be	AUX
ejpam-6940	443	4	not	not	PART
ejpam-6940	443	5	difficult	difficult	ADJ
ejpam-6940	443	6	to	to	PART
ejpam-6940	443	7	check	check	VERB
ejpam-6940	443	8	that	that	SCONJ
ejpam-6940	443	9	all	all	PRON
ejpam-6940	443	10	above	above	ADP
ejpam-6940	443	11	tables	table	NOUN
ejpam-6940	443	12	a	a	DET
ejpam-6940	443	13	,	,	PUNCT
ejpam-6940	443	14	b	b	NOUN
ejpam-6940	443	15	,	,	PUNCT
ejpam-6940	443	16	c	c	PROPN
ejpam-6940	443	17	and	and	CCONJ
ejpam-6940	443	18	d	d	PROPN
ejpam-6940	443	19	are	be	AUX
ejpam-6940	443	20	q	q	NOUN
ejpam-6940	443	21	-	-	PUNCT
ejpam-6940	443	22	algebras	algebras	ADJ
ejpam-6940	443	23	with	with	ADP
ejpam-6940	443	24	x1	x1	PROPN
ejpam-6940	443	25	acts	act	NOUN
ejpam-6940	443	26	as	as	ADP
ejpam-6940	443	27	a	a	DET
ejpam-6940	443	28	constant	constant	ADJ
ejpam-6940	443	29	0	0	NUM
ejpam-6940	443	30	.	.	PUNCT
ejpam-6940	444	1	let	let	AUX
ejpam-6940	444	2	consider	consider	VERB
ejpam-6940	444	3	the	the	DET
ejpam-6940	444	4	table	table	NOUN
ejpam-6940	444	5	a.	a.	NOUN
ejpam-6940	444	6	since	since	SCONJ
ejpam-6940	444	7	x1x3	x1x3	PROPN
ejpam-6940	444	8	=	=	PUNCT
ejpam-6940	444	9	{	{	PUNCT
ejpam-6940	444	10	x1	x1	PROPN
ejpam-6940	444	11	}	}	PUNCT
ejpam-6940	444	12	,	,	PUNCT
ejpam-6940	444	13	x2x3	x2x3	X
ejpam-6940	444	14	=	=	PUNCT
ejpam-6940	444	15	{	{	PUNCT
ejpam-6940	444	16	x1	x1	PROPN
ejpam-6940	444	17	,	,	PUNCT
ejpam-6940	444	18	x2	x2	PROPN
ejpam-6940	444	19	}	}	PUNCT
ejpam-6940	444	20	and	and	CCONJ
ejpam-6940	444	21	x3x3	x3x3	X
ejpam-6940	444	22	=	=	PRON
ejpam-6940	444	23	{	{	PUNCT
ejpam-6940	444	24	x1	x1	PROPN
ejpam-6940	444	25	,	,	PUNCT
ejpam-6940	444	26	x3	x3	ADJ
ejpam-6940	444	27	}	}	PUNCT
ejpam-6940	444	28	,	,	PUNCT
ejpam-6940	444	29	x3	x3	PROPN
ejpam-6940	444	30	is	be	AUX
ejpam-6940	444	31	an	an	DET
ejpam-6940	444	32	edge	edge	NOUN
ejpam-6940	444	33	q	q	NOUN
ejpam-6940	444	34	-	-	NOUN
ejpam-6940	444	35	algebra	algebra	NOUN
ejpam-6940	444	36	.	.	PUNCT
ejpam-6940	445	1	similarly	similarly	ADV
ejpam-6940	445	2	,	,	PUNCT
ejpam-6940	445	3	we	we	PRON
ejpam-6940	445	4	get	get	VERB
ejpam-6940	445	5	that	that	DET
ejpam-6940	445	6	tables	table	NOUN
ejpam-6940	445	7	b	b	NOUN
ejpam-6940	445	8	,	,	PUNCT
ejpam-6940	445	9	c	c	PROPN
ejpam-6940	445	10	and	and	CCONJ
ejpam-6940	445	11	d	d	PROPN
ejpam-6940	445	12	are	be	AUX
ejpam-6940	445	13	edge	edge	NOUN
ejpam-6940	445	14	q	q	NOUN
ejpam-6940	445	15	-	-	PUNCT
ejpam-6940	445	16	algebras	algebras	X
ejpam-6940	445	17	.	.	PUNCT
ejpam-6940	446	1	moreover	moreover	ADV
ejpam-6940	446	2	,	,	PUNCT
ejpam-6940	446	3	|eq(x3)|	|eq(x3)|	PROPN
ejpam-6940	446	4	=	=	NOUN
ejpam-6940	446	5	23	23	NUM
ejpam-6940	446	6	2−3(3)+2	2−3(3)+2	NUM
ejpam-6940	446	7	=	=	SYM
ejpam-6940	446	8	22	22	NUM
ejpam-6940	446	9	=	=	SYM
ejpam-6940	446	10	4	4	NUM
ejpam-6940	446	11	and	and	CCONJ
ejpam-6940	446	12	eq(x3	eq(x3	NOUN
ejpam-6940	446	13	)	)	PUNCT
ejpam-6940	446	14	=	=	PRON
ejpam-6940	446	15	{	{	PUNCT
ejpam-6940	446	16	a	a	DET
ejpam-6940	446	17	,	,	PUNCT
ejpam-6940	446	18	b	b	NOUN
ejpam-6940	446	19	,	,	PUNCT
ejpam-6940	446	20	c	c	X
ejpam-6940	446	21	,	,	PUNCT
ejpam-6940	446	22	d	d	NOUN
ejpam-6940	446	23	}	}	PUNCT
ejpam-6940	446	24	.	.	PUNCT
ejpam-6940	447	1	next	next	ADJ
ejpam-6940	447	2	proposition	proposition	NOUN
ejpam-6940	447	3	reveals	reveal	VERB
ejpam-6940	447	4	that	that	SCONJ
ejpam-6940	447	5	any	any	DET
ejpam-6940	447	6	algebraic	algebraic	ADJ
ejpam-6940	447	7	structure	structure	NOUN
ejpam-6940	447	8	obtained	obtain	VERB
ejpam-6940	447	9	from	from	ADP
ejpam-6940	447	10	the	the	DET
ejpam-6940	447	11	construction	construction	NOUN
ejpam-6940	447	12	(	(	PUNCT
ejpam-6940	447	13	∗	∗	NOUN
ejpam-6940	447	14	)	)	PUNCT
ejpam-6940	447	15	is	be	AUX
ejpam-6940	447	16	an	an	DET
ejpam-6940	447	17	edge	edge	NOUN
ejpam-6940	447	18	q	q	NOUN
ejpam-6940	447	19	-	-	NOUN
ejpam-6940	447	20	algebra	algebra	NOUN
ejpam-6940	447	21	.	.	PUNCT
ejpam-6940	448	1	proposition	proposition	NOUN
ejpam-6940	448	2	16	16	NUM
ejpam-6940	448	3	.	.	PUNCT
ejpam-6940	449	1	let	let	VERB
ejpam-6940	449	2	xn	xn	PUNCT
ejpam-6940	450	1	=	=	PUNCT
ejpam-6940	450	2	{	{	PUNCT
ejpam-6940	450	3	x1	x1	PROPN
ejpam-6940	450	4	,	,	PUNCT
ejpam-6940	450	5	x2	x2	PROPN
ejpam-6940	450	6	,	,	PUNCT
ejpam-6940	450	7	.	.	PUNCT
ejpam-6940	450	8	.	.	PUNCT
ejpam-6940	450	9	.	.	PUNCT
ejpam-6940	451	1	,	,	PUNCT
ejpam-6940	451	2	xn	xn	PROPN
ejpam-6940	451	3	}	}	PUNCT
ejpam-6940	451	4	.	.	PUNCT
ejpam-6940	452	1	for	for	ADP
ejpam-6940	452	2	any	any	DET
ejpam-6940	452	3	a	a	DET
ejpam-6940	452	4	∈	∈	PROPN
ejpam-6940	452	5	eq(xn	eq(xn	PROPN
ejpam-6940	452	6	)	)	PUNCT
ejpam-6940	452	7	,	,	PUNCT
ejpam-6940	452	8	a	a	PRON
ejpam-6940	452	9	is	be	AUX
ejpam-6940	452	10	an	an	DET
ejpam-6940	452	11	edge	edge	NOUN
ejpam-6940	452	12	qalgebra	qalgebra	NOUN
ejpam-6940	452	13	.	.	PUNCT
ejpam-6940	453	1	proof	proof	NOUN
ejpam-6940	453	2	.	.	PUNCT
ejpam-6940	454	1	let	let	VERB
ejpam-6940	454	2	a	a	DET
ejpam-6940	454	3	be	be	AUX
ejpam-6940	454	4	any	any	DET
ejpam-6940	454	5	algebra	algebra	NOUN
ejpam-6940	454	6	in	in	ADP
ejpam-6940	454	7	eq(xn	eq(xn	PROPN
ejpam-6940	454	8	)	)	PUNCT
ejpam-6940	454	9	.	.	PUNCT
ejpam-6940	455	1	let	let	VERB
ejpam-6940	455	2	xi	xi	PROPN
ejpam-6940	455	3	,	,	PUNCT
ejpam-6940	455	4	xj	xj	PROPN
ejpam-6940	455	5	,	,	PUNCT
ejpam-6940	455	6	xk	xk	PROPN
ejpam-6940	455	7	∈	∈	PROPN
ejpam-6940	456	1	xn	xn	PROPN
ejpam-6940	456	2	.	.	PUNCT
ejpam-6940	457	1	then	then	ADV
ejpam-6940	457	2	we	we	PRON
ejpam-6940	457	3	get	get	VERB
ejpam-6940	457	4	xix1	xix1	NOUN
ejpam-6940	457	5	=	=	PRON
ejpam-6940	457	6	xi	xi	PROPN
ejpam-6940	457	7	and	and	CCONJ
ejpam-6940	457	8	xixi	xixi	PROPN
ejpam-6940	457	9	=	=	SYM
ejpam-6940	457	10	x1	x1	PROPN
ejpam-6940	457	11	.	.	PUNCT
ejpam-6940	458	1	therefore	therefore	ADV
ejpam-6940	458	2	,	,	PUNCT
ejpam-6940	458	3	the	the	DET
ejpam-6940	458	4	conditions	condition	NOUN
ejpam-6940	458	5	(	(	PUNCT
ejpam-6940	458	6	q1	q1	PROPN
ejpam-6940	458	7	)	)	PUNCT
ejpam-6940	458	8	and	and	CCONJ
ejpam-6940	458	9	(	(	PUNCT
ejpam-6940	458	10	q2	q2	NOUN
ejpam-6940	458	11	)	)	PUNCT
ejpam-6940	458	12	hold	hold	VERB
ejpam-6940	458	13	and	and	CCONJ
ejpam-6940	458	14	x1	x1	NOUN
ejpam-6940	458	15	acts	act	VERB
ejpam-6940	458	16	as	as	ADP
ejpam-6940	458	17	a	a	DET
ejpam-6940	458	18	constant	constant	ADJ
ejpam-6940	458	19	0	0	NUM
ejpam-6940	458	20	.	.	PUNCT
ejpam-6940	459	1	next	next	ADJ
ejpam-6940	459	2	,	,	PUNCT
ejpam-6940	459	3	we	we	PRON
ejpam-6940	459	4	calculate	calculate	VERB
ejpam-6940	459	5	(	(	PUNCT
ejpam-6940	459	6	xixj)xk	xixj)xk	NUM
ejpam-6940	459	7	and	and	CCONJ
ejpam-6940	459	8	(	(	PUNCT
ejpam-6940	459	9	xixk)xj	xixk)xj	X
ejpam-6940	459	10	.	.	PUNCT
ejpam-6940	460	1	observe	observe	VERB
ejpam-6940	460	2	that	that	SCONJ
ejpam-6940	460	3	xixj	xixj	PROPN
ejpam-6940	460	4	∈	∈	PROPN
ejpam-6940	460	5	{	{	PUNCT
ejpam-6940	460	6	x1	x1	PROPN
ejpam-6940	460	7	,	,	PUNCT
ejpam-6940	460	8	xi	xi	ADJ
ejpam-6940	460	9	}	}	PUNCT
ejpam-6940	460	10	and	and	CCONJ
ejpam-6940	460	11	xixk	xixk	PROPN
ejpam-6940	460	12	∈	∈	PROPN
ejpam-6940	460	13	{	{	PUNCT
ejpam-6940	460	14	x1	x1	PROPN
ejpam-6940	460	15	,	,	PUNCT
ejpam-6940	460	16	xi	xi	ADJ
ejpam-6940	460	17	}	}	PUNCT
ejpam-6940	460	18	.	.	PUNCT
ejpam-6940	461	1	a.	a.	PROPN
ejpam-6940	461	2	anantayasethi	anantayasethi	PROPN
ejpam-6940	461	3	et	et	PROPN
ejpam-6940	461	4	al	al	PROPN
ejpam-6940	461	5	.	.	PUNCT
ejpam-6940	461	6	/	/	SYM
ejpam-6940	461	7	eur	eur	PROPN
ejpam-6940	461	8	.	.	PUNCT
ejpam-6940	462	1	j.	j.	PROPN
ejpam-6940	462	2	pure	pure	PROPN
ejpam-6940	462	3	appl	appl	PROPN
ejpam-6940	462	4	.	.	PROPN
ejpam-6940	462	5	math	math	PROPN
ejpam-6940	462	6	,	,	PUNCT
ejpam-6940	462	7	18	18	NUM
ejpam-6940	462	8	(	(	PUNCT
ejpam-6940	462	9	4	4	NUM
ejpam-6940	462	10	)	)	PUNCT
ejpam-6940	462	11	(	(	PUNCT
ejpam-6940	462	12	2025	2025	NUM
ejpam-6940	462	13	)	)	PUNCT
ejpam-6940	462	14	,	,	PUNCT
ejpam-6940	462	15	6940	6940	NUM
ejpam-6940	462	16	12	12	NUM
ejpam-6940	462	17	of	of	ADP
ejpam-6940	462	18	14	14	NUM
ejpam-6940	462	19	if	if	SCONJ
ejpam-6940	462	20	xixj	xixj	PROPN
ejpam-6940	462	21	=	=	SYM
ejpam-6940	462	22	x1	x1	PROPN
ejpam-6940	462	23	,	,	PUNCT
ejpam-6940	462	24	then	then	ADV
ejpam-6940	462	25	(	(	PUNCT
ejpam-6940	462	26	xixj)xk	xixj)xk	PUNCT
ejpam-6940	462	27	=	=	PUNCT
ejpam-6940	462	28	x1xk	x1xk	PUNCT
ejpam-6940	463	1	=	=	PUNCT
ejpam-6940	463	2	x1	x1	PROPN
ejpam-6940	463	3	and	and	CCONJ
ejpam-6940	463	4	(	(	PUNCT
ejpam-6940	463	5	xixk)xj	xixk)xj	NUM
ejpam-6940	463	6	∈	∈	PROPN
ejpam-6940	463	7	{	{	PUNCT
ejpam-6940	463	8	x1xj	x1xj	PROPN
ejpam-6940	463	9	,	,	PUNCT
ejpam-6940	463	10	xixj	xixj	PROPN
ejpam-6940	463	11	}	}	PUNCT
ejpam-6940	463	12	=	=	PUNCT
ejpam-6940	463	13	{	{	PUNCT
ejpam-6940	463	14	x1	x1	PROPN
ejpam-6940	463	15	}	}	PUNCT
ejpam-6940	463	16	.	.	PUNCT
ejpam-6940	464	1	it	it	PRON
ejpam-6940	464	2	follows	follow	VERB
ejpam-6940	464	3	that	that	PRON
ejpam-6940	464	4	(	(	PUNCT
ejpam-6940	464	5	xixj)xk	xixj)xk	PUNCT
ejpam-6940	464	6	=	=	PUNCT
ejpam-6940	465	1	x1	x1	PROPN
ejpam-6940	465	2	=	=	SYM
ejpam-6940	465	3	(	(	PUNCT
ejpam-6940	465	4	xixk)xj	xixk)xj	NUM
ejpam-6940	465	5	.	.	PUNCT
ejpam-6940	466	1	if	if	SCONJ
ejpam-6940	466	2	xixj	xixj	PROPN
ejpam-6940	466	3	=	=	SYM
ejpam-6940	466	4	xi	xi	PROPN
ejpam-6940	466	5	,	,	PUNCT
ejpam-6940	466	6	then	then	ADV
ejpam-6940	466	7	(	(	PUNCT
ejpam-6940	466	8	xixj)xk	xixj)xk	PUNCT
ejpam-6940	466	9	=	=	PUNCT
ejpam-6940	466	10	xixk	xixk	PROPN
ejpam-6940	466	11	.	.	PUNCT
ejpam-6940	467	1	if	if	SCONJ
ejpam-6940	467	2	xixk	xixk	PROPN
ejpam-6940	467	3	=	=	SYM
ejpam-6940	467	4	x1	x1	PROPN
ejpam-6940	467	5	,	,	PUNCT
ejpam-6940	467	6	then	then	ADV
ejpam-6940	467	7	(	(	PUNCT
ejpam-6940	467	8	xixj)xk	xixj)xk	PUNCT
ejpam-6940	467	9	=	=	SYM
ejpam-6940	467	10	xixk	xixk	PROPN
ejpam-6940	468	1	=	=	SYM
ejpam-6940	468	2	x1	x1	PROPN
ejpam-6940	468	3	=	=	PUNCT
ejpam-6940	468	4	x1xj	x1xj	PUNCT
ejpam-6940	469	1	=	=	SYM
ejpam-6940	469	2	(	(	PUNCT
ejpam-6940	469	3	xixk)xj	xixk)xj	NUM
ejpam-6940	469	4	.	.	PUNCT
ejpam-6940	470	1	if	if	SCONJ
ejpam-6940	470	2	xixk	xixk	PROPN
ejpam-6940	470	3	=	=	SYM
ejpam-6940	470	4	xi	xi	PROPN
ejpam-6940	470	5	,	,	PUNCT
ejpam-6940	470	6	then	then	ADV
ejpam-6940	470	7	(	(	PUNCT
ejpam-6940	470	8	xixj)xk	xixj)xk	PUNCT
ejpam-6940	470	9	=	=	SYM
ejpam-6940	470	10	xixk	xixk	PROPN
ejpam-6940	471	1	=	=	PRON
ejpam-6940	471	2	xi	xi	PROPN
ejpam-6940	471	3	=	=	PUNCT
ejpam-6940	471	4	xixj	xixj	PROPN
ejpam-6940	471	5	=	=	SYM
ejpam-6940	471	6	(	(	PUNCT
ejpam-6940	471	7	xixk)xj	xixk)xj	PROPN
ejpam-6940	471	8	.	.	PUNCT
ejpam-6940	472	1	altogether	altogether	ADV
ejpam-6940	472	2	,	,	PUNCT
ejpam-6940	472	3	(	(	PUNCT
ejpam-6940	472	4	q3	q3	PROPN
ejpam-6940	472	5	)	)	PUNCT
ejpam-6940	472	6	is	be	AUX
ejpam-6940	472	7	fulfilled	fulfil	VERB
ejpam-6940	472	8	.	.	PUNCT
ejpam-6940	473	1	since	since	SCONJ
ejpam-6940	473	2	xixn	xixn	ADJ
ejpam-6940	473	3	=	=	SYM
ejpam-6940	473	4	{	{	PUNCT
ejpam-6940	473	5	x1	x1	PROPN
ejpam-6940	473	6	,	,	PUNCT
ejpam-6940	473	7	xi	xi	ADP
ejpam-6940	473	8	}	}	PUNCT
ejpam-6940	473	9	for	for	ADP
ejpam-6940	473	10	all	all	PRON
ejpam-6940	473	11	xi	xi	ADP
ejpam-6940	473	12	∈	∈	PROPN
ejpam-6940	473	13	xn	xn	PROPN
ejpam-6940	473	14	,	,	PUNCT
ejpam-6940	473	15	then	then	ADV
ejpam-6940	473	16	the	the	DET
ejpam-6940	473	17	edge	edge	NOUN
ejpam-6940	473	18	property	property	NOUN
ejpam-6940	473	19	is	be	AUX
ejpam-6940	473	20	hold	hold	NOUN
ejpam-6940	473	21	.	.	PUNCT
ejpam-6940	474	1	altogether	altogether	ADV
ejpam-6940	474	2	,	,	PUNCT
ejpam-6940	474	3	a	a	PRON
ejpam-6940	474	4	is	be	AUX
ejpam-6940	474	5	an	an	DET
ejpam-6940	474	6	edge	edge	NOUN
ejpam-6940	474	7	q	q	NOUN
ejpam-6940	474	8	-	-	NOUN
ejpam-6940	474	9	algebra	algebra	NOUN
ejpam-6940	474	10	.	.	PUNCT
ejpam-6940	475	1	from	from	ADP
ejpam-6940	475	2	example	example	NOUN
ejpam-6940	475	3	7	7	NUM
ejpam-6940	475	4	,	,	PUNCT
ejpam-6940	475	5	if	if	SCONJ
ejpam-6940	475	6	we	we	PRON
ejpam-6940	475	7	replace	replace	VERB
ejpam-6940	475	8	x1	x1	NUM
ejpam-6940	475	9	by	by	ADP
ejpam-6940	475	10	0	0	NUM
ejpam-6940	475	11	,	,	PUNCT
ejpam-6940	475	12	then	then	ADV
ejpam-6940	475	13	we	we	PRON
ejpam-6940	475	14	get	get	VERB
ejpam-6940	475	15	the	the	DET
ejpam-6940	475	16	following	follow	VERB
ejpam-6940	475	17	edge	edge	NOUN
ejpam-6940	475	18	q	q	NOUN
ejpam-6940	475	19	-	-	PUNCT
ejpam-6940	475	20	algebras	algebras	X
ejpam-6940	475	21	:	:	PUNCT
ejpam-6940	476	1	0	0	NUM
ejpam-6940	476	2	x2	x2	NOUN
ejpam-6940	476	3	x3	x3	ADJ
ejpam-6940	476	4	0	0	NUM
ejpam-6940	476	5	0	0	NUM
ejpam-6940	476	6	0	0	NUM
ejpam-6940	476	7	0	0	NUM
ejpam-6940	477	1	x2	x2	NOUN
ejpam-6940	477	2	x2	x2	NOUN
ejpam-6940	477	3	0	0	NUM
ejpam-6940	477	4	0	0	NUM
ejpam-6940	478	1	x3	x3	ADJ
ejpam-6940	478	2	x3	x3	ADJ
ejpam-6940	478	3	0	0	NUM
ejpam-6940	478	4	0	0	NUM
ejpam-6940	478	5	0	0	NUM
ejpam-6940	479	1	x2	x2	NOUN
ejpam-6940	479	2	x3	x3	ADJ
ejpam-6940	479	3	0	0	NUM
ejpam-6940	479	4	0	0	NUM
ejpam-6940	479	5	0	0	NUM
ejpam-6940	479	6	0	0	NUM
ejpam-6940	480	1	x2	x2	NOUN
ejpam-6940	480	2	x2	x2	NOUN
ejpam-6940	480	3	0	0	NUM
ejpam-6940	480	4	0	0	NUM
ejpam-6940	481	1	x3	x3	ADJ
ejpam-6940	481	2	x3	x3	ADJ
ejpam-6940	481	3	x3	x3	ADJ
ejpam-6940	481	4	0	0	PUNCT
ejpam-6940	482	1	a	a	DET
ejpam-6940	482	2	b	b	PROPN
ejpam-6940	482	3	0	0	NUM
ejpam-6940	482	4	x2	x2	NOUN
ejpam-6940	482	5	x3	x3	ADJ
ejpam-6940	482	6	0	0	NUM
ejpam-6940	482	7	0	0	NUM
ejpam-6940	482	8	0	0	NUM
ejpam-6940	482	9	0	0	NUM
ejpam-6940	483	1	x2	x2	NOUN
ejpam-6940	483	2	x2	x2	NOUN
ejpam-6940	483	3	0	0	PUNCT
ejpam-6940	484	1	x2	x2	INTJ
ejpam-6940	484	2	x3	x3	ADJ
ejpam-6940	484	3	x3	x3	VERB
ejpam-6940	484	4	0	0	NUM
ejpam-6940	484	5	0	0	NUM
ejpam-6940	484	6	0	0	NUM
ejpam-6940	485	1	x2	x2	NOUN
ejpam-6940	485	2	x3	x3	ADJ
ejpam-6940	485	3	0	0	NUM
ejpam-6940	485	4	0	0	NUM
ejpam-6940	485	5	0	0	NUM
ejpam-6940	485	6	0	0	NUM
ejpam-6940	486	1	x2	x2	NOUN
ejpam-6940	486	2	x2	x2	NOUN
ejpam-6940	486	3	0	0	PUNCT
ejpam-6940	487	1	x2	x2	INTJ
ejpam-6940	487	2	x3	x3	ADJ
ejpam-6940	487	3	x3	x3	ADJ
ejpam-6940	487	4	x3	x3	ADJ
ejpam-6940	487	5	0	0	PUNCT
ejpam-6940	488	1	c	c	NOUN
ejpam-6940	488	2	d	d	NOUN
ejpam-6940	488	3	let	let	AUX
ejpam-6940	488	4	consider	consider	VERB
ejpam-6940	488	5	an	an	DET
ejpam-6940	488	6	edge	edge	NOUN
ejpam-6940	488	7	q	q	NOUN
ejpam-6940	488	8	algebra	algebra	NOUN
ejpam-6940	488	9	x	x	PUNCT
ejpam-6940	488	10	of	of	ADP
ejpam-6940	488	11	order	order	NOUN
ejpam-6940	488	12	n	n	CCONJ
ejpam-6940	488	13	,	,	PUNCT
ejpam-6940	488	14	for	for	ADP
ejpam-6940	488	15	any	any	DET
ejpam-6940	488	16	positive	positive	ADJ
ejpam-6940	488	17	integer	integer	NOUN
ejpam-6940	488	18	n.	n.	NOUN
ejpam-6940	488	19	for	for	ADP
ejpam-6940	488	20	any	any	DET
ejpam-6940	488	21	element	element	NOUN
ejpam-6940	488	22	a	a	DET
ejpam-6940	488	23	∈	∈	PROPN
ejpam-6940	488	24	x	x	NOUN
ejpam-6940	488	25	,	,	PUNCT
ejpam-6940	488	26	ax	ax	NOUN
ejpam-6940	488	27	=	=	SYM
ejpam-6940	488	28	{	{	PUNCT
ejpam-6940	488	29	0	0	NUM
ejpam-6940	488	30	,	,	PUNCT
ejpam-6940	488	31	a	a	PRON
ejpam-6940	488	32	}	}	PUNCT
ejpam-6940	488	33	.	.	PUNCT
ejpam-6940	489	1	there	there	PRON
ejpam-6940	489	2	follows	follow	VERB
ejpam-6940	489	3	that	that	SCONJ
ejpam-6940	489	4	ax	ax	NOUN
ejpam-6940	489	5	∈	∈	PROPN
ejpam-6940	489	6	{	{	PUNCT
ejpam-6940	489	7	0	0	NUM
ejpam-6940	489	8	,	,	PUNCT
ejpam-6940	489	9	a	a	PRON
ejpam-6940	489	10	}	}	PUNCT
ejpam-6940	489	11	for	for	ADP
ejpam-6940	489	12	all	all	PRON
ejpam-6940	489	13	x	x	SYM
ejpam-6940	489	14	∈	∈	ADJ
ejpam-6940	489	15	x.	x.	NOUN
ejpam-6940	489	16	hence	hence	ADV
ejpam-6940	489	17	,	,	PUNCT
ejpam-6940	489	18	there	there	PRON
ejpam-6940	489	19	is	be	VERB
ejpam-6940	489	20	an	an	DET
ejpam-6940	489	21	algebraic	algebraic	ADJ
ejpam-6940	489	22	structure	structure	NOUN
ejpam-6940	489	23	y	y	PROPN
ejpam-6940	489	24	∈	∈	PROPN
ejpam-6940	489	25	eq(xn	eq(xn	PROPN
ejpam-6940	489	26	)	)	PUNCT
ejpam-6940	489	27	which	which	PRON
ejpam-6940	489	28	is	be	AUX
ejpam-6940	489	29	coinciding	coincide	VERB
ejpam-6940	489	30	to	to	PART
ejpam-6940	489	31	x.	x.	NOUN
ejpam-6940	489	32	proposition	proposition	PROPN
ejpam-6940	489	33	17	17	NUM
ejpam-6940	489	34	.	.	PUNCT
ejpam-6940	490	1	let	let	VERB
ejpam-6940	490	2	n	n	PRON
ejpam-6940	490	3	be	be	AUX
ejpam-6940	490	4	a	a	DET
ejpam-6940	490	5	positive	positive	ADJ
ejpam-6940	490	6	integer	integer	NOUN
ejpam-6940	490	7	and	and	CCONJ
ejpam-6940	490	8	let	let	VERB
ejpam-6940	490	9	y	y	PRON
ejpam-6940	490	10	be	be	AUX
ejpam-6940	490	11	an	an	DET
ejpam-6940	490	12	edge	edge	NOUN
ejpam-6940	490	13	q	q	NOUN
ejpam-6940	490	14	-	-	NOUN
ejpam-6940	490	15	algebra	algebra	NOUN
ejpam-6940	490	16	of	of	ADP
ejpam-6940	490	17	order	order	NOUN
ejpam-6940	490	18	n.	n.	NOUN
ejpam-6940	490	19	then	then	ADV
ejpam-6940	490	20	y	y	PROPN
ejpam-6940	490	21	is	be	AUX
ejpam-6940	490	22	isomorphic	isomorphic	ADJ
ejpam-6940	490	23	to	to	ADP
ejpam-6940	490	24	x	x	PUNCT
ejpam-6940	490	25	for	for	ADP
ejpam-6940	490	26	some	some	DET
ejpam-6940	490	27	x	x	SYM
ejpam-6940	490	28	∈	∈	PROPN
ejpam-6940	490	29	eq(xn	eq(xn	PROPN
ejpam-6940	490	30	)	)	PUNCT
ejpam-6940	490	31	.	.	PUNCT
ejpam-6940	491	1	combining	combine	VERB
ejpam-6940	491	2	proposition	proposition	NOUN
ejpam-6940	491	3	16	16	NUM
ejpam-6940	491	4	and	and	CCONJ
ejpam-6940	491	5	proposition	proposition	NOUN
ejpam-6940	491	6	17	17	NUM
ejpam-6940	491	7	we	we	PRON
ejpam-6940	491	8	get	get	AUX
ejpam-6940	491	9	:	:	PUNCT
ejpam-6940	491	10	theorem	theorem	NOUN
ejpam-6940	491	11	1	1	NUM
ejpam-6940	491	12	.	.	PUNCT
ejpam-6940	491	13	eq(xn	eq(xn	NOUN
ejpam-6940	491	14	)	)	PUNCT
ejpam-6940	491	15	is	be	AUX
ejpam-6940	491	16	the	the	DET
ejpam-6940	491	17	set	set	NOUN
ejpam-6940	491	18	of	of	ADP
ejpam-6940	491	19	all	all	DET
ejpam-6940	491	20	edge	edge	NOUN
ejpam-6940	491	21	q	q	NOUN
ejpam-6940	491	22	-	-	PUNCT
ejpam-6940	491	23	algebras	algebra	NOUN
ejpam-6940	491	24	of	of	ADP
ejpam-6940	491	25	order	order	NOUN
ejpam-6940	491	26	n	n	NOUN
ejpam-6940	491	27	and	and	CCONJ
ejpam-6940	491	28	hence	hence	ADV
ejpam-6940	491	29	,	,	PUNCT
ejpam-6940	491	30	there	there	PRON
ejpam-6940	491	31	are	be	VERB
ejpam-6940	491	32	precisely	precisely	ADV
ejpam-6940	491	33	2n	2n	NUM
ejpam-6940	491	34	2−3n+2	2−3n+2	NUM
ejpam-6940	491	35	different	different	ADJ
ejpam-6940	491	36	edge	edge	NOUN
ejpam-6940	491	37	q	q	NOUN
ejpam-6940	491	38	-	-	PUNCT
ejpam-6940	491	39	algebras	algebra	NOUN
ejpam-6940	491	40	of	of	ADP
ejpam-6940	491	41	order	order	NOUN
ejpam-6940	491	42	n.	n.	NOUN
ejpam-6940	491	43	4	4	X
ejpam-6940	491	44	.	.	PUNCT
ejpam-6940	491	45	conclusion	conclusion	NOUN
ejpam-6940	491	46	we	we	PRON
ejpam-6940	491	47	have	have	AUX
ejpam-6940	491	48	introduced	introduce	VERB
ejpam-6940	491	49	the	the	DET
ejpam-6940	491	50	concept	concept	NOUN
ejpam-6940	491	51	of	of	ADP
ejpam-6940	491	52	edge	edge	NOUN
ejpam-6940	491	53	q	q	NOUN
ejpam-6940	491	54	-	-	PUNCT
ejpam-6940	491	55	algebras	algebra	VERB
ejpam-6940	491	56	and	and	CCONJ
ejpam-6940	491	57	explored	explore	VERB
ejpam-6940	491	58	their	their	PRON
ejpam-6940	491	59	properties	property	NOUN
ejpam-6940	491	60	.	.	PUNCT
ejpam-6940	492	1	we	we	PRON
ejpam-6940	492	2	obtained	obtain	VERB
ejpam-6940	492	3	some	some	DET
ejpam-6940	492	4	results	result	NOUN
ejpam-6940	492	5	related	relate	VERB
ejpam-6940	492	6	to	to	ADP
ejpam-6940	492	7	the	the	DET
ejpam-6940	492	8	concepts	concept	NOUN
ejpam-6940	492	9	of	of	ADP
ejpam-6940	492	10	subalgebras	subalgebra	NOUN
ejpam-6940	492	11	and	and	CCONJ
ejpam-6940	492	12	ideals	ideal	NOUN
ejpam-6940	492	13	.	.	PUNCT
ejpam-6940	493	1	the	the	DET
ejpam-6940	493	2	product	product	NOUN
ejpam-6940	493	3	of	of	ADP
ejpam-6940	493	4	subalgebras	subalgebra	NOUN
ejpam-6940	493	5	of	of	ADP
ejpam-6940	493	6	an	an	DET
ejpam-6940	493	7	edge	edge	NOUN
ejpam-6940	493	8	q	q	NOUN
ejpam-6940	493	9	-	-	PUNCT
ejpam-6940	493	10	algebra	algebra	NOUN
ejpam-6940	493	11	is	be	AUX
ejpam-6940	493	12	again	again	ADV
ejpam-6940	493	13	a	a	DET
ejpam-6940	493	14	subalgebra	subalgebra	NOUN
ejpam-6940	493	15	,	,	PUNCT
ejpam-6940	493	16	offered	offer	VERB
ejpam-6940	493	17	in	in	ADP
ejpam-6940	493	18	corollary	corollary	ADJ
ejpam-6940	493	19	1	1	NUM
ejpam-6940	493	20	.	.	PUNCT
ejpam-6940	493	21	similarly	similarly	ADV
ejpam-6940	493	22	,	,	PUNCT
ejpam-6940	493	23	the	the	DET
ejpam-6940	493	24	product	product	NOUN
ejpam-6940	493	25	of	of	ADP
ejpam-6940	493	26	ideals	ideal	NOUN
ejpam-6940	493	27	of	of	ADP
ejpam-6940	493	28	edge	edge	NOUN
ejpam-6940	493	29	q	q	NOUN
ejpam-6940	493	30	-	-	PUNCT
ejpam-6940	493	31	algebra	algebra	NOUN
ejpam-6940	493	32	is	be	AUX
ejpam-6940	493	33	also	also	ADV
ejpam-6940	493	34	an	an	DET
ejpam-6940	493	35	ideal	ideal	NOUN
ejpam-6940	493	36	.	.	PUNCT
ejpam-6940	494	1	moreover	moreover	ADV
ejpam-6940	494	2	,	,	PUNCT
ejpam-6940	494	3	the	the	DET
ejpam-6940	494	4	set	set	NOUN
ejpam-6940	494	5	of	of	ADP
ejpam-6940	494	6	all	all	DET
ejpam-6940	494	7	subalgebras	subalgebra	NOUN
ejpam-6940	494	8	of	of	ADP
ejpam-6940	494	9	an	an	DET
ejpam-6940	494	10	edge	edge	NOUN
ejpam-6940	494	11	q	q	NOUN
ejpam-6940	494	12	-	-	PUNCT
ejpam-6940	494	13	algebra	algebra	NOUN
ejpam-6940	494	14	x	x	NOUN
ejpam-6940	494	15	,	,	PUNCT
ejpam-6940	494	16	sub(x	sub(x	PROPN
ejpam-6940	494	17	)	)	PUNCT
ejpam-6940	494	18	,	,	PUNCT
ejpam-6940	494	19	and	and	CCONJ
ejpam-6940	494	20	the	the	DET
ejpam-6940	494	21	set	set	NOUN
ejpam-6940	494	22	of	of	ADP
ejpam-6940	494	23	all	all	DET
ejpam-6940	494	24	ideals	ideal	NOUN
ejpam-6940	494	25	,	,	PUNCT
ejpam-6940	494	26	i(x	i(x	NOUN
ejpam-6940	494	27	)	)	PUNCT
ejpam-6940	494	28	form	form	VERB
ejpam-6940	494	29	a	a	DET
ejpam-6940	494	30	semigroup	semigroup	NOUN
ejpam-6940	494	31	as	as	SCONJ
ejpam-6940	494	32	shown	show	VERB
ejpam-6940	494	33	in	in	ADP
ejpam-6940	494	34	proposition	proposition	NOUN
ejpam-6940	494	35	9	9	NUM
ejpam-6940	494	36	and	and	CCONJ
ejpam-6940	494	37	proposition	proposition	NOUN
ejpam-6940	494	38	12	12	NUM
ejpam-6940	494	39	,	,	PUNCT
ejpam-6940	494	40	respectively	respectively	ADV
ejpam-6940	494	41	.	.	PUNCT
ejpam-6940	495	1	we	we	PRON
ejpam-6940	495	2	enumerated	enumerate	VERB
ejpam-6940	495	3	all	all	PRON
ejpam-6940	495	4	of	of	ADP
ejpam-6940	495	5	subalgebras	subalgebra	NOUN
ejpam-6940	495	6	of	of	ADP
ejpam-6940	495	7	edge	edge	NOUN
ejpam-6940	495	8	q	q	NOUN
ejpam-6940	495	9	-	-	PUNCT
ejpam-6940	495	10	algebra	algebra	ADJ
ejpam-6940	495	11	x.	x.	NOUN
ejpam-6940	495	12	proposition	proposition	NOUN
ejpam-6940	495	13	8	8	NUM
ejpam-6940	495	14	shows	show	VERB
ejpam-6940	495	15	the	the	DET
ejpam-6940	495	16	total	total	ADJ
ejpam-6940	495	17	number	number	NOUN
ejpam-6940	495	18	of	of	ADP
ejpam-6940	495	19	all	all	DET
ejpam-6940	495	20	subalgebras	subalgebra	NOUN
ejpam-6940	495	21	of	of	ADP
ejpam-6940	495	22	x	x	X
ejpam-6940	495	23	and	and	CCONJ
ejpam-6940	495	24	|sub(x)|	|sub(x)|	NOUN
ejpam-6940	495	25	=	=	SYM
ejpam-6940	496	1	2|x|−1	2|x|−1	NOUN
ejpam-6940	496	2	.	.	PUNCT
ejpam-6940	497	1	the	the	DET
ejpam-6940	497	2	construction	construction	NOUN
ejpam-6940	497	3	of	of	ADP
ejpam-6940	497	4	edge	edge	NOUN
ejpam-6940	497	5	q	q	NOUN
ejpam-6940	497	6	-	-	PUNCT
ejpam-6940	497	7	algebras	algebras	PROPN
ejpam-6940	497	8	is	be	AUX
ejpam-6940	497	9	presented	present	VERB
ejpam-6940	497	10	.	.	PUNCT
ejpam-6940	498	1	we	we	PRON
ejpam-6940	498	2	also	also	ADV
ejpam-6940	498	3	showed	show	VERB
ejpam-6940	498	4	the	the	DET
ejpam-6940	498	5	total	total	ADJ
ejpam-6940	498	6	number	number	NOUN
ejpam-6940	498	7	of	of	ADP
ejpam-6940	498	8	all	all	DET
ejpam-6940	498	9	structures	structure	NOUN
ejpam-6940	498	10	of	of	ADP
ejpam-6940	498	11	edge	edge	NOUN
ejpam-6940	498	12	q	q	NOUN
ejpam-6940	498	13	-	-	PUNCT
ejpam-6940	498	14	algebras	algebra	NOUN
ejpam-6940	498	15	,	,	PUNCT
ejpam-6940	498	16	as	as	ADP
ejpam-6940	498	17	in	in	ADP
ejpam-6940	498	18	theorem	theorem	NOUN
ejpam-6940	498	19	1	1	X
ejpam-6940	498	20	.	.	X
ejpam-6940	499	1	there	there	PRON
ejpam-6940	499	2	are	be	VERB
ejpam-6940	499	3	2n	2n	NUM
ejpam-6940	499	4	2−3n+2	2−3n+2	NUM
ejpam-6940	499	5	structures	structure	NOUN
ejpam-6940	499	6	of	of	ADP
ejpam-6940	499	7	edge	edge	NOUN
ejpam-6940	499	8	q	q	NOUN
ejpam-6940	499	9	-	-	PUNCT
ejpam-6940	499	10	algebras	algebra	NOUN
ejpam-6940	499	11	of	of	ADP
ejpam-6940	499	12	order	order	NOUN
ejpam-6940	499	13	n.	n.	NOUN
ejpam-6940	499	14	for	for	ADP
ejpam-6940	499	15	further	further	ADJ
ejpam-6940	499	16	study	study	NOUN
ejpam-6940	499	17	,	,	PUNCT
ejpam-6940	499	18	one	one	PRON
ejpam-6940	499	19	can	can	AUX
ejpam-6940	499	20	a.	a.	NOUN
ejpam-6940	499	21	anantayasethi	anantayasethi	PROPN
ejpam-6940	499	22	et	et	PROPN
ejpam-6940	499	23	al	al	PROPN
ejpam-6940	499	24	.	.	PUNCT
ejpam-6940	499	25	/	/	SYM
ejpam-6940	499	26	eur	eur	PROPN
ejpam-6940	499	27	.	.	PUNCT
ejpam-6940	500	1	j.	j.	PROPN
ejpam-6940	500	2	pure	pure	PROPN
ejpam-6940	500	3	appl	appl	PROPN
ejpam-6940	500	4	.	.	PROPN
ejpam-6940	500	5	math	math	PROPN
ejpam-6940	500	6	,	,	PUNCT
ejpam-6940	500	7	18	18	NUM
ejpam-6940	500	8	(	(	PUNCT
ejpam-6940	500	9	4	4	NUM
ejpam-6940	500	10	)	)	PUNCT
ejpam-6940	500	11	(	(	PUNCT
ejpam-6940	500	12	2025	2025	NUM
ejpam-6940	500	13	)	)	PUNCT
ejpam-6940	500	14	,	,	PUNCT
ejpam-6940	500	15	6940	6940	NUM
ejpam-6940	500	16	13	13	NUM
ejpam-6940	500	17	of	of	ADP
ejpam-6940	500	18	14	14	NUM
ejpam-6940	500	19	consider	consider	VERB
ejpam-6940	500	20	edge	edge	VERB
ejpam-6940	500	21	q	q	NOUN
ejpam-6940	500	22	-	-	PUNCT
ejpam-6940	500	23	algebras	algebras	PROPN
ejpam-6940	500	24	based	base	VERB
ejpam-6940	500	25	on	on	ADP
ejpam-6940	500	26	the	the	DET
ejpam-6940	500	27	following	follow	VERB
ejpam-6940	500	28	concepts	concept	NOUN
ejpam-6940	500	29	:	:	PUNCT
ejpam-6940	500	30	hyper	hyper	ADJ
ejpam-6940	500	31	algebras	algebra	NOUN
ejpam-6940	500	32	;	;	PUNCT
ejpam-6940	500	33	filters	filter	NOUN
ejpam-6940	500	34	and	and	CCONJ
ejpam-6940	500	35	another	another	DET
ejpam-6940	500	36	kinds	kind	NOUN
ejpam-6940	500	37	of	of	ADP
ejpam-6940	500	38	ideals	ideal	NOUN
ejpam-6940	500	39	;	;	PUNCT
ejpam-6940	500	40	fuzzy	fuzzy	ADJ
ejpam-6940	500	41	subalgebras	subalgebra	NOUN
ejpam-6940	500	42	,	,	PUNCT
ejpam-6940	500	43	fuzzy	fuzzy	ADJ
ejpam-6940	500	44	ideals	ideal	NOUN
ejpam-6940	500	45	;	;	PUNCT
ejpam-6940	500	46	homomorphisms	homomorphism	NOUN
ejpam-6940	500	47	and	and	CCONJ
ejpam-6940	500	48	isomorphisms	isomorphism	NOUN
ejpam-6940	500	49	;	;	PUNCT
ejpam-6940	500	50	connections	connection	NOUN
ejpam-6940	500	51	with	with	ADP
ejpam-6940	500	52	related	relate	VERB
ejpam-6940	500	53	algebras	algebra	NOUN
ejpam-6940	500	54	.	.	PUNCT
ejpam-6940	501	1	acknowledgements	acknowledgement	NOUN
ejpam-6940	501	2	this	this	DET
ejpam-6940	501	3	research	research	NOUN
ejpam-6940	501	4	project	project	NOUN
ejpam-6940	501	5	was	be	AUX
ejpam-6940	501	6	financially	financially	ADV
ejpam-6940	501	7	supported	support	VERB
ejpam-6940	501	8	by	by	ADP
ejpam-6940	501	9	mahasarakham	mahasarakham	PROPN
ejpam-6940	501	10	university	university	PROPN
ejpam-6940	501	11	,	,	PUNCT
ejpam-6940	501	12	thailand	thailand	PROPN
ejpam-6940	501	13	.	.	PUNCT
ejpam-6940	502	1	references	reference	NOUN
ejpam-6940	502	2	[	[	X
ejpam-6940	502	3	1	1	NUM
ejpam-6940	502	4	]	]	X
ejpam-6940	502	5	y.	y.	PROPN
ejpam-6940	502	6	imai	imai	PROPN
ejpam-6940	502	7	and	and	CCONJ
ejpam-6940	502	8	k.	k.	PROPN
ejpam-6940	502	9	iseki	iseki	PROPN
ejpam-6940	502	10	.	.	PUNCT
ejpam-6940	503	1	on	on	ADP
ejpam-6940	503	2	axiom	axiom	NOUN
ejpam-6940	503	3	system	system	NOUN
ejpam-6940	503	4	of	of	ADP
ejpam-6940	503	5	proposaitional	proposaitional	ADJ
ejpam-6940	503	6	calculi	calculi	PROPN
ejpam-6940	503	7	.	.	PUNCT
ejpam-6940	504	1	xiv	xiv	PROPN
ejpam-6940	504	2	.	.	PUNCT
ejpam-6940	505	1	proceedings	proceeding	NOUN
ejpam-6940	505	2	of	of	ADP
ejpam-6940	505	3	the	the	DET
ejpam-6940	505	4	japan	japan	PROPN
ejpam-6940	505	5	academy	academy	PROPN
ejpam-6940	505	6	,	,	PUNCT
ejpam-6940	505	7	42:19–22	42:19–22	NUM
ejpam-6940	505	8	,	,	PUNCT
ejpam-6940	505	9	1966	1966	NUM
ejpam-6940	505	10	.	.	PUNCT
ejpam-6940	506	1	[	[	X
ejpam-6940	506	2	2	2	NUM
ejpam-6940	506	3	]	]	PUNCT
ejpam-6940	506	4	k.	k.	PROPN
ejpam-6940	506	5	iseki	iseki	PROPN
ejpam-6940	506	6	.	.	PUNCT
ejpam-6940	507	1	an	an	DET
ejpam-6940	507	2	algebra	algebra	NOUN
ejpam-6940	507	3	related	relate	VERB
ejpam-6940	507	4	with	with	ADP
ejpam-6940	507	5	a	a	DET
ejpam-6940	507	6	propositional	propositional	ADJ
ejpam-6940	507	7	calculus	calculus	NOUN
ejpam-6940	507	8	.	.	PUNCT
ejpam-6940	508	1	proceedings	proceeding	NOUN
ejpam-6940	508	2	of	of	ADP
ejpam-6940	508	3	the	the	DET
ejpam-6940	508	4	japan	japan	PROPN
ejpam-6940	508	5	academy	academy	PROPN
ejpam-6940	508	6	,	,	PUNCT
ejpam-6940	508	7	42:26–29	42:26–29	PROPN
ejpam-6940	508	8	,	,	PUNCT
ejpam-6940	508	9	1966	1966	NUM
ejpam-6940	508	10	.	.	PUNCT
ejpam-6940	509	1	[	[	X
ejpam-6940	509	2	3	3	X
ejpam-6940	509	3	]	]	PUNCT
ejpam-6940	509	4	k.	k.	PROPN
ejpam-6940	509	5	iseki	iseki	PROPN
ejpam-6940	509	6	and	and	CCONJ
ejpam-6940	509	7	s.	s.	PROPN
ejpam-6940	509	8	tanaka	tanaka	PROPN
ejpam-6940	509	9	.	.	PUNCT
ejpam-6940	510	1	an	an	DET
ejpam-6940	510	2	introduction	introduction	NOUN
ejpam-6940	510	3	to	to	ADP
ejpam-6940	510	4	theory	theory	NOUN
ejpam-6940	510	5	of	of	ADP
ejpam-6940	510	6	bck	bck	NOUN
ejpam-6940	510	7	-	-	PUNCT
ejpam-6940	510	8	algebra	algebra	NOUN
ejpam-6940	510	9	.	.	PUNCT
ejpam-6940	511	1	mathematica	mathematica	PROPN
ejpam-6940	511	2	japonica	japonica	PROPN
ejpam-6940	511	3	,	,	PUNCT
ejpam-6940	511	4	23:1–26	23:1–26	NUM
ejpam-6940	511	5	,	,	PUNCT
ejpam-6940	511	6	1978	1978	NUM
ejpam-6940	511	7	.	.	PUNCT
ejpam-6940	512	1	[	[	X
ejpam-6940	512	2	4	4	NUM
ejpam-6940	512	3	]	]	PUNCT
ejpam-6940	512	4	q.	q.	PROPN
ejpam-6940	512	5	p.	p.	PROPN
ejpam-6940	512	6	hu	hu	PROPN
ejpam-6940	513	1	and	and	CCONJ
ejpam-6940	514	1	x.	x.	PROPN
ejpam-6940	514	2	li	li	PROPN
ejpam-6940	514	3	.	.	PROPN
ejpam-6940	515	1	on	on	ADP
ejpam-6940	515	2	bch	bch	PROPN
ejpam-6940	515	3	-	-	PUNCT
ejpam-6940	515	4	algebras	algebras	PROPN
ejpam-6940	515	5	.	.	PUNCT
ejpam-6940	516	1	mathematics	mathematic	NOUN
ejpam-6940	516	2	seminar	seminar	NOUN
ejpam-6940	516	3	notes	note	VERB
ejpam-6940	516	4	kobe	kobe	PROPN
ejpam-6940	516	5	university	university	PROPN
ejpam-6940	516	6	,	,	PUNCT
ejpam-6940	516	7	2:313–320	2:313–320	NUM
ejpam-6940	516	8	,	,	PUNCT
ejpam-6940	516	9	1983	1983	NUM
ejpam-6940	516	10	.	.	PUNCT
ejpam-6940	517	1	[	[	X
ejpam-6940	517	2	5	5	NUM
ejpam-6940	517	3	]	]	PUNCT
ejpam-6940	517	4	q.	q.	PROPN
ejpam-6940	517	5	p.	p.	PROPN
ejpam-6940	517	6	hu	hu	PROPN
ejpam-6940	518	1	and	and	CCONJ
ejpam-6940	519	1	x.	x.	PROPN
ejpam-6940	519	2	li	li	PROPN
ejpam-6940	519	3	.	.	PROPN
ejpam-6940	520	1	on	on	ADP
ejpam-6940	520	2	proper	proper	ADJ
ejpam-6940	520	3	bch	bch	NOUN
ejpam-6940	520	4	-	-	PUNCT
ejpam-6940	520	5	algebras	algebras	PROPN
ejpam-6940	520	6	.	.	PUNCT
ejpam-6940	521	1	mathematica	mathematica	PROPN
ejpam-6940	521	2	japonica	japonica	PROPN
ejpam-6940	521	3	,	,	PUNCT
ejpam-6940	521	4	4:659–661	4:659–661	NUM
ejpam-6940	521	5	,	,	PUNCT
ejpam-6940	521	6	1983	1983	NUM
ejpam-6940	521	7	.	.	PUNCT
ejpam-6940	522	1	[	[	X
ejpam-6940	522	2	6	6	NUM
ejpam-6940	522	3	]	]	X
ejpam-6940	522	4	y.	y.	PROPN
ejpam-6940	522	5	komori	komori	PROPN
ejpam-6940	522	6	.	.	PUNCT
ejpam-6940	523	1	the	the	DET
ejpam-6940	523	2	class	class	NOUN
ejpam-6940	523	3	of	of	ADP
ejpam-6940	523	4	bcc	bcc	PROPN
ejpam-6940	523	5	-	-	PUNCT
ejpam-6940	523	6	algebras	algebras	PROPN
ejpam-6940	523	7	is	be	AUX
ejpam-6940	523	8	not	not	PART
ejpam-6940	523	9	a	a	DET
ejpam-6940	523	10	variety	variety	NOUN
ejpam-6940	523	11	.	.	PUNCT
ejpam-6940	524	1	mathematica	mathematica	PROPN
ejpam-6940	524	2	japonica	japonica	PROPN
ejpam-6940	524	3	,	,	PUNCT
ejpam-6940	524	4	29:391	29:391	NUM
ejpam-6940	524	5	–	–	PUNCT
ejpam-6940	524	6	394	394	NUM
ejpam-6940	524	7	,	,	PUNCT
ejpam-6940	524	8	1984	1984	NUM
ejpam-6940	524	9	.	.	PUNCT
ejpam-6940	525	1	[	[	X
ejpam-6940	525	2	7	7	X
ejpam-6940	525	3	]	]	X
ejpam-6940	525	4	w.	w.	PROPN
ejpam-6940	525	5	dudek	dudek	PROPN
ejpam-6940	525	6	.	.	PUNCT
ejpam-6940	526	1	on	on	ADP
ejpam-6940	526	2	proper	proper	ADJ
ejpam-6940	526	3	bcc	bcc	PROPN
ejpam-6940	526	4	-	-	PUNCT
ejpam-6940	526	5	algebras	algebras	PROPN
ejpam-6940	526	6	.	.	PUNCT
ejpam-6940	527	1	bulletin	bulletin	NOUN
ejpam-6940	527	2	of	of	ADP
ejpam-6940	527	3	the	the	DET
ejpam-6940	527	4	institute	institute	PROPN
ejpam-6940	527	5	of	of	ADP
ejpam-6940	527	6	mathematics	mathematics	PROPN
ejpam-6940	527	7	academia	academia	PROPN
ejpam-6940	527	8	sinica	sinica	PROPN
ejpam-6940	527	9	,	,	PUNCT
ejpam-6940	527	10	20:137–150	20:137–150	PROPN
ejpam-6940	527	11	,	,	PUNCT
ejpam-6940	527	12	1992	1992	NUM
ejpam-6940	527	13	.	.	PUNCT
ejpam-6940	528	1	[	[	X
ejpam-6940	528	2	8	8	X
ejpam-6940	528	3	]	]	X
ejpam-6940	528	4	j.	j.	PROPN
ejpam-6940	528	5	neggers	neggers	PROPN
ejpam-6940	528	6	and	and	CCONJ
ejpam-6940	528	7	h.	h.	PROPN
ejpam-6940	528	8	s.	s.	PROPN
ejpam-6940	528	9	kim	kim	PROPN
ejpam-6940	528	10	.	.	PUNCT
ejpam-6940	529	1	on	on	ADP
ejpam-6940	529	2	d	d	PROPN
ejpam-6940	529	3	-	-	PUNCT
ejpam-6940	529	4	algebras	algebras	PROPN
ejpam-6940	529	5	.	.	PUNCT
ejpam-6940	530	1	mathematica	mathematica	PROPN
ejpam-6940	530	2	slovaca	slovaca	PROPN
ejpam-6940	530	3	,	,	PUNCT
ejpam-6940	530	4	49:19–26	49:19–26	NOUN
ejpam-6940	530	5	,	,	PUNCT
ejpam-6940	530	6	1999	1999	NUM
ejpam-6940	530	7	.	.	PUNCT
ejpam-6940	531	1	[	[	X
ejpam-6940	531	2	9	9	NUM
ejpam-6940	531	3	]	]	X
ejpam-6940	531	4	j.	j.	PROPN
ejpam-6940	531	5	neggers	neggers	PROPN
ejpam-6940	531	6	,	,	PUNCT
ejpam-6940	531	7	s.	s.	PROPN
ejpam-6940	531	8	ahn	ahn	PROPN
ejpam-6940	531	9	,	,	PUNCT
ejpam-6940	531	10	and	and	CCONJ
ejpam-6940	531	11	h.	h.	PROPN
ejpam-6940	531	12	s.	s.	PROPN
ejpam-6940	531	13	kim	kim	PROPN
ejpam-6940	531	14	.	.	PUNCT
ejpam-6940	532	1	on	on	ADP
ejpam-6940	532	2	q	q	NOUN
ejpam-6940	532	3	-	-	PUNCT
ejpam-6940	532	4	algebras	algebra	NOUN
ejpam-6940	532	5	.	.	PUNCT
ejpam-6940	533	1	international	international	ADJ
ejpam-6940	533	2	journal	journal	PROPN
ejpam-6940	533	3	of	of	ADP
ejpam-6940	533	4	mathematics	mathematics	PROPN
ejpam-6940	533	5	and	and	CCONJ
ejpam-6940	533	6	mathematical	mathematical	ADJ
ejpam-6940	533	7	sciences	science	NOUN
ejpam-6940	533	8	,	,	PUNCT
ejpam-6940	533	9	27:749–757	27:749–757	NOUN
ejpam-6940	533	10	,	,	PUNCT
ejpam-6940	533	11	2001	2001	NUM
ejpam-6940	533	12	.	.	PUNCT
ejpam-6940	534	1	[	[	X
ejpam-6940	534	2	10	10	NUM
ejpam-6940	534	3	]	]	PUNCT
ejpam-6940	534	4	s.	s.	PROPN
ejpam-6940	534	5	ahn	ahn	PROPN
ejpam-6940	534	6	,	,	PUNCT
ejpam-6940	534	7	h.	h.	PROPN
ejpam-6940	534	8	s.	s.	PROPN
ejpam-6940	534	9	kim	kim	PROPN
ejpam-6940	534	10	,	,	PUNCT
ejpam-6940	534	11	and	and	CCONJ
ejpam-6940	534	12	h.	h.	PROPN
ejpam-6940	534	13	d.	d.	PROPN
ejpam-6940	534	14	lee	lee	PROPN
ejpam-6940	534	15	.	.	PUNCT
ejpam-6940	535	1	r	r	X
ejpam-6940	535	2	-	-	PUNCT
ejpam-6940	535	3	maps	map	NOUN
ejpam-6940	535	4	and	and	CCONJ
ejpam-6940	535	5	l	l	NOUN
ejpam-6940	535	6	-	-	NOUN
ejpam-6940	535	7	map	map	NOUN
ejpam-6940	535	8	in	in	ADP
ejpam-6940	535	9	q	q	NOUN
ejpam-6940	535	10	-	-	PUNCT
ejpam-6940	535	11	algebras	algebra	NOUN
ejpam-6940	535	12	.	.	PUNCT
ejpam-6940	536	1	international	international	ADJ
ejpam-6940	536	2	journal	journal	PROPN
ejpam-6940	536	3	of	of	ADP
ejpam-6940	536	4	pure	pure	ADJ
ejpam-6940	536	5	and	and	CCONJ
ejpam-6940	536	6	applied	applied	ADJ
ejpam-6940	536	7	mathematics	mathematic	NOUN
ejpam-6940	536	8	,	,	PUNCT
ejpam-6940	536	9	12:419–425	12:419–425	NUM
ejpam-6940	536	10	,	,	PUNCT
ejpam-6940	536	11	2004	2004	NUM
ejpam-6940	536	12	.	.	PUNCT
ejpam-6940	537	1	[	[	X
ejpam-6940	537	2	11	11	NUM
ejpam-6940	537	3	]	]	PUNCT
ejpam-6940	537	4	s.	s.	PROPN
ejpam-6940	537	5	m.	m.	PROPN
ejpam-6940	537	6	lee	lee	PROPN
ejpam-6940	537	7	and	and	CCONJ
ejpam-6940	537	8	k.	k.	PROPN
ejpam-6940	537	9	h.	h.	PROPN
ejpam-6940	537	10	kim	kim	PROPN
ejpam-6940	537	11	.	.	PUNCT
ejpam-6940	538	1	on	on	ADP
ejpam-6940	538	2	right	right	ADJ
ejpam-6940	538	3	fixed	fix	VERB
ejpam-6940	538	4	maps	map	NOUN
ejpam-6940	538	5	of	of	ADP
ejpam-6940	538	6	q	q	NOUN
ejpam-6940	538	7	-	-	PUNCT
ejpam-6940	538	8	algebras	algebra	NOUN
ejpam-6940	538	9	.	.	PUNCT
ejpam-6940	539	1	international	international	PROPN
ejpam-6940	539	2	mathematical	mathematical	PROPN
ejpam-6940	539	3	forum	forum	PROPN
ejpam-6940	539	4	,	,	PUNCT
ejpam-6940	539	5	6:31–37	6:31–37	NOUN
ejpam-6940	539	6	,	,	PUNCT
ejpam-6940	539	7	2011	2011	NUM
ejpam-6940	539	8	.	.	PUNCT
ejpam-6940	540	1	[	[	X
ejpam-6940	540	2	12	12	NUM
ejpam-6940	540	3	]	]	PUNCT
ejpam-6940	540	4	s.	s.	PROPN
ejpam-6940	540	5	ahn	ahn	PROPN
ejpam-6940	540	6	and	and	CCONJ
ejpam-6940	540	7	s.	s.	PROPN
ejpam-6940	540	8	e.	e.	PROPN
ejpam-6940	540	9	kang	kang	PROPN
ejpam-6940	540	10	.	.	PUNCT
ejpam-6940	541	1	the	the	DET
ejpam-6940	541	2	role	role	NOUN
ejpam-6940	541	3	of	of	ADP
ejpam-6940	541	4	t(x	t(x	PROPN
ejpam-6940	541	5	)	)	PUNCT
ejpam-6940	541	6	in	in	ADP
ejpam-6940	541	7	the	the	DET
ejpam-6940	541	8	ideal	ideal	ADJ
ejpam-6940	541	9	theory	theory	NOUN
ejpam-6940	541	10	of	of	ADP
ejpam-6940	541	11	q	q	NOUN
ejpam-6940	541	12	-	-	PUNCT
ejpam-6940	541	13	algebras	algebras	X
ejpam-6940	541	14	.	.	PUNCT
ejpam-6940	542	1	honam	honam	PROPN
ejpam-6940	542	2	mathematical	mathematical	PROPN
ejpam-6940	542	3	journal	journal	PROPN
ejpam-6940	542	4	,	,	PUNCT
ejpam-6940	542	5	32:515–523	32:515–523	NUM
ejpam-6940	542	6	,	,	PUNCT
ejpam-6940	542	7	2010	2010	NUM
ejpam-6940	542	8	.	.	PUNCT
ejpam-6940	543	1	[	[	X
ejpam-6940	543	2	13	13	NUM
ejpam-6940	543	3	]	]	PUNCT
ejpam-6940	543	4	a.	a.	NOUN
ejpam-6940	543	5	anantayasethi	anantayasethi	PROPN
ejpam-6940	543	6	,	,	PUNCT
ejpam-6940	543	7	t.	t.	PROPN
ejpam-6940	543	8	kunawat	kunawat	PROPN
ejpam-6940	543	9	,	,	PUNCT
ejpam-6940	543	10	and	and	CCONJ
ejpam-6940	543	11	p.	p.	PROPN
ejpam-6940	543	12	moonipa	moonipa	NOUN
ejpam-6940	543	13	.	.	PUNCT
ejpam-6940	544	1	relations	relation	NOUN
ejpam-6940	544	2	between	between	ADP
ejpam-6940	544	3	g	g	NOUN
ejpam-6940	544	4	-	-	PUNCT
ejpam-6940	544	5	part	part	NOUN
ejpam-6940	544	6	and	and	CCONJ
ejpam-6940	544	7	atoms	atom	NOUN
ejpam-6940	544	8	in	in	ADP
ejpam-6940	544	9	q	q	NOUN
ejpam-6940	544	10	-	-	PUNCT
ejpam-6940	544	11	algebras	algebras	ADJ
ejpam-6940	544	12	.	.	PUNCT
ejpam-6940	545	1	european	european	PROPN
ejpam-6940	545	2	journal	journal	PROPN
ejpam-6940	545	3	of	of	ADP
ejpam-6940	545	4	pure	pure	ADJ
ejpam-6940	545	5	and	and	CCONJ
ejpam-6940	545	6	applied	applied	ADJ
ejpam-6940	545	7	mathematics	mathematic	NOUN
ejpam-6940	545	8	,	,	PUNCT
ejpam-6940	545	9	17:3268–3276	17:3268–3276	NUM
ejpam-6940	545	10	,	,	PUNCT
ejpam-6940	545	11	2024	2024	NUM
ejpam-6940	545	12	.	.	PUNCT
ejpam-6940	546	1	[	[	X
ejpam-6940	546	2	14	14	NUM
ejpam-6940	546	3	]	]	X
ejpam-6940	546	4	d.	d.	PROPN
ejpam-6940	546	5	sun	sun	PROPN
ejpam-6940	546	6	.	.	PUNCT
ejpam-6940	547	1	on	on	ADP
ejpam-6940	547	2	atoms	atom	NOUN
ejpam-6940	547	3	of	of	ADP
ejpam-6940	547	4	bck	bck	NOUN
ejpam-6940	547	5	-	-	PUNCT
ejpam-6940	547	6	algebras	algebras	PROPN
ejpam-6940	547	7	.	.	PUNCT
ejpam-6940	548	1	scientiae	scientiae	PROPN
ejpam-6940	548	2	mathematicae	mathematicae	VERB
ejpam-6940	548	3	japonicae	japonicae	PROPN
ejpam-6940	548	4	online	online	NOUN
ejpam-6940	548	5	,	,	PUNCT
ejpam-6940	548	6	2:115	2:115	NUM
ejpam-6940	548	7	–	–	PUNCT
ejpam-6940	548	8	124	124	NUM
ejpam-6940	548	9	,	,	PUNCT
ejpam-6940	548	10	2001	2001	NUM
ejpam-6940	548	11	.	.	PUNCT
ejpam-6940	549	1	[	[	X
ejpam-6940	549	2	15	15	NUM
ejpam-6940	549	3	]	]	X
ejpam-6940	549	4	s.	s.	PROPN
ejpam-6940	549	5	m.	m.	PROPN
ejpam-6940	549	6	mostafa	mostafa	PROPN
ejpam-6940	549	7	,	,	PUNCT
ejpam-6940	549	8	m.	m.	NOUN
ejpam-6940	549	9	a.	a.	PROPN
ejpam-6940	549	10	naby	naby	PROPN
ejpam-6940	549	11	,	,	PUNCT
ejpam-6940	549	12	and	and	CCONJ
ejpam-6940	549	13	o.	o.	PROPN
ejpam-6940	549	14	r.	r.	PROPN
ejpam-6940	549	15	elgendy	elgendy	PROPN
ejpam-6940	549	16	.	.	PUNCT
ejpam-6940	550	1	fuzzy	fuzzy	ADJ
ejpam-6940	550	2	q	q	NOUN
ejpam-6940	550	3	-	-	NOUN
ejpam-6940	550	4	ideals	ideal	NOUN
ejpam-6940	550	5	in	in	ADP
ejpam-6940	550	6	q	q	NOUN
ejpam-6940	550	7	-	-	PUNCT
ejpam-6940	550	8	algebras	algebra	NOUN
ejpam-6940	550	9	.	.	PUNCT
ejpam-6940	551	1	world	world	NOUN
ejpam-6940	551	2	applied	apply	VERB
ejpam-6940	551	3	programming	programming	NOUN
ejpam-6940	551	4	,	,	PUNCT
ejpam-6940	551	5	2:69–80	2:69–80	PROPN
ejpam-6940	551	6	,	,	PUNCT
ejpam-6940	551	7	2012	2012	NUM
ejpam-6940	551	8	.	.	PUNCT
ejpam-6940	552	1	a.	a.	NOUN
ejpam-6940	552	2	anantayasethi	anantayasethi	PROPN
ejpam-6940	552	3	et	et	PROPN
ejpam-6940	552	4	al	al	PROPN
ejpam-6940	552	5	.	.	PUNCT
ejpam-6940	552	6	/	/	SYM
ejpam-6940	552	7	eur	eur	PROPN
ejpam-6940	552	8	.	.	PUNCT
ejpam-6940	553	1	j.	j.	PROPN
ejpam-6940	553	2	pure	pure	PROPN
ejpam-6940	553	3	appl	appl	PROPN
ejpam-6940	553	4	.	.	PROPN
ejpam-6940	553	5	math	math	PROPN
ejpam-6940	553	6	,	,	PUNCT
ejpam-6940	553	7	18	18	NUM
ejpam-6940	553	8	(	(	PUNCT
ejpam-6940	553	9	4	4	NUM
ejpam-6940	553	10	)	)	PUNCT
ejpam-6940	553	11	(	(	PUNCT
ejpam-6940	553	12	2025	2025	NUM
ejpam-6940	553	13	)	)	PUNCT
ejpam-6940	553	14	,	,	PUNCT
ejpam-6940	553	15	6940	6940	NUM
ejpam-6940	553	16	14	14	NUM
ejpam-6940	553	17	of	of	ADP
ejpam-6940	553	18	14	14	NUM
ejpam-6940	553	19	[	[	X
ejpam-6940	553	20	16	16	NUM
ejpam-6940	553	21	]	]	PUNCT
ejpam-6940	553	22	h.	h.	PROPN
ejpam-6940	553	23	k.	k.	PROPN
ejpam-6940	553	24	abdullah	abdullah	PROPN
ejpam-6940	553	25	and	and	CCONJ
ejpam-6940	553	26	m.	m.	NOUN
ejpam-6940	553	27	tach	tach	PROPN
ejpam-6940	553	28	.	.	PUNCT
ejpam-6940	554	1	intuitionistic	intuitionistic	ADJ
ejpam-6940	554	2	fuzzy	fuzzy	ADJ
ejpam-6940	554	3	prime	prime	ADJ
ejpam-6940	554	4	ideal	ideal	NOUN
ejpam-6940	554	5	on	on	ADP
ejpam-6940	554	6	q	q	NOUN
ejpam-6940	554	7	-	-	PUNCT
ejpam-6940	554	8	algebras	algebra	NOUN
ejpam-6940	554	9	.	.	PUNCT
ejpam-6940	555	1	international	international	ADJ
ejpam-6940	555	2	journal	journal	PROPN
ejpam-6940	555	3	of	of	ADP
ejpam-6940	555	4	academic	academic	ADJ
ejpam-6940	555	5	and	and	CCONJ
ejpam-6940	555	6	applied	applied	ADJ
ejpam-6940	555	7	research	research	NOUN
ejpam-6940	555	8	,	,	PUNCT
ejpam-6940	555	9	4:66–78	4:66–78	NUM
ejpam-6940	555	10	,	,	PUNCT
ejpam-6940	555	11	2020	2020	NUM
ejpam-6940	555	12	.	.	PUNCT
ejpam-6940	556	1	[	[	X
ejpam-6940	556	2	17	17	NUM
ejpam-6940	556	3	]	]	PUNCT
ejpam-6940	556	4	a.	a.	NOUN
ejpam-6940	556	5	anantayasethi	anantayasethi	PROPN
ejpam-6940	556	6	and	and	CCONJ
ejpam-6940	556	7	j.	j.	PROPN
ejpam-6940	556	8	koppitz	koppitz	PROPN
ejpam-6940	556	9	.	.	PUNCT
ejpam-6940	557	1	characterization	characterization	NOUN
ejpam-6940	557	2	of	of	ADP
ejpam-6940	557	3	ideals	ideal	NOUN
ejpam-6940	557	4	of	of	ADP
ejpam-6940	557	5	q	q	NOUN
ejpam-6940	557	6	-	-	PUNCT
ejpam-6940	557	7	algebras	algebras	ADV
ejpam-6940	557	8	related	relate	VERB
ejpam-6940	557	9	to	to	ADP
ejpam-6940	557	10	its	its	PRON
ejpam-6940	557	11	g	g	NOUN
ejpam-6940	557	12	-	-	PUNCT
ejpam-6940	557	13	part	part	NOUN
ejpam-6940	557	14	.	.	PUNCT
ejpam-6940	558	1	journal	journal	NOUN
ejpam-6940	558	2	of	of	ADP
ejpam-6940	558	3	discrete	discrete	ADJ
ejpam-6940	558	4	mathematical	mathematical	ADJ
ejpam-6940	558	5	sciences	science	NOUN
ejpam-6940	558	6	and	and	CCONJ
ejpam-6940	558	7	cryptography	cryptography	NOUN
ejpam-6940	558	8	,	,	PUNCT
ejpam-6940	558	9	28:131–141	28:131–141	PROPN
ejpam-6940	558	10	,	,	PUNCT
ejpam-6940	558	11	2025	2025	NUM
ejpam-6940	558	12	.	.	PUNCT
ejpam-6940	559	1	[	[	X
ejpam-6940	559	2	18	18	NUM
ejpam-6940	559	3	]	]	PUNCT
ejpam-6940	559	4	j.	j.	PROPN
ejpam-6940	559	5	howie	howie	PROPN
ejpam-6940	559	6	.	.	PUNCT
ejpam-6940	560	1	fundamentals	fundamental	NOUN
ejpam-6940	560	2	of	of	ADP
ejpam-6940	560	3	semigroup	semigroup	PROPN
ejpam-6940	560	4	theory	theory	NOUN
ejpam-6940	560	5	.	.	PUNCT
ejpam-6940	561	1	oxford	oxford	PROPN
ejpam-6940	561	2	university	university	PROPN
ejpam-6940	561	3	press	press	PROPN
ejpam-6940	561	4	inc	inc	PROPN
ejpam-6940	561	5	.	.	PROPN
ejpam-6940	561	6	,	,	PUNCT
ejpam-6940	561	7	uas	uas	PROPN
ejpam-6940	561	8	,	,	PUNCT
ejpam-6940	561	9	1995	1995	NUM
ejpam-6940	561	10	.	.	PUNCT