id	sid	tid	token	lemma	pos
ejpam-6792	1	1	european	european	PROPN
ejpam-6792	1	2	journal	journal	PROPN
ejpam-6792	1	3	of	of	ADP
ejpam-6792	1	4	pure	pure	ADJ
ejpam-6792	1	5	and	and	CCONJ
ejpam-6792	1	6	applied	applied	ADJ
ejpam-6792	1	7	mathematics	mathematic	NOUN
ejpam-6792	1	8	2025	2025	NUM
ejpam-6792	1	9	,	,	PUNCT
ejpam-6792	1	10	vol	vol	NOUN
ejpam-6792	1	11	.	.	PROPN
ejpam-6792	1	12	18	18	NUM
ejpam-6792	1	13	,	,	PUNCT
ejpam-6792	1	14	issue	issue	NOUN
ejpam-6792	1	15	4	4	NUM
ejpam-6792	1	16	,	,	PUNCT
ejpam-6792	1	17	article	article	NOUN
ejpam-6792	1	18	number	number	NOUN
ejpam-6792	1	19	6792	6792	NUM
ejpam-6792	1	20	issn	issn	PROPN
ejpam-6792	1	21	1307	1307	NUM
ejpam-6792	1	22	-	-	SYM
ejpam-6792	1	23	5543	5543	NUM
ejpam-6792	1	24	–	–	PUNCT
ejpam-6792	1	25	ejpam.com	ejpam.com	X
ejpam-6792	1	26	published	publish	VERB
ejpam-6792	1	27	by	by	ADP
ejpam-6792	1	28	new	new	PROPN
ejpam-6792	1	29	york	york	PROPN
ejpam-6792	1	30	business	business	PROPN
ejpam-6792	1	31	global	global	ADJ
ejpam-6792	1	32	direct	direct	ADJ
ejpam-6792	1	33	product	product	NOUN
ejpam-6792	1	34	of	of	ADP
ejpam-6792	1	35	q	q	NOUN
ejpam-6792	1	36	-	-	PUNCT
ejpam-6792	1	37	algebras	algebras	ADJ
ejpam-6792	1	38	ananya	ananya	PROPN
ejpam-6792	1	39	anantayasethi1	anantayasethi1	PROPN
ejpam-6792	1	40	,	,	PUNCT
ejpam-6792	1	41	kittisak	kittisak	PROPN
ejpam-6792	1	42	sangsura1	sangsura1	PROPN
ejpam-6792	1	43	,	,	PUNCT
ejpam-6792	1	44	napaporn	napaporn	PROPN
ejpam-6792	1	45	sarasit2,∗	sarasit2,∗	ADJ
ejpam-6792	1	46	1	1	NUM
ejpam-6792	1	47	department	department	NOUN
ejpam-6792	1	48	of	of	ADP
ejpam-6792	1	49	mathematics	mathematic	NOUN
ejpam-6792	1	50	,	,	PUNCT
ejpam-6792	1	51	science	science	NOUN
ejpam-6792	1	52	faculty	faculty	NOUN
ejpam-6792	1	53	,	,	PUNCT
ejpam-6792	1	54	mahasarakham	mahasarakham	PROPN
ejpam-6792	1	55	university	university	PROPN
ejpam-6792	1	56	,	,	PUNCT
ejpam-6792	1	57	mahasarakham	mahasarakham	PROPN
ejpam-6792	1	58	,	,	PUNCT
ejpam-6792	1	59	44150	44150	NUM
ejpam-6792	1	60	,	,	PUNCT
ejpam-6792	1	61	thailand	thailand	PROPN
ejpam-6792	1	62	2	2	NUM
ejpam-6792	1	63	division	division	NOUN
ejpam-6792	1	64	of	of	ADP
ejpam-6792	1	65	mathematics	mathematic	NOUN
ejpam-6792	1	66	,	,	PUNCT
ejpam-6792	1	67	faculty	faculty	NOUN
ejpam-6792	1	68	of	of	ADP
ejpam-6792	1	69	engineering	engineering	NOUN
ejpam-6792	1	70	,	,	PUNCT
ejpam-6792	1	71	rajamangala	rajamangala	PROPN
ejpam-6792	1	72	university	university	PROPN
ejpam-6792	1	73	of	of	ADP
ejpam-6792	1	74	technology	technology	PROPN
ejpam-6792	1	75	isan	isan	PROPN
ejpam-6792	1	76	,	,	PUNCT
ejpam-6792	1	77	khon	khon	PROPN
ejpam-6792	1	78	kaen	kaen	PROPN
ejpam-6792	1	79	40000	40000	NUM
ejpam-6792	1	80	,	,	PUNCT
ejpam-6792	1	81	thailand	thailand	PROPN
ejpam-6792	1	82	abstract	abstract	NOUN
ejpam-6792	1	83	.	.	PUNCT
ejpam-6792	2	1	in	in	ADP
ejpam-6792	2	2	this	this	DET
ejpam-6792	2	3	work	work	NOUN
ejpam-6792	2	4	,	,	PUNCT
ejpam-6792	2	5	the	the	DET
ejpam-6792	2	6	direct	direct	ADJ
ejpam-6792	2	7	product	product	NOUN
ejpam-6792	2	8	of	of	ADP
ejpam-6792	2	9	q	q	NOUN
ejpam-6792	2	10	-	-	PUNCT
ejpam-6792	2	11	algebras	algebras	PROPN
ejpam-6792	2	12	is	be	AUX
ejpam-6792	2	13	discussed	discuss	VERB
ejpam-6792	2	14	.	.	PUNCT
ejpam-6792	3	1	we	we	PRON
ejpam-6792	3	2	investigate	investigate	VERB
ejpam-6792	3	3	some	some	DET
ejpam-6792	3	4	properties	property	NOUN
ejpam-6792	3	5	of	of	ADP
ejpam-6792	3	6	this	this	DET
ejpam-6792	3	7	direct	direct	ADJ
ejpam-6792	3	8	product	product	NOUN
ejpam-6792	3	9	related	relate	VERB
ejpam-6792	3	10	to	to	ADP
ejpam-6792	3	11	concepts	concept	NOUN
ejpam-6792	3	12	of	of	ADP
ejpam-6792	3	13	subalgebra	subalgebra	NOUN
ejpam-6792	3	14	,	,	PUNCT
ejpam-6792	3	15	ideal	ideal	ADJ
ejpam-6792	3	16	and	and	CCONJ
ejpam-6792	3	17	g	g	NOUN
ejpam-6792	3	18	-	-	PUNCT
ejpam-6792	3	19	part	part	NOUN
ejpam-6792	3	20	of	of	ADP
ejpam-6792	3	21	q	q	NOUN
ejpam-6792	3	22	-	-	PUNCT
ejpam-6792	3	23	algebras	algebras	X
ejpam-6792	3	24	.	.	PUNCT
ejpam-6792	4	1	we	we	PRON
ejpam-6792	4	2	obtain	obtain	VERB
ejpam-6792	4	3	that	that	SCONJ
ejpam-6792	4	4	the	the	DET
ejpam-6792	4	5	set	set	NOUN
ejpam-6792	4	6	g	g	NOUN
ejpam-6792	4	7	-	-	PUNCT
ejpam-6792	4	8	part	part	NOUN
ejpam-6792	4	9	of	of	ADP
ejpam-6792	4	10	the	the	DET
ejpam-6792	4	11	direct	direct	ADJ
ejpam-6792	4	12	product	product	NOUN
ejpam-6792	4	13	of	of	ADP
ejpam-6792	4	14	q	q	NOUN
ejpam-6792	4	15	-	-	PUNCT
ejpam-6792	4	16	algebras	algebras	PROPN
ejpam-6792	4	17	is	be	AUX
ejpam-6792	4	18	an	an	DET
ejpam-6792	4	19	exactly	exactly	ADV
ejpam-6792	4	20	the	the	DET
ejpam-6792	4	21	direct	direct	ADJ
ejpam-6792	4	22	product	product	NOUN
ejpam-6792	4	23	of	of	ADP
ejpam-6792	4	24	sets	set	NOUN
ejpam-6792	4	25	of	of	ADP
ejpam-6792	4	26	g	g	NOUN
ejpam-6792	4	27	-	-	PUNCT
ejpam-6792	4	28	part	part	NOUN
ejpam-6792	4	29	of	of	ADP
ejpam-6792	4	30	q	q	NOUN
ejpam-6792	4	31	-	-	PUNCT
ejpam-6792	4	32	algebras	algebras	X
ejpam-6792	4	33	.	.	PUNCT
ejpam-6792	5	1	we	we	PRON
ejpam-6792	5	2	also	also	ADV
ejpam-6792	5	3	provide	provide	VERB
ejpam-6792	5	4	the	the	DET
ejpam-6792	5	5	characterization	characterization	NOUN
ejpam-6792	5	6	of	of	ADP
ejpam-6792	5	7	two	two	NUM
ejpam-6792	5	8	elements	element	NOUN
ejpam-6792	5	9	subsets	subset	NOUN
ejpam-6792	5	10	of	of	ADP
ejpam-6792	5	11	the	the	DET
ejpam-6792	5	12	direct	direct	ADJ
ejpam-6792	5	13	product	product	NOUN
ejpam-6792	5	14	of	of	ADP
ejpam-6792	5	15	q	q	NOUN
ejpam-6792	5	16	-	-	PUNCT
ejpam-6792	5	17	algebras	algebras	PROPN
ejpam-6792	5	18	to	to	PART
ejpam-6792	5	19	be	be	AUX
ejpam-6792	5	20	subalgebras	subalgebras	PROPN
ejpam-6792	5	21	.	.	PUNCT
ejpam-6792	6	1	moreover	moreover	ADV
ejpam-6792	6	2	,	,	PUNCT
ejpam-6792	6	3	we	we	PRON
ejpam-6792	6	4	show	show	VERB
ejpam-6792	6	5	the	the	DET
ejpam-6792	6	6	connection	connection	NOUN
ejpam-6792	6	7	between	between	ADP
ejpam-6792	6	8	the	the	DET
ejpam-6792	6	9	direct	direct	ADJ
ejpam-6792	6	10	product	product	NOUN
ejpam-6792	6	11	of	of	ADP
ejpam-6792	6	12	q	q	NOUN
ejpam-6792	6	13	-	-	PUNCT
ejpam-6792	6	14	algebras	algebras	PROPN
ejpam-6792	6	15	and	and	CCONJ
ejpam-6792	6	16	ci	ci	NOUN
ejpam-6792	6	17	-	-	PUNCT
ejpam-6792	6	18	algebras	algebras	PROPN
ejpam-6792	6	19	.	.	PUNCT
ejpam-6792	7	1	2020	2020	NUM
ejpam-6792	7	2	mathematics	mathematics	PROPN
ejpam-6792	7	3	subject	subject	NOUN
ejpam-6792	7	4	classifications	classification	NOUN
ejpam-6792	7	5	:	:	PUNCT
ejpam-6792	7	6	03g25	03g25	NUM
ejpam-6792	7	7	,	,	PUNCT
ejpam-6792	7	8	06f35	06f35	NUM
ejpam-6792	7	9	,	,	PUNCT
ejpam-6792	7	10	20k05	20k05	NUM
ejpam-6792	7	11	key	key	ADJ
ejpam-6792	7	12	words	word	NOUN
ejpam-6792	7	13	and	and	CCONJ
ejpam-6792	7	14	phrases	phrase	NOUN
ejpam-6792	7	15	:	:	PUNCT
ejpam-6792	7	16	q	q	X
ejpam-6792	7	17	-	-	PUNCT
ejpam-6792	7	18	algebra	algebra	NOUN
ejpam-6792	7	19	,	,	PUNCT
ejpam-6792	7	20	direct	direct	ADJ
ejpam-6792	7	21	product	product	NOUN
ejpam-6792	7	22	,	,	PUNCT
ejpam-6792	7	23	subalgebra	subalgebra	NOUN
ejpam-6792	7	24	,	,	PUNCT
ejpam-6792	7	25	g	g	NOUN
ejpam-6792	7	26	-	-	PUNCT
ejpam-6792	7	27	part	part	NOUN
ejpam-6792	7	28	,	,	PUNCT
ejpam-6792	7	29	atom	atom	NOUN
ejpam-6792	7	30	,	,	PUNCT
ejpam-6792	7	31	ci	ci	NOUN
ejpam-6792	7	32	-	-	NOUN
ejpam-6792	7	33	algebra	algebra	NOUN
ejpam-6792	7	34	1	1	NUM
ejpam-6792	7	35	.	.	PUNCT
ejpam-6792	7	36	introduction	introduction	NOUN
ejpam-6792	7	37	and	and	CCONJ
ejpam-6792	7	38	preliminaries	preliminary	NOUN
ejpam-6792	7	39	back	back	ADV
ejpam-6792	7	40	to	to	ADP
ejpam-6792	7	41	the	the	DET
ejpam-6792	7	42	year	year	NOUN
ejpam-6792	7	43	1966	1966	NUM
ejpam-6792	7	44	,	,	PUNCT
ejpam-6792	7	45	y.	y.	PROPN
ejpam-6792	7	46	imai	imai	PROPN
ejpam-6792	7	47	and	and	CCONJ
ejpam-6792	7	48	k.	k.	PROPN
ejpam-6792	7	49	iseki	iseki	PROPN
ejpam-6792	7	50	created	create	VERB
ejpam-6792	7	51	a	a	DET
ejpam-6792	7	52	new	new	ADJ
ejpam-6792	7	53	logical	logical	ADJ
ejpam-6792	7	54	algebra	algebra	NOUN
ejpam-6792	7	55	which	which	PRON
ejpam-6792	7	56	is	be	AUX
ejpam-6792	7	57	called	call	VERB
ejpam-6792	7	58	a	a	DET
ejpam-6792	7	59	bck	bck	NOUN
ejpam-6792	7	60	-	-	PUNCT
ejpam-6792	7	61	algebra	algebra	NOUN
ejpam-6792	7	62	[	[	X
ejpam-6792	7	63	1	1	NUM
ejpam-6792	7	64	]	]	PUNCT
ejpam-6792	7	65	.	.	PUNCT
ejpam-6792	8	1	in	in	ADP
ejpam-6792	8	2	the	the	DET
ejpam-6792	8	3	same	same	ADJ
ejpam-6792	8	4	year	year	NOUN
ejpam-6792	8	5	k.	k.	PROPN
ejpam-6792	8	6	iseki	iseki	PROPN
ejpam-6792	8	7	announced	announce	VERB
ejpam-6792	8	8	a	a	DET
ejpam-6792	8	9	notion	notion	NOUN
ejpam-6792	8	10	of	of	ADP
ejpam-6792	8	11	a	a	DET
ejpam-6792	8	12	bcialgebra	bcialgebra	NOUN
ejpam-6792	8	13	which	which	PRON
ejpam-6792	8	14	is	be	AUX
ejpam-6792	8	15	a	a	DET
ejpam-6792	8	16	generalization	generalization	NOUN
ejpam-6792	8	17	of	of	ADP
ejpam-6792	8	18	a	a	DET
ejpam-6792	8	19	bck	bck	NOUN
ejpam-6792	8	20	-	-	PUNCT
ejpam-6792	8	21	algebra	algebra	NOUN
ejpam-6792	8	22	in	in	ADP
ejpam-6792	8	23	[	[	X
ejpam-6792	8	24	2	2	NUM
ejpam-6792	8	25	]	]	PUNCT
ejpam-6792	8	26	.	.	PUNCT
ejpam-6792	9	1	for	for	ADP
ejpam-6792	9	2	more	more	ADJ
ejpam-6792	9	3	insight	insight	NOUN
ejpam-6792	9	4	of	of	ADP
ejpam-6792	9	5	bck	bck	PROPN
ejpam-6792	9	6	,	,	PUNCT
ejpam-6792	9	7	bci	bci	NOUN
ejpam-6792	9	8	-	-	PUNCT
ejpam-6792	9	9	algebras	algebra	NOUN
ejpam-6792	9	10	see	see	VERB
ejpam-6792	9	11	also	also	ADV
ejpam-6792	9	12	[	[	X
ejpam-6792	9	13	3–5	3–5	NOUN
ejpam-6792	9	14	]	]	PUNCT
ejpam-6792	9	15	.	.	PUNCT
ejpam-6792	10	1	since	since	SCONJ
ejpam-6792	10	2	then	then	ADV
ejpam-6792	10	3	many	many	ADJ
ejpam-6792	10	4	algebraic	algebraic	ADJ
ejpam-6792	10	5	structures	structure	NOUN
ejpam-6792	10	6	are	be	AUX
ejpam-6792	10	7	invented	invent	VERB
ejpam-6792	10	8	,	,	PUNCT
ejpam-6792	10	9	most	most	ADJ
ejpam-6792	10	10	of	of	ADP
ejpam-6792	10	11	them	they	PRON
ejpam-6792	10	12	are	be	AUX
ejpam-6792	10	13	generalization	generalization	NOUN
ejpam-6792	10	14	of	of	ADP
ejpam-6792	10	15	bck	bck	PROPN
ejpam-6792	10	16	,	,	PUNCT
ejpam-6792	10	17	bci	bci	NOUN
ejpam-6792	10	18	-	-	PUNCT
ejpam-6792	10	19	algebras	algebras	X
ejpam-6792	10	20	.	.	PUNCT
ejpam-6792	11	1	q.	q.	PROPN
ejpam-6792	11	2	hu	hu	PROPN
ejpam-6792	12	1	and	and	CCONJ
ejpam-6792	12	2	x.	x.	PROPN
ejpam-6792	12	3	li	li	PROPN
ejpam-6792	12	4	introduced	introduce	VERB
ejpam-6792	12	5	the	the	DET
ejpam-6792	12	6	notion	notion	NOUN
ejpam-6792	12	7	of	of	ADP
ejpam-6792	12	8	bch	bch	NOUN
ejpam-6792	12	9	-	-	PUNCT
ejpam-6792	12	10	algebra	algebra	NOUN
ejpam-6792	12	11	in	in	ADP
ejpam-6792	12	12	1983	1983	NUM
ejpam-6792	12	13	[	[	X
ejpam-6792	12	14	6	6	NUM
ejpam-6792	12	15	]	]	PUNCT
ejpam-6792	12	16	,	,	PUNCT
ejpam-6792	12	17	some	some	DET
ejpam-6792	12	18	basic	basic	ADJ
ejpam-6792	12	19	properties	property	NOUN
ejpam-6792	12	20	of	of	ADP
ejpam-6792	12	21	bch	bch	NOUN
ejpam-6792	12	22	-	-	PUNCT
ejpam-6792	12	23	algebras	algebra	NOUN
ejpam-6792	12	24	are	be	AUX
ejpam-6792	12	25	explored	explore	VERB
ejpam-6792	12	26	.	.	PUNCT
ejpam-6792	13	1	in	in	ADP
ejpam-6792	13	2	1984	1984	NUM
ejpam-6792	13	3	,	,	PUNCT
ejpam-6792	13	4	y.	y.	PROPN
ejpam-6792	13	5	komori	komori	PROPN
ejpam-6792	13	6	introduced	introduce	VERB
ejpam-6792	13	7	the	the	DET
ejpam-6792	13	8	notion	notion	NOUN
ejpam-6792	13	9	of	of	ADP
ejpam-6792	13	10	bcc	bcc	PROPN
ejpam-6792	13	11	-	-	PUNCT
ejpam-6792	13	12	algebras	algebras	X
ejpam-6792	14	1	[	[	X
ejpam-6792	14	2	7	7	NUM
ejpam-6792	14	3	]	]	PUNCT
ejpam-6792	14	4	.	.	PUNCT
ejpam-6792	15	1	in	in	ADP
ejpam-6792	15	2	[	[	X
ejpam-6792	15	3	8	8	NUM
ejpam-6792	15	4	]	]	PUNCT
ejpam-6792	15	5	,	,	PUNCT
ejpam-6792	15	6	y.	y.	PROPN
ejpam-6792	15	7	b.	b.	PROPN
ejpam-6792	15	8	jun	jun	PROPN
ejpam-6792	15	9	,	,	PUNCT
ejpam-6792	15	10	e.	e.	PROPN
ejpam-6792	15	11	h.	h.	PROPN
ejpam-6792	15	12	roh	roh	PROPN
ejpam-6792	15	13	and	and	CCONJ
ejpam-6792	15	14	h.	h.	PROPN
ejpam-6792	15	15	s.	s.	PROPN
ejpam-6792	15	16	kim	kim	PROPN
ejpam-6792	15	17	introduced	introduce	VERB
ejpam-6792	15	18	another	another	DET
ejpam-6792	15	19	generalization	generalization	NOUN
ejpam-6792	15	20	of	of	ADP
ejpam-6792	15	21	bck	bck	PROPN
ejpam-6792	15	22	,	,	PUNCT
ejpam-6792	15	23	bci	bci	PROPN
ejpam-6792	15	24	,	,	PUNCT
ejpam-6792	15	25	bcc	bcc	PROPN
ejpam-6792	15	26	-	-	PUNCT
ejpam-6792	15	27	algebras	algebras	PROPN
ejpam-6792	15	28	in	in	ADP
ejpam-6792	15	29	1998	1998	NUM
ejpam-6792	15	30	,	,	PUNCT
ejpam-6792	15	31	which	which	PRON
ejpam-6792	15	32	called	call	VERB
ejpam-6792	15	33	bh	bh	NOUN
ejpam-6792	15	34	-	-	NOUN
ejpam-6792	15	35	algebra	algebra	NOUN
ejpam-6792	15	36	.	.	PUNCT
ejpam-6792	16	1	the	the	DET
ejpam-6792	16	2	authors	author	NOUN
ejpam-6792	16	3	of	of	ADP
ejpam-6792	16	4	[	[	X
ejpam-6792	16	5	8	8	NUM
ejpam-6792	16	6	]	]	PUNCT
ejpam-6792	16	7	provided	provide	VERB
ejpam-6792	16	8	that	that	SCONJ
ejpam-6792	16	9	every	every	DET
ejpam-6792	16	10	bounded	bounded	ADJ
ejpam-6792	16	11	bh	bh	NOUN
ejpam-6792	16	12	-	-	NOUN
ejpam-6792	16	13	algebra	algebra	NOUN
ejpam-6792	16	14	contains	contain	VERB
ejpam-6792	16	15	a	a	DET
ejpam-6792	16	16	maximal	maximal	ADJ
ejpam-6792	16	17	ideal	ideal	NOUN
ejpam-6792	16	18	.	.	PUNCT
ejpam-6792	17	1	in	in	ADP
ejpam-6792	17	2	2001	2001	NUM
ejpam-6792	17	3	,	,	PUNCT
ejpam-6792	17	4	j.	j.	PROPN
ejpam-6792	17	5	neggers	neggers	PROPN
ejpam-6792	17	6	,	,	PUNCT
ejpam-6792	17	7	s.	s.	PROPN
ejpam-6792	17	8	s.	s.	PROPN
ejpam-6792	17	9	ahn	ahn	PROPN
ejpam-6792	17	10	and	and	CCONJ
ejpam-6792	17	11	h.	h.	PROPN
ejpam-6792	17	12	s.	s.	PROPN
ejpam-6792	17	13	kim	kim	PROPN
ejpam-6792	17	14	introduced	introduce	VERB
ejpam-6792	17	15	the	the	DET
ejpam-6792	17	16	notion	notion	NOUN
ejpam-6792	17	17	of	of	ADP
ejpam-6792	17	18	q	q	NOUN
ejpam-6792	17	19	-	-	PUNCT
ejpam-6792	17	20	algebra	algebra	NOUN
ejpam-6792	17	21	which	which	PRON
ejpam-6792	17	22	is	be	AUX
ejpam-6792	17	23	a	a	DET
ejpam-6792	17	24	generalization	generalization	NOUN
ejpam-6792	17	25	of	of	ADP
ejpam-6792	17	26	bck	bck	PROPN
ejpam-6792	17	27	,	,	PUNCT
ejpam-6792	17	28	bci	bci	PROPN
ejpam-6792	17	29	,	,	PUNCT
ejpam-6792	17	30	bch	bch	PROPN
ejpam-6792	17	31	,	,	PUNCT
ejpam-6792	17	32	bh	bh	NOUN
ejpam-6792	17	33	-	-	PUNCT
ejpam-6792	17	34	algebras	algebras	X
ejpam-6792	18	1	[	[	X
ejpam-6792	18	2	9	9	NUM
ejpam-6792	18	3	]	]	PUNCT
ejpam-6792	18	4	.	.	PUNCT
ejpam-6792	19	1	a	a	DET
ejpam-6792	19	2	q	q	NOUN
ejpam-6792	19	3	-	-	PUNCT
ejpam-6792	19	4	algebra	algebra	NOUN
ejpam-6792	19	5	consists	consist	VERB
ejpam-6792	19	6	of	of	ADP
ejpam-6792	19	7	a	a	DET
ejpam-6792	19	8	non	non	ADJ
ejpam-6792	19	9	-	-	ADJ
ejpam-6792	19	10	empty	empty	ADJ
ejpam-6792	19	11	set	set	NOUN
ejpam-6792	19	12	x	x	PUNCT
ejpam-6792	19	13	and	and	CCONJ
ejpam-6792	19	14	a	a	DET
ejpam-6792	19	15	constant	constant	ADJ
ejpam-6792	19	16	0	0	NUM
ejpam-6792	19	17	∈	∈	NOUN
ejpam-6792	19	18	x	x	NOUN
ejpam-6792	19	19	,	,	PUNCT
ejpam-6792	19	20	with	with	ADP
ejpam-6792	19	21	a	a	DET
ejpam-6792	19	22	binary	binary	ADJ
ejpam-6792	19	23	operation	operation	NOUN
ejpam-6792	19	24	∗	∗	NOUN
ejpam-6792	19	25	on	on	ADP
ejpam-6792	19	26	x	x	PUNCT
ejpam-6792	19	27	that	that	PRON
ejpam-6792	19	28	satisfies	satisfy	VERB
ejpam-6792	19	29	the	the	DET
ejpam-6792	19	30	following	follow	VERB
ejpam-6792	19	31	three	three	NUM
ejpam-6792	19	32	conditions	condition	NOUN
ejpam-6792	19	33	:	:	PUNCT
ejpam-6792	19	34	for	for	ADP
ejpam-6792	19	35	any	any	DET
ejpam-6792	19	36	x	x	NOUN
ejpam-6792	19	37	,	,	PUNCT
ejpam-6792	19	38	y	y	PROPN
ejpam-6792	19	39	,	,	PUNCT
ejpam-6792	19	40	z	z	NOUN
ejpam-6792	19	41	∈	∈	PROPN
ejpam-6792	19	42	x	x	SYM
ejpam-6792	19	43	(	(	PUNCT
ejpam-6792	19	44	q1	q1	PROPN
ejpam-6792	19	45	)	)	PUNCT
ejpam-6792	19	46	x	x	SYM
ejpam-6792	19	47	∗	∗	NOUN
ejpam-6792	19	48	x	x	SYM
ejpam-6792	19	49	=	=	SYM
ejpam-6792	19	50	0	0	NUM
ejpam-6792	19	51	,	,	PUNCT
ejpam-6792	19	52	∗corresponding	∗corresponde	VERB
ejpam-6792	19	53	author	author	NOUN
ejpam-6792	19	54	.	.	PUNCT
ejpam-6792	20	1	doi	doi	NOUN
ejpam-6792	20	2	:	:	PUNCT
ejpam-6792	20	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6792	https://doi.org/10.29020/nybg.ejpam.v18i4.6792	ADJ
ejpam-6792	20	4	email	email	NOUN
ejpam-6792	20	5	addresses	address	VERB
ejpam-6792	20	6	:	:	PUNCT
ejpam-6792	20	7	ananya.a@msu.ac.th	ananya.a@msu.ac.th	ADP
ejpam-6792	20	8	(	(	PUNCT
ejpam-6792	20	9	a.	a.	NOUN
ejpam-6792	20	10	anantayasethi	anantayasethi	PROPN
ejpam-6792	20	11	)	)	PUNCT
ejpam-6792	20	12	,	,	PUNCT
ejpam-6792	21	1	kittisak.s@msu.ac.th	kittisak.s@msu.ac.th	PROPN
ejpam-6792	21	2	(	(	PUNCT
ejpam-6792	21	3	k.	k.	PROPN
ejpam-6792	21	4	saengsura	saengsura	PROPN
ejpam-6792	21	5	)	)	PUNCT
ejpam-6792	21	6	,	,	PUNCT
ejpam-6792	21	7	napaporn.sr@rmuti.ac.th	napaporn.sr@rmuti.ac.th	PROPN
ejpam-6792	21	8	(	(	PUNCT
ejpam-6792	21	9	n.	n.	NOUN
ejpam-6792	21	10	sarasit	sarasit	PROPN
ejpam-6792	21	11	)	)	PUNCT
ejpam-6792	21	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6792	21	13	1	1	NUM
ejpam-6792	21	14	copyright	copyright	NOUN
ejpam-6792	21	15	:	:	PUNCT
ejpam-6792	21	16	©	©	PROPN
ejpam-6792	21	17	2025	2025	NUM
ejpam-6792	21	18	the	the	DET
ejpam-6792	21	19	author(s	author(s	NOUN
ejpam-6792	21	20	)	)	PUNCT
ejpam-6792	21	21	.	.	PUNCT
ejpam-6792	22	1	(	(	PUNCT
ejpam-6792	22	2	cc	cc	NOUN
ejpam-6792	22	3	by	by	ADP
ejpam-6792	22	4	-	-	PUNCT
ejpam-6792	22	5	nc	nc	PROPN
ejpam-6792	22	6	4.0	4.0	NUM
ejpam-6792	22	7	)	)	PUNCT
ejpam-6792	22	8	a.	a.	NOUN
ejpam-6792	22	9	anantayasethi	anantayasethi	PROPN
ejpam-6792	22	10	,	,	PUNCT
ejpam-6792	22	11	k.	k.	PROPN
ejpam-6792	22	12	saengsura	saengsura	PROPN
ejpam-6792	22	13	,	,	PUNCT
ejpam-6792	22	14	n.	n.	NOUN
ejpam-6792	22	15	sarasit	sarasit	PROPN
ejpam-6792	22	16	/	/	SYM
ejpam-6792	22	17	eur	eur	PROPN
ejpam-6792	22	18	.	.	PUNCT
ejpam-6792	23	1	j.	j.	PROPN
ejpam-6792	23	2	pure	pure	PROPN
ejpam-6792	23	3	appl	appl	PROPN
ejpam-6792	23	4	.	.	PROPN
ejpam-6792	23	5	math	math	PROPN
ejpam-6792	23	6	,	,	PUNCT
ejpam-6792	23	7	18	18	NUM
ejpam-6792	23	8	(	(	PUNCT
ejpam-6792	23	9	4	4	NUM
ejpam-6792	23	10	)	)	PUNCT
ejpam-6792	23	11	(	(	PUNCT
ejpam-6792	23	12	2025	2025	NUM
ejpam-6792	23	13	)	)	PUNCT
ejpam-6792	23	14	,	,	PUNCT
ejpam-6792	23	15	6792	6792	NUM
ejpam-6792	23	16	2	2	NUM
ejpam-6792	23	17	of	of	ADP
ejpam-6792	23	18	10	10	NUM
ejpam-6792	23	19	(	(	PUNCT
ejpam-6792	23	20	q2	q2	NOUN
ejpam-6792	23	21	)	)	PUNCT
ejpam-6792	23	22	x	x	SYM
ejpam-6792	24	1	∗	∗	NOUN
ejpam-6792	24	2	0	0	NUM
ejpam-6792	25	1	=	=	SYM
ejpam-6792	25	2	x	x	NOUN
ejpam-6792	25	3	,	,	PUNCT
ejpam-6792	25	4	(	(	PUNCT
ejpam-6792	25	5	q3	q3	PROPN
ejpam-6792	25	6	)	)	PUNCT
ejpam-6792	25	7	(	(	PUNCT
ejpam-6792	25	8	x	x	SYM
ejpam-6792	25	9	∗	∗	PROPN
ejpam-6792	25	10	y	y	NOUN
ejpam-6792	25	11	)	)	PUNCT
ejpam-6792	25	12	∗	∗	NOUN
ejpam-6792	25	13	z	z	NOUN
ejpam-6792	25	14	=	=	SYM
ejpam-6792	25	15	(	(	PUNCT
ejpam-6792	25	16	x	x	X
ejpam-6792	25	17	∗	∗	PROPN
ejpam-6792	25	18	z	z	NOUN
ejpam-6792	25	19	)	)	PUNCT
ejpam-6792	25	20	∗	∗	NOUN
ejpam-6792	25	21	y.	y.	NOUN
ejpam-6792	26	1	we	we	PRON
ejpam-6792	26	2	will	will	AUX
ejpam-6792	26	3	omit	omit	VERB
ejpam-6792	26	4	the	the	DET
ejpam-6792	26	5	symbol	symbol	NOUN
ejpam-6792	26	6	of	of	ADP
ejpam-6792	26	7	a	a	DET
ejpam-6792	26	8	binary	binary	ADJ
ejpam-6792	26	9	operation	operation	NOUN
ejpam-6792	26	10	,	,	PUNCT
ejpam-6792	26	11	we	we	PRON
ejpam-6792	26	12	write	write	VERB
ejpam-6792	26	13	xy	xy	PROPN
ejpam-6792	26	14	instead	instead	ADV
ejpam-6792	26	15	of	of	ADP
ejpam-6792	26	16	x∗y	x∗y	X
ejpam-6792	26	17	for	for	ADP
ejpam-6792	26	18	convenient	convenient	ADJ
ejpam-6792	26	19	reason	reason	NOUN
ejpam-6792	26	20	.	.	PUNCT
ejpam-6792	27	1	there	there	PRON
ejpam-6792	27	2	are	be	VERB
ejpam-6792	27	3	various	various	ADJ
ejpam-6792	27	4	literature	literature	NOUN
ejpam-6792	27	5	discussed	discuss	VERB
ejpam-6792	27	6	the	the	DET
ejpam-6792	27	7	notion	notion	NOUN
ejpam-6792	27	8	of	of	ADP
ejpam-6792	27	9	q	q	NOUN
ejpam-6792	27	10	-	-	PUNCT
ejpam-6792	27	11	algebra	algebra	NOUN
ejpam-6792	27	12	.	.	PUNCT
ejpam-6792	28	1	in	in	ADP
ejpam-6792	28	2	[	[	X
ejpam-6792	28	3	9	9	NUM
ejpam-6792	28	4	]	]	PUNCT
ejpam-6792	28	5	,	,	PUNCT
ejpam-6792	28	6	some	some	DET
ejpam-6792	28	7	basic	basic	ADJ
ejpam-6792	28	8	properties	property	NOUN
ejpam-6792	28	9	of	of	ADP
ejpam-6792	28	10	q	q	NOUN
ejpam-6792	28	11	-	-	PUNCT
ejpam-6792	28	12	algebra	algebra	NOUN
ejpam-6792	28	13	are	be	AUX
ejpam-6792	28	14	investigated	investigate	VERB
ejpam-6792	28	15	.	.	PUNCT
ejpam-6792	29	1	the	the	DET
ejpam-6792	29	2	authors	author	NOUN
ejpam-6792	29	3	also	also	ADV
ejpam-6792	29	4	presented	present	VERB
ejpam-6792	29	5	the	the	DET
ejpam-6792	29	6	concepts	concept	NOUN
ejpam-6792	29	7	of	of	ADP
ejpam-6792	29	8	ideal	ideal	ADJ
ejpam-6792	29	9	and	and	CCONJ
ejpam-6792	29	10	quadratic	quadratic	ADJ
ejpam-6792	29	11	q	q	NOUN
ejpam-6792	29	12	-	-	PUNCT
ejpam-6792	29	13	algebras	algebras	X
ejpam-6792	29	14	.	.	PUNCT
ejpam-6792	30	1	a	a	DET
ejpam-6792	30	2	non	non	ADJ
ejpam-6792	30	3	-	-	ADJ
ejpam-6792	30	4	empty	empty	ADJ
ejpam-6792	30	5	subset	subset	NOUN
ejpam-6792	30	6	i	i	PRON
ejpam-6792	30	7	of	of	ADP
ejpam-6792	30	8	a	a	DET
ejpam-6792	30	9	q	q	NOUN
ejpam-6792	30	10	-	-	NOUN
ejpam-6792	30	11	algebra	algebra	NOUN
ejpam-6792	30	12	x	x	PUNCT
ejpam-6792	30	13	is	be	AUX
ejpam-6792	30	14	an	an	DET
ejpam-6792	30	15	ideal	ideal	NOUN
ejpam-6792	30	16	if	if	SCONJ
ejpam-6792	30	17	i	i	PRON
ejpam-6792	30	18	satisfies	satisfy	VERB
ejpam-6792	30	19	the	the	DET
ejpam-6792	30	20	conditions	condition	NOUN
ejpam-6792	30	21	(	(	PUNCT
ejpam-6792	30	22	i1	i1	PROPN
ejpam-6792	30	23	)	)	PUNCT
ejpam-6792	30	24	and	and	CCONJ
ejpam-6792	30	25	(	(	PUNCT
ejpam-6792	30	26	i2	i2	PROPN
ejpam-6792	30	27	)	)	PUNCT
ejpam-6792	30	28	as	as	ADP
ejpam-6792	30	29	following	follow	VERB
ejpam-6792	30	30	:	:	PUNCT
ejpam-6792	30	31	for	for	ADP
ejpam-6792	30	32	all	all	DET
ejpam-6792	30	33	x	x	NOUN
ejpam-6792	30	34	,	,	PUNCT
ejpam-6792	30	35	y	y	PROPN
ejpam-6792	30	36	∈	∈	PROPN
ejpam-6792	30	37	x	x	X
ejpam-6792	30	38	,	,	PUNCT
ejpam-6792	30	39	(	(	PUNCT
ejpam-6792	30	40	i1	i1	PROPN
ejpam-6792	30	41	)	)	PUNCT
ejpam-6792	30	42	0	0	PUNCT
ejpam-6792	31	1	∈	∈	PROPN
ejpam-6792	32	1	i	i	PRON
ejpam-6792	32	2	,	,	PUNCT
ejpam-6792	32	3	(	(	PUNCT
ejpam-6792	32	4	i2	i2	PROPN
ejpam-6792	32	5	)	)	PUNCT
ejpam-6792	32	6	xy	xy	PROPN
ejpam-6792	33	1	∈	∈	PROPN
ejpam-6792	34	1	i	i	PRON
ejpam-6792	34	2	and	and	CCONJ
ejpam-6792	34	3	y	y	PROPN
ejpam-6792	34	4	∈	∈	PROPN
ejpam-6792	35	1	i	i	PRON
ejpam-6792	35	2	imply	imply	VERB
ejpam-6792	35	3	x	x	X
ejpam-6792	35	4	∈	∈	PROPN
ejpam-6792	35	5	i.	i.	NOUN
ejpam-6792	35	6	the	the	DET
ejpam-6792	35	7	authors	author	NOUN
ejpam-6792	35	8	in	in	ADP
ejpam-6792	35	9	[	[	X
ejpam-6792	35	10	9	9	NUM
ejpam-6792	35	11	]	]	PUNCT
ejpam-6792	35	12	showed	show	VERB
ejpam-6792	35	13	that	that	SCONJ
ejpam-6792	35	14	the	the	DET
ejpam-6792	35	15	subset	subset	NOUN
ejpam-6792	35	16	b(x	b(x	NOUN
ejpam-6792	35	17	)	)	PUNCT
ejpam-6792	35	18	=	=	PRON
ejpam-6792	35	19	{	{	PUNCT
ejpam-6792	35	20	x	x	PUNCT
ejpam-6792	35	21	∈	∈	NOUN
ejpam-6792	35	22	x|	x|	X
ejpam-6792	35	23	0x	0x	NOUN
ejpam-6792	35	24	=	=	SYM
ejpam-6792	35	25	0	0	NUM
ejpam-6792	35	26	}	}	PUNCT
ejpam-6792	35	27	of	of	ADP
ejpam-6792	35	28	a	a	DET
ejpam-6792	35	29	q	q	NOUN
ejpam-6792	35	30	-	-	NOUN
ejpam-6792	35	31	algebra	algebra	NOUN
ejpam-6792	35	32	x	x	PUNCT
ejpam-6792	35	33	is	be	AUX
ejpam-6792	35	34	an	an	DET
ejpam-6792	35	35	ideal	ideal	NOUN
ejpam-6792	35	36	.	.	PUNCT
ejpam-6792	36	1	they	they	PRON
ejpam-6792	36	2	also	also	ADV
ejpam-6792	36	3	studied	study	VERB
ejpam-6792	36	4	the	the	DET
ejpam-6792	36	5	set	set	ADJ
ejpam-6792	36	6	g(x	g(x	NOUN
ejpam-6792	36	7	)	)	PUNCT
ejpam-6792	37	1	=	=	PRON
ejpam-6792	37	2	{	{	PUNCT
ejpam-6792	37	3	x	x	PUNCT
ejpam-6792	37	4	∈	∈	NOUN
ejpam-6792	37	5	x|	x|	X
ejpam-6792	37	6	0x	0x	NOUN
ejpam-6792	37	7	=	=	SYM
ejpam-6792	37	8	x	x	SYM
ejpam-6792	37	9	}	}	PUNCT
ejpam-6792	37	10	of	of	ADP
ejpam-6792	37	11	x	x	X
ejpam-6792	37	12	,	,	PUNCT
ejpam-6792	37	13	which	which	PRON
ejpam-6792	37	14	is	be	AUX
ejpam-6792	37	15	called	call	VERB
ejpam-6792	37	16	the	the	DET
ejpam-6792	37	17	g	g	NOUN
ejpam-6792	37	18	-	-	PUNCT
ejpam-6792	37	19	part	part	NOUN
ejpam-6792	37	20	of	of	ADP
ejpam-6792	37	21	x.	x.	NOUN
ejpam-6792	37	22	it	it	PRON
ejpam-6792	37	23	is	be	AUX
ejpam-6792	37	24	easy	easy	ADJ
ejpam-6792	37	25	to	to	PART
ejpam-6792	37	26	see	see	VERB
ejpam-6792	37	27	that	that	DET
ejpam-6792	37	28	0	0	NUM
ejpam-6792	37	29	∈	∈	PROPN
ejpam-6792	37	30	g(x	g(x	NOUN
ejpam-6792	37	31	)	)	PUNCT
ejpam-6792	37	32	.	.	PUNCT
ejpam-6792	38	1	hence	hence	ADV
ejpam-6792	38	2	,	,	PUNCT
ejpam-6792	38	3	g(x	g(x	NOUN
ejpam-6792	38	4	)	)	PUNCT
ejpam-6792	38	5	̸=	̸=	PROPN
ejpam-6792	38	6	∅.	∅.	X
ejpam-6792	38	7	in	in	ADP
ejpam-6792	38	8	general	general	ADJ
ejpam-6792	38	9	,	,	PUNCT
ejpam-6792	38	10	the	the	DET
ejpam-6792	38	11	set	set	ADJ
ejpam-6792	38	12	g(x	g(x	NOUN
ejpam-6792	38	13	)	)	PUNCT
ejpam-6792	38	14	need	need	VERB
ejpam-6792	38	15	not	not	PART
ejpam-6792	38	16	to	to	PART
ejpam-6792	38	17	be	be	AUX
ejpam-6792	38	18	an	an	DET
ejpam-6792	38	19	ideal	ideal	NOUN
ejpam-6792	38	20	of	of	ADP
ejpam-6792	38	21	x.	x.	NOUN
ejpam-6792	38	22	the	the	DET
ejpam-6792	38	23	authors	author	NOUN
ejpam-6792	38	24	proved	prove	VERB
ejpam-6792	38	25	that	that	SCONJ
ejpam-6792	38	26	the	the	DET
ejpam-6792	38	27	set	set	NOUN
ejpam-6792	38	28	g(x	g(x	NOUN
ejpam-6792	38	29	)	)	PUNCT
ejpam-6792	38	30	is	be	AUX
ejpam-6792	38	31	an	an	DET
ejpam-6792	38	32	ideal	ideal	NOUN
ejpam-6792	39	1	whenever	whenever	SCONJ
ejpam-6792	39	2	|x|	|x|	PROPN
ejpam-6792	39	3	=	=	SYM
ejpam-6792	39	4	2	2	NUM
ejpam-6792	39	5	and	and	CCONJ
ejpam-6792	39	6	|x|	|x|	PROPN
ejpam-6792	39	7	=	=	SYM
ejpam-6792	39	8	3	3	X
ejpam-6792	39	9	.	.	PUNCT
ejpam-6792	40	1	in	in	ADP
ejpam-6792	40	2	[	[	X
ejpam-6792	40	3	10	10	NUM
ejpam-6792	40	4	]	]	PUNCT
ejpam-6792	40	5	,	,	PUNCT
ejpam-6792	40	6	[	[	X
ejpam-6792	40	7	11	11	NUM
ejpam-6792	40	8	]	]	PUNCT
ejpam-6792	40	9	and	and	CCONJ
ejpam-6792	40	10	[	[	X
ejpam-6792	40	11	12	12	NUM
ejpam-6792	40	12	]	]	PUNCT
ejpam-6792	40	13	,	,	PUNCT
ejpam-6792	40	14	the	the	DET
ejpam-6792	40	15	authors	author	NOUN
ejpam-6792	40	16	explored	explore	VERB
ejpam-6792	40	17	more	more	ADJ
ejpam-6792	40	18	properties	property	NOUN
ejpam-6792	40	19	of	of	ADP
ejpam-6792	40	20	q	q	NOUN
ejpam-6792	40	21	-	-	PUNCT
ejpam-6792	40	22	algebras	algebra	NOUN
ejpam-6792	40	23	concerning	concern	VERB
ejpam-6792	40	24	to	to	ADP
ejpam-6792	40	25	the	the	DET
ejpam-6792	40	26	concepts	concept	NOUN
ejpam-6792	40	27	of	of	ADP
ejpam-6792	40	28	ideal	ideal	ADJ
ejpam-6792	40	29	,	,	PUNCT
ejpam-6792	40	30	g	g	NOUN
ejpam-6792	40	31	-	-	PUNCT
ejpam-6792	40	32	part	part	NOUN
ejpam-6792	40	33	and	and	CCONJ
ejpam-6792	40	34	atom	atom	NOUN
ejpam-6792	40	35	.	.	PUNCT
ejpam-6792	41	1	the	the	DET
ejpam-6792	41	2	characterization	characterization	NOUN
ejpam-6792	41	3	of	of	ADP
ejpam-6792	41	4	ideals	ideal	NOUN
ejpam-6792	41	5	and	and	CCONJ
ejpam-6792	41	6	some	some	DET
ejpam-6792	41	7	properties	property	NOUN
ejpam-6792	41	8	of	of	ADP
ejpam-6792	41	9	the	the	DET
ejpam-6792	41	10	set	set	NOUN
ejpam-6792	41	11	g	g	NOUN
ejpam-6792	41	12	-	-	PUNCT
ejpam-6792	41	13	part	part	NOUN
ejpam-6792	41	14	were	be	AUX
ejpam-6792	41	15	presented	present	VERB
ejpam-6792	41	16	in	in	ADP
ejpam-6792	41	17	[	[	X
ejpam-6792	41	18	11	11	NUM
ejpam-6792	41	19	]	]	PUNCT
ejpam-6792	41	20	.	.	PUNCT
ejpam-6792	42	1	the	the	DET
ejpam-6792	42	2	authors	author	NOUN
ejpam-6792	42	3	showed	show	VERB
ejpam-6792	42	4	that	that	SCONJ
ejpam-6792	42	5	if	if	SCONJ
ejpam-6792	42	6	the	the	DET
ejpam-6792	42	7	set	set	NOUN
ejpam-6792	42	8	g	g	NOUN
ejpam-6792	42	9	-	-	PUNCT
ejpam-6792	42	10	part	part	NOUN
ejpam-6792	42	11	is	be	AUX
ejpam-6792	42	12	an	an	DET
ejpam-6792	42	13	ideal	ideal	NOUN
ejpam-6792	42	14	,	,	PUNCT
ejpam-6792	42	15	it	it	PRON
ejpam-6792	42	16	is	be	AUX
ejpam-6792	42	17	an	an	DET
ejpam-6792	42	18	abelian	abelian	ADJ
ejpam-6792	42	19	group	group	NOUN
ejpam-6792	42	20	.	.	PUNCT
ejpam-6792	43	1	in	in	ADP
ejpam-6792	43	2	[	[	X
ejpam-6792	43	3	10	10	NUM
ejpam-6792	43	4	]	]	PUNCT
ejpam-6792	43	5	,	,	PUNCT
ejpam-6792	43	6	the	the	DET
ejpam-6792	43	7	concepts	concept	NOUN
ejpam-6792	43	8	of	of	ADP
ejpam-6792	43	9	atom	atom	NOUN
ejpam-6792	43	10	and	and	CCONJ
ejpam-6792	43	11	strong	strong	ADJ
ejpam-6792	43	12	atom	atom	NOUN
ejpam-6792	43	13	in	in	ADP
ejpam-6792	43	14	q	q	NOUN
ejpam-6792	43	15	-	-	PUNCT
ejpam-6792	43	16	algebras	algebra	NOUN
ejpam-6792	43	17	are	be	AUX
ejpam-6792	43	18	offered	offer	VERB
ejpam-6792	43	19	.	.	PUNCT
ejpam-6792	44	1	an	an	DET
ejpam-6792	44	2	element	element	NOUN
ejpam-6792	44	3	a	a	DET
ejpam-6792	44	4	∈	∈	NOUN
ejpam-6792	44	5	x	x	PUNCT
ejpam-6792	44	6	is	be	AUX
ejpam-6792	44	7	an	an	DET
ejpam-6792	44	8	atom	atom	NOUN
ejpam-6792	44	9	of	of	ADP
ejpam-6792	44	10	x	x	NOUN
ejpam-6792	44	11	if	if	SCONJ
ejpam-6792	44	12	xa	xa	PROPN
ejpam-6792	44	13	=	=	SYM
ejpam-6792	44	14	0	0	PROPN
ejpam-6792	44	15	implies	imply	VERB
ejpam-6792	44	16	x	x	PUNCT
ejpam-6792	44	17	=	=	PUNCT
ejpam-6792	44	18	a	a	PRON
ejpam-6792	44	19	for	for	ADP
ejpam-6792	44	20	x	x	SYM
ejpam-6792	44	21	∈	∈	PROPN
ejpam-6792	44	22	x.	x.	NOUN
ejpam-6792	45	1	if	if	SCONJ
ejpam-6792	45	2	0	0	NUM
ejpam-6792	45	3	̸=	̸=	PROPN
ejpam-6792	45	4	a	a	PRON
ejpam-6792	45	5	is	be	AUX
ejpam-6792	45	6	an	an	DET
ejpam-6792	45	7	atom	atom	NOUN
ejpam-6792	45	8	and	and	CCONJ
ejpam-6792	45	9	ax	ax	NOUN
ejpam-6792	45	10	=	=	PUNCT
ejpam-6792	45	11	a	a	PRON
ejpam-6792	45	12	for	for	ADP
ejpam-6792	45	13	all	all	PRON
ejpam-6792	45	14	x	x	SYM
ejpam-6792	45	15	∈	∈	PROPN
ejpam-6792	45	16	x	x	SYM
ejpam-6792	45	17	\	\	X
ejpam-6792	45	18	{	{	PUNCT
ejpam-6792	45	19	a	a	X
ejpam-6792	45	20	}	}	PUNCT
ejpam-6792	45	21	,	,	PUNCT
ejpam-6792	45	22	then	then	ADV
ejpam-6792	45	23	we	we	PRON
ejpam-6792	45	24	call	call	VERB
ejpam-6792	45	25	that	that	SCONJ
ejpam-6792	45	26	a	a	PRON
ejpam-6792	45	27	is	be	AUX
ejpam-6792	45	28	a	a	DET
ejpam-6792	45	29	strong	strong	ADJ
ejpam-6792	45	30	atom	atom	NOUN
ejpam-6792	45	31	.	.	PUNCT
ejpam-6792	46	1	in	in	ADP
ejpam-6792	46	2	[	[	X
ejpam-6792	46	3	10	10	NUM
ejpam-6792	46	4	]	]	PUNCT
ejpam-6792	46	5	,	,	PUNCT
ejpam-6792	46	6	the	the	DET
ejpam-6792	46	7	authors	author	NOUN
ejpam-6792	46	8	obtained	obtain	VERB
ejpam-6792	46	9	that	that	SCONJ
ejpam-6792	46	10	the	the	DET
ejpam-6792	46	11	set	set	NOUN
ejpam-6792	46	12	of	of	ADP
ejpam-6792	46	13	all	all	DET
ejpam-6792	46	14	strong	strong	ADJ
ejpam-6792	46	15	atoms	atom	NOUN
ejpam-6792	46	16	together	together	ADV
ejpam-6792	46	17	with	with	ADP
ejpam-6792	46	18	a	a	DET
ejpam-6792	46	19	constant	constant	ADJ
ejpam-6792	46	20	0	0	NUM
ejpam-6792	46	21	is	be	AUX
ejpam-6792	46	22	a	a	DET
ejpam-6792	46	23	subalgebra	subalgebra	NOUN
ejpam-6792	46	24	.	.	PUNCT
ejpam-6792	47	1	a	a	DET
ejpam-6792	47	2	non	non	ADJ
ejpam-6792	47	3	-	-	ADJ
ejpam-6792	47	4	empty	empty	ADJ
ejpam-6792	47	5	subset	subset	NOUN
ejpam-6792	47	6	s	s	NOUN
ejpam-6792	47	7	of	of	ADP
ejpam-6792	47	8	a	a	DET
ejpam-6792	47	9	q	q	NOUN
ejpam-6792	47	10	-	-	NOUN
ejpam-6792	47	11	algebra	algebra	NOUN
ejpam-6792	47	12	x	x	PUNCT
ejpam-6792	47	13	is	be	AUX
ejpam-6792	47	14	called	call	VERB
ejpam-6792	47	15	a	a	DET
ejpam-6792	47	16	subalgebra	subalgebra	NOUN
ejpam-6792	47	17	if	if	SCONJ
ejpam-6792	47	18	xy	xy	PROPN
ejpam-6792	47	19	∈	∈	PROPN
ejpam-6792	47	20	s	s	VERB
ejpam-6792	47	21	for	for	ADP
ejpam-6792	47	22	any	any	DET
ejpam-6792	47	23	x	x	NOUN
ejpam-6792	47	24	,	,	PUNCT
ejpam-6792	47	25	y	y	PROPN
ejpam-6792	47	26	∈	∈	PROPN
ejpam-6792	47	27	s.	s.	PROPN
ejpam-6792	47	28	moreover	moreover	ADV
ejpam-6792	47	29	,	,	PUNCT
ejpam-6792	47	30	they	they	PRON
ejpam-6792	47	31	showed	show	VERB
ejpam-6792	47	32	that	that	SCONJ
ejpam-6792	47	33	if	if	SCONJ
ejpam-6792	47	34	g(x	g(x	NOUN
ejpam-6792	47	35	)	)	PUNCT
ejpam-6792	47	36	is	be	AUX
ejpam-6792	47	37	an	an	DET
ejpam-6792	47	38	ideal	ideal	NOUN
ejpam-6792	47	39	,	,	PUNCT
ejpam-6792	47	40	then	then	ADV
ejpam-6792	47	41	all	all	DET
ejpam-6792	47	42	elements	element	NOUN
ejpam-6792	47	43	in	in	ADP
ejpam-6792	47	44	g(x	g(x	NOUN
ejpam-6792	47	45	)	)	PUNCT
ejpam-6792	47	46	are	be	AUX
ejpam-6792	47	47	atoms	atom	NOUN
ejpam-6792	47	48	.	.	PUNCT
ejpam-6792	48	1	in	in	ADP
ejpam-6792	48	2	[	[	X
ejpam-6792	48	3	12	12	NUM
ejpam-6792	48	4	]	]	PUNCT
ejpam-6792	48	5	,	,	PUNCT
ejpam-6792	48	6	the	the	DET
ejpam-6792	48	7	authors	author	NOUN
ejpam-6792	48	8	proved	prove	VERB
ejpam-6792	48	9	that	that	SCONJ
ejpam-6792	48	10	if	if	SCONJ
ejpam-6792	48	11	an	an	DET
ejpam-6792	48	12	ideal	ideal	NOUN
ejpam-6792	48	13	i	i	PRON
ejpam-6792	48	14	of	of	ADP
ejpam-6792	48	15	a	a	DET
ejpam-6792	48	16	q	q	NOUN
ejpam-6792	48	17	-	-	NOUN
ejpam-6792	48	18	algebra	algebra	NOUN
ejpam-6792	48	19	x	x	PUNCT
ejpam-6792	48	20	contains	contain	VERB
ejpam-6792	48	21	a	a	DET
ejpam-6792	48	22	strong	strong	ADJ
ejpam-6792	48	23	atom	atom	NOUN
ejpam-6792	48	24	,	,	PUNCT
ejpam-6792	48	25	then	then	ADV
ejpam-6792	48	26	i	i	PRON
ejpam-6792	48	27	=	=	PUNCT
ejpam-6792	48	28	x.	x.	NOUN
ejpam-6792	48	29	there	there	PRON
ejpam-6792	48	30	is	be	VERB
ejpam-6792	48	31	discussion	discussion	NOUN
ejpam-6792	48	32	of	of	ADP
ejpam-6792	48	33	another	another	DET
ejpam-6792	48	34	kind	kind	NOUN
ejpam-6792	48	35	of	of	ADP
ejpam-6792	48	36	ideals	ideal	NOUN
ejpam-6792	48	37	in	in	ADP
ejpam-6792	48	38	[	[	X
ejpam-6792	48	39	13	13	NUM
ejpam-6792	48	40	]	]	PUNCT
ejpam-6792	48	41	.	.	PUNCT
ejpam-6792	49	1	s.	s.	PROPN
ejpam-6792	49	2	m.	m.	PROPN
ejpam-6792	49	3	mootafa	mootafa	PROPN
ejpam-6792	49	4	et	et	PROPN
ejpam-6792	49	5	al	al	PROPN
ejpam-6792	49	6	discussed	discuss	VERB
ejpam-6792	49	7	the	the	DET
ejpam-6792	49	8	concepts	concept	NOUN
ejpam-6792	49	9	of	of	ADP
ejpam-6792	49	10	q	q	NOUN
ejpam-6792	49	11	-	-	PUNCT
ejpam-6792	49	12	ideal	ideal	ADJ
ejpam-6792	49	13	and	and	CCONJ
ejpam-6792	49	14	fuzzy	fuzzy	ADJ
ejpam-6792	49	15	q	q	NOUN
ejpam-6792	49	16	-	-	NOUN
ejpam-6792	49	17	ideal	ideal	NOUN
ejpam-6792	49	18	in	in	ADP
ejpam-6792	49	19	2012	2012	NUM
ejpam-6792	49	20	.	.	PUNCT
ejpam-6792	50	1	they	they	PRON
ejpam-6792	50	2	founded	found	VERB
ejpam-6792	50	3	that	that	SCONJ
ejpam-6792	50	4	any	any	DET
ejpam-6792	50	5	q	q	NOUN
ejpam-6792	50	6	-	-	PUNCT
ejpam-6792	50	7	ideal	ideal	NOUN
ejpam-6792	50	8	is	be	AUX
ejpam-6792	50	9	an	an	DET
ejpam-6792	50	10	ideal	ideal	NOUN
ejpam-6792	50	11	.	.	PUNCT
ejpam-6792	51	1	there	there	PRON
ejpam-6792	51	2	are	be	VERB
ejpam-6792	51	3	other	other	ADJ
ejpam-6792	51	4	discussions	discussion	NOUN
ejpam-6792	51	5	of	of	ADP
ejpam-6792	51	6	q	q	NOUN
ejpam-6792	51	7	-	-	PUNCT
ejpam-6792	51	8	algebras	algebras	X
ejpam-6792	51	9	.	.	PUNCT
ejpam-6792	52	1	in	in	ADP
ejpam-6792	52	2	[	[	X
ejpam-6792	52	3	14	14	NUM
ejpam-6792	52	4	,	,	PUNCT
ejpam-6792	52	5	15	15	NUM
ejpam-6792	52	6	]	]	PUNCT
ejpam-6792	52	7	h.	h.	PROPN
ejpam-6792	52	8	k.	k.	PROPN
ejpam-6792	52	9	abdullah	abdullah	PROPN
ejpam-6792	52	10	and	and	CCONJ
ejpam-6792	52	11	m.	m.	NOUN
ejpam-6792	52	12	tach	tach	NOUN
ejpam-6792	52	13	introduced	introduce	VERB
ejpam-6792	52	14	the	the	DET
ejpam-6792	52	15	notion	notion	NOUN
ejpam-6792	52	16	of	of	ADP
ejpam-6792	52	17	prime	prime	ADJ
ejpam-6792	52	18	ideal	ideal	ADJ
ejpam-6792	52	19	and	and	CCONJ
ejpam-6792	52	20	fuzzy	fuzzy	ADJ
ejpam-6792	52	21	prime	prime	ADJ
ejpam-6792	52	22	ideal	ideal	NOUN
ejpam-6792	52	23	.	.	PUNCT
ejpam-6792	53	1	the	the	DET
ejpam-6792	53	2	authors	author	NOUN
ejpam-6792	53	3	showed	show	VERB
ejpam-6792	53	4	that	that	SCONJ
ejpam-6792	53	5	an	an	DET
ejpam-6792	53	6	ideal	ideal	NOUN
ejpam-6792	53	7	i	i	PRON
ejpam-6792	53	8	of	of	ADP
ejpam-6792	53	9	a	a	DET
ejpam-6792	53	10	q	q	NOUN
ejpam-6792	53	11	-	-	NOUN
ejpam-6792	53	12	algebra	algebra	NOUN
ejpam-6792	53	13	x	x	PUNCT
ejpam-6792	53	14	with	with	ADP
ejpam-6792	53	15	|i|	|i|	PROPN
ejpam-6792	53	16	=	=	SYM
ejpam-6792	53	17	|x|	|x|	PROPN
ejpam-6792	53	18	−	−	NOUN
ejpam-6792	53	19	1	1	NUM
ejpam-6792	53	20	is	be	AUX
ejpam-6792	53	21	a	a	DET
ejpam-6792	53	22	prime	prime	ADJ
ejpam-6792	53	23	ideal	ideal	NOUN
ejpam-6792	53	24	.	.	PUNCT
ejpam-6792	54	1	the	the	DET
ejpam-6792	54	2	other	other	ADJ
ejpam-6792	54	3	approaches	approach	NOUN
ejpam-6792	54	4	of	of	ADP
ejpam-6792	54	5	discussion	discussion	NOUN
ejpam-6792	54	6	in	in	ADP
ejpam-6792	54	7	q	q	NOUN
ejpam-6792	54	8	-	-	PUNCT
ejpam-6792	54	9	algebras	algebra	NOUN
ejpam-6792	54	10	are	be	AUX
ejpam-6792	54	11	morphisms	morphism	NOUN
ejpam-6792	54	12	and	and	CCONJ
ejpam-6792	54	13	mappings	mapping	NOUN
ejpam-6792	54	14	,	,	PUNCT
ejpam-6792	54	15	more	more	ADJ
ejpam-6792	54	16	details	detail	NOUN
ejpam-6792	54	17	of	of	ADP
ejpam-6792	54	18	these	these	DET
ejpam-6792	54	19	topics	topic	NOUN
ejpam-6792	54	20	can	can	AUX
ejpam-6792	54	21	be	be	AUX
ejpam-6792	54	22	found	find	VERB
ejpam-6792	54	23	in	in	ADP
ejpam-6792	54	24	[	[	X
ejpam-6792	54	25	16	16	NUM
ejpam-6792	54	26	,	,	PUNCT
ejpam-6792	54	27	17	17	NUM
ejpam-6792	54	28	]	]	PUNCT
ejpam-6792	54	29	.	.	PUNCT
ejpam-6792	55	1	the	the	DET
ejpam-6792	55	2	concept	concept	NOUN
ejpam-6792	55	3	of	of	ADP
ejpam-6792	55	4	direct	direct	ADJ
ejpam-6792	55	5	products	product	NOUN
ejpam-6792	55	6	was	be	AUX
ejpam-6792	55	7	discussed	discuss	VERB
ejpam-6792	55	8	in	in	ADP
ejpam-6792	55	9	various	various	ADJ
ejpam-6792	55	10	kind	kind	NOUN
ejpam-6792	55	11	of	of	ADP
ejpam-6792	55	12	algebras	algebra	NOUN
ejpam-6792	55	13	,	,	PUNCT
ejpam-6792	55	14	for	for	ADP
ejpam-6792	55	15	instant	instant	ADJ
ejpam-6792	55	16	groups	group	NOUN
ejpam-6792	55	17	,	,	PUNCT
ejpam-6792	55	18	rings	ring	NOUN
ejpam-6792	55	19	and	and	CCONJ
ejpam-6792	55	20	modules	module	NOUN
ejpam-6792	55	21	.	.	PUNCT
ejpam-6792	56	1	the	the	DET
ejpam-6792	56	2	direct	direct	ADJ
ejpam-6792	56	3	product	product	NOUN
ejpam-6792	56	4	forms	form	NOUN
ejpam-6792	56	5	by	by	ADP
ejpam-6792	56	6	taking	take	VERB
ejpam-6792	56	7	the	the	DET
ejpam-6792	56	8	cartesian	cartesian	ADJ
ejpam-6792	56	9	product	product	NOUN
ejpam-6792	56	10	of	of	ADP
ejpam-6792	56	11	their	their	PRON
ejpam-6792	56	12	base	base	NOUN
ejpam-6792	56	13	sets	set	NOUN
ejpam-6792	56	14	of	of	ADP
ejpam-6792	56	15	algebras	algebra	NOUN
ejpam-6792	56	16	as	as	SCONJ
ejpam-6792	56	17	the	the	DET
ejpam-6792	56	18	carrier	carrier	NOUN
ejpam-6792	56	19	set	set	VERB
ejpam-6792	56	20	and	and	CCONJ
ejpam-6792	56	21	defining	define	VERB
ejpam-6792	56	22	operations	operation	NOUN
ejpam-6792	56	23	component	component	NOUN
ejpam-6792	56	24	-	-	PUNCT
ejpam-6792	56	25	wise	wise	ADJ
ejpam-6792	56	26	.	.	PUNCT
ejpam-6792	57	1	in	in	ADP
ejpam-6792	57	2	2016	2016	NUM
ejpam-6792	57	3	,	,	PUNCT
ejpam-6792	57	4	j.	j.	PROPN
ejpam-6792	57	5	a.	a.	PROPN
ejpam-6792	57	6	v.	v.	PROPN
ejpam-6792	57	7	lingcong	lingcong	PROPN
ejpam-6792	57	8	and	and	CCONJ
ejpam-6792	57	9	j.	j.	PROPN
ejpam-6792	57	10	c.	c.	PROPN
ejpam-6792	57	11	endam	endam	PROPN
ejpam-6792	57	12	considered	consider	VERB
ejpam-6792	57	13	the	the	DET
ejpam-6792	57	14	direct	direct	ADJ
ejpam-6792	57	15	product	product	NOUN
ejpam-6792	57	16	of	of	ADP
ejpam-6792	57	17	b	b	NOUN
ejpam-6792	57	18	-	-	PUNCT
ejpam-6792	57	19	algebras	algebras	X
ejpam-6792	57	20	[	[	X
ejpam-6792	57	21	18	18	NUM
ejpam-6792	57	22	]	]	PUNCT
ejpam-6792	57	23	,	,	PUNCT
ejpam-6792	57	24	some	some	DET
ejpam-6792	57	25	basic	basic	ADJ
ejpam-6792	57	26	properties	property	NOUN
ejpam-6792	57	27	were	be	AUX
ejpam-6792	57	28	investigated	investigate	VERB
ejpam-6792	57	29	.	.	PUNCT
ejpam-6792	58	1	they	they	PRON
ejpam-6792	58	2	also	also	ADV
ejpam-6792	58	3	studied	study	VERB
ejpam-6792	58	4	isomorphisms	isomorphism	NOUN
ejpam-6792	58	5	among	among	ADP
ejpam-6792	58	6	the	the	DET
ejpam-6792	58	7	direct	direct	ADJ
ejpam-6792	58	8	product	product	NOUN
ejpam-6792	58	9	of	of	ADP
ejpam-6792	58	10	b	b	NOUN
ejpam-6792	58	11	-	-	PUNCT
ejpam-6792	58	12	algebras	algebras	PROPN
ejpam-6792	58	13	and	and	CCONJ
ejpam-6792	58	14	obtained	obtain	VERB
ejpam-6792	58	15	a	a	DET
ejpam-6792	58	16	necessary	necessary	ADJ
ejpam-6792	58	17	condition	condition	NOUN
ejpam-6792	58	18	for	for	SCONJ
ejpam-6792	58	19	a	a	DET
ejpam-6792	58	20	mapping	mapping	NOUN
ejpam-6792	58	21	to	to	PART
ejpam-6792	58	22	be	be	AUX
ejpam-6792	58	23	an	an	DET
ejpam-6792	58	24	isomorphism	isomorphism	NOUN
ejpam-6792	58	25	.	.	PUNCT
ejpam-6792	59	1	in	in	ADP
ejpam-6792	59	2	2019	2019	NUM
ejpam-6792	59	3	the	the	DET
ejpam-6792	59	4	direct	direct	ADJ
ejpam-6792	59	5	product	product	NOUN
ejpam-6792	59	6	in	in	ADP
ejpam-6792	59	7	bg	bg	PROPN
ejpam-6792	59	8	-	-	PUNCT
ejpam-6792	59	9	algebras	algebras	PROPN
ejpam-6792	59	10	was	be	AUX
ejpam-6792	59	11	discussed	discuss	VERB
ejpam-6792	59	12	by	by	ADP
ejpam-6792	59	13	s.	s.	PROPN
ejpam-6792	59	14	widianto	widianto	PROPN
ejpam-6792	59	15	et	et	NOUN
ejpam-6792	59	16	al.[19	al.[19	NOUN
ejpam-6792	59	17	]	]	PUNCT
ejpam-6792	59	18	.	.	PUNCT
ejpam-6792	60	1	they	they	PRON
ejpam-6792	60	2	showed	show	VERB
ejpam-6792	60	3	that	that	SCONJ
ejpam-6792	60	4	the	the	DET
ejpam-6792	60	5	direct	direct	ADJ
ejpam-6792	60	6	product	product	NOUN
ejpam-6792	60	7	of	of	ADP
ejpam-6792	60	8	commutative	commutative	ADJ
ejpam-6792	60	9	bg	bg	PROPN
ejpam-6792	60	10	-	-	PUNCT
ejpam-6792	60	11	algebras	algebras	PROPN
ejpam-6792	60	12	is	be	AUX
ejpam-6792	60	13	again	again	ADV
ejpam-6792	60	14	a	a	DET
ejpam-6792	60	15	commutative	commutative	ADJ
ejpam-6792	60	16	bgalgebra	bgalgebra	NOUN
ejpam-6792	60	17	.	.	PUNCT
ejpam-6792	61	1	later	later	ADV
ejpam-6792	61	2	,	,	PUNCT
ejpam-6792	61	3	the	the	DET
ejpam-6792	61	4	direct	direct	ADJ
ejpam-6792	61	5	product	product	NOUN
ejpam-6792	61	6	of	of	ADP
ejpam-6792	61	7	bp	bp	PROPN
ejpam-6792	61	8	-algebras	-algebras	PROPN
ejpam-6792	61	9	and	and	CCONJ
ejpam-6792	61	10	gk	gk	PROPN
ejpam-6792	61	11	-	-	PUNCT
ejpam-6792	61	12	algebras	algebras	PROPN
ejpam-6792	61	13	were	be	AUX
ejpam-6792	61	14	presented	present	VERB
ejpam-6792	61	15	in	in	ADP
ejpam-6792	61	16	[	[	X
ejpam-6792	61	17	20	20	NUM
ejpam-6792	61	18	]	]	PUNCT
ejpam-6792	61	19	and	and	CCONJ
ejpam-6792	62	1	[	[	X
ejpam-6792	62	2	21	21	NUM
ejpam-6792	62	3	]	]	X
ejpam-6792	62	4	,	,	PUNCT
ejpam-6792	62	5	respectively	respectively	ADV
ejpam-6792	62	6	.	.	PUNCT
ejpam-6792	63	1	in	in	ADP
ejpam-6792	63	2	[	[	X
ejpam-6792	63	3	22	22	NUM
ejpam-6792	63	4	]	]	PUNCT
ejpam-6792	63	5	,	,	PUNCT
ejpam-6792	63	6	c.	c.	PROPN
ejpam-6792	63	7	chanmanee	chanmanee	PROPN
ejpam-6792	63	8	et	et	PROPN
ejpam-6792	63	9	al	al	PROPN
ejpam-6792	63	10	.	.	PROPN
ejpam-6792	63	11	introduced	introduce	VERB
ejpam-6792	63	12	the	the	DET
ejpam-6792	63	13	concept	concept	NOUN
ejpam-6792	63	14	of	of	ADP
ejpam-6792	63	15	the	the	DET
ejpam-6792	63	16	direct	direct	ADJ
ejpam-6792	63	17	product	product	NOUN
ejpam-6792	63	18	of	of	ADP
ejpam-6792	63	19	infinite	infinite	ADJ
ejpam-6792	63	20	family	family	NOUN
ejpam-6792	63	21	of	of	ADP
ejpam-6792	63	22	b	b	NOUN
ejpam-6792	63	23	-	-	PUNCT
ejpam-6792	63	24	algebras	algebras	PROPN
ejpam-6792	63	25	in	in	ADP
ejpam-6792	63	26	2022	2022	NUM
ejpam-6792	63	27	.	.	PUNCT
ejpam-6792	64	1	the	the	DET
ejpam-6792	64	2	authors	author	NOUN
ejpam-6792	64	3	called	call	VERB
ejpam-6792	64	4	this	this	DET
ejpam-6792	64	5	product	product	NOUN
ejpam-6792	64	6	by	by	ADP
ejpam-6792	64	7	”	"	PUNCT
ejpam-6792	64	8	the	the	DET
ejpam-6792	64	9	external	external	ADJ
ejpam-6792	64	10	direct	direct	ADJ
ejpam-6792	64	11	product	product	NOUN
ejpam-6792	64	12	”	"	PUNCT
ejpam-6792	64	13	as	as	SCONJ
ejpam-6792	64	14	it	it	PRON
ejpam-6792	64	15	is	be	AUX
ejpam-6792	64	16	a	a	DET
ejpam-6792	64	17	generalization	generalization	NOUN
ejpam-6792	64	18	of	of	ADP
ejpam-6792	64	19	the	the	DET
ejpam-6792	64	20	direct	direct	ADJ
ejpam-6792	64	21	product	product	NOUN
ejpam-6792	64	22	.	.	PUNCT
ejpam-6792	65	1	in	in	ADP
ejpam-6792	65	2	2023	2023	NUM
ejpam-6792	65	3	-	-	SYM
ejpam-6792	65	4	2024	2024	NUM
ejpam-6792	65	5	,	,	PUNCT
ejpam-6792	65	6	the	the	DET
ejpam-6792	65	7	concept	concept	NOUN
ejpam-6792	65	8	of	of	ADP
ejpam-6792	65	9	an	an	DET
ejpam-6792	65	10	external	external	ADJ
ejpam-6792	65	11	direct	direct	ADJ
ejpam-6792	65	12	product	product	NOUN
ejpam-6792	65	13	was	be	AUX
ejpam-6792	65	14	considered	consider	VERB
ejpam-6792	65	15	in	in	ADP
ejpam-6792	65	16	iup	iup	PROPN
ejpam-6792	65	17	-algebras	-algebras	PROPN
ejpam-6792	65	18	,	,	PUNCT
ejpam-6792	65	19	dual	dual	ADJ
ejpam-6792	65	20	up	up	ADP
ejpam-6792	65	21	-algebras	-algebras	PROPN
ejpam-6792	65	22	a.	a.	NOUN
ejpam-6792	65	23	anantayasethi	anantayasethi	PROPN
ejpam-6792	65	24	,	,	PUNCT
ejpam-6792	65	25	k.	k.	PROPN
ejpam-6792	65	26	saengsura	saengsura	PROPN
ejpam-6792	65	27	,	,	PUNCT
ejpam-6792	65	28	n.	n.	NOUN
ejpam-6792	65	29	sarasit	sarasit	PROPN
ejpam-6792	65	30	/	/	SYM
ejpam-6792	65	31	eur	eur	PROPN
ejpam-6792	65	32	.	.	PUNCT
ejpam-6792	66	1	j.	j.	PROPN
ejpam-6792	66	2	pure	pure	PROPN
ejpam-6792	66	3	appl	appl	PROPN
ejpam-6792	66	4	.	.	PROPN
ejpam-6792	66	5	math	math	PROPN
ejpam-6792	66	6	,	,	PUNCT
ejpam-6792	66	7	18	18	NUM
ejpam-6792	66	8	(	(	PUNCT
ejpam-6792	66	9	4	4	NUM
ejpam-6792	66	10	)	)	PUNCT
ejpam-6792	66	11	(	(	PUNCT
ejpam-6792	66	12	2025	2025	NUM
ejpam-6792	66	13	)	)	PUNCT
ejpam-6792	66	14	,	,	PUNCT
ejpam-6792	66	15	6792	6792	NUM
ejpam-6792	66	16	3	3	NUM
ejpam-6792	66	17	of	of	ADP
ejpam-6792	66	18	10	10	NUM
ejpam-6792	66	19	and	and	CCONJ
ejpam-6792	66	20	ju	ju	NOUN
ejpam-6792	66	21	-algebras	-algebras	PROPN
ejpam-6792	66	22	,	,	PUNCT
ejpam-6792	66	23	for	for	ADP
ejpam-6792	66	24	more	more	ADJ
ejpam-6792	66	25	details	detail	NOUN
ejpam-6792	66	26	see	see	VERB
ejpam-6792	66	27	[	[	X
ejpam-6792	66	28	22	22	NUM
ejpam-6792	66	29	]	]	PUNCT
ejpam-6792	66	30	,	,	PUNCT
ejpam-6792	66	31	[	[	X
ejpam-6792	66	32	23	23	NUM
ejpam-6792	66	33	]	]	PUNCT
ejpam-6792	66	34	and	and	CCONJ
ejpam-6792	66	35	[	[	X
ejpam-6792	66	36	24	24	NUM
ejpam-6792	66	37	]	]	PUNCT
ejpam-6792	66	38	,	,	PUNCT
ejpam-6792	66	39	respectively	respectively	ADV
ejpam-6792	66	40	.	.	PUNCT
ejpam-6792	67	1	in	in	ADP
ejpam-6792	67	2	this	this	DET
ejpam-6792	67	3	work	work	NOUN
ejpam-6792	67	4	,	,	PUNCT
ejpam-6792	67	5	we	we	PRON
ejpam-6792	67	6	discuss	discuss	VERB
ejpam-6792	67	7	a	a	DET
ejpam-6792	67	8	concept	concept	NOUN
ejpam-6792	67	9	of	of	ADP
ejpam-6792	67	10	the	the	DET
ejpam-6792	67	11	direct	direct	ADJ
ejpam-6792	67	12	product	product	NOUN
ejpam-6792	67	13	of	of	ADP
ejpam-6792	67	14	q	q	NOUN
ejpam-6792	67	15	-	-	PUNCT
ejpam-6792	67	16	algebras	algebras	X
ejpam-6792	67	17	.	.	PUNCT
ejpam-6792	68	1	we	we	PRON
ejpam-6792	68	2	explore	explore	VERB
ejpam-6792	68	3	some	some	DET
ejpam-6792	68	4	properties	property	NOUN
ejpam-6792	68	5	of	of	ADP
ejpam-6792	68	6	direct	direct	ADJ
ejpam-6792	68	7	product	product	NOUN
ejpam-6792	68	8	of	of	ADP
ejpam-6792	68	9	q	q	NOUN
ejpam-6792	68	10	-	-	PUNCT
ejpam-6792	68	11	algebras	algebras	X
ejpam-6792	68	12	.	.	PUNCT
ejpam-6792	69	1	the	the	DET
ejpam-6792	69	2	connection	connection	NOUN
ejpam-6792	69	3	of	of	ADP
ejpam-6792	69	4	the	the	DET
ejpam-6792	69	5	g	g	NOUN
ejpam-6792	69	6	-	-	PUNCT
ejpam-6792	69	7	part	part	NOUN
ejpam-6792	69	8	of	of	ADP
ejpam-6792	69	9	q	q	NOUN
ejpam-6792	69	10	-	-	PUNCT
ejpam-6792	69	11	algebras	algebra	NOUN
ejpam-6792	69	12	and	and	CCONJ
ejpam-6792	69	13	the	the	DET
ejpam-6792	69	14	g	g	NOUN
ejpam-6792	69	15	-	-	PUNCT
ejpam-6792	69	16	part	part	NOUN
ejpam-6792	69	17	of	of	ADP
ejpam-6792	69	18	its	its	PRON
ejpam-6792	69	19	direct	direct	ADJ
ejpam-6792	69	20	product	product	NOUN
ejpam-6792	69	21	is	be	AUX
ejpam-6792	69	22	investigated	investigate	VERB
ejpam-6792	69	23	.	.	PUNCT
ejpam-6792	70	1	moreover	moreover	ADV
ejpam-6792	70	2	,	,	PUNCT
ejpam-6792	70	3	we	we	PRON
ejpam-6792	70	4	provide	provide	VERB
ejpam-6792	70	5	a	a	DET
ejpam-6792	70	6	condition	condition	NOUN
ejpam-6792	70	7	for	for	ADP
ejpam-6792	70	8	the	the	DET
ejpam-6792	70	9	direct	direct	ADJ
ejpam-6792	70	10	product	product	NOUN
ejpam-6792	70	11	of	of	ADP
ejpam-6792	70	12	q	q	NOUN
ejpam-6792	70	13	-	-	PUNCT
ejpam-6792	70	14	algebras	algebras	PROPN
ejpam-6792	70	15	to	to	PART
ejpam-6792	70	16	be	be	AUX
ejpam-6792	70	17	a	a	DET
ejpam-6792	70	18	ci	ci	NOUN
ejpam-6792	70	19	-	-	NOUN
ejpam-6792	70	20	algebra	algebra	NOUN
ejpam-6792	70	21	.	.	PUNCT
ejpam-6792	71	1	now	now	ADV
ejpam-6792	71	2	we	we	PRON
ejpam-6792	71	3	recall	recall	VERB
ejpam-6792	71	4	some	some	DET
ejpam-6792	71	5	properties	property	NOUN
ejpam-6792	71	6	that	that	PRON
ejpam-6792	71	7	we	we	PRON
ejpam-6792	71	8	will	will	AUX
ejpam-6792	71	9	use	use	VERB
ejpam-6792	71	10	later	later	ADV
ejpam-6792	71	11	.	.	PUNCT
ejpam-6792	72	1	proposition	proposition	NOUN
ejpam-6792	72	2	1	1	NUM
ejpam-6792	72	3	.	.	PUNCT
ejpam-6792	73	1	[	[	X
ejpam-6792	73	2	25	25	NUM
ejpam-6792	73	3	]	]	PUNCT
ejpam-6792	73	4	every	every	DET
ejpam-6792	73	5	q	q	NOUN
ejpam-6792	73	6	-	-	NOUN
ejpam-6792	73	7	algebra	algebra	NOUN
ejpam-6792	73	8	x	x	SYM
ejpam-6792	73	9	satisfies	satisfy	VERB
ejpam-6792	73	10	the	the	DET
ejpam-6792	73	11	following	follow	VERB
ejpam-6792	73	12	property	property	NOUN
ejpam-6792	73	13	:	:	PUNCT
ejpam-6792	73	14	0(xy	0(xy	NUM
ejpam-6792	73	15	)	)	PUNCT
ejpam-6792	74	1	=	=	PRON
ejpam-6792	74	2	(	(	PUNCT
ejpam-6792	74	3	0x)(0y	0x)(0y	NOUN
ejpam-6792	74	4	)	)	PUNCT
ejpam-6792	74	5	for	for	ADP
ejpam-6792	74	6	all	all	DET
ejpam-6792	74	7	x	x	NOUN
ejpam-6792	74	8	,	,	PUNCT
ejpam-6792	74	9	y	y	PROPN
ejpam-6792	74	10	∈	∈	PROPN
ejpam-6792	74	11	x.	x.	NOUN
ejpam-6792	74	12	proposition	proposition	NOUN
ejpam-6792	74	13	2	2	NUM
ejpam-6792	74	14	.	.	PUNCT
ejpam-6792	75	1	[	[	X
ejpam-6792	75	2	5	5	X
ejpam-6792	75	3	]	]	PUNCT
ejpam-6792	75	4	let	let	VERB
ejpam-6792	75	5	x	x	PRON
ejpam-6792	75	6	be	be	AUX
ejpam-6792	75	7	a	a	DET
ejpam-6792	75	8	q	q	NOUN
ejpam-6792	75	9	-	-	NOUN
ejpam-6792	75	10	algebra	algebra	NOUN
ejpam-6792	75	11	and	and	CCONJ
ejpam-6792	75	12	let	let	VERB
ejpam-6792	75	13	a	a	DET
ejpam-6792	75	14	and	and	CCONJ
ejpam-6792	75	15	b	b	NOUN
ejpam-6792	75	16	elements	element	NOUN
ejpam-6792	75	17	of	of	ADP
ejpam-6792	75	18	g(x	g(x	NOUN
ejpam-6792	75	19	)	)	PUNCT
ejpam-6792	75	20	.	.	PUNCT
ejpam-6792	76	1	then	then	ADV
ejpam-6792	76	2	ab	ab	PROPN
ejpam-6792	76	3	=	=	SYM
ejpam-6792	76	4	ba	ba	PROPN
ejpam-6792	76	5	.	.	PUNCT
ejpam-6792	76	6	corollary	corollary	ADJ
ejpam-6792	77	1	1	1	NUM
ejpam-6792	77	2	.	.	PUNCT
ejpam-6792	78	1	[	[	X
ejpam-6792	78	2	5	5	X
ejpam-6792	78	3	]	]	PUNCT
ejpam-6792	78	4	let	let	VERB
ejpam-6792	78	5	x	x	PRON
ejpam-6792	78	6	be	be	AUX
ejpam-6792	78	7	a	a	DET
ejpam-6792	78	8	q	q	NOUN
ejpam-6792	78	9	-	-	NOUN
ejpam-6792	78	10	algebra	algebra	NOUN
ejpam-6792	78	11	.	.	PUNCT
ejpam-6792	79	1	a	a	DET
ejpam-6792	79	2	left	left	ADJ
ejpam-6792	79	3	cancellation	cancellation	NOUN
ejpam-6792	79	4	law	law	NOUN
ejpam-6792	79	5	holds	hold	VERB
ejpam-6792	79	6	in	in	ADP
ejpam-6792	79	7	g(x	g(x	NOUN
ejpam-6792	79	8	)	)	PUNCT
ejpam-6792	79	9	,	,	PUNCT
ejpam-6792	79	10	i.e.	i.e.	X
ejpam-6792	79	11	for	for	ADP
ejpam-6792	79	12	all	all	DET
ejpam-6792	79	13	a	a	DET
ejpam-6792	79	14	,	,	PUNCT
ejpam-6792	79	15	b	b	NOUN
ejpam-6792	79	16	,	,	PUNCT
ejpam-6792	79	17	c	c	PROPN
ejpam-6792	79	18	∈	∈	PROPN
ejpam-6792	79	19	g(x	g(x	PROPN
ejpam-6792	79	20	)	)	PUNCT
ejpam-6792	79	21	,	,	PUNCT
ejpam-6792	79	22	ab	ab	PROPN
ejpam-6792	79	23	=	=	PUNCT
ejpam-6792	79	24	ac	ac	PROPN
ejpam-6792	79	25	implies	imply	VERB
ejpam-6792	79	26	b	b	PROPN
ejpam-6792	79	27	=	=	SYM
ejpam-6792	79	28	c.	c.	NOUN
ejpam-6792	79	29	proposition	proposition	NOUN
ejpam-6792	79	30	3	3	NUM
ejpam-6792	79	31	.	.	PUNCT
ejpam-6792	80	1	[	[	X
ejpam-6792	80	2	11	11	NUM
ejpam-6792	80	3	]	]	PUNCT
ejpam-6792	80	4	let	let	VERB
ejpam-6792	80	5	x	x	PRON
ejpam-6792	80	6	be	be	AUX
ejpam-6792	80	7	a	a	DET
ejpam-6792	80	8	q	q	NOUN
ejpam-6792	80	9	-	-	NOUN
ejpam-6792	80	10	algebra	algebra	NOUN
ejpam-6792	80	11	and	and	CCONJ
ejpam-6792	80	12	let	let	VERB
ejpam-6792	80	13	0	0	NUM
ejpam-6792	80	14	̸=	̸=	PROPN
ejpam-6792	80	15	a	a	PRON
ejpam-6792	80	16	,	,	PUNCT
ejpam-6792	80	17	b	b	NOUN
ejpam-6792	80	18	,	,	PUNCT
ejpam-6792	80	19	c	c	PROPN
ejpam-6792	80	20	∈	∈	PROPN
ejpam-6792	80	21	g(x	g(x	PROPN
ejpam-6792	80	22	)	)	PUNCT
ejpam-6792	80	23	.	.	PUNCT
ejpam-6792	81	1	then	then	ADV
ejpam-6792	81	2	the	the	DET
ejpam-6792	81	3	following	follow	VERB
ejpam-6792	81	4	three	three	NUM
ejpam-6792	81	5	properties	property	NOUN
ejpam-6792	81	6	hold	hold	VERB
ejpam-6792	81	7	:	:	PUNCT
ejpam-6792	81	8	(	(	PUNCT
ejpam-6792	81	9	i	i	NOUN
ejpam-6792	81	10	)	)	PUNCT
ejpam-6792	81	11	a	a	DET
ejpam-6792	81	12	̸=	̸=	PROPN
ejpam-6792	81	13	b	b	PROPN
ejpam-6792	81	14	implies	imply	VERB
ejpam-6792	81	15	ab	ab	PROPN
ejpam-6792	81	16	̸∈	̸∈	PROPN
ejpam-6792	81	17	{	{	PUNCT
ejpam-6792	81	18	0	0	PROPN
ejpam-6792	81	19	,	,	PUNCT
ejpam-6792	81	20	a	a	PRON
ejpam-6792	81	21	,	,	PUNCT
ejpam-6792	81	22	b	b	NOUN
ejpam-6792	81	23	}	}	PUNCT
ejpam-6792	81	24	.	.	PUNCT
ejpam-6792	82	1	(	(	PUNCT
ejpam-6792	82	2	ii	ii	NOUN
ejpam-6792	82	3	)	)	PUNCT
ejpam-6792	82	4	if	if	SCONJ
ejpam-6792	82	5	ab	ab	PROPN
ejpam-6792	82	6	=	=	SYM
ejpam-6792	82	7	c	c	PROPN
ejpam-6792	82	8	,	,	PUNCT
ejpam-6792	82	9	then	then	ADV
ejpam-6792	82	10	ac	ac	PROPN
ejpam-6792	82	11	=	=	PROPN
ejpam-6792	82	12	b	b	PROPN
ejpam-6792	82	13	and	and	CCONJ
ejpam-6792	82	14	bc	bc	PROPN
ejpam-6792	82	15	=	=	SYM
ejpam-6792	82	16	a.	a.	PROPN
ejpam-6792	82	17	(	(	PUNCT
ejpam-6792	82	18	iii	iii	NOUN
ejpam-6792	82	19	)	)	PUNCT
ejpam-6792	82	20	for	for	ADP
ejpam-6792	82	21	any	any	DET
ejpam-6792	82	22	x	x	SYM
ejpam-6792	82	23	∈	∈	PROPN
ejpam-6792	82	24	x	x	NOUN
ejpam-6792	82	25	,	,	PUNCT
ejpam-6792	82	26	xa	xa	PROPN
ejpam-6792	82	27	̸=	̸=	PROPN
ejpam-6792	82	28	x.	x.	NOUN
ejpam-6792	82	29	proposition	proposition	NOUN
ejpam-6792	82	30	4	4	NUM
ejpam-6792	82	31	.	.	PUNCT
ejpam-6792	83	1	[	[	X
ejpam-6792	83	2	10	10	NUM
ejpam-6792	83	3	]	]	PUNCT
ejpam-6792	83	4	every	every	DET
ejpam-6792	83	5	element	element	NOUN
ejpam-6792	83	6	of	of	ADP
ejpam-6792	83	7	a	a	DET
ejpam-6792	83	8	q	q	NOUN
ejpam-6792	83	9	-	-	NOUN
ejpam-6792	83	10	algebra	algebra	NOUN
ejpam-6792	83	11	x	x	PUNCT
ejpam-6792	83	12	is	be	AUX
ejpam-6792	83	13	an	an	DET
ejpam-6792	83	14	atom	atom	NOUN
ejpam-6792	83	15	if	if	SCONJ
ejpam-6792	83	16	and	and	CCONJ
ejpam-6792	83	17	only	only	ADV
ejpam-6792	83	18	if	if	SCONJ
ejpam-6792	83	19	a(xb	a(xb	PROPN
ejpam-6792	83	20	)	)	PUNCT
ejpam-6792	83	21	=	=	SYM
ejpam-6792	83	22	b(xa	b(xa	PROPN
ejpam-6792	83	23	)	)	PUNCT
ejpam-6792	83	24	for	for	ADP
ejpam-6792	83	25	all	all	DET
ejpam-6792	83	26	a	a	DET
ejpam-6792	83	27	,	,	PUNCT
ejpam-6792	83	28	b	b	NOUN
ejpam-6792	83	29	,	,	PUNCT
ejpam-6792	83	30	x	x	SYM
ejpam-6792	83	31	∈	∈	NOUN
ejpam-6792	83	32	x.	x.	NOUN
ejpam-6792	84	1	2	2	X
ejpam-6792	84	2	.	.	X
ejpam-6792	84	3	direct	direct	ADJ
ejpam-6792	84	4	product	product	NOUN
ejpam-6792	84	5	q	q	NOUN
ejpam-6792	84	6	-	-	PUNCT
ejpam-6792	84	7	algebras	algebras	ADV
ejpam-6792	84	8	we	we	PRON
ejpam-6792	84	9	will	will	AUX
ejpam-6792	84	10	define	define	VERB
ejpam-6792	84	11	a	a	DET
ejpam-6792	84	12	direct	direct	ADJ
ejpam-6792	84	13	product	product	NOUN
ejpam-6792	84	14	on	on	ADP
ejpam-6792	84	15	q	q	NOUN
ejpam-6792	84	16	-	-	PUNCT
ejpam-6792	84	17	algebras	algebras	ADJ
ejpam-6792	84	18	in	in	ADP
ejpam-6792	84	19	a	a	DET
ejpam-6792	84	20	usual	usual	ADJ
ejpam-6792	84	21	way	way	NOUN
ejpam-6792	84	22	.	.	PUNCT
ejpam-6792	85	1	definition	definition	NOUN
ejpam-6792	85	2	1	1	NUM
ejpam-6792	85	3	.	.	PUNCT
ejpam-6792	86	1	let	let	VERB
ejpam-6792	86	2	{	{	PUNCT
ejpam-6792	86	3	(	(	PUNCT
ejpam-6792	86	4	xi	xi	INTJ
ejpam-6792	86	5	;	;	PUNCT
ejpam-6792	86	6	∗i	∗i	PROPN
ejpam-6792	86	7	,	,	PUNCT
ejpam-6792	86	8	0i)|	0i)|	VERB
ejpam-6792	87	1	i	i	NOUN
ejpam-6792	87	2	∈	∈	PROPN
ejpam-6792	88	1	i	i	PRON
ejpam-6792	88	2	}	}	PUNCT
ejpam-6792	88	3	be	be	VERB
ejpam-6792	88	4	a	a	DET
ejpam-6792	88	5	non	non	ADJ
ejpam-6792	88	6	-	-	ADJ
ejpam-6792	88	7	empty	empty	ADJ
ejpam-6792	88	8	family	family	NOUN
ejpam-6792	88	9	of	of	ADP
ejpam-6792	88	10	q	q	NOUN
ejpam-6792	88	11	-	-	PUNCT
ejpam-6792	88	12	algebras	algebras	X
ejpam-6792	88	13	.	.	PUNCT
ejpam-6792	89	1	let	let	VERB
ejpam-6792	89	2	∏	∏	PROPN
ejpam-6792	89	3	i∈i	i∈i	ADJ
ejpam-6792	89	4	xi	xi	AUX
ejpam-6792	89	5	be	be	AUX
ejpam-6792	89	6	the	the	DET
ejpam-6792	89	7	cartesian	cartesian	ADJ
ejpam-6792	89	8	product	product	NOUN
ejpam-6792	89	9	of	of	ADP
ejpam-6792	89	10	the	the	DET
ejpam-6792	89	11	set	set	NOUN
ejpam-6792	89	12	xi	xi	PROPN
ejpam-6792	89	13	,	,	PUNCT
ejpam-6792	89	14	i	i	PRON
ejpam-6792	89	15	∈	∈	PROPN
ejpam-6792	89	16	i:∏	i:∏	NUM
ejpam-6792	89	17	i∈i	i∈i	ADJ
ejpam-6792	89	18	xi	xi	X
ejpam-6792	89	19	=	=	PUNCT
ejpam-6792	89	20	{	{	PUNCT
ejpam-6792	89	21	(	(	PUNCT
ejpam-6792	89	22	xi)i∈i	xi)i∈i	NUM
ejpam-6792	89	23	|xi	|xi	X
ejpam-6792	89	24	∈	∈	PROPN
ejpam-6792	89	25	xi	xi	PROPN
ejpam-6792	89	26	,	,	PUNCT
ejpam-6792	89	27	i	i	PRON
ejpam-6792	89	28	∈	∈	PROPN
ejpam-6792	89	29	i	i	PRON
ejpam-6792	89	30	}	}	PUNCT
ejpam-6792	89	31	.	.	PUNCT
ejpam-6792	90	1	we	we	PRON
ejpam-6792	90	2	define	define	VERB
ejpam-6792	90	3	a	a	DET
ejpam-6792	90	4	binary	binary	ADJ
ejpam-6792	90	5	operation	operation	NOUN
ejpam-6792	90	6	⊛	⊛	NUM
ejpam-6792	90	7	on	on	ADP
ejpam-6792	90	8	∏	∏	PROPN
ejpam-6792	90	9	i∈i	i∈i	NOUN
ejpam-6792	90	10	xi	xi	INTJ
ejpam-6792	90	11	as	as	ADP
ejpam-6792	90	12	following	follow	VERB
ejpam-6792	90	13	:	:	PUNCT
ejpam-6792	90	14	for	for	ADP
ejpam-6792	90	15	(	(	PUNCT
ejpam-6792	90	16	xi)i∈i	xi)i∈i	NUM
ejpam-6792	90	17	,	,	PUNCT
ejpam-6792	90	18	(	(	PUNCT
ejpam-6792	90	19	yi)i∈i	yi)i∈i	NUM
ejpam-6792	90	20	∈	∈	PROPN
ejpam-6792	90	21	∏	∏	PROPN
ejpam-6792	90	22	i∈i	i∈i	ADJ
ejpam-6792	90	23	xi	xi	PROPN
ejpam-6792	90	24	,	,	PUNCT
ejpam-6792	90	25	(	(	PUNCT
ejpam-6792	90	26	xi)i∈i	xi)i∈i	X
ejpam-6792	90	27	⊛	⊛	NUM
ejpam-6792	90	28	(	(	PUNCT
ejpam-6792	90	29	yi)i∈i	yi)i∈i	NUM
ejpam-6792	90	30	=	=	SYM
ejpam-6792	90	31	(	(	PUNCT
ejpam-6792	90	32	xi	xi	X
ejpam-6792	90	33	∗	∗	PROPN
ejpam-6792	90	34	yi)i∈i	yi)i∈i	NUM
ejpam-6792	90	35	.	.	PUNCT
ejpam-6792	91	1	then	then	ADV
ejpam-6792	91	2	we	we	PRON
ejpam-6792	91	3	get	get	VERB
ejpam-6792	91	4	that	that	PRON
ejpam-6792	91	5	(	(	PUNCT
ejpam-6792	91	6	∏	∏	PROPN
ejpam-6792	91	7	i∈i	i∈i	ADJ
ejpam-6792	91	8	xi;⊛	xi;⊛	PROPN
ejpam-6792	91	9	,	,	PUNCT
ejpam-6792	91	10	(	(	PUNCT
ejpam-6792	91	11	0i)∈i	0i)∈i	NUM
ejpam-6792	91	12	)	)	PUNCT
ejpam-6792	91	13	is	be	AUX
ejpam-6792	91	14	a	a	DET
ejpam-6792	91	15	direct	direct	ADJ
ejpam-6792	91	16	product	product	NOUN
ejpam-6792	91	17	q	q	NOUN
ejpam-6792	91	18	-	-	NOUN
ejpam-6792	91	19	algebra	algebra	NOUN
ejpam-6792	91	20	as	as	SCONJ
ejpam-6792	91	21	shown	show	VERB
ejpam-6792	91	22	in	in	ADP
ejpam-6792	91	23	the	the	DET
ejpam-6792	91	24	following	follow	VERB
ejpam-6792	91	25	proposition	proposition	NOUN
ejpam-6792	91	26	.	.	PUNCT
ejpam-6792	92	1	proposition	proposition	NOUN
ejpam-6792	92	2	5	5	NUM
ejpam-6792	92	3	.	.	PUNCT
ejpam-6792	93	1	let	let	VERB
ejpam-6792	93	2	{	{	PUNCT
ejpam-6792	93	3	(	(	PUNCT
ejpam-6792	93	4	xi	xi	INTJ
ejpam-6792	93	5	;	;	PUNCT
ejpam-6792	93	6	∗i	∗i	PROPN
ejpam-6792	93	7	,	,	PUNCT
ejpam-6792	93	8	0i)|	0i)|	VERB
ejpam-6792	94	1	i	i	NOUN
ejpam-6792	94	2	∈	∈	PROPN
ejpam-6792	95	1	i	i	PRON
ejpam-6792	95	2	}	}	PUNCT
ejpam-6792	95	3	be	be	VERB
ejpam-6792	95	4	a	a	DET
ejpam-6792	95	5	non	non	ADJ
ejpam-6792	95	6	-	-	ADJ
ejpam-6792	95	7	empty	empty	ADJ
ejpam-6792	95	8	family	family	NOUN
ejpam-6792	95	9	of	of	ADP
ejpam-6792	95	10	q	q	NOUN
ejpam-6792	95	11	-	-	PUNCT
ejpam-6792	95	12	algebras	algebras	X
ejpam-6792	95	13	.	.	PUNCT
ejpam-6792	96	1	then	then	ADV
ejpam-6792	96	2	(	(	PUNCT
ejpam-6792	96	3	∏	∏	PROPN
ejpam-6792	96	4	i∈i	i∈i	ADJ
ejpam-6792	96	5	xi;⊛	xi;⊛	PROPN
ejpam-6792	96	6	,	,	PUNCT
ejpam-6792	96	7	(	(	PUNCT
ejpam-6792	96	8	0i)∈i	0i)∈i	NUM
ejpam-6792	96	9	)	)	PUNCT
ejpam-6792	96	10	is	be	AUX
ejpam-6792	96	11	a	a	DET
ejpam-6792	96	12	q	q	NOUN
ejpam-6792	96	13	-	-	PUNCT
ejpam-6792	96	14	algebra	algebra	NOUN
ejpam-6792	96	15	.	.	PUNCT
ejpam-6792	97	1	a.	a.	NOUN
ejpam-6792	97	2	anantayasethi	anantayasethi	PROPN
ejpam-6792	97	3	,	,	PUNCT
ejpam-6792	97	4	k.	k.	PROPN
ejpam-6792	97	5	saengsura	saengsura	PROPN
ejpam-6792	97	6	,	,	PUNCT
ejpam-6792	97	7	n.	n.	NOUN
ejpam-6792	97	8	sarasit	sarasit	PROPN
ejpam-6792	97	9	/	/	SYM
ejpam-6792	97	10	eur	eur	PROPN
ejpam-6792	97	11	.	.	PUNCT
ejpam-6792	98	1	j.	j.	PROPN
ejpam-6792	98	2	pure	pure	PROPN
ejpam-6792	98	3	appl	appl	PROPN
ejpam-6792	98	4	.	.	PROPN
ejpam-6792	98	5	math	math	PROPN
ejpam-6792	98	6	,	,	PUNCT
ejpam-6792	98	7	18	18	NUM
ejpam-6792	98	8	(	(	PUNCT
ejpam-6792	98	9	4	4	NUM
ejpam-6792	98	10	)	)	PUNCT
ejpam-6792	98	11	(	(	PUNCT
ejpam-6792	98	12	2025	2025	NUM
ejpam-6792	98	13	)	)	PUNCT
ejpam-6792	98	14	,	,	PUNCT
ejpam-6792	98	15	6792	6792	NUM
ejpam-6792	98	16	4	4	NUM
ejpam-6792	98	17	of	of	ADP
ejpam-6792	98	18	10	10	NUM
ejpam-6792	98	19	proof	proof	NOUN
ejpam-6792	98	20	.	.	PUNCT
ejpam-6792	99	1	since	since	SCONJ
ejpam-6792	99	2	0i	0i	PROPN
ejpam-6792	99	3	∈	∈	PROPN
ejpam-6792	99	4	xi	xi	PROPN
ejpam-6792	99	5	for	for	ADP
ejpam-6792	99	6	all	all	PRON
ejpam-6792	99	7	i	i	PRON
ejpam-6792	99	8	∈	∈	PROPN
ejpam-6792	100	1	i	i	PRON
ejpam-6792	100	2	,	,	PUNCT
ejpam-6792	100	3	then	then	ADV
ejpam-6792	100	4	(	(	PUNCT
ejpam-6792	100	5	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	100	6	∈	∈	PROPN
ejpam-6792	100	7	∏	∏	PROPN
ejpam-6792	100	8	i∈i	i∈i	ADJ
ejpam-6792	100	9	xi	xi	PROPN
ejpam-6792	100	10	.	.	PUNCT
ejpam-6792	101	1	let	let	VERB
ejpam-6792	101	2	(	(	PUNCT
ejpam-6792	101	3	xi)i∈i	xi)i∈i	NUM
ejpam-6792	101	4	,	,	PUNCT
ejpam-6792	101	5	(	(	PUNCT
ejpam-6792	101	6	yi)i∈i	yi)i∈i	NUM
ejpam-6792	101	7	,	,	PUNCT
ejpam-6792	101	8	(	(	PUNCT
ejpam-6792	101	9	zi)i∈i	zi)i∈i	NUM
ejpam-6792	101	10	∈∏	∈∏	NOUN
ejpam-6792	101	11	i∈i	i∈i	NOUN
ejpam-6792	101	12	xi	xi	PROPN
ejpam-6792	101	13	.	.	PUNCT
ejpam-6792	102	1	since	since	SCONJ
ejpam-6792	102	2	(	(	PUNCT
ejpam-6792	102	3	xi)i∈i	xi)i∈i	NUM
ejpam-6792	102	4	⊛	⊛	NUM
ejpam-6792	102	5	(	(	PUNCT
ejpam-6792	102	6	0i)i∈i	0i)i∈i	NUM
ejpam-6792	102	7	=	=	SYM
ejpam-6792	102	8	(	(	PUNCT
ejpam-6792	102	9	xi	xi	X
ejpam-6792	102	10	∗i	∗i	PROPN
ejpam-6792	102	11	0i)i∈i	0i)i∈i	NUM
ejpam-6792	102	12	=	=	SYM
ejpam-6792	102	13	(	(	PUNCT
ejpam-6792	102	14	xi)i∈i	xi)i∈i	NUM
ejpam-6792	102	15	then	then	ADV
ejpam-6792	102	16	the	the	DET
ejpam-6792	102	17	condition	condition	NOUN
ejpam-6792	102	18	(	(	PUNCT
ejpam-6792	102	19	q2	q2	NOUN
ejpam-6792	102	20	)	)	PUNCT
ejpam-6792	102	21	is	be	AUX
ejpam-6792	102	22	satisfied	satisfied	ADJ
ejpam-6792	102	23	.	.	PUNCT
ejpam-6792	103	1	moreover	moreover	ADV
ejpam-6792	103	2	,	,	PUNCT
ejpam-6792	103	3	(	(	PUNCT
ejpam-6792	103	4	xi)i∈i	xi)i∈i	X
ejpam-6792	103	5	⊛	⊛	NUM
ejpam-6792	103	6	(	(	PUNCT
ejpam-6792	103	7	xi)i∈i	xi)i∈i	NUM
ejpam-6792	103	8	=	=	SYM
ejpam-6792	103	9	(	(	PUNCT
ejpam-6792	103	10	xi	xi	ADP
ejpam-6792	103	11	∗i	∗i	PROPN
ejpam-6792	103	12	xi)i∈i	xi)i∈i	NUM
ejpam-6792	103	13	=	=	SYM
ejpam-6792	103	14	(	(	PUNCT
ejpam-6792	103	15	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	103	16	there	there	PRON
ejpam-6792	103	17	follows	follow	VERB
ejpam-6792	103	18	(	(	PUNCT
ejpam-6792	103	19	q1	q1	PROPN
ejpam-6792	103	20	)	)	PUNCT
ejpam-6792	103	21	is	be	AUX
ejpam-6792	103	22	fulfilled	fulfil	VERB
ejpam-6792	103	23	.	.	PUNCT
ejpam-6792	104	1	let	let	VERB
ejpam-6792	104	2	consider	consider	VERB
ejpam-6792	104	3	(	(	PUNCT
ejpam-6792	104	4	(	(	PUNCT
ejpam-6792	104	5	xi)i∈i	xi)i∈i	X
ejpam-6792	104	6	⊛	⊛	NUM
ejpam-6792	104	7	(	(	PUNCT
ejpam-6792	104	8	yi)i∈i)⊛	yi)i∈i)⊛	VERB
ejpam-6792	104	9	(	(	PUNCT
ejpam-6792	104	10	zi)i∈i	zi)i∈i	NUM
ejpam-6792	104	11	=	=	SYM
ejpam-6792	104	12	(	(	PUNCT
ejpam-6792	104	13	xi	xi	ADP
ejpam-6792	104	14	∗i	∗i	PROPN
ejpam-6792	104	15	yi)i∈i	yi)i∈i	X
ejpam-6792	104	16	⊛	⊛	NUM
ejpam-6792	104	17	(	(	PUNCT
ejpam-6792	104	18	zi)i∈i	zi)i∈i	NUM
ejpam-6792	104	19	=	=	SYM
ejpam-6792	104	20	(	(	PUNCT
ejpam-6792	104	21	(	(	PUNCT
ejpam-6792	104	22	xi	xi	ADP
ejpam-6792	104	23	∗i	∗i	PROPN
ejpam-6792	104	24	yi	yi	PROPN
ejpam-6792	104	25	)	)	PUNCT
ejpam-6792	104	26	∗i	∗i	PROPN
ejpam-6792	104	27	zi)i∈i	zi)i∈i	NUM
ejpam-6792	104	28	=	=	SYM
ejpam-6792	104	29	(	(	PUNCT
ejpam-6792	104	30	(	(	PUNCT
ejpam-6792	104	31	xi	xi	X
ejpam-6792	104	32	∗i	∗i	PROPN
ejpam-6792	104	33	zi	zi	PROPN
ejpam-6792	104	34	)	)	PUNCT
ejpam-6792	104	35	∗i	∗i	PROPN
ejpam-6792	104	36	yi)i∈i	yi)i∈i	NUM
ejpam-6792	104	37	=	=	SYM
ejpam-6792	104	38	(	(	PUNCT
ejpam-6792	104	39	xi	xi	ADP
ejpam-6792	104	40	∗i	∗i	PROPN
ejpam-6792	104	41	zi)i∈i	zi)i∈i	NUM
ejpam-6792	104	42	⊛	⊛	NUM
ejpam-6792	104	43	(	(	PUNCT
ejpam-6792	104	44	yi)i∈i	yi)i∈i	NUM
ejpam-6792	104	45	=	=	SYM
ejpam-6792	104	46	(	(	PUNCT
ejpam-6792	104	47	(	(	PUNCT
ejpam-6792	104	48	xi)i∈i	xi)i∈i	X
ejpam-6792	104	49	⊛	⊛	NUM
ejpam-6792	104	50	(	(	PUNCT
ejpam-6792	104	51	zi)i∈i)⊛	zi)i∈i)⊛	X
ejpam-6792	104	52	(	(	PUNCT
ejpam-6792	104	53	yi)i∈i	yi)i∈i	NUM
ejpam-6792	104	54	.	.	PUNCT
ejpam-6792	105	1	thus	thus	ADV
ejpam-6792	105	2	,	,	PUNCT
ejpam-6792	105	3	the	the	DET
ejpam-6792	105	4	condition	condition	NOUN
ejpam-6792	105	5	(	(	PUNCT
ejpam-6792	105	6	q3	q3	PROPN
ejpam-6792	105	7	)	)	PUNCT
ejpam-6792	105	8	is	be	AUX
ejpam-6792	105	9	satisfied	satisfied	ADJ
ejpam-6792	105	10	.	.	PUNCT
ejpam-6792	106	1	hence	hence	ADV
ejpam-6792	106	2	,	,	PUNCT
ejpam-6792	106	3	(	(	PUNCT
ejpam-6792	106	4	∏	∏	X
ejpam-6792	106	5	i∈i	i∈i	ADJ
ejpam-6792	106	6	xi;⊛	xi;⊛	PROPN
ejpam-6792	106	7	,	,	PUNCT
ejpam-6792	106	8	(	(	PUNCT
ejpam-6792	106	9	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	106	10	)	)	PUNCT
ejpam-6792	106	11	is	be	AUX
ejpam-6792	106	12	a	a	DET
ejpam-6792	106	13	q	q	NOUN
ejpam-6792	106	14	-	-	PUNCT
ejpam-6792	106	15	algebra	algebra	NOUN
ejpam-6792	106	16	.	.	PUNCT
ejpam-6792	106	17	example	example	NOUN
ejpam-6792	107	1	1	1	NUM
ejpam-6792	107	2	.	.	PUNCT
ejpam-6792	107	3	let	let	VERB
ejpam-6792	107	4	x1	x1	PROPN
ejpam-6792	107	5	=	=	PUNCT
ejpam-6792	107	6	{	{	PUNCT
ejpam-6792	107	7	01	01	NUM
ejpam-6792	107	8	,	,	PUNCT
ejpam-6792	107	9	a	a	PRON
ejpam-6792	107	10	}	}	PUNCT
ejpam-6792	107	11	and	and	CCONJ
ejpam-6792	107	12	x2	x2	PROPN
ejpam-6792	107	13	=	=	PRON
ejpam-6792	107	14	{	{	PUNCT
ejpam-6792	107	15	02	02	NUM
ejpam-6792	107	16	,	,	PUNCT
ejpam-6792	107	17	x	x	NOUN
ejpam-6792	107	18	,	,	PUNCT
ejpam-6792	107	19	y	y	PROPN
ejpam-6792	107	20	,	,	PUNCT
ejpam-6792	107	21	z	z	NOUN
ejpam-6792	107	22	}	}	PUNCT
ejpam-6792	107	23	be	be	AUX
ejpam-6792	107	24	the	the	DET
ejpam-6792	107	25	sets	set	NOUN
ejpam-6792	107	26	with	with	ADP
ejpam-6792	107	27	the	the	DET
ejpam-6792	107	28	binary	binary	ADJ
ejpam-6792	107	29	operations	operation	NOUN
ejpam-6792	107	30	∗1	∗1	PROPN
ejpam-6792	107	31	and	and	CCONJ
ejpam-6792	107	32	∗2	∗2	PROPN
ejpam-6792	107	33	defined	define	VERB
ejpam-6792	107	34	on	on	ADP
ejpam-6792	107	35	x1	x1	PROPN
ejpam-6792	107	36	and	and	CCONJ
ejpam-6792	107	37	x2	x2	PROPN
ejpam-6792	107	38	,	,	PUNCT
ejpam-6792	107	39	respectively	respectively	ADV
ejpam-6792	107	40	.	.	PUNCT
ejpam-6792	108	1	∗1	∗1	PROPN
ejpam-6792	108	2	01	01	NUM
ejpam-6792	109	1	a	a	DET
ejpam-6792	109	2	01	01	NUM
ejpam-6792	109	3	01	01	NUM
ejpam-6792	109	4	a	a	PRON
ejpam-6792	109	5	a	a	DET
ejpam-6792	109	6	a	a	DET
ejpam-6792	109	7	01	01	NUM
ejpam-6792	109	8	∗2	∗2	NOUN
ejpam-6792	109	9	02	02	NUM
ejpam-6792	109	10	x	x	SYM
ejpam-6792	109	11	y	y	PROPN
ejpam-6792	109	12	z	z	PROPN
ejpam-6792	109	13	02	02	NUM
ejpam-6792	109	14	02	02	NUM
ejpam-6792	110	1	x	x	PROPN
ejpam-6792	110	2	z	z	NOUN
ejpam-6792	110	3	y	y	NOUN
ejpam-6792	110	4	x	x	PUNCT
ejpam-6792	110	5	x	x	X
ejpam-6792	110	6	02	02	NUM
ejpam-6792	110	7	y	y	PROPN
ejpam-6792	110	8	z	z	PROPN
ejpam-6792	110	9	y	y	PROPN
ejpam-6792	110	10	y	y	PROPN
ejpam-6792	110	11	z	z	PROPN
ejpam-6792	110	12	02	02	NUM
ejpam-6792	110	13	x	x	X
ejpam-6792	110	14	z	z	NOUN
ejpam-6792	110	15	z	z	NOUN
ejpam-6792	110	16	y	y	NOUN
ejpam-6792	110	17	x	x	PROPN
ejpam-6792	110	18	02	02	NUM
ejpam-6792	110	19	.	.	PUNCT
ejpam-6792	111	1	then	then	ADV
ejpam-6792	111	2	(	(	PUNCT
ejpam-6792	111	3	x1	x1	NOUN
ejpam-6792	111	4	;	;	PUNCT
ejpam-6792	111	5	∗1	∗1	PROPN
ejpam-6792	111	6	,	,	PUNCT
ejpam-6792	111	7	01	01	NUM
ejpam-6792	111	8	)	)	PUNCT
ejpam-6792	111	9	and	and	CCONJ
ejpam-6792	111	10	(	(	PUNCT
ejpam-6792	111	11	x2	x2	PROPN
ejpam-6792	111	12	;	;	PUNCT
ejpam-6792	111	13	∗2	∗2	PROPN
ejpam-6792	111	14	,	,	PUNCT
ejpam-6792	111	15	02	02	NUM
ejpam-6792	111	16	)	)	PUNCT
ejpam-6792	111	17	are	be	AUX
ejpam-6792	111	18	q	q	NOUN
ejpam-6792	111	19	-	-	PUNCT
ejpam-6792	111	20	algebras	algebras	X
ejpam-6792	111	21	.	.	PUNCT
ejpam-6792	112	1	from	from	ADP
ejpam-6792	112	2	proposition	proposition	NOUN
ejpam-6792	112	3	5	5	NUM
ejpam-6792	112	4	,	,	PUNCT
ejpam-6792	112	5	(	(	PUNCT
ejpam-6792	112	6	x1×x2;⊛	x1×x2;⊛	PROPN
ejpam-6792	112	7	,	,	PUNCT
ejpam-6792	112	8	(	(	PUNCT
ejpam-6792	112	9	01	01	NUM
ejpam-6792	112	10	,	,	PUNCT
ejpam-6792	112	11	02	02	NUM
ejpam-6792	112	12	)	)	PUNCT
ejpam-6792	112	13	)	)	PUNCT
ejpam-6792	112	14	is	be	AUX
ejpam-6792	112	15	a	a	DET
ejpam-6792	112	16	q	q	NOUN
ejpam-6792	112	17	-	-	PUNCT
ejpam-6792	112	18	algebra	algebra	NOUN
ejpam-6792	112	19	,	,	PUNCT
ejpam-6792	112	20	illustrated	illustrate	VERB
ejpam-6792	112	21	as	as	ADP
ejpam-6792	112	22	the	the	DET
ejpam-6792	112	23	following	follow	VERB
ejpam-6792	112	24	table	table	NOUN
ejpam-6792	112	25	.	.	PUNCT
ejpam-6792	113	1	⊛	⊛	NUM
ejpam-6792	113	2	(	(	PUNCT
ejpam-6792	113	3	01	01	NUM
ejpam-6792	113	4	,	,	PUNCT
ejpam-6792	113	5	02	02	NUM
ejpam-6792	113	6	)	)	PUNCT
ejpam-6792	113	7	(	(	PUNCT
ejpam-6792	113	8	01	01	NUM
ejpam-6792	113	9	,	,	PUNCT
ejpam-6792	113	10	x	x	NOUN
ejpam-6792	113	11	)	)	PUNCT
ejpam-6792	113	12	(	(	PUNCT
ejpam-6792	113	13	01	01	NUM
ejpam-6792	113	14	,	,	PUNCT
ejpam-6792	113	15	y	y	NOUN
ejpam-6792	113	16	)	)	PUNCT
ejpam-6792	113	17	(	(	PUNCT
ejpam-6792	113	18	01	01	NUM
ejpam-6792	113	19	,	,	PUNCT
ejpam-6792	113	20	z	z	NOUN
ejpam-6792	113	21	)	)	PUNCT
ejpam-6792	113	22	(	(	PUNCT
ejpam-6792	113	23	a	a	PRON
ejpam-6792	113	24	,	,	PUNCT
ejpam-6792	113	25	02	02	NUM
ejpam-6792	113	26	)	)	PUNCT
ejpam-6792	113	27	(	(	PUNCT
ejpam-6792	113	28	a	a	PRON
ejpam-6792	113	29	,	,	PUNCT
ejpam-6792	113	30	x	x	NOUN
ejpam-6792	113	31	)	)	PUNCT
ejpam-6792	113	32	(	(	PUNCT
ejpam-6792	113	33	a	a	DET
ejpam-6792	113	34	,	,	PUNCT
ejpam-6792	113	35	y	y	NOUN
ejpam-6792	113	36	)	)	PUNCT
ejpam-6792	113	37	(	(	PUNCT
ejpam-6792	113	38	a	a	DET
ejpam-6792	113	39	,	,	PUNCT
ejpam-6792	113	40	z	z	NOUN
ejpam-6792	113	41	)	)	PUNCT
ejpam-6792	113	42	(	(	PUNCT
ejpam-6792	113	43	01	01	NUM
ejpam-6792	113	44	,	,	PUNCT
ejpam-6792	113	45	02	02	NUM
ejpam-6792	113	46	)	)	PUNCT
ejpam-6792	113	47	(	(	PUNCT
ejpam-6792	113	48	01	01	NUM
ejpam-6792	113	49	,	,	PUNCT
ejpam-6792	113	50	02	02	NUM
ejpam-6792	113	51	)	)	PUNCT
ejpam-6792	113	52	(	(	PUNCT
ejpam-6792	113	53	01	01	NUM
ejpam-6792	113	54	,	,	PUNCT
ejpam-6792	113	55	x	x	NOUN
ejpam-6792	113	56	)	)	PUNCT
ejpam-6792	113	57	(	(	PUNCT
ejpam-6792	113	58	01	01	NUM
ejpam-6792	113	59	,	,	PUNCT
ejpam-6792	113	60	z	z	NOUN
ejpam-6792	113	61	)	)	PUNCT
ejpam-6792	113	62	(	(	PUNCT
ejpam-6792	113	63	01	01	NUM
ejpam-6792	113	64	,	,	PUNCT
ejpam-6792	113	65	y	y	PROPN
ejpam-6792	113	66	)	)	PUNCT
ejpam-6792	113	67	(	(	PUNCT
ejpam-6792	113	68	a	a	PRON
ejpam-6792	113	69	,	,	PUNCT
ejpam-6792	113	70	02	02	NUM
ejpam-6792	113	71	)	)	PUNCT
ejpam-6792	113	72	(	(	PUNCT
ejpam-6792	113	73	a	a	PRON
ejpam-6792	113	74	,	,	PUNCT
ejpam-6792	113	75	x	x	NOUN
ejpam-6792	113	76	)	)	PUNCT
ejpam-6792	113	77	(	(	PUNCT
ejpam-6792	113	78	a	a	DET
ejpam-6792	113	79	,	,	PUNCT
ejpam-6792	113	80	z	z	NOUN
ejpam-6792	113	81	)	)	PUNCT
ejpam-6792	113	82	(	(	PUNCT
ejpam-6792	113	83	a	a	DET
ejpam-6792	113	84	,	,	PUNCT
ejpam-6792	113	85	y	y	NOUN
ejpam-6792	113	86	)	)	PUNCT
ejpam-6792	113	87	(	(	PUNCT
ejpam-6792	113	88	01	01	NUM
ejpam-6792	113	89	,	,	PUNCT
ejpam-6792	113	90	x	x	NOUN
ejpam-6792	113	91	)	)	PUNCT
ejpam-6792	113	92	(	(	PUNCT
ejpam-6792	113	93	01	01	NUM
ejpam-6792	113	94	,	,	PUNCT
ejpam-6792	113	95	x	x	NOUN
ejpam-6792	113	96	)	)	PUNCT
ejpam-6792	113	97	(	(	PUNCT
ejpam-6792	113	98	01	01	NUM
ejpam-6792	113	99	,	,	PUNCT
ejpam-6792	113	100	02	02	NUM
ejpam-6792	113	101	)	)	PUNCT
ejpam-6792	113	102	(	(	PUNCT
ejpam-6792	113	103	01	01	NUM
ejpam-6792	113	104	,	,	PUNCT
ejpam-6792	113	105	y	y	NOUN
ejpam-6792	113	106	)	)	PUNCT
ejpam-6792	113	107	(	(	PUNCT
ejpam-6792	113	108	01	01	NUM
ejpam-6792	113	109	,	,	PUNCT
ejpam-6792	113	110	z	z	NOUN
ejpam-6792	113	111	)	)	PUNCT
ejpam-6792	113	112	(	(	PUNCT
ejpam-6792	113	113	a	a	PRON
ejpam-6792	113	114	,	,	PUNCT
ejpam-6792	113	115	x	x	NOUN
ejpam-6792	113	116	)	)	PUNCT
ejpam-6792	113	117	(	(	PUNCT
ejpam-6792	113	118	a	a	PRON
ejpam-6792	113	119	,	,	PUNCT
ejpam-6792	113	120	02	02	NUM
ejpam-6792	113	121	)	)	PUNCT
ejpam-6792	113	122	(	(	PUNCT
ejpam-6792	113	123	a	a	DET
ejpam-6792	113	124	,	,	PUNCT
ejpam-6792	113	125	y	y	NOUN
ejpam-6792	113	126	)	)	PUNCT
ejpam-6792	113	127	(	(	PUNCT
ejpam-6792	113	128	a	a	DET
ejpam-6792	113	129	,	,	PUNCT
ejpam-6792	113	130	z	z	NOUN
ejpam-6792	113	131	)	)	PUNCT
ejpam-6792	113	132	(	(	PUNCT
ejpam-6792	113	133	01	01	NUM
ejpam-6792	113	134	,	,	PUNCT
ejpam-6792	113	135	y	y	NOUN
ejpam-6792	113	136	)	)	PUNCT
ejpam-6792	113	137	(	(	PUNCT
ejpam-6792	113	138	01	01	NUM
ejpam-6792	113	139	,	,	PUNCT
ejpam-6792	113	140	y	y	NOUN
ejpam-6792	113	141	)	)	PUNCT
ejpam-6792	113	142	(	(	PUNCT
ejpam-6792	113	143	01	01	NUM
ejpam-6792	113	144	,	,	PUNCT
ejpam-6792	113	145	z	z	NOUN
ejpam-6792	113	146	)	)	PUNCT
ejpam-6792	113	147	(	(	PUNCT
ejpam-6792	113	148	01	01	NUM
ejpam-6792	113	149	,	,	PUNCT
ejpam-6792	113	150	02	02	NUM
ejpam-6792	113	151	)	)	PUNCT
ejpam-6792	113	152	(	(	PUNCT
ejpam-6792	113	153	01	01	NUM
ejpam-6792	113	154	,	,	PUNCT
ejpam-6792	113	155	x	x	NOUN
ejpam-6792	113	156	)	)	PUNCT
ejpam-6792	113	157	(	(	PUNCT
ejpam-6792	113	158	a	a	DET
ejpam-6792	113	159	,	,	PUNCT
ejpam-6792	113	160	y	y	NOUN
ejpam-6792	113	161	)	)	PUNCT
ejpam-6792	113	162	(	(	PUNCT
ejpam-6792	113	163	a	a	PRON
ejpam-6792	113	164	,	,	PUNCT
ejpam-6792	113	165	z	z	NOUN
ejpam-6792	113	166	)	)	PUNCT
ejpam-6792	113	167	(	(	PUNCT
ejpam-6792	113	168	a	a	PRON
ejpam-6792	113	169	,	,	PUNCT
ejpam-6792	113	170	02	02	NUM
ejpam-6792	113	171	)	)	PUNCT
ejpam-6792	113	172	(	(	PUNCT
ejpam-6792	113	173	a	a	PRON
ejpam-6792	113	174	,	,	PUNCT
ejpam-6792	113	175	x	x	NOUN
ejpam-6792	113	176	)	)	PUNCT
ejpam-6792	113	177	(	(	PUNCT
ejpam-6792	113	178	01	01	NUM
ejpam-6792	113	179	,	,	PUNCT
ejpam-6792	113	180	z	z	NOUN
ejpam-6792	113	181	)	)	PUNCT
ejpam-6792	113	182	(	(	PUNCT
ejpam-6792	113	183	01	01	NUM
ejpam-6792	113	184	,	,	PUNCT
ejpam-6792	113	185	z	z	NOUN
ejpam-6792	113	186	)	)	PUNCT
ejpam-6792	113	187	(	(	PUNCT
ejpam-6792	113	188	01	01	NUM
ejpam-6792	113	189	,	,	PUNCT
ejpam-6792	113	190	y	y	NOUN
ejpam-6792	113	191	)	)	PUNCT
ejpam-6792	113	192	(	(	PUNCT
ejpam-6792	113	193	01	01	NUM
ejpam-6792	113	194	,	,	PUNCT
ejpam-6792	113	195	x	x	NOUN
ejpam-6792	113	196	)	)	PUNCT
ejpam-6792	113	197	(	(	PUNCT
ejpam-6792	113	198	01	01	NUM
ejpam-6792	113	199	,	,	PUNCT
ejpam-6792	113	200	02	02	NUM
ejpam-6792	113	201	)	)	PUNCT
ejpam-6792	113	202	(	(	PUNCT
ejpam-6792	113	203	a	a	DET
ejpam-6792	113	204	,	,	PUNCT
ejpam-6792	113	205	z	z	NOUN
ejpam-6792	113	206	)	)	PUNCT
ejpam-6792	113	207	(	(	PUNCT
ejpam-6792	113	208	a	a	DET
ejpam-6792	113	209	,	,	PUNCT
ejpam-6792	113	210	y	y	NOUN
ejpam-6792	113	211	)	)	PUNCT
ejpam-6792	113	212	(	(	PUNCT
ejpam-6792	113	213	a	a	PRON
ejpam-6792	113	214	,	,	PUNCT
ejpam-6792	113	215	x	x	NOUN
ejpam-6792	113	216	)	)	PUNCT
ejpam-6792	113	217	(	(	PUNCT
ejpam-6792	113	218	a	a	PRON
ejpam-6792	113	219	,	,	PUNCT
ejpam-6792	113	220	02	02	NUM
ejpam-6792	113	221	)	)	PUNCT
ejpam-6792	113	222	(	(	PUNCT
ejpam-6792	113	223	a	a	PRON
ejpam-6792	113	224	,	,	PUNCT
ejpam-6792	113	225	02	02	NUM
ejpam-6792	113	226	)	)	PUNCT
ejpam-6792	113	227	(	(	PUNCT
ejpam-6792	113	228	a	a	PRON
ejpam-6792	113	229	,	,	PUNCT
ejpam-6792	113	230	02	02	NUM
ejpam-6792	113	231	)	)	PUNCT
ejpam-6792	113	232	(	(	PUNCT
ejpam-6792	113	233	a	a	PRON
ejpam-6792	113	234	,	,	PUNCT
ejpam-6792	113	235	x	x	NOUN
ejpam-6792	113	236	)	)	PUNCT
ejpam-6792	113	237	(	(	PUNCT
ejpam-6792	113	238	a	a	DET
ejpam-6792	113	239	,	,	PUNCT
ejpam-6792	113	240	z	z	NOUN
ejpam-6792	113	241	)	)	PUNCT
ejpam-6792	113	242	(	(	PUNCT
ejpam-6792	113	243	a	a	DET
ejpam-6792	113	244	,	,	PUNCT
ejpam-6792	113	245	y	y	NOUN
ejpam-6792	113	246	)	)	PUNCT
ejpam-6792	113	247	(	(	PUNCT
ejpam-6792	113	248	01	01	NUM
ejpam-6792	113	249	,	,	PUNCT
ejpam-6792	113	250	02	02	NUM
ejpam-6792	113	251	)	)	PUNCT
ejpam-6792	113	252	(	(	PUNCT
ejpam-6792	113	253	01	01	NUM
ejpam-6792	113	254	,	,	PUNCT
ejpam-6792	113	255	x	x	NOUN
ejpam-6792	113	256	)	)	PUNCT
ejpam-6792	113	257	(	(	PUNCT
ejpam-6792	113	258	01	01	NUM
ejpam-6792	113	259	,	,	PUNCT
ejpam-6792	113	260	z	z	NOUN
ejpam-6792	113	261	)	)	PUNCT
ejpam-6792	113	262	(	(	PUNCT
ejpam-6792	113	263	01	01	NUM
ejpam-6792	113	264	,	,	PUNCT
ejpam-6792	113	265	y	y	PROPN
ejpam-6792	113	266	)	)	PUNCT
ejpam-6792	113	267	(	(	PUNCT
ejpam-6792	113	268	a	a	PRON
ejpam-6792	113	269	,	,	PUNCT
ejpam-6792	113	270	x	x	NOUN
ejpam-6792	113	271	)	)	PUNCT
ejpam-6792	113	272	(	(	PUNCT
ejpam-6792	113	273	a	a	PRON
ejpam-6792	113	274	,	,	PUNCT
ejpam-6792	113	275	x	x	NOUN
ejpam-6792	113	276	)	)	PUNCT
ejpam-6792	113	277	(	(	PUNCT
ejpam-6792	113	278	a	a	PRON
ejpam-6792	113	279	,	,	PUNCT
ejpam-6792	113	280	02	02	NUM
ejpam-6792	113	281	)	)	PUNCT
ejpam-6792	113	282	(	(	PUNCT
ejpam-6792	113	283	a	a	DET
ejpam-6792	113	284	,	,	PUNCT
ejpam-6792	113	285	y	y	NOUN
ejpam-6792	113	286	)	)	PUNCT
ejpam-6792	113	287	(	(	PUNCT
ejpam-6792	113	288	a	a	DET
ejpam-6792	113	289	,	,	PUNCT
ejpam-6792	113	290	z	z	NOUN
ejpam-6792	113	291	)	)	PUNCT
ejpam-6792	113	292	(	(	PUNCT
ejpam-6792	113	293	01	01	NUM
ejpam-6792	113	294	,	,	PUNCT
ejpam-6792	113	295	x	x	NOUN
ejpam-6792	113	296	)	)	PUNCT
ejpam-6792	113	297	(	(	PUNCT
ejpam-6792	113	298	01	01	NUM
ejpam-6792	113	299	,	,	PUNCT
ejpam-6792	113	300	02	02	NUM
ejpam-6792	113	301	)	)	PUNCT
ejpam-6792	113	302	(	(	PUNCT
ejpam-6792	113	303	01	01	NUM
ejpam-6792	113	304	,	,	PUNCT
ejpam-6792	113	305	y	y	NOUN
ejpam-6792	113	306	)	)	PUNCT
ejpam-6792	113	307	(	(	PUNCT
ejpam-6792	113	308	01	01	NUM
ejpam-6792	113	309	,	,	PUNCT
ejpam-6792	113	310	z	z	NOUN
ejpam-6792	113	311	)	)	PUNCT
ejpam-6792	113	312	(	(	PUNCT
ejpam-6792	113	313	a	a	DET
ejpam-6792	113	314	,	,	PUNCT
ejpam-6792	113	315	y	y	NOUN
ejpam-6792	113	316	)	)	PUNCT
ejpam-6792	113	317	(	(	PUNCT
ejpam-6792	113	318	a	a	DET
ejpam-6792	113	319	,	,	PUNCT
ejpam-6792	113	320	y	y	NOUN
ejpam-6792	113	321	)	)	PUNCT
ejpam-6792	113	322	(	(	PUNCT
ejpam-6792	113	323	a	a	DET
ejpam-6792	113	324	,	,	PUNCT
ejpam-6792	113	325	z	z	NOUN
ejpam-6792	113	326	)	)	PUNCT
ejpam-6792	113	327	(	(	PUNCT
ejpam-6792	113	328	a	a	PRON
ejpam-6792	113	329	,	,	PUNCT
ejpam-6792	113	330	02	02	NUM
ejpam-6792	113	331	)	)	PUNCT
ejpam-6792	113	332	(	(	PUNCT
ejpam-6792	113	333	a	a	PRON
ejpam-6792	113	334	,	,	PUNCT
ejpam-6792	113	335	x	x	NOUN
ejpam-6792	113	336	)	)	PUNCT
ejpam-6792	113	337	(	(	PUNCT
ejpam-6792	113	338	01	01	NUM
ejpam-6792	113	339	,	,	PUNCT
ejpam-6792	113	340	y	y	NOUN
ejpam-6792	113	341	)	)	PUNCT
ejpam-6792	113	342	(	(	PUNCT
ejpam-6792	113	343	01	01	NUM
ejpam-6792	113	344	,	,	PUNCT
ejpam-6792	113	345	z	z	NOUN
ejpam-6792	113	346	)	)	PUNCT
ejpam-6792	113	347	(	(	PUNCT
ejpam-6792	113	348	01	01	NUM
ejpam-6792	113	349	,	,	PUNCT
ejpam-6792	113	350	02	02	NUM
ejpam-6792	113	351	)	)	PUNCT
ejpam-6792	113	352	(	(	PUNCT
ejpam-6792	113	353	01	01	NUM
ejpam-6792	113	354	,	,	PUNCT
ejpam-6792	113	355	x	x	NOUN
ejpam-6792	113	356	)	)	PUNCT
ejpam-6792	113	357	(	(	PUNCT
ejpam-6792	113	358	a	a	DET
ejpam-6792	113	359	,	,	PUNCT
ejpam-6792	113	360	z	z	NOUN
ejpam-6792	113	361	)	)	PUNCT
ejpam-6792	113	362	(	(	PUNCT
ejpam-6792	113	363	a	a	PRON
ejpam-6792	113	364	,	,	PUNCT
ejpam-6792	113	365	z	z	NOUN
ejpam-6792	113	366	)	)	PUNCT
ejpam-6792	113	367	(	(	PUNCT
ejpam-6792	113	368	a	a	DET
ejpam-6792	113	369	,	,	PUNCT
ejpam-6792	113	370	y	y	NOUN
ejpam-6792	113	371	)	)	PUNCT
ejpam-6792	113	372	(	(	PUNCT
ejpam-6792	113	373	a	a	PRON
ejpam-6792	113	374	,	,	PUNCT
ejpam-6792	113	375	x	x	NOUN
ejpam-6792	113	376	)	)	PUNCT
ejpam-6792	113	377	(	(	PUNCT
ejpam-6792	113	378	a	a	PRON
ejpam-6792	113	379	,	,	PUNCT
ejpam-6792	113	380	02	02	NUM
ejpam-6792	113	381	)	)	PUNCT
ejpam-6792	113	382	(	(	PUNCT
ejpam-6792	113	383	01	01	NUM
ejpam-6792	113	384	,	,	PUNCT
ejpam-6792	113	385	z	z	NOUN
ejpam-6792	113	386	)	)	PUNCT
ejpam-6792	113	387	(	(	PUNCT
ejpam-6792	113	388	01	01	NUM
ejpam-6792	113	389	,	,	PUNCT
ejpam-6792	113	390	y	y	NOUN
ejpam-6792	113	391	)	)	PUNCT
ejpam-6792	113	392	(	(	PUNCT
ejpam-6792	113	393	01	01	NUM
ejpam-6792	113	394	,	,	PUNCT
ejpam-6792	113	395	x	x	NOUN
ejpam-6792	113	396	)	)	PUNCT
ejpam-6792	113	397	(	(	PUNCT
ejpam-6792	113	398	01	01	NUM
ejpam-6792	113	399	,	,	PUNCT
ejpam-6792	113	400	02	02	NUM
ejpam-6792	113	401	)	)	PUNCT
ejpam-6792	113	402	.	.	PUNCT
ejpam-6792	114	1	it	it	PRON
ejpam-6792	114	2	is	be	AUX
ejpam-6792	114	3	easy	easy	ADJ
ejpam-6792	114	4	to	to	PART
ejpam-6792	114	5	see	see	VERB
ejpam-6792	114	6	that	that	SCONJ
ejpam-6792	114	7	g(x1	g(x1	ADJ
ejpam-6792	114	8	×x2	×x2	NOUN
ejpam-6792	114	9	)	)	PUNCT
ejpam-6792	114	10	=	=	PRON
ejpam-6792	114	11	{	{	PUNCT
ejpam-6792	114	12	(	(	PUNCT
ejpam-6792	114	13	01	01	NUM
ejpam-6792	114	14	,	,	PUNCT
ejpam-6792	114	15	02	02	NUM
ejpam-6792	114	16	)	)	PUNCT
ejpam-6792	114	17	,	,	PUNCT
ejpam-6792	114	18	(	(	PUNCT
ejpam-6792	114	19	01	01	NUM
ejpam-6792	114	20	,	,	PUNCT
ejpam-6792	114	21	x	x	NOUN
ejpam-6792	114	22	)	)	PUNCT
ejpam-6792	114	23	,	,	PUNCT
ejpam-6792	114	24	(	(	PUNCT
ejpam-6792	114	25	a	a	PRON
ejpam-6792	114	26	,	,	PUNCT
ejpam-6792	114	27	02	02	NUM
ejpam-6792	114	28	)	)	PUNCT
ejpam-6792	114	29	,	,	PUNCT
ejpam-6792	114	30	(	(	PUNCT
ejpam-6792	114	31	a	a	PRON
ejpam-6792	114	32	,	,	PUNCT
ejpam-6792	114	33	x	x	NOUN
ejpam-6792	114	34	)	)	PUNCT
ejpam-6792	114	35	}	}	PUNCT
ejpam-6792	114	36	.	.	PUNCT
ejpam-6792	115	1	it	it	PRON
ejpam-6792	115	2	is	be	AUX
ejpam-6792	115	3	a	a	DET
ejpam-6792	115	4	routine	routine	NOUN
ejpam-6792	115	5	to	to	PART
ejpam-6792	115	6	verify	verify	VERB
ejpam-6792	115	7	that	that	SCONJ
ejpam-6792	115	8	the	the	DET
ejpam-6792	115	9	set	set	NOUN
ejpam-6792	115	10	s	s	PART
ejpam-6792	115	11	=	=	X
ejpam-6792	115	12	{	{	PUNCT
ejpam-6792	115	13	(	(	PUNCT
ejpam-6792	115	14	01	01	NUM
ejpam-6792	115	15	,	,	PUNCT
ejpam-6792	115	16	02	02	NUM
ejpam-6792	115	17	)	)	PUNCT
ejpam-6792	115	18	,	,	PUNCT
ejpam-6792	115	19	(	(	PUNCT
ejpam-6792	115	20	01	01	NUM
ejpam-6792	115	21	,	,	PUNCT
ejpam-6792	115	22	x	x	NOUN
ejpam-6792	115	23	)	)	PUNCT
ejpam-6792	115	24	,	,	PUNCT
ejpam-6792	115	25	(	(	PUNCT
ejpam-6792	115	26	a	a	DET
ejpam-6792	115	27	,	,	PUNCT
ejpam-6792	115	28	02	02	NUM
ejpam-6792	115	29	)	)	PUNCT
ejpam-6792	115	30	,	,	PUNCT
ejpam-6792	115	31	(	(	PUNCT
ejpam-6792	115	32	a	a	DET
ejpam-6792	115	33	,	,	PUNCT
ejpam-6792	115	34	x	x	NOUN
ejpam-6792	115	35	)	)	PUNCT
ejpam-6792	115	36	}	}	PUNCT
ejpam-6792	115	37	is	be	AUX
ejpam-6792	115	38	a	a	DET
ejpam-6792	115	39	subalgebra	subalgebra	NOUN
ejpam-6792	115	40	of	of	ADP
ejpam-6792	115	41	x1×x2	x1×x2	PROPN
ejpam-6792	115	42	.	.	PUNCT
ejpam-6792	116	1	moreover	moreover	ADV
ejpam-6792	116	2	,	,	PUNCT
ejpam-6792	116	3	we	we	PRON
ejpam-6792	116	4	get	get	VERB
ejpam-6792	116	5	that	that	DET
ejpam-6792	116	6	b1	b1	NOUN
ejpam-6792	116	7	=	=	SYM
ejpam-6792	116	8	{	{	PUNCT
ejpam-6792	116	9	01	01	NUM
ejpam-6792	116	10	,	,	PUNCT
ejpam-6792	116	11	a	a	PRON
ejpam-6792	116	12	}	}	PUNCT
ejpam-6792	116	13	is	be	AUX
ejpam-6792	116	14	a	a	DET
ejpam-6792	116	15	subalgebra	subalgebra	NOUN
ejpam-6792	116	16	of	of	ADP
ejpam-6792	116	17	x1	x1	PROPN
ejpam-6792	116	18	,	,	PUNCT
ejpam-6792	116	19	b2	b2	NOUN
ejpam-6792	116	20	=	=	SYM
ejpam-6792	116	21	{	{	PUNCT
ejpam-6792	116	22	02	02	NUM
ejpam-6792	116	23	,	,	PUNCT
ejpam-6792	116	24	x	x	PRON
ejpam-6792	116	25	}	}	PUNCT
ejpam-6792	116	26	is	be	AUX
ejpam-6792	116	27	a	a	DET
ejpam-6792	116	28	subalgebra	subalgebra	NOUN
ejpam-6792	116	29	of	of	ADP
ejpam-6792	116	30	x2	x2	PROPN
ejpam-6792	116	31	and	and	CCONJ
ejpam-6792	116	32	s	s	NOUN
ejpam-6792	116	33	=	=	ADJ
ejpam-6792	116	34	b1×b2	b1×b2	PROPN
ejpam-6792	116	35	.	.	PUNCT
ejpam-6792	117	1	in	in	ADP
ejpam-6792	117	2	general	general	ADJ
ejpam-6792	117	3	,	,	PUNCT
ejpam-6792	117	4	a	a	DET
ejpam-6792	117	5	sub	sub	ADJ
ejpam-6792	117	6	-	-	ADJ
ejpam-6792	117	7	direct	direct	ADJ
ejpam-6792	117	8	product	product	NOUN
ejpam-6792	117	9	∏	∏	NUM
ejpam-6792	117	10	i∈i	i∈i	ADJ
ejpam-6792	117	11	bi	bi	NOUN
ejpam-6792	117	12	of	of	ADP
ejpam-6792	117	13	a	a	DET
ejpam-6792	117	14	direct	direct	ADJ
ejpam-6792	117	15	product	product	NOUN
ejpam-6792	117	16	∏	∏	PROPN
ejpam-6792	117	17	i∈i	i∈i	ADV
ejpam-6792	117	18	xi	xi	ADP
ejpam-6792	117	19	of	of	ADP
ejpam-6792	117	20	q	q	NOUN
ejpam-6792	117	21	-	-	PUNCT
ejpam-6792	117	22	algebras	algebras	PROPN
ejpam-6792	117	23	is	be	AUX
ejpam-6792	117	24	a	a	DET
ejpam-6792	117	25	subalgebra	subalgebra	NOUN
ejpam-6792	117	26	whenever	whenever	SCONJ
ejpam-6792	117	27	bi	bi	NOUN
ejpam-6792	117	28	is	be	AUX
ejpam-6792	117	29	a	a	DET
ejpam-6792	117	30	subalgebra	subalgebra	NOUN
ejpam-6792	117	31	of	of	ADP
ejpam-6792	117	32	xi	xi	PROPN
ejpam-6792	117	33	for	for	ADP
ejpam-6792	117	34	all	all	DET
ejpam-6792	117	35	i	i	PRON
ejpam-6792	117	36	∈	∈	PROPN
ejpam-6792	117	37	i.	i.	NOUN
ejpam-6792	117	38	this	this	DET
ejpam-6792	117	39	fact	fact	NOUN
ejpam-6792	117	40	can	can	AUX
ejpam-6792	117	41	be	be	AUX
ejpam-6792	117	42	seen	see	VERB
ejpam-6792	117	43	in	in	ADP
ejpam-6792	117	44	the	the	DET
ejpam-6792	117	45	following	follow	VERB
ejpam-6792	117	46	proposition	proposition	NOUN
ejpam-6792	117	47	.	.	PUNCT
ejpam-6792	118	1	a.	a.	PROPN
ejpam-6792	118	2	anantayasethi	anantayasethi	PROPN
ejpam-6792	118	3	,	,	PUNCT
ejpam-6792	118	4	k.	k.	PROPN
ejpam-6792	118	5	saengsura	saengsura	PROPN
ejpam-6792	118	6	,	,	PUNCT
ejpam-6792	118	7	n.	n.	NOUN
ejpam-6792	118	8	sarasit	sarasit	PROPN
ejpam-6792	118	9	/	/	SYM
ejpam-6792	118	10	eur	eur	PROPN
ejpam-6792	118	11	.	.	PUNCT
ejpam-6792	119	1	j.	j.	PROPN
ejpam-6792	119	2	pure	pure	PROPN
ejpam-6792	119	3	appl	appl	PROPN
ejpam-6792	119	4	.	.	PROPN
ejpam-6792	119	5	math	math	PROPN
ejpam-6792	119	6	,	,	PUNCT
ejpam-6792	119	7	18	18	NUM
ejpam-6792	119	8	(	(	PUNCT
ejpam-6792	119	9	4	4	NUM
ejpam-6792	119	10	)	)	PUNCT
ejpam-6792	119	11	(	(	PUNCT
ejpam-6792	119	12	2025	2025	NUM
ejpam-6792	119	13	)	)	PUNCT
ejpam-6792	119	14	,	,	PUNCT
ejpam-6792	119	15	6792	6792	NUM
ejpam-6792	119	16	5	5	NUM
ejpam-6792	119	17	of	of	ADP
ejpam-6792	119	18	10	10	NUM
ejpam-6792	119	19	proposition	proposition	NOUN
ejpam-6792	119	20	6	6	NUM
ejpam-6792	119	21	.	.	PUNCT
ejpam-6792	120	1	let	let	VERB
ejpam-6792	120	2	∏	∏	PROPN
ejpam-6792	120	3	i∈i	i∈i	ADJ
ejpam-6792	120	4	xi	xi	AUX
ejpam-6792	120	5	be	be	AUX
ejpam-6792	120	6	the	the	DET
ejpam-6792	120	7	direct	direct	ADJ
ejpam-6792	120	8	product	product	NOUN
ejpam-6792	120	9	of	of	ADP
ejpam-6792	120	10	q	q	NOUN
ejpam-6792	120	11	-	-	PUNCT
ejpam-6792	120	12	algebras	algebra	VERB
ejpam-6792	120	13	and	and	CCONJ
ejpam-6792	120	14	let	let	VERB
ejpam-6792	120	15	∅	∅	NOUN
ejpam-6792	120	16	̸=	̸=	PROPN
ejpam-6792	120	17	bi	bi	NOUN
ejpam-6792	120	18	⊆	⊆	NUM
ejpam-6792	120	19	xi	xi	PROPN
ejpam-6792	120	20	for	for	ADP
ejpam-6792	120	21	all	all	DET
ejpam-6792	120	22	i	i	PRON
ejpam-6792	120	23	∈	∈	PROPN
ejpam-6792	120	24	i.	i.	NOUN
ejpam-6792	120	25	then	then	ADV
ejpam-6792	120	26	bi	bi	PROPN
ejpam-6792	120	27	is	be	AUX
ejpam-6792	120	28	a	a	DET
ejpam-6792	120	29	subalgebra	subalgebra	NOUN
ejpam-6792	120	30	of	of	ADP
ejpam-6792	120	31	xi	xi	PROPN
ejpam-6792	120	32	for	for	ADP
ejpam-6792	120	33	all	all	PRON
ejpam-6792	120	34	i	i	PRON
ejpam-6792	120	35	∈	∈	VERB
ejpam-6792	121	1	i	i	PRON
ejpam-6792	121	2	if	if	SCONJ
ejpam-6792	121	3	and	and	CCONJ
ejpam-6792	121	4	only	only	ADV
ejpam-6792	121	5	if	if	SCONJ
ejpam-6792	121	6	∏	∏	PROPN
ejpam-6792	121	7	i∈i	i∈i	ADJ
ejpam-6792	121	8	bi	bi	NOUN
ejpam-6792	121	9	is	be	AUX
ejpam-6792	121	10	a	a	DET
ejpam-6792	121	11	subalgebra	subalgebra	NOUN
ejpam-6792	121	12	of∏	of∏	NOUN
ejpam-6792	121	13	i∈i	i∈i	ADJ
ejpam-6792	121	14	xi	xi	PROPN
ejpam-6792	121	15	.	.	PUNCT
ejpam-6792	122	1	proof	proof	NOUN
ejpam-6792	122	2	.	.	PUNCT
ejpam-6792	123	1	assume	assume	VERB
ejpam-6792	123	2	bi	bi	PROPN
ejpam-6792	123	3	is	be	AUX
ejpam-6792	123	4	a	a	DET
ejpam-6792	123	5	subalgebra	subalgebra	NOUN
ejpam-6792	123	6	of	of	ADP
ejpam-6792	123	7	xi	xi	PROPN
ejpam-6792	123	8	for	for	ADP
ejpam-6792	123	9	all	all	PRON
ejpam-6792	123	10	i	i	PRON
ejpam-6792	123	11	∈	∈	PROPN
ejpam-6792	123	12	i.	i.	NOUN
ejpam-6792	123	13	let	let	VERB
ejpam-6792	123	14	(	(	PUNCT
ejpam-6792	123	15	xi)i∈i	xi)i∈i	NUM
ejpam-6792	123	16	,	,	PUNCT
ejpam-6792	123	17	(	(	PUNCT
ejpam-6792	123	18	yi)i∈i	yi)i∈i	NUM
ejpam-6792	123	19	∈	∈	PROPN
ejpam-6792	123	20	∏	∏	PROPN
ejpam-6792	123	21	i∈i	i∈i	ADJ
ejpam-6792	123	22	bi	bi	NOUN
ejpam-6792	123	23	.	.	PUNCT
ejpam-6792	124	1	then	then	ADV
ejpam-6792	124	2	xi	xi	PROPN
ejpam-6792	124	3	∗	∗	PROPN
ejpam-6792	124	4	yi	yi	PROPN
ejpam-6792	124	5	∈	∈	PROPN
ejpam-6792	124	6	bi	bi	NOUN
ejpam-6792	124	7	for	for	ADP
ejpam-6792	124	8	all	all	PRON
ejpam-6792	124	9	i	i	PRON
ejpam-6792	124	10	∈	∈	PROPN
ejpam-6792	125	1	i	i	PRON
ejpam-6792	125	2	,	,	PUNCT
ejpam-6792	125	3	there	there	PRON
ejpam-6792	125	4	follows	follow	VERB
ejpam-6792	125	5	that	that	SCONJ
ejpam-6792	125	6	(	(	PUNCT
ejpam-6792	125	7	xi)i∈i	xi)i∈i	X
ejpam-6792	125	8	⊛	⊛	NUM
ejpam-6792	125	9	(	(	PUNCT
ejpam-6792	125	10	yi)i∈i	yi)i∈i	NUM
ejpam-6792	125	11	=	=	SYM
ejpam-6792	125	12	(	(	PUNCT
ejpam-6792	125	13	xi	xi	ADP
ejpam-6792	125	14	∗i	∗i	PROPN
ejpam-6792	125	15	yi)i∈i	yi)i∈i	NUM
ejpam-6792	125	16	∈	∈	PROPN
ejpam-6792	125	17	∏	∏	PROPN
ejpam-6792	125	18	i∈i	i∈i	ADJ
ejpam-6792	125	19	bi	bi	NOUN
ejpam-6792	125	20	.	.	PUNCT
ejpam-6792	126	1	thus	thus	ADV
ejpam-6792	126	2	,	,	PUNCT
ejpam-6792	126	3	∏	∏	PROPN
ejpam-6792	126	4	i∈i	i∈i	ADJ
ejpam-6792	126	5	bi	bi	NOUN
ejpam-6792	126	6	is	be	AUX
ejpam-6792	126	7	a	a	DET
ejpam-6792	126	8	subalgebra	subalgebra	NOUN
ejpam-6792	126	9	.	.	PUNCT
ejpam-6792	127	1	conversely	conversely	ADV
ejpam-6792	127	2	,	,	PUNCT
ejpam-6792	127	3	let	let	VERB
ejpam-6792	127	4	xi	xi	NOUN
ejpam-6792	127	5	,	,	PUNCT
ejpam-6792	127	6	yi	yi	PROPN
ejpam-6792	127	7	∈	∈	PROPN
ejpam-6792	127	8	bi	bi	NOUN
ejpam-6792	127	9	for	for	ADP
ejpam-6792	127	10	each	each	DET
ejpam-6792	127	11	i	i	PROPN
ejpam-6792	127	12	∈	∈	PROPN
ejpam-6792	127	13	i.	i.	NOUN
ejpam-6792	127	14	since	since	SCONJ
ejpam-6792	127	15	∏	∏	PROPN
ejpam-6792	127	16	i∈i	i∈i	ADJ
ejpam-6792	127	17	bi	bi	NOUN
ejpam-6792	127	18	is	be	AUX
ejpam-6792	127	19	a	a	DET
ejpam-6792	127	20	subalgebra	subalgebra	NOUN
ejpam-6792	127	21	then	then	ADV
ejpam-6792	127	22	(	(	PUNCT
ejpam-6792	127	23	xi	xi	ADP
ejpam-6792	127	24	∗i	∗i	PROPN
ejpam-6792	127	25	yi)i∈i	yi)i∈i	NUM
ejpam-6792	127	26	=	=	SYM
ejpam-6792	127	27	(	(	PUNCT
ejpam-6792	127	28	xi)i∈i	xi)i∈i	X
ejpam-6792	127	29	⊛	⊛	NUM
ejpam-6792	127	30	(	(	PUNCT
ejpam-6792	127	31	yi)i∈i	yi)i∈i	NUM
ejpam-6792	127	32	∈	∈	PROPN
ejpam-6792	127	33	∏	∏	PROPN
ejpam-6792	127	34	i∈i	i∈i	ADJ
ejpam-6792	127	35	bi	bi	NOUN
ejpam-6792	127	36	.	.	PUNCT
ejpam-6792	128	1	therefore	therefore	ADV
ejpam-6792	128	2	,	,	PUNCT
ejpam-6792	128	3	xi	xi	ADP
ejpam-6792	128	4	∗i	∗i	PROPN
ejpam-6792	128	5	yi	yi	PROPN
ejpam-6792	128	6	∈	∈	PROPN
ejpam-6792	128	7	bi	bi	NOUN
ejpam-6792	128	8	and	and	CCONJ
ejpam-6792	128	9	there	there	PRON
ejpam-6792	128	10	follows	follow	VERB
ejpam-6792	128	11	bi	bi	NOUN
ejpam-6792	128	12	is	be	AUX
ejpam-6792	128	13	a	a	DET
ejpam-6792	128	14	subalgebra	subalgebra	NOUN
ejpam-6792	128	15	of	of	ADP
ejpam-6792	128	16	xi	xi	PROPN
ejpam-6792	128	17	for	for	ADP
ejpam-6792	128	18	all	all	PRON
ejpam-6792	128	19	i	i	PRON
ejpam-6792	128	20	∈	∈	PROPN
ejpam-6792	128	21	i.	i.	NOUN
ejpam-6792	128	22	example	example	NOUN
ejpam-6792	128	23	2	2	X
ejpam-6792	128	24	.	.	X
ejpam-6792	128	25	consider	consider	VERB
ejpam-6792	128	26	q	q	NOUN
ejpam-6792	128	27	-	-	PUNCT
ejpam-6792	128	28	algebras	algebras	X
ejpam-6792	128	29	(	(	PUNCT
ejpam-6792	128	30	x1	x1	PROPN
ejpam-6792	128	31	;	;	PUNCT
ejpam-6792	128	32	∗1	∗1	PROPN
ejpam-6792	128	33	,	,	PUNCT
ejpam-6792	128	34	01	01	NUM
ejpam-6792	128	35	)	)	PUNCT
ejpam-6792	128	36	and	and	CCONJ
ejpam-6792	128	37	(	(	PUNCT
ejpam-6792	128	38	x2	x2	PROPN
ejpam-6792	128	39	;	;	PUNCT
ejpam-6792	128	40	∗2	∗2	PROPN
ejpam-6792	128	41	,	,	PUNCT
ejpam-6792	128	42	02	02	NUM
ejpam-6792	128	43	)	)	PUNCT
ejpam-6792	128	44	from	from	ADP
ejpam-6792	128	45	example	example	NOUN
ejpam-6792	128	46	1	1	X
ejpam-6792	128	47	.	.	PUNCT
ejpam-6792	129	1	we	we	PRON
ejpam-6792	129	2	see	see	VERB
ejpam-6792	129	3	that	that	DET
ejpam-6792	129	4	g(x1	g(x1	NOUN
ejpam-6792	129	5	)	)	PUNCT
ejpam-6792	129	6	=	=	PUNCT
ejpam-6792	129	7	{	{	PUNCT
ejpam-6792	129	8	01	01	NUM
ejpam-6792	129	9	,	,	PUNCT
ejpam-6792	129	10	a	a	DET
ejpam-6792	129	11	}	}	PUNCT
ejpam-6792	129	12	,	,	PUNCT
ejpam-6792	129	13	g(x2	g(x2	NOUN
ejpam-6792	129	14	)	)	PUNCT
ejpam-6792	129	15	=	=	PUNCT
ejpam-6792	129	16	{	{	PUNCT
ejpam-6792	129	17	02	02	NUM
ejpam-6792	129	18	,	,	PUNCT
ejpam-6792	129	19	x	x	NOUN
ejpam-6792	129	20	}	}	PUNCT
ejpam-6792	129	21	and	and	CCONJ
ejpam-6792	129	22	g(x1	g(x1	ADJ
ejpam-6792	129	23	)	)	PUNCT
ejpam-6792	129	24	×	×	NOUN
ejpam-6792	129	25	g(x2	g(x2	NOUN
ejpam-6792	129	26	)	)	PUNCT
ejpam-6792	129	27	=	=	PUNCT
ejpam-6792	129	28	{	{	PUNCT
ejpam-6792	129	29	01	01	NUM
ejpam-6792	129	30	,	,	PUNCT
ejpam-6792	129	31	a	a	DET
ejpam-6792	129	32	}	}	PUNCT
ejpam-6792	129	33	×	×	NOUN
ejpam-6792	129	34	{	{	PUNCT
ejpam-6792	129	35	02	02	NUM
ejpam-6792	129	36	,	,	PUNCT
ejpam-6792	129	37	x	x	NOUN
ejpam-6792	129	38	}	}	PUNCT
ejpam-6792	129	39	=	=	SYM
ejpam-6792	129	40	{	{	PUNCT
ejpam-6792	129	41	(	(	PUNCT
ejpam-6792	129	42	01	01	NUM
ejpam-6792	129	43	,	,	PUNCT
ejpam-6792	129	44	02	02	NUM
ejpam-6792	129	45	)	)	PUNCT
ejpam-6792	129	46	,	,	PUNCT
ejpam-6792	129	47	(	(	PUNCT
ejpam-6792	129	48	01	01	NUM
ejpam-6792	129	49	,	,	PUNCT
ejpam-6792	129	50	x	x	NOUN
ejpam-6792	129	51	)	)	PUNCT
ejpam-6792	129	52	,	,	PUNCT
ejpam-6792	129	53	(	(	PUNCT
ejpam-6792	129	54	a	a	PRON
ejpam-6792	129	55	,	,	PUNCT
ejpam-6792	129	56	02	02	NUM
ejpam-6792	129	57	)	)	PUNCT
ejpam-6792	129	58	,	,	PUNCT
ejpam-6792	129	59	(	(	PUNCT
ejpam-6792	129	60	a	a	PRON
ejpam-6792	129	61	,	,	PUNCT
ejpam-6792	129	62	x	x	NOUN
ejpam-6792	129	63	)	)	PUNCT
ejpam-6792	129	64	}	}	PUNCT
ejpam-6792	130	1	=	=	SYM
ejpam-6792	130	2	g(x1	g(x1	ADJ
ejpam-6792	130	3	×x2	×x2	NOUN
ejpam-6792	130	4	)	)	PUNCT
ejpam-6792	130	5	.	.	PUNCT
ejpam-6792	131	1	example	example	NOUN
ejpam-6792	131	2	2	2	NUM
ejpam-6792	131	3	shows	show	VERB
ejpam-6792	131	4	a	a	DET
ejpam-6792	131	5	relation	relation	NOUN
ejpam-6792	131	6	of	of	ADP
ejpam-6792	131	7	the	the	DET
ejpam-6792	131	8	set	set	NOUN
ejpam-6792	131	9	g	g	NOUN
ejpam-6792	131	10	-	-	PUNCT
ejpam-6792	131	11	part	part	NOUN
ejpam-6792	131	12	of	of	ADP
ejpam-6792	131	13	the	the	DET
ejpam-6792	131	14	direct	direct	ADJ
ejpam-6792	131	15	product	product	NOUN
ejpam-6792	131	16	of	of	ADP
ejpam-6792	131	17	q	q	NOUN
ejpam-6792	131	18	-	-	PUNCT
ejpam-6792	131	19	algebras	algebras	X
ejpam-6792	131	20	,	,	PUNCT
ejpam-6792	131	21	g	g	PROPN
ejpam-6792	131	22	(	(	PUNCT
ejpam-6792	131	23	∏	∏	PROPN
ejpam-6792	131	24	i∈i	i∈i	ADJ
ejpam-6792	131	25	xi	xi	PROPN
ejpam-6792	131	26	)	)	PUNCT
ejpam-6792	131	27	,	,	PUNCT
ejpam-6792	131	28	and	and	CCONJ
ejpam-6792	131	29	a	a	DET
ejpam-6792	131	30	direct	direct	ADJ
ejpam-6792	131	31	product	product	NOUN
ejpam-6792	131	32	of	of	ADP
ejpam-6792	131	33	the	the	DET
ejpam-6792	131	34	set	set	NOUN
ejpam-6792	131	35	g	g	NOUN
ejpam-6792	131	36	-	-	PUNCT
ejpam-6792	131	37	part	part	NOUN
ejpam-6792	131	38	of	of	ADP
ejpam-6792	131	39	xi	xi	PROPN
ejpam-6792	131	40	,	,	PUNCT
ejpam-6792	131	41	i	i	PRON
ejpam-6792	131	42	∈	∈	PROPN
ejpam-6792	131	43	i	i	PRON
ejpam-6792	131	44	,	,	PUNCT
ejpam-6792	131	45	∏	∏	PROPN
ejpam-6792	131	46	i∈i	i∈i	PROPN
ejpam-6792	131	47	g(xi	g(xi	PROPN
ejpam-6792	131	48	)	)	PUNCT
ejpam-6792	131	49	.	.	PUNCT
ejpam-6792	132	1	both	both	DET
ejpam-6792	132	2	sets	set	NOUN
ejpam-6792	132	3	are	be	AUX
ejpam-6792	132	4	coincide	coincide	ADJ
ejpam-6792	132	5	as	as	SCONJ
ejpam-6792	132	6	shown	show	VERB
ejpam-6792	132	7	in	in	ADP
ejpam-6792	132	8	the	the	DET
ejpam-6792	132	9	following	follow	VERB
ejpam-6792	132	10	proposition	proposition	NOUN
ejpam-6792	132	11	.	.	PUNCT
ejpam-6792	133	1	proposition	proposition	NOUN
ejpam-6792	133	2	7	7	NUM
ejpam-6792	133	3	.	.	PUNCT
ejpam-6792	134	1	g	g	NOUN
ejpam-6792	134	2	(	(	PUNCT
ejpam-6792	134	3	∏	∏	PROPN
ejpam-6792	134	4	i∈i	i∈i	ADJ
ejpam-6792	134	5	xi	xi	NOUN
ejpam-6792	134	6	)	)	PUNCT
ejpam-6792	134	7	=	=	SYM
ejpam-6792	134	8	∏	∏	PROPN
ejpam-6792	134	9	i∈i	i∈i	PROPN
ejpam-6792	134	10	g(xi	g(xi	PROPN
ejpam-6792	134	11	)	)	PUNCT
ejpam-6792	134	12	.	.	PUNCT
ejpam-6792	135	1	proof	proof	NOUN
ejpam-6792	135	2	.	.	PUNCT
ejpam-6792	136	1	let	let	VERB
ejpam-6792	136	2	(	(	PUNCT
ejpam-6792	136	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	136	4	∈	∈	PROPN
ejpam-6792	136	5	g	g	NOUN
ejpam-6792	136	6	(	(	PUNCT
ejpam-6792	136	7	∏	∏	PROPN
ejpam-6792	136	8	i∈i	i∈i	ADJ
ejpam-6792	136	9	xi	xi	PROPN
ejpam-6792	136	10	)	)	PUNCT
ejpam-6792	136	11	.	.	PUNCT
ejpam-6792	137	1	then	then	ADV
ejpam-6792	137	2	(	(	PUNCT
ejpam-6792	137	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	137	4	=	=	SYM
ejpam-6792	137	5	(	(	PUNCT
ejpam-6792	137	6	0i)i∈i	0i)i∈i	NUM
ejpam-6792	137	7	⊛	⊛	NUM
ejpam-6792	137	8	(	(	PUNCT
ejpam-6792	137	9	ai)i∈i	ai)i∈i	NUM
ejpam-6792	137	10	=	=	SYM
ejpam-6792	137	11	(	(	PUNCT
ejpam-6792	137	12	0i	0i	X
ejpam-6792	137	13	∗i	∗i	PROPN
ejpam-6792	137	14	ai)i∈i	ai)i∈i	NUM
ejpam-6792	137	15	.	.	PUNCT
ejpam-6792	138	1	it	it	PRON
ejpam-6792	138	2	follows	follow	VERB
ejpam-6792	138	3	that	that	SCONJ
ejpam-6792	138	4	ai	ai	VERB
ejpam-6792	138	5	=	=	NOUN
ejpam-6792	138	6	0i	0i	NOUN
ejpam-6792	138	7	∗i	∗i	PROPN
ejpam-6792	138	8	ai	ai	VERB
ejpam-6792	138	9	for	for	ADP
ejpam-6792	138	10	all	all	DET
ejpam-6792	138	11	i	i	PRON
ejpam-6792	138	12	∈	∈	PROPN
ejpam-6792	138	13	i.	i.	NOUN
ejpam-6792	138	14	hence	hence	ADV
ejpam-6792	138	15	,	,	PUNCT
ejpam-6792	138	16	ai	ai	VERB
ejpam-6792	138	17	∈	∈	PROPN
ejpam-6792	138	18	g(xi	g(xi	PROPN
ejpam-6792	138	19	)	)	PUNCT
ejpam-6792	138	20	for	for	ADP
ejpam-6792	138	21	all	all	PRON
ejpam-6792	138	22	i	i	PRON
ejpam-6792	138	23	∈	∈	PROPN
ejpam-6792	138	24	i.	i.	NOUN
ejpam-6792	138	25	therefore	therefore	ADV
ejpam-6792	138	26	,	,	PUNCT
ejpam-6792	138	27	(	(	PUNCT
ejpam-6792	138	28	ai)i∈i	ai)i∈i	NUM
ejpam-6792	138	29	∈∏	∈∏	ADJ
ejpam-6792	138	30	i∈i	i∈i	ADJ
ejpam-6792	138	31	g(xi	g(xi	PROPN
ejpam-6792	138	32	)	)	PUNCT
ejpam-6792	138	33	,	,	PUNCT
ejpam-6792	138	34	i.e.	i.e.	X
ejpam-6792	138	35	g	g	PROPN
ejpam-6792	138	36	(	(	PUNCT
ejpam-6792	138	37	∏	∏	PROPN
ejpam-6792	138	38	i∈i	i∈i	ADJ
ejpam-6792	138	39	xi	xi	NOUN
ejpam-6792	138	40	)	)	PUNCT
ejpam-6792	138	41	⊆	⊆	NUM
ejpam-6792	138	42	∏	∏	PROPN
ejpam-6792	138	43	i∈i	i∈i	ADJ
ejpam-6792	138	44	g(xi	g(xi	PROPN
ejpam-6792	138	45	)	)	PUNCT
ejpam-6792	138	46	.	.	PUNCT
ejpam-6792	139	1	for	for	ADP
ejpam-6792	139	2	the	the	DET
ejpam-6792	139	3	opposite	opposite	ADJ
ejpam-6792	139	4	inclusion	inclusion	NOUN
ejpam-6792	139	5	,	,	PUNCT
ejpam-6792	139	6	let	let	VERB
ejpam-6792	139	7	(	(	PUNCT
ejpam-6792	139	8	ci)i∈i	ci)i∈i	NUM
ejpam-6792	139	9	∈	∈	PROPN
ejpam-6792	139	10	∏	∏	PROPN
ejpam-6792	139	11	i∈i	i∈i	PROPN
ejpam-6792	139	12	g(xi	g(xi	PROPN
ejpam-6792	139	13	)	)	PUNCT
ejpam-6792	139	14	.	.	PUNCT
ejpam-6792	140	1	then	then	ADV
ejpam-6792	140	2	for	for	ADP
ejpam-6792	140	3	all	all	PRON
ejpam-6792	140	4	i	i	PRON
ejpam-6792	140	5	∈	∈	PROPN
ejpam-6792	141	1	i	i	PRON
ejpam-6792	141	2	,	,	PUNCT
ejpam-6792	141	3	ci	ci	PROPN
ejpam-6792	141	4	∈	∈	PROPN
ejpam-6792	141	5	g(xi	g(xi	PROPN
ejpam-6792	141	6	)	)	PUNCT
ejpam-6792	141	7	.	.	PUNCT
ejpam-6792	142	1	it	it	PRON
ejpam-6792	142	2	follows	follow	VERB
ejpam-6792	142	3	that	that	SCONJ
ejpam-6792	142	4	0i	0i	NOUN
ejpam-6792	142	5	∗i	∗i	PROPN
ejpam-6792	142	6	ci	ci	PROPN
ejpam-6792	142	7	=	=	PROPN
ejpam-6792	142	8	ci	ci	PROPN
ejpam-6792	143	1	and	and	CCONJ
ejpam-6792	143	2	then	then	ADV
ejpam-6792	143	3	(	(	PUNCT
ejpam-6792	143	4	ci)i∈i	ci)i∈i	NUM
ejpam-6792	143	5	=	=	SYM
ejpam-6792	143	6	(	(	PUNCT
ejpam-6792	143	7	0i	0i	X
ejpam-6792	143	8	∗i	∗i	VERB
ejpam-6792	143	9	ci)i∈i	ci)i∈i	NUM
ejpam-6792	143	10	=	=	SYM
ejpam-6792	143	11	(	(	PUNCT
ejpam-6792	143	12	0i)i∈i	0i)i∈i	NUM
ejpam-6792	143	13	⊛	⊛	NUM
ejpam-6792	143	14	(	(	PUNCT
ejpam-6792	143	15	ci)i∈i	ci)i∈i	NUM
ejpam-6792	143	16	.	.	PUNCT
ejpam-6792	144	1	thus	thus	ADV
ejpam-6792	144	2	,	,	PUNCT
ejpam-6792	144	3	(	(	PUNCT
ejpam-6792	144	4	ci)i∈i	ci)i∈i	PROPN
ejpam-6792	144	5	∈	∈	NOUN
ejpam-6792	144	6	g	g	NOUN
ejpam-6792	144	7	(	(	PUNCT
ejpam-6792	144	8	∏	∏	PROPN
ejpam-6792	144	9	i∈i	i∈i	ADJ
ejpam-6792	144	10	xi	xi	PROPN
ejpam-6792	144	11	)	)	PUNCT
ejpam-6792	144	12	.	.	PUNCT
ejpam-6792	145	1	hence	hence	ADV
ejpam-6792	145	2	,	,	PUNCT
ejpam-6792	145	3	∏	∏	PROPN
ejpam-6792	145	4	i∈i	i∈i	PROPN
ejpam-6792	145	5	g(xi	g(xi	PROPN
ejpam-6792	145	6	)	)	PUNCT
ejpam-6792	145	7	⊆	⊆	NUM
ejpam-6792	145	8	g	g	NOUN
ejpam-6792	145	9	(	(	PUNCT
ejpam-6792	145	10	∏	∏	PROPN
ejpam-6792	145	11	i∈i	i∈i	ADJ
ejpam-6792	145	12	xi	xi	PROPN
ejpam-6792	145	13	)	)	PUNCT
ejpam-6792	145	14	.	.	PUNCT
ejpam-6792	146	1	for	for	ADP
ejpam-6792	146	2	any	any	DET
ejpam-6792	146	3	element	element	NOUN
ejpam-6792	146	4	(	(	PUNCT
ejpam-6792	146	5	ai)i∈i	ai)i∈i	NUM
ejpam-6792	146	6	∈	∈	PROPN
ejpam-6792	146	7	g	g	NOUN
ejpam-6792	146	8	(	(	PUNCT
ejpam-6792	146	9	∏	∏	PROPN
ejpam-6792	146	10	i∈i	i∈i	ADJ
ejpam-6792	146	11	xi	xi	PROPN
ejpam-6792	146	12	)	)	PUNCT
ejpam-6792	146	13	,	,	PUNCT
ejpam-6792	146	14	(	(	PUNCT
ejpam-6792	146	15	0i)i∈i	0i)i∈i	NUM
ejpam-6792	146	16	⊛	⊛	NUM
ejpam-6792	146	17	(	(	PUNCT
ejpam-6792	146	18	ai)i∈i	ai)i∈i	NUM
ejpam-6792	146	19	=	=	SYM
ejpam-6792	146	20	(	(	PUNCT
ejpam-6792	146	21	ai)i∈i	ai)i∈i	NUM
ejpam-6792	146	22	.	.	PUNCT
ejpam-6792	147	1	combining	combine	VERB
ejpam-6792	147	2	this	this	DET
ejpam-6792	147	3	fact	fact	NOUN
ejpam-6792	147	4	and	and	CCONJ
ejpam-6792	147	5	conditions	condition	NOUN
ejpam-6792	147	6	(	(	PUNCT
ejpam-6792	147	7	q1	q1	PROPN
ejpam-6792	147	8	)	)	PUNCT
ejpam-6792	147	9	,	,	PUNCT
ejpam-6792	147	10	(	(	PUNCT
ejpam-6792	147	11	q2	q2	NOUN
ejpam-6792	147	12	)	)	PUNCT
ejpam-6792	147	13	we	we	PRON
ejpam-6792	147	14	get	get	VERB
ejpam-6792	147	15	the	the	DET
ejpam-6792	147	16	following	follow	VERB
ejpam-6792	147	17	proposition	proposition	NOUN
ejpam-6792	147	18	.	.	PUNCT
ejpam-6792	148	1	proposition	proposition	NOUN
ejpam-6792	148	2	8	8	NUM
ejpam-6792	148	3	.	.	PUNCT
ejpam-6792	149	1	a	a	DET
ejpam-6792	149	2	subset	subset	NOUN
ejpam-6792	149	3	s	s	X
ejpam-6792	149	4	=	=	X
ejpam-6792	149	5	{	{	PUNCT
ejpam-6792	149	6	(	(	PUNCT
ejpam-6792	149	7	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	149	8	,	,	PUNCT
ejpam-6792	149	9	(	(	PUNCT
ejpam-6792	149	10	ai)i∈i	ai)i∈i	NUM
ejpam-6792	149	11	}	}	PUNCT
ejpam-6792	149	12	is	be	AUX
ejpam-6792	149	13	a	a	DET
ejpam-6792	149	14	subalgebra	subalgebra	NOUN
ejpam-6792	149	15	of	of	ADP
ejpam-6792	149	16	∏	∏	PROPN
ejpam-6792	149	17	i∈i	i∈i	NOUN
ejpam-6792	149	18	xi	xi	NOUN
ejpam-6792	149	19	for	for	ADP
ejpam-6792	149	20	all	all	PRON
ejpam-6792	149	21	(	(	PUNCT
ejpam-6792	149	22	ai)i∈i	ai)i∈i	NUM
ejpam-6792	149	23	∈	∈	PROPN
ejpam-6792	149	24	g	g	NOUN
ejpam-6792	149	25	(	(	PUNCT
ejpam-6792	149	26	∏	∏	PROPN
ejpam-6792	149	27	i∈i	i∈i	ADJ
ejpam-6792	149	28	xi	xi	PROPN
ejpam-6792	149	29	)	)	PUNCT
ejpam-6792	149	30	.	.	PUNCT
ejpam-6792	150	1	proof	proof	NOUN
ejpam-6792	150	2	.	.	PUNCT
ejpam-6792	151	1	let	let	VERB
ejpam-6792	151	2	(	(	PUNCT
ejpam-6792	151	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	151	4	∈	∈	PROPN
ejpam-6792	151	5	g	g	NOUN
ejpam-6792	151	6	(	(	PUNCT
ejpam-6792	151	7	∏	∏	PROPN
ejpam-6792	151	8	i∈i	i∈i	ADJ
ejpam-6792	151	9	xi	xi	PROPN
ejpam-6792	151	10	)	)	PUNCT
ejpam-6792	151	11	and	and	CCONJ
ejpam-6792	151	12	let	let	VERB
ejpam-6792	151	13	s	s	PRON
ejpam-6792	151	14	=	=	VERB
ejpam-6792	151	15	{	{	PUNCT
ejpam-6792	151	16	(	(	PUNCT
ejpam-6792	151	17	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	151	18	,	,	PUNCT
ejpam-6792	151	19	(	(	PUNCT
ejpam-6792	151	20	ai)i∈i	ai)i∈i	NUM
ejpam-6792	151	21	}	}	PUNCT
ejpam-6792	151	22	.	.	PUNCT
ejpam-6792	152	1	properties	property	NOUN
ejpam-6792	152	2	(	(	PUNCT
ejpam-6792	152	3	q1	q1	PROPN
ejpam-6792	152	4	)	)	PUNCT
ejpam-6792	152	5	and	and	CCONJ
ejpam-6792	152	6	(	(	PUNCT
ejpam-6792	152	7	q2	q2	NOUN
ejpam-6792	152	8	)	)	PUNCT
ejpam-6792	152	9	imply	imply	VERB
ejpam-6792	152	10	that	that	SCONJ
ejpam-6792	152	11	(	(	PUNCT
ejpam-6792	152	12	0i)i∈i	0i)i∈i	NUM
ejpam-6792	152	13	⊛	⊛	NUM
ejpam-6792	152	14	(	(	PUNCT
ejpam-6792	152	15	0i)i∈i	0i)i∈i	NUM
ejpam-6792	152	16	=	=	SYM
ejpam-6792	152	17	(	(	PUNCT
ejpam-6792	152	18	0i)i∈i	0i)i∈i	NUM
ejpam-6792	152	19	∈	∈	PROPN
ejpam-6792	152	20	s	s	NOUN
ejpam-6792	152	21	,	,	PUNCT
ejpam-6792	152	22	(	(	PUNCT
ejpam-6792	152	23	ai)i∈i	ai)i∈i	X
ejpam-6792	152	24	⊛	⊛	NUM
ejpam-6792	152	25	(	(	PUNCT
ejpam-6792	152	26	ai)i∈i	ai)i∈i	NUM
ejpam-6792	152	27	=	=	SYM
ejpam-6792	152	28	(	(	PUNCT
ejpam-6792	152	29	0i)i∈i	0i)i∈i	NUM
ejpam-6792	152	30	∈	∈	PROPN
ejpam-6792	152	31	s	s	PART
ejpam-6792	152	32	)	)	PUNCT
ejpam-6792	152	33	and	and	CCONJ
ejpam-6792	152	34	(	(	PUNCT
ejpam-6792	152	35	ai)i∈i	ai)i∈i	NUM
ejpam-6792	152	36	⊛	⊛	NUM
ejpam-6792	152	37	(	(	PUNCT
ejpam-6792	152	38	0i)i∈i	0i)i∈i	NUM
ejpam-6792	152	39	=	=	SYM
ejpam-6792	152	40	(	(	PUNCT
ejpam-6792	152	41	ai)i∈i	ai)i∈i	NUM
ejpam-6792	152	42	∈	∈	PROPN
ejpam-6792	152	43	s.	s.	PROPN
ejpam-6792	152	44	since	since	SCONJ
ejpam-6792	152	45	(	(	PUNCT
ejpam-6792	152	46	ai)i∈i	ai)i∈i	NUM
ejpam-6792	152	47	∈	∈	PROPN
ejpam-6792	152	48	g	g	NOUN
ejpam-6792	152	49	(	(	PUNCT
ejpam-6792	152	50	∏	∏	PROPN
ejpam-6792	152	51	i∈i	i∈i	ADJ
ejpam-6792	152	52	xi	xi	PROPN
ejpam-6792	152	53	)	)	PUNCT
ejpam-6792	152	54	,	,	PUNCT
ejpam-6792	152	55	then	then	ADV
ejpam-6792	152	56	(	(	PUNCT
ejpam-6792	152	57	0i)i∈i	0i)i∈i	NUM
ejpam-6792	152	58	⊛	⊛	NUM
ejpam-6792	152	59	(	(	PUNCT
ejpam-6792	152	60	ai)i∈i	ai)i∈i	NUM
ejpam-6792	152	61	=	=	SYM
ejpam-6792	152	62	(	(	PUNCT
ejpam-6792	152	63	ai)i∈i	ai)i∈i	NUM
ejpam-6792	152	64	.	.	PUNCT
ejpam-6792	153	1	therefore	therefore	ADV
ejpam-6792	153	2	,	,	PUNCT
ejpam-6792	153	3	(	(	PUNCT
ejpam-6792	153	4	0i)i∈i	0i)i∈i	NUM
ejpam-6792	153	5	⊛	⊛	NUM
ejpam-6792	153	6	(	(	PUNCT
ejpam-6792	153	7	ai)i∈i	ai)i∈i	NUM
ejpam-6792	153	8	∈	∈	PROPN
ejpam-6792	153	9	s.	s.	PROPN
ejpam-6792	153	10	altogether	altogether	ADV
ejpam-6792	153	11	,	,	PUNCT
ejpam-6792	153	12	s	s	VERB
ejpam-6792	153	13	=	=	X
ejpam-6792	153	14	{	{	PUNCT
ejpam-6792	153	15	(	(	PUNCT
ejpam-6792	153	16	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	153	17	,	,	PUNCT
ejpam-6792	153	18	(	(	PUNCT
ejpam-6792	153	19	ai)i∈i	ai)i∈i	NUM
ejpam-6792	153	20	}	}	PUNCT
ejpam-6792	153	21	is	be	AUX
ejpam-6792	153	22	closed	close	VERB
ejpam-6792	153	23	.	.	PUNCT
ejpam-6792	154	1	hence	hence	ADV
ejpam-6792	154	2	,	,	PUNCT
ejpam-6792	154	3	s	s	PART
ejpam-6792	154	4	=	=	X
ejpam-6792	154	5	{	{	PUNCT
ejpam-6792	154	6	(	(	PUNCT
ejpam-6792	154	7	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	154	8	,	,	PUNCT
ejpam-6792	154	9	(	(	PUNCT
ejpam-6792	154	10	ai)i∈i	ai)i∈i	NUM
ejpam-6792	154	11	}	}	PUNCT
ejpam-6792	154	12	is	be	AUX
ejpam-6792	154	13	a	a	DET
ejpam-6792	154	14	subalgebra	subalgebra	NOUN
ejpam-6792	154	15	of	of	ADP
ejpam-6792	154	16	∏	∏	NUM
ejpam-6792	154	17	i∈i	i∈i	ADJ
ejpam-6792	154	18	xi	xi	PROPN
ejpam-6792	154	19	.	.	PUNCT
ejpam-6792	154	20	a.	a.	PROPN
ejpam-6792	154	21	anantayasethi	anantayasethi	PROPN
ejpam-6792	154	22	,	,	PUNCT
ejpam-6792	154	23	k.	k.	PROPN
ejpam-6792	154	24	saengsura	saengsura	PROPN
ejpam-6792	154	25	,	,	PUNCT
ejpam-6792	154	26	n.	n.	NOUN
ejpam-6792	154	27	sarasit	sarasit	PROPN
ejpam-6792	154	28	/	/	SYM
ejpam-6792	154	29	eur	eur	PROPN
ejpam-6792	154	30	.	.	PUNCT
ejpam-6792	155	1	j.	j.	PROPN
ejpam-6792	155	2	pure	pure	PROPN
ejpam-6792	155	3	appl	appl	PROPN
ejpam-6792	155	4	.	.	PROPN
ejpam-6792	155	5	math	math	PROPN
ejpam-6792	155	6	,	,	PUNCT
ejpam-6792	155	7	18	18	NUM
ejpam-6792	155	8	(	(	PUNCT
ejpam-6792	155	9	4	4	NUM
ejpam-6792	155	10	)	)	PUNCT
ejpam-6792	155	11	(	(	PUNCT
ejpam-6792	155	12	2025	2025	NUM
ejpam-6792	155	13	)	)	PUNCT
ejpam-6792	155	14	,	,	PUNCT
ejpam-6792	155	15	6792	6792	NUM
ejpam-6792	155	16	6	6	NUM
ejpam-6792	155	17	of	of	ADP
ejpam-6792	155	18	10	10	NUM
ejpam-6792	155	19	next	next	ADJ
ejpam-6792	155	20	proposition	proposition	NOUN
ejpam-6792	155	21	shows	show	VERB
ejpam-6792	155	22	necessary	necessary	ADJ
ejpam-6792	155	23	and	and	CCONJ
ejpam-6792	155	24	sufficient	sufficient	ADJ
ejpam-6792	155	25	conditions	condition	NOUN
ejpam-6792	155	26	for	for	ADP
ejpam-6792	155	27	a	a	DET
ejpam-6792	155	28	two	two	NUM
ejpam-6792	155	29	-	-	PUNCT
ejpam-6792	155	30	element	element	NOUN
ejpam-6792	155	31	subset	subset	NOUN
ejpam-6792	155	32	of	of	ADP
ejpam-6792	155	33	∏	∏	PROPN
ejpam-6792	155	34	i∈i	i∈i	ADJ
ejpam-6792	155	35	xi	xi	X
ejpam-6792	155	36	containing	contain	VERB
ejpam-6792	155	37	a	a	DET
ejpam-6792	155	38	constant	constant	ADJ
ejpam-6792	155	39	(	(	PUNCT
ejpam-6792	155	40	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	155	41	to	to	PART
ejpam-6792	155	42	be	be	AUX
ejpam-6792	155	43	a	a	DET
ejpam-6792	155	44	subalgebra	subalgebra	NOUN
ejpam-6792	155	45	of	of	ADP
ejpam-6792	155	46	∏	∏	NUM
ejpam-6792	155	47	i∈i	i∈i	ADJ
ejpam-6792	155	48	xi	xi	PROPN
ejpam-6792	155	49	.	.	PUNCT
ejpam-6792	156	1	proposition	proposition	NOUN
ejpam-6792	156	2	9	9	NUM
ejpam-6792	156	3	.	.	PUNCT
ejpam-6792	157	1	let	let	VERB
ejpam-6792	157	2	(	(	PUNCT
ejpam-6792	157	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	157	4	∈	∈	PROPN
ejpam-6792	157	5	∏	∏	PROPN
ejpam-6792	157	6	i∈i	i∈i	NOUN
ejpam-6792	157	7	xi	xi	X
ejpam-6792	157	8	and	and	CCONJ
ejpam-6792	157	9	let	let	VERB
ejpam-6792	157	10	s	s	PRON
ejpam-6792	157	11	=	=	VERB
ejpam-6792	157	12	{	{	PUNCT
ejpam-6792	157	13	(	(	PUNCT
ejpam-6792	157	14	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	157	15	,	,	PUNCT
ejpam-6792	157	16	(	(	PUNCT
ejpam-6792	157	17	ai)i∈i	ai)i∈i	NUM
ejpam-6792	157	18	}	}	PUNCT
ejpam-6792	157	19	.	.	PUNCT
ejpam-6792	158	1	then	then	ADV
ejpam-6792	158	2	s	s	VERB
ejpam-6792	158	3	is	be	AUX
ejpam-6792	158	4	a	a	DET
ejpam-6792	158	5	subalgebra	subalgebra	NOUN
ejpam-6792	158	6	of	of	ADP
ejpam-6792	158	7	∏	∏	NUM
ejpam-6792	158	8	i∈i	i∈i	ADJ
ejpam-6792	158	9	xi	xi	INTJ
ejpam-6792	159	1	if	if	SCONJ
ejpam-6792	159	2	and	and	CCONJ
ejpam-6792	159	3	only	only	ADV
ejpam-6792	159	4	if	if	SCONJ
ejpam-6792	159	5	(	(	PUNCT
ejpam-6792	159	6	ai)i∈i	ai)i∈i	NUM
ejpam-6792	159	7	∈	∈	PROPN
ejpam-6792	159	8	g	g	NOUN
ejpam-6792	159	9	(	(	PUNCT
ejpam-6792	159	10	∏	∏	PROPN
ejpam-6792	159	11	i∈i	i∈i	ADJ
ejpam-6792	159	12	xi	xi	NOUN
ejpam-6792	159	13	)	)	PUNCT
ejpam-6792	159	14	or	or	CCONJ
ejpam-6792	159	15	(	(	PUNCT
ejpam-6792	159	16	ai)i∈i	ai)i∈i	NUM
ejpam-6792	159	17	∈	∈	PROPN
ejpam-6792	159	18	b	b	NOUN
ejpam-6792	159	19	(	(	PUNCT
ejpam-6792	159	20	∏	∏	PROPN
ejpam-6792	159	21	i∈i	i∈i	NOUN
ejpam-6792	159	22	xi	xi	PROPN
ejpam-6792	159	23	)	)	PUNCT
ejpam-6792	159	24	.	.	PUNCT
ejpam-6792	160	1	proof	proof	NOUN
ejpam-6792	160	2	.	.	PUNCT
ejpam-6792	161	1	(	(	PUNCT
ejpam-6792	161	2	⇒	⇒	NOUN
ejpam-6792	161	3	)	)	PUNCT
ejpam-6792	161	4	assume	assume	VERB
ejpam-6792	161	5	s	s	X
ejpam-6792	161	6	=	=	X
ejpam-6792	161	7	{	{	PUNCT
ejpam-6792	161	8	(	(	PUNCT
ejpam-6792	161	9	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	161	10	,	,	PUNCT
ejpam-6792	161	11	(	(	PUNCT
ejpam-6792	161	12	ai)i∈i	ai)i∈i	NUM
ejpam-6792	161	13	}	}	PUNCT
ejpam-6792	161	14	is	be	AUX
ejpam-6792	161	15	a	a	DET
ejpam-6792	161	16	subalgebra	subalgebra	NOUN
ejpam-6792	161	17	of	of	ADP
ejpam-6792	161	18	∏	∏	NUM
ejpam-6792	161	19	i∈i	i∈i	ADJ
ejpam-6792	162	1	xi	xi	PROPN
ejpam-6792	162	2	.	.	PUNCT
ejpam-6792	163	1	then	then	ADV
ejpam-6792	163	2	by	by	ADP
ejpam-6792	163	3	closure	closure	NOUN
ejpam-6792	163	4	property	property	NOUN
ejpam-6792	163	5	of	of	ADP
ejpam-6792	163	6	s	s	PROPN
ejpam-6792	163	7	,	,	PUNCT
ejpam-6792	163	8	(	(	PUNCT
ejpam-6792	163	9	0i)i∈i	0i)i∈i	NUM
ejpam-6792	163	10	⊛	⊛	NUM
ejpam-6792	163	11	(	(	PUNCT
ejpam-6792	163	12	ai)i∈i	ai)i∈i	NUM
ejpam-6792	163	13	∈	∈	PROPN
ejpam-6792	163	14	s	s	NOUN
ejpam-6792	163	15	,	,	PUNCT
ejpam-6792	163	16	there	there	PRON
ejpam-6792	163	17	follows	follow	VERB
ejpam-6792	163	18	(	(	PUNCT
ejpam-6792	163	19	0i)i∈i	0i)i∈i	NUM
ejpam-6792	163	20	⊛	⊛	NUM
ejpam-6792	163	21	(	(	PUNCT
ejpam-6792	163	22	ai)i∈i	ai)i∈i	NUM
ejpam-6792	163	23	=	=	SYM
ejpam-6792	163	24	(	(	PUNCT
ejpam-6792	163	25	ai)i∈i	ai)i∈i	NUM
ejpam-6792	163	26	or	or	CCONJ
ejpam-6792	163	27	(	(	PUNCT
ejpam-6792	163	28	0i)i∈i	0i)i∈i	NUM
ejpam-6792	163	29	⊛	⊛	NUM
ejpam-6792	163	30	(	(	PUNCT
ejpam-6792	163	31	ai)i∈i	ai)i∈i	NUM
ejpam-6792	163	32	=	=	SYM
ejpam-6792	163	33	(	(	PUNCT
ejpam-6792	163	34	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	163	35	.	.	PUNCT
ejpam-6792	164	1	hence	hence	ADV
ejpam-6792	164	2	,	,	PUNCT
ejpam-6792	164	3	(	(	PUNCT
ejpam-6792	164	4	ai)i∈i	ai)i∈i	NUM
ejpam-6792	164	5	∈	∈	PROPN
ejpam-6792	164	6	g	g	NOUN
ejpam-6792	164	7	(	(	PUNCT
ejpam-6792	164	8	∏	∏	PROPN
ejpam-6792	164	9	i∈i	i∈i	ADJ
ejpam-6792	164	10	xi	xi	NOUN
ejpam-6792	164	11	)	)	PUNCT
ejpam-6792	164	12	or	or	CCONJ
ejpam-6792	164	13	(	(	PUNCT
ejpam-6792	164	14	ai)i∈i	ai)i∈i	NUM
ejpam-6792	164	15	∈	∈	PROPN
ejpam-6792	164	16	b	b	NOUN
ejpam-6792	164	17	(	(	PUNCT
ejpam-6792	164	18	∏	∏	PROPN
ejpam-6792	164	19	i∈i	i∈i	NOUN
ejpam-6792	164	20	xi	xi	PROPN
ejpam-6792	164	21	)	)	PUNCT
ejpam-6792	164	22	.	.	PUNCT
ejpam-6792	165	1	(	(	PUNCT
ejpam-6792	165	2	⇐	⇐	NOUN
ejpam-6792	165	3	)	)	PUNCT
ejpam-6792	165	4	if	if	SCONJ
ejpam-6792	165	5	(	(	PUNCT
ejpam-6792	165	6	ai)i∈i	ai)i∈i	NUM
ejpam-6792	165	7	∈	∈	PROPN
ejpam-6792	165	8	g	g	NOUN
ejpam-6792	165	9	(	(	PUNCT
ejpam-6792	165	10	∏	∏	PROPN
ejpam-6792	165	11	i∈i	i∈i	ADJ
ejpam-6792	165	12	xi	xi	PROPN
ejpam-6792	165	13	)	)	PUNCT
ejpam-6792	165	14	,	,	PUNCT
ejpam-6792	165	15	then	then	ADV
ejpam-6792	165	16	s	s	AUX
ejpam-6792	165	17	=	=	PUNCT
ejpam-6792	165	18	{	{	PUNCT
ejpam-6792	165	19	(	(	PUNCT
ejpam-6792	165	20	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	165	21	,	,	PUNCT
ejpam-6792	165	22	(	(	PUNCT
ejpam-6792	165	23	ai)i∈i	ai)i∈i	NUM
ejpam-6792	165	24	}	}	PUNCT
ejpam-6792	165	25	is	be	AUX
ejpam-6792	165	26	a	a	DET
ejpam-6792	165	27	subalgebra	subalgebra	NOUN
ejpam-6792	165	28	by	by	ADP
ejpam-6792	165	29	proposition	proposition	NOUN
ejpam-6792	165	30	8	8	NUM
ejpam-6792	165	31	.	.	PUNCT
ejpam-6792	166	1	if	if	SCONJ
ejpam-6792	166	2	(	(	PUNCT
ejpam-6792	166	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	166	4	∈	∈	PROPN
ejpam-6792	166	5	b	b	NOUN
ejpam-6792	166	6	(	(	PUNCT
ejpam-6792	166	7	∏	∏	PROPN
ejpam-6792	166	8	i∈i	i∈i	NOUN
ejpam-6792	166	9	xi	xi	PROPN
ejpam-6792	166	10	)	)	PUNCT
ejpam-6792	166	11	,	,	PUNCT
ejpam-6792	166	12	then	then	ADV
ejpam-6792	166	13	(	(	PUNCT
ejpam-6792	166	14	0i)i∈i	0i)i∈i	NUM
ejpam-6792	166	15	⊛	⊛	NUM
ejpam-6792	166	16	(	(	PUNCT
ejpam-6792	166	17	ai)i∈i	ai)i∈i	NUM
ejpam-6792	166	18	=	=	SYM
ejpam-6792	166	19	(	(	PUNCT
ejpam-6792	166	20	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	166	21	.	.	PUNCT
ejpam-6792	167	1	from	from	ADP
ejpam-6792	167	2	this	this	DET
ejpam-6792	167	3	fact	fact	NOUN
ejpam-6792	167	4	and	and	CCONJ
ejpam-6792	167	5	(	(	PUNCT
ejpam-6792	167	6	q1	q1	PROPN
ejpam-6792	167	7	)	)	PUNCT
ejpam-6792	167	8	,	,	PUNCT
ejpam-6792	167	9	(	(	PUNCT
ejpam-6792	167	10	q2	q2	NOUN
ejpam-6792	167	11	)	)	PUNCT
ejpam-6792	167	12	we	we	PRON
ejpam-6792	167	13	can	can	AUX
ejpam-6792	167	14	conclude	conclude	VERB
ejpam-6792	167	15	that	that	DET
ejpam-6792	167	16	s	s	VERB
ejpam-6792	167	17	=	=	X
ejpam-6792	167	18	{	{	PUNCT
ejpam-6792	167	19	(	(	PUNCT
ejpam-6792	167	20	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	167	21	,	,	PUNCT
ejpam-6792	167	22	(	(	PUNCT
ejpam-6792	167	23	ai)i∈i	ai)i∈i	NUM
ejpam-6792	167	24	}	}	PUNCT
ejpam-6792	167	25	is	be	AUX
ejpam-6792	167	26	closed	closed	ADJ
ejpam-6792	167	27	.	.	PUNCT
ejpam-6792	168	1	therefore	therefore	ADV
ejpam-6792	168	2	,	,	PUNCT
ejpam-6792	168	3	s	s	VERB
ejpam-6792	168	4	is	be	AUX
ejpam-6792	168	5	a	a	DET
ejpam-6792	168	6	subalgebra	subalgebra	NOUN
ejpam-6792	168	7	of	of	ADP
ejpam-6792	168	8	∏	∏	NUM
ejpam-6792	168	9	i∈i	i∈i	ADJ
ejpam-6792	168	10	xi	xi	PROPN
ejpam-6792	168	11	.	.	PUNCT
ejpam-6792	169	1	moreover	moreover	ADV
ejpam-6792	169	2	,	,	PUNCT
ejpam-6792	169	3	the	the	DET
ejpam-6792	169	4	set	set	NOUN
ejpam-6792	169	5	g	g	NOUN
ejpam-6792	169	6	(	(	PUNCT
ejpam-6792	169	7	∏	∏	PROPN
ejpam-6792	169	8	i∈i	i∈i	NOUN
ejpam-6792	169	9	xi	xi	NOUN
ejpam-6792	169	10	)	)	PUNCT
ejpam-6792	169	11	itself	itself	PRON
ejpam-6792	169	12	is	be	AUX
ejpam-6792	169	13	also	also	ADV
ejpam-6792	169	14	a	a	DET
ejpam-6792	169	15	subalgebra	subalgebra	NOUN
ejpam-6792	169	16	.	.	PUNCT
ejpam-6792	170	1	proposition	proposition	NOUN
ejpam-6792	170	2	10	10	NUM
ejpam-6792	170	3	.	.	PUNCT
ejpam-6792	171	1	g	g	NOUN
ejpam-6792	171	2	(	(	PUNCT
ejpam-6792	171	3	∏	∏	PROPN
ejpam-6792	171	4	i∈i	i∈i	NOUN
ejpam-6792	171	5	xi	xi	NOUN
ejpam-6792	171	6	)	)	PUNCT
ejpam-6792	171	7	is	be	AUX
ejpam-6792	171	8	a	a	DET
ejpam-6792	171	9	subalgebra	subalgebra	NOUN
ejpam-6792	171	10	of	of	ADP
ejpam-6792	171	11	∏	∏	NUM
ejpam-6792	171	12	i∈i	i∈i	ADJ
ejpam-6792	171	13	xi	xi	PROPN
ejpam-6792	171	14	.	.	PUNCT
ejpam-6792	172	1	proof	proof	NOUN
ejpam-6792	172	2	.	.	PUNCT
ejpam-6792	173	1	since	since	SCONJ
ejpam-6792	173	2	0i	0i	PROPN
ejpam-6792	173	3	∈	∈	PROPN
ejpam-6792	173	4	g(xi	g(xi	PROPN
ejpam-6792	173	5	)	)	PUNCT
ejpam-6792	173	6	for	for	ADP
ejpam-6792	173	7	all	all	PRON
ejpam-6792	173	8	i	i	PRON
ejpam-6792	173	9	∈	∈	VERB
ejpam-6792	174	1	i	i	PRON
ejpam-6792	174	2	and	and	CCONJ
ejpam-6792	174	3	by	by	ADP
ejpam-6792	174	4	proposition	proposition	NOUN
ejpam-6792	174	5	7	7	NUM
ejpam-6792	174	6	,	,	PUNCT
ejpam-6792	174	7	then	then	ADV
ejpam-6792	174	8	(	(	PUNCT
ejpam-6792	174	9	0)i∈i	0)i∈i	NUM
ejpam-6792	174	10	∈	∈	PROPN
ejpam-6792	174	11	g	g	PROPN
ejpam-6792	174	12	(	(	PUNCT
ejpam-6792	174	13	∏	∏	PROPN
ejpam-6792	174	14	i∈i	i∈i	ADJ
ejpam-6792	174	15	xi	xi	PROPN
ejpam-6792	174	16	)	)	PUNCT
ejpam-6792	174	17	.	.	PUNCT
ejpam-6792	175	1	therefore	therefore	ADV
ejpam-6792	175	2	,	,	PUNCT
ejpam-6792	175	3	g	g	PROPN
ejpam-6792	175	4	(	(	PUNCT
ejpam-6792	175	5	∏	∏	PROPN
ejpam-6792	175	6	i∈i	i∈i	ADJ
ejpam-6792	175	7	xi	xi	NOUN
ejpam-6792	175	8	)	)	PUNCT
ejpam-6792	175	9	̸=	̸=	PROPN
ejpam-6792	175	10	∅.	∅.	ADV
ejpam-6792	175	11	let	let	VERB
ejpam-6792	175	12	(	(	PUNCT
ejpam-6792	175	13	ai)i∈i	ai)i∈i	NUM
ejpam-6792	175	14	,	,	PUNCT
ejpam-6792	175	15	(	(	PUNCT
ejpam-6792	175	16	bi)i∈i	bi)i∈i	NUM
ejpam-6792	175	17	∈	∈	PROPN
ejpam-6792	175	18	g	g	PROPN
ejpam-6792	175	19	(	(	PUNCT
ejpam-6792	175	20	∏	∏	PROPN
ejpam-6792	175	21	i∈i	i∈i	ADJ
ejpam-6792	175	22	xi	xi	PROPN
ejpam-6792	175	23	)	)	PUNCT
ejpam-6792	175	24	.	.	PUNCT
ejpam-6792	176	1	then	then	ADV
ejpam-6792	176	2	(	(	PUNCT
ejpam-6792	176	3	0i)i∈i	0i)i∈i	NUM
ejpam-6792	176	4	⊛	⊛	NUM
ejpam-6792	176	5	(	(	PUNCT
ejpam-6792	176	6	ai)i∈i	ai)i∈i	NUM
ejpam-6792	176	7	=	=	SYM
ejpam-6792	176	8	(	(	PUNCT
ejpam-6792	176	9	ai)i∈i	ai)i∈i	NUM
ejpam-6792	176	10	and	and	CCONJ
ejpam-6792	176	11	(	(	PUNCT
ejpam-6792	176	12	0i)i∈i	0i)i∈i	NUM
ejpam-6792	176	13	⊛	⊛	NUM
ejpam-6792	176	14	(	(	PUNCT
ejpam-6792	176	15	bi)i∈i	bi)i∈i	NUM
ejpam-6792	176	16	=	=	SYM
ejpam-6792	176	17	(	(	PUNCT
ejpam-6792	176	18	bi)i∈i	bi)i∈i	NOUN
ejpam-6792	176	19	.	.	PUNCT
ejpam-6792	177	1	by	by	ADP
ejpam-6792	177	2	proposition	proposition	NOUN
ejpam-6792	177	3	1	1	NUM
ejpam-6792	177	4	we	we	PRON
ejpam-6792	177	5	get	get	VERB
ejpam-6792	177	6	(	(	PUNCT
ejpam-6792	177	7	0i)i∈i	0i)i∈i	NOUN
ejpam-6792	177	8	⊛	⊛	X
ejpam-6792	178	1	[	[	X
ejpam-6792	178	2	(	(	PUNCT
ejpam-6792	178	3	ai)i∈i	ai)i∈i	NOUN
ejpam-6792	178	4	⊛	⊛	NUM
ejpam-6792	178	5	(	(	PUNCT
ejpam-6792	178	6	bi)i∈i	bi)i∈i	X
ejpam-6792	178	7	]	]	PUNCT
ejpam-6792	178	8	=	=	PUNCT
ejpam-6792	179	1	[	[	X
ejpam-6792	179	2	(	(	PUNCT
ejpam-6792	179	3	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	179	4	⊛	⊛	NUM
ejpam-6792	179	5	(	(	PUNCT
ejpam-6792	179	6	ai)i∈i	ai)i∈i	NUM
ejpam-6792	179	7	]	]	PUNCT
ejpam-6792	179	8	⊛	⊛	NUM
ejpam-6792	179	9	[	[	X
ejpam-6792	179	10	(	(	PUNCT
ejpam-6792	179	11	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	179	12	⊛	⊛	NUM
ejpam-6792	179	13	(	(	PUNCT
ejpam-6792	179	14	bi)i∈i	bi)i∈i	X
ejpam-6792	179	15	]	]	PUNCT
ejpam-6792	179	16	=	=	PUNCT
ejpam-6792	179	17	(	(	PUNCT
ejpam-6792	179	18	ai)i∈i	ai)i∈i	X
ejpam-6792	179	19	⊛	⊛	NUM
ejpam-6792	179	20	(	(	PUNCT
ejpam-6792	179	21	bi)i∈i	bi)i∈i	NUM
ejpam-6792	179	22	.	.	PUNCT
ejpam-6792	180	1	it	it	PRON
ejpam-6792	180	2	follows	follow	VERB
ejpam-6792	180	3	that	that	SCONJ
ejpam-6792	180	4	(	(	PUNCT
ejpam-6792	180	5	ai)i∈i	ai)i∈i	NOUN
ejpam-6792	180	6	⊛	⊛	NUM
ejpam-6792	180	7	(	(	PUNCT
ejpam-6792	180	8	bi)i∈i	bi)i∈i	NUM
ejpam-6792	180	9	∈	∈	PROPN
ejpam-6792	180	10	g	g	PROPN
ejpam-6792	180	11	(	(	PUNCT
ejpam-6792	180	12	∏	∏	PROPN
ejpam-6792	180	13	i∈i	i∈i	ADJ
ejpam-6792	180	14	xi	xi	PROPN
ejpam-6792	180	15	)	)	PUNCT
ejpam-6792	180	16	.	.	PUNCT
ejpam-6792	181	1	therefore	therefore	ADV
ejpam-6792	181	2	,	,	PUNCT
ejpam-6792	181	3	g	g	PROPN
ejpam-6792	181	4	(	(	PUNCT
ejpam-6792	181	5	∏	∏	PROPN
ejpam-6792	181	6	i∈i	i∈i	ADJ
ejpam-6792	181	7	xi	xi	NOUN
ejpam-6792	181	8	)	)	PUNCT
ejpam-6792	181	9	is	be	AUX
ejpam-6792	181	10	closed	close	VERB
ejpam-6792	181	11	.	.	PUNCT
ejpam-6792	182	1	proposition	proposition	NOUN
ejpam-6792	182	2	11	11	NUM
ejpam-6792	182	3	.	.	PUNCT
ejpam-6792	183	1	g	g	NOUN
ejpam-6792	183	2	(	(	PUNCT
ejpam-6792	183	3	∏	∏	PROPN
ejpam-6792	183	4	i∈i	i∈i	ADJ
ejpam-6792	183	5	xi	xi	NOUN
ejpam-6792	183	6	)	)	PUNCT
ejpam-6792	183	7	is	be	AUX
ejpam-6792	183	8	an	an	DET
ejpam-6792	183	9	abelian	abelian	ADJ
ejpam-6792	183	10	group	group	NOUN
ejpam-6792	183	11	.	.	PUNCT
ejpam-6792	184	1	proof	proof	NOUN
ejpam-6792	184	2	.	.	PUNCT
ejpam-6792	185	1	let	let	VERB
ejpam-6792	185	2	(	(	PUNCT
ejpam-6792	185	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	185	4	,	,	PUNCT
ejpam-6792	185	5	(	(	PUNCT
ejpam-6792	185	6	bi)i∈i	bi)i∈i	NUM
ejpam-6792	185	7	,	,	PUNCT
ejpam-6792	185	8	(	(	PUNCT
ejpam-6792	185	9	ci)i∈i	ci)i∈i	NUM
ejpam-6792	185	10	∈	∈	NOUN
ejpam-6792	185	11	g	g	NOUN
ejpam-6792	185	12	(	(	PUNCT
ejpam-6792	185	13	∏	∏	PROPN
ejpam-6792	185	14	i∈i	i∈i	ADJ
ejpam-6792	185	15	xi	xi	PROPN
ejpam-6792	185	16	)	)	PUNCT
ejpam-6792	185	17	.	.	PUNCT
ejpam-6792	186	1	then	then	ADV
ejpam-6792	186	2	(	(	PUNCT
ejpam-6792	186	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	186	4	⊛	⊛	NUM
ejpam-6792	186	5	(	(	PUNCT
ejpam-6792	186	6	bi)i∈i	bi)i∈i	NUM
ejpam-6792	186	7	∈	∈	PROPN
ejpam-6792	186	8	g	g	PROPN
ejpam-6792	186	9	(	(	PUNCT
ejpam-6792	186	10	∏	∏	PROPN
ejpam-6792	186	11	i∈i	i∈i	ADJ
ejpam-6792	186	12	xi	xi	PROPN
ejpam-6792	186	13	)	)	PUNCT
ejpam-6792	186	14	by	by	ADP
ejpam-6792	186	15	proposition	proposition	NOUN
ejpam-6792	186	16	10	10	NUM
ejpam-6792	186	17	.	.	PUNCT
ejpam-6792	187	1	the	the	DET
ejpam-6792	187	2	commutative	commutative	ADJ
ejpam-6792	187	3	property	property	NOUN
ejpam-6792	187	4	follows	follow	VERB
ejpam-6792	187	5	from	from	ADP
ejpam-6792	187	6	proposition	proposition	NOUN
ejpam-6792	187	7	2	2	NUM
ejpam-6792	187	8	.	.	PUNCT
ejpam-6792	188	1	by	by	ADP
ejpam-6792	188	2	(	(	PUNCT
ejpam-6792	188	3	q3	q3	PROPN
ejpam-6792	188	4	)	)	PUNCT
ejpam-6792	188	5	and	and	CCONJ
ejpam-6792	188	6	the	the	DET
ejpam-6792	188	7	commutative	commutative	ADJ
ejpam-6792	188	8	property	property	NOUN
ejpam-6792	188	9	,	,	PUNCT
ejpam-6792	188	10	there	there	PRON
ejpam-6792	188	11	follows	follow	VERB
ejpam-6792	188	12	[	[	X
ejpam-6792	188	13	(	(	PUNCT
ejpam-6792	188	14	ai)i∈i	ai)i∈i	NOUN
ejpam-6792	188	15	⊛	⊛	NUM
ejpam-6792	188	16	(	(	PUNCT
ejpam-6792	188	17	bi)i∈i	bi)i∈i	NOUN
ejpam-6792	188	18	]	]	PUNCT
ejpam-6792	188	19	⊛	⊛	NUM
ejpam-6792	188	20	(	(	PUNCT
ejpam-6792	188	21	ci)i∈i	ci)i∈i	NUM
ejpam-6792	188	22	=	=	SYM
ejpam-6792	189	1	[	[	X
ejpam-6792	189	2	(	(	PUNCT
ejpam-6792	189	3	ai)i∈i	ai)i∈i	NOUN
ejpam-6792	189	4	⊛	⊛	NUM
ejpam-6792	189	5	(	(	PUNCT
ejpam-6792	189	6	ci)i∈i	ci)i∈i	NUM
ejpam-6792	189	7	]	]	PUNCT
ejpam-6792	189	8	⊛	⊛	NUM
ejpam-6792	189	9	(	(	PUNCT
ejpam-6792	189	10	bi)i∈i	bi)i∈i	X
ejpam-6792	189	11	=	=	PUNCT
ejpam-6792	190	1	[	[	X
ejpam-6792	190	2	(	(	PUNCT
ejpam-6792	190	3	ci)i∈i	ci)i∈i	NUM
ejpam-6792	190	4	⊛	⊛	NUM
ejpam-6792	190	5	(	(	PUNCT
ejpam-6792	190	6	ai)i∈i	ai)i∈i	NUM
ejpam-6792	190	7	]	]	PUNCT
ejpam-6792	190	8	⊛	⊛	NUM
ejpam-6792	190	9	(	(	PUNCT
ejpam-6792	190	10	bi)i∈i	bi)i∈i	X
ejpam-6792	190	11	=	=	PUNCT
ejpam-6792	191	1	[	[	X
ejpam-6792	191	2	(	(	PUNCT
ejpam-6792	191	3	ci)i∈i	ci)i∈i	NUM
ejpam-6792	191	4	⊛	⊛	NUM
ejpam-6792	191	5	(	(	PUNCT
ejpam-6792	191	6	bi)i∈i	bi)i∈i	NOUN
ejpam-6792	191	7	]	]	PUNCT
ejpam-6792	191	8	⊛	⊛	NUM
ejpam-6792	191	9	(	(	PUNCT
ejpam-6792	191	10	ai)i∈i	ai)i∈i	NUM
ejpam-6792	191	11	=	=	PUNCT
ejpam-6792	191	12	[	[	X
ejpam-6792	191	13	(	(	PUNCT
ejpam-6792	191	14	bi)i∈i	bi)i∈i	NOUN
ejpam-6792	191	15	⊛	⊛	NUM
ejpam-6792	191	16	(	(	PUNCT
ejpam-6792	191	17	ci)i∈i	ci)i∈i	NUM
ejpam-6792	191	18	]	]	PUNCT
ejpam-6792	191	19	⊛	⊛	NUM
ejpam-6792	191	20	(	(	PUNCT
ejpam-6792	191	21	ai)i∈i	ai)i∈i	NUM
ejpam-6792	191	22	=	=	SYM
ejpam-6792	191	23	(	(	PUNCT
ejpam-6792	191	24	ai)i∈i	ai)i∈i	NOUN
ejpam-6792	191	25	⊛	⊛	PUNCT
ejpam-6792	191	26	[	[	X
ejpam-6792	191	27	(	(	PUNCT
ejpam-6792	191	28	bi)i∈i	bi)i∈i	NOUN
ejpam-6792	191	29	⊛	⊛	NUM
ejpam-6792	191	30	(	(	PUNCT
ejpam-6792	191	31	ci)i∈i	ci)i∈i	NUM
ejpam-6792	191	32	]	]	PUNCT
ejpam-6792	191	33	.	.	PUNCT
ejpam-6792	192	1	therefore	therefore	ADV
ejpam-6792	192	2	,	,	PUNCT
ejpam-6792	192	3	an	an	DET
ejpam-6792	192	4	associative	associative	ADJ
ejpam-6792	192	5	law	law	NOUN
ejpam-6792	192	6	is	be	AUX
ejpam-6792	192	7	fulfilled	fulfil	VERB
ejpam-6792	192	8	.	.	PUNCT
ejpam-6792	193	1	since	since	SCONJ
ejpam-6792	193	2	(	(	PUNCT
ejpam-6792	193	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	193	4	∈	∈	PROPN
ejpam-6792	193	5	g	g	NOUN
ejpam-6792	193	6	(	(	PUNCT
ejpam-6792	193	7	∏	∏	PROPN
ejpam-6792	193	8	i∈i	i∈i	ADJ
ejpam-6792	193	9	xi	xi	PROPN
ejpam-6792	193	10	)	)	PUNCT
ejpam-6792	193	11	and	and	CCONJ
ejpam-6792	193	12	by	by	ADP
ejpam-6792	193	13	(	(	PUNCT
ejpam-6792	193	14	q2	q2	NOUN
ejpam-6792	193	15	)	)	PUNCT
ejpam-6792	193	16	,	,	PUNCT
ejpam-6792	193	17	then	then	ADV
ejpam-6792	193	18	(	(	PUNCT
ejpam-6792	193	19	0i)i∈i	0i)i∈i	NUM
ejpam-6792	193	20	⊛	⊛	NUM
ejpam-6792	193	21	(	(	PUNCT
ejpam-6792	193	22	ai)i∈i	ai)i∈i	NUM
ejpam-6792	193	23	=	=	SYM
ejpam-6792	193	24	(	(	PUNCT
ejpam-6792	193	25	ai)i∈i	ai)i∈i	NUM
ejpam-6792	193	26	=	=	SYM
ejpam-6792	193	27	(	(	PUNCT
ejpam-6792	193	28	ai)i∈i	ai)i∈i	NOUN
ejpam-6792	193	29	⊛	⊛	NUM
ejpam-6792	193	30	(	(	PUNCT
ejpam-6792	193	31	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	193	32	.	.	PUNCT
ejpam-6792	194	1	therefore	therefore	ADV
ejpam-6792	194	2	,	,	PUNCT
ejpam-6792	194	3	(	(	PUNCT
ejpam-6792	194	4	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	194	5	is	be	AUX
ejpam-6792	194	6	an	an	DET
ejpam-6792	194	7	identity	identity	NOUN
ejpam-6792	194	8	element	element	NOUN
ejpam-6792	194	9	.	.	PUNCT
ejpam-6792	195	1	a.	a.	NOUN
ejpam-6792	195	2	anantayasethi	anantayasethi	PROPN
ejpam-6792	195	3	,	,	PUNCT
ejpam-6792	195	4	k.	k.	PROPN
ejpam-6792	195	5	saengsura	saengsura	PROPN
ejpam-6792	195	6	,	,	PUNCT
ejpam-6792	195	7	n.	n.	NOUN
ejpam-6792	195	8	sarasit	sarasit	PROPN
ejpam-6792	195	9	/	/	SYM
ejpam-6792	195	10	eur	eur	PROPN
ejpam-6792	195	11	.	.	PUNCT
ejpam-6792	196	1	j.	j.	PROPN
ejpam-6792	196	2	pure	pure	PROPN
ejpam-6792	196	3	appl	appl	PROPN
ejpam-6792	196	4	.	.	PROPN
ejpam-6792	196	5	math	math	PROPN
ejpam-6792	196	6	,	,	PUNCT
ejpam-6792	196	7	18	18	NUM
ejpam-6792	196	8	(	(	PUNCT
ejpam-6792	196	9	4	4	NUM
ejpam-6792	196	10	)	)	PUNCT
ejpam-6792	196	11	(	(	PUNCT
ejpam-6792	196	12	2025	2025	NUM
ejpam-6792	196	13	)	)	PUNCT
ejpam-6792	196	14	,	,	PUNCT
ejpam-6792	196	15	6792	6792	NUM
ejpam-6792	196	16	7	7	NUM
ejpam-6792	196	17	of	of	ADP
ejpam-6792	196	18	10	10	NUM
ejpam-6792	196	19	since	since	SCONJ
ejpam-6792	196	20	(	(	PUNCT
ejpam-6792	196	21	ai)i∈i	ai)i∈i	NUM
ejpam-6792	196	22	⊛	⊛	NUM
ejpam-6792	196	23	(	(	PUNCT
ejpam-6792	196	24	ai)i∈i	ai)i∈i	NUM
ejpam-6792	196	25	=	=	SYM
ejpam-6792	196	26	(	(	PUNCT
ejpam-6792	196	27	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	196	28	by	by	ADP
ejpam-6792	196	29	(	(	PUNCT
ejpam-6792	196	30	q2	q2	NOUN
ejpam-6792	196	31	)	)	PUNCT
ejpam-6792	196	32	,	,	PUNCT
ejpam-6792	196	33	then	then	ADV
ejpam-6792	196	34	(	(	PUNCT
ejpam-6792	196	35	ai)i∈i	ai)i∈i	NUM
ejpam-6792	196	36	is	be	AUX
ejpam-6792	196	37	an	an	DET
ejpam-6792	196	38	inverse	inverse	NOUN
ejpam-6792	196	39	of	of	ADP
ejpam-6792	196	40	itself	itself	PRON
ejpam-6792	196	41	.	.	PUNCT
ejpam-6792	197	1	altogether	altogether	ADV
ejpam-6792	197	2	,	,	PUNCT
ejpam-6792	197	3	g	g	PROPN
ejpam-6792	197	4	(	(	PUNCT
ejpam-6792	197	5	∏	∏	PROPN
ejpam-6792	197	6	i∈i	i∈i	NOUN
ejpam-6792	197	7	xi	xi	NOUN
ejpam-6792	197	8	)	)	PUNCT
ejpam-6792	197	9	is	be	AUX
ejpam-6792	197	10	an	an	DET
ejpam-6792	197	11	abelian	abelian	ADJ
ejpam-6792	197	12	group	group	NOUN
ejpam-6792	197	13	.	.	PUNCT
ejpam-6792	198	1	it	it	PRON
ejpam-6792	198	2	is	be	AUX
ejpam-6792	198	3	known	know	VERB
ejpam-6792	198	4	that	that	SCONJ
ejpam-6792	198	5	an	an	DET
ejpam-6792	198	6	abelian	abelian	ADJ
ejpam-6792	198	7	group	group	NOUN
ejpam-6792	198	8	g	g	PROPN
ejpam-6792	198	9	such	such	ADJ
ejpam-6792	198	10	that	that	SCONJ
ejpam-6792	198	11	every	every	DET
ejpam-6792	198	12	element	element	NOUN
ejpam-6792	198	13	(	(	PUNCT
ejpam-6792	198	14	except	except	SCONJ
ejpam-6792	198	15	an	an	DET
ejpam-6792	198	16	identity	identity	NOUN
ejpam-6792	198	17	element	element	NOUN
ejpam-6792	198	18	e	e	NOUN
ejpam-6792	198	19	)	)	PUNCT
ejpam-6792	198	20	has	have	VERB
ejpam-6792	198	21	order	order	NOUN
ejpam-6792	198	22	2	2	NUM
ejpam-6792	198	23	,	,	PUNCT
ejpam-6792	198	24	g	g	PROPN
ejpam-6792	198	25	has	have	VERB
ejpam-6792	198	26	an	an	DET
ejpam-6792	198	27	order	order	NOUN
ejpam-6792	198	28	2k	2k	NOUN
ejpam-6792	198	29	for	for	ADP
ejpam-6792	198	30	some	some	DET
ejpam-6792	198	31	positive	positive	ADJ
ejpam-6792	198	32	integer	integer	NOUN
ejpam-6792	198	33	k.	k.	INTJ
ejpam-6792	198	34	by	by	ADP
ejpam-6792	198	35	this	this	DET
ejpam-6792	198	36	fact	fact	NOUN
ejpam-6792	198	37	and	and	CCONJ
ejpam-6792	198	38	proposition	proposition	NOUN
ejpam-6792	198	39	11	11	NUM
ejpam-6792	198	40	we	we	PRON
ejpam-6792	198	41	obtain	obtain	VERB
ejpam-6792	198	42	the	the	DET
ejpam-6792	198	43	following	follow	VERB
ejpam-6792	198	44	proposition	proposition	NOUN
ejpam-6792	198	45	.	.	PUNCT
ejpam-6792	199	1	proposition	proposition	NOUN
ejpam-6792	199	2	12	12	NUM
ejpam-6792	199	3	.	.	PUNCT
ejpam-6792	200	1	if	if	SCONJ
ejpam-6792	200	2	1	1	NUM
ejpam-6792	200	3	̸=	̸=	PROPN
ejpam-6792	200	4	|	|	ADV
ejpam-6792	200	5	∏	∏	NUM
ejpam-6792	200	6	i∈i	i∈i	ADJ
ejpam-6792	200	7	xi|	xi|	PROPN
ejpam-6792	201	1	=	=	PUNCT
ejpam-6792	202	1	k	k	PROPN
ejpam-6792	202	2	for	for	ADP
ejpam-6792	202	3	some	some	DET
ejpam-6792	202	4	odd	odd	ADJ
ejpam-6792	202	5	number	number	NOUN
ejpam-6792	202	6	k	k	NOUN
ejpam-6792	202	7	,	,	PUNCT
ejpam-6792	202	8	then	then	ADV
ejpam-6792	202	9	g	g	PROPN
ejpam-6792	202	10	(	(	PUNCT
ejpam-6792	202	11	∏	∏	PROPN
ejpam-6792	202	12	i∈i	i∈i	ADJ
ejpam-6792	202	13	xi	xi	NOUN
ejpam-6792	202	14	)	)	PUNCT
ejpam-6792	202	15	̸=	̸=	PROPN
ejpam-6792	202	16	∏	∏	NUM
ejpam-6792	202	17	i∈i	i∈i	ADJ
ejpam-6792	202	18	xi	xi	PROPN
ejpam-6792	202	19	.	.	PUNCT
ejpam-6792	203	1	proof	proof	NOUN
ejpam-6792	203	2	.	.	PUNCT
ejpam-6792	204	1	assume	assume	VERB
ejpam-6792	204	2	that	that	SCONJ
ejpam-6792	204	3	|	|	ADV
ejpam-6792	204	4	∏	∏	NUM
ejpam-6792	204	5	i∈i	i∈i	ADJ
ejpam-6792	204	6	xi|	xi|	PROPN
ejpam-6792	204	7	=	=	PUNCT
ejpam-6792	205	1	k	k	PROPN
ejpam-6792	205	2	for	for	ADP
ejpam-6792	205	3	some	some	DET
ejpam-6792	205	4	odd	odd	ADJ
ejpam-6792	205	5	number	number	NOUN
ejpam-6792	205	6	k	k	PROPN
ejpam-6792	205	7	>	>	X
ejpam-6792	205	8	1	1	X
ejpam-6792	205	9	.	.	PUNCT
ejpam-6792	205	10	suppose	suppose	VERB
ejpam-6792	205	11	g	g	PROPN
ejpam-6792	205	12	(	(	PUNCT
ejpam-6792	205	13	∏	∏	X
ejpam-6792	205	14	i∈i	i∈i	ADJ
ejpam-6792	205	15	xi	xi	NOUN
ejpam-6792	205	16	)	)	PUNCT
ejpam-6792	206	1	=	=	NOUN
ejpam-6792	206	2	∏	∏	PROPN
ejpam-6792	206	3	i∈i	i∈i	ADJ
ejpam-6792	206	4	xi	xi	PROPN
ejpam-6792	206	5	.	.	PUNCT
ejpam-6792	207	1	then	then	ADV
ejpam-6792	207	2	|g	|g	PROPN
ejpam-6792	207	3	(	(	PUNCT
ejpam-6792	207	4	∏	∏	X
ejpam-6792	207	5	i∈i	i∈i	ADJ
ejpam-6792	207	6	xi)|	xi)|	PUNCT
ejpam-6792	207	7	=	=	SYM
ejpam-6792	208	1	k.	k.	PROPN
ejpam-6792	208	2	there	there	PRON
ejpam-6792	208	3	follows	follow	VERB
ejpam-6792	208	4	|g	|g	NOUN
ejpam-6792	208	5	(	(	PUNCT
ejpam-6792	208	6	∏	∏	X
ejpam-6792	208	7	i∈i	i∈i	NOUN
ejpam-6792	208	8	xi)|	xi)|	PROPN
ejpam-6792	208	9	is	be	AUX
ejpam-6792	208	10	odd	odd	ADJ
ejpam-6792	208	11	,	,	PUNCT
ejpam-6792	208	12	this	this	PRON
ejpam-6792	208	13	is	be	AUX
ejpam-6792	208	14	impossible	impossible	ADJ
ejpam-6792	208	15	since	since	SCONJ
ejpam-6792	208	16	g	g	PROPN
ejpam-6792	208	17	(	(	PUNCT
ejpam-6792	208	18	∏	∏	PROPN
ejpam-6792	208	19	i∈i	i∈i	NOUN
ejpam-6792	208	20	xi	xi	NOUN
ejpam-6792	208	21	)	)	PUNCT
ejpam-6792	208	22	is	be	AUX
ejpam-6792	208	23	an	an	DET
ejpam-6792	208	24	abelian	abelian	ADJ
ejpam-6792	208	25	group	group	NOUN
ejpam-6792	208	26	by	by	ADP
ejpam-6792	208	27	proposition	proposition	NOUN
ejpam-6792	208	28	11	11	NUM
ejpam-6792	208	29	.	.	PUNCT
ejpam-6792	209	1	hence	hence	ADV
ejpam-6792	209	2	,	,	PUNCT
ejpam-6792	209	3	g	g	PROPN
ejpam-6792	209	4	(	(	PUNCT
ejpam-6792	209	5	∏	∏	PROPN
ejpam-6792	209	6	i∈i	i∈i	ADJ
ejpam-6792	209	7	xi	xi	NOUN
ejpam-6792	209	8	)	)	PUNCT
ejpam-6792	209	9	̸=	̸=	PROPN
ejpam-6792	209	10	∏	∏	NUM
ejpam-6792	209	11	i∈i	i∈i	ADJ
ejpam-6792	209	12	xi	xi	PROPN
ejpam-6792	209	13	.	.	PUNCT
ejpam-6792	210	1	proposition	proposition	NOUN
ejpam-6792	210	2	13	13	NUM
ejpam-6792	210	3	.	.	PUNCT
ejpam-6792	211	1	if	if	SCONJ
ejpam-6792	211	2	g	g	PROPN
ejpam-6792	211	3	(	(	PUNCT
ejpam-6792	211	4	∏	∏	PROPN
ejpam-6792	211	5	i∈i	i∈i	ADJ
ejpam-6792	211	6	xi	xi	NOUN
ejpam-6792	211	7	)	)	PUNCT
ejpam-6792	211	8	=	=	SYM
ejpam-6792	211	9	∏	∏	PROPN
ejpam-6792	211	10	i∈i	i∈i	ADJ
ejpam-6792	211	11	xi	xi	PROPN
ejpam-6792	211	12	,	,	PUNCT
ejpam-6792	211	13	then	then	ADV
ejpam-6792	211	14	every	every	DET
ejpam-6792	211	15	element	element	NOUN
ejpam-6792	211	16	of	of	ADP
ejpam-6792	211	17	∏	∏	PROPN
ejpam-6792	211	18	i∈i	i∈i	ADJ
ejpam-6792	211	19	xi	xi	VERB
ejpam-6792	211	20	is	be	AUX
ejpam-6792	211	21	an	an	DET
ejpam-6792	211	22	atom	atom	NOUN
ejpam-6792	211	23	.	.	PUNCT
ejpam-6792	212	1	proof	proof	NOUN
ejpam-6792	212	2	.	.	PUNCT
ejpam-6792	213	1	assume	assume	VERB
ejpam-6792	213	2	g	g	PROPN
ejpam-6792	213	3	(	(	PUNCT
ejpam-6792	213	4	∏	∏	X
ejpam-6792	213	5	i∈i	i∈i	ADJ
ejpam-6792	213	6	xi	xi	NOUN
ejpam-6792	213	7	)	)	PUNCT
ejpam-6792	214	1	=	=	SYM
ejpam-6792	214	2	∏	∏	PROPN
ejpam-6792	214	3	i∈i	i∈i	ADJ
ejpam-6792	214	4	xi	xi	PROPN
ejpam-6792	214	5	.	.	PUNCT
ejpam-6792	215	1	then	then	ADV
ejpam-6792	215	2	by	by	ADP
ejpam-6792	215	3	proposition	proposition	NOUN
ejpam-6792	215	4	11	11	NUM
ejpam-6792	215	5	,	,	PUNCT
ejpam-6792	215	6	∏	∏	PROPN
ejpam-6792	215	7	i∈i	i∈i	ADV
ejpam-6792	215	8	xi	xi	VERB
ejpam-6792	215	9	is	be	AUX
ejpam-6792	215	10	an	an	DET
ejpam-6792	215	11	abelian	abelian	ADJ
ejpam-6792	215	12	group	group	NOUN
ejpam-6792	215	13	.	.	PUNCT
ejpam-6792	216	1	let	let	VERB
ejpam-6792	216	2	(	(	PUNCT
ejpam-6792	216	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	216	4	∈	∈	PROPN
ejpam-6792	216	5	∏	∏	PROPN
ejpam-6792	216	6	i∈i	i∈i	ADJ
ejpam-6792	216	7	xi	xi	PROPN
ejpam-6792	216	8	.	.	PROPN
ejpam-6792	217	1	assume	assume	VERB
ejpam-6792	217	2	(	(	PUNCT
ejpam-6792	217	3	xi)i∈i	xi)i∈i	X
ejpam-6792	217	4	⊛	⊛	NUM
ejpam-6792	217	5	(	(	PUNCT
ejpam-6792	217	6	ai)i∈i	ai)i∈i	NUM
ejpam-6792	217	7	=	=	SYM
ejpam-6792	217	8	(	(	PUNCT
ejpam-6792	217	9	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	217	10	.	.	PUNCT
ejpam-6792	218	1	since	since	SCONJ
ejpam-6792	218	2	∏	∏	PROPN
ejpam-6792	218	3	i∈i	i∈i	NOUN
ejpam-6792	218	4	xi	xi	VERB
ejpam-6792	218	5	is	be	AUX
ejpam-6792	218	6	an	an	DET
ejpam-6792	218	7	abelian	abelian	ADJ
ejpam-6792	218	8	group	group	NOUN
ejpam-6792	218	9	,	,	PUNCT
ejpam-6792	218	10	then	then	ADV
ejpam-6792	218	11	(	(	PUNCT
ejpam-6792	218	12	ai)i∈i	ai)i∈i	NUM
ejpam-6792	218	13	⊛	⊛	NUM
ejpam-6792	218	14	(	(	PUNCT
ejpam-6792	218	15	xi)i∈i	xi)i∈i	NUM
ejpam-6792	218	16	=	=	SYM
ejpam-6792	218	17	(	(	PUNCT
ejpam-6792	218	18	xi)i∈i	xi)i∈i	X
ejpam-6792	218	19	⊛	⊛	NUM
ejpam-6792	218	20	(	(	PUNCT
ejpam-6792	218	21	ai)i∈i	ai)i∈i	NUM
ejpam-6792	218	22	=	=	SYM
ejpam-6792	218	23	(	(	PUNCT
ejpam-6792	218	24	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	218	25	.	.	PUNCT
ejpam-6792	219	1	from	from	ADP
ejpam-6792	219	2	(	(	PUNCT
ejpam-6792	219	3	ai)i∈i	ai)i∈i	NUM
ejpam-6792	219	4	⊛	⊛	NUM
ejpam-6792	219	5	(	(	PUNCT
ejpam-6792	219	6	xi)i∈i	xi)i∈i	NUM
ejpam-6792	219	7	=	=	SYM
ejpam-6792	219	8	(	(	PUNCT
ejpam-6792	219	9	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	219	10	and	and	CCONJ
ejpam-6792	219	11	by	by	ADP
ejpam-6792	219	12	(	(	PUNCT
ejpam-6792	219	13	q1	q1	PROPN
ejpam-6792	219	14	)	)	PUNCT
ejpam-6792	219	15	(	(	PUNCT
ejpam-6792	219	16	ai)i∈i	ai)i∈i	NUM
ejpam-6792	219	17	⊛	⊛	NUM
ejpam-6792	219	18	(	(	PUNCT
ejpam-6792	219	19	ai)i∈i	ai)i∈i	NUM
ejpam-6792	219	20	=	=	SYM
ejpam-6792	219	21	(	(	PUNCT
ejpam-6792	219	22	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	219	23	,	,	PUNCT
ejpam-6792	219	24	applying	apply	VERB
ejpam-6792	219	25	corollary	corollary	NOUN
ejpam-6792	219	26	1	1	NUM
ejpam-6792	219	27	we	we	PRON
ejpam-6792	219	28	get	get	VERB
ejpam-6792	219	29	(	(	PUNCT
ejpam-6792	219	30	ai)i∈i	ai)i∈i	NUM
ejpam-6792	219	31	=	=	SYM
ejpam-6792	219	32	(	(	PUNCT
ejpam-6792	219	33	xi)i∈i	xi)i∈i	NUM
ejpam-6792	219	34	.	.	PUNCT
ejpam-6792	220	1	therefore	therefore	ADV
ejpam-6792	220	2	,	,	PUNCT
ejpam-6792	220	3	(	(	PUNCT
ejpam-6792	220	4	ai)i∈i	ai)i∈i	NUM
ejpam-6792	220	5	is	be	AUX
ejpam-6792	220	6	an	an	DET
ejpam-6792	220	7	atom	atom	NOUN
ejpam-6792	220	8	for	for	ADP
ejpam-6792	220	9	all	all	DET
ejpam-6792	220	10	(	(	PUNCT
ejpam-6792	220	11	ai)i∈i	ai)i∈i	NUM
ejpam-6792	220	12	∈	∈	PROPN
ejpam-6792	220	13	∏	∏	PROPN
ejpam-6792	220	14	i∈i	i∈i	ADJ
ejpam-6792	220	15	xi	xi	PROPN
ejpam-6792	220	16	.	.	PUNCT
ejpam-6792	221	1	the	the	DET
ejpam-6792	221	2	converse	converse	NOUN
ejpam-6792	221	3	of	of	ADP
ejpam-6792	221	4	proposition	proposition	NOUN
ejpam-6792	221	5	13	13	NUM
ejpam-6792	221	6	is	be	AUX
ejpam-6792	221	7	not	not	PART
ejpam-6792	221	8	true	true	ADJ
ejpam-6792	221	9	.	.	PUNCT
ejpam-6792	222	1	let	let	AUX
ejpam-6792	222	2	consider	consider	VERB
ejpam-6792	222	3	a	a	DET
ejpam-6792	222	4	q	q	NOUN
ejpam-6792	222	5	-	-	NOUN
ejpam-6792	222	6	algebra	algebra	NOUN
ejpam-6792	222	7	(	(	PUNCT
ejpam-6792	222	8	x1	x1	PROPN
ejpam-6792	222	9	×x2;⊛	×x2;⊛	NOUN
ejpam-6792	222	10	,	,	PUNCT
ejpam-6792	222	11	(	(	PUNCT
ejpam-6792	222	12	01	01	NUM
ejpam-6792	222	13	,	,	PUNCT
ejpam-6792	222	14	02	02	NUM
ejpam-6792	222	15	)	)	PUNCT
ejpam-6792	222	16	)	)	PUNCT
ejpam-6792	222	17	from	from	ADP
ejpam-6792	222	18	example	example	NOUN
ejpam-6792	222	19	1	1	NUM
ejpam-6792	222	20	.	.	NUM
ejpam-6792	222	21	⊛	⊛	NUM
ejpam-6792	222	22	(	(	PUNCT
ejpam-6792	222	23	01	01	NUM
ejpam-6792	222	24	,	,	PUNCT
ejpam-6792	222	25	02	02	NUM
ejpam-6792	222	26	)	)	PUNCT
ejpam-6792	222	27	(	(	PUNCT
ejpam-6792	222	28	01	01	NUM
ejpam-6792	222	29	,	,	PUNCT
ejpam-6792	222	30	x	x	NOUN
ejpam-6792	222	31	)	)	PUNCT
ejpam-6792	222	32	(	(	PUNCT
ejpam-6792	222	33	01	01	NUM
ejpam-6792	222	34	,	,	PUNCT
ejpam-6792	222	35	y	y	NOUN
ejpam-6792	222	36	)	)	PUNCT
ejpam-6792	222	37	(	(	PUNCT
ejpam-6792	222	38	01	01	NUM
ejpam-6792	222	39	,	,	PUNCT
ejpam-6792	222	40	z	z	NOUN
ejpam-6792	222	41	)	)	PUNCT
ejpam-6792	222	42	(	(	PUNCT
ejpam-6792	222	43	a	a	PRON
ejpam-6792	222	44	,	,	PUNCT
ejpam-6792	222	45	02	02	NUM
ejpam-6792	222	46	)	)	PUNCT
ejpam-6792	222	47	(	(	PUNCT
ejpam-6792	222	48	a	a	PRON
ejpam-6792	222	49	,	,	PUNCT
ejpam-6792	222	50	x	x	NOUN
ejpam-6792	222	51	)	)	PUNCT
ejpam-6792	222	52	(	(	PUNCT
ejpam-6792	222	53	a	a	DET
ejpam-6792	222	54	,	,	PUNCT
ejpam-6792	222	55	y	y	NOUN
ejpam-6792	222	56	)	)	PUNCT
ejpam-6792	222	57	(	(	PUNCT
ejpam-6792	222	58	a	a	DET
ejpam-6792	222	59	,	,	PUNCT
ejpam-6792	222	60	z	z	NOUN
ejpam-6792	222	61	)	)	PUNCT
ejpam-6792	222	62	(	(	PUNCT
ejpam-6792	222	63	01	01	NUM
ejpam-6792	222	64	,	,	PUNCT
ejpam-6792	222	65	02	02	NUM
ejpam-6792	222	66	)	)	PUNCT
ejpam-6792	222	67	(	(	PUNCT
ejpam-6792	222	68	01	01	NUM
ejpam-6792	222	69	,	,	PUNCT
ejpam-6792	222	70	02	02	NUM
ejpam-6792	222	71	)	)	PUNCT
ejpam-6792	222	72	(	(	PUNCT
ejpam-6792	222	73	01	01	NUM
ejpam-6792	222	74	,	,	PUNCT
ejpam-6792	222	75	x	x	NOUN
ejpam-6792	222	76	)	)	PUNCT
ejpam-6792	222	77	(	(	PUNCT
ejpam-6792	222	78	01	01	NUM
ejpam-6792	222	79	,	,	PUNCT
ejpam-6792	222	80	z	z	NOUN
ejpam-6792	222	81	)	)	PUNCT
ejpam-6792	222	82	(	(	PUNCT
ejpam-6792	222	83	01	01	NUM
ejpam-6792	222	84	,	,	PUNCT
ejpam-6792	222	85	y	y	PROPN
ejpam-6792	222	86	)	)	PUNCT
ejpam-6792	222	87	(	(	PUNCT
ejpam-6792	222	88	a	a	PRON
ejpam-6792	222	89	,	,	PUNCT
ejpam-6792	222	90	02	02	NUM
ejpam-6792	222	91	)	)	PUNCT
ejpam-6792	222	92	(	(	PUNCT
ejpam-6792	222	93	a	a	PRON
ejpam-6792	222	94	,	,	PUNCT
ejpam-6792	222	95	x	x	NOUN
ejpam-6792	222	96	)	)	PUNCT
ejpam-6792	222	97	(	(	PUNCT
ejpam-6792	222	98	a	a	DET
ejpam-6792	222	99	,	,	PUNCT
ejpam-6792	222	100	z	z	NOUN
ejpam-6792	222	101	)	)	PUNCT
ejpam-6792	222	102	(	(	PUNCT
ejpam-6792	222	103	a	a	DET
ejpam-6792	222	104	,	,	PUNCT
ejpam-6792	222	105	y	y	NOUN
ejpam-6792	222	106	)	)	PUNCT
ejpam-6792	222	107	(	(	PUNCT
ejpam-6792	222	108	01	01	NUM
ejpam-6792	222	109	,	,	PUNCT
ejpam-6792	222	110	x	x	NOUN
ejpam-6792	222	111	)	)	PUNCT
ejpam-6792	222	112	(	(	PUNCT
ejpam-6792	222	113	01	01	NUM
ejpam-6792	222	114	,	,	PUNCT
ejpam-6792	222	115	x	x	NOUN
ejpam-6792	222	116	)	)	PUNCT
ejpam-6792	222	117	(	(	PUNCT
ejpam-6792	222	118	01	01	NUM
ejpam-6792	222	119	,	,	PUNCT
ejpam-6792	222	120	02	02	NUM
ejpam-6792	222	121	)	)	PUNCT
ejpam-6792	222	122	(	(	PUNCT
ejpam-6792	222	123	01	01	NUM
ejpam-6792	222	124	,	,	PUNCT
ejpam-6792	222	125	y	y	NOUN
ejpam-6792	222	126	)	)	PUNCT
ejpam-6792	222	127	(	(	PUNCT
ejpam-6792	222	128	01	01	NUM
ejpam-6792	222	129	,	,	PUNCT
ejpam-6792	222	130	z	z	NOUN
ejpam-6792	222	131	)	)	PUNCT
ejpam-6792	222	132	(	(	PUNCT
ejpam-6792	222	133	a	a	PRON
ejpam-6792	222	134	,	,	PUNCT
ejpam-6792	222	135	x	x	NOUN
ejpam-6792	222	136	)	)	PUNCT
ejpam-6792	222	137	(	(	PUNCT
ejpam-6792	222	138	a	a	PRON
ejpam-6792	222	139	,	,	PUNCT
ejpam-6792	222	140	02	02	NUM
ejpam-6792	222	141	)	)	PUNCT
ejpam-6792	222	142	(	(	PUNCT
ejpam-6792	222	143	a	a	DET
ejpam-6792	222	144	,	,	PUNCT
ejpam-6792	222	145	y	y	NOUN
ejpam-6792	222	146	)	)	PUNCT
ejpam-6792	222	147	(	(	PUNCT
ejpam-6792	222	148	a	a	DET
ejpam-6792	222	149	,	,	PUNCT
ejpam-6792	222	150	z	z	NOUN
ejpam-6792	222	151	)	)	PUNCT
ejpam-6792	222	152	(	(	PUNCT
ejpam-6792	222	153	01	01	NUM
ejpam-6792	222	154	,	,	PUNCT
ejpam-6792	222	155	y	y	NOUN
ejpam-6792	222	156	)	)	PUNCT
ejpam-6792	222	157	(	(	PUNCT
ejpam-6792	222	158	01	01	NUM
ejpam-6792	222	159	,	,	PUNCT
ejpam-6792	222	160	y	y	NOUN
ejpam-6792	222	161	)	)	PUNCT
ejpam-6792	222	162	(	(	PUNCT
ejpam-6792	222	163	01	01	NUM
ejpam-6792	222	164	,	,	PUNCT
ejpam-6792	222	165	z	z	NOUN
ejpam-6792	222	166	)	)	PUNCT
ejpam-6792	222	167	(	(	PUNCT
ejpam-6792	222	168	01	01	NUM
ejpam-6792	222	169	,	,	PUNCT
ejpam-6792	222	170	02	02	NUM
ejpam-6792	222	171	)	)	PUNCT
ejpam-6792	222	172	(	(	PUNCT
ejpam-6792	222	173	01	01	NUM
ejpam-6792	222	174	,	,	PUNCT
ejpam-6792	222	175	x	x	NOUN
ejpam-6792	222	176	)	)	PUNCT
ejpam-6792	222	177	(	(	PUNCT
ejpam-6792	222	178	a	a	DET
ejpam-6792	222	179	,	,	PUNCT
ejpam-6792	222	180	y	y	NOUN
ejpam-6792	222	181	)	)	PUNCT
ejpam-6792	222	182	(	(	PUNCT
ejpam-6792	222	183	a	a	DET
ejpam-6792	222	184	,	,	PUNCT
ejpam-6792	222	185	z	z	NOUN
ejpam-6792	222	186	)	)	PUNCT
ejpam-6792	222	187	(	(	PUNCT
ejpam-6792	222	188	a	a	PRON
ejpam-6792	222	189	,	,	PUNCT
ejpam-6792	222	190	02	02	NUM
ejpam-6792	222	191	)	)	PUNCT
ejpam-6792	222	192	(	(	PUNCT
ejpam-6792	222	193	a	a	PRON
ejpam-6792	222	194	,	,	PUNCT
ejpam-6792	222	195	x	x	NOUN
ejpam-6792	222	196	)	)	PUNCT
ejpam-6792	222	197	(	(	PUNCT
ejpam-6792	222	198	01	01	NUM
ejpam-6792	222	199	,	,	PUNCT
ejpam-6792	222	200	z	z	NOUN
ejpam-6792	222	201	)	)	PUNCT
ejpam-6792	222	202	(	(	PUNCT
ejpam-6792	222	203	01	01	NUM
ejpam-6792	222	204	,	,	PUNCT
ejpam-6792	222	205	z	z	NOUN
ejpam-6792	222	206	)	)	PUNCT
ejpam-6792	222	207	(	(	PUNCT
ejpam-6792	222	208	01	01	NUM
ejpam-6792	222	209	,	,	PUNCT
ejpam-6792	222	210	y	y	NOUN
ejpam-6792	222	211	)	)	PUNCT
ejpam-6792	222	212	(	(	PUNCT
ejpam-6792	222	213	01	01	NUM
ejpam-6792	222	214	,	,	PUNCT
ejpam-6792	222	215	x	x	NOUN
ejpam-6792	222	216	)	)	PUNCT
ejpam-6792	222	217	(	(	PUNCT
ejpam-6792	222	218	01	01	NUM
ejpam-6792	222	219	,	,	PUNCT
ejpam-6792	222	220	02	02	NUM
ejpam-6792	222	221	)	)	PUNCT
ejpam-6792	222	222	(	(	PUNCT
ejpam-6792	222	223	a	a	DET
ejpam-6792	222	224	,	,	PUNCT
ejpam-6792	222	225	z	z	NOUN
ejpam-6792	222	226	)	)	PUNCT
ejpam-6792	222	227	(	(	PUNCT
ejpam-6792	222	228	a	a	DET
ejpam-6792	222	229	,	,	PUNCT
ejpam-6792	222	230	y	y	NOUN
ejpam-6792	222	231	)	)	PUNCT
ejpam-6792	222	232	(	(	PUNCT
ejpam-6792	222	233	a	a	PRON
ejpam-6792	222	234	,	,	PUNCT
ejpam-6792	222	235	x	x	NOUN
ejpam-6792	222	236	)	)	PUNCT
ejpam-6792	222	237	(	(	PUNCT
ejpam-6792	222	238	a	a	PRON
ejpam-6792	222	239	,	,	PUNCT
ejpam-6792	222	240	02	02	NUM
ejpam-6792	222	241	)	)	PUNCT
ejpam-6792	222	242	(	(	PUNCT
ejpam-6792	222	243	a	a	PRON
ejpam-6792	222	244	,	,	PUNCT
ejpam-6792	222	245	02	02	NUM
ejpam-6792	222	246	)	)	PUNCT
ejpam-6792	222	247	(	(	PUNCT
ejpam-6792	222	248	a	a	PRON
ejpam-6792	222	249	,	,	PUNCT
ejpam-6792	222	250	02	02	NUM
ejpam-6792	222	251	)	)	PUNCT
ejpam-6792	222	252	(	(	PUNCT
ejpam-6792	222	253	a	a	PRON
ejpam-6792	222	254	,	,	PUNCT
ejpam-6792	222	255	x	x	NOUN
ejpam-6792	222	256	)	)	PUNCT
ejpam-6792	222	257	(	(	PUNCT
ejpam-6792	222	258	a	a	DET
ejpam-6792	222	259	,	,	PUNCT
ejpam-6792	222	260	z	z	NOUN
ejpam-6792	222	261	)	)	PUNCT
ejpam-6792	222	262	(	(	PUNCT
ejpam-6792	222	263	a	a	DET
ejpam-6792	222	264	,	,	PUNCT
ejpam-6792	222	265	y	y	NOUN
ejpam-6792	222	266	)	)	PUNCT
ejpam-6792	222	267	(	(	PUNCT
ejpam-6792	222	268	01	01	NUM
ejpam-6792	222	269	,	,	PUNCT
ejpam-6792	222	270	02	02	NUM
ejpam-6792	222	271	)	)	PUNCT
ejpam-6792	222	272	(	(	PUNCT
ejpam-6792	222	273	01	01	NUM
ejpam-6792	222	274	,	,	PUNCT
ejpam-6792	222	275	x	x	NOUN
ejpam-6792	222	276	)	)	PUNCT
ejpam-6792	222	277	(	(	PUNCT
ejpam-6792	222	278	01	01	NUM
ejpam-6792	222	279	,	,	PUNCT
ejpam-6792	222	280	z	z	NOUN
ejpam-6792	222	281	)	)	PUNCT
ejpam-6792	222	282	(	(	PUNCT
ejpam-6792	222	283	01	01	NUM
ejpam-6792	222	284	,	,	PUNCT
ejpam-6792	222	285	y	y	PROPN
ejpam-6792	222	286	)	)	PUNCT
ejpam-6792	222	287	(	(	PUNCT
ejpam-6792	222	288	a	a	PRON
ejpam-6792	222	289	,	,	PUNCT
ejpam-6792	222	290	x	x	NOUN
ejpam-6792	222	291	)	)	PUNCT
ejpam-6792	222	292	(	(	PUNCT
ejpam-6792	222	293	a	a	PRON
ejpam-6792	222	294	,	,	PUNCT
ejpam-6792	222	295	x	x	NOUN
ejpam-6792	222	296	)	)	PUNCT
ejpam-6792	222	297	(	(	PUNCT
ejpam-6792	222	298	a	a	PRON
ejpam-6792	222	299	,	,	PUNCT
ejpam-6792	222	300	02	02	NUM
ejpam-6792	222	301	)	)	PUNCT
ejpam-6792	222	302	(	(	PUNCT
ejpam-6792	222	303	a	a	DET
ejpam-6792	222	304	,	,	PUNCT
ejpam-6792	222	305	y	y	NOUN
ejpam-6792	222	306	)	)	PUNCT
ejpam-6792	222	307	(	(	PUNCT
ejpam-6792	222	308	a	a	DET
ejpam-6792	222	309	,	,	PUNCT
ejpam-6792	222	310	z	z	NOUN
ejpam-6792	222	311	)	)	PUNCT
ejpam-6792	222	312	(	(	PUNCT
ejpam-6792	222	313	01	01	NUM
ejpam-6792	222	314	,	,	PUNCT
ejpam-6792	222	315	x	x	NOUN
ejpam-6792	222	316	)	)	PUNCT
ejpam-6792	222	317	(	(	PUNCT
ejpam-6792	222	318	01	01	NUM
ejpam-6792	222	319	,	,	PUNCT
ejpam-6792	222	320	02	02	NUM
ejpam-6792	222	321	)	)	PUNCT
ejpam-6792	222	322	(	(	PUNCT
ejpam-6792	222	323	01	01	NUM
ejpam-6792	222	324	,	,	PUNCT
ejpam-6792	222	325	y	y	NOUN
ejpam-6792	222	326	)	)	PUNCT
ejpam-6792	222	327	(	(	PUNCT
ejpam-6792	222	328	01	01	NUM
ejpam-6792	222	329	,	,	PUNCT
ejpam-6792	222	330	z	z	NOUN
ejpam-6792	222	331	)	)	PUNCT
ejpam-6792	222	332	(	(	PUNCT
ejpam-6792	222	333	a	a	DET
ejpam-6792	222	334	,	,	PUNCT
ejpam-6792	222	335	y	y	NOUN
ejpam-6792	222	336	)	)	PUNCT
ejpam-6792	222	337	(	(	PUNCT
ejpam-6792	222	338	a	a	DET
ejpam-6792	222	339	,	,	PUNCT
ejpam-6792	222	340	y	y	NOUN
ejpam-6792	222	341	)	)	PUNCT
ejpam-6792	222	342	(	(	PUNCT
ejpam-6792	222	343	a	a	DET
ejpam-6792	222	344	,	,	PUNCT
ejpam-6792	222	345	z	z	NOUN
ejpam-6792	222	346	)	)	PUNCT
ejpam-6792	222	347	(	(	PUNCT
ejpam-6792	222	348	a	a	PRON
ejpam-6792	222	349	,	,	PUNCT
ejpam-6792	222	350	02	02	NUM
ejpam-6792	222	351	)	)	PUNCT
ejpam-6792	222	352	(	(	PUNCT
ejpam-6792	222	353	a	a	PRON
ejpam-6792	222	354	,	,	PUNCT
ejpam-6792	222	355	x	x	NOUN
ejpam-6792	222	356	)	)	PUNCT
ejpam-6792	222	357	(	(	PUNCT
ejpam-6792	222	358	01	01	NUM
ejpam-6792	222	359	,	,	PUNCT
ejpam-6792	222	360	y	y	NOUN
ejpam-6792	222	361	)	)	PUNCT
ejpam-6792	222	362	(	(	PUNCT
ejpam-6792	222	363	01	01	NUM
ejpam-6792	222	364	,	,	PUNCT
ejpam-6792	222	365	z	z	NOUN
ejpam-6792	222	366	)	)	PUNCT
ejpam-6792	222	367	(	(	PUNCT
ejpam-6792	222	368	01	01	NUM
ejpam-6792	222	369	,	,	PUNCT
ejpam-6792	222	370	02	02	NUM
ejpam-6792	222	371	)	)	PUNCT
ejpam-6792	222	372	(	(	PUNCT
ejpam-6792	222	373	01	01	NUM
ejpam-6792	222	374	,	,	PUNCT
ejpam-6792	222	375	x	x	NOUN
ejpam-6792	222	376	)	)	PUNCT
ejpam-6792	222	377	(	(	PUNCT
ejpam-6792	222	378	a	a	DET
ejpam-6792	222	379	,	,	PUNCT
ejpam-6792	222	380	z	z	NOUN
ejpam-6792	222	381	)	)	PUNCT
ejpam-6792	222	382	(	(	PUNCT
ejpam-6792	222	383	a	a	DET
ejpam-6792	222	384	,	,	PUNCT
ejpam-6792	222	385	z	z	NOUN
ejpam-6792	222	386	)	)	PUNCT
ejpam-6792	222	387	(	(	PUNCT
ejpam-6792	222	388	a	a	DET
ejpam-6792	222	389	,	,	PUNCT
ejpam-6792	222	390	y	y	NOUN
ejpam-6792	222	391	)	)	PUNCT
ejpam-6792	222	392	(	(	PUNCT
ejpam-6792	222	393	a	a	PRON
ejpam-6792	222	394	,	,	PUNCT
ejpam-6792	222	395	x	x	NOUN
ejpam-6792	222	396	)	)	PUNCT
ejpam-6792	222	397	(	(	PUNCT
ejpam-6792	222	398	a	a	PRON
ejpam-6792	222	399	,	,	PUNCT
ejpam-6792	222	400	02	02	NUM
ejpam-6792	222	401	)	)	PUNCT
ejpam-6792	222	402	(	(	PUNCT
ejpam-6792	222	403	01	01	NUM
ejpam-6792	222	404	,	,	PUNCT
ejpam-6792	222	405	z	z	NOUN
ejpam-6792	222	406	)	)	PUNCT
ejpam-6792	222	407	(	(	PUNCT
ejpam-6792	222	408	01	01	NUM
ejpam-6792	222	409	,	,	PUNCT
ejpam-6792	222	410	y	y	NOUN
ejpam-6792	222	411	)	)	PUNCT
ejpam-6792	222	412	(	(	PUNCT
ejpam-6792	222	413	01	01	NUM
ejpam-6792	222	414	,	,	PUNCT
ejpam-6792	222	415	x	x	NOUN
ejpam-6792	222	416	)	)	PUNCT
ejpam-6792	222	417	(	(	PUNCT
ejpam-6792	222	418	01	01	NUM
ejpam-6792	222	419	,	,	PUNCT
ejpam-6792	222	420	02	02	NUM
ejpam-6792	222	421	)	)	PUNCT
ejpam-6792	222	422	.	.	PUNCT
ejpam-6792	223	1	it	it	PRON
ejpam-6792	223	2	is	be	AUX
ejpam-6792	223	3	not	not	PART
ejpam-6792	223	4	difficult	difficult	ADJ
ejpam-6792	223	5	to	to	PART
ejpam-6792	223	6	verify	verify	VERB
ejpam-6792	223	7	that	that	SCONJ
ejpam-6792	223	8	all	all	DET
ejpam-6792	223	9	elements	element	NOUN
ejpam-6792	223	10	of	of	ADP
ejpam-6792	223	11	x1	x1	ADJ
ejpam-6792	223	12	×x2	×x2	NOUN
ejpam-6792	223	13	are	be	AUX
ejpam-6792	223	14	atoms	atom	NOUN
ejpam-6792	223	15	and	and	CCONJ
ejpam-6792	223	16	g(x1	g(x1	NOUN
ejpam-6792	223	17	×x2	×x2	NOUN
ejpam-6792	223	18	)	)	PUNCT
ejpam-6792	223	19	=	=	PRON
ejpam-6792	223	20	{	{	PUNCT
ejpam-6792	223	21	(	(	PUNCT
ejpam-6792	223	22	01	01	NUM
ejpam-6792	223	23	,	,	PUNCT
ejpam-6792	223	24	02	02	NUM
ejpam-6792	223	25	)	)	PUNCT
ejpam-6792	223	26	,	,	PUNCT
ejpam-6792	223	27	(	(	PUNCT
ejpam-6792	223	28	01	01	NUM
ejpam-6792	223	29	,	,	PUNCT
ejpam-6792	223	30	x	x	NOUN
ejpam-6792	223	31	)	)	PUNCT
ejpam-6792	223	32	,	,	PUNCT
ejpam-6792	223	33	(	(	PUNCT
ejpam-6792	223	34	a	a	PRON
ejpam-6792	223	35	,	,	PUNCT
ejpam-6792	223	36	02	02	NUM
ejpam-6792	223	37	)	)	PUNCT
ejpam-6792	223	38	,	,	PUNCT
ejpam-6792	223	39	(	(	PUNCT
ejpam-6792	223	40	a	a	PRON
ejpam-6792	223	41	,	,	PUNCT
ejpam-6792	223	42	x	x	NOUN
ejpam-6792	223	43	)	)	PUNCT
ejpam-6792	223	44	}	}	PUNCT
ejpam-6792	223	45	.	.	PUNCT
ejpam-6792	224	1	therefore	therefore	ADV
ejpam-6792	224	2	,	,	PUNCT
ejpam-6792	224	3	g(x1	g(x1	ADJ
ejpam-6792	224	4	×x2	×x2	NOUN
ejpam-6792	224	5	)	)	PUNCT
ejpam-6792	224	6	̸=	̸=	PROPN
ejpam-6792	224	7	x1	x1	PUNCT
ejpam-6792	224	8	×x2	×x2	PROPN
ejpam-6792	224	9	.	.	PUNCT
ejpam-6792	225	1	corollary	corollary	ADJ
ejpam-6792	225	2	2	2	NUM
ejpam-6792	225	3	.	.	PUNCT
ejpam-6792	226	1	if	if	SCONJ
ejpam-6792	226	2	g	g	PROPN
ejpam-6792	226	3	(	(	PUNCT
ejpam-6792	226	4	∏	∏	PROPN
ejpam-6792	226	5	i∈i	i∈i	ADJ
ejpam-6792	226	6	xi	xi	NOUN
ejpam-6792	226	7	)	)	PUNCT
ejpam-6792	226	8	=	=	SYM
ejpam-6792	226	9	∏	∏	PROPN
ejpam-6792	226	10	i∈i	i∈i	ADJ
ejpam-6792	226	11	xi	xi	PROPN
ejpam-6792	226	12	,	,	PUNCT
ejpam-6792	226	13	then	then	ADV
ejpam-6792	226	14	(	(	PUNCT
ejpam-6792	226	15	xi)i∈i	xi)i∈i	X
ejpam-6792	226	16	⊛	⊛	NUM
ejpam-6792	227	1	[	[	X
ejpam-6792	227	2	(	(	PUNCT
ejpam-6792	227	3	yi)i∈i	yi)i∈i	NUM
ejpam-6792	227	4	⊛	⊛	NUM
ejpam-6792	227	5	(	(	PUNCT
ejpam-6792	227	6	zi)i∈i	zi)i∈i	NUM
ejpam-6792	227	7	]	]	PUNCT
ejpam-6792	227	8	=	=	PUNCT
ejpam-6792	227	9	(	(	PUNCT
ejpam-6792	227	10	zi)i∈i	zi)i∈i	NUM
ejpam-6792	227	11	⊛	⊛	PART
ejpam-6792	228	1	[	[	X
ejpam-6792	228	2	(	(	PUNCT
ejpam-6792	228	3	yi)i∈i	yi)i∈i	NUM
ejpam-6792	228	4	⊛	⊛	NUM
ejpam-6792	228	5	(	(	PUNCT
ejpam-6792	228	6	xi)i∈i	xi)i∈i	NUM
ejpam-6792	228	7	]	]	PUNCT
ejpam-6792	228	8	for	for	ADP
ejpam-6792	228	9	all	all	DET
ejpam-6792	228	10	(	(	PUNCT
ejpam-6792	228	11	xi)i∈i	xi)i∈i	NUM
ejpam-6792	228	12	,	,	PUNCT
ejpam-6792	228	13	(	(	PUNCT
ejpam-6792	228	14	yi)i∈i	yi)i∈i	NUM
ejpam-6792	228	15	,	,	PUNCT
ejpam-6792	228	16	(	(	PUNCT
ejpam-6792	228	17	zi)i∈i	zi)i∈i	NUM
ejpam-6792	228	18	∈	∈	PROPN
ejpam-6792	228	19	∏	∏	PROPN
ejpam-6792	228	20	i∈i	i∈i	ADJ
ejpam-6792	228	21	xi	xi	PROPN
ejpam-6792	228	22	.	.	PUNCT
ejpam-6792	228	23	a.	a.	PROPN
ejpam-6792	228	24	anantayasethi	anantayasethi	PROPN
ejpam-6792	228	25	,	,	PUNCT
ejpam-6792	228	26	k.	k.	PROPN
ejpam-6792	228	27	saengsura	saengsura	PROPN
ejpam-6792	228	28	,	,	PUNCT
ejpam-6792	228	29	n.	n.	NOUN
ejpam-6792	228	30	sarasit	sarasit	PROPN
ejpam-6792	228	31	/	/	SYM
ejpam-6792	228	32	eur	eur	PROPN
ejpam-6792	228	33	.	.	PUNCT
ejpam-6792	229	1	j.	j.	PROPN
ejpam-6792	229	2	pure	pure	PROPN
ejpam-6792	229	3	appl	appl	PROPN
ejpam-6792	229	4	.	.	PROPN
ejpam-6792	229	5	math	math	PROPN
ejpam-6792	229	6	,	,	PUNCT
ejpam-6792	229	7	18	18	NUM
ejpam-6792	229	8	(	(	PUNCT
ejpam-6792	229	9	4	4	NUM
ejpam-6792	229	10	)	)	PUNCT
ejpam-6792	229	11	(	(	PUNCT
ejpam-6792	229	12	2025	2025	NUM
ejpam-6792	229	13	)	)	PUNCT
ejpam-6792	229	14	,	,	PUNCT
ejpam-6792	229	15	6792	6792	NUM
ejpam-6792	229	16	8	8	NUM
ejpam-6792	229	17	of	of	ADP
ejpam-6792	229	18	10	10	NUM
ejpam-6792	229	19	proof	proof	NOUN
ejpam-6792	229	20	.	.	PUNCT
ejpam-6792	230	1	assume	assume	VERB
ejpam-6792	230	2	that	that	SCONJ
ejpam-6792	230	3	g	g	PROPN
ejpam-6792	230	4	(	(	PUNCT
ejpam-6792	230	5	∏	∏	X
ejpam-6792	230	6	i∈i	i∈i	ADJ
ejpam-6792	230	7	xi	xi	NOUN
ejpam-6792	230	8	)	)	PUNCT
ejpam-6792	231	1	=	=	SYM
ejpam-6792	231	2	∏	∏	PROPN
ejpam-6792	231	3	i∈i	i∈i	ADJ
ejpam-6792	231	4	xi	xi	PROPN
ejpam-6792	231	5	.	.	PUNCT
ejpam-6792	232	1	then	then	ADV
ejpam-6792	232	2	by	by	ADP
ejpam-6792	232	3	proposition	proposition	NOUN
ejpam-6792	232	4	13	13	NUM
ejpam-6792	232	5	all	all	DET
ejpam-6792	232	6	elements	element	NOUN
ejpam-6792	232	7	of	of	ADP
ejpam-6792	232	8	∏	∏	PROPN
ejpam-6792	232	9	i∈i	i∈i	NOUN
ejpam-6792	232	10	xi	xi	X
ejpam-6792	232	11	are	be	AUX
ejpam-6792	232	12	atoms	atom	NOUN
ejpam-6792	232	13	.	.	PUNCT
ejpam-6792	233	1	by	by	ADP
ejpam-6792	233	2	proposition	proposition	NOUN
ejpam-6792	233	3	4	4	NUM
ejpam-6792	233	4	we	we	PRON
ejpam-6792	233	5	get	get	VERB
ejpam-6792	233	6	(	(	PUNCT
ejpam-6792	233	7	xi)i∈i⊛[(yi)i∈i⊛(zi)i∈i	xi)i∈i⊛[(yi)i∈i⊛(zi)i∈i	NOUN
ejpam-6792	233	8	]	]	X
ejpam-6792	234	1	=	=	SYM
ejpam-6792	234	2	(	(	PUNCT
ejpam-6792	234	3	zi)i∈i)⊛[(yi)i∈i⊛(xi)i∈i	zi)i∈i)⊛[(yi)i∈i⊛(xi)i∈i	X
ejpam-6792	234	4	]	]	PUNCT
ejpam-6792	234	5	for	for	ADP
ejpam-6792	234	6	all	all	DET
ejpam-6792	234	7	(	(	PUNCT
ejpam-6792	234	8	xi)i∈i	xi)i∈i	NUM
ejpam-6792	234	9	,	,	PUNCT
ejpam-6792	234	10	(	(	PUNCT
ejpam-6792	234	11	yi)i∈i	yi)i∈i	NUM
ejpam-6792	234	12	,	,	PUNCT
ejpam-6792	234	13	(	(	PUNCT
ejpam-6792	234	14	zi)i∈i	zi)i∈i	NUM
ejpam-6792	234	15	∈	∈	PROPN
ejpam-6792	234	16	∏	∏	PROPN
ejpam-6792	234	17	i∈i	i∈i	ADJ
ejpam-6792	234	18	xi	xi	PROPN
ejpam-6792	234	19	.	.	PUNCT
ejpam-6792	235	1	next	next	ADJ
ejpam-6792	235	2	proposition	proposition	NOUN
ejpam-6792	235	3	shows	show	VERB
ejpam-6792	235	4	a	a	DET
ejpam-6792	235	5	condition	condition	NOUN
ejpam-6792	235	6	for	for	ADP
ejpam-6792	235	7	the	the	DET
ejpam-6792	235	8	direct	direct	ADJ
ejpam-6792	235	9	product	product	NOUN
ejpam-6792	235	10	of	of	ADP
ejpam-6792	235	11	q	q	NOUN
ejpam-6792	235	12	-	-	PUNCT
ejpam-6792	235	13	algebras	algebras	PROPN
ejpam-6792	235	14	to	to	PART
ejpam-6792	235	15	be	be	AUX
ejpam-6792	235	16	a	a	DET
ejpam-6792	235	17	ci	ci	NOUN
ejpam-6792	235	18	-	-	NOUN
ejpam-6792	235	19	algebra	algebra	NOUN
ejpam-6792	235	20	.	.	PUNCT
ejpam-6792	236	1	a	a	DET
ejpam-6792	236	2	sysytem	sysytem	NOUN
ejpam-6792	236	3	(	(	PUNCT
ejpam-6792	236	4	x	x	NOUN
ejpam-6792	236	5	;	;	PUNCT
ejpam-6792	236	6	∗	∗	NOUN
ejpam-6792	236	7	,	,	PUNCT
ejpam-6792	236	8	0	0	NUM
ejpam-6792	236	9	)	)	PUNCT
ejpam-6792	236	10	consists	consist	VERB
ejpam-6792	236	11	of	of	ADP
ejpam-6792	236	12	a	a	DET
ejpam-6792	236	13	non	non	ADJ
ejpam-6792	236	14	-	-	ADJ
ejpam-6792	236	15	empty	empty	ADJ
ejpam-6792	236	16	set	set	NOUN
ejpam-6792	236	17	x	x	NOUN
ejpam-6792	236	18	,	,	PUNCT
ejpam-6792	236	19	together	together	ADV
ejpam-6792	236	20	with	with	ADP
ejpam-6792	236	21	a	a	DET
ejpam-6792	236	22	binary	binary	ADJ
ejpam-6792	236	23	operation	operation	NOUN
ejpam-6792	236	24	∗	∗	NOUN
ejpam-6792	236	25	defined	define	VERB
ejpam-6792	236	26	on	on	ADP
ejpam-6792	236	27	x	x	PUNCT
ejpam-6792	236	28	and	and	CCONJ
ejpam-6792	236	29	a	a	DET
ejpam-6792	236	30	constant	constant	ADJ
ejpam-6792	236	31	0	0	NUM
ejpam-6792	236	32	∈	∈	NOUN
ejpam-6792	236	33	x	x	AUX
ejpam-6792	236	34	is	be	AUX
ejpam-6792	236	35	called	call	VERB
ejpam-6792	236	36	a	a	DET
ejpam-6792	236	37	ci	ci	NOUN
ejpam-6792	236	38	-	-	NOUN
ejpam-6792	236	39	algebra	algebra	NOUN
ejpam-6792	236	40	if	if	SCONJ
ejpam-6792	236	41	(	(	PUNCT
ejpam-6792	236	42	ci1	ci1	PROPN
ejpam-6792	236	43	)	)	PUNCT
ejpam-6792	236	44	xx	xx	NUM
ejpam-6792	236	45	=	=	SYM
ejpam-6792	236	46	0	0	NUM
ejpam-6792	236	47	,	,	PUNCT
ejpam-6792	236	48	(	(	PUNCT
ejpam-6792	236	49	ci2	ci2	NOUN
ejpam-6792	236	50	)	)	PUNCT
ejpam-6792	236	51	0x	0x	NOUN
ejpam-6792	236	52	=	=	PUNCT
ejpam-6792	236	53	x	x	X
ejpam-6792	236	54	and	and	CCONJ
ejpam-6792	236	55	(	(	PUNCT
ejpam-6792	236	56	ci3	ci3	NOUN
ejpam-6792	236	57	)	)	PUNCT
ejpam-6792	236	58	x(yz	x(yz	NUM
ejpam-6792	236	59	)	)	PUNCT
ejpam-6792	236	60	=	=	SYM
ejpam-6792	237	1	y(xz	y(xz	NOUN
ejpam-6792	237	2	)	)	PUNCT
ejpam-6792	237	3	for	for	ADP
ejpam-6792	237	4	all	all	DET
ejpam-6792	237	5	x	x	NOUN
ejpam-6792	237	6	,	,	PUNCT
ejpam-6792	237	7	y	y	PROPN
ejpam-6792	237	8	,	,	PUNCT
ejpam-6792	237	9	z	z	PROPN
ejpam-6792	237	10	∈	∈	PROPN
ejpam-6792	237	11	x	x	X
ejpam-6792	237	12	,	,	PUNCT
ejpam-6792	237	13	are	be	AUX
ejpam-6792	237	14	satisfied	satisfied	ADJ
ejpam-6792	237	15	.	.	PUNCT
ejpam-6792	238	1	proposition	proposition	NOUN
ejpam-6792	238	2	14	14	NUM
ejpam-6792	238	3	.	.	PUNCT
ejpam-6792	239	1	g	g	NOUN
ejpam-6792	239	2	(	(	PUNCT
ejpam-6792	239	3	∏	∏	PROPN
ejpam-6792	239	4	i∈i	i∈i	ADJ
ejpam-6792	239	5	xi	xi	NOUN
ejpam-6792	239	6	)	)	PUNCT
ejpam-6792	239	7	=	=	SYM
ejpam-6792	239	8	∏	∏	PROPN
ejpam-6792	239	9	i∈i	i∈i	ADV
ejpam-6792	239	10	xi	xi	INTJ
ejpam-6792	240	1	if	if	SCONJ
ejpam-6792	240	2	and	and	CCONJ
ejpam-6792	240	3	only	only	ADV
ejpam-6792	240	4	if	if	SCONJ
ejpam-6792	240	5	∏	∏	PROPN
ejpam-6792	240	6	i∈i	i∈i	VERB
ejpam-6792	240	7	xi	xi	VERB
ejpam-6792	240	8	is	be	AUX
ejpam-6792	240	9	a	a	DET
ejpam-6792	240	10	ci	ci	NOUN
ejpam-6792	240	11	-	-	NOUN
ejpam-6792	240	12	algebra	algebra	NOUN
ejpam-6792	240	13	.	.	PUNCT
ejpam-6792	241	1	proof	proof	NOUN
ejpam-6792	241	2	.	.	PUNCT
ejpam-6792	242	1	assume	assume	VERB
ejpam-6792	242	2	that	that	SCONJ
ejpam-6792	242	3	g	g	PROPN
ejpam-6792	242	4	(	(	PUNCT
ejpam-6792	242	5	∏	∏	X
ejpam-6792	242	6	i∈i	i∈i	ADJ
ejpam-6792	242	7	xi	xi	NOUN
ejpam-6792	242	8	)	)	PUNCT
ejpam-6792	243	1	=	=	SYM
ejpam-6792	243	2	∏	∏	PROPN
ejpam-6792	243	3	i∈i	i∈i	ADJ
ejpam-6792	243	4	xi	xi	PROPN
ejpam-6792	243	5	.	.	PUNCT
ejpam-6792	244	1	then	then	ADV
ejpam-6792	244	2	∏	∏	PROPN
ejpam-6792	244	3	i∈i	i∈i	ADV
ejpam-6792	244	4	xi	xi	VERB
ejpam-6792	244	5	is	be	AUX
ejpam-6792	244	6	an	an	DET
ejpam-6792	244	7	abelian	abelian	ADJ
ejpam-6792	244	8	group	group	NOUN
ejpam-6792	244	9	by	by	ADP
ejpam-6792	244	10	proposition	proposition	NOUN
ejpam-6792	244	11	11	11	NUM
ejpam-6792	244	12	.	.	PUNCT
ejpam-6792	245	1	we	we	PRON
ejpam-6792	245	2	want	want	VERB
ejpam-6792	245	3	to	to	PART
ejpam-6792	245	4	show	show	VERB
ejpam-6792	245	5	that	that	SCONJ
ejpam-6792	245	6	∏	∏	PROPN
ejpam-6792	245	7	i∈i	i∈i	NOUN
ejpam-6792	245	8	xi	xi	VERB
ejpam-6792	245	9	is	be	AUX
ejpam-6792	245	10	a	a	DET
ejpam-6792	245	11	ci	ci	NOUN
ejpam-6792	245	12	-	-	NOUN
ejpam-6792	245	13	algebra	algebra	NOUN
ejpam-6792	245	14	.	.	PUNCT
ejpam-6792	246	1	it	it	PRON
ejpam-6792	246	2	is	be	AUX
ejpam-6792	246	3	enough	enough	ADJ
ejpam-6792	246	4	to	to	PART
ejpam-6792	246	5	show	show	VERB
ejpam-6792	246	6	only	only	ADV
ejpam-6792	246	7	the	the	DET
ejpam-6792	246	8	condition	condition	NOUN
ejpam-6792	246	9	(	(	PUNCT
ejpam-6792	246	10	ci3	ci3	NOUN
ejpam-6792	246	11	)	)	PUNCT
ejpam-6792	246	12	.	.	PUNCT
ejpam-6792	247	1	let	let	VERB
ejpam-6792	247	2	(	(	PUNCT
ejpam-6792	247	3	xi)i∈i	xi)i∈i	NUM
ejpam-6792	247	4	,	,	PUNCT
ejpam-6792	247	5	(	(	PUNCT
ejpam-6792	247	6	yi)i∈i	yi)i∈i	NUM
ejpam-6792	247	7	,	,	PUNCT
ejpam-6792	247	8	(	(	PUNCT
ejpam-6792	247	9	zi)i∈i	zi)i∈i	NUM
ejpam-6792	247	10	∈	∈	PROPN
ejpam-6792	247	11	∏	∏	PROPN
ejpam-6792	247	12	i∈i	i∈i	ADJ
ejpam-6792	247	13	xi	xi	PROPN
ejpam-6792	247	14	.	.	PUNCT
ejpam-6792	248	1	from	from	ADP
ejpam-6792	248	2	a	a	DET
ejpam-6792	248	3	commutative	commutative	ADJ
ejpam-6792	248	4	property	property	NOUN
ejpam-6792	248	5	and	and	CCONJ
ejpam-6792	248	6	(	(	PUNCT
ejpam-6792	248	7	q3	q3	PROPN
ejpam-6792	248	8	)	)	PUNCT
ejpam-6792	248	9	we	we	PRON
ejpam-6792	248	10	can	can	AUX
ejpam-6792	248	11	calculate	calculate	VERB
ejpam-6792	248	12	that	that	SCONJ
ejpam-6792	248	13	(	(	PUNCT
ejpam-6792	248	14	xi)i∈i	xi)i∈i	X
ejpam-6792	248	15	⊛	⊛	X
ejpam-6792	248	16	[	[	X
ejpam-6792	248	17	(	(	PUNCT
ejpam-6792	248	18	yi)i∈i	yi)i∈i	NOUN
ejpam-6792	248	19	⊛(zi)i∈i	⊛(zi)i∈i	NOUN
ejpam-6792	248	20	]	]	PUNCT
ejpam-6792	248	21	=	=	SYM
ejpam-6792	248	22	(	(	PUNCT
ejpam-6792	248	23	xi)i∈i	xi)i∈i	X
ejpam-6792	248	24	⊛	⊛	X
ejpam-6792	248	25	[	[	X
ejpam-6792	248	26	(	(	PUNCT
ejpam-6792	248	27	zi)i∈i	zi)i∈i	NUM
ejpam-6792	248	28	⊛(yi)i∈i	⊛(yi)i∈i	PRON
ejpam-6792	248	29	]	]	PUNCT
ejpam-6792	248	30	=	=	PUNCT
ejpam-6792	249	1	[	[	X
ejpam-6792	249	2	(	(	PUNCT
ejpam-6792	249	3	zi)i∈i	zi)i∈i	NUM
ejpam-6792	249	4	⊛(yi)i∈i	⊛(yi)i∈i	NUM
ejpam-6792	249	5	]	]	PUNCT
ejpam-6792	249	6	⊛	⊛	NUM
ejpam-6792	249	7	(	(	PUNCT
ejpam-6792	249	8	xi)i∈i	xi)i∈i	X
ejpam-6792	249	9	=	=	PUNCT
ejpam-6792	249	10	[	[	X
ejpam-6792	249	11	(	(	PUNCT
ejpam-6792	249	12	zi)i∈i⊛(xi)i∈i	zi)i∈i⊛(xi)i∈i	X
ejpam-6792	249	13	]	]	X
ejpam-6792	249	14	⊛(yi)i∈i	⊛(yi)i∈i	NOUN
ejpam-6792	249	15	=	=	SYM
ejpam-6792	249	16	(	(	PUNCT
ejpam-6792	249	17	yi)i∈i⊛[(zi)i∈i⊛(xi)i∈i	yi)i∈i⊛[(zi)i∈i⊛(xi)i∈i	X
ejpam-6792	249	18	]	]	X
ejpam-6792	249	19	=	=	SYM
ejpam-6792	249	20	(	(	PUNCT
ejpam-6792	249	21	yi)i∈i⊛[(xi)i∈i⊛(zi)i∈i	yi)i∈i⊛[(xi)i∈i⊛(zi)i∈i	X
ejpam-6792	249	22	]	]	PUNCT
ejpam-6792	249	23	.	.	PUNCT
ejpam-6792	250	1	hence	hence	ADV
ejpam-6792	250	2	,	,	PUNCT
ejpam-6792	250	3	(	(	PUNCT
ejpam-6792	250	4	ci3	ci3	ADJ
ejpam-6792	250	5	)	)	PUNCT
ejpam-6792	250	6	is	be	AUX
ejpam-6792	250	7	satisfied	satisfied	ADJ
ejpam-6792	250	8	.	.	PUNCT
ejpam-6792	251	1	therefore	therefore	ADV
ejpam-6792	251	2	,	,	PUNCT
ejpam-6792	251	3	∏	∏	PROPN
ejpam-6792	251	4	i∈i	i∈i	ADV
ejpam-6792	251	5	xi	xi	VERB
ejpam-6792	251	6	is	be	AUX
ejpam-6792	251	7	a	a	DET
ejpam-6792	251	8	ci	ci	NOUN
ejpam-6792	251	9	-	-	NOUN
ejpam-6792	251	10	algebra	algebra	NOUN
ejpam-6792	251	11	.	.	PUNCT
ejpam-6792	252	1	the	the	DET
ejpam-6792	252	2	converse	converse	NOUN
ejpam-6792	252	3	direction	direction	NOUN
ejpam-6792	252	4	follows	follow	VERB
ejpam-6792	252	5	from	from	ADP
ejpam-6792	252	6	the	the	DET
ejpam-6792	252	7	condition	condition	NOUN
ejpam-6792	252	8	(	(	PUNCT
ejpam-6792	252	9	ci2	ci2	NOUN
ejpam-6792	252	10	)	)	PUNCT
ejpam-6792	252	11	.	.	PUNCT
ejpam-6792	253	1	3	3	X
ejpam-6792	253	2	.	.	X
ejpam-6792	253	3	conclusion	conclusion	NOUN
ejpam-6792	253	4	we	we	PRON
ejpam-6792	253	5	discussed	discuss	VERB
ejpam-6792	253	6	the	the	DET
ejpam-6792	253	7	direct	direct	ADJ
ejpam-6792	253	8	product	product	NOUN
ejpam-6792	253	9	of	of	ADP
ejpam-6792	253	10	q	q	NOUN
ejpam-6792	253	11	-	-	PUNCT
ejpam-6792	253	12	algebras	algebras	X
ejpam-6792	253	13	.	.	PUNCT
ejpam-6792	254	1	we	we	PRON
ejpam-6792	254	2	obtained	obtain	VERB
ejpam-6792	254	3	that	that	SCONJ
ejpam-6792	254	4	the	the	DET
ejpam-6792	254	5	direct	direct	ADJ
ejpam-6792	254	6	product	product	NOUN
ejpam-6792	254	7	of	of	ADP
ejpam-6792	254	8	q	q	NOUN
ejpam-6792	254	9	-	-	PUNCT
ejpam-6792	254	10	algebras	algebras	PROPN
ejpam-6792	254	11	is	be	AUX
ejpam-6792	254	12	again	again	ADV
ejpam-6792	254	13	a	a	DET
ejpam-6792	254	14	q	q	NOUN
ejpam-6792	254	15	-	-	NOUN
ejpam-6792	254	16	algebra	algebra	NOUN
ejpam-6792	254	17	.	.	PUNCT
ejpam-6792	255	1	several	several	ADJ
ejpam-6792	255	2	basic	basic	ADJ
ejpam-6792	255	3	properties	property	NOUN
ejpam-6792	255	4	of	of	ADP
ejpam-6792	255	5	the	the	DET
ejpam-6792	255	6	direct	direct	ADJ
ejpam-6792	255	7	product	product	NOUN
ejpam-6792	255	8	of	of	ADP
ejpam-6792	255	9	q	q	NOUN
ejpam-6792	255	10	-	-	PUNCT
ejpam-6792	255	11	algebras	algebra	NOUN
ejpam-6792	255	12	are	be	AUX
ejpam-6792	255	13	presented	present	VERB
ejpam-6792	255	14	.	.	PUNCT
ejpam-6792	256	1	we	we	PRON
ejpam-6792	256	2	also	also	ADV
ejpam-6792	256	3	studied	study	VERB
ejpam-6792	256	4	this	this	DET
ejpam-6792	256	5	topic	topic	NOUN
ejpam-6792	256	6	related	relate	VERB
ejpam-6792	256	7	to	to	ADP
ejpam-6792	256	8	many	many	ADJ
ejpam-6792	256	9	concepts	concept	NOUN
ejpam-6792	256	10	in	in	ADP
ejpam-6792	256	11	qalgebras	qalgebra	NOUN
ejpam-6792	256	12	,	,	PUNCT
ejpam-6792	256	13	for	for	ADP
ejpam-6792	256	14	instant	instant	NOUN
ejpam-6792	256	15	,	,	PUNCT
ejpam-6792	256	16	subalgebra	subalgebra	NOUN
ejpam-6792	256	17	,	,	PUNCT
ejpam-6792	256	18	g	g	NOUN
ejpam-6792	256	19	-	-	PUNCT
ejpam-6792	256	20	part	part	NOUN
ejpam-6792	256	21	and	and	CCONJ
ejpam-6792	256	22	atom	atom	NOUN
ejpam-6792	256	23	.	.	PUNCT
ejpam-6792	257	1	we	we	PRON
ejpam-6792	257	2	showed	show	VERB
ejpam-6792	257	3	necessary	necessary	ADJ
ejpam-6792	257	4	and	and	CCONJ
ejpam-6792	257	5	sufficient	sufficient	ADJ
ejpam-6792	257	6	conditions	condition	NOUN
ejpam-6792	257	7	for	for	ADP
ejpam-6792	257	8	a	a	DET
ejpam-6792	257	9	two	two	NUM
ejpam-6792	257	10	-	-	PUNCT
ejpam-6792	257	11	element	element	NOUN
ejpam-6792	257	12	subset	subset	NOUN
ejpam-6792	257	13	of	of	ADP
ejpam-6792	257	14	the	the	DET
ejpam-6792	257	15	direct	direct	ADJ
ejpam-6792	257	16	product	product	NOUN
ejpam-6792	257	17	of	of	ADP
ejpam-6792	257	18	q	q	NOUN
ejpam-6792	257	19	-	-	PUNCT
ejpam-6792	257	20	algebras	algebra	NOUN
ejpam-6792	257	21	,	,	PUNCT
ejpam-6792	257	22	∏	∏	PROPN
ejpam-6792	257	23	i∈i	i∈i	ADJ
ejpam-6792	257	24	xi	xi	PROPN
ejpam-6792	257	25	,	,	PUNCT
ejpam-6792	257	26	containing	contain	VERB
ejpam-6792	257	27	a	a	DET
ejpam-6792	257	28	constant	constant	ADJ
ejpam-6792	257	29	(	(	PUNCT
ejpam-6792	257	30	0i)i∈i	0i)i∈i	ADJ
ejpam-6792	257	31	to	to	PART
ejpam-6792	257	32	be	be	AUX
ejpam-6792	257	33	a	a	DET
ejpam-6792	257	34	subalgebra	subalgebra	NOUN
ejpam-6792	257	35	of	of	ADP
ejpam-6792	257	36	∏	∏	NUM
ejpam-6792	257	37	i∈i	i∈i	ADJ
ejpam-6792	257	38	xi	xi	PROPN
ejpam-6792	257	39	.	.	PUNCT
ejpam-6792	258	1	we	we	PRON
ejpam-6792	258	2	also	also	ADV
ejpam-6792	258	3	provided	provide	VERB
ejpam-6792	258	4	some	some	DET
ejpam-6792	258	5	properties	property	NOUN
ejpam-6792	258	6	of	of	ADP
ejpam-6792	258	7	the	the	DET
ejpam-6792	258	8	set	set	NOUN
ejpam-6792	258	9	g	g	NOUN
ejpam-6792	258	10	-	-	PUNCT
ejpam-6792	258	11	part	part	NOUN
ejpam-6792	258	12	,	,	PUNCT
ejpam-6792	258	13	g	g	PROPN
ejpam-6792	258	14	(	(	PUNCT
ejpam-6792	258	15	∏	∏	PROPN
ejpam-6792	258	16	i∈i	i∈i	ADJ
ejpam-6792	258	17	xi	xi	PROPN
ejpam-6792	258	18	)	)	PUNCT
ejpam-6792	258	19	,	,	PUNCT
ejpam-6792	258	20	of	of	ADP
ejpam-6792	258	21	a	a	DET
ejpam-6792	258	22	q	q	NOUN
ejpam-6792	258	23	-	-	PUNCT
ejpam-6792	258	24	algebra	algebra	NOUN
ejpam-6792	258	25	∏	∏	NUM
ejpam-6792	258	26	i∈i	i∈i	ADJ
ejpam-6792	258	27	xi	xi	PROPN
ejpam-6792	258	28	.	.	PUNCT
ejpam-6792	259	1	we	we	PRON
ejpam-6792	259	2	proved	prove	VERB
ejpam-6792	259	3	that	that	SCONJ
ejpam-6792	259	4	the	the	DET
ejpam-6792	259	5	set	set	NOUN
ejpam-6792	259	6	g	g	NOUN
ejpam-6792	259	7	(	(	PUNCT
ejpam-6792	259	8	∏	∏	PROPN
ejpam-6792	259	9	i∈i	i∈i	NOUN
ejpam-6792	259	10	xi	xi	NOUN
ejpam-6792	259	11	)	)	PUNCT
ejpam-6792	259	12	is	be	AUX
ejpam-6792	259	13	a	a	DET
ejpam-6792	259	14	subalgebra	subalgebra	NOUN
ejpam-6792	259	15	,	,	PUNCT
ejpam-6792	259	16	moreover	moreover	ADV
ejpam-6792	259	17	,	,	PUNCT
ejpam-6792	259	18	it	it	PRON
ejpam-6792	259	19	is	be	AUX
ejpam-6792	259	20	an	an	DET
ejpam-6792	259	21	abelian	abelian	ADJ
ejpam-6792	259	22	group	group	NOUN
ejpam-6792	259	23	.	.	PUNCT
ejpam-6792	260	1	we	we	PRON
ejpam-6792	260	2	also	also	ADV
ejpam-6792	260	3	showed	show	VERB
ejpam-6792	260	4	that	that	SCONJ
ejpam-6792	260	5	every	every	DET
ejpam-6792	260	6	element	element	NOUN
ejpam-6792	260	7	of	of	ADP
ejpam-6792	260	8	∏	∏	PROPN
ejpam-6792	260	9	i∈i	i∈i	ADJ
ejpam-6792	260	10	xi	xi	VERB
ejpam-6792	260	11	is	be	AUX
ejpam-6792	260	12	an	an	DET
ejpam-6792	260	13	atom	atom	NOUN
ejpam-6792	260	14	whenever	whenever	SCONJ
ejpam-6792	260	15	the	the	DET
ejpam-6792	260	16	set	set	NOUN
ejpam-6792	260	17	g	g	NOUN
ejpam-6792	260	18	(	(	PUNCT
ejpam-6792	260	19	∏	∏	PROPN
ejpam-6792	260	20	i∈i	i∈i	NOUN
ejpam-6792	260	21	xi	xi	NOUN
ejpam-6792	260	22	)	)	PUNCT
ejpam-6792	260	23	is	be	AUX
ejpam-6792	260	24	equal	equal	ADJ
ejpam-6792	260	25	to	to	ADP
ejpam-6792	260	26	∏	∏	PROPN
ejpam-6792	260	27	i∈i	i∈i	ADJ
ejpam-6792	260	28	xi	xi	PROPN
ejpam-6792	260	29	.	.	PUNCT
ejpam-6792	261	1	for	for	ADP
ejpam-6792	261	2	further	further	ADJ
ejpam-6792	261	3	study	study	NOUN
ejpam-6792	261	4	,	,	PUNCT
ejpam-6792	261	5	one	one	PRON
ejpam-6792	261	6	can	can	AUX
ejpam-6792	261	7	consider	consider	VERB
ejpam-6792	261	8	the	the	DET
ejpam-6792	261	9	direct	direct	ADJ
ejpam-6792	261	10	product	product	NOUN
ejpam-6792	261	11	of	of	ADP
ejpam-6792	261	12	q	q	NOUN
ejpam-6792	261	13	-	-	PUNCT
ejpam-6792	261	14	algebras	algebras	ADJ
ejpam-6792	261	15	in	in	ADP
ejpam-6792	261	16	the	the	DET
ejpam-6792	261	17	direction	direction	NOUN
ejpam-6792	261	18	that	that	PRON
ejpam-6792	261	19	related	relate	VERB
ejpam-6792	261	20	to	to	ADP
ejpam-6792	261	21	the	the	DET
ejpam-6792	261	22	following	follow	VERB
ejpam-6792	261	23	topics	topic	NOUN
ejpam-6792	261	24	:	:	PUNCT
ejpam-6792	261	25	ideals	ideal	NOUN
ejpam-6792	261	26	and	and	CCONJ
ejpam-6792	261	27	filters	filter	NOUN
ejpam-6792	261	28	;	;	PUNCT
ejpam-6792	261	29	fuzzy	fuzzy	ADJ
ejpam-6792	261	30	subalgebras	subalgebra	NOUN
ejpam-6792	261	31	,	,	PUNCT
ejpam-6792	261	32	fuzzy	fuzzy	ADJ
ejpam-6792	261	33	ideals	ideal	NOUN
ejpam-6792	261	34	;	;	PUNCT
ejpam-6792	261	35	homomorphisms	homomorphism	NOUN
ejpam-6792	261	36	and	and	CCONJ
ejpam-6792	261	37	isomorphisms	isomorphism	NOUN
ejpam-6792	261	38	;	;	PUNCT
ejpam-6792	261	39	more	more	ADJ
ejpam-6792	261	40	insight	insight	NOUN
ejpam-6792	261	41	of	of	ADP
ejpam-6792	261	42	atoms	atom	NOUN
ejpam-6792	261	43	and	and	CCONJ
ejpam-6792	261	44	strong	strong	ADJ
ejpam-6792	261	45	atoms	atom	NOUN
ejpam-6792	261	46	.	.	PUNCT
ejpam-6792	262	1	another	another	DET
ejpam-6792	262	2	direction	direction	NOUN
ejpam-6792	262	3	of	of	ADP
ejpam-6792	262	4	study	study	NOUN
ejpam-6792	262	5	is	be	AUX
ejpam-6792	262	6	a	a	DET
ejpam-6792	262	7	concept	concept	NOUN
ejpam-6792	262	8	of	of	ADP
ejpam-6792	262	9	a	a	DET
ejpam-6792	262	10	generalization	generalization	NOUN
ejpam-6792	262	11	of	of	ADP
ejpam-6792	262	12	the	the	DET
ejpam-6792	262	13	direct	direct	ADJ
ejpam-6792	262	14	product	product	NOUN
ejpam-6792	262	15	,	,	PUNCT
ejpam-6792	262	16	i.e.	i.e.	X
ejpam-6792	262	17	the	the	DET
ejpam-6792	262	18	external	external	ADJ
ejpam-6792	262	19	direct	direct	ADJ
ejpam-6792	262	20	product	product	NOUN
ejpam-6792	262	21	of	of	ADP
ejpam-6792	262	22	q	q	NOUN
ejpam-6792	262	23	-	-	PUNCT
ejpam-6792	262	24	algebras	algebras	X
ejpam-6792	262	25	.	.	PUNCT
ejpam-6792	263	1	a.	a.	PROPN
ejpam-6792	263	2	anantayasethi	anantayasethi	PROPN
ejpam-6792	263	3	,	,	PUNCT
ejpam-6792	263	4	k.	k.	PROPN
ejpam-6792	263	5	saengsura	saengsura	PROPN
ejpam-6792	263	6	,	,	PUNCT
ejpam-6792	263	7	n.	n.	NOUN
ejpam-6792	263	8	sarasit	sarasit	PROPN
ejpam-6792	263	9	/	/	SYM
ejpam-6792	263	10	eur	eur	PROPN
ejpam-6792	263	11	.	.	PUNCT
ejpam-6792	264	1	j.	j.	PROPN
ejpam-6792	264	2	pure	pure	PROPN
ejpam-6792	264	3	appl	appl	PROPN
ejpam-6792	264	4	.	.	PROPN
ejpam-6792	264	5	math	math	PROPN
ejpam-6792	264	6	,	,	PUNCT
ejpam-6792	264	7	18	18	NUM
ejpam-6792	264	8	(	(	PUNCT
ejpam-6792	264	9	4	4	NUM
ejpam-6792	264	10	)	)	PUNCT
ejpam-6792	264	11	(	(	PUNCT
ejpam-6792	264	12	2025	2025	NUM
ejpam-6792	264	13	)	)	PUNCT
ejpam-6792	264	14	,	,	PUNCT
ejpam-6792	264	15	6792	6792	NUM
ejpam-6792	264	16	9	9	NUM
ejpam-6792	264	17	of	of	ADP
ejpam-6792	264	18	10	10	NUM
ejpam-6792	264	19	4	4	NUM
ejpam-6792	264	20	.	.	PUNCT
ejpam-6792	265	1	acknowledgements	acknowledgement	NOUN
ejpam-6792	265	2	this	this	DET
ejpam-6792	265	3	research	research	NOUN
ejpam-6792	265	4	project	project	NOUN
ejpam-6792	265	5	was	be	AUX
ejpam-6792	265	6	financially	financially	ADV
ejpam-6792	265	7	supported	support	VERB
ejpam-6792	265	8	by	by	ADP
ejpam-6792	265	9	mahasarakham	mahasarakham	PROPN
ejpam-6792	265	10	university	university	PROPN
ejpam-6792	265	11	,	,	PUNCT
ejpam-6792	265	12	thailand	thailand	PROPN
ejpam-6792	265	13	.	.	PUNCT
ejpam-6792	266	1	references	reference	NOUN
ejpam-6792	266	2	[	[	X
ejpam-6792	266	3	1	1	NUM
ejpam-6792	266	4	]	]	X
ejpam-6792	266	5	y.	y.	PROPN
ejpam-6792	266	6	imai	imai	PROPN
ejpam-6792	266	7	and	and	CCONJ
ejpam-6792	266	8	k.	k.	PROPN
ejpam-6792	266	9	iseki	iseki	PROPN
ejpam-6792	266	10	.	.	PUNCT
ejpam-6792	267	1	on	on	ADP
ejpam-6792	267	2	axiom	axiom	NOUN
ejpam-6792	267	3	system	system	NOUN
ejpam-6792	267	4	of	of	ADP
ejpam-6792	267	5	propositional	propositional	ADJ
ejpam-6792	267	6	calculi	calculi	PROPN
ejpam-6792	267	7	xiv	xiv	PROPN
ejpam-6792	267	8	.	.	PUNCT
ejpam-6792	268	1	proceeding	proceeding	PROPN
ejpam-6792	268	2	of	of	ADP
ejpam-6792	268	3	japan	japan	PROPN
ejpam-6792	268	4	academy	academy	PROPN
ejpam-6792	268	5	,	,	PUNCT
ejpam-6792	268	6	42(1):19–22	42(1):19–22	NUM
ejpam-6792	268	7	,	,	PUNCT
ejpam-6792	268	8	1966	1966	NUM
ejpam-6792	268	9	.	.	PUNCT
ejpam-6792	269	1	[	[	X
ejpam-6792	269	2	2	2	NUM
ejpam-6792	269	3	]	]	PUNCT
ejpam-6792	269	4	k.	k.	PROPN
ejpam-6792	269	5	iseki	iseki	PROPN
ejpam-6792	269	6	.	.	PUNCT
ejpam-6792	270	1	an	an	DET
ejpam-6792	270	2	algebra	algebra	NOUN
ejpam-6792	270	3	related	relate	VERB
ejpam-6792	270	4	with	with	ADP
ejpam-6792	270	5	a	a	DET
ejpam-6792	270	6	propositional	propositional	ADJ
ejpam-6792	270	7	calculus	calculus	NOUN
ejpam-6792	270	8	.	.	PUNCT
ejpam-6792	271	1	proceeding	proceeding	NOUN
ejpam-6792	271	2	of	of	ADP
ejpam-6792	271	3	japan	japan	PROPN
ejpam-6792	271	4	academy	academy	PROPN
ejpam-6792	271	5	,	,	PUNCT
ejpam-6792	271	6	42(1):26–29	42(1):26–29	NUM
ejpam-6792	271	7	,	,	PUNCT
ejpam-6792	271	8	1966	1966	NUM
ejpam-6792	271	9	.	.	PUNCT
ejpam-6792	272	1	[	[	X
ejpam-6792	272	2	3	3	NUM
ejpam-6792	272	3	]	]	X
ejpam-6792	272	4	y.	y.	PROPN
ejpam-6792	272	5	arai	arai	PROPN
ejpam-6792	272	6	,	,	PUNCT
ejpam-6792	272	7	k.iseki	k.iseki	NOUN
ejpam-6792	272	8	,	,	PUNCT
ejpam-6792	272	9	and	and	CCONJ
ejpam-6792	272	10	s.	s.	PROPN
ejpam-6792	272	11	tanaka	tanaka	PROPN
ejpam-6792	272	12	.	.	PUNCT
ejpam-6792	273	1	characterizations	characterization	NOUN
ejpam-6792	273	2	of	of	ADP
ejpam-6792	273	3	bci	bci	PROPN
ejpam-6792	273	4	,	,	PUNCT
ejpam-6792	273	5	bck	bck	NOUN
ejpam-6792	273	6	-	-	PUNCT
ejpam-6792	273	7	algebra	algebra	NOUN
ejpam-6792	273	8	.	.	PUNCT
ejpam-6792	274	1	proceeding	proceeding	NOUN
ejpam-6792	274	2	of	of	ADP
ejpam-6792	274	3	japan	japan	PROPN
ejpam-6792	274	4	academy	academy	PROPN
ejpam-6792	274	5	,	,	PUNCT
ejpam-6792	274	6	42:105–107	42:105–107	PROPN
ejpam-6792	274	7	,	,	PUNCT
ejpam-6792	274	8	1966	1966	NUM
ejpam-6792	274	9	.	.	PUNCT
ejpam-6792	275	1	[	[	X
ejpam-6792	275	2	4	4	X
ejpam-6792	275	3	]	]	PUNCT
ejpam-6792	275	4	k.	k.	PROPN
ejpam-6792	275	5	iseki	iseki	PROPN
ejpam-6792	275	6	and	and	CCONJ
ejpam-6792	275	7	s.	s.	PROPN
ejpam-6792	275	8	tanaka	tanaka	PROPN
ejpam-6792	275	9	.	.	PUNCT
ejpam-6792	276	1	an	an	DET
ejpam-6792	276	2	introduction	introduction	NOUN
ejpam-6792	276	3	to	to	ADP
ejpam-6792	276	4	theory	theory	NOUN
ejpam-6792	276	5	of	of	ADP
ejpam-6792	276	6	bck	bck	NOUN
ejpam-6792	276	7	-	-	PUNCT
ejpam-6792	276	8	algebra	algebra	NOUN
ejpam-6792	276	9	.	.	PUNCT
ejpam-6792	277	1	mathematics	mathematic	NOUN
ejpam-6792	277	2	japonica	japonica	PROPN
ejpam-6792	277	3	.	.	PUNCT
ejpam-6792	277	4	,	,	PUNCT
ejpam-6792	277	5	23:1–26	23:1–26	NUM
ejpam-6792	277	6	,	,	PUNCT
ejpam-6792	277	7	1978	1978	NUM
ejpam-6792	277	8	.	.	PUNCT
ejpam-6792	278	1	[	[	X
ejpam-6792	278	2	5	5	X
ejpam-6792	278	3	]	]	X
ejpam-6792	278	4	y.	y.	PROPN
ejpam-6792	278	5	b.	b.	PROPN
ejpam-6792	278	6	jun	jun	PROPN
ejpam-6792	278	7	,	,	PUNCT
ejpam-6792	278	8	m.	m.	PROPN
ejpam-6792	278	9	hong	hong	PROPN
ejpam-6792	278	10	,	,	PUNCT
ejpam-6792	278	11	e.	e.	PROPN
ejpam-6792	278	12	h.	h.	PROPN
ejpam-6792	278	13	roh	roh	PROPN
ejpam-6792	278	14	,	,	PUNCT
ejpam-6792	278	15	and	and	CCONJ
ejpam-6792	278	16	j.	j.	PROPN
ejpam-6792	278	17	meng	meng	PROPN
ejpam-6792	278	18	.	.	PUNCT
ejpam-6792	279	1	on	on	ADP
ejpam-6792	279	2	the	the	DET
ejpam-6792	279	3	bci	bci	PROPN
ejpam-6792	279	4	-	-	ADJ
ejpam-6792	279	5	g	g	ADJ
ejpam-6792	279	6	part	part	NOUN
ejpam-6792	279	7	of	of	ADP
ejpam-6792	279	8	bci	bci	NOUN
ejpam-6792	279	9	-	-	PUNCT
ejpam-6792	279	10	algebras	algebras	X
ejpam-6792	279	11	(	(	PUNCT
ejpam-6792	279	12	iii	iii	NOUN
ejpam-6792	279	13	)	)	PUNCT
ejpam-6792	279	14	.	.	PUNCT
ejpam-6792	280	1	communications	communication	NOUN
ejpam-6792	280	2	of	of	ADP
ejpam-6792	280	3	the	the	DET
ejpam-6792	280	4	korean	korean	ADJ
ejpam-6792	280	5	mathematical	mathematical	ADJ
ejpam-6792	280	6	society	society	NOUN
ejpam-6792	280	7	,	,	PUNCT
ejpam-6792	280	8	9(3):531–538	9(3):531–538	NUM
ejpam-6792	280	9	,	,	PUNCT
ejpam-6792	280	10	1994	1994	NUM
ejpam-6792	280	11	.	.	PUNCT
ejpam-6792	281	1	[	[	X
ejpam-6792	281	2	6	6	NUM
ejpam-6792	281	3	]	]	PUNCT
ejpam-6792	281	4	q.	q.	PROPN
ejpam-6792	281	5	hu	hu	PROPN
ejpam-6792	281	6	and	and	CCONJ
ejpam-6792	281	7	x.	x.	PROPN
ejpam-6792	281	8	li	li	PROPN
ejpam-6792	281	9	.	.	PROPN
ejpam-6792	282	1	on	on	ADP
ejpam-6792	282	2	bch	bch	PROPN
ejpam-6792	282	3	-	-	PUNCT
ejpam-6792	282	4	algebras	algebras	PROPN
ejpam-6792	282	5	.	.	PUNCT
ejpam-6792	283	1	mathematics	mathematic	NOUN
ejpam-6792	283	2	seminar	seminar	NOUN
ejpam-6792	283	3	note	note	PROPN
ejpam-6792	283	4	kobe	kobe	PROPN
ejpam-6792	283	5	university	university	PROPN
ejpam-6792	283	6	,	,	PUNCT
ejpam-6792	283	7	2(11):313–320	2(11):313–320	NUM
ejpam-6792	283	8	,	,	PUNCT
ejpam-6792	283	9	1983	1983	NUM
ejpam-6792	283	10	.	.	PUNCT
ejpam-6792	284	1	[	[	X
ejpam-6792	284	2	7	7	X
ejpam-6792	284	3	]	]	X
ejpam-6792	284	4	y.	y.	PROPN
ejpam-6792	284	5	komori	komori	PROPN
ejpam-6792	284	6	.	.	PUNCT
ejpam-6792	285	1	the	the	DET
ejpam-6792	285	2	class	class	NOUN
ejpam-6792	285	3	of	of	ADP
ejpam-6792	285	4	bcc	bcc	PROPN
ejpam-6792	285	5	-	-	PUNCT
ejpam-6792	285	6	algebras	algebras	PROPN
ejpam-6792	285	7	is	be	AUX
ejpam-6792	285	8	not	not	PART
ejpam-6792	285	9	a	a	DET
ejpam-6792	285	10	variety	variety	NOUN
ejpam-6792	285	11	.	.	PUNCT
ejpam-6792	286	1	mathematica	mathematica	PROPN
ejpam-6792	286	2	japonica	japonica	PROPN
ejpam-6792	286	3	,	,	PUNCT
ejpam-6792	286	4	29:391	29:391	NUM
ejpam-6792	286	5	–	–	PUNCT
ejpam-6792	286	6	394	394	NUM
ejpam-6792	286	7	,	,	PUNCT
ejpam-6792	286	8	1984	1984	NUM
ejpam-6792	286	9	.	.	PUNCT
ejpam-6792	287	1	[	[	X
ejpam-6792	287	2	8	8	NUM
ejpam-6792	287	3	]	]	X
ejpam-6792	287	4	y.	y.	PROPN
ejpam-6792	287	5	b.	b.	PROPN
ejpam-6792	287	6	jun	jun	PROPN
ejpam-6792	287	7	,	,	PUNCT
ejpam-6792	287	8	e.	e.	PROPN
ejpam-6792	287	9	h.	h.	PROPN
ejpam-6792	287	10	roh	roh	PROPN
ejpam-6792	287	11	,	,	PUNCT
ejpam-6792	287	12	and	and	CCONJ
ejpam-6792	287	13	h.	h.	PROPN
ejpam-6792	287	14	s.	s.	PROPN
ejpam-6792	287	15	kim	kim	PROPN
ejpam-6792	287	16	.	.	PUNCT
ejpam-6792	288	1	on	on	ADP
ejpam-6792	288	2	bh	bh	NOUN
ejpam-6792	288	3	-	-	PUNCT
ejpam-6792	288	4	algebras	algebras	PROPN
ejpam-6792	288	5	.	.	PUNCT
ejpam-6792	289	1	scientiae	scientiae	PROPN
ejpam-6792	289	2	mathematicae	mathematicae	PROPN
ejpam-6792	289	3	,	,	PUNCT
ejpam-6792	289	4	1(3):347–354	1(3):347–354	NUM
ejpam-6792	289	5	,	,	PUNCT
ejpam-6792	289	6	1998	1998	NUM
ejpam-6792	289	7	.	.	PUNCT
ejpam-6792	290	1	[	[	X
ejpam-6792	290	2	9	9	NUM
ejpam-6792	290	3	]	]	X
ejpam-6792	290	4	j.	j.	PROPN
ejpam-6792	290	5	neggers	neggers	PROPN
ejpam-6792	290	6	,	,	PUNCT
ejpam-6792	290	7	s.	s.	PROPN
ejpam-6792	290	8	ahn	ahn	PROPN
ejpam-6792	290	9	,	,	PUNCT
ejpam-6792	290	10	and	and	CCONJ
ejpam-6792	290	11	h.s	h.s	PROPN
ejpam-6792	290	12	.	.	PROPN
ejpam-6792	290	13	kim	kim	PROPN
ejpam-6792	290	14	.	.	PUNCT
ejpam-6792	291	1	on	on	ADP
ejpam-6792	291	2	q	q	NOUN
ejpam-6792	291	3	-	-	PUNCT
ejpam-6792	291	4	algebras	algebra	NOUN
ejpam-6792	291	5	.	.	PUNCT
ejpam-6792	292	1	international	international	ADJ
ejpam-6792	292	2	journal	journal	PROPN
ejpam-6792	292	3	of	of	ADP
ejpam-6792	292	4	mathematics	mathematics	PROPN
ejpam-6792	292	5	and	and	CCONJ
ejpam-6792	292	6	mathematical	mathematical	ADJ
ejpam-6792	292	7	sciences	science	NOUN
ejpam-6792	292	8	,	,	PUNCT
ejpam-6792	292	9	27(12):749–757	27(12):749–757	NOUN
ejpam-6792	292	10	,	,	PUNCT
ejpam-6792	292	11	2001	2001	NUM
ejpam-6792	292	12	.	.	PUNCT
ejpam-6792	293	1	[	[	X
ejpam-6792	293	2	10	10	NUM
ejpam-6792	293	3	]	]	X
ejpam-6792	293	4	a.	a.	NOUN
ejpam-6792	293	5	anantayasethi	anantayasethi	PROPN
ejpam-6792	293	6	,	,	PUNCT
ejpam-6792	293	7	t.	t.	PROPN
ejpam-6792	293	8	kunawat	kunawat	PROPN
ejpam-6792	293	9	,	,	PUNCT
ejpam-6792	293	10	and	and	CCONJ
ejpam-6792	293	11	p.	p.	PROPN
ejpam-6792	293	12	moonnipa	moonnipa	PROPN
ejpam-6792	293	13	.	.	PUNCT
ejpam-6792	294	1	relation	relation	NOUN
ejpam-6792	294	2	between	between	ADP
ejpam-6792	294	3	g	g	NOUN
ejpam-6792	294	4	-	-	PUNCT
ejpam-6792	294	5	part	part	NOUN
ejpam-6792	294	6	and	and	CCONJ
ejpam-6792	294	7	atoms	atom	NOUN
ejpam-6792	294	8	in	in	ADP
ejpam-6792	294	9	q	q	NOUN
ejpam-6792	294	10	-	-	PUNCT
ejpam-6792	294	11	algebras	algebras	ADJ
ejpam-6792	294	12	.	.	PUNCT
ejpam-6792	295	1	european	european	PROPN
ejpam-6792	295	2	journal	journal	PROPN
ejpam-6792	295	3	of	of	ADP
ejpam-6792	295	4	pure	pure	ADJ
ejpam-6792	295	5	and	and	CCONJ
ejpam-6792	295	6	applied	applied	ADJ
ejpam-6792	295	7	mathematics	mathematic	NOUN
ejpam-6792	295	8	,	,	PUNCT
ejpam-6792	295	9	17(4):3268–3276	17(4):3268–3276	NUM
ejpam-6792	295	10	,	,	PUNCT
ejpam-6792	295	11	2024	2024	NUM
ejpam-6792	295	12	.	.	PUNCT
ejpam-6792	296	1	[	[	X
ejpam-6792	296	2	11	11	NUM
ejpam-6792	296	3	]	]	PUNCT
ejpam-6792	296	4	a.	a.	NOUN
ejpam-6792	296	5	anantayasethi	anantayasethi	PROPN
ejpam-6792	296	6	and	and	CCONJ
ejpam-6792	296	7	j.	j.	PROPN
ejpam-6792	296	8	koppitz	koppitz	PROPN
ejpam-6792	296	9	.	.	PUNCT
ejpam-6792	297	1	characterization	characterization	NOUN
ejpam-6792	297	2	of	of	ADP
ejpam-6792	297	3	ideals	ideal	NOUN
ejpam-6792	297	4	of	of	ADP
ejpam-6792	297	5	q	q	NOUN
ejpam-6792	297	6	-	-	PUNCT
ejpam-6792	297	7	algebras	algebras	ADV
ejpam-6792	297	8	related	relate	VERB
ejpam-6792	297	9	to	to	ADP
ejpam-6792	297	10	its	its	PRON
ejpam-6792	297	11	g	g	NOUN
ejpam-6792	297	12	-	-	PUNCT
ejpam-6792	297	13	part	part	NOUN
ejpam-6792	297	14	.	.	PUNCT
ejpam-6792	298	1	journal	journal	NOUN
ejpam-6792	298	2	of	of	ADP
ejpam-6792	298	3	discrete	discrete	ADJ
ejpam-6792	298	4	mathematical	mathematical	ADJ
ejpam-6792	298	5	sciences	science	NOUN
ejpam-6792	298	6	and	and	CCONJ
ejpam-6792	298	7	cryptography	cryptography	NOUN
ejpam-6792	298	8	,	,	PUNCT
ejpam-6792	298	9	28(1):131	28(1):131	NUM
ejpam-6792	298	10	–	–	PUNCT
ejpam-6792	298	11	141	141	NUM
ejpam-6792	298	12	,	,	PUNCT
ejpam-6792	298	13	2025	2025	NUM
ejpam-6792	298	14	.	.	PUNCT
ejpam-6792	299	1	[	[	X
ejpam-6792	299	2	12	12	NUM
ejpam-6792	299	3	]	]	PUNCT
ejpam-6792	299	4	k.	k.	PROPN
ejpam-6792	299	5	saengsura	saengsura	PROPN
ejpam-6792	299	6	,	,	PUNCT
ejpam-6792	299	7	n.	n.	NOUN
ejpam-6792	299	8	sarasit	sarasit	NOUN
ejpam-6792	299	9	,	,	PUNCT
ejpam-6792	299	10	and	and	CCONJ
ejpam-6792	299	11	a.	a.	NOUN
ejpam-6792	299	12	anantayasethi	anantayasethi	PROPN
ejpam-6792	299	13	.	.	PUNCT
ejpam-6792	300	1	on	on	ADP
ejpam-6792	300	2	the	the	DET
ejpam-6792	300	3	strong	strong	ADJ
ejpam-6792	300	4	atoms	atom	NOUN
ejpam-6792	300	5	of	of	ADP
ejpam-6792	300	6	q	q	NOUN
ejpam-6792	300	7	-	-	PUNCT
ejpam-6792	300	8	algebra	algebra	NOUN
ejpam-6792	300	9	.	.	PUNCT
ejpam-6792	301	1	axioms	axiom	NOUN
ejpam-6792	301	2	,	,	PUNCT
ejpam-6792	301	3	14:271	14:271	NUM
ejpam-6792	301	4	,	,	PUNCT
ejpam-6792	301	5	2005	2005	NUM
ejpam-6792	301	6	.	.	PUNCT
ejpam-6792	302	1	[	[	X
ejpam-6792	302	2	13	13	NUM
ejpam-6792	302	3	]	]	PUNCT
ejpam-6792	302	4	s.	s.	PROPN
ejpam-6792	302	5	m.	m.	PROPN
ejpam-6792	302	6	mostafa	mostafa	PROPN
ejpam-6792	302	7	,	,	PUNCT
ejpam-6792	302	8	m.	m.	NOUN
ejpam-6792	302	9	a.	a.	PROPN
ejpam-6792	302	10	naby	naby	PROPN
ejpam-6792	302	11	,	,	PUNCT
ejpam-6792	302	12	and	and	CCONJ
ejpam-6792	302	13	o.	o.	PROPN
ejpam-6792	302	14	r.	r.	PROPN
ejpam-6792	302	15	elgendy	elgendy	PROPN
ejpam-6792	302	16	.	.	PUNCT
ejpam-6792	303	1	fuzzy	fuzzy	ADJ
ejpam-6792	303	2	q	q	NOUN
ejpam-6792	303	3	-	-	NOUN
ejpam-6792	303	4	ideals	ideal	NOUN
ejpam-6792	303	5	in	in	ADP
ejpam-6792	303	6	q	q	NOUN
ejpam-6792	303	7	-	-	PUNCT
ejpam-6792	303	8	algebras	algebra	NOUN
ejpam-6792	303	9	.	.	PUNCT
ejpam-6792	304	1	world	world	NOUN
ejpam-6792	304	2	applied	apply	VERB
ejpam-6792	304	3	programming	programming	NOUN
ejpam-6792	304	4	,	,	PUNCT
ejpam-6792	304	5	2(2):69–80	2(2):69–80	NUM
ejpam-6792	304	6	,	,	PUNCT
ejpam-6792	304	7	2012	2012	NUM
ejpam-6792	304	8	.	.	PUNCT
ejpam-6792	305	1	[	[	X
ejpam-6792	305	2	14	14	NUM
ejpam-6792	305	3	]	]	X
ejpam-6792	305	4	h.	h.	PROPN
ejpam-6792	305	5	k.	k.	PROPN
ejpam-6792	305	6	abdullah	abdullah	PROPN
ejpam-6792	305	7	and	and	CCONJ
ejpam-6792	305	8	m.	m.	NOUN
ejpam-6792	305	9	tach	tach	PROPN
ejpam-6792	305	10	.	.	PUNCT
ejpam-6792	306	1	intuitionistic	intuitionistic	ADJ
ejpam-6792	306	2	fuzzy	fuzzy	ADJ
ejpam-6792	306	3	prime	prime	ADJ
ejpam-6792	306	4	ideal	ideal	NOUN
ejpam-6792	306	5	on	on	ADP
ejpam-6792	306	6	q	q	NOUN
ejpam-6792	306	7	-	-	PUNCT
ejpam-6792	306	8	algebras	algebra	NOUN
ejpam-6792	306	9	.	.	PUNCT
ejpam-6792	307	1	international	international	ADJ
ejpam-6792	307	2	journal	journal	PROPN
ejpam-6792	307	3	of	of	ADP
ejpam-6792	307	4	academic	academic	ADJ
ejpam-6792	307	5	and	and	CCONJ
ejpam-6792	307	6	applied	applied	ADJ
ejpam-6792	307	7	research	research	NOUN
ejpam-6792	307	8	,	,	PUNCT
ejpam-6792	307	9	4(10):66–78	4(10):66–78	NOUN
ejpam-6792	307	10	,	,	PUNCT
ejpam-6792	307	11	2020	2020	NUM
ejpam-6792	307	12	.	.	PUNCT
ejpam-6792	308	1	[	[	X
ejpam-6792	308	2	15	15	NUM
ejpam-6792	308	3	]	]	X
ejpam-6792	308	4	h.	h.	PROPN
ejpam-6792	308	5	k.	k.	PROPN
ejpam-6792	308	6	abdullah	abdullah	PROPN
ejpam-6792	308	7	and	and	CCONJ
ejpam-6792	308	8	m.	m.	NOUN
ejpam-6792	308	9	tach	tach	PROPN
ejpam-6792	308	10	.	.	PUNCT
ejpam-6792	309	1	prime	prime	PROPN
ejpam-6792	309	2	ideal	ideal	NOUN
ejpam-6792	309	3	in	in	ADP
ejpam-6792	309	4	q	q	NOUN
ejpam-6792	309	5	-	-	NOUN
ejpam-6792	309	6	algebra	algebra	NOUN
ejpam-6792	309	7	.	.	PUNCT
ejpam-6792	310	1	international	international	ADJ
ejpam-6792	310	2	journal	journal	NOUN
ejpam-6792	310	3	of	of	ADP
ejpam-6792	310	4	academic	academic	ADJ
ejpam-6792	310	5	and	and	CCONJ
ejpam-6792	310	6	applied	applied	ADJ
ejpam-6792	310	7	research	research	NOUN
ejpam-6792	310	8	,	,	PUNCT
ejpam-6792	310	9	4(10):79–87	4(10):79–87	NUM
ejpam-6792	310	10	,	,	PUNCT
ejpam-6792	310	11	2020	2020	NUM
ejpam-6792	310	12	.	.	PUNCT
ejpam-6792	311	1	[	[	X
ejpam-6792	311	2	16	16	NUM
ejpam-6792	311	3	]	]	PUNCT
ejpam-6792	311	4	s.	s.	PROPN
ejpam-6792	311	5	s.	s.	PROPN
ejpam-6792	311	6	ahn	ahn	PROPN
ejpam-6792	311	7	,	,	PUNCT
ejpam-6792	311	8	h.	h.	PROPN
ejpam-6792	311	9	s.	s.	PROPN
ejpam-6792	311	10	kim	kim	PROPN
ejpam-6792	311	11	,	,	PUNCT
ejpam-6792	311	12	and	and	CCONJ
ejpam-6792	311	13	h.	h.	PROPN
ejpam-6792	311	14	d.	d.	PROPN
ejpam-6792	311	15	lee	lee	PROPN
ejpam-6792	311	16	.	.	PUNCT
ejpam-6792	312	1	r	r	X
ejpam-6792	312	2	-	-	PUNCT
ejpam-6792	312	3	maps	map	NOUN
ejpam-6792	312	4	and	and	CCONJ
ejpam-6792	312	5	l	l	NOUN
ejpam-6792	312	6	-	-	NOUN
ejpam-6792	312	7	map	map	NOUN
ejpam-6792	312	8	in	in	ADP
ejpam-6792	312	9	q	q	NOUN
ejpam-6792	312	10	-	-	PUNCT
ejpam-6792	312	11	algebras	algebra	NOUN
ejpam-6792	312	12	.	.	PUNCT
ejpam-6792	312	13	ijpam	ijpam	PROPN
ejpam-6792	312	14	.	.	PROPN
ejpam-6792	312	15	,	,	PUNCT
ejpam-6792	312	16	12(4):419–425	12(4):419–425	PROPN
ejpam-6792	312	17	,	,	PUNCT
ejpam-6792	312	18	2004	2004	NUM
ejpam-6792	312	19	.	.	PUNCT
ejpam-6792	313	1	[	[	X
ejpam-6792	313	2	17	17	NUM
ejpam-6792	313	3	]	]	X
ejpam-6792	313	4	s.	s.	PROPN
ejpam-6792	313	5	m.	m.	PROPN
ejpam-6792	313	6	lee	lee	PROPN
ejpam-6792	313	7	and	and	CCONJ
ejpam-6792	313	8	k.	k.	PROPN
ejpam-6792	313	9	h.	h.	PROPN
ejpam-6792	313	10	kim	kim	PROPN
ejpam-6792	313	11	.	.	PUNCT
ejpam-6792	314	1	on	on	ADP
ejpam-6792	314	2	right	right	ADJ
ejpam-6792	314	3	fixed	fix	VERB
ejpam-6792	314	4	maps	map	NOUN
ejpam-6792	314	5	of	of	ADP
ejpam-6792	314	6	q	q	NOUN
ejpam-6792	314	7	-	-	PUNCT
ejpam-6792	314	8	algebras	algebras	ADJ
ejpam-6792	314	9	.	.	PUNCT
ejpam-6792	315	1	internayinal	internayinal	PROPN
ejpam-6792	315	2	mathematical	mathematical	PROPN
ejpam-6792	315	3	forum	forum	PROPN
ejpam-6792	315	4	,	,	PUNCT
ejpam-6792	315	5	6(1):31–37	6(1):31–37	NUM
ejpam-6792	315	6	,	,	PUNCT
ejpam-6792	315	7	2011	2011	NUM
ejpam-6792	315	8	.	.	PUNCT
ejpam-6792	316	1	a.	a.	PROPN
ejpam-6792	316	2	anantayasethi	anantayasethi	PROPN
ejpam-6792	316	3	,	,	PUNCT
ejpam-6792	316	4	k.	k.	PROPN
ejpam-6792	316	5	saengsura	saengsura	PROPN
ejpam-6792	316	6	,	,	PUNCT
ejpam-6792	316	7	n.	n.	NOUN
ejpam-6792	316	8	sarasit	sarasit	PROPN
ejpam-6792	316	9	/	/	SYM
ejpam-6792	316	10	eur	eur	PROPN
ejpam-6792	316	11	.	.	PUNCT
ejpam-6792	317	1	j.	j.	PROPN
ejpam-6792	317	2	pure	pure	PROPN
ejpam-6792	317	3	appl	appl	PROPN
ejpam-6792	317	4	.	.	PROPN
ejpam-6792	317	5	math	math	PROPN
ejpam-6792	317	6	,	,	PUNCT
ejpam-6792	317	7	18	18	NUM
ejpam-6792	317	8	(	(	PUNCT
ejpam-6792	317	9	4	4	NUM
ejpam-6792	317	10	)	)	PUNCT
ejpam-6792	317	11	(	(	PUNCT
ejpam-6792	317	12	2025	2025	NUM
ejpam-6792	317	13	)	)	PUNCT
ejpam-6792	317	14	,	,	PUNCT
ejpam-6792	317	15	6792	6792	NUM
ejpam-6792	317	16	10	10	NUM
ejpam-6792	317	17	of	of	ADP
ejpam-6792	317	18	10	10	NUM
ejpam-6792	318	1	[	[	SYM
ejpam-6792	318	2	18	18	NUM
ejpam-6792	318	3	]	]	PUNCT
ejpam-6792	318	4	j.	j.	PROPN
ejpam-6792	318	5	a.	a.	PROPN
ejpam-6792	318	6	v.	v.	PROPN
ejpam-6792	318	7	lingcong	lingcong	PROPN
ejpam-6792	318	8	and	and	CCONJ
ejpam-6792	318	9	j.	j.	PROPN
ejpam-6792	318	10	c.	c.	PROPN
ejpam-6792	318	11	endam	endam	PROPN
ejpam-6792	318	12	.	.	PUNCT
ejpam-6792	319	1	direct	direct	ADJ
ejpam-6792	319	2	product	product	NOUN
ejpam-6792	319	3	of	of	ADP
ejpam-6792	319	4	b	b	NOUN
ejpam-6792	319	5	-	-	PUNCT
ejpam-6792	319	6	algebras	algebras	PROPN
ejpam-6792	319	7	.	.	PUNCT
ejpam-6792	320	1	international	international	PROPN
ejpam-6792	320	2	of	of	ADP
ejpam-6792	320	3	algebra	algebra	PROPN
ejpam-6792	320	4	,	,	PUNCT
ejpam-6792	320	5	10(1):33–40	10(1):33–40	NUM
ejpam-6792	320	6	,	,	PUNCT
ejpam-6792	320	7	2016	2016	NUM
ejpam-6792	320	8	.	.	PUNCT
ejpam-6792	321	1	[	[	X
ejpam-6792	321	2	19	19	NUM
ejpam-6792	321	3	]	]	X
ejpam-6792	321	4	s.	s.	PROPN
ejpam-6792	321	5	widianto	widianto	PROPN
ejpam-6792	321	6	,	,	PUNCT
ejpam-6792	321	7	s.	s.	PROPN
ejpam-6792	321	8	gemawati	gemawati	PROPN
ejpam-6792	321	9	,	,	PUNCT
ejpam-6792	321	10	and	and	CCONJ
ejpam-6792	321	11	kartini	kartini	NOUN
ejpam-6792	321	12	.	.	PUNCT
ejpam-6792	322	1	direct	direct	ADJ
ejpam-6792	322	2	product	product	NOUN
ejpam-6792	322	3	of	of	ADP
ejpam-6792	322	4	bg	bg	PROPN
ejpam-6792	322	5	-	-	PUNCT
ejpam-6792	322	6	algebras	algebras	PROPN
ejpam-6792	322	7	.	.	PUNCT
ejpam-6792	323	1	international	international	ADJ
ejpam-6792	323	2	journal	journal	PROPN
ejpam-6792	323	3	of	of	ADP
ejpam-6792	323	4	algebra	algebra	PROPN
ejpam-6792	323	5	,	,	PUNCT
ejpam-6792	323	6	13(5):239–247	13(5):239–247	PROPN
ejpam-6792	323	7	,	,	PUNCT
ejpam-6792	323	8	2019	2019	NUM
ejpam-6792	323	9	.	.	PUNCT
ejpam-6792	324	1	[	[	X
ejpam-6792	324	2	20	20	NUM
ejpam-6792	324	3	]	]	PUNCT
ejpam-6792	324	4	a.	a.	NOUN
ejpam-6792	324	5	setian	setian	PROPN
ejpam-6792	324	6	,	,	PUNCT
ejpam-6792	324	7	s.	s.	PROPN
ejpam-6792	324	8	gemawati	gemawati	PROPN
ejpam-6792	324	9	,	,	PUNCT
ejpam-6792	324	10	and	and	CCONJ
ejpam-6792	324	11	l.	l.	PROPN
ejpam-6792	324	12	deswita	deswita	PROPN
ejpam-6792	324	13	.	.	PUNCT
ejpam-6792	325	1	direct	direct	ADJ
ejpam-6792	325	2	product	product	NOUN
ejpam-6792	325	3	of	of	ADP
ejpam-6792	325	4	bp	bp	PROPN
ejpam-6792	325	5	-	-	PUNCT
ejpam-6792	325	6	algebras	algebras	PROPN
ejpam-6792	325	7	.	.	PUNCT
ejpam-6792	326	1	international	international	ADJ
ejpam-6792	326	2	journal	journal	PROPN
ejpam-6792	326	3	of	of	ADP
ejpam-6792	326	4	mathematics	mathematics	NOUN
ejpam-6792	326	5	trends	trend	NOUN
ejpam-6792	326	6	and	and	CCONJ
ejpam-6792	326	7	tcehnology	tcehnology	NOUN
ejpam-6792	326	8	,	,	PUNCT
ejpam-6792	326	9	66(10):63–66	66(10):63–66	NOUN
ejpam-6792	326	10	,	,	PUNCT
ejpam-6792	326	11	2020	2020	NUM
ejpam-6792	326	12	.	.	PUNCT
ejpam-6792	327	1	[	[	X
ejpam-6792	327	2	21	21	NUM
ejpam-6792	327	3	]	]	X
ejpam-6792	327	4	j.	j.	PROPN
ejpam-6792	327	5	kavitha	kavitha	PROPN
ejpam-6792	327	6	and	and	CCONJ
ejpam-6792	327	7	r.	r.	PROPN
ejpam-6792	327	8	gowri	gowri	PROPN
ejpam-6792	327	9	.	.	PUNCT
ejpam-6792	328	1	direct	direct	ADJ
ejpam-6792	328	2	product	product	NOUN
ejpam-6792	328	3	of	of	ADP
ejpam-6792	328	4	gk	gk	PROPN
ejpam-6792	328	5	-	-	PUNCT
ejpam-6792	328	6	algebras	algebras	PROPN
ejpam-6792	328	7	.	.	PUNCT
ejpam-6792	329	1	indian	indian	PROPN
ejpam-6792	329	2	journal	journal	PROPN
ejpam-6792	329	3	of	of	ADP
ejpam-6792	329	4	science	science	NOUN
ejpam-6792	329	5	and	and	CCONJ
ejpam-6792	329	6	tchnology	tchnology	NOUN
ejpam-6792	329	7	,	,	PUNCT
ejpam-6792	329	8	14(35):2802–2805	14(35):2802–2805	NUM
ejpam-6792	329	9	,	,	PUNCT
ejpam-6792	329	10	2021	2021	NUM
ejpam-6792	329	11	.	.	PUNCT
ejpam-6792	330	1	[	[	X
ejpam-6792	330	2	22	22	NUM
ejpam-6792	330	3	]	]	PUNCT
ejpam-6792	330	4	c.	c.	PROPN
ejpam-6792	330	5	chanmanee	chanmanee	PROPN
ejpam-6792	330	6	,	,	PUNCT
ejpam-6792	330	7	r.	r.	PROPN
ejpam-6792	330	8	prasertpong	prasertpong	PROPN
ejpam-6792	330	9	,	,	PUNCT
ejpam-6792	330	10	p.	p.	PROPN
ejpam-6792	330	11	julatha	julatha	PROPN
ejpam-6792	330	12	,	,	PUNCT
ejpam-6792	330	13	n.	n.	PROPN
ejpam-6792	330	14	lekkoksung	lekkoksung	PROPN
ejpam-6792	330	15	,	,	PUNCT
ejpam-6792	330	16	and	and	CCONJ
ejpam-6792	330	17	a.	a.	NOUN
ejpam-6792	330	18	iampan	iampan	PROPN
ejpam-6792	330	19	.	.	PUNCT
ejpam-6792	331	1	on	on	ADP
ejpam-6792	331	2	external	external	ADJ
ejpam-6792	331	3	direct	direct	ADJ
ejpam-6792	331	4	products	product	NOUN
ejpam-6792	331	5	of	of	ADP
ejpam-6792	331	6	iup	iup	NOUN
ejpam-6792	331	7	-	-	PUNCT
ejpam-6792	331	8	algebras	algebras	PROPN
ejpam-6792	331	9	.	.	PUNCT
ejpam-6792	332	1	journal	journal	PROPN
ejpam-6792	332	2	of	of	ADP
ejpam-6792	332	3	innovative	innovative	ADJ
ejpam-6792	332	4	computing	computing	NOUN
ejpam-6792	332	5	,	,	PUNCT
ejpam-6792	332	6	information	information	NOUN
ejpam-6792	332	7	and	and	CCONJ
ejpam-6792	332	8	control	control	NOUN
ejpam-6792	332	9	,	,	PUNCT
ejpam-6792	332	10	19(3):775–787	19(3):775–787	NUM
ejpam-6792	332	11	,	,	PUNCT
ejpam-6792	332	12	2023	2023	NUM
ejpam-6792	332	13	.	.	PUNCT
ejpam-6792	333	1	[	[	X
ejpam-6792	333	2	23	23	NUM
ejpam-6792	333	3	]	]	X
ejpam-6792	333	4	c.	c.	PROPN
ejpam-6792	333	5	chanmanee	chanmanee	PROPN
ejpam-6792	333	6	,	,	PUNCT
ejpam-6792	333	7	r.	r.	PROPN
ejpam-6792	333	8	chinram	chinram	PROPN
ejpam-6792	333	9	,	,	PUNCT
ejpam-6792	333	10	r.	r.	PROPN
ejpam-6792	333	11	prasertpong	prasertpong	PROPN
ejpam-6792	333	12	,	,	PUNCT
ejpam-6792	333	13	p.	p.	PROPN
ejpam-6792	333	14	julatha	julatha	PROPN
ejpam-6792	333	15	,	,	PUNCT
ejpam-6792	333	16	and	and	CCONJ
ejpam-6792	333	17	a.	a.	NOUN
ejpam-6792	333	18	iampan	iampan	PROPN
ejpam-6792	333	19	.	.	PUNCT
ejpam-6792	334	1	external	external	ADJ
ejpam-6792	334	2	direct	direct	ADJ
ejpam-6792	334	3	products	product	NOUN
ejpam-6792	334	4	on	on	ADP
ejpam-6792	334	5	dual	dual	ADJ
ejpam-6792	334	6	up	up	ADV
ejpam-6792	334	7	(	(	PUNCT
ejpam-6792	334	8	bcc)-algebras	bcc)-algebras	PROPN
ejpam-6792	334	9	.	.	PROPN
ejpam-6792	334	10	journal	journal	NOUN
ejpam-6792	334	11	of	of	ADP
ejpam-6792	334	12	mathematics	mathematic	NOUN
ejpam-6792	334	13	and	and	CCONJ
ejpam-6792	334	14	computer	computer	NOUN
ejpam-6792	334	15	science	science	NOUN
ejpam-6792	334	16	,	,	PUNCT
ejpam-6792	334	17	29(2):175–191	29(2):175–191	PROPN
ejpam-6792	334	18	,	,	PUNCT
ejpam-6792	334	19	2023	2023	NUM
ejpam-6792	334	20	.	.	PUNCT
ejpam-6792	335	1	[	[	X
ejpam-6792	335	2	24	24	NUM
ejpam-6792	335	3	]	]	PUNCT
ejpam-6792	335	4	c.	c.	PROPN
ejpam-6792	335	5	chanmanee	chanmanee	PROPN
ejpam-6792	335	6	,	,	PUNCT
ejpam-6792	335	7	p.	p.	PROPN
ejpam-6792	335	8	julatha	julatha	PROPN
ejpam-6792	335	9	,	,	PUNCT
ejpam-6792	335	10	w.	w.	PROPN
ejpam-6792	335	11	nakkhasen	nakkhasen	PROPN
ejpam-6792	335	12	,	,	PUNCT
ejpam-6792	335	13	r.	r.	PROPN
ejpam-6792	335	14	prasertpong	prasertpong	PROPN
ejpam-6792	335	15	,	,	PUNCT
ejpam-6792	335	16	,	,	PUNCT
ejpam-6792	335	17	and	and	CCONJ
ejpam-6792	335	18	a.	a.	NOUN
ejpam-6792	335	19	iampan	iampan	PROPN
ejpam-6792	335	20	.	.	PUNCT
ejpam-6792	336	1	external	external	ADJ
ejpam-6792	336	2	direct	direct	ADJ
ejpam-6792	336	3	products	product	NOUN
ejpam-6792	336	4	ju	ju	NOUN
ejpam-6792	336	5	-	-	PUNCT
ejpam-6792	336	6	algebras	algebras	PROPN
ejpam-6792	336	7	.	.	PUNCT
ejpam-6792	337	1	international	international	ADJ
ejpam-6792	337	2	journal	journal	NOUN
ejpam-6792	337	3	of	of	ADP
ejpam-6792	337	4	analysis	analysis	NOUN
ejpam-6792	337	5	and	and	CCONJ
ejpam-6792	337	6	applications	application	NOUN
ejpam-6792	337	7	,	,	PUNCT
ejpam-6792	337	8	22:183	22:183	NUM
ejpam-6792	337	9	,	,	PUNCT
ejpam-6792	337	10	2024	2024	NUM
ejpam-6792	337	11	.	.	PUNCT
ejpam-6792	338	1	[	[	X
ejpam-6792	338	2	25	25	NUM
ejpam-6792	338	3	]	]	PUNCT
ejpam-6792	338	4	s.	s.	PROPN
ejpam-6792	338	5	ahn	ahn	PROPN
ejpam-6792	338	6	and	and	CCONJ
ejpam-6792	338	7	s.	s.	PROPN
ejpam-6792	338	8	e.	e.	PROPN
ejpam-6792	338	9	kang	kang	PROPN
ejpam-6792	338	10	.	.	PUNCT
ejpam-6792	339	1	the	the	DET
ejpam-6792	339	2	role	role	NOUN
ejpam-6792	339	3	of	of	ADP
ejpam-6792	339	4	t(x	t(x	PROPN
ejpam-6792	339	5	)	)	PUNCT
ejpam-6792	339	6	in	in	ADP
ejpam-6792	339	7	the	the	DET
ejpam-6792	339	8	ideal	ideal	ADJ
ejpam-6792	339	9	theory	theory	NOUN
ejpam-6792	339	10	of	of	ADP
ejpam-6792	339	11	q	q	NOUN
ejpam-6792	339	12	-	-	PUNCT
ejpam-6792	339	13	algebras	algebras	X
ejpam-6792	339	14	.	.	PUNCT
ejpam-6792	340	1	honam	honam	PROPN
ejpam-6792	340	2	mathematical	mathematical	PROPN
ejpam-6792	340	3	j.	j.	PROPN
ejpam-6792	340	4	,	,	PUNCT
ejpam-6792	340	5	32(3):515–523	32(3):515–523	PROPN
ejpam-6792	340	6	,	,	PUNCT
ejpam-6792	340	7	2010	2010	NUM
ejpam-6792	340	8	.	.	PUNCT