id	sid	tid	token	lemma	pos
ejpam-4738	1	1	european	european	PROPN
ejpam-4738	1	2	journal	journal	PROPN
ejpam-4738	1	3	of	of	ADP
ejpam-4738	1	4	pure	pure	ADJ
ejpam-4738	1	5	and	and	CCONJ
ejpam-4738	1	6	applied	apply	VERB
ejpam-4738	1	7	mathematics	mathematic	NOUN
ejpam-4738	1	8	vol	vol	NOUN
ejpam-4738	1	9	.	.	PUNCT
ejpam-4738	2	1	16	16	NUM
ejpam-4738	2	2	,	,	PUNCT
ejpam-4738	2	3	no	no	INTJ
ejpam-4738	2	4	.	.	NOUN
ejpam-4738	2	5	2	2	NUM
ejpam-4738	2	6	,	,	PUNCT
ejpam-4738	2	7	2023	2023	NUM
ejpam-4738	2	8	,	,	PUNCT
ejpam-4738	2	9	997	997	NUM
ejpam-4738	2	10	-	-	SYM
ejpam-4738	2	11	1004	1004	NUM
ejpam-4738	2	12	issn	issn	PROPN
ejpam-4738	2	13	1307	1307	NUM
ejpam-4738	2	14	-	-	SYM
ejpam-4738	2	15	5543	5543	NUM
ejpam-4738	2	16	–	–	PUNCT
ejpam-4738	3	1	ejpam.com	ejpam.com	X
ejpam-4738	3	2	published	publish	VERB
ejpam-4738	3	3	by	by	ADP
ejpam-4738	3	4	new	new	PROPN
ejpam-4738	3	5	york	york	PROPN
ejpam-4738	3	6	business	business	PROPN
ejpam-4738	3	7	global	global	ADJ
ejpam-4738	3	8	on	on	ADP
ejpam-4738	3	9	direct	direct	ADJ
ejpam-4738	3	10	product	product	NOUN
ejpam-4738	3	11	of	of	ADP
ejpam-4738	3	12	d	d	NOUN
ejpam-4738	3	13	-	-	PUNCT
ejpam-4738	3	14	algebras	algebras	X
ejpam-4738	3	15	maliwan	maliwan	PROPN
ejpam-4738	3	16	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	3	17	department	department	PROPN
ejpam-4738	3	18	of	of	ADP
ejpam-4738	3	19	mathematics	mathematic	NOUN
ejpam-4738	3	20	,	,	PUNCT
ejpam-4738	3	21	faculty	faculty	NOUN
ejpam-4738	3	22	of	of	ADP
ejpam-4738	3	23	science	science	NOUN
ejpam-4738	3	24	,	,	PUNCT
ejpam-4738	3	25	mahasarakham	mahasarakham	PROPN
ejpam-4738	3	26	university	university	PROPN
ejpam-4738	3	27	,	,	PUNCT
ejpam-4738	3	28	maha	maha	PROPN
ejpam-4738	3	29	sarakham	sarakham	PROPN
ejpam-4738	3	30	44150	44150	NUM
ejpam-4738	3	31	,	,	PUNCT
ejpam-4738	3	32	thailand	thailand	PROPN
ejpam-4738	3	33	abstract	abstract	PROPN
ejpam-4738	3	34	.	.	PUNCT
ejpam-4738	4	1	the	the	DET
ejpam-4738	4	2	main	main	ADJ
ejpam-4738	4	3	aim	aim	NOUN
ejpam-4738	4	4	of	of	ADP
ejpam-4738	4	5	this	this	DET
ejpam-4738	4	6	work	work	NOUN
ejpam-4738	4	7	is	be	AUX
ejpam-4738	4	8	to	to	PART
ejpam-4738	4	9	introduce	introduce	VERB
ejpam-4738	4	10	and	and	CCONJ
ejpam-4738	4	11	study	study	VERB
ejpam-4738	4	12	the	the	DET
ejpam-4738	4	13	notions	notion	NOUN
ejpam-4738	4	14	of	of	ADP
ejpam-4738	4	15	ideal	ideal	ADJ
ejpam-4738	4	16	direct	direct	ADJ
ejpam-4738	4	17	product	product	NOUN
ejpam-4738	4	18	d	d	NOUN
ejpam-4738	4	19	-	-	PUNCT
ejpam-4738	4	20	algebras	algebra	NOUN
ejpam-4738	4	21	,	,	PUNCT
ejpam-4738	4	22	d	d	ADJ
ejpam-4738	4	23	-	-	ADJ
ejpam-4738	4	24	ideal	ideal	ADJ
ejpam-4738	4	25	direct	direct	ADJ
ejpam-4738	4	26	product	product	NOUN
ejpam-4738	4	27	d	d	NOUN
ejpam-4738	4	28	-	-	PUNCT
ejpam-4738	4	29	algebras	algebra	NOUN
ejpam-4738	4	30	,	,	PUNCT
ejpam-4738	4	31	sub	sub	ADJ
ejpam-4738	4	32	-	-	ADJ
ejpam-4738	4	33	direct	direct	ADJ
ejpam-4738	4	34	product	product	NOUN
ejpam-4738	4	35	d	d	NOUN
ejpam-4738	4	36	-	-	PUNCT
ejpam-4738	4	37	algebras	algebra	NOUN
ejpam-4738	4	38	,	,	PUNCT
ejpam-4738	4	39	edge	edge	VERB
ejpam-4738	4	40	direct	direct	ADJ
ejpam-4738	4	41	product	product	NOUN
ejpam-4738	4	42	and	and	CCONJ
ejpam-4738	4	43	positive	positive	ADJ
ejpam-4738	4	44	implicative	implicative	ADJ
ejpam-4738	4	45	direct	direct	ADJ
ejpam-4738	4	46	product	product	NOUN
ejpam-4738	4	47	d	d	NOUN
ejpam-4738	4	48	-	-	PUNCT
ejpam-4738	4	49	algebras	algebras	PROPN
ejpam-4738	4	50	and	and	CCONJ
ejpam-4738	4	51	investigate	investigate	VERB
ejpam-4738	4	52	their	their	PRON
ejpam-4738	4	53	characterizations	characterization	NOUN
ejpam-4738	4	54	.	.	PUNCT
ejpam-4738	5	1	2020	2020	NUM
ejpam-4738	5	2	mathematics	mathematic	NOUN
ejpam-4738	5	3	subject	subject	NOUN
ejpam-4738	5	4	classifications	classification	NOUN
ejpam-4738	5	5	:	:	PUNCT
ejpam-4738	5	6	06f35	06f35	NUM
ejpam-4738	5	7	key	key	ADJ
ejpam-4738	5	8	words	word	NOUN
ejpam-4738	5	9	and	and	CCONJ
ejpam-4738	5	10	phrases	phrase	NOUN
ejpam-4738	5	11	:	:	PUNCT
ejpam-4738	5	12	d	d	X
ejpam-4738	5	13	-	-	PUNCT
ejpam-4738	5	14	algebras	algebras	ADV
ejpam-4738	5	15	,	,	PUNCT
ejpam-4738	5	16	direct	direct	ADJ
ejpam-4738	5	17	product	product	NOUN
ejpam-4738	5	18	d	d	NOUN
ejpam-4738	5	19	-	-	PUNCT
ejpam-4738	5	20	algebras	algebra	NOUN
ejpam-4738	5	21	,	,	PUNCT
ejpam-4738	5	22	ideal	ideal	ADJ
ejpam-4738	5	23	direct	direct	ADJ
ejpam-4738	5	24	product	product	NOUN
ejpam-4738	5	25	d	d	NOUN
ejpam-4738	5	26	-	-	PUNCT
ejpam-4738	5	27	algebras	algebra	NOUN
ejpam-4738	5	28	,	,	PUNCT
ejpam-4738	5	29	d	d	ADJ
ejpam-4738	5	30	-	-	ADJ
ejpam-4738	5	31	ideal	ideal	ADJ
ejpam-4738	5	32	direct	direct	ADJ
ejpam-4738	5	33	product	product	NOUN
ejpam-4738	5	34	d	d	NOUN
ejpam-4738	5	35	-	-	PUNCT
ejpam-4738	5	36	algebras	algebras	ADJ
ejpam-4738	5	37	,	,	PUNCT
ejpam-4738	5	38	ideal	ideal	ADJ
ejpam-4738	5	39	,	,	PUNCT
ejpam-4738	5	40	edge	edge	VERB
ejpam-4738	5	41	direct	direct	ADJ
ejpam-4738	5	42	product	product	NOUN
ejpam-4738	5	43	d	d	NOUN
ejpam-4738	5	44	-	-	PUNCT
ejpam-4738	5	45	algebras	algebras	ADJ
ejpam-4738	5	46	,	,	PUNCT
ejpam-4738	5	47	positive	positive	ADJ
ejpam-4738	5	48	implicative	implicative	ADJ
ejpam-4738	5	49	,	,	PUNCT
ejpam-4738	5	50	direct	direct	ADJ
ejpam-4738	5	51	product	product	NOUN
ejpam-4738	5	52	d	d	NOUN
ejpam-4738	5	53	-	-	PUNCT
ejpam-4738	5	54	algebra	algebra	ADJ
ejpam-4738	5	55	1	1	NUM
ejpam-4738	5	56	.	.	PUNCT
ejpam-4738	6	1	introduction	introduction	NOUN
ejpam-4738	6	2	the	the	DET
ejpam-4738	6	3	concept	concept	NOUN
ejpam-4738	6	4	of	of	ADP
ejpam-4738	6	5	d	d	PROPN
ejpam-4738	6	6	-	-	PUNCT
ejpam-4738	6	7	algebras	algebras	PROPN
ejpam-4738	6	8	was	be	AUX
ejpam-4738	6	9	first	first	ADV
ejpam-4738	6	10	introduced	introduce	VERB
ejpam-4738	6	11	by	by	ADP
ejpam-4738	6	12	j.	j.	PROPN
ejpam-4738	6	13	neggers	neggers	PROPN
ejpam-4738	6	14	and	and	CCONJ
ejpam-4738	6	15	h.	h.	PROPN
ejpam-4738	6	16	s.	s.	PROPN
ejpam-4738	6	17	kim	kim	PROPN
ejpam-4738	7	1	(	(	PUNCT
ejpam-4738	7	2	[	[	X
ejpam-4738	7	3	9	9	NUM
ejpam-4738	7	4	]	]	NUM
ejpam-4738	7	5	)	)	PUNCT
ejpam-4738	7	6	.	.	PUNCT
ejpam-4738	8	1	a	a	DET
ejpam-4738	8	2	d	d	NOUN
ejpam-4738	8	3	-	-	NOUN
ejpam-4738	8	4	algebrax	algebrax	NOUN
ejpam-4738	8	5	=	=	SYM
ejpam-4738	8	6	(	(	PUNCT
ejpam-4738	8	7	x	x	X
ejpam-4738	8	8	,	,	PUNCT
ejpam-4738	8	9	∗	∗	NOUN
ejpam-4738	8	10	,	,	PUNCT
ejpam-4738	8	11	0	0	NUM
ejpam-4738	8	12	)	)	PUNCT
ejpam-4738	8	13	is	be	AUX
ejpam-4738	8	14	an	an	DET
ejpam-4738	8	15	algebra	algebra	NOUN
ejpam-4738	8	16	of	of	ADP
ejpam-4738	8	17	type	type	NOUN
ejpam-4738	8	18	(	(	PUNCT
ejpam-4738	8	19	2	2	NUM
ejpam-4738	8	20	,	,	PUNCT
ejpam-4738	8	21	0	0	NUM
ejpam-4738	8	22	)	)	PUNCT
ejpam-4738	8	23	,	,	PUNCT
ejpam-4738	8	24	that	that	ADV
ejpam-4738	8	25	is	is	ADV
ejpam-4738	8	26	,	,	PUNCT
ejpam-4738	8	27	a	a	DET
ejpam-4738	8	28	nonempty	nonempty	ADJ
ejpam-4738	8	29	setx	setx	NOUN
ejpam-4738	8	30	together	together	ADV
ejpam-4738	8	31	with	with	ADP
ejpam-4738	8	32	a	a	DET
ejpam-4738	8	33	binary	binary	ADJ
ejpam-4738	8	34	operation	operation	NOUN
ejpam-4738	8	35	∗	∗	NOUN
ejpam-4738	8	36	and	and	CCONJ
ejpam-4738	8	37	a	a	DET
ejpam-4738	8	38	constant	constant	ADJ
ejpam-4738	8	39	0	0	NUM
ejpam-4738	8	40	satisfying	satisfy	VERB
ejpam-4738	8	41	some	some	DET
ejpam-4738	8	42	axioms	axiom	NOUN
ejpam-4738	8	43	in	in	ADP
ejpam-4738	8	44	[	[	X
ejpam-4738	8	45	1	1	NUM
ejpam-4738	8	46	]	]	PUNCT
ejpam-4738	8	47	,	,	PUNCT
ejpam-4738	8	48	they	they	PRON
ejpam-4738	8	49	introduced	introduce	VERB
ejpam-4738	8	50	and	and	CCONJ
ejpam-4738	8	51	investigated	investigate	VERB
ejpam-4738	8	52	several	several	ADJ
ejpam-4738	8	53	relations	relation	NOUN
ejpam-4738	8	54	between	between	ADP
ejpam-4738	8	55	d	d	NOUN
ejpam-4738	8	56	-	-	PUNCT
ejpam-4738	8	57	algebras	algebra	NOUN
ejpam-4738	8	58	and	and	CCONJ
ejpam-4738	8	59	bck	bck	NOUN
ejpam-4738	8	60	-	-	PUNCT
ejpam-4738	8	61	algebras	algebras	PROPN
ejpam-4738	8	62	and	and	CCONJ
ejpam-4738	8	63	showed	show	VERB
ejpam-4738	8	64	that	that	SCONJ
ejpam-4738	8	65	the	the	DET
ejpam-4738	8	66	class	class	NOUN
ejpam-4738	8	67	of	of	ADP
ejpam-4738	8	68	oriented	orient	VERB
ejpam-4738	8	69	digraphs	digraphs	ADJ
ejpam-4738	8	70	corresponds	correspond	NOUN
ejpam-4738	8	71	in	in	ADP
ejpam-4738	8	72	a	a	DET
ejpam-4738	8	73	simple	simple	ADJ
ejpam-4738	8	74	way	way	NOUN
ejpam-4738	8	75	to	to	ADP
ejpam-4738	8	76	the	the	DET
ejpam-4738	8	77	class	class	NOUN
ejpam-4738	8	78	of	of	ADP
ejpam-4738	8	79	edge	edge	NOUN
ejpam-4738	8	80	d	d	NOUN
ejpam-4738	8	81	-	-	PUNCT
ejpam-4738	8	82	algebras	algebra	NOUN
ejpam-4738	8	83	and	and	CCONJ
ejpam-4738	8	84	that	that	SCONJ
ejpam-4738	8	85	arbitrary	arbitrary	ADJ
ejpam-4738	8	86	d	d	X
ejpam-4738	8	87	-	-	PUNCT
ejpam-4738	8	88	algebras	algebras	PROPN
ejpam-4738	8	89	also	also	ADV
ejpam-4738	8	90	determine	determine	VERB
ejpam-4738	8	91	unique	unique	ADJ
ejpam-4738	8	92	edge	edge	NOUN
ejpam-4738	8	93	d	d	NOUN
ejpam-4738	8	94	-	-	PUNCT
ejpam-4738	8	95	algebras	algebras	ADJ
ejpam-4738	8	96	in	in	ADP
ejpam-4738	8	97	a	a	DET
ejpam-4738	8	98	natural	natural	ADJ
ejpam-4738	8	99	manner	manner	NOUN
ejpam-4738	8	100	.	.	PUNCT
ejpam-4738	9	1	in	in	ADP
ejpam-4738	9	2	1999	1999	NUM
ejpam-4738	9	3	,	,	PUNCT
ejpam-4738	9	4	j.	j.	PROPN
ejpam-4738	9	5	neggers	neggers	PROPN
ejpam-4738	9	6	,	,	PUNCT
ejpam-4738	9	7	y.	y.	PROPN
ejpam-4738	9	8	b.	b.	PROPN
ejpam-4738	9	9	jun	jun	PROPN
ejpam-4738	9	10	and	and	CCONJ
ejpam-4738	9	11	h.	h.	PROPN
ejpam-4738	9	12	s.	s.	PROPN
ejpam-4738	9	13	kim	kim	PROPN
ejpam-4738	9	14	(	(	PUNCT
ejpam-4738	9	15	[	[	X
ejpam-4738	9	16	8	8	NUM
ejpam-4738	9	17	]	]	NUM
ejpam-4738	9	18	)	)	PUNCT
ejpam-4738	9	19	,	,	PUNCT
ejpam-4738	9	20	introduced	introduce	VERB
ejpam-4738	9	21	the	the	DET
ejpam-4738	9	22	notions	notion	NOUN
ejpam-4738	9	23	of	of	ADP
ejpam-4738	9	24	a	a	DET
ejpam-4738	9	25	d	d	NOUN
ejpam-4738	9	26	-	-	PUNCT
ejpam-4738	9	27	subalgebra	subalgebra	ADJ
ejpam-4738	9	28	,	,	PUNCT
ejpam-4738	9	29	d	d	NOUN
ejpam-4738	9	30	-	-	NOUN
ejpam-4738	9	31	ideal	ideal	ADJ
ejpam-4738	9	32	,	,	PUNCT
ejpam-4738	9	33	and	and	CCONJ
ejpam-4738	9	34	a	a	DET
ejpam-4738	9	35	d∗-ideal	d∗-ideal	NOUN
ejpam-4738	9	36	in	in	ADP
ejpam-4738	9	37	d	d	NOUN
ejpam-4738	9	38	-	-	PUNCT
ejpam-4738	9	39	algebras	algebras	X
ejpam-4738	9	40	,	,	PUNCT
ejpam-4738	9	41	and	and	CCONJ
ejpam-4738	9	42	investigated	investigate	VERB
ejpam-4738	9	43	relations	relation	NOUN
ejpam-4738	9	44	among	among	ADP
ejpam-4738	9	45	them	they	PRON
ejpam-4738	9	46	.	.	PUNCT
ejpam-4738	10	1	furthermore	furthermore	ADV
ejpam-4738	10	2	,	,	PUNCT
ejpam-4738	10	3	they	they	PRON
ejpam-4738	10	4	are	be	AUX
ejpam-4738	10	5	able	able	ADJ
ejpam-4738	10	6	to	to	PART
ejpam-4738	10	7	define	define	VERB
ejpam-4738	10	8	the	the	DET
ejpam-4738	10	9	ideal	ideal	NOUN
ejpam-4738	10	10	of	of	ADP
ejpam-4738	10	11	a	a	DET
ejpam-4738	10	12	quotient	quotient	NOUN
ejpam-4738	10	13	d	d	NOUN
ejpam-4738	10	14	-	-	PUNCT
ejpam-4738	10	15	algebra	algebra	NOUN
ejpam-4738	10	16	and	and	CCONJ
ejpam-4738	10	17	to	to	PART
ejpam-4738	10	18	prove	prove	VERB
ejpam-4738	10	19	a	a	DET
ejpam-4738	10	20	fundamental	fundamental	ADJ
ejpam-4738	10	21	theorem	theorem	NOUN
ejpam-4738	10	22	of	of	ADP
ejpam-4738	10	23	d	d	NOUN
ejpam-4738	10	24	-	-	PUNCT
ejpam-4738	10	25	morphisms	morphism	VERB
ejpam-4738	10	26	for	for	ADP
ejpam-4738	10	27	d	d	NOUN
ejpam-4738	10	28	-	-	PUNCT
ejpam-4738	10	29	algebras	algebras	PROPN
ejpam-4738	10	30	as	as	ADP
ejpam-4738	10	31	a	a	DET
ejpam-4738	10	32	consequence	consequence	NOUN
ejpam-4738	10	33	.	.	PUNCT
ejpam-4738	11	1	s.	s.	PROPN
ejpam-4738	11	2	s.	s.	PROPN
ejpam-4738	11	3	ahn	ahn	PROPN
ejpam-4738	11	4	and	and	CCONJ
ejpam-4738	11	5	k.	k.	PROPN
ejpam-4738	11	6	s.	s.	PROPN
ejpam-4738	12	1	so	so	ADV
ejpam-4738	12	2	(	(	PUNCT
ejpam-4738	12	3	[	[	X
ejpam-4738	12	4	1	1	NUM
ejpam-4738	12	5	]	]	PUNCT
ejpam-4738	12	6	,	,	PUNCT
ejpam-4738	12	7	defined	define	VERB
ejpam-4738	12	8	left	left	ADJ
ejpam-4738	12	9	-	-	PUNCT
ejpam-4738	12	10	regular	regular	ADJ
ejpam-4738	12	11	maps	map	NOUN
ejpam-4738	12	12	on	on	ADP
ejpam-4738	12	13	d-algebras.these	d-algebras.these	ADJ
ejpam-4738	12	14	mappings	mapping	NOUN
ejpam-4738	12	15	show	show	NOUN
ejpam-4738	12	16	behaviors	behavior	NOUN
ejpam-4738	12	17	reminiscent	reminiscent	ADJ
ejpam-4738	12	18	or	or	CCONJ
ejpam-4738	12	19	homomorphisms	homomorphism	NOUN
ejpam-4738	12	20	on	on	ADP
ejpam-4738	12	21	d	d	NOUN
ejpam-4738	12	22	-	-	PUNCT
ejpam-4738	12	23	algebras	algebras	X
ejpam-4738	12	24	.	.	PUNCT
ejpam-4738	13	1	in	in	ADP
ejpam-4738	13	2	particular	particular	ADJ
ejpam-4738	13	3	,	,	PUNCT
ejpam-4738	13	4	they	they	PRON
ejpam-4738	13	5	have	have	AUX
ejpam-4738	13	6	introduced	introduce	VERB
ejpam-4738	13	7	the	the	DET
ejpam-4738	13	8	kernels	kernel	NOUN
ejpam-4738	13	9	,	,	PUNCT
ejpam-4738	13	10	annihilators	annihilators	PROPN
ejpam-4738	13	11	,	,	PUNCT
ejpam-4738	13	12	co	co	NOUN
ejpam-4738	13	13	-	-	NOUN
ejpam-4738	13	14	annihilators	annihilator	NOUN
ejpam-4738	13	15	and	and	CCONJ
ejpam-4738	13	16	some	some	PRON
ejpam-4738	13	17	of	of	ADP
ejpam-4738	13	18	their	their	PRON
ejpam-4738	13	19	properties	property	NOUN
ejpam-4738	13	20	for	for	ADP
ejpam-4738	13	21	these	these	DET
ejpam-4738	13	22	mappings	mapping	NOUN
ejpam-4738	13	23	,	,	PUNCT
ejpam-4738	13	24	especially	especially	ADV
ejpam-4738	13	25	in	in	ADP
ejpam-4738	13	26	the	the	DET
ejpam-4738	13	27	setting	setting	NOUN
ejpam-4738	13	28	of	of	ADP
ejpam-4738	13	29	positive	positive	ADJ
ejpam-4738	13	30	implicative	implicative	ADJ
ejpam-4738	13	31	d	d	NOUN
ejpam-4738	13	32	-	-	PUNCT
ejpam-4738	13	33	algebras	algebras	X
ejpam-4738	13	34	.	.	PUNCT
ejpam-4738	14	1	the	the	DET
ejpam-4738	14	2	study	study	NOUN
ejpam-4738	14	3	of	of	ADP
ejpam-4738	14	4	multipliers	multiplier	NOUN
ejpam-4738	14	5	have	have	AUX
ejpam-4738	14	6	been	be	AUX
ejpam-4738	14	7	made	make	VERB
ejpam-4738	14	8	by	by	ADP
ejpam-4738	14	9	various	various	ADJ
ejpam-4738	14	10	researchers	researcher	NOUN
ejpam-4738	14	11	in	in	ADP
ejpam-4738	14	12	the	the	DET
ejpam-4738	14	13	context	context	NOUN
ejpam-4738	14	14	of	of	ADP
ejpam-4738	14	15	c*-algebras	c*-algebra	NOUN
ejpam-4738	14	16	,	,	PUNCT
ejpam-4738	14	17	rings	ring	NOUN
ejpam-4738	14	18	and	and	CCONJ
ejpam-4738	14	19	semigroups	semigroup	NOUN
ejpam-4738	14	20	in	in	ADP
ejpam-4738	14	21	(	(	PUNCT
ejpam-4738	14	22	[	[	X
ejpam-4738	14	23	6	6	NUM
ejpam-4738	14	24	]	]	NUM
ejpam-4738	14	25	)	)	PUNCT
ejpam-4738	14	26	.	.	PUNCT
ejpam-4738	15	1	in	in	ADP
ejpam-4738	15	2	2012	2012	NUM
ejpam-4738	15	3	,	,	PUNCT
ejpam-4738	15	4	m.	m.	NOUN
ejpam-4738	15	5	a.	a.	PROPN
ejpam-4738	15	6	chaudhry	chaudhry	PROPN
ejpam-4738	15	7	and	and	CCONJ
ejpam-4738	15	8	f.	f.	PROPN
ejpam-4738	15	9	ali	ali	PROPN
ejpam-4738	15	10	(	(	PUNCT
ejpam-4738	15	11	[	[	X
ejpam-4738	15	12	3	3	NUM
ejpam-4738	15	13	]	]	PUNCT
ejpam-4738	15	14	)	)	PUNCT
ejpam-4738	15	15	introduced	introduce	VERB
ejpam-4738	15	16	the	the	DET
ejpam-4738	15	17	concept	concept	NOUN
ejpam-4738	15	18	of	of	ADP
ejpam-4738	15	19	a	a	DET
ejpam-4738	15	20	multiplier	multiplier	NOUN
ejpam-4738	15	21	on	on	ADP
ejpam-4738	15	22	d	d	NOUN
ejpam-4738	15	23	-	-	PUNCT
ejpam-4738	15	24	algebra	algebra	NOUN
ejpam-4738	15	25	and	and	CCONJ
ejpam-4738	15	26	obtain	obtain	VERB
ejpam-4738	15	27	some	some	DET
ejpam-4738	15	28	properties	property	NOUN
ejpam-4738	15	29	of	of	ADP
ejpam-4738	15	30	multipliers	multiplier	NOUN
ejpam-4738	15	31	of	of	ADP
ejpam-4738	15	32	d	d	PROPN
ejpam-4738	15	33	-	-	PUNCT
ejpam-4738	15	34	algebras	algebra	NOUN
ejpam-4738	15	35	.	.	PUNCT
ejpam-4738	16	1	doi	doi	NOUN
ejpam-4738	16	2	:	:	PUNCT
ejpam-4738	16	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4738	https://doi.org/10.29020/nybg.ejpam.v16i2.4738	VERB
ejpam-4738	16	4	email	email	NOUN
ejpam-4738	16	5	address	address	NOUN
ejpam-4738	16	6	:	:	PUNCT
ejpam-4738	16	7	maliwan.t@msu.ac.th	maliwan.t@msu.ac.th	PROPN
ejpam-4738	16	8	(	(	PUNCT
ejpam-4738	16	9	m.	m.	NOUN
ejpam-4738	16	10	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	16	11	)	)	PUNCT
ejpam-4738	16	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4738	17	1	997	997	NUM
ejpam-4738	18	1	©	©	PROPN
ejpam-4738	18	2	2023	2023	NUM
ejpam-4738	18	3	ejpam	ejpam	NOUN
ejpam-4738	18	4	all	all	DET
ejpam-4738	18	5	rights	right	NOUN
ejpam-4738	18	6	reserved	reserve	VERB
ejpam-4738	18	7	.	.	PUNCT
ejpam-4738	19	1	m.	m.	PROPN
ejpam-4738	19	2	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	19	3	/	/	SYM
ejpam-4738	19	4	eur	eur	PROPN
ejpam-4738	19	5	.	.	PUNCT
ejpam-4738	20	1	j.	j.	PROPN
ejpam-4738	20	2	pure	pure	PROPN
ejpam-4738	20	3	appl	appl	PROPN
ejpam-4738	20	4	.	.	PROPN
ejpam-4738	20	5	math	math	PROPN
ejpam-4738	20	6	,	,	PUNCT
ejpam-4738	20	7	16	16	NUM
ejpam-4738	20	8	(	(	PUNCT
ejpam-4738	20	9	2	2	NUM
ejpam-4738	20	10	)	)	PUNCT
ejpam-4738	20	11	(	(	PUNCT
ejpam-4738	20	12	2023	2023	NUM
ejpam-4738	20	13	)	)	PUNCT
ejpam-4738	20	14	,	,	PUNCT
ejpam-4738	20	15	997	997	NUM
ejpam-4738	20	16	-	-	SYM
ejpam-4738	20	17	1004	1004	NUM
ejpam-4738	20	18	998	998	NUM
ejpam-4738	20	19	the	the	DET
ejpam-4738	20	20	concept	concept	NOUN
ejpam-4738	20	21	of	of	ADP
ejpam-4738	20	22	the	the	DET
ejpam-4738	20	23	direct	direct	ADJ
ejpam-4738	20	24	product	product	NOUN
ejpam-4738	20	25	,	,	PUNCT
ejpam-4738	20	26	was	be	AUX
ejpam-4738	20	27	first	first	ADV
ejpam-4738	20	28	defined	define	VERB
ejpam-4738	20	29	in	in	ADP
ejpam-4738	20	30	groups	group	NOUN
ejpam-4738	20	31	and	and	CCONJ
ejpam-4738	20	32	obtained	obtain	VERB
ejpam-4738	20	33	the	the	DET
ejpam-4738	20	34	properties	property	NOUN
ejpam-4738	20	35	that	that	PRON
ejpam-4738	20	36	a	a	DET
ejpam-4738	20	37	direct	direct	ADJ
ejpam-4738	20	38	product	product	NOUN
ejpam-4738	20	39	of	of	ADP
ejpam-4738	20	40	groups	group	NOUN
ejpam-4738	20	41	is	be	AUX
ejpam-4738	20	42	also	also	ADV
ejpam-4738	20	43	a	a	DET
ejpam-4738	20	44	group	group	NOUN
ejpam-4738	20	45	.	.	PUNCT
ejpam-4738	21	1	in	in	ADP
ejpam-4738	21	2	1999	1999	NUM
ejpam-4738	21	3	,	,	PUNCT
ejpam-4738	21	4	j.	j.	PROPN
ejpam-4738	21	5	neggers	neggers	PROPN
ejpam-4738	21	6	and	and	CCONJ
ejpam-4738	21	7	h.	h.	PROPN
ejpam-4738	21	8	s.	s.	PROPN
ejpam-4738	21	9	kim	kim	PROPN
ejpam-4738	22	1	(	(	PUNCT
ejpam-4738	22	2	[	[	X
ejpam-4738	22	3	9	9	NUM
ejpam-4738	22	4	]	]	PUNCT
ejpam-4738	22	5	)	)	PUNCT
ejpam-4738	22	6	introduced	introduce	VERB
ejpam-4738	22	7	the	the	DET
ejpam-4738	22	8	concept	concept	NOUN
ejpam-4738	22	9	of	of	ADP
ejpam-4738	22	10	a	a	DET
ejpam-4738	22	11	direct	direct	ADJ
ejpam-4738	22	12	product	product	NOUN
ejpam-4738	22	13	of	of	ADP
ejpam-4738	22	14	d	d	NOUN
ejpam-4738	22	15	-	-	PUNCT
ejpam-4738	22	16	algebras	algebras	X
ejpam-4738	22	17	,	,	PUNCT
ejpam-4738	22	18	they	they	PRON
ejpam-4738	22	19	investigate	investigate	VERB
ejpam-4738	22	20	several	several	ADJ
ejpam-4738	22	21	relations	relation	NOUN
ejpam-4738	22	22	between	between	ADP
ejpam-4738	22	23	projection	projection	NOUN
ejpam-4738	22	24	mappings	mapping	NOUN
ejpam-4738	22	25	and	and	CCONJ
ejpam-4738	22	26	d	d	NOUN
ejpam-4738	22	27	-	-	PUNCT
ejpam-4738	22	28	morphisms	morphism	VERB
ejpam-4738	22	29	on	on	ADP
ejpam-4738	22	30	a	a	DET
ejpam-4738	22	31	direct	direct	ADJ
ejpam-4738	22	32	sum	sum	NOUN
ejpam-4738	22	33	of	of	ADP
ejpam-4738	22	34	edge	edge	NOUN
ejpam-4738	22	35	d	d	NOUN
ejpam-4738	22	36	-	-	PUNCT
ejpam-4738	22	37	algebras	algebra	NOUN
ejpam-4738	22	38	,	,	PUNCT
ejpam-4738	22	39	in	in	ADP
ejpam-4738	22	40	2020	2020	NUM
ejpam-4738	22	41	,	,	PUNCT
ejpam-4738	22	42	a.	a.	NOUN
ejpam-4738	22	43	setiani	setiani	PROPN
ejpam-4738	22	44	,	,	PUNCT
ejpam-4738	22	45	s.	s.	PROPN
ejpam-4738	22	46	gemawati	gemawati	PROPN
ejpam-4738	22	47	and	and	CCONJ
ejpam-4738	22	48	l.	l.	PROPN
ejpam-4738	22	49	deswita	deswita	PROPN
ejpam-4738	23	1	(	(	PUNCT
ejpam-4738	23	2	[	[	X
ejpam-4738	23	3	10	10	NUM
ejpam-4738	23	4	]	]	PUNCT
ejpam-4738	23	5	)	)	PUNCT
ejpam-4738	23	6	introduced	introduce	VERB
ejpam-4738	23	7	the	the	DET
ejpam-4738	23	8	notions	notion	NOUN
ejpam-4738	23	9	of	of	ADP
ejpam-4738	23	10	a	a	DET
ejpam-4738	23	11	direct	direct	ADJ
ejpam-4738	23	12	product	product	NOUN
ejpam-4738	23	13	of	of	ADP
ejpam-4738	23	14	bp	bp	PROPN
ejpam-4738	23	15	-	-	PUNCT
ejpam-4738	23	16	algebra	algebra	PROPN
ejpam-4738	23	17	and	and	CCONJ
ejpam-4738	23	18	some	some	PRON
ejpam-4738	23	19	of	of	ADP
ejpam-4738	23	20	related	related	ADJ
ejpam-4738	23	21	properties	property	NOUN
ejpam-4738	23	22	are	be	AUX
ejpam-4738	23	23	investigated	investigate	VERB
ejpam-4738	23	24	.	.	PUNCT
ejpam-4738	24	1	also	also	ADV
ejpam-4738	24	2	,	,	PUNCT
ejpam-4738	24	3	the	the	DET
ejpam-4738	24	4	notion	notion	NOUN
ejpam-4738	24	5	of	of	ADP
ejpam-4738	24	6	direct	direct	ADJ
ejpam-4738	24	7	product	product	NOUN
ejpam-4738	24	8	of	of	ADP
ejpam-4738	24	9	0	0	NUM
ejpam-4738	24	10	-	-	PUNCT
ejpam-4738	24	11	commutative	commutative	ADJ
ejpam-4738	24	12	bp	bp	PROPN
ejpam-4738	24	13	-	-	PUNCT
ejpam-4738	24	14	algebra	algebra	PROPN
ejpam-4738	24	15	and	and	CCONJ
ejpam-4738	24	16	bp	bp	NOUN
ejpam-4738	24	17	-	-	PUNCT
ejpam-4738	24	18	homomorphisms	homomorphism	NOUN
ejpam-4738	24	19	were	be	AUX
ejpam-4738	24	20	studied	study	VERB
ejpam-4738	24	21	.	.	PUNCT
ejpam-4738	25	1	in	in	ADP
ejpam-4738	25	2	2022	2022	NUM
ejpam-4738	25	3	,	,	PUNCT
ejpam-4738	25	4	c.	c.	PROPN
ejpam-4738	25	5	chanmanee	chanmanee	PROPN
ejpam-4738	25	6	,	,	PUNCT
ejpam-4738	25	7	r.	r.	PROPN
ejpam-4738	25	8	chinram	chinram	PROPN
ejpam-4738	25	9	,	,	PUNCT
ejpam-4738	25	10	r.	r.	PROPN
ejpam-4738	25	11	prasertpong	prasertpong	PROPN
ejpam-4738	25	12	,	,	PUNCT
ejpam-4738	25	13	p.	p.	PROPN
ejpam-4738	25	14	julatha	julatha	PROPN
ejpam-4738	25	15	,	,	PUNCT
ejpam-4738	25	16	and	and	CCONJ
ejpam-4738	25	17	a.	a.	NOUN
ejpam-4738	25	18	iampan	iampan	NOUN
ejpam-4738	25	19	(	(	PUNCT
ejpam-4738	25	20	[	[	X
ejpam-4738	25	21	2	2	NUM
ejpam-4738	25	22	]	]	PUNCT
ejpam-4738	25	23	)	)	PUNCT
ejpam-4738	25	24	gave	give	VERB
ejpam-4738	25	25	the	the	DET
ejpam-4738	25	26	concept	concept	NOUN
ejpam-4738	25	27	an	an	DET
ejpam-4738	25	28	external	external	ADJ
ejpam-4738	25	29	direct	direct	ADJ
ejpam-4738	25	30	produc	produc	NOUN
ejpam-4738	25	31	and	and	CCONJ
ejpam-4738	25	32	a	a	DET
ejpam-4738	25	33	weak	weak	ADJ
ejpam-4738	25	34	direct	direct	ADJ
ejpam-4738	25	35	product	product	NOUN
ejpam-4738	25	36	of	of	ADP
ejpam-4738	25	37	b	b	NOUN
ejpam-4738	25	38	-	-	PUNCT
ejpam-4738	25	39	algebras	algebras	PROPN
ejpam-4738	25	40	and	and	CCONJ
ejpam-4738	25	41	they	they	PRON
ejpam-4738	25	42	provided	provide	VERB
ejpam-4738	25	43	several	several	ADJ
ejpam-4738	25	44	fundamental	fundamental	ADJ
ejpam-4738	25	45	theorems	theorem	NOUN
ejpam-4738	25	46	of	of	ADP
ejpam-4738	25	47	(	(	PUNCT
ejpam-4738	25	48	anti-)b	anti-)b	ADV
ejpam-4738	25	49	-	-	PUNCT
ejpam-4738	25	50	homomorphisms	homomorphism	NOUN
ejpam-4738	25	51	in	in	ADP
ejpam-4738	25	52	view	view	NOUN
ejpam-4738	25	53	of	of	ADP
ejpam-4738	25	54	the	the	DET
ejpam-4738	25	55	external	external	ADJ
ejpam-4738	25	56	direct	direct	ADJ
ejpam-4738	25	57	product	product	NOUN
ejpam-4738	25	58	b	b	NOUN
ejpam-4738	25	59	-	-	PUNCT
ejpam-4738	25	60	algebras	algebras	X
ejpam-4738	25	61	.	.	PUNCT
ejpam-4738	26	1	in	in	ADP
ejpam-4738	26	2	this	this	DET
ejpam-4738	26	3	paper	paper	NOUN
ejpam-4738	26	4	,	,	PUNCT
ejpam-4738	26	5	we	we	PRON
ejpam-4738	26	6	introduce	introduce	VERB
ejpam-4738	26	7	the	the	DET
ejpam-4738	26	8	concept	concept	NOUN
ejpam-4738	26	9	of	of	ADP
ejpam-4738	26	10	an	an	DET
ejpam-4738	26	11	ideal	ideal	ADJ
ejpam-4738	26	12	direct	direct	ADJ
ejpam-4738	26	13	product	product	NOUN
ejpam-4738	26	14	d	d	NOUN
ejpam-4738	26	15	-	-	PUNCT
ejpam-4738	26	16	algebra	algebra	NOUN
ejpam-4738	26	17	,	,	PUNCT
ejpam-4738	26	18	a	a	DET
ejpam-4738	26	19	d	d	ADJ
ejpam-4738	26	20	-	-	ADJ
ejpam-4738	26	21	ideal	ideal	ADJ
ejpam-4738	26	22	direct	direct	ADJ
ejpam-4738	26	23	product	product	NOUN
ejpam-4738	26	24	d	d	NOUN
ejpam-4738	26	25	-	-	PUNCT
ejpam-4738	26	26	algebra	algebra	ADJ
ejpam-4738	26	27	,	,	PUNCT
ejpam-4738	26	28	sub	sub	ADJ
ejpam-4738	26	29	-	-	ADJ
ejpam-4738	26	30	direct	direct	ADJ
ejpam-4738	26	31	product	product	NOUN
ejpam-4738	26	32	d	d	NOUN
ejpam-4738	26	33	-	-	PUNCT
ejpam-4738	26	34	algebra	algebra	NOUN
ejpam-4738	26	35	,	,	PUNCT
ejpam-4738	26	36	an	an	DET
ejpam-4738	26	37	edge	edge	NOUN
ejpam-4738	26	38	direct	direct	ADJ
ejpam-4738	26	39	product	product	NOUN
ejpam-4738	26	40	and	and	CCONJ
ejpam-4738	26	41	a	a	DET
ejpam-4738	26	42	positive	positive	ADJ
ejpam-4738	26	43	implicative	implicative	ADJ
ejpam-4738	26	44	direct	direct	ADJ
ejpam-4738	26	45	product	product	NOUN
ejpam-4738	26	46	d	d	NOUN
ejpam-4738	26	47	-	-	PUNCT
ejpam-4738	26	48	algebra	algebra	NOUN
ejpam-4738	26	49	.	.	PUNCT
ejpam-4738	27	1	2	2	X
ejpam-4738	27	2	.	.	X
ejpam-4738	27	3	preliminaries	preliminary	NOUN
ejpam-4738	27	4	first	first	ADV
ejpam-4738	27	5	,	,	PUNCT
ejpam-4738	27	6	we	we	PRON
ejpam-4738	27	7	will	will	AUX
ejpam-4738	27	8	review	review	VERB
ejpam-4738	27	9	some	some	DET
ejpam-4738	27	10	essential	essential	ADJ
ejpam-4738	27	11	notations	notation	NOUN
ejpam-4738	27	12	and	and	CCONJ
ejpam-4738	27	13	definitions	definition	NOUN
ejpam-4738	27	14	of	of	ADP
ejpam-4738	27	15	d	d	NOUN
ejpam-4738	27	16	-	-	PUNCT
ejpam-4738	27	17	algebras	algebras	ADJ
ejpam-4738	27	18	and	and	CCONJ
ejpam-4738	27	19	ordinary	ordinary	ADJ
ejpam-4738	27	20	senses	sense	NOUN
ejpam-4738	27	21	that	that	PRON
ejpam-4738	27	22	are	be	AUX
ejpam-4738	27	23	needed	need	VERB
ejpam-4738	27	24	for	for	ADP
ejpam-4738	27	25	this	this	DET
ejpam-4738	27	26	study	study	NOUN
ejpam-4738	27	27	in	in	ADP
ejpam-4738	27	28	this	this	DET
ejpam-4738	27	29	section	section	NOUN
ejpam-4738	27	30	.	.	PUNCT
ejpam-4738	28	1	definition	definition	NOUN
ejpam-4738	28	2	1	1	NUM
ejpam-4738	28	3	.	.	PUNCT
ejpam-4738	29	1	[	[	X
ejpam-4738	29	2	9	9	NUM
ejpam-4738	29	3	]	]	PUNCT
ejpam-4738	29	4	a	a	DET
ejpam-4738	29	5	d	d	X
ejpam-4738	29	6	-	-	PUNCT
ejpam-4738	29	7	algebras	algebras	PROPN
ejpam-4738	29	8	is	be	AUX
ejpam-4738	29	9	a	a	DET
ejpam-4738	29	10	non	non	ADJ
ejpam-4738	29	11	-	-	ADJ
ejpam-4738	29	12	empty	empty	ADJ
ejpam-4738	29	13	set	set	NOUN
ejpam-4738	29	14	x	x	PUNCT
ejpam-4738	29	15	with	with	ADP
ejpam-4738	29	16	a	a	DET
ejpam-4738	29	17	constant	constant	ADJ
ejpam-4738	29	18	0	0	NUM
ejpam-4738	29	19	and	and	CCONJ
ejpam-4738	29	20	a	a	DET
ejpam-4738	29	21	binary	binary	ADJ
ejpam-4738	29	22	operation	operation	NOUN
ejpam-4738	29	23	∗	∗	NOUN
ejpam-4738	29	24	satisfying	satisfy	VERB
ejpam-4738	29	25	the	the	DET
ejpam-4738	29	26	following	follow	VERB
ejpam-4738	29	27	axioms	axiom	NOUN
ejpam-4738	29	28	:	:	PUNCT
ejpam-4738	29	29	(	(	PUNCT
ejpam-4738	29	30	i	i	NOUN
ejpam-4738	29	31	)	)	PUNCT
ejpam-4738	30	1	x	x	SYM
ejpam-4738	30	2	∗	∗	NOUN
ejpam-4738	30	3	x	x	SYM
ejpam-4738	30	4	=	=	SYM
ejpam-4738	30	5	0	0	NUM
ejpam-4738	30	6	,	,	PUNCT
ejpam-4738	30	7	(	(	PUNCT
ejpam-4738	30	8	ii	ii	NOUN
ejpam-4738	30	9	)	)	PUNCT
ejpam-4738	30	10	0	0	NUM
ejpam-4738	30	11	∗	∗	NOUN
ejpam-4738	30	12	x	x	X
ejpam-4738	30	13	=	=	SYM
ejpam-4738	30	14	0	0	NUM
ejpam-4738	30	15	,	,	PUNCT
ejpam-4738	30	16	(	(	PUNCT
ejpam-4738	30	17	iii	iii	NOUN
ejpam-4738	30	18	)	)	PUNCT
ejpam-4738	30	19	x	x	SYM
ejpam-4738	30	20	∗	∗	NOUN
ejpam-4738	30	21	y	y	NOUN
ejpam-4738	30	22	=	=	SYM
ejpam-4738	30	23	0	0	PROPN
ejpam-4738	31	1	and	and	CCONJ
ejpam-4738	31	2	y	y	PROPN
ejpam-4738	31	3	∗	∗	NOUN
ejpam-4738	31	4	x	x	PUNCT
ejpam-4738	32	1	=	=	SYM
ejpam-4738	32	2	0	0	NUM
ejpam-4738	32	3	imply	imply	VERB
ejpam-4738	32	4	x	x	X
ejpam-4738	32	5	=	=	SYM
ejpam-4738	32	6	y	y	PROPN
ejpam-4738	32	7	for	for	ADP
ejpam-4738	32	8	all	all	DET
ejpam-4738	32	9	x	x	NOUN
ejpam-4738	32	10	,	,	PUNCT
ejpam-4738	32	11	y	y	PROPN
ejpam-4738	32	12	∈	∈	PROPN
ejpam-4738	32	13	x.	x.	NOUN
ejpam-4738	32	14	a	a	DET
ejpam-4738	32	15	nonempty	nonempty	NOUN
ejpam-4738	32	16	subset	subset	VERB
ejpam-4738	32	17	s	s	NOUN
ejpam-4738	32	18	of	of	ADP
ejpam-4738	32	19	a	a	DET
ejpam-4738	32	20	d	d	NOUN
ejpam-4738	32	21	-	-	NOUN
ejpam-4738	32	22	algebra	algebra	NOUN
ejpam-4738	32	23	x	x	PUNCT
ejpam-4738	32	24	is	be	AUX
ejpam-4738	32	25	said	say	VERB
ejpam-4738	32	26	to	to	PART
ejpam-4738	32	27	be	be	AUX
ejpam-4738	32	28	a	a	DET
ejpam-4738	32	29	sub	sub	NOUN
ejpam-4738	32	30	-	-	NOUN
ejpam-4738	32	31	algebra	algebra	NOUN
ejpam-4738	32	32	of	of	ADP
ejpam-4738	32	33	x	x	SYM
ejpam-4738	32	34	if	if	SCONJ
ejpam-4738	32	35	x	x	PROPN
ejpam-4738	32	36	∗	∗	VERB
ejpam-4738	32	37	y	y	PROPN
ejpam-4738	32	38	∈	∈	PROPN
ejpam-4738	32	39	s	s	PROPN
ejpam-4738	32	40	for	for	ADP
ejpam-4738	32	41	all	all	DET
ejpam-4738	32	42	x	x	NOUN
ejpam-4738	32	43	,	,	PUNCT
ejpam-4738	32	44	y	y	PROPN
ejpam-4738	32	45	∈	∈	PROPN
ejpam-4738	32	46	s.	s.	PROPN
ejpam-4738	32	47	definition	definition	NOUN
ejpam-4738	32	48	2	2	NUM
ejpam-4738	32	49	.	.	PUNCT
ejpam-4738	33	1	[	[	X
ejpam-4738	33	2	1	1	X
ejpam-4738	33	3	]	]	PUNCT
ejpam-4738	33	4	a	a	DET
ejpam-4738	33	5	d	d	NOUN
ejpam-4738	33	6	-	-	PUNCT
ejpam-4738	33	7	algebras	algebras	X
ejpam-4738	33	8	(	(	PUNCT
ejpam-4738	33	9	x	x	X
ejpam-4738	33	10	,	,	PUNCT
ejpam-4738	33	11	∗	∗	NOUN
ejpam-4738	33	12	,	,	PUNCT
ejpam-4738	33	13	0	0	NUM
ejpam-4738	33	14	)	)	PUNCT
ejpam-4738	33	15	is	be	AUX
ejpam-4738	33	16	said	say	VERB
ejpam-4738	33	17	to	to	PART
ejpam-4738	33	18	be	be	AUX
ejpam-4738	33	19	a	a	DET
ejpam-4738	33	20	positive	positive	ADJ
ejpam-4738	33	21	implicative	implicative	NOUN
ejpam-4738	33	22	if	if	SCONJ
ejpam-4738	33	23	(	(	PUNCT
ejpam-4738	33	24	x	x	PROPN
ejpam-4738	33	25	∗	∗	PROPN
ejpam-4738	33	26	y	y	NOUN
ejpam-4738	33	27	)	)	PUNCT
ejpam-4738	33	28	∗	∗	NOUN
ejpam-4738	33	29	z	z	NOUN
ejpam-4738	33	30	=	=	SYM
ejpam-4738	33	31	(	(	PUNCT
ejpam-4738	33	32	x	x	X
ejpam-4738	33	33	∗	∗	PROPN
ejpam-4738	33	34	z	z	NOUN
ejpam-4738	33	35	)	)	PUNCT
ejpam-4738	33	36	∗	∗	NOUN
ejpam-4738	33	37	(	(	PUNCT
ejpam-4738	33	38	y	y	PROPN
ejpam-4738	33	39	∗	∗	PROPN
ejpam-4738	33	40	z	z	PROPN
ejpam-4738	33	41	)	)	PUNCT
ejpam-4738	33	42	for	for	ADP
ejpam-4738	33	43	all	all	DET
ejpam-4738	33	44	x	x	NOUN
ejpam-4738	33	45	,	,	PUNCT
ejpam-4738	33	46	y	y	PROPN
ejpam-4738	33	47	,	,	PUNCT
ejpam-4738	33	48	z	z	PROPN
ejpam-4738	33	49	∈	∈	PROPN
ejpam-4738	33	50	x.	x.	NOUN
ejpam-4738	33	51	example	example	NOUN
ejpam-4738	34	1	1	1	NUM
ejpam-4738	34	2	.	.	PUNCT
ejpam-4738	35	1	[	[	X
ejpam-4738	35	2	1	1	X
ejpam-4738	35	3	]	]	PUNCT
ejpam-4738	35	4	let	let	VERB
ejpam-4738	35	5	x	x	PUNCT
ejpam-4738	35	6	=	=	PUNCT
ejpam-4738	35	7	{	{	PUNCT
ejpam-4738	35	8	0	0	NUM
ejpam-4738	35	9	,	,	PUNCT
ejpam-4738	35	10	a	a	PRON
ejpam-4738	35	11	,	,	PUNCT
ejpam-4738	35	12	b	b	NOUN
ejpam-4738	35	13	,	,	PUNCT
ejpam-4738	35	14	c	c	AUX
ejpam-4738	35	15	}	}	PUNCT
ejpam-4738	35	16	be	be	AUX
ejpam-4738	35	17	a	a	DET
ejpam-4738	35	18	set	set	NOUN
ejpam-4738	35	19	with	with	ADP
ejpam-4738	35	20	a	a	DET
ejpam-4738	35	21	binary	binary	ADJ
ejpam-4738	35	22	operation	operation	NOUN
ejpam-4738	35	23	∗	∗	NOUN
ejpam-4738	35	24	on	on	ADP
ejpam-4738	35	25	x	x	PUNCT
ejpam-4738	35	26	defined	define	VERB
ejpam-4738	35	27	by	by	ADP
ejpam-4738	35	28	the	the	DET
ejpam-4738	35	29	following	follow	VERB
ejpam-4738	35	30	table	table	NOUN
ejpam-4738	35	31	:	:	PUNCT
ejpam-4738	35	32	∗	∗	NOUN
ejpam-4738	35	33	0	0	PUNCT
ejpam-4738	36	1	a	a	DET
ejpam-4738	36	2	b	b	NOUN
ejpam-4738	36	3	c	c	NOUN
ejpam-4738	36	4	0	0	NUM
ejpam-4738	36	5	0	0	NUM
ejpam-4738	36	6	0	0	NUM
ejpam-4738	36	7	0	0	NUM
ejpam-4738	36	8	0	0	NUM
ejpam-4738	36	9	a	a	DET
ejpam-4738	36	10	a	a	DET
ejpam-4738	36	11	0	0	NUM
ejpam-4738	36	12	a	a	DET
ejpam-4738	36	13	0	0	NUM
ejpam-4738	36	14	b	b	PROPN
ejpam-4738	36	15	b	b	PROPN
ejpam-4738	36	16	b	b	PROPN
ejpam-4738	36	17	0	0	NUM
ejpam-4738	36	18	0	0	NUM
ejpam-4738	36	19	c	c	NOUN
ejpam-4738	36	20	c	c	NOUN
ejpam-4738	36	21	c	c	NOUN
ejpam-4738	36	22	c	c	PROPN
ejpam-4738	36	23	0	0	NUM
ejpam-4738	36	24	m.	m.	NOUN
ejpam-4738	36	25	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	36	26	/	/	SYM
ejpam-4738	36	27	eur	eur	PROPN
ejpam-4738	36	28	.	.	PUNCT
ejpam-4738	37	1	j.	j.	PROPN
ejpam-4738	37	2	pure	pure	PROPN
ejpam-4738	37	3	appl	appl	PROPN
ejpam-4738	37	4	.	.	PROPN
ejpam-4738	37	5	math	math	PROPN
ejpam-4738	37	6	,	,	PUNCT
ejpam-4738	37	7	16	16	NUM
ejpam-4738	37	8	(	(	PUNCT
ejpam-4738	37	9	2	2	NUM
ejpam-4738	37	10	)	)	PUNCT
ejpam-4738	37	11	(	(	PUNCT
ejpam-4738	37	12	2023	2023	NUM
ejpam-4738	37	13	)	)	PUNCT
ejpam-4738	37	14	,	,	PUNCT
ejpam-4738	37	15	997	997	NUM
ejpam-4738	37	16	-	-	SYM
ejpam-4738	37	17	1004	1004	NUM
ejpam-4738	37	18	999	999	NUM
ejpam-4738	37	19	then	then	ADV
ejpam-4738	37	20	(	(	PUNCT
ejpam-4738	37	21	x	x	X
ejpam-4738	37	22	,	,	PUNCT
ejpam-4738	37	23	∗	∗	NOUN
ejpam-4738	37	24	,	,	PUNCT
ejpam-4738	37	25	0	0	NUM
ejpam-4738	37	26	)	)	PUNCT
ejpam-4738	37	27	is	be	AUX
ejpam-4738	37	28	a	a	DET
ejpam-4738	37	29	positive	positive	ADJ
ejpam-4738	37	30	implicative	implicative	ADJ
ejpam-4738	37	31	d	d	NOUN
ejpam-4738	37	32	-	-	NOUN
ejpam-4738	37	33	algebra	algebra	NOUN
ejpam-4738	37	34	.	.	PUNCT
ejpam-4738	37	35	example	example	NOUN
ejpam-4738	38	1	2	2	NUM
ejpam-4738	38	2	.	.	PUNCT
ejpam-4738	39	1	[	[	X
ejpam-4738	39	2	5	5	X
ejpam-4738	39	3	]	]	PUNCT
ejpam-4738	39	4	let	let	VERB
ejpam-4738	39	5	x	x	PUNCT
ejpam-4738	39	6	=	=	PUNCT
ejpam-4738	39	7	{	{	PUNCT
ejpam-4738	39	8	0	0	NUM
ejpam-4738	39	9	,	,	PUNCT
ejpam-4738	39	10	a	a	DET
ejpam-4738	39	11	,	,	PUNCT
ejpam-4738	39	12	b	b	NOUN
ejpam-4738	39	13	,	,	PUNCT
ejpam-4738	39	14	c	c	AUX
ejpam-4738	39	15	}	}	PUNCT
ejpam-4738	39	16	be	be	AUX
ejpam-4738	39	17	a	a	DET
ejpam-4738	39	18	set	set	NOUN
ejpam-4738	39	19	with	with	ADP
ejpam-4738	39	20	a	a	DET
ejpam-4738	39	21	binary	binary	ADJ
ejpam-4738	39	22	operation	operation	NOUN
ejpam-4738	39	23	∗	∗	NOUN
ejpam-4738	39	24	on	on	ADP
ejpam-4738	39	25	x	x	PUNCT
ejpam-4738	39	26	defined	define	VERB
ejpam-4738	39	27	by	by	ADP
ejpam-4738	39	28	the	the	DET
ejpam-4738	39	29	following	follow	VERB
ejpam-4738	39	30	table	table	NOUN
ejpam-4738	39	31	:	:	PUNCT
ejpam-4738	39	32	∗	∗	NOUN
ejpam-4738	39	33	0	0	PUNCT
ejpam-4738	40	1	a	a	DET
ejpam-4738	40	2	b	b	NOUN
ejpam-4738	40	3	c	c	NOUN
ejpam-4738	40	4	0	0	NUM
ejpam-4738	40	5	0	0	NUM
ejpam-4738	40	6	0	0	NUM
ejpam-4738	40	7	0	0	NUM
ejpam-4738	40	8	0	0	NUM
ejpam-4738	40	9	a	a	DET
ejpam-4738	40	10	a	a	DET
ejpam-4738	40	11	0	0	NUM
ejpam-4738	40	12	0	0	NUM
ejpam-4738	40	13	b	b	PROPN
ejpam-4738	40	14	b	b	PROPN
ejpam-4738	40	15	b	b	PROPN
ejpam-4738	40	16	b	b	PROPN
ejpam-4738	40	17	0	0	NUM
ejpam-4738	40	18	0	0	NUM
ejpam-4738	40	19	c	c	NOUN
ejpam-4738	40	20	c	c	NOUN
ejpam-4738	40	21	c	c	NOUN
ejpam-4738	40	22	c	c	NOUN
ejpam-4738	40	23	0	0	PUNCT
ejpam-4738	41	1	then	then	ADV
ejpam-4738	41	2	(	(	PUNCT
ejpam-4738	41	3	x	x	X
ejpam-4738	41	4	,	,	PUNCT
ejpam-4738	41	5	∗	∗	NOUN
ejpam-4738	41	6	,	,	PUNCT
ejpam-4738	41	7	0	0	NUM
ejpam-4738	41	8	)	)	PUNCT
ejpam-4738	41	9	is	be	AUX
ejpam-4738	41	10	a	a	DET
ejpam-4738	41	11	d	d	NOUN
ejpam-4738	41	12	-	-	PUNCT
ejpam-4738	41	13	algebra	algebra	NOUN
ejpam-4738	41	14	but	but	CCONJ
ejpam-4738	41	15	not	not	PART
ejpam-4738	41	16	positive	positive	ADJ
ejpam-4738	41	17	implicative	implicative	NOUN
ejpam-4738	41	18	because	because	SCONJ
ejpam-4738	41	19	(	(	PUNCT
ejpam-4738	41	20	a∗b)∗c	a∗b)∗c	X
ejpam-4738	41	21	=	=	SYM
ejpam-4738	41	22	0∗c	0∗c	NUM
ejpam-4738	41	23	=	=	SYM
ejpam-4738	41	24	0	0	NUM
ejpam-4738	42	1	̸=	̸=	PROPN
ejpam-4738	42	2	b	b	PROPN
ejpam-4738	42	3	=	=	SYM
ejpam-4738	42	4	b	b	PROPN
ejpam-4738	42	5	∗	∗	NOUN
ejpam-4738	42	6	0	0	NUM
ejpam-4738	43	1	=	=	SYM
ejpam-4738	43	2	(	(	PUNCT
ejpam-4738	43	3	a	a	DET
ejpam-4738	43	4	∗	∗	NOUN
ejpam-4738	43	5	c	c	NOUN
ejpam-4738	43	6	)	)	PUNCT
ejpam-4738	43	7	∗	∗	NOUN
ejpam-4738	43	8	(	(	PUNCT
ejpam-4738	43	9	b	b	NOUN
ejpam-4738	43	10	∗	∗	NOUN
ejpam-4738	43	11	c	c	NOUN
ejpam-4738	43	12	)	)	PUNCT
ejpam-4738	43	13	.	.	PUNCT
ejpam-4738	44	1	the	the	DET
ejpam-4738	44	2	set	set	NOUN
ejpam-4738	44	3	s1	s1	NOUN
ejpam-4738	44	4	=	=	PUNCT
ejpam-4738	44	5	{	{	PUNCT
ejpam-4738	44	6	b	b	PROPN
ejpam-4738	44	7	,	,	PUNCT
ejpam-4738	44	8	c	c	NOUN
ejpam-4738	44	9	}	}	PUNCT
ejpam-4738	44	10	is	be	AUX
ejpam-4738	44	11	not	not	PART
ejpam-4738	44	12	a	a	DET
ejpam-4738	44	13	sub	sub	NOUN
ejpam-4738	44	14	-	-	NOUN
ejpam-4738	44	15	algebra	algebra	NOUN
ejpam-4738	44	16	of	of	ADP
ejpam-4738	44	17	x	x	SYM
ejpam-4738	44	18	whereas	whereas	SCONJ
ejpam-4738	44	19	s2	s2	VERB
ejpam-4738	44	20	=	=	PUNCT
ejpam-4738	44	21	{	{	PUNCT
ejpam-4738	44	22	0	0	NUM
ejpam-4738	44	23	,	,	PUNCT
ejpam-4738	44	24	a	a	DET
ejpam-4738	44	25	,	,	PUNCT
ejpam-4738	44	26	b	b	NOUN
ejpam-4738	44	27	}	}	PUNCT
ejpam-4738	44	28	is	be	AUX
ejpam-4738	44	29	a	a	DET
ejpam-4738	44	30	sub	sub	NOUN
ejpam-4738	44	31	-	-	NOUN
ejpam-4738	44	32	algebra	algebra	NOUN
ejpam-4738	44	33	of	of	ADP
ejpam-4738	44	34	x.	x.	NOUN
ejpam-4738	44	35	definition	definition	NOUN
ejpam-4738	44	36	3	3	NUM
ejpam-4738	44	37	.	.	PUNCT
ejpam-4738	45	1	[	[	X
ejpam-4738	45	2	7	7	X
ejpam-4738	45	3	]	]	X
ejpam-4738	45	4	let	let	VERB
ejpam-4738	45	5	(	(	PUNCT
ejpam-4738	45	6	x	x	X
ejpam-4738	45	7	,	,	PUNCT
ejpam-4738	45	8	∗	∗	NOUN
ejpam-4738	45	9	,	,	PUNCT
ejpam-4738	45	10	0	0	NUM
ejpam-4738	45	11	)	)	PUNCT
ejpam-4738	45	12	be	be	AUX
ejpam-4738	45	13	a	a	DET
ejpam-4738	45	14	d	d	NOUN
ejpam-4738	45	15	-	-	PUNCT
ejpam-4738	45	16	algebra	algebra	NOUN
ejpam-4738	45	17	and	and	CCONJ
ejpam-4738	45	18	x	x	NOUN
ejpam-4738	45	19	∈	∈	PROPN
ejpam-4738	45	20	x	x	X
ejpam-4738	45	21	.	.	PUNCT
ejpam-4738	46	1	define	define	VERB
ejpam-4738	46	2	x	x	NOUN
ejpam-4738	46	3	∗x	∗x	NOUN
ejpam-4738	46	4	:	:	PUNCT
ejpam-4738	46	5	=	=	NUM
ejpam-4738	46	6	{	{	PUNCT
ejpam-4738	46	7	x	x	X
ejpam-4738	46	8	∗	∗	NOUN
ejpam-4738	46	9	a	a	DET
ejpam-4738	46	10	|a	|a	NOUN
ejpam-4738	46	11	∈	∈	NOUN
ejpam-4738	46	12	x	x	PRON
ejpam-4738	46	13	}	}	PUNCT
ejpam-4738	46	14	.	.	PUNCT
ejpam-4738	47	1	we	we	PRON
ejpam-4738	47	2	say	say	VERB
ejpam-4738	47	3	that	that	SCONJ
ejpam-4738	47	4	x	x	PRON
ejpam-4738	47	5	is	be	AUX
ejpam-4738	47	6	edge	edge	NOUN
ejpam-4738	47	7	if	if	SCONJ
ejpam-4738	47	8	x	x	VERB
ejpam-4738	47	9	∗x	∗x	NOUN
ejpam-4738	47	10	=	=	SYM
ejpam-4738	47	11	{	{	PUNCT
ejpam-4738	47	12	x	x	NOUN
ejpam-4738	47	13	,	,	PUNCT
ejpam-4738	47	14	0	0	NUM
ejpam-4738	47	15	}	}	PUNCT
ejpam-4738	47	16	for	for	ADP
ejpam-4738	47	17	all	all	DET
ejpam-4738	47	18	x	x	SYM
ejpam-4738	47	19	∈	∈	PROPN
ejpam-4738	47	20	x.	x.	NOUN
ejpam-4738	47	21	example	example	NOUN
ejpam-4738	48	1	3	3	NUM
ejpam-4738	48	2	.	.	PUNCT
ejpam-4738	49	1	[	[	X
ejpam-4738	49	2	7	7	X
ejpam-4738	49	3	]	]	X
ejpam-4738	49	4	let	let	NOUN
ejpam-4738	49	5	x	x	PUNCT
ejpam-4738	49	6	=	=	PUNCT
ejpam-4738	49	7	{	{	PUNCT
ejpam-4738	49	8	0	0	NUM
ejpam-4738	49	9	,	,	PUNCT
ejpam-4738	49	10	1	1	NUM
ejpam-4738	49	11	,	,	PUNCT
ejpam-4738	49	12	2	2	NUM
ejpam-4738	49	13	,	,	PUNCT
ejpam-4738	49	14	3	3	NUM
ejpam-4738	49	15	}	}	PUNCT
ejpam-4738	49	16	be	be	AUX
ejpam-4738	49	17	a	a	DET
ejpam-4738	49	18	set	set	NOUN
ejpam-4738	49	19	with	with	ADP
ejpam-4738	49	20	the	the	DET
ejpam-4738	49	21	binary	binary	PROPN
ejpam-4738	49	22	operation	operation	NOUN
ejpam-4738	49	23	∗	∗	NOUN
ejpam-4738	49	24	on	on	ADP
ejpam-4738	49	25	x	x	PUNCT
ejpam-4738	49	26	defined	define	VERB
ejpam-4738	49	27	by	by	ADP
ejpam-4738	49	28	the	the	DET
ejpam-4738	49	29	following	follow	VERB
ejpam-4738	49	30	table	table	NOUN
ejpam-4738	49	31	:	:	PUNCT
ejpam-4738	49	32	∗	∗	NOUN
ejpam-4738	49	33	0	0	NUM
ejpam-4738	50	1	1	1	NUM
ejpam-4738	50	2	2	2	NUM
ejpam-4738	50	3	3	3	NUM
ejpam-4738	50	4	0	0	NUM
ejpam-4738	50	5	0	0	NUM
ejpam-4738	50	6	0	0	NUM
ejpam-4738	50	7	0	0	NUM
ejpam-4738	50	8	0	0	NUM
ejpam-4738	50	9	1	1	NUM
ejpam-4738	50	10	1	1	NUM
ejpam-4738	50	11	0	0	NUM
ejpam-4738	50	12	0	0	NUM
ejpam-4738	50	13	1	1	NUM
ejpam-4738	50	14	2	2	NUM
ejpam-4738	50	15	2	2	NUM
ejpam-4738	50	16	2	2	NUM
ejpam-4738	50	17	0	0	NUM
ejpam-4738	50	18	0	0	NUM
ejpam-4738	50	19	3	3	NUM
ejpam-4738	50	20	3	3	NUM
ejpam-4738	50	21	3	3	NUM
ejpam-4738	50	22	3	3	NUM
ejpam-4738	50	23	0	0	NUM
ejpam-4738	50	24	then	then	ADV
ejpam-4738	50	25	(	(	PUNCT
ejpam-4738	50	26	x	x	X
ejpam-4738	50	27	,	,	PUNCT
ejpam-4738	50	28	∗	∗	NOUN
ejpam-4738	50	29	,	,	PUNCT
ejpam-4738	50	30	0	0	NUM
ejpam-4738	50	31	)	)	PUNCT
ejpam-4738	50	32	is	be	AUX
ejpam-4738	50	33	an	an	DET
ejpam-4738	50	34	edge	edge	NOUN
ejpam-4738	50	35	d	d	NOUN
ejpam-4738	50	36	-	-	NOUN
ejpam-4738	50	37	algebra	algebra	NOUN
ejpam-4738	50	38	.	.	PUNCT
ejpam-4738	50	39	example	example	NOUN
ejpam-4738	51	1	4	4	NUM
ejpam-4738	51	2	.	.	PUNCT
ejpam-4738	52	1	[	[	X
ejpam-4738	52	2	4	4	X
ejpam-4738	52	3	]	]	PUNCT
ejpam-4738	52	4	let	let	VERB
ejpam-4738	52	5	x	x	PUNCT
ejpam-4738	52	6	=	=	PUNCT
ejpam-4738	52	7	{	{	PUNCT
ejpam-4738	52	8	0	0	NUM
ejpam-4738	52	9	,	,	PUNCT
ejpam-4738	52	10	1	1	NUM
ejpam-4738	52	11	,	,	PUNCT
ejpam-4738	52	12	2	2	NUM
ejpam-4738	52	13	,	,	PUNCT
ejpam-4738	52	14	3	3	NUM
ejpam-4738	52	15	}	}	PUNCT
ejpam-4738	52	16	be	be	AUX
ejpam-4738	52	17	a	a	DET
ejpam-4738	52	18	set	set	NOUN
ejpam-4738	52	19	with	with	ADP
ejpam-4738	52	20	the	the	DET
ejpam-4738	52	21	following	follow	VERB
ejpam-4738	52	22	table	table	NOUN
ejpam-4738	52	23	:	:	PUNCT
ejpam-4738	52	24	∗	∗	NOUN
ejpam-4738	52	25	0	0	NUM
ejpam-4738	53	1	1	1	NUM
ejpam-4738	53	2	2	2	NUM
ejpam-4738	53	3	3	3	NUM
ejpam-4738	53	4	0	0	NUM
ejpam-4738	53	5	0	0	NUM
ejpam-4738	53	6	0	0	NUM
ejpam-4738	53	7	0	0	NUM
ejpam-4738	53	8	0	0	NUM
ejpam-4738	53	9	1	1	NUM
ejpam-4738	53	10	1	1	NUM
ejpam-4738	53	11	0	0	NUM
ejpam-4738	53	12	1	1	NUM
ejpam-4738	53	13	0	0	NUM
ejpam-4738	53	14	2	2	NUM
ejpam-4738	53	15	2	2	NUM
ejpam-4738	53	16	2	2	NUM
ejpam-4738	53	17	0	0	NUM
ejpam-4738	53	18	0	0	NUM
ejpam-4738	53	19	3	3	NUM
ejpam-4738	53	20	3	3	NUM
ejpam-4738	53	21	3	3	NUM
ejpam-4738	53	22	1	1	NUM
ejpam-4738	53	23	0	0	NUM
ejpam-4738	53	24	since	since	SCONJ
ejpam-4738	53	25	3	3	NUM
ejpam-4738	53	26	∗x	∗x	NOUN
ejpam-4738	53	27	=	=	PUNCT
ejpam-4738	53	28	{	{	PUNCT
ejpam-4738	53	29	3	3	NUM
ejpam-4738	53	30	,	,	PUNCT
ejpam-4738	53	31	1	1	NUM
ejpam-4738	53	32	,	,	PUNCT
ejpam-4738	53	33	0	0	NUM
ejpam-4738	53	34	}	}	PUNCT
ejpam-4738	53	35	=	=	NOUN
ejpam-4738	53	36	̸	̸	NUM
ejpam-4738	53	37	{	{	PUNCT
ejpam-4738	53	38	3	3	NUM
ejpam-4738	53	39	,	,	PUNCT
ejpam-4738	53	40	0	0	NUM
ejpam-4738	53	41	}	}	PUNCT
ejpam-4738	53	42	,	,	PUNCT
ejpam-4738	53	43	then	then	ADV
ejpam-4738	53	44	(	(	PUNCT
ejpam-4738	53	45	x	x	X
ejpam-4738	53	46	,	,	PUNCT
ejpam-4738	53	47	∗	∗	NOUN
ejpam-4738	53	48	,	,	PUNCT
ejpam-4738	53	49	0	0	NUM
ejpam-4738	53	50	)	)	PUNCT
ejpam-4738	53	51	is	be	AUX
ejpam-4738	53	52	not	not	PART
ejpam-4738	53	53	an	an	DET
ejpam-4738	53	54	edge	edge	NOUN
ejpam-4738	53	55	d	d	NOUN
ejpam-4738	53	56	-	-	NOUN
ejpam-4738	53	57	algebra	algebra	NOUN
ejpam-4738	53	58	.	.	PUNCT
ejpam-4738	54	1	m.	m.	NOUN
ejpam-4738	54	2	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	54	3	/	/	SYM
ejpam-4738	54	4	eur	eur	PROPN
ejpam-4738	54	5	.	.	PUNCT
ejpam-4738	55	1	j.	j.	PROPN
ejpam-4738	55	2	pure	pure	PROPN
ejpam-4738	55	3	appl	appl	PROPN
ejpam-4738	55	4	.	.	PROPN
ejpam-4738	55	5	math	math	PROPN
ejpam-4738	55	6	,	,	PUNCT
ejpam-4738	55	7	16	16	NUM
ejpam-4738	55	8	(	(	PUNCT
ejpam-4738	55	9	2	2	NUM
ejpam-4738	55	10	)	)	PUNCT
ejpam-4738	55	11	(	(	PUNCT
ejpam-4738	55	12	2023	2023	NUM
ejpam-4738	55	13	)	)	PUNCT
ejpam-4738	55	14	,	,	PUNCT
ejpam-4738	55	15	997	997	NUM
ejpam-4738	55	16	-	-	SYM
ejpam-4738	55	17	1004	1004	NUM
ejpam-4738	55	18	1000	1000	NUM
ejpam-4738	55	19	theorem	theorem	NOUN
ejpam-4738	55	20	1	1	NUM
ejpam-4738	55	21	.	.	PUNCT
ejpam-4738	56	1	[	[	X
ejpam-4738	56	2	7	7	X
ejpam-4738	56	3	]	]	X
ejpam-4738	56	4	let	let	VERB
ejpam-4738	56	5	(	(	PUNCT
ejpam-4738	56	6	x	x	X
ejpam-4738	56	7	,	,	PUNCT
ejpam-4738	56	8	∗	∗	NOUN
ejpam-4738	56	9	,	,	PUNCT
ejpam-4738	56	10	0	0	NUM
ejpam-4738	56	11	)	)	PUNCT
ejpam-4738	56	12	be	be	AUX
ejpam-4738	56	13	an	an	DET
ejpam-4738	56	14	edge	edge	NOUN
ejpam-4738	56	15	d	d	NOUN
ejpam-4738	56	16	-	-	NOUN
ejpam-4738	56	17	algebra	algebra	NOUN
ejpam-4738	56	18	.	.	PUNCT
ejpam-4738	57	1	then	then	ADV
ejpam-4738	57	2	the	the	DET
ejpam-4738	57	3	following	follow	VERB
ejpam-4738	57	4	conditions	condition	NOUN
ejpam-4738	57	5	are	be	AUX
ejpam-4738	57	6	satisfiesd	satisfiesd	NOUN
ejpam-4738	57	7	:	:	PUNCT
ejpam-4738	57	8	(	(	PUNCT
ejpam-4738	57	9	i	i	NOUN
ejpam-4738	57	10	)	)	PUNCT
ejpam-4738	57	11	x	x	SYM
ejpam-4738	58	1	∗	∗	NOUN
ejpam-4738	58	2	0	0	NUM
ejpam-4738	59	1	=	=	SYM
ejpam-4738	59	2	x	x	NOUN
ejpam-4738	59	3	,	,	PUNCT
ejpam-4738	59	4	(	(	PUNCT
ejpam-4738	59	5	ii	ii	NOUN
ejpam-4738	59	6	)	)	PUNCT
ejpam-4738	59	7	(	(	PUNCT
ejpam-4738	59	8	x	x	SYM
ejpam-4738	59	9	∗	∗	PROPN
ejpam-4738	59	10	y	y	NOUN
ejpam-4738	59	11	)	)	PUNCT
ejpam-4738	59	12	∗	∗	NOUN
ejpam-4738	59	13	z	z	NOUN
ejpam-4738	59	14	=	=	SYM
ejpam-4738	59	15	(	(	PUNCT
ejpam-4738	59	16	x	x	X
ejpam-4738	59	17	∗	∗	PROPN
ejpam-4738	59	18	z	z	NOUN
ejpam-4738	59	19	)	)	PUNCT
ejpam-4738	59	20	∗	∗	PROPN
ejpam-4738	59	21	y	y	PROPN
ejpam-4738	59	22	,	,	PUNCT
ejpam-4738	59	23	(	(	PUNCT
ejpam-4738	59	24	iii	iii	NOUN
ejpam-4738	59	25	)	)	PUNCT
ejpam-4738	59	26	x	x	SYM
ejpam-4738	59	27	∗	∗	NOUN
ejpam-4738	59	28	(	(	PUNCT
ejpam-4738	59	29	x	x	X
ejpam-4738	59	30	∗	∗	NOUN
ejpam-4738	59	31	y	y	NOUN
ejpam-4738	59	32	)	)	PUNCT
ejpam-4738	60	1	=	=	SYM
ejpam-4738	60	2	y	y	PROPN
ejpam-4738	60	3	,	,	PUNCT
ejpam-4738	60	4	for	for	ADP
ejpam-4738	60	5	any	any	DET
ejpam-4738	60	6	x	x	NOUN
ejpam-4738	60	7	,	,	PUNCT
ejpam-4738	60	8	y	y	PROPN
ejpam-4738	60	9	,	,	PUNCT
ejpam-4738	60	10	z	z	PROPN
ejpam-4738	60	11	∈	∈	PROPN
ejpam-4738	60	12	x.	x.	NOUN
ejpam-4738	60	13	definition	definition	NOUN
ejpam-4738	60	14	4	4	NUM
ejpam-4738	60	15	.	.	PUNCT
ejpam-4738	61	1	[	[	X
ejpam-4738	61	2	3	3	X
ejpam-4738	61	3	]	]	X
ejpam-4738	61	4	let	let	VERB
ejpam-4738	61	5	(	(	PUNCT
ejpam-4738	61	6	x	x	NOUN
ejpam-4738	61	7	,	,	PUNCT
ejpam-4738	61	8	∗	∗	NOUN
ejpam-4738	61	9	,	,	PUNCT
ejpam-4738	61	10	0	0	NUM
ejpam-4738	61	11	)	)	PUNCT
ejpam-4738	61	12	be	be	AUX
ejpam-4738	61	13	a	a	DET
ejpam-4738	61	14	d	d	NOUN
ejpam-4738	61	15	-	-	PUNCT
ejpam-4738	61	16	algebra	algebra	NOUN
ejpam-4738	61	17	and	and	CCONJ
ejpam-4738	61	18	i	i	PRON
ejpam-4738	61	19	a	a	DET
ejpam-4738	61	20	subset	subset	NOUN
ejpam-4738	61	21	of	of	ADP
ejpam-4738	61	22	x	x	PRON
ejpam-4738	61	23	,	,	PUNCT
ejpam-4738	61	24	then	then	ADV
ejpam-4738	61	25	i	i	PRON
ejpam-4738	61	26	is	be	AUX
ejpam-4738	61	27	called	call	VERB
ejpam-4738	61	28	an	an	DET
ejpam-4738	61	29	ideal	ideal	NOUN
ejpam-4738	61	30	of	of	ADP
ejpam-4738	61	31	x	x	PRON
ejpam-4738	61	32	if	if	SCONJ
ejpam-4738	61	33	it	it	PRON
ejpam-4738	61	34	satisfies	satisfy	VERB
ejpam-4738	61	35	the	the	DET
ejpam-4738	61	36	following	follow	VERB
ejpam-4738	61	37	conditions	condition	NOUN
ejpam-4738	61	38	:	:	PUNCT
ejpam-4738	61	39	(	(	PUNCT
ejpam-4738	61	40	i	i	NOUN
ejpam-4738	61	41	)	)	PUNCT
ejpam-4738	61	42	0	0	PUNCT
ejpam-4738	62	1	∈	∈	PROPN
ejpam-4738	63	1	i	i	PRON
ejpam-4738	63	2	,	,	PUNCT
ejpam-4738	63	3	(	(	PUNCT
ejpam-4738	63	4	ii	ii	NOUN
ejpam-4738	63	5	)	)	PUNCT
ejpam-4738	63	6	if	if	SCONJ
ejpam-4738	63	7	x	x	PROPN
ejpam-4738	63	8	∗	∗	VERB
ejpam-4738	63	9	y	y	NOUN
ejpam-4738	63	10	∈	∈	PROPN
ejpam-4738	64	1	i	i	PRON
ejpam-4738	64	2	and	and	CCONJ
ejpam-4738	64	3	y	y	PROPN
ejpam-4738	64	4	∈	∈	PROPN
ejpam-4738	65	1	i	i	PRON
ejpam-4738	65	2	imply	imply	VERB
ejpam-4738	65	3	x	x	X
ejpam-4738	65	4	∈	∈	PROPN
ejpam-4738	65	5	i.	i.	NOUN
ejpam-4738	65	6	definition	definition	NOUN
ejpam-4738	65	7	5	5	NUM
ejpam-4738	65	8	.	.	PUNCT
ejpam-4738	66	1	[	[	X
ejpam-4738	66	2	3	3	X
ejpam-4738	66	3	]	]	X
ejpam-4738	66	4	let	let	VERB
ejpam-4738	66	5	(	(	PUNCT
ejpam-4738	66	6	x	x	NOUN
ejpam-4738	66	7	,	,	PUNCT
ejpam-4738	66	8	∗	∗	NOUN
ejpam-4738	66	9	,	,	PUNCT
ejpam-4738	66	10	0	0	NUM
ejpam-4738	66	11	)	)	PUNCT
ejpam-4738	66	12	be	be	AUX
ejpam-4738	66	13	a	a	DET
ejpam-4738	66	14	d	d	NOUN
ejpam-4738	66	15	-	-	PUNCT
ejpam-4738	66	16	algebra	algebra	NOUN
ejpam-4738	66	17	and	and	CCONJ
ejpam-4738	66	18	i	i	PRON
ejpam-4738	66	19	a	a	DET
ejpam-4738	66	20	nonempty	nonempty	NOUN
ejpam-4738	66	21	subset	subset	NOUN
ejpam-4738	66	22	of	of	ADP
ejpam-4738	66	23	x	x	PRON
ejpam-4738	66	24	,	,	PUNCT
ejpam-4738	66	25	then	then	ADV
ejpam-4738	66	26	i	i	PRON
ejpam-4738	66	27	is	be	AUX
ejpam-4738	66	28	called	call	VERB
ejpam-4738	66	29	a	a	DET
ejpam-4738	66	30	d	d	NOUN
ejpam-4738	66	31	-	-	NOUN
ejpam-4738	66	32	ideal	ideal	NOUN
ejpam-4738	66	33	of	of	ADP
ejpam-4738	66	34	x	x	PRON
ejpam-4738	66	35	if	if	SCONJ
ejpam-4738	66	36	it	it	PRON
ejpam-4738	66	37	satisfies	satisfy	VERB
ejpam-4738	66	38	the	the	DET
ejpam-4738	66	39	following	follow	VERB
ejpam-4738	66	40	conditions	condition	NOUN
ejpam-4738	66	41	:	:	PUNCT
ejpam-4738	66	42	(	(	PUNCT
ejpam-4738	66	43	i	i	NOUN
ejpam-4738	66	44	)	)	PUNCT
ejpam-4738	66	45	if	if	SCONJ
ejpam-4738	66	46	x	x	PROPN
ejpam-4738	66	47	∗	∗	VERB
ejpam-4738	66	48	y	y	NOUN
ejpam-4738	66	49	∈	∈	PROPN
ejpam-4738	67	1	i	i	PRON
ejpam-4738	67	2	and	and	CCONJ
ejpam-4738	67	3	y	y	PROPN
ejpam-4738	67	4	∈	∈	PROPN
ejpam-4738	68	1	i	i	PRON
ejpam-4738	68	2	imply	imply	VERB
ejpam-4738	68	3	x	x	X
ejpam-4738	68	4	∈	∈	PROPN
ejpam-4738	69	1	i	i	PRON
ejpam-4738	69	2	,	,	PUNCT
ejpam-4738	69	3	(	(	PUNCT
ejpam-4738	69	4	ii	ii	NOUN
ejpam-4738	69	5	)	)	PUNCT
ejpam-4738	69	6	if	if	SCONJ
ejpam-4738	69	7	x	x	SYM
ejpam-4738	69	8	∈	∈	PROPN
ejpam-4738	69	9	i	i	PRON
ejpam-4738	69	10	and	and	CCONJ
ejpam-4738	69	11	y	y	PROPN
ejpam-4738	69	12	∈	∈	PROPN
ejpam-4738	69	13	x	x	X
ejpam-4738	69	14	imply	imply	VERB
ejpam-4738	69	15	x	x	PUNCT
ejpam-4738	69	16	∗	∗	NOUN
ejpam-4738	69	17	y	y	PROPN
ejpam-4738	69	18	∈	∈	PROPN
ejpam-4738	69	19	i.	i.	PROPN
ejpam-4738	69	20	clealy	clealy	PROPN
ejpam-4738	69	21	,	,	PUNCT
ejpam-4738	69	22	if	if	SCONJ
ejpam-4738	69	23	i	i	PRON
ejpam-4738	69	24	is	be	AUX
ejpam-4738	69	25	a	a	DET
ejpam-4738	69	26	d	d	NOUN
ejpam-4738	69	27	-	-	NOUN
ejpam-4738	69	28	ideal	ideal	NOUN
ejpam-4738	69	29	of	of	ADP
ejpam-4738	69	30	a	a	DET
ejpam-4738	69	31	d	d	NOUN
ejpam-4738	69	32	-	-	NOUN
ejpam-4738	69	33	algebra	algebra	NOUN
ejpam-4738	69	34	x	x	NOUN
ejpam-4738	69	35	,	,	PUNCT
ejpam-4738	69	36	then	then	ADV
ejpam-4738	69	37	x	x	X
ejpam-4738	69	38	∗	∗	NOUN
ejpam-4738	69	39	x	x	X
ejpam-4738	69	40	=	=	SYM
ejpam-4738	69	41	0	0	NUM
ejpam-4738	69	42	∈	∈	PROPN
ejpam-4738	69	43	i	i	PRON
ejpam-4738	69	44	for	for	ADP
ejpam-4738	69	45	any	any	DET
ejpam-4738	69	46	x	x	SYM
ejpam-4738	69	47	∈	∈	PROPN
ejpam-4738	70	1	i	i	PRON
ejpam-4738	70	2	and	and	CCONJ
ejpam-4738	70	3	then	then	ADV
ejpam-4738	70	4	i	i	PRON
ejpam-4738	70	5	is	be	AUX
ejpam-4738	70	6	an	an	DET
ejpam-4738	70	7	ideal	ideal	NOUN
ejpam-4738	70	8	of	of	ADP
ejpam-4738	70	9	x	x	PRON
ejpam-4738	70	10	,	,	PUNCT
ejpam-4738	70	11	but	but	CCONJ
ejpam-4738	70	12	the	the	DET
ejpam-4738	70	13	converse	converse	NOUN
ejpam-4738	70	14	need	need	AUX
ejpam-4738	70	15	not	not	PART
ejpam-4738	70	16	be	be	AUX
ejpam-4738	70	17	true	true	ADJ
ejpam-4738	70	18	as	as	ADP
ejpam-4738	70	19	the	the	DET
ejpam-4738	70	20	following	follow	VERB
ejpam-4738	70	21	example	example	NOUN
ejpam-4738	70	22	:	:	PUNCT
ejpam-4738	70	23	example	example	NOUN
ejpam-4738	70	24	5	5	NUM
ejpam-4738	70	25	.	.	PUNCT
ejpam-4738	71	1	[	[	X
ejpam-4738	71	2	9	9	NUM
ejpam-4738	71	3	]	]	PUNCT
ejpam-4738	71	4	let	let	VERB
ejpam-4738	71	5	x	x	PUNCT
ejpam-4738	71	6	=	=	PUNCT
ejpam-4738	71	7	{	{	PUNCT
ejpam-4738	71	8	0	0	NUM
ejpam-4738	71	9	,	,	PUNCT
ejpam-4738	71	10	a	a	DET
ejpam-4738	71	11	,	,	PUNCT
ejpam-4738	71	12	b	b	NOUN
ejpam-4738	71	13	,	,	PUNCT
ejpam-4738	71	14	c	c	AUX
ejpam-4738	71	15	}	}	PUNCT
ejpam-4738	71	16	be	be	AUX
ejpam-4738	71	17	a	a	DET
ejpam-4738	71	18	set	set	NOUN
ejpam-4738	71	19	with	with	ADP
ejpam-4738	71	20	binary	binary	ADJ
ejpam-4738	71	21	operation	operation	NOUN
ejpam-4738	71	22	∗	∗	NOUN
ejpam-4738	71	23	on	on	ADP
ejpam-4738	71	24	x	x	PUNCT
ejpam-4738	71	25	defined	define	VERB
ejpam-4738	71	26	by	by	ADP
ejpam-4738	71	27	the	the	DET
ejpam-4738	71	28	following	follow	VERB
ejpam-4738	71	29	table	table	NOUN
ejpam-4738	71	30	:	:	PUNCT
ejpam-4738	71	31	∗	∗	NOUN
ejpam-4738	71	32	0	0	PUNCT
ejpam-4738	72	1	a	a	DET
ejpam-4738	72	2	b	b	NOUN
ejpam-4738	72	3	c	c	NOUN
ejpam-4738	72	4	0	0	NUM
ejpam-4738	72	5	0	0	NUM
ejpam-4738	72	6	0	0	NUM
ejpam-4738	72	7	0	0	NUM
ejpam-4738	72	8	0	0	NUM
ejpam-4738	72	9	a	a	DET
ejpam-4738	72	10	a	a	DET
ejpam-4738	72	11	0	0	NUM
ejpam-4738	72	12	0	0	NUM
ejpam-4738	72	13	b	b	PROPN
ejpam-4738	72	14	b	b	PROPN
ejpam-4738	72	15	b	b	PROPN
ejpam-4738	72	16	b	b	PROPN
ejpam-4738	72	17	0	0	NUM
ejpam-4738	72	18	0	0	NUM
ejpam-4738	72	19	c	c	NOUN
ejpam-4738	72	20	c	c	NOUN
ejpam-4738	72	21	c	c	NOUN
ejpam-4738	72	22	c	c	NOUN
ejpam-4738	72	23	0	0	PUNCT
ejpam-4738	73	1	then	then	ADV
ejpam-4738	73	2	(	(	PUNCT
ejpam-4738	73	3	x	x	X
ejpam-4738	73	4	,	,	PUNCT
ejpam-4738	73	5	∗	∗	NOUN
ejpam-4738	73	6	,	,	PUNCT
ejpam-4738	73	7	0	0	NUM
ejpam-4738	73	8	)	)	PUNCT
ejpam-4738	73	9	is	be	AUX
ejpam-4738	73	10	a	a	DET
ejpam-4738	73	11	d	d	NOUN
ejpam-4738	73	12	-	-	PUNCT
ejpam-4738	73	13	algebra	algebra	NOUN
ejpam-4738	73	14	and	and	CCONJ
ejpam-4738	73	15	i	i	PRON
ejpam-4738	73	16	:	:	PUNCT
ejpam-4738	73	17	=	=	X
ejpam-4738	73	18	{	{	PUNCT
ejpam-4738	73	19	0	0	NUM
ejpam-4738	73	20	,	,	PUNCT
ejpam-4738	73	21	a	a	DET
ejpam-4738	73	22	}	}	PUNCT
ejpam-4738	73	23	is	be	AUX
ejpam-4738	73	24	an	an	DET
ejpam-4738	73	25	ideal	ideal	NOUN
ejpam-4738	73	26	of	of	ADP
ejpam-4738	73	27	x	x	PRON
ejpam-4738	73	28	,	,	PUNCT
ejpam-4738	73	29	but	but	CCONJ
ejpam-4738	73	30	not	not	PART
ejpam-4738	73	31	a	a	DET
ejpam-4738	73	32	dideal	dideal	NOUN
ejpam-4738	73	33	of	of	ADP
ejpam-4738	73	34	x	x	PRON
ejpam-4738	73	35	,	,	PUNCT
ejpam-4738	73	36	since	since	SCONJ
ejpam-4738	73	37	a	a	DET
ejpam-4738	73	38	∗	∗	NOUN
ejpam-4738	73	39	c	c	NOUN
ejpam-4738	73	40	=	=	SYM
ejpam-4738	73	41	b	b	PROPN
ejpam-4738	73	42	/∈	/∈	PROPN
ejpam-4738	73	43	i.	i.	PROPN
ejpam-4738	73	44	theorem	theorem	VERB
ejpam-4738	73	45	2	2	NUM
ejpam-4738	73	46	.	.	PUNCT
ejpam-4738	74	1	[	[	X
ejpam-4738	74	2	9	9	NUM
ejpam-4738	74	3	]	]	PUNCT
ejpam-4738	74	4	let	let	VERB
ejpam-4738	74	5	i	i	PRON
ejpam-4738	74	6	be	be	AUX
ejpam-4738	74	7	a	a	DET
ejpam-4738	74	8	d	d	NOUN
ejpam-4738	74	9	-	-	NOUN
ejpam-4738	74	10	ideal	ideal	NOUN
ejpam-4738	74	11	of	of	ADP
ejpam-4738	74	12	a	a	DET
ejpam-4738	74	13	d	d	NOUN
ejpam-4738	74	14	-	-	NOUN
ejpam-4738	74	15	algebra	algebra	ADJ
ejpam-4738	74	16	x.	x.	NOUN
ejpam-4738	75	1	if	if	SCONJ
ejpam-4738	75	2	x	x	SYM
ejpam-4738	75	3	∈	∈	PROPN
ejpam-4738	75	4	i	i	PRON
ejpam-4738	75	5	and	and	CCONJ
ejpam-4738	75	6	y	y	PROPN
ejpam-4738	75	7	∈	∈	PROPN
ejpam-4738	75	8	x	x	PUNCT
ejpam-4738	75	9	such	such	ADJ
ejpam-4738	75	10	that	that	SCONJ
ejpam-4738	75	11	y	y	PROPN
ejpam-4738	75	12	∗	∗	NOUN
ejpam-4738	75	13	x	x	PUNCT
ejpam-4738	76	1	=	=	SYM
ejpam-4738	76	2	0	0	NUM
ejpam-4738	76	3	,	,	PUNCT
ejpam-4738	76	4	then	then	ADV
ejpam-4738	76	5	y	y	PROPN
ejpam-4738	76	6	∈	∈	PROPN
ejpam-4738	76	7	i.	i.	NOUN
ejpam-4738	76	8	3	3	X
ejpam-4738	76	9	.	.	PUNCT
ejpam-4738	77	1	direct	direct	ADJ
ejpam-4738	77	2	product	product	NOUN
ejpam-4738	77	3	d	d	NOUN
ejpam-4738	77	4	-	-	PUNCT
ejpam-4738	77	5	algebras	algebras	PROPN
ejpam-4738	77	6	j.	j.	PROPN
ejpam-4738	77	7	neggers	neggers	PROPN
ejpam-4738	77	8	and	and	CCONJ
ejpam-4738	77	9	h.	h.	PROPN
ejpam-4738	77	10	s.	s.	PROPN
ejpam-4738	77	11	kim	kim	PROPN
ejpam-4738	78	1	(	(	PUNCT
ejpam-4738	78	2	[	[	X
ejpam-4738	78	3	9	9	NUM
ejpam-4738	78	4	]	]	PUNCT
ejpam-4738	78	5	)	)	PUNCT
ejpam-4738	78	6	introduced	introduce	VERB
ejpam-4738	78	7	the	the	DET
ejpam-4738	78	8	concept	concept	NOUN
ejpam-4738	78	9	of	of	ADP
ejpam-4738	78	10	a	a	DET
ejpam-4738	78	11	direct	direct	ADJ
ejpam-4738	78	12	product	product	NOUN
ejpam-4738	78	13	of	of	ADP
ejpam-4738	78	14	d	d	NOUN
ejpam-4738	78	15	-	-	PUNCT
ejpam-4738	78	16	algebras	algebras	ADJ
ejpam-4738	78	17	as	as	SCONJ
ejpam-4738	78	18	follows	follow	VERB
ejpam-4738	78	19	.	.	PUNCT
ejpam-4738	79	1	let	let	VERB
ejpam-4738	79	2	{	{	PUNCT
ejpam-4738	79	3	(	(	PUNCT
ejpam-4738	79	4	xi	xi	PROPN
ejpam-4738	79	5	,	,	PUNCT
ejpam-4738	79	6	∗	∗	NOUN
ejpam-4738	79	7	,	,	PUNCT
ejpam-4738	79	8	0	0	NUM
ejpam-4738	79	9	)	)	PUNCT
ejpam-4738	80	1	|	|	ADV
ejpam-4738	80	2	i	i	PRON
ejpam-4738	80	3	∈	∈	VERB
ejpam-4738	80	4	i	i	PRON
ejpam-4738	80	5	}	}	PUNCT
ejpam-4738	80	6	be	be	VERB
ejpam-4738	80	7	a	a	DET
ejpam-4738	80	8	non	non	ADJ
ejpam-4738	80	9	-	-	ADJ
ejpam-4738	80	10	empty	empty	ADJ
ejpam-4738	80	11	family	family	NOUN
ejpam-4738	80	12	of	of	ADP
ejpam-4738	80	13	d	d	NOUN
ejpam-4738	80	14	-	-	PUNCT
ejpam-4738	80	15	algebras	algebra	NOUN
ejpam-4738	80	16	and	and	CCONJ
ejpam-4738	80	17	∏	∏	NUM
ejpam-4738	80	18	i∈i	i∈i	ADJ
ejpam-4738	80	19	xi=	xi=	PROPN
ejpam-4738	80	20	{	{	PUNCT
ejpam-4738	80	21	(	(	PUNCT
ejpam-4738	80	22	xi)i∈i	xi)i∈i	NUM
ejpam-4738	80	23	|	|	ADV
ejpam-4738	80	24	xi	xi	X
ejpam-4738	80	25	∈	∈	PROPN
ejpam-4738	80	26	xi	xi	ADP
ejpam-4738	80	27	}	}	PUNCT
ejpam-4738	80	28	.	.	PUNCT
ejpam-4738	81	1	then	then	ADV
ejpam-4738	81	2	(	(	PUNCT
ejpam-4738	81	3	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	81	4	where	where	SCONJ
ejpam-4738	81	5	0i	0i	PROPN
ejpam-4738	81	6	∈	∈	PROPN
ejpam-4738	81	7	xi	xi	PROPN
ejpam-4738	81	8	.	.	PROPN
ejpam-4738	81	9	serves	serve	VERB
ejpam-4738	81	10	as	as	ADP
ejpam-4738	81	11	0	0	NUM
ejpam-4738	81	12	of	of	ADP
ejpam-4738	81	13	∏	∏	PROPN
ejpam-4738	81	14	i∈i	i∈i	ADJ
ejpam-4738	81	15	xi	xi	PROPN
ejpam-4738	81	16	.	.	PUNCT
ejpam-4738	81	17	define	define	VERB
ejpam-4738	81	18	a	a	DET
ejpam-4738	81	19	binary	binary	ADJ
ejpam-4738	81	20	m.	m.	NOUN
ejpam-4738	81	21	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	81	22	/	/	SYM
ejpam-4738	81	23	eur	eur	PROPN
ejpam-4738	81	24	.	.	PUNCT
ejpam-4738	82	1	j.	j.	PROPN
ejpam-4738	82	2	pure	pure	PROPN
ejpam-4738	82	3	appl	appl	PROPN
ejpam-4738	82	4	.	.	PROPN
ejpam-4738	82	5	math	math	PROPN
ejpam-4738	82	6	,	,	PUNCT
ejpam-4738	82	7	16	16	NUM
ejpam-4738	82	8	(	(	PUNCT
ejpam-4738	82	9	2	2	NUM
ejpam-4738	82	10	)	)	PUNCT
ejpam-4738	82	11	(	(	PUNCT
ejpam-4738	82	12	2023	2023	NUM
ejpam-4738	82	13	)	)	PUNCT
ejpam-4738	82	14	,	,	PUNCT
ejpam-4738	82	15	997	997	NUM
ejpam-4738	82	16	-	-	SYM
ejpam-4738	82	17	1004	1004	NUM
ejpam-4738	82	18	1001	1001	NUM
ejpam-4738	82	19	operation	operation	NOUN
ejpam-4738	82	20	⊙	⊙	NOUN
ejpam-4738	82	21	on	on	ADP
ejpam-4738	82	22	∏	∏	PROPN
ejpam-4738	82	23	i∈i	i∈i	NOUN
ejpam-4738	82	24	xi	xi	INTJ
ejpam-4738	82	25	by	by	ADP
ejpam-4738	82	26	(	(	PUNCT
ejpam-4738	82	27	xi)i∈i	xi)i∈i	PROPN
ejpam-4738	82	28	⊙	⊙	X
ejpam-4738	82	29	(	(	PUNCT
ejpam-4738	82	30	yi)i∈i	yi)i∈i	NUM
ejpam-4738	82	31	=	=	SYM
ejpam-4738	82	32	(	(	PUNCT
ejpam-4738	82	33	xi	xi	X
ejpam-4738	82	34	∗	∗	NOUN
ejpam-4738	82	35	yi)i∈i	yi)i∈i	NUM
ejpam-4738	82	36	for	for	ADP
ejpam-4738	82	37	all	all	PRON
ejpam-4738	82	38	(	(	PUNCT
ejpam-4738	82	39	xi)i∈i	xi)i∈i	NUM
ejpam-4738	82	40	,	,	PUNCT
ejpam-4738	82	41	(	(	PUNCT
ejpam-4738	82	42	yi)i∈i	yi)i∈i	NUM
ejpam-4738	82	43	∈	∈	PROPN
ejpam-4738	82	44	∏	∏	PROPN
ejpam-4738	82	45	i∈i	i∈i	ADJ
ejpam-4738	82	46	xi	xi	PROPN
ejpam-4738	82	47	.	.	PUNCT
ejpam-4738	83	1	then	then	ADV
ejpam-4738	83	2	(	(	PUNCT
ejpam-4738	83	3	∏	∏	PROPN
ejpam-4738	83	4	i∈i	i∈i	ADJ
ejpam-4738	83	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	83	6	,	,	PUNCT
ejpam-4738	83	7	(	(	PUNCT
ejpam-4738	83	8	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	83	9	)	)	PUNCT
ejpam-4738	83	10	is	be	AUX
ejpam-4738	83	11	a	a	DET
ejpam-4738	83	12	d	d	NOUN
ejpam-4738	83	13	-	-	NOUN
ejpam-4738	83	14	algebra	algebra	NOUN
ejpam-4738	83	15	,	,	PUNCT
ejpam-4738	83	16	called	call	VERB
ejpam-4738	83	17	a	a	DET
ejpam-4738	83	18	direct	direct	ADJ
ejpam-4738	83	19	product	product	NOUN
ejpam-4738	83	20	d	d	NOUN
ejpam-4738	83	21	-	-	PUNCT
ejpam-4738	83	22	algebra	algebra	NOUN
ejpam-4738	83	23	.	.	PUNCT
ejpam-4738	84	1	that	that	PRON
ejpam-4738	84	2	is	be	AUX
ejpam-4738	84	3	a	a	DET
ejpam-4738	84	4	direct	direct	ADJ
ejpam-4738	84	5	product	product	NOUN
ejpam-4738	84	6	d	d	NOUN
ejpam-4738	84	7	-	-	PUNCT
ejpam-4738	84	8	algebra	algebra	NOUN
ejpam-4738	84	9	(	(	PUNCT
ejpam-4738	84	10	∏	∏	PROPN
ejpam-4738	84	11	i∈i	i∈i	ADJ
ejpam-4738	84	12	xi,⊙	xi,⊙	PROPN
ejpam-4738	84	13	,	,	PUNCT
ejpam-4738	84	14	(	(	PUNCT
ejpam-4738	84	15	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	84	16	)	)	PUNCT
ejpam-4738	84	17	is	be	AUX
ejpam-4738	84	18	satisfies	satisfie	NOUN
ejpam-4738	84	19	the	the	DET
ejpam-4738	84	20	following	follow	VERB
ejpam-4738	84	21	conditions	condition	NOUN
ejpam-4738	84	22	:	:	PUNCT
ejpam-4738	84	23	(	(	PUNCT
ejpam-4738	84	24	i	i	NOUN
ejpam-4738	84	25	)	)	PUNCT
ejpam-4738	84	26	(	(	PUNCT
ejpam-4738	84	27	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	28	⊙	⊙	X
ejpam-4738	84	29	(	(	PUNCT
ejpam-4738	84	30	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	31	=	=	SYM
ejpam-4738	84	32	(	(	PUNCT
ejpam-4738	84	33	0i)i∈i	0i)i∈i	NUM
ejpam-4738	84	34	,	,	PUNCT
ejpam-4738	84	35	(	(	PUNCT
ejpam-4738	84	36	ii	ii	NOUN
ejpam-4738	84	37	)	)	PUNCT
ejpam-4738	84	38	(	(	PUNCT
ejpam-4738	84	39	0i)i∈i)⊙	0i)i∈i)⊙	NOUN
ejpam-4738	84	40	(	(	PUNCT
ejpam-4738	84	41	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	42	=	=	SYM
ejpam-4738	84	43	(	(	PUNCT
ejpam-4738	84	44	0i)i∈i	0i)i∈i	NUM
ejpam-4738	84	45	,	,	PUNCT
ejpam-4738	84	46	(	(	PUNCT
ejpam-4738	84	47	iii	iii	NOUN
ejpam-4738	84	48	)	)	PUNCT
ejpam-4738	84	49	(	(	PUNCT
ejpam-4738	84	50	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	51	⊙	⊙	X
ejpam-4738	84	52	(	(	PUNCT
ejpam-4738	84	53	yi)i∈i	yi)i∈i	NUM
ejpam-4738	84	54	=	=	SYM
ejpam-4738	84	55	(	(	PUNCT
ejpam-4738	84	56	0i)i∈i	0i)i∈i	NUM
ejpam-4738	84	57	and	and	CCONJ
ejpam-4738	84	58	(	(	PUNCT
ejpam-4738	84	59	yi)i∈i	yi)i∈i	NUM
ejpam-4738	84	60	⊙	⊙	NOUN
ejpam-4738	84	61	(	(	PUNCT
ejpam-4738	84	62	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	63	=	=	SYM
ejpam-4738	84	64	(	(	PUNCT
ejpam-4738	84	65	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	84	66	implies	imply	VERB
ejpam-4738	84	67	(	(	PUNCT
ejpam-4738	84	68	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	69	=	=	SYM
ejpam-4738	84	70	(	(	PUNCT
ejpam-4738	84	71	yi)i∈i	yi)i∈i	NUM
ejpam-4738	84	72	for	for	ADP
ejpam-4738	84	73	all	all	PRON
ejpam-4738	84	74	(	(	PUNCT
ejpam-4738	84	75	xi)i∈i	xi)i∈i	NUM
ejpam-4738	84	76	,	,	PUNCT
ejpam-4738	84	77	(	(	PUNCT
ejpam-4738	84	78	yi)i∈i	yi)i∈i	NUM
ejpam-4738	84	79	∈	∈	PROPN
ejpam-4738	84	80	∏	∏	PROPN
ejpam-4738	84	81	i∈i	i∈i	ADJ
ejpam-4738	84	82	xi	xi	PROPN
ejpam-4738	84	83	.	.	PUNCT
ejpam-4738	85	1	definition	definition	NOUN
ejpam-4738	85	2	6	6	NUM
ejpam-4738	85	3	.	.	PUNCT
ejpam-4738	86	1	let	let	VERB
ejpam-4738	86	2	(	(	PUNCT
ejpam-4738	86	3	∏	∏	PROPN
ejpam-4738	86	4	i∈i	i∈i	ADJ
ejpam-4738	86	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	86	6	,	,	PUNCT
ejpam-4738	86	7	(	(	PUNCT
ejpam-4738	86	8	0i)i∈i	0i)i∈i	X
ejpam-4738	86	9	)	)	PUNCT
ejpam-4738	86	10	be	be	AUX
ejpam-4738	86	11	a	a	DET
ejpam-4738	86	12	direct	direct	ADJ
ejpam-4738	86	13	product	product	NOUN
ejpam-4738	86	14	d	d	NOUN
ejpam-4738	86	15	-	-	PUNCT
ejpam-4738	86	16	algebra	algebra	NOUN
ejpam-4738	86	17	.	.	PUNCT
ejpam-4738	87	1	a	a	DET
ejpam-4738	87	2	non	non	ADJ
ejpam-4738	87	3	-	-	ADJ
ejpam-4738	87	4	empty	empty	ADJ
ejpam-4738	87	5	subset∏	subset∏	NOUN
ejpam-4738	87	6	i∈i	i∈i	ADJ
ejpam-4738	87	7	ni	ni	PROPN
ejpam-4738	87	8	of	of	ADP
ejpam-4738	87	9	∏	∏	PROPN
ejpam-4738	87	10	i∈i	i∈i	NOUN
ejpam-4738	87	11	xi	xi	INTJ
ejpam-4738	87	12	is	be	AUX
ejpam-4738	87	13	said	say	VERB
ejpam-4738	87	14	to	to	PART
ejpam-4738	87	15	be	be	AUX
ejpam-4738	87	16	an	an	DET
ejpam-4738	87	17	ideal	ideal	ADJ
ejpam-4738	87	18	direct	direct	ADJ
ejpam-4738	87	19	product	product	NOUN
ejpam-4738	87	20	d	d	NOUN
ejpam-4738	87	21	-	-	PUNCT
ejpam-4738	87	22	algebra	algebra	NOUN
ejpam-4738	87	23	if	if	SCONJ
ejpam-4738	87	24	it	it	PRON
ejpam-4738	87	25	satisfies	satisfy	VERB
ejpam-4738	87	26	the	the	DET
ejpam-4738	87	27	following	follow	VERB
ejpam-4738	87	28	conditions	condition	NOUN
ejpam-4738	87	29	:	:	PUNCT
ejpam-4738	87	30	(	(	PUNCT
ejpam-4738	87	31	i1	i1	PROPN
ejpam-4738	87	32	)	)	PUNCT
ejpam-4738	87	33	(	(	PUNCT
ejpam-4738	87	34	0i)i∈i	0i)i∈i	NUM
ejpam-4738	87	35	∈	∈	PROPN
ejpam-4738	87	36	∏	∏	PROPN
ejpam-4738	87	37	i∈i	i∈i	PROPN
ejpam-4738	87	38	ni	ni	PROPN
ejpam-4738	87	39	,	,	PUNCT
ejpam-4738	87	40	(	(	PUNCT
ejpam-4738	87	41	i2	i2	PROPN
ejpam-4738	87	42	)	)	PUNCT
ejpam-4738	87	43	(	(	PUNCT
ejpam-4738	87	44	xi)i∈i	xi)i∈i	NUM
ejpam-4738	87	45	∗	∗	NOUN
ejpam-4738	87	46	(	(	PUNCT
ejpam-4738	87	47	yi)i∈i	yi)i∈i	NUM
ejpam-4738	87	48	∈	∈	PROPN
ejpam-4738	87	49	∏	∏	PROPN
ejpam-4738	87	50	i∈i	i∈i	PROPN
ejpam-4738	87	51	ni	ni	PROPN
ejpam-4738	87	52	and	and	CCONJ
ejpam-4738	87	53	(	(	PUNCT
ejpam-4738	87	54	yi)i∈i	yi)i∈i	NUM
ejpam-4738	87	55	∈	∈	PROPN
ejpam-4738	87	56	∏	∏	PROPN
ejpam-4738	87	57	i∈i	i∈i	ADJ
ejpam-4738	87	58	ni	ni	PROPN
ejpam-4738	87	59	implies	imply	VERB
ejpam-4738	87	60	(	(	PUNCT
ejpam-4738	87	61	xi)i∈i	xi)i∈i	NUM
ejpam-4738	87	62	∈	∈	PROPN
ejpam-4738	87	63	∏	∏	PROPN
ejpam-4738	87	64	i∈i	i∈i	PROPN
ejpam-4738	87	65	ni	ni	PROPN
ejpam-4738	87	66	.	.	PROPN
ejpam-4738	87	67	definition	definition	NOUN
ejpam-4738	87	68	7	7	NUM
ejpam-4738	87	69	.	.	PUNCT
ejpam-4738	88	1	let	let	VERB
ejpam-4738	88	2	(	(	PUNCT
ejpam-4738	88	3	∏	∏	PROPN
ejpam-4738	88	4	i∈i	i∈i	ADJ
ejpam-4738	88	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	88	6	,	,	PUNCT
ejpam-4738	88	7	(	(	PUNCT
ejpam-4738	88	8	0i)i∈i	0i)i∈i	X
ejpam-4738	88	9	)	)	PUNCT
ejpam-4738	88	10	be	be	AUX
ejpam-4738	88	11	a	a	DET
ejpam-4738	88	12	direct	direct	ADJ
ejpam-4738	88	13	product	product	NOUN
ejpam-4738	88	14	d	d	NOUN
ejpam-4738	88	15	-	-	PUNCT
ejpam-4738	88	16	algebra	algebra	NOUN
ejpam-4738	88	17	.	.	PUNCT
ejpam-4738	89	1	a	a	DET
ejpam-4738	89	2	non	non	ADJ
ejpam-4738	89	3	-	-	ADJ
ejpam-4738	89	4	empty	empty	ADJ
ejpam-4738	89	5	subset∏	subset∏	NOUN
ejpam-4738	89	6	i∈i	i∈i	ADJ
ejpam-4738	89	7	ni	ni	PROPN
ejpam-4738	89	8	of	of	ADP
ejpam-4738	89	9	∏	∏	PROPN
ejpam-4738	89	10	i∈i	i∈i	NOUN
ejpam-4738	89	11	xi	xi	INTJ
ejpam-4738	89	12	is	be	AUX
ejpam-4738	89	13	said	say	VERB
ejpam-4738	89	14	to	to	PART
ejpam-4738	89	15	be	be	AUX
ejpam-4738	89	16	a	a	DET
ejpam-4738	89	17	d	d	ADJ
ejpam-4738	89	18	-	-	PUNCT
ejpam-4738	89	19	ideal	ideal	ADJ
ejpam-4738	89	20	direct	direct	ADJ
ejpam-4738	89	21	product	product	NOUN
ejpam-4738	89	22	d	d	NOUN
ejpam-4738	89	23	-	-	PUNCT
ejpam-4738	89	24	algebras	algebras	X
ejpam-4738	89	25	if	if	SCONJ
ejpam-4738	89	26	it	it	PRON
ejpam-4738	89	27	satisfies	satisfy	VERB
ejpam-4738	89	28	the	the	DET
ejpam-4738	89	29	following	follow	VERB
ejpam-4738	89	30	conditions	condition	NOUN
ejpam-4738	89	31	:	:	PUNCT
ejpam-4738	89	32	(	(	PUNCT
ejpam-4738	89	33	d1	d1	NOUN
ejpam-4738	89	34	)	)	PUNCT
ejpam-4738	89	35	(	(	PUNCT
ejpam-4738	89	36	xi)i∈i	xi)i∈i	NUM
ejpam-4738	89	37	⊙	⊙	X
ejpam-4738	89	38	(	(	PUNCT
ejpam-4738	89	39	yi)i∈i	yi)i∈i	NUM
ejpam-4738	89	40	∈	∈	PROPN
ejpam-4738	89	41	∏	∏	PROPN
ejpam-4738	89	42	i∈i	i∈i	PROPN
ejpam-4738	89	43	ni	ni	PROPN
ejpam-4738	89	44	and	and	CCONJ
ejpam-4738	89	45	(	(	PUNCT
ejpam-4738	89	46	yi)i∈i	yi)i∈i	NUM
ejpam-4738	89	47	∈	∈	PROPN
ejpam-4738	89	48	∏	∏	PROPN
ejpam-4738	89	49	i∈i	i∈i	ADJ
ejpam-4738	89	50	ni	ni	PROPN
ejpam-4738	89	51	implies	imply	VERB
ejpam-4738	89	52	(	(	PUNCT
ejpam-4738	89	53	xi)i∈i	xi)i∈i	NUM
ejpam-4738	89	54	∈	∈	PROPN
ejpam-4738	89	55	∏	∏	PROPN
ejpam-4738	89	56	i∈i	i∈i	PROPN
ejpam-4738	89	57	ni	ni	PROPN
ejpam-4738	89	58	,	,	PUNCT
ejpam-4738	89	59	(	(	PUNCT
ejpam-4738	89	60	d2	d2	PROPN
ejpam-4738	89	61	)	)	PUNCT
ejpam-4738	89	62	(	(	PUNCT
ejpam-4738	89	63	xi)i∈i	xi)i∈i	NUM
ejpam-4738	89	64	∈	∈	PROPN
ejpam-4738	89	65	∏	∏	PROPN
ejpam-4738	89	66	i∈i	i∈i	PROPN
ejpam-4738	89	67	ni	ni	PROPN
ejpam-4738	89	68	and	and	CCONJ
ejpam-4738	89	69	(	(	PUNCT
ejpam-4738	89	70	yi)i∈i	yi)i∈i	NUM
ejpam-4738	89	71	∈	∈	PROPN
ejpam-4738	89	72	∏	∏	PROPN
ejpam-4738	89	73	i∈i	i∈i	ADV
ejpam-4738	89	74	xi	xi	PROPN
ejpam-4738	89	75	implies	imply	VERB
ejpam-4738	89	76	(	(	PUNCT
ejpam-4738	89	77	xi)i∈i	xi)i∈i	NUM
ejpam-4738	89	78	⊙	⊙	X
ejpam-4738	89	79	(	(	PUNCT
ejpam-4738	89	80	yi)i∈i	yi)i∈i	NUM
ejpam-4738	89	81	∈	∈	PROPN
ejpam-4738	89	82	∏	∏	PROPN
ejpam-4738	89	83	i∈i	i∈i	PROPN
ejpam-4738	89	84	ni	ni	PROPN
ejpam-4738	89	85	.	.	PUNCT
ejpam-4738	90	1	example	example	NOUN
ejpam-4738	91	1	6	6	NUM
ejpam-4738	91	2	.	.	PUNCT
ejpam-4738	92	1	[	[	X
ejpam-4738	92	2	1	1	NUM
ejpam-4738	92	3	]	]	PUNCT
ejpam-4738	92	4	,	,	PUNCT
ejpam-4738	92	5	[	[	X
ejpam-4738	92	6	9	9	NUM
ejpam-4738	92	7	]	]	PUNCT
ejpam-4738	92	8	let	let	VERB
ejpam-4738	92	9	x1	x1	PROPN
ejpam-4738	92	10	=	=	PUNCT
ejpam-4738	92	11	{	{	PUNCT
ejpam-4738	92	12	0	0	NUM
ejpam-4738	92	13	,	,	PUNCT
ejpam-4738	92	14	1	1	NUM
ejpam-4738	92	15	,	,	PUNCT
ejpam-4738	92	16	2	2	NUM
ejpam-4738	92	17	,	,	PUNCT
ejpam-4738	92	18	3	3	NUM
ejpam-4738	92	19	}	}	PUNCT
ejpam-4738	92	20	and	and	CCONJ
ejpam-4738	92	21	x2	x2	NOUN
ejpam-4738	92	22	=	=	PRON
ejpam-4738	92	23	{	{	PUNCT
ejpam-4738	92	24	0′	0′	NUM
ejpam-4738	92	25	,	,	PUNCT
ejpam-4738	92	26	a	a	DET
ejpam-4738	92	27	,	,	PUNCT
ejpam-4738	92	28	b	b	NOUN
ejpam-4738	92	29	,	,	PUNCT
ejpam-4738	92	30	c	c	NOUN
ejpam-4738	92	31	}	}	PUNCT
ejpam-4738	92	32	.	.	PUNCT
ejpam-4738	93	1	define	define	VERB
ejpam-4738	93	2	binary	binary	ADJ
ejpam-4738	93	3	operations	operation	NOUN
ejpam-4738	93	4	∗	∗	NOUN
ejpam-4738	93	5	on	on	ADP
ejpam-4738	93	6	x1	x1	PROPN
ejpam-4738	93	7	and	and	CCONJ
ejpam-4738	93	8	∗′	∗′	PROPN
ejpam-4738	93	9	on	on	ADP
ejpam-4738	93	10	x2	x2	PROPN
ejpam-4738	93	11	.	.	PUNCT
ejpam-4738	94	1	defined	define	VERB
ejpam-4738	94	2	by	by	ADP
ejpam-4738	94	3	the	the	DET
ejpam-4738	94	4	following	follow	VERB
ejpam-4738	94	5	two	two	NUM
ejpam-4738	94	6	tables	table	NOUN
ejpam-4738	94	7	,	,	PUNCT
ejpam-4738	94	8	respectively	respectively	ADV
ejpam-4738	94	9	.	.	PUNCT
ejpam-4738	95	1	∗	∗	NOUN
ejpam-4738	95	2	0	0	NUM
ejpam-4738	95	3	1	1	NUM
ejpam-4738	95	4	2	2	NUM
ejpam-4738	95	5	3	3	NUM
ejpam-4738	95	6	0	0	NUM
ejpam-4738	95	7	0	0	NUM
ejpam-4738	95	8	0	0	NUM
ejpam-4738	95	9	0	0	NUM
ejpam-4738	95	10	0	0	NUM
ejpam-4738	95	11	1	1	NUM
ejpam-4738	95	12	1	1	NUM
ejpam-4738	95	13	0	0	NUM
ejpam-4738	95	14	1	1	NUM
ejpam-4738	95	15	0	0	NUM
ejpam-4738	95	16	2	2	NUM
ejpam-4738	95	17	2	2	NUM
ejpam-4738	95	18	2	2	NUM
ejpam-4738	95	19	0	0	NUM
ejpam-4738	95	20	0	0	NUM
ejpam-4738	95	21	3	3	NUM
ejpam-4738	95	22	3	3	NUM
ejpam-4738	95	23	3	3	NUM
ejpam-4738	95	24	3	3	NUM
ejpam-4738	95	25	0	0	NUM
ejpam-4738	95	26	∗′	∗′	NUM
ejpam-4738	95	27	0	0	NUM
ejpam-4738	95	28	′	′	NUM
ejpam-4738	95	29	a	a	DET
ejpam-4738	95	30	b	b	NOUN
ejpam-4738	95	31	c	c	NOUN
ejpam-4738	95	32	0	0	NUM
ejpam-4738	96	1	′	′	NUM
ejpam-4738	96	2	0	0	NUM
ejpam-4738	97	1	′	′	NUM
ejpam-4738	97	2	0	0	NUM
ejpam-4738	98	1	′	′	NUM
ejpam-4738	98	2	0	0	NUM
ejpam-4738	99	1	′	′	NUM
ejpam-4738	99	2	0	0	NUM
ejpam-4738	100	1	′	′	NUM
ejpam-4738	100	2	a	a	DET
ejpam-4738	100	3	a	a	DET
ejpam-4738	100	4	0	0	NUM
ejpam-4738	100	5	′	′	NUM
ejpam-4738	100	6	0	0	NUM
ejpam-4738	101	1	′	′	NUM
ejpam-4738	101	2	b	b	PROPN
ejpam-4738	101	3	b	b	X
ejpam-4738	101	4	b	b	PROPN
ejpam-4738	101	5	b	b	PROPN
ejpam-4738	101	6	0	0	NUM
ejpam-4738	101	7	′	′	NUM
ejpam-4738	101	8	0	0	NUM
ejpam-4738	102	1	′	′	NUM
ejpam-4738	103	1	c	c	NOUN
ejpam-4738	103	2	c	c	NOUN
ejpam-4738	103	3	c	c	NOUN
ejpam-4738	103	4	c	c	NOUN
ejpam-4738	103	5	0	0	NUM
ejpam-4738	103	6	′	′	NUM
ejpam-4738	103	7	by	by	ADP
ejpam-4738	103	8	example	example	NOUN
ejpam-4738	103	9	4	4	NUM
ejpam-4738	103	10	and	and	CCONJ
ejpam-4738	103	11	example	example	NOUN
ejpam-4738	103	12	5	5	NUM
ejpam-4738	103	13	,	,	PUNCT
ejpam-4738	103	14	(	(	PUNCT
ejpam-4738	103	15	x1	x1	PROPN
ejpam-4738	103	16	,	,	PUNCT
ejpam-4738	103	17	∗	∗	NOUN
ejpam-4738	103	18	,	,	PUNCT
ejpam-4738	103	19	0	0	NUM
ejpam-4738	103	20	)	)	PUNCT
ejpam-4738	103	21	and	and	CCONJ
ejpam-4738	103	22	(	(	PUNCT
ejpam-4738	103	23	x2	x2	PROPN
ejpam-4738	103	24	,	,	PUNCT
ejpam-4738	103	25	∗′	∗′	PROPN
ejpam-4738	103	26	,	,	PUNCT
ejpam-4738	103	27	0	0	NUM
ejpam-4738	103	28	′	′	NUM
ejpam-4738	103	29	)	)	PUNCT
ejpam-4738	103	30	are	be	AUX
ejpam-4738	103	31	d	d	NOUN
ejpam-4738	103	32	-	-	PUNCT
ejpam-4738	103	33	algebras	algebras	X
ejpam-4738	103	34	.	.	PUNCT
ejpam-4738	104	1	consider	consider	VERB
ejpam-4738	104	2	an	an	DET
ejpam-4738	104	3	ideal	ideal	ADJ
ejpam-4738	104	4	n1	n1	NOUN
ejpam-4738	104	5	=	=	SYM
ejpam-4738	104	6	{	{	PUNCT
ejpam-4738	104	7	0	0	NUM
ejpam-4738	104	8	,	,	PUNCT
ejpam-4738	104	9	1	1	NUM
ejpam-4738	104	10	}	}	PUNCT
ejpam-4738	104	11	of	of	ADP
ejpam-4738	104	12	(	(	PUNCT
ejpam-4738	104	13	x1	x1	ADJ
ejpam-4738	104	14	and	and	CCONJ
ejpam-4738	104	15	ideal	ideal	ADJ
ejpam-4738	104	16	n2	n2	NOUN
ejpam-4738	104	17	=	=	SYM
ejpam-4738	104	18	{	{	PUNCT
ejpam-4738	104	19	0′	0′	PROPN
ejpam-4738	104	20	,	,	PUNCT
ejpam-4738	104	21	a	a	PRON
ejpam-4738	104	22	}	}	PUNCT
ejpam-4738	104	23	of	of	ADP
ejpam-4738	104	24	x2	x2	PROPN
ejpam-4738	104	25	,	,	PUNCT
ejpam-4738	104	26	we	we	PRON
ejpam-4738	104	27	have	have	VERB
ejpam-4738	104	28	n1	n1	ADJ
ejpam-4738	104	29	×	×	PROPN
ejpam-4738	104	30	n2	n2	NOUN
ejpam-4738	104	31	=	=	PUNCT
ejpam-4738	104	32	{	{	PUNCT
ejpam-4738	104	33	(	(	PUNCT
ejpam-4738	104	34	0	0	NUM
ejpam-4738	104	35	,	,	PUNCT
ejpam-4738	104	36	0′	0′	NUM
ejpam-4738	104	37	)	)	PUNCT
ejpam-4738	104	38	,	,	PUNCT
ejpam-4738	104	39	(	(	PUNCT
ejpam-4738	104	40	0	0	NUM
ejpam-4738	104	41	,	,	PUNCT
ejpam-4738	104	42	a	a	PRON
ejpam-4738	104	43	)	)	PUNCT
ejpam-4738	104	44	,	,	PUNCT
ejpam-4738	104	45	(	(	PUNCT
ejpam-4738	104	46	1	1	NUM
ejpam-4738	104	47	,	,	PUNCT
ejpam-4738	104	48	0	0	NUM
ejpam-4738	104	49	′	′	NUM
ejpam-4738	104	50	)	)	PUNCT
ejpam-4738	104	51	}	}	PUNCT
ejpam-4738	104	52	,	,	PUNCT
ejpam-4738	104	53	(	(	PUNCT
ejpam-4738	104	54	1	1	NUM
ejpam-4738	104	55	,	,	PUNCT
ejpam-4738	104	56	a	a	NOUN
ejpam-4738	104	57	)	)	PUNCT
ejpam-4738	104	58	,	,	PUNCT
ejpam-4738	104	59	}	}	PUNCT
ejpam-4738	104	60	is	be	AUX
ejpam-4738	104	61	an	an	DET
ejpam-4738	104	62	ideal	ideal	ADJ
ejpam-4738	104	63	direct	direct	ADJ
ejpam-4738	104	64	product	product	NOUN
ejpam-4738	104	65	d	d	NOUN
ejpam-4738	104	66	-	-	PUNCT
ejpam-4738	104	67	algebra	algebra	NOUN
ejpam-4738	104	68	but	but	CCONJ
ejpam-4738	104	69	not	not	PART
ejpam-4738	104	70	a	a	DET
ejpam-4738	104	71	d	d	ADJ
ejpam-4738	104	72	-	-	ADJ
ejpam-4738	104	73	ideal	ideal	ADJ
ejpam-4738	104	74	direct	direct	ADJ
ejpam-4738	104	75	product	product	NOUN
ejpam-4738	104	76	d	d	NOUN
ejpam-4738	104	77	-	-	PUNCT
ejpam-4738	104	78	algebra	algebra	NOUN
ejpam-4738	104	79	,	,	PUNCT
ejpam-4738	104	80	since	since	SCONJ
ejpam-4738	104	81	then	then	ADV
ejpam-4738	104	82	(	(	PUNCT
ejpam-4738	104	83	0	0	NUM
ejpam-4738	104	84	,	,	PUNCT
ejpam-4738	104	85	a)⊙	a)⊙	PROPN
ejpam-4738	104	86	(	(	PUNCT
ejpam-4738	104	87	2	2	NUM
ejpam-4738	104	88	,	,	PUNCT
ejpam-4738	104	89	c	c	NOUN
ejpam-4738	104	90	)	)	PUNCT
ejpam-4738	104	91	=	=	SYM
ejpam-4738	104	92	(	(	PUNCT
ejpam-4738	104	93	0∗2	0∗2	PROPN
ejpam-4738	104	94	,	,	PUNCT
ejpam-4738	104	95	a∗′	a∗′	NUM
ejpam-4738	104	96	c	c	X
ejpam-4738	104	97	)	)	PUNCT
ejpam-4738	104	98	=	=	SYM
ejpam-4738	104	99	(	(	PUNCT
ejpam-4738	104	100	0	0	NUM
ejpam-4738	104	101	,	,	PUNCT
ejpam-4738	104	102	b	b	NOUN
ejpam-4738	104	103	)	)	PUNCT
ejpam-4738	104	104	/∈	/∈	PUNCT
ejpam-4738	105	1	n1×n2	n1×n2	NOUN
ejpam-4738	105	2	.	.	PUNCT
ejpam-4738	105	3	m.	m.	PROPN
ejpam-4738	105	4	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4738	105	5	/	/	SYM
ejpam-4738	105	6	eur	eur	PROPN
ejpam-4738	105	7	.	.	PUNCT
ejpam-4738	106	1	j.	j.	PROPN
ejpam-4738	106	2	pure	pure	PROPN
ejpam-4738	106	3	appl	appl	PROPN
ejpam-4738	106	4	.	.	PROPN
ejpam-4738	106	5	math	math	PROPN
ejpam-4738	106	6	,	,	PUNCT
ejpam-4738	106	7	16	16	NUM
ejpam-4738	106	8	(	(	PUNCT
ejpam-4738	106	9	2	2	NUM
ejpam-4738	106	10	)	)	PUNCT
ejpam-4738	106	11	(	(	PUNCT
ejpam-4738	106	12	2023	2023	NUM
ejpam-4738	106	13	)	)	PUNCT
ejpam-4738	106	14	,	,	PUNCT
ejpam-4738	106	15	997	997	NUM
ejpam-4738	106	16	-	-	SYM
ejpam-4738	106	17	1004	1004	NUM
ejpam-4738	106	18	1002	1002	NUM
ejpam-4738	106	19	definition	definition	NOUN
ejpam-4738	106	20	8	8	NUM
ejpam-4738	106	21	.	.	PUNCT
ejpam-4738	107	1	let	let	VERB
ejpam-4738	107	2	(	(	PUNCT
ejpam-4738	107	3	∏	∏	PROPN
ejpam-4738	107	4	i∈i	i∈i	ADJ
ejpam-4738	107	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	107	6	,	,	PUNCT
ejpam-4738	107	7	(	(	PUNCT
ejpam-4738	107	8	0i)i∈i	0i)i∈i	X
ejpam-4738	107	9	)	)	PUNCT
ejpam-4738	107	10	be	be	AUX
ejpam-4738	107	11	a	a	DET
ejpam-4738	107	12	direct	direct	ADJ
ejpam-4738	107	13	product	product	NOUN
ejpam-4738	107	14	d	d	NOUN
ejpam-4738	107	15	-	-	PUNCT
ejpam-4738	107	16	algebra	algebra	NOUN
ejpam-4738	107	17	,	,	PUNCT
ejpam-4738	108	1	a	a	DET
ejpam-4738	108	2	nonempty	nonempty	ADJ
ejpam-4738	108	3	subset∏	subset∏	AUX
ejpam-4738	108	4	i∈i	i∈i	ADJ
ejpam-4738	108	5	ni	ni	PROPN
ejpam-4738	108	6	of	of	ADP
ejpam-4738	108	7	∏	∏	PROPN
ejpam-4738	108	8	i∈i	i∈i	NOUN
ejpam-4738	108	9	xi	xi	INTJ
ejpam-4738	108	10	is	be	AUX
ejpam-4738	108	11	said	say	VERB
ejpam-4738	108	12	to	to	PART
ejpam-4738	108	13	be	be	AUX
ejpam-4738	108	14	a	a	DET
ejpam-4738	108	15	sub	sub	ADJ
ejpam-4738	108	16	-	-	ADJ
ejpam-4738	108	17	direct	direct	ADJ
ejpam-4738	108	18	product	product	NOUN
ejpam-4738	108	19	of	of	ADP
ejpam-4738	108	20	∏	∏	PROPN
ejpam-4738	108	21	i∈i	i∈i	ADJ
ejpam-4738	108	22	xi	xi	INTJ
ejpam-4738	108	23	if	if	SCONJ
ejpam-4738	108	24	(	(	PUNCT
ejpam-4738	108	25	xi)i∈i	xi)i∈i	NUM
ejpam-4738	108	26	⊙	⊙	X
ejpam-4738	108	27	(	(	PUNCT
ejpam-4738	108	28	yi)i∈i	yi)i∈i	NUM
ejpam-4738	108	29	∈	∈	PROPN
ejpam-4738	108	30	∏	∏	PROPN
ejpam-4738	108	31	i∈i	i∈i	PROPN
ejpam-4738	108	32	ni	ni	PROPN
ejpam-4738	108	33	for	for	ADP
ejpam-4738	108	34	all	all	PRON
ejpam-4738	108	35	(	(	PUNCT
ejpam-4738	108	36	xi)i∈i	xi)i∈i	NUM
ejpam-4738	108	37	,	,	PUNCT
ejpam-4738	108	38	(	(	PUNCT
ejpam-4738	108	39	yi)i∈i	yi)i∈i	NUM
ejpam-4738	108	40	∈	∈	PROPN
ejpam-4738	108	41	∏	∏	PROPN
ejpam-4738	108	42	i∈i	i∈i	PROPN
ejpam-4738	108	43	ni	ni	PROPN
ejpam-4738	108	44	.	.	PROPN
ejpam-4738	108	45	theorem	theorem	VERB
ejpam-4738	108	46	3	3	NUM
ejpam-4738	108	47	.	.	PUNCT
ejpam-4738	109	1	every	every	DET
ejpam-4738	109	2	d	d	ADJ
ejpam-4738	109	3	-	-	ADJ
ejpam-4738	109	4	ideal	ideal	ADJ
ejpam-4738	109	5	direct	direct	ADJ
ejpam-4738	109	6	product	product	NOUN
ejpam-4738	109	7	d	d	X
ejpam-4738	109	8	-	-	PUNCT
ejpam-4738	109	9	algebra	algebra	NOUN
ejpam-4738	109	10	is	be	AUX
ejpam-4738	109	11	an	an	DET
ejpam-4738	109	12	ideal	ideal	ADJ
ejpam-4738	109	13	direct	direct	ADJ
ejpam-4738	109	14	product	product	NOUN
ejpam-4738	109	15	d	d	NOUN
ejpam-4738	109	16	-	-	PUNCT
ejpam-4738	109	17	algebra	algebra	NOUN
ejpam-4738	109	18	.	.	PUNCT
ejpam-4738	110	1	proof	proof	NOUN
ejpam-4738	110	2	.	.	PUNCT
ejpam-4738	111	1	let	let	VERB
ejpam-4738	111	2	(	(	PUNCT
ejpam-4738	111	3	∏	∏	PROPN
ejpam-4738	111	4	i∈i	i∈i	ADJ
ejpam-4738	111	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	111	6	,	,	PUNCT
ejpam-4738	111	7	(	(	PUNCT
ejpam-4738	111	8	0i)i∈i	0i)i∈i	X
ejpam-4738	111	9	)	)	PUNCT
ejpam-4738	111	10	be	be	AUX
ejpam-4738	111	11	a	a	DET
ejpam-4738	111	12	direct	direct	ADJ
ejpam-4738	111	13	product	product	NOUN
ejpam-4738	111	14	d	d	NOUN
ejpam-4738	111	15	-	-	PUNCT
ejpam-4738	111	16	algebra	algebra	NOUN
ejpam-4738	111	17	and	and	CCONJ
ejpam-4738	111	18	∏	∏	PROPN
ejpam-4738	111	19	i∈i	i∈i	NOUN
ejpam-4738	111	20	ni	ni	PROPN
ejpam-4738	111	21	be	be	AUX
ejpam-4738	111	22	a	a	DET
ejpam-4738	111	23	d	d	ADJ
ejpam-4738	111	24	-	-	PUNCT
ejpam-4738	111	25	ideal	ideal	ADJ
ejpam-4738	111	26	of∏	of∏	NOUN
ejpam-4738	111	27	i∈i	i∈i	ADJ
ejpam-4738	111	28	xi	xi	PROPN
ejpam-4738	111	29	.	.	PUNCT
ejpam-4738	112	1	since	since	SCONJ
ejpam-4738	112	2	xi	xi	PROPN
ejpam-4738	112	3	∗	∗	NOUN
ejpam-4738	112	4	xi	xi	PUNCT
ejpam-4738	113	1	=	=	NOUN
ejpam-4738	113	2	0i	0i	PROPN
ejpam-4738	113	3	for	for	ADP
ejpam-4738	113	4	all	all	PRON
ejpam-4738	113	5	i	i	PRON
ejpam-4738	113	6	∈	∈	PROPN
ejpam-4738	113	7	i	i	PRON
ejpam-4738	113	8	implies	imply	VERB
ejpam-4738	113	9	that	that	SCONJ
ejpam-4738	113	10	(	(	PUNCT
ejpam-4738	113	11	xi)i∈i	xi)i∈i	NUM
ejpam-4738	113	12	⊙	⊙	X
ejpam-4738	113	13	(	(	PUNCT
ejpam-4738	113	14	xi)i∈i	xi)i∈i	NUM
ejpam-4738	113	15	=	=	SYM
ejpam-4738	113	16	(	(	PUNCT
ejpam-4738	113	17	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	113	18	∈	∈	PROPN
ejpam-4738	113	19	∏	∏	PROPN
ejpam-4738	113	20	i∈i	i∈i	PROPN
ejpam-4738	113	21	ni	ni	PROPN
ejpam-4738	113	22	for	for	ADP
ejpam-4738	113	23	any	any	DET
ejpam-4738	113	24	(	(	PUNCT
ejpam-4738	113	25	xi)i∈i	xi)i∈i	NUM
ejpam-4738	113	26	∈	∈	PROPN
ejpam-4738	113	27	∏	∏	PROPN
ejpam-4738	113	28	i∈i	i∈i	PROPN
ejpam-4738	113	29	ni	ni	PROPN
ejpam-4738	113	30	.	.	PUNCT
ejpam-4738	114	1	thus	thus	ADV
ejpam-4738	114	2	∏	∏	NUM
ejpam-4738	114	3	i∈i	i∈i	NOUN
ejpam-4738	114	4	ni	ni	PROPN
ejpam-4738	114	5	is	be	AUX
ejpam-4738	114	6	an	an	DET
ejpam-4738	114	7	ideal	ideal	NOUN
ejpam-4738	114	8	of	of	ADP
ejpam-4738	114	9	∏	∏	PROPN
ejpam-4738	114	10	i∈i	i∈i	ADJ
ejpam-4738	114	11	xi	xi	PROPN
ejpam-4738	114	12	.	.	PUNCT
ejpam-4738	114	13	theorem	theorem	VERB
ejpam-4738	114	14	4	4	NUM
ejpam-4738	114	15	.	.	PUNCT
ejpam-4738	115	1	every	every	DET
ejpam-4738	115	2	d	d	NOUN
ejpam-4738	115	3	-	-	PUNCT
ejpam-4738	115	4	ideal	ideal	VERB
ejpam-4738	115	5	a	a	DET
ejpam-4738	115	6	direct	direct	ADJ
ejpam-4738	115	7	product	product	NOUN
ejpam-4738	115	8	d	d	NOUN
ejpam-4738	115	9	-	-	PUNCT
ejpam-4738	115	10	algebra	algebra	NOUN
ejpam-4738	115	11	is	be	AUX
ejpam-4738	115	12	a	a	DET
ejpam-4738	115	13	sub	sub	ADJ
ejpam-4738	115	14	-	-	ADJ
ejpam-4738	115	15	direct	direct	ADJ
ejpam-4738	115	16	product	product	NOUN
ejpam-4738	115	17	d	d	NOUN
ejpam-4738	115	18	-	-	PUNCT
ejpam-4738	115	19	algebra	algebra	NOUN
ejpam-4738	115	20	.	.	PUNCT
ejpam-4738	116	1	proof	proof	NOUN
ejpam-4738	116	2	.	.	PUNCT
ejpam-4738	117	1	it	it	PRON
ejpam-4738	117	2	is	be	AUX
ejpam-4738	117	3	clear	clear	ADJ
ejpam-4738	117	4	by	by	ADP
ejpam-4738	117	5	definition	definition	NOUN
ejpam-4738	117	6	7	7	NUM
ejpam-4738	117	7	and	and	CCONJ
ejpam-4738	117	8	8	8	NUM
ejpam-4738	117	9	definition	definition	NOUN
ejpam-4738	117	10	9	9	NUM
ejpam-4738	117	11	.	.	PUNCT
ejpam-4738	118	1	let	let	AUX
ejpam-4738	118	2	(	(	PUNCT
ejpam-4738	118	3	∏	∏	PROPN
ejpam-4738	118	4	i∈i	i∈i	ADJ
ejpam-4738	118	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	118	6	,	,	PUNCT
ejpam-4738	118	7	(	(	PUNCT
ejpam-4738	118	8	0i)i∈i	0i)i∈i	X
ejpam-4738	118	9	)	)	PUNCT
ejpam-4738	118	10	be	be	AUX
ejpam-4738	118	11	a	a	DET
ejpam-4738	118	12	direct	direct	ADJ
ejpam-4738	118	13	product	product	NOUN
ejpam-4738	118	14	d	d	NOUN
ejpam-4738	118	15	-	-	PUNCT
ejpam-4738	118	16	algebra	algebra	NOUN
ejpam-4738	118	17	and	and	CCONJ
ejpam-4738	118	18	(	(	PUNCT
ejpam-4738	118	19	ai)i∈i	ai)i∈i	NUM
ejpam-4738	118	20	∈	∈	PROPN
ejpam-4738	118	21	∏	∏	PROPN
ejpam-4738	118	22	i∈i	i∈i	ADJ
ejpam-4738	118	23	xi	xi	PROPN
ejpam-4738	118	24	.	.	PUNCT
ejpam-4738	118	25	define	define	VERB
ejpam-4738	118	26	the	the	DET
ejpam-4738	118	27	set	set	NOUN
ejpam-4738	118	28	(	(	PUNCT
ejpam-4738	118	29	ai)i∈i	ai)i∈i	NOUN
ejpam-4738	118	30	⊙	⊙	X
ejpam-4738	118	31	∏	∏	PROPN
ejpam-4738	118	32	i∈i	i∈i	ADV
ejpam-4738	118	33	xi:=	xi:=	PUNCT
ejpam-4738	118	34	{	{	PUNCT
ejpam-4738	118	35	(	(	PUNCT
ejpam-4738	118	36	ai)i∈i	ai)i∈i	NUM
ejpam-4738	118	37	⊙	⊙	NOUN
ejpam-4738	118	38	(	(	PUNCT
ejpam-4738	118	39	xi)i∈i	xi)i∈i	NUM
ejpam-4738	118	40	|	|	CCONJ
ejpam-4738	118	41	(	(	PUNCT
ejpam-4738	118	42	xi)i∈i	xi)i∈i	NUM
ejpam-4738	118	43	∈	∈	PROPN
ejpam-4738	118	44	∏	∏	PROPN
ejpam-4738	118	45	i∈i	i∈i	ADJ
ejpam-4738	118	46	xi	xi	ADP
ejpam-4738	118	47	}	}	PUNCT
ejpam-4738	118	48	.	.	PUNCT
ejpam-4738	119	1	we	we	PRON
ejpam-4738	119	2	say	say	VERB
ejpam-4738	119	3	that	that	SCONJ
ejpam-4738	119	4	∏	∏	PROPN
ejpam-4738	119	5	i∈i	i∈i	NOUN
ejpam-4738	119	6	xi	xi	INTJ
ejpam-4738	119	7	is	be	AUX
ejpam-4738	119	8	to	to	PART
ejpam-4738	119	9	be	be	AUX
ejpam-4738	119	10	an	an	DET
ejpam-4738	119	11	edge	edge	NOUN
ejpam-4738	119	12	direct	direct	ADJ
ejpam-4738	119	13	product	product	NOUN
ejpam-4738	119	14	of	of	ADP
ejpam-4738	119	15	d	d	NOUN
ejpam-4738	119	16	-	-	PUNCT
ejpam-4738	119	17	algebra	algebra	NOUN
ejpam-4738	119	18	if	if	SCONJ
ejpam-4738	119	19	(	(	PUNCT
ejpam-4738	119	20	ai)i∈i	ai)i∈i	NOUN
ejpam-4738	119	21	⊙	⊙	X
ejpam-4738	119	22	∏	∏	PROPN
ejpam-4738	119	23	i∈i	i∈i	ADV
ejpam-4738	119	24	xi:=	xi:=	PUNCT
ejpam-4738	119	25	{	{	PUNCT
ejpam-4738	119	26	(	(	PUNCT
ejpam-4738	119	27	ai)i∈i	ai)i∈i	NUM
ejpam-4738	119	28	,	,	PUNCT
ejpam-4738	119	29	(	(	PUNCT
ejpam-4738	119	30	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	119	31	}	}	PUNCT
ejpam-4738	119	32	.	.	PUNCT
ejpam-4738	120	1	example	example	NOUN
ejpam-4738	121	1	7	7	NUM
ejpam-4738	121	2	.	.	PUNCT
ejpam-4738	122	1	[	[	X
ejpam-4738	122	2	9],[7	9],[7	X
ejpam-4738	122	3	]	]	PUNCT
ejpam-4738	122	4	let	let	VERB
ejpam-4738	122	5	x1	x1	PROPN
ejpam-4738	122	6	=	=	PUNCT
ejpam-4738	122	7	{	{	PUNCT
ejpam-4738	122	8	0	0	NUM
ejpam-4738	122	9	,	,	PUNCT
ejpam-4738	122	10	1	1	NUM
ejpam-4738	122	11	,	,	PUNCT
ejpam-4738	122	12	2	2	NUM
ejpam-4738	122	13	,	,	PUNCT
ejpam-4738	122	14	3	3	NUM
ejpam-4738	122	15	}	}	PUNCT
ejpam-4738	122	16	and	and	CCONJ
ejpam-4738	122	17	x2	x2	NOUN
ejpam-4738	122	18	=	=	PRON
ejpam-4738	122	19	{	{	PUNCT
ejpam-4738	122	20	0′	0′	NUM
ejpam-4738	122	21	,	,	PUNCT
ejpam-4738	122	22	a	a	DET
ejpam-4738	122	23	,	,	PUNCT
ejpam-4738	122	24	b	b	NOUN
ejpam-4738	122	25	,	,	PUNCT
ejpam-4738	122	26	c	c	AUX
ejpam-4738	122	27	}	}	PUNCT
ejpam-4738	122	28	be	be	AUX
ejpam-4738	122	29	the	the	DET
ejpam-4738	122	30	set	set	NOUN
ejpam-4738	122	31	with	with	ADP
ejpam-4738	122	32	a	a	DET
ejpam-4738	122	33	binary	binary	ADJ
ejpam-4738	122	34	operation	operation	NOUN
ejpam-4738	122	35	∗	∗	NOUN
ejpam-4738	122	36	and	and	CCONJ
ejpam-4738	122	37	∗′	∗′	NOUN
ejpam-4738	122	38	respectively	respectively	ADV
ejpam-4738	122	39	that	that	SCONJ
ejpam-4738	122	40	following	follow	VERB
ejpam-4738	122	41	2	2	NUM
ejpam-4738	122	42	of	of	ADP
ejpam-4738	122	43	tables	table	NOUN
ejpam-4738	122	44	:	:	PUNCT
ejpam-4738	122	45	∗	∗	NOUN
ejpam-4738	122	46	0	0	NUM
ejpam-4738	122	47	1	1	NUM
ejpam-4738	122	48	2	2	NUM
ejpam-4738	122	49	3	3	NUM
ejpam-4738	122	50	0	0	NUM
ejpam-4738	122	51	0	0	NUM
ejpam-4738	122	52	0	0	NUM
ejpam-4738	122	53	0	0	NUM
ejpam-4738	122	54	0	0	NUM
ejpam-4738	122	55	1	1	NUM
ejpam-4738	122	56	1	1	NUM
ejpam-4738	122	57	0	0	NUM
ejpam-4738	122	58	0	0	NUM
ejpam-4738	122	59	1	1	NUM
ejpam-4738	122	60	2	2	NUM
ejpam-4738	122	61	2	2	NUM
ejpam-4738	122	62	2	2	NUM
ejpam-4738	122	63	0	0	NUM
ejpam-4738	122	64	0	0	NUM
ejpam-4738	122	65	3	3	NUM
ejpam-4738	122	66	3	3	NUM
ejpam-4738	122	67	3	3	NUM
ejpam-4738	122	68	3	3	NUM
ejpam-4738	122	69	0	0	NUM
ejpam-4738	122	70	∗′	∗′	NUM
ejpam-4738	122	71	0	0	NUM
ejpam-4738	122	72	′	′	NUM
ejpam-4738	122	73	a	a	DET
ejpam-4738	122	74	b	b	NOUN
ejpam-4738	122	75	c	c	NOUN
ejpam-4738	122	76	0	0	NUM
ejpam-4738	123	1	′	′	NUM
ejpam-4738	123	2	0	0	NUM
ejpam-4738	124	1	′	′	NUM
ejpam-4738	124	2	0	0	NUM
ejpam-4738	125	1	′	′	NUM
ejpam-4738	125	2	0	0	NUM
ejpam-4738	126	1	′	′	NUM
ejpam-4738	126	2	0	0	NUM
ejpam-4738	127	1	′	′	NUM
ejpam-4738	127	2	a	a	DET
ejpam-4738	127	3	a	a	DET
ejpam-4738	127	4	0	0	NUM
ejpam-4738	127	5	′	′	NUM
ejpam-4738	127	6	0	0	NUM
ejpam-4738	128	1	′	′	NUM
ejpam-4738	128	2	b	b	PROPN
ejpam-4738	128	3	b	b	X
ejpam-4738	128	4	b	b	PROPN
ejpam-4738	128	5	b	b	PROPN
ejpam-4738	128	6	0	0	NUM
ejpam-4738	128	7	′	′	NUM
ejpam-4738	128	8	0	0	NUM
ejpam-4738	129	1	′	′	NUM
ejpam-4738	130	1	c	c	NOUN
ejpam-4738	130	2	c	c	NOUN
ejpam-4738	130	3	c	c	NOUN
ejpam-4738	130	4	c	c	NOUN
ejpam-4738	130	5	0	0	NUM
ejpam-4738	131	1	′	′	NUM
ejpam-4738	131	2	then	then	ADV
ejpam-4738	131	3	(	(	PUNCT
ejpam-4738	131	4	x1	x1	PROPN
ejpam-4738	131	5	,	,	PUNCT
ejpam-4738	131	6	∗	∗	NOUN
ejpam-4738	131	7	,	,	PUNCT
ejpam-4738	131	8	0	0	NUM
ejpam-4738	131	9	)	)	PUNCT
ejpam-4738	131	10	and	and	CCONJ
ejpam-4738	131	11	(	(	PUNCT
ejpam-4738	131	12	x2	x2	PROPN
ejpam-4738	131	13	,	,	PUNCT
ejpam-4738	131	14	∗	∗	NOUN
ejpam-4738	131	15	′	′	NUM
ejpam-4738	131	16	,	,	PUNCT
ejpam-4738	131	17	0	0	NUM
ejpam-4738	131	18	)	)	PUNCT
ejpam-4738	131	19	are	be	AUX
ejpam-4738	131	20	edge	edge	NOUN
ejpam-4738	131	21	d	d	NOUN
ejpam-4738	131	22	-	-	PUNCT
ejpam-4738	131	23	algebras	algebras	X
ejpam-4738	131	24	.	.	PUNCT
ejpam-4738	132	1	but	but	CCONJ
ejpam-4738	132	2	x1	x1	DET
ejpam-4738	132	3	×x2	×x2	NOUN
ejpam-4738	132	4	is	be	AUX
ejpam-4738	132	5	not	not	PART
ejpam-4738	132	6	an	an	DET
ejpam-4738	132	7	edge	edge	NOUN
ejpam-4738	132	8	direct	direct	ADJ
ejpam-4738	132	9	product	product	NOUN
ejpam-4738	132	10	d	d	NOUN
ejpam-4738	132	11	-	-	PUNCT
ejpam-4738	132	12	algebra	algebra	NOUN
ejpam-4738	132	13	,	,	PUNCT
ejpam-4738	132	14	because	because	SCONJ
ejpam-4738	132	15	of	of	ADP
ejpam-4738	132	16	(	(	PUNCT
ejpam-4738	132	17	2	2	NUM
ejpam-4738	132	18	,	,	PUNCT
ejpam-4738	132	19	a)⊙(x1×x2	a)⊙(x1×x2	NOUN
ejpam-4738	132	20	)	)	PUNCT
ejpam-4738	132	21	=	=	SYM
ejpam-4738	132	22	{	{	PUNCT
ejpam-4738	132	23	(	(	PUNCT
ejpam-4738	132	24	2	2	NUM
ejpam-4738	132	25	,	,	PUNCT
ejpam-4738	132	26	a	a	NOUN
ejpam-4738	132	27	)	)	PUNCT
ejpam-4738	132	28	,	,	PUNCT
ejpam-4738	132	29	(	(	PUNCT
ejpam-4738	132	30	2	2	NUM
ejpam-4738	132	31	,	,	PUNCT
ejpam-4738	132	32	0	0	NUM
ejpam-4738	132	33	)	)	PUNCT
ejpam-4738	132	34	,	,	PUNCT
ejpam-4738	132	35	(	(	PUNCT
ejpam-4738	132	36	0	0	NUM
ejpam-4738	132	37	,	,	PUNCT
ejpam-4738	132	38	a	a	PRON
ejpam-4738	132	39	)	)	PUNCT
ejpam-4738	132	40	,	,	PUNCT
ejpam-4738	132	41	(	(	PUNCT
ejpam-4738	132	42	0	0	NUM
ejpam-4738	132	43	,	,	PUNCT
ejpam-4738	132	44	0′	0′	NUM
ejpam-4738	132	45	)	)	PUNCT
ejpam-4738	132	46	}	}	PUNCT
ejpam-4738	133	1	=	=	SYM
ejpam-4738	133	2	̸	̸	NUM
ejpam-4738	133	3	{	{	PUNCT
ejpam-4738	133	4	(	(	PUNCT
ejpam-4738	133	5	0	0	NUM
ejpam-4738	133	6	,	,	PUNCT
ejpam-4738	133	7	0′	0′	NUM
ejpam-4738	133	8	)	)	PUNCT
ejpam-4738	133	9	,	,	PUNCT
ejpam-4738	133	10	(	(	PUNCT
ejpam-4738	133	11	2	2	NUM
ejpam-4738	133	12	,	,	PUNCT
ejpam-4738	133	13	a	a	NOUN
ejpam-4738	133	14	)	)	PUNCT
ejpam-4738	133	15	}	}	PUNCT
ejpam-4738	133	16	.	.	PUNCT
ejpam-4738	134	1	theorem	theorem	NOUN
ejpam-4738	134	2	5	5	NUM
ejpam-4738	134	3	.	.	PUNCT
ejpam-4738	135	1	let	let	VERB
ejpam-4738	135	2	(	(	PUNCT
ejpam-4738	135	3	∏	∏	PROPN
ejpam-4738	135	4	i∈i	i∈i	ADJ
ejpam-4738	135	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	135	6	,	,	PUNCT
ejpam-4738	135	7	(	(	PUNCT
ejpam-4738	135	8	0i)i∈i	0i)i∈i	X
ejpam-4738	135	9	)	)	PUNCT
ejpam-4738	135	10	be	be	VERB
ejpam-4738	135	11	an	an	DET
ejpam-4738	135	12	edge	edge	NOUN
ejpam-4738	135	13	direct	direct	ADJ
ejpam-4738	135	14	product	product	NOUN
ejpam-4738	135	15	d	d	NOUN
ejpam-4738	135	16	-	-	PUNCT
ejpam-4738	135	17	algebra	algebra	NOUN
ejpam-4738	135	18	and	and	CCONJ
ejpam-4738	135	19	∏	∏	PROPN
ejpam-4738	135	20	i∈i	i∈i	NOUN
ejpam-4738	135	21	ni	ni	PROPN
ejpam-4738	135	22	be	be	AUX
ejpam-4738	135	23	an	an	DET
ejpam-4738	135	24	ideal	ideal	ADJ
ejpam-4738	135	25	direct	direct	ADJ
ejpam-4738	135	26	product	product	NOUN
ejpam-4738	135	27	of	of	ADP
ejpam-4738	135	28	∏	∏	PROPN
ejpam-4738	135	29	i∈i	i∈i	ADJ
ejpam-4738	135	30	xi	xi	INTJ
ejpam-4738	135	31	.	.	PUNCT
ejpam-4738	136	1	if	if	SCONJ
ejpam-4738	136	2	(	(	PUNCT
ejpam-4738	136	3	ni)i∈i	ni)i∈i	NUM
ejpam-4738	136	4	∈	∈	PROPN
ejpam-4738	136	5	∏	∏	PROPN
ejpam-4738	136	6	i∈i	i∈i	ADJ
ejpam-4738	136	7	ni	ni	PROPN
ejpam-4738	136	8	and	and	CCONJ
ejpam-4738	136	9	(	(	PUNCT
ejpam-4738	136	10	xi)i∈i	xi)i∈i	NUM
ejpam-4738	136	11	∈	∈	PROPN
ejpam-4738	136	12	∏	∏	PROPN
ejpam-4738	136	13	i∈i	i∈i	ADJ
ejpam-4738	136	14	xi	xi	PROPN
ejpam-4738	136	15	,	,	PUNCT
ejpam-4738	136	16	then	then	ADV
ejpam-4738	136	17	(	(	PUNCT
ejpam-4738	136	18	xi)i∈i	xi)i∈i	PROPN
ejpam-4738	136	19	⊙	⊙	X
ejpam-4738	136	20	(	(	PUNCT
ejpam-4738	136	21	(	(	PUNCT
ejpam-4738	136	22	xi)i∈i	xi)i∈i	NUM
ejpam-4738	136	23	⊙	⊙	X
ejpam-4738	136	24	(	(	PUNCT
ejpam-4738	136	25	ni)i∈i	ni)i∈i	NUM
ejpam-4738	136	26	)	)	PUNCT
ejpam-4738	136	27	∈	∈	PROPN
ejpam-4738	136	28	∏	∏	PROPN
ejpam-4738	136	29	i∈i	i∈i	PROPN
ejpam-4738	136	30	ni	ni	PROPN
ejpam-4738	136	31	.	.	PROPN
ejpam-4738	136	32	proof	proof	NOUN
ejpam-4738	136	33	.	.	PUNCT
ejpam-4738	137	1	consider	consider	VERB
ejpam-4738	137	2	(	(	PUNCT
ejpam-4738	137	3	(	(	PUNCT
ejpam-4738	137	4	xi)i∈i⊙((xi)i∈i⊙(ni)i∈i))⊙(ni)i∈i	xi)i∈i⊙((xi)i∈i⊙(ni)i∈i))⊙(ni)i∈i	NOUN
ejpam-4738	137	5	=	=	SYM
ejpam-4738	137	6	(	(	PUNCT
ejpam-4738	137	7	(	(	PUNCT
ejpam-4738	137	8	xi)i∈i⊙(ni)i∈i))⊙((xi)i∈i⊙	xi)i∈i⊙(ni)i∈i))⊙((xi)i∈i⊙	PROPN
ejpam-4738	137	9	(	(	PUNCT
ejpam-4738	137	10	ni)i∈i	ni)i∈i	NUM
ejpam-4738	137	11	)	)	PUNCT
ejpam-4738	137	12	)	)	PUNCT
ejpam-4738	138	1	=	=	SYM
ejpam-4738	138	2	(	(	PUNCT
ejpam-4738	138	3	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	138	4	,	,	PUNCT
ejpam-4738	138	5	by	by	ADP
ejpam-4738	138	6	definition	definition	NOUN
ejpam-4738	138	7	7	7	NUM
ejpam-4738	138	8	and	and	CCONJ
ejpam-4738	138	9	theorem	theorem	VERB
ejpam-4738	138	10	1	1	NUM
ejpam-4738	138	11	,	,	PUNCT
ejpam-4738	138	12	(	(	PUNCT
ejpam-4738	138	13	xi)i∈i	xi)i∈i	NUM
ejpam-4738	138	14	⊙	⊙	X
ejpam-4738	138	15	(	(	PUNCT
ejpam-4738	138	16	(	(	PUNCT
ejpam-4738	138	17	xi)i∈i	xi)i∈i	NUM
ejpam-4738	138	18	⊙	⊙	X
ejpam-4738	138	19	(	(	PUNCT
ejpam-4738	138	20	ni)i∈i	ni)i∈i	NUM
ejpam-4738	138	21	)	)	PUNCT
ejpam-4738	138	22	∈	∈	PROPN
ejpam-4738	138	23	∏	∏	PROPN
ejpam-4738	138	24	i∈i	i∈i	PROPN
ejpam-4738	138	25	ni	ni	PROPN
ejpam-4738	138	26	.	.	PROPN
ejpam-4738	138	27	definition	definition	NOUN
ejpam-4738	138	28	10	10	NUM
ejpam-4738	138	29	.	.	PUNCT
ejpam-4738	139	1	a	a	DET
ejpam-4738	139	2	direct	direct	ADJ
ejpam-4738	139	3	product	product	NOUN
ejpam-4738	139	4	d	d	NOUN
ejpam-4738	139	5	-	-	PUNCT
ejpam-4738	139	6	algebra	algebra	NOUN
ejpam-4738	139	7	(	(	PUNCT
ejpam-4738	139	8	∏	∏	PROPN
ejpam-4738	139	9	i∈i	i∈i	ADJ
ejpam-4738	139	10	xi,⊙	xi,⊙	PROPN
ejpam-4738	139	11	,	,	PUNCT
ejpam-4738	139	12	(	(	PUNCT
ejpam-4738	139	13	0i)i∈i	0i)i∈i	NUM
ejpam-4738	139	14	)	)	PUNCT
ejpam-4738	139	15	is	be	AUX
ejpam-4738	139	16	said	say	VERB
ejpam-4738	139	17	to	to	PART
ejpam-4738	139	18	be	be	AUX
ejpam-4738	139	19	positive	positive	ADJ
ejpam-4738	139	20	implicative	implicative	ADJ
ejpam-4738	139	21	if	if	SCONJ
ejpam-4738	139	22	(	(	PUNCT
ejpam-4738	139	23	(	(	PUNCT
ejpam-4738	139	24	xi)i∈i	xi)i∈i	NUM
ejpam-4738	139	25	⊙	⊙	X
ejpam-4738	139	26	(	(	PUNCT
ejpam-4738	139	27	yi)i∈i	yi)i∈i	NUM
ejpam-4738	139	28	)	)	PUNCT
ejpam-4738	139	29	⊙	⊙	NOUN
ejpam-4738	139	30	(	(	PUNCT
ejpam-4738	139	31	zi)i∈i	zi)i∈i	NUM
ejpam-4738	139	32	=	=	SYM
ejpam-4738	139	33	(	(	PUNCT
ejpam-4738	139	34	(	(	PUNCT
ejpam-4738	139	35	xi)i∈i	xi)i∈i	NUM
ejpam-4738	139	36	⊙	⊙	X
ejpam-4738	139	37	(	(	PUNCT
ejpam-4738	139	38	zi)i∈i	zi)i∈i	NUM
ejpam-4738	139	39	)	)	PUNCT
ejpam-4738	139	40	⊙	⊙	NOUN
ejpam-4738	139	41	(	(	PUNCT
ejpam-4738	139	42	(	(	PUNCT
ejpam-4738	139	43	yi)i∈i	yi)i∈i	NUM
ejpam-4738	139	44	⊙	⊙	NOUN
ejpam-4738	139	45	(	(	PUNCT
ejpam-4738	139	46	zi)i∈i	zi)i∈i	NUM
ejpam-4738	139	47	)	)	PUNCT
ejpam-4738	139	48	for	for	ADP
ejpam-4738	139	49	all	all	PRON
ejpam-4738	139	50	(	(	PUNCT
ejpam-4738	139	51	xi)i∈i	xi)i∈i	NUM
ejpam-4738	139	52	,	,	PUNCT
ejpam-4738	139	53	(	(	PUNCT
ejpam-4738	139	54	yi)i∈i	yi)i∈i	NUM
ejpam-4738	139	55	,	,	PUNCT
ejpam-4738	139	56	(	(	PUNCT
ejpam-4738	139	57	zi)i∈i	zi)i∈i	NUM
ejpam-4738	139	58	∈	∈	PROPN
ejpam-4738	139	59	∏	∏	PROPN
ejpam-4738	139	60	i∈i	i∈i	ADJ
ejpam-4738	139	61	xi	xi	PROPN
ejpam-4738	139	62	.	.	PUNCT
ejpam-4738	140	1	references	reference	NOUN
ejpam-4738	140	2	1003	1003	NUM
ejpam-4738	140	3	theorem	theorem	VERB
ejpam-4738	140	4	6	6	NUM
ejpam-4738	140	5	.	.	PUNCT
ejpam-4738	141	1	let	let	VERB
ejpam-4738	141	2	{	{	PUNCT
ejpam-4738	141	3	(	(	PUNCT
ejpam-4738	141	4	xi	xi	PROPN
ejpam-4738	141	5	,	,	PUNCT
ejpam-4738	141	6	∗	∗	NOUN
ejpam-4738	141	7	,	,	PUNCT
ejpam-4738	141	8	0i	0i	NOUN
ejpam-4738	141	9	)	)	PUNCT
ejpam-4738	142	1	|	|	ADV
ejpam-4738	142	2	i	i	PRON
ejpam-4738	142	3	∈	∈	VERB
ejpam-4738	142	4	i	i	PRON
ejpam-4738	142	5	}	}	PUNCT
ejpam-4738	142	6	be	be	VERB
ejpam-4738	142	7	a	a	DET
ejpam-4738	142	8	non	non	ADJ
ejpam-4738	142	9	-	-	ADJ
ejpam-4738	142	10	empty	empty	ADJ
ejpam-4738	142	11	family	family	NOUN
ejpam-4738	142	12	of	of	ADP
ejpam-4738	142	13	positive	positive	ADJ
ejpam-4738	142	14	implicative	implicative	ADJ
ejpam-4738	142	15	d	d	NOUN
ejpam-4738	142	16	-	-	PUNCT
ejpam-4738	142	17	algebra	algebra	NOUN
ejpam-4738	142	18	,	,	PUNCT
ejpam-4738	142	19	then	then	ADV
ejpam-4738	142	20	(	(	PUNCT
ejpam-4738	142	21	∏	∏	PROPN
ejpam-4738	142	22	i∈i	i∈i	ADJ
ejpam-4738	142	23	xi,⊙	xi,⊙	PROPN
ejpam-4738	142	24	,	,	PUNCT
ejpam-4738	142	25	(	(	PUNCT
ejpam-4738	142	26	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	142	27	)	)	PUNCT
ejpam-4738	142	28	is	be	AUX
ejpam-4738	142	29	a	a	DET
ejpam-4738	142	30	positive	positive	ADJ
ejpam-4738	142	31	implicative	implicative	ADJ
ejpam-4738	142	32	direct	direct	ADJ
ejpam-4738	142	33	product	product	NOUN
ejpam-4738	142	34	d	d	NOUN
ejpam-4738	142	35	-	-	PUNCT
ejpam-4738	142	36	algebra	algebra	NOUN
ejpam-4738	142	37	.	.	PUNCT
ejpam-4738	143	1	proof	proof	NOUN
ejpam-4738	143	2	.	.	PUNCT
ejpam-4738	144	1	let	let	VERB
ejpam-4738	144	2	(	(	PUNCT
ejpam-4738	144	3	xi)i∈i	xi)i∈i	NUM
ejpam-4738	144	4	,	,	PUNCT
ejpam-4738	144	5	(	(	PUNCT
ejpam-4738	144	6	yi)i∈i	yi)i∈i	NUM
ejpam-4738	144	7	,	,	PUNCT
ejpam-4738	144	8	(	(	PUNCT
ejpam-4738	144	9	zi)i∈i	zi)i∈i	NUM
ejpam-4738	144	10	∈	∈	PROPN
ejpam-4738	144	11	∏	∏	PROPN
ejpam-4738	144	12	i∈i	i∈i	ADJ
ejpam-4738	144	13	xi	xi	PROPN
ejpam-4738	144	14	.	.	PUNCT
ejpam-4738	145	1	then	then	ADV
ejpam-4738	145	2	(	(	PUNCT
ejpam-4738	145	3	(	(	PUNCT
ejpam-4738	145	4	xi)i∈i	xi)i∈i	NUM
ejpam-4738	145	5	⊙	⊙	X
ejpam-4738	145	6	(	(	PUNCT
ejpam-4738	145	7	yi)i∈i)⊙	yi)i∈i)⊙	PROPN
ejpam-4738	145	8	(	(	PUNCT
ejpam-4738	145	9	zi)i∈i	zi)i∈i	NUM
ejpam-4738	145	10	=	=	SYM
ejpam-4738	145	11	(	(	PUNCT
ejpam-4738	145	12	xi	xi	X
ejpam-4738	145	13	∗	∗	PROPN
ejpam-4738	145	14	yi)i∈i	yi)i∈i	NUM
ejpam-4738	145	15	∗	∗	NOUN
ejpam-4738	145	16	(	(	PUNCT
ejpam-4738	145	17	zi))i∈i	zi))i∈i	NUM
ejpam-4738	145	18	=	=	SYM
ejpam-4738	145	19	(	(	PUNCT
ejpam-4738	145	20	xi	xi	X
ejpam-4738	145	21	∗	∗	NOUN
ejpam-4738	145	22	zi)i∈i	zi)i∈i	NUM
ejpam-4738	145	23	∗	∗	NOUN
ejpam-4738	145	24	(	(	PUNCT
ejpam-4738	145	25	yi	yi	PROPN
ejpam-4738	145	26	∗	∗	VERB
ejpam-4738	145	27	zi)i∈i	zi)i∈i	NUM
ejpam-4738	145	28	=	=	SYM
ejpam-4738	145	29	(	(	PUNCT
ejpam-4738	145	30	(	(	PUNCT
ejpam-4738	145	31	xi)i∈i	xi)i∈i	NUM
ejpam-4738	145	32	⊙	⊙	X
ejpam-4738	145	33	(	(	PUNCT
ejpam-4738	145	34	zi)i∈i)⊙	zi)i∈i)⊙	NUM
ejpam-4738	145	35	(	(	PUNCT
ejpam-4738	145	36	(	(	PUNCT
ejpam-4738	145	37	yi)i∈i	yi)i∈i	NUM
ejpam-4738	145	38	⊙	⊙	NOUN
ejpam-4738	145	39	(	(	PUNCT
ejpam-4738	145	40	zi)i∈i	zi)i∈i	NUM
ejpam-4738	145	41	)	)	PUNCT
ejpam-4738	145	42	.	.	PUNCT
ejpam-4738	146	1	thus	thus	ADV
ejpam-4738	146	2	∏	∏	NUM
ejpam-4738	146	3	i∈i	i∈i	NOUN
ejpam-4738	146	4	xi	xi	VERB
ejpam-4738	146	5	is	be	AUX
ejpam-4738	146	6	a	a	DET
ejpam-4738	146	7	positive	positive	ADJ
ejpam-4738	146	8	implicative	implicative	ADJ
ejpam-4738	146	9	direct	direct	ADJ
ejpam-4738	146	10	product	product	NOUN
ejpam-4738	146	11	d	d	NOUN
ejpam-4738	146	12	-	-	PUNCT
ejpam-4738	146	13	algebra	algebra	NOUN
ejpam-4738	146	14	.	.	PUNCT
ejpam-4738	147	1	theorem	theorem	VERB
ejpam-4738	147	2	7	7	NUM
ejpam-4738	147	3	.	.	PUNCT
ejpam-4738	148	1	every	every	DET
ejpam-4738	148	2	ideal	ideal	NOUN
ejpam-4738	148	3	of	of	ADP
ejpam-4738	148	4	a	a	DET
ejpam-4738	148	5	positive	positive	ADJ
ejpam-4738	148	6	implicative	implicative	ADJ
ejpam-4738	148	7	direct	direct	ADJ
ejpam-4738	148	8	product	product	NOUN
ejpam-4738	148	9	d	d	X
ejpam-4738	148	10	-	-	PUNCT
ejpam-4738	148	11	algebra	algebra	NOUN
ejpam-4738	148	12	is	be	AUX
ejpam-4738	148	13	a	a	DET
ejpam-4738	148	14	d	d	ADJ
ejpam-4738	148	15	-	-	PUNCT
ejpam-4738	148	16	ideal	ideal	ADJ
ejpam-4738	148	17	direct	direct	ADJ
ejpam-4738	148	18	product	product	NOUN
ejpam-4738	148	19	d	d	NOUN
ejpam-4738	148	20	-	-	PUNCT
ejpam-4738	148	21	algebra	algebra	NOUN
ejpam-4738	148	22	.	.	PUNCT
ejpam-4738	149	1	proof	proof	NOUN
ejpam-4738	149	2	.	.	PUNCT
ejpam-4738	150	1	let	let	VERB
ejpam-4738	150	2	(	(	PUNCT
ejpam-4738	150	3	∏	∏	PROPN
ejpam-4738	150	4	i∈i	i∈i	ADJ
ejpam-4738	150	5	xi,⊙	xi,⊙	PROPN
ejpam-4738	150	6	,	,	PUNCT
ejpam-4738	150	7	(	(	PUNCT
ejpam-4738	150	8	0i)i∈i	0i)i∈i	X
ejpam-4738	150	9	)	)	PUNCT
ejpam-4738	150	10	be	be	VERB
ejpam-4738	150	11	a	a	DET
ejpam-4738	150	12	positive	positive	ADJ
ejpam-4738	150	13	implicative	implicative	ADJ
ejpam-4738	150	14	direct	direct	ADJ
ejpam-4738	150	15	product	product	NOUN
ejpam-4738	150	16	d	d	NOUN
ejpam-4738	150	17	-	-	PUNCT
ejpam-4738	150	18	algebra	algebra	NOUN
ejpam-4738	150	19	and∏	and∏	PROPN
ejpam-4738	150	20	i∈i	i∈i	ADJ
ejpam-4738	150	21	ni	ni	PROPN
ejpam-4738	150	22	is	be	AUX
ejpam-4738	150	23	an	an	DET
ejpam-4738	150	24	ideal	ideal	NOUN
ejpam-4738	150	25	of	of	ADP
ejpam-4738	150	26	∏	∏	PROPN
ejpam-4738	150	27	i∈i	i∈i	ADJ
ejpam-4738	150	28	xi	xi	PROPN
ejpam-4738	150	29	.	.	PUNCT
ejpam-4738	151	1	by	by	ADP
ejpam-4738	151	2	definition	definition	NOUN
ejpam-4738	151	3	10	10	NUM
ejpam-4738	151	4	,	,	PUNCT
ejpam-4738	151	5	we	we	PRON
ejpam-4738	151	6	have	have	VERB
ejpam-4738	151	7	(	(	PUNCT
ejpam-4738	151	8	ni)i∈i	ni)i∈i	NUM
ejpam-4738	151	9	⊙	⊙	NOUN
ejpam-4738	151	10	(	(	PUNCT
ejpam-4738	151	11	xi)i∈i)⊙	xi)i∈i)⊙	NUM
ejpam-4738	151	12	(	(	PUNCT
ejpam-4738	151	13	ni)i∈i	ni)i∈i	NUM
ejpam-4738	151	14	=	=	SYM
ejpam-4738	151	15	(	(	PUNCT
ejpam-4738	151	16	ni	ni	PROPN
ejpam-4738	151	17	∗	∗	PROPN
ejpam-4738	151	18	xi)i∈i	xi)i∈i	NUM
ejpam-4738	151	19	⊙	⊙	NOUN
ejpam-4738	151	20	(	(	PUNCT
ejpam-4738	151	21	ni)i∈i	ni)i∈i	NUM
ejpam-4738	151	22	=	=	SYM
ejpam-4738	151	23	(	(	PUNCT
ejpam-4738	151	24	(	(	PUNCT
ejpam-4738	151	25	ni	ni	PROPN
ejpam-4738	151	26	∗	∗	PROPN
ejpam-4738	151	27	xi	xi	NOUN
ejpam-4738	151	28	)	)	PUNCT
ejpam-4738	151	29	∗	∗	NOUN
ejpam-4738	151	30	(	(	PUNCT
ejpam-4738	151	31	(	(	PUNCT
ejpam-4738	151	32	ni))i∈i	ni))i∈i	NOUN
ejpam-4738	151	33	=	=	SYM
ejpam-4738	151	34	(	(	PUNCT
ejpam-4738	151	35	(	(	PUNCT
ejpam-4738	151	36	ni	ni	PROPN
ejpam-4738	151	37	∗	∗	PROPN
ejpam-4738	151	38	ni	ni	PROPN
ejpam-4738	151	39	)	)	PUNCT
ejpam-4738	151	40	∗	∗	NOUN
ejpam-4738	151	41	(	(	PUNCT
ejpam-4738	151	42	xi	xi	X
ejpam-4738	151	43	∗	∗	NOUN
ejpam-4738	151	44	ni))i∈i	ni))i∈i	NOUN
ejpam-4738	151	45	=	=	PUNCT
ejpam-4738	151	46	(	(	PUNCT
ejpam-4738	151	47	0i	0i	PROPN
ejpam-4738	151	48	∗	∗	PROPN
ejpam-4738	151	49	(	(	PUNCT
ejpam-4738	151	50	xi	xi	X
ejpam-4738	151	51	∗	∗	NOUN
ejpam-4738	151	52	ni))i∈i	ni))i∈i	NOUN
ejpam-4738	151	53	=	=	PUNCT
ejpam-4738	151	54	(	(	PUNCT
ejpam-4738	151	55	0i)i∈i	0i)i∈i	ADJ
ejpam-4738	151	56	∈	∈	PROPN
ejpam-4738	151	57	i.	i.	NOUN
ejpam-4738	151	58	hence	hence	ADV
ejpam-4738	151	59	(	(	PUNCT
ejpam-4738	151	60	(	(	PUNCT
ejpam-4738	151	61	ni)i∈i	ni)i∈i	NUM
ejpam-4738	151	62	⊙	⊙	NOUN
ejpam-4738	151	63	(	(	PUNCT
ejpam-4738	151	64	xi)i∈i	xi)i∈i	NUM
ejpam-4738	151	65	)	)	PUNCT
ejpam-4738	151	66	∈	∈	PROPN
ejpam-4738	152	1	i	i	PROPN
ejpam-4738	152	2	,	,	PUNCT
ejpam-4738	152	3	implies	imply	VERB
ejpam-4738	152	4	that	that	SCONJ
ejpam-4738	152	5	∏	∏	PROPN
ejpam-4738	152	6	i∈i	i∈i	NOUN
ejpam-4738	152	7	ni	ni	PROPN
ejpam-4738	152	8	is	be	AUX
ejpam-4738	152	9	a	a	DET
ejpam-4738	152	10	d	d	ADJ
ejpam-4738	152	11	-	-	PUNCT
ejpam-4738	152	12	ideal	ideal	ADJ
ejpam-4738	152	13	direct	direct	ADJ
ejpam-4738	152	14	product	product	NOUN
ejpam-4738	152	15	of	of	ADP
ejpam-4738	152	16	d	d	NOUN
ejpam-4738	152	17	-	-	PUNCT
ejpam-4738	152	18	algebras	algebra	NOUN
ejpam-4738	152	19	.	.	PUNCT
ejpam-4738	153	1	4	4	X
ejpam-4738	153	2	.	.	X
ejpam-4738	153	3	conclusion	conclusion	NOUN
ejpam-4738	153	4	in	in	ADP
ejpam-4738	153	5	this	this	DET
ejpam-4738	153	6	paper	paper	NOUN
ejpam-4738	153	7	,	,	PUNCT
ejpam-4738	153	8	we	we	PRON
ejpam-4738	153	9	give	give	VERB
ejpam-4738	153	10	the	the	DET
ejpam-4738	153	11	concept	concept	NOUN
ejpam-4738	153	12	of	of	ADP
ejpam-4738	153	13	ideal	ideal	ADJ
ejpam-4738	153	14	,	,	PUNCT
ejpam-4738	153	15	d	d	NOUN
ejpam-4738	153	16	-	-	PUNCT
ejpam-4738	153	17	ideal	ideal	ADJ
ejpam-4738	153	18	,	,	PUNCT
ejpam-4738	153	19	sub	sub	ADJ
ejpam-4738	153	20	-	-	ADJ
ejpam-4738	153	21	direct	direct	ADJ
ejpam-4738	153	22	product	product	NOUN
ejpam-4738	153	23	and	and	CCONJ
ejpam-4738	153	24	edge	edge	NOUN
ejpam-4738	153	25	in	in	ADP
ejpam-4738	153	26	a	a	DET
ejpam-4738	153	27	direct	direct	ADJ
ejpam-4738	153	28	product	product	NOUN
ejpam-4738	153	29	d	d	NOUN
ejpam-4738	153	30	-	-	PUNCT
ejpam-4738	153	31	algebra	algebra	NOUN
ejpam-4738	153	32	and	and	CCONJ
ejpam-4738	153	33	we	we	PRON
ejpam-4738	153	34	prove	prove	VERB
ejpam-4738	153	35	relationship	relationship	NOUN
ejpam-4738	153	36	between	between	ADP
ejpam-4738	153	37	ideal	ideal	ADJ
ejpam-4738	153	38	direct	direct	ADJ
ejpam-4738	153	39	product	product	NOUN
ejpam-4738	153	40	and	and	CCONJ
ejpam-4738	153	41	d	d	NOUN
ejpam-4738	153	42	-	-	ADJ
ejpam-4738	153	43	ideal	ideal	ADJ
ejpam-4738	153	44	direct	direct	ADJ
ejpam-4738	153	45	product	product	NOUN
ejpam-4738	153	46	of	of	ADP
ejpam-4738	153	47	d	d	NOUN
ejpam-4738	153	48	-	-	PUNCT
ejpam-4738	153	49	algebras	algebras	X
ejpam-4738	153	50	.	.	PUNCT
ejpam-4738	154	1	moreover	moreover	ADV
ejpam-4738	154	2	,	,	PUNCT
ejpam-4738	154	3	we	we	PRON
ejpam-4738	154	4	shown	show	VERB
ejpam-4738	154	5	that	that	SCONJ
ejpam-4738	154	6	a	a	DET
ejpam-4738	154	7	direct	direct	ADJ
ejpam-4738	154	8	product	product	NOUN
ejpam-4738	154	9	of	of	ADP
ejpam-4738	154	10	edge	edge	NOUN
ejpam-4738	154	11	d	d	NOUN
ejpam-4738	154	12	-	-	PUNCT
ejpam-4738	154	13	algebras	algebras	PROPN
ejpam-4738	154	14	is	be	AUX
ejpam-4738	154	15	not	not	PART
ejpam-4738	154	16	an	an	DET
ejpam-4738	154	17	edge	edge	NOUN
ejpam-4738	154	18	direct	direct	ADJ
ejpam-4738	154	19	product	product	NOUN
ejpam-4738	154	20	d	d	NOUN
ejpam-4738	154	21	-	-	PUNCT
ejpam-4738	154	22	algebra	algebra	NOUN
ejpam-4738	154	23	.	.	PUNCT
ejpam-4738	155	1	acknowledgements	acknowledgement	NOUN
ejpam-4738	155	2	this	this	DET
ejpam-4738	155	3	research	research	NOUN
ejpam-4738	155	4	project	project	NOUN
ejpam-4738	155	5	was	be	AUX
ejpam-4738	155	6	financially	financially	ADV
ejpam-4738	155	7	supported	support	VERB
ejpam-4738	155	8	by	by	ADP
ejpam-4738	155	9	mahasarakham	mahasarakham	PROPN
ejpam-4738	155	10	university	university	PROPN
ejpam-4738	155	11	.	.	PUNCT
ejpam-4738	156	1	references	reference	NOUN
ejpam-4738	156	2	[	[	X
ejpam-4738	156	3	1	1	X
ejpam-4738	156	4	]	]	PUNCT
ejpam-4738	156	5	s.	s.	PROPN
ejpam-4738	156	6	s.	s.	PROPN
ejpam-4738	156	7	ahn	ahn	PROPN
ejpam-4738	156	8	and	and	CCONJ
ejpam-4738	156	9	k.	k.	PROPN
ejpam-4738	156	10	s.	s.	PROPN
ejpam-4738	157	1	so	so	ADV
ejpam-4738	157	2	.	.	PUNCT
ejpam-4738	158	1	on	on	ADP
ejpam-4738	158	2	kernels	kernel	NOUN
ejpam-4738	158	3	and	and	CCONJ
ejpam-4738	158	4	annihilators	annihilator	NOUN
ejpam-4738	158	5	of	of	ADP
ejpam-4738	158	6	left	left	ADJ
ejpam-4738	158	7	-	-	PUNCT
ejpam-4738	158	8	regular	regular	ADJ
ejpam-4738	158	9	mappings	mapping	NOUN
ejpam-4738	158	10	in	in	ADP
ejpam-4738	158	11	d	d	NOUN
ejpam-4738	158	12	-	-	PUNCT
ejpam-4738	158	13	algebras	algebras	X
ejpam-4738	158	14	.	.	PUNCT
ejpam-4738	159	1	honam	honam	PROPN
ejpam-4738	159	2	mathematical	mathematical	PROPN
ejpam-4738	159	3	journal	journal	PROPN
ejpam-4738	159	4	,	,	PUNCT
ejpam-4738	159	5	30(4):645–658	30(4):645–658	PROPN
ejpam-4738	159	6	,	,	PUNCT
ejpam-4738	159	7	2008	2008	NUM
ejpam-4738	159	8	.	.	PUNCT
ejpam-4738	160	1	references	reference	NOUN
ejpam-4738	160	2	1004	1004	NUM
ejpam-4738	160	3	[	[	X
ejpam-4738	160	4	2	2	NUM
ejpam-4738	160	5	]	]	PUNCT
ejpam-4738	160	6	c.	c.	PROPN
ejpam-4738	160	7	chanmanee	chanmanee	PROPN
ejpam-4738	160	8	,	,	PUNCT
ejpam-4738	160	9	r.	r.	PROPN
ejpam-4738	160	10	chinram	chinram	PROPN
ejpam-4738	160	11	,	,	PUNCT
ejpam-4738	160	12	r.	r.	PROPN
ejpam-4738	160	13	prasertpong	prasertpong	PROPN
ejpam-4738	160	14	,	,	PUNCT
ejpam-4738	160	15	p.	p.	PROPN
ejpam-4738	160	16	julatha	julatha	PROPN
ejpam-4738	160	17	,	,	PUNCT
ejpam-4738	160	18	and	and	CCONJ
ejpam-4738	160	19	a.	a.	NOUN
ejpam-4738	160	20	iampan	iampan	PROPN
ejpam-4738	160	21	.	.	PUNCT
ejpam-4738	161	1	direct	direct	ADJ
ejpam-4738	161	2	product	product	NOUN
ejpam-4738	161	3	of	of	ADP
ejpam-4738	161	4	infinite	infinite	ADJ
ejpam-4738	161	5	family	family	NOUN
ejpam-4738	161	6	of	of	ADP
ejpam-4738	161	7	b	b	PROPN
ejpam-4738	161	8	-	-	PUNCT
ejpam-4738	161	9	algebras	algebras	PROPN
ejpam-4738	161	10	.	.	PUNCT
ejpam-4738	162	1	european	european	PROPN
ejpam-4738	162	2	journal	journal	PROPN
ejpam-4738	162	3	of	of	ADP
ejpam-4738	162	4	pure	pure	ADJ
ejpam-4738	162	5	and	and	CCONJ
ejpam-4738	162	6	applied	applied	ADJ
ejpam-4738	162	7	mathematics	mathematic	NOUN
ejpam-4738	162	8	,	,	PUNCT
ejpam-4738	162	9	15:999–1014	15:999–1014	NUM
ejpam-4738	162	10	,	,	PUNCT
ejpam-4738	162	11	2022	2022	NUM
ejpam-4738	162	12	.	.	PUNCT
ejpam-4738	163	1	[	[	X
ejpam-4738	163	2	3	3	X
ejpam-4738	163	3	]	]	X
ejpam-4738	163	4	m.	m.	NOUN
ejpam-4738	163	5	a.	a.	PROPN
ejpam-4738	163	6	chaudhry	chaudhry	PROPN
ejpam-4738	163	7	and	and	CCONJ
ejpam-4738	163	8	f.	f.	PROPN
ejpam-4738	163	9	ali	ali	PROPN
ejpam-4738	163	10	.	.	PROPN
ejpam-4738	163	11	multipliers	multiplier	NOUN
ejpam-4738	163	12	in	in	ADP
ejpam-4738	163	13	d	d	PROPN
ejpam-4738	163	14	-	-	PUNCT
ejpam-4738	163	15	algebras	algebras	PROPN
ejpam-4738	163	16	.	.	PUNCT
ejpam-4738	164	1	world	world	PROPN
ejpam-4738	164	2	world	world	PROPN
ejpam-4738	164	3	applied	apply	VERB
ejpam-4738	164	4	sciences	science	NOUN
ejpam-4738	164	5	journal	journal	NOUN
ejpam-4738	164	6	,	,	PUNCT
ejpam-4738	164	7	18:1649–1653	18:1649–1653	PROPN
ejpam-4738	164	8	,	,	PUNCT
ejpam-4738	164	9	2012	2012	NUM
ejpam-4738	164	10	.	.	PUNCT
ejpam-4738	165	1	[	[	X
ejpam-4738	165	2	4	4	X
ejpam-4738	165	3	]	]	PUNCT
ejpam-4738	165	4	s.	s.	PROPN
ejpam-4738	165	5	r.	r.	PROPN
ejpam-4738	165	6	kakumanu	kakumanu	PROPN
ejpam-4738	165	7	.	.	PUNCT
ejpam-4738	166	1	sl	sl	PROPN
ejpam-4738	166	2	and	and	CCONJ
ejpam-4738	166	3	sr	sr	PROPN
ejpam-4738	166	4	ideal	ideal	NOUN
ejpam-4738	166	5	on	on	ADP
ejpam-4738	166	6	d	d	NOUN
ejpam-4738	166	7	-	-	PUNCT
ejpam-4738	166	8	algebras	algebra	NOUN
ejpam-4738	166	9	.	.	PUNCT
ejpam-4738	167	1	international	international	ADJ
ejpam-4738	167	2	journal	journal	PROPN
ejpam-4738	167	3	of	of	ADP
ejpam-4738	167	4	advanced	advanced	ADJ
ejpam-4738	167	5	in	in	ADP
ejpam-4738	167	6	management	management	NOUN
ejpam-4738	167	7	technology	technology	NOUN
ejpam-4738	167	8	and	and	CCONJ
ejpam-4738	167	9	engineering	engineering	NOUN
ejpam-4738	167	10	sciences	science	NOUN
ejpam-4738	167	11	,	,	PUNCT
ejpam-4738	167	12	12:15–19	12:15–19	NUM
ejpam-4738	167	13	,	,	PUNCT
ejpam-4738	167	14	2017	2017	NUM
ejpam-4738	167	15	.	.	PUNCT
ejpam-4738	168	1	[	[	X
ejpam-4738	168	2	5	5	X
ejpam-4738	168	3	]	]	PUNCT
ejpam-4738	168	4	k.	k.	PROPN
ejpam-4738	168	5	h.	h.	PROPN
ejpam-4738	168	6	kim	kim	PROPN
ejpam-4738	168	7	.	.	PUNCT
ejpam-4738	169	1	on	on	ADP
ejpam-4738	169	2	fuzzy	fuzzy	ADJ
ejpam-4738	169	3	dot	dot	NOUN
ejpam-4738	169	4	subalgebras	subalgebra	NOUN
ejpam-4738	169	5	of	of	ADP
ejpam-4738	169	6	d	d	PROPN
ejpam-4738	169	7	-	-	PUNCT
ejpam-4738	169	8	algebras	algebra	NOUN
ejpam-4738	169	9	.	.	PUNCT
ejpam-4738	170	1	international	international	PROPN
ejpam-4738	170	2	mathematical	mathematical	PROPN
ejpam-4738	170	3	forum	forum	PROPN
ejpam-4738	170	4	,	,	PUNCT
ejpam-4738	170	5	13:645–651	13:645–651	NUM
ejpam-4738	170	6	,	,	PUNCT
ejpam-4738	170	7	2009	2009	NUM
ejpam-4738	170	8	.	.	PUNCT
ejpam-4738	171	1	[	[	X
ejpam-4738	171	2	6	6	NUM
ejpam-4738	171	3	]	]	PUNCT
ejpam-4738	171	4	b.	b.	PROPN
ejpam-4738	171	5	larsen	larsen	PROPN
ejpam-4738	171	6	.	.	PUNCT
ejpam-4738	172	1	an	an	DET
ejpam-4738	172	2	introduction	introduction	NOUN
ejpam-4738	172	3	to	to	ADP
ejpam-4738	172	4	the	the	DET
ejpam-4738	172	5	theory	theory	NOUN
ejpam-4738	172	6	of	of	ADP
ejpam-4738	172	7	multipliers	multiplier	NOUN
ejpam-4738	172	8	.	.	PUNCT
ejpam-4738	173	1	springer	springer	NOUN
ejpam-4738	173	2	-	-	PUNCT
ejpam-4738	173	3	verlag	verlag	PROPN
ejpam-4738	173	4	,	,	PUNCT
ejpam-4738	173	5	berlin	berlin	PROPN
ejpam-4738	173	6	,	,	PUNCT
ejpam-4738	173	7	1971	1971	NUM
ejpam-4738	173	8	.	.	PUNCT
ejpam-4738	174	1	[	[	X
ejpam-4738	174	2	7	7	X
ejpam-4738	174	3	]	]	X
ejpam-4738	174	4	j.	j.	PROPN
ejpam-4738	174	5	neggers	neggers	PROPN
ejpam-4738	174	6	.	.	PUNCT
ejpam-4738	175	1	on	on	ADP
ejpam-4738	175	2	d	d	PROPN
ejpam-4738	175	3	-	-	PUNCT
ejpam-4738	175	4	algebras	algebras	PROPN
ejpam-4738	175	5	.	.	PUNCT
ejpam-4738	176	1	mathematica	mathematica	PROPN
ejpam-4738	176	2	slovaca	slovaca	PROPN
ejpam-4738	176	3	,	,	PUNCT
ejpam-4738	176	4	49(1):19–26	49(1):19–26	NUM
ejpam-4738	176	5	,	,	PUNCT
ejpam-4738	176	6	1996	1996	NUM
ejpam-4738	176	7	.	.	PUNCT
ejpam-4738	177	1	[	[	X
ejpam-4738	177	2	8	8	NUM
ejpam-4738	177	3	]	]	X
ejpam-4738	177	4	j.	j.	PROPN
ejpam-4738	177	5	neggers	neggers	PROPN
ejpam-4738	177	6	,	,	PUNCT
ejpam-4738	177	7	y.	y.	PROPN
ejpam-4738	177	8	b.	b.	PROPN
ejpam-4738	177	9	jun	jun	PROPN
ejpam-4738	177	10	,	,	PUNCT
ejpam-4738	177	11	and	and	CCONJ
ejpam-4738	177	12	h.	h.	PROPN
ejpam-4738	177	13	s.	s.	PROPN
ejpam-4738	177	14	kim	kim	PROPN
ejpam-4738	177	15	.	.	PUNCT
ejpam-4738	178	1	on	on	ADP
ejpam-4738	178	2	d	d	NOUN
ejpam-4738	178	3	-	-	NOUN
ejpam-4738	178	4	ideals	ideal	NOUN
ejpam-4738	178	5	in	in	ADP
ejpam-4738	178	6	d	d	NOUN
ejpam-4738	178	7	-	-	PUNCT
ejpam-4738	178	8	algebras	algebras	PROPN
ejpam-4738	178	9	.	.	PUNCT
ejpam-4738	178	10	mathematica	mathematica	PROPN
ejpam-4738	178	11	slovaca	slovaca	PROPN
ejpam-4738	178	12	,	,	PUNCT
ejpam-4738	178	13	49(3):243–251	49(3):243–251	PROPN
ejpam-4738	178	14	,	,	PUNCT
ejpam-4738	178	15	1999	1999	NUM
ejpam-4738	178	16	.	.	PUNCT
ejpam-4738	179	1	[	[	X
ejpam-4738	179	2	9	9	NUM
ejpam-4738	179	3	]	]	X
ejpam-4738	179	4	j.	j.	PROPN
ejpam-4738	179	5	neggers	neggers	PROPN
ejpam-4738	179	6	and	and	CCONJ
ejpam-4738	179	7	h.	h.	PROPN
ejpam-4738	179	8	s.	s.	PROPN
ejpam-4738	179	9	kim	kim	PROPN
ejpam-4738	179	10	.	.	PUNCT
ejpam-4738	180	1	on	on	ADP
ejpam-4738	180	2	d	d	PROPN
ejpam-4738	180	3	-	-	PUNCT
ejpam-4738	180	4	algebras	algebras	PROPN
ejpam-4738	180	5	.	.	PUNCT
ejpam-4738	181	1	mathematica	mathematica	PROPN
ejpam-4738	181	2	slovaca	slovaca	PROPN
ejpam-4738	181	3	,	,	PUNCT
ejpam-4738	181	4	49(1):19–26	49(1):19–26	NUM
ejpam-4738	181	5	,	,	PUNCT
ejpam-4738	181	6	1999	1999	NUM
ejpam-4738	181	7	.	.	PUNCT
ejpam-4738	182	1	[	[	X
ejpam-4738	182	2	10	10	NUM
ejpam-4738	182	3	]	]	PUNCT
ejpam-4738	182	4	a.	a.	NOUN
ejpam-4738	182	5	setiani	setiani	PROPN
ejpam-4738	182	6	,	,	PUNCT
ejpam-4738	182	7	s.	s.	PROPN
ejpam-4738	182	8	gemawati	gemawati	PROPN
ejpam-4738	182	9	,	,	PUNCT
ejpam-4738	182	10	and	and	CCONJ
ejpam-4738	182	11	l.	l.	PROPN
ejpam-4738	182	12	deswita	deswita	PROPN
ejpam-4738	182	13	.	.	PUNCT
ejpam-4738	183	1	direct	direct	ADJ
ejpam-4738	183	2	product	product	NOUN
ejpam-4738	183	3	of	of	ADP
ejpam-4738	183	4	bp	bp	PROPN
ejpam-4738	183	5	-	-	PUNCT
ejpam-4738	183	6	algebra	algebra	NOUN
ejpam-4738	183	7	.	.	PUNCT
ejpam-4738	184	1	international	international	ADJ
ejpam-4738	184	2	journal	journal	PROPN
ejpam-4738	184	3	of	of	ADP
ejpam-4738	184	4	mathematics	mathematics	NOUN
ejpam-4738	184	5	trends	trend	NOUN
ejpam-4738	184	6	and	and	CCONJ
ejpam-4738	184	7	technology	technology	NOUN
ejpam-4738	184	8	,	,	PUNCT
ejpam-4738	184	9	66:63–69	66:63–69	NUM
ejpam-4738	184	10	,	,	PUNCT
ejpam-4738	184	11	2020	2020	NUM
ejpam-4738	184	12	.	.	PUNCT