id	sid	tid	token	lemma	pos
ejpam-4977	1	1	european	european	PROPN
ejpam-4977	1	2	journal	journal	PROPN
ejpam-4977	1	3	of	of	ADP
ejpam-4977	1	4	pure	pure	ADJ
ejpam-4977	1	5	and	and	CCONJ
ejpam-4977	1	6	applied	apply	VERB
ejpam-4977	1	7	mathematics	mathematic	NOUN
ejpam-4977	1	8	vol	vol	NOUN
ejpam-4977	1	9	.	.	PROPN
ejpam-4977	2	1	17	17	NUM
ejpam-4977	2	2	,	,	PUNCT
ejpam-4977	2	3	no	no	INTJ
ejpam-4977	2	4	.	.	NOUN
ejpam-4977	2	5	1	1	NUM
ejpam-4977	2	6	,	,	PUNCT
ejpam-4977	2	7	2024	2024	NUM
ejpam-4977	2	8	,	,	PUNCT
ejpam-4977	2	9	116	116	NUM
ejpam-4977	2	10	-	-	SYM
ejpam-4977	2	11	123	123	NUM
ejpam-4977	2	12	issn	issn	PROPN
ejpam-4977	2	13	1307	1307	NUM
ejpam-4977	2	14	-	-	SYM
ejpam-4977	2	15	5543	5543	NUM
ejpam-4977	2	16	–	–	PUNCT
ejpam-4977	2	17	ejpam.com	ejpam.com	X
ejpam-4977	2	18	published	publish	VERB
ejpam-4977	2	19	by	by	ADP
ejpam-4977	2	20	new	new	PROPN
ejpam-4977	2	21	york	york	PROPN
ejpam-4977	2	22	business	business	PROPN
ejpam-4977	2	23	global	global	ADJ
ejpam-4977	2	24	on	on	ADP
ejpam-4977	2	25	filter	filter	NOUN
ejpam-4977	2	26	of	of	ADP
ejpam-4977	2	27	cyclic	cyclic	PROPN
ejpam-4977	2	28	b	b	NOUN
ejpam-4977	2	29	-	-	PUNCT
ejpam-4977	2	30	algebras	algebras	X
ejpam-4977	2	31	maliwan	maliwan	PROPN
ejpam-4977	2	32	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	2	33	department	department	PROPN
ejpam-4977	2	34	of	of	ADP
ejpam-4977	2	35	mathematics	mathematic	NOUN
ejpam-4977	2	36	,	,	PUNCT
ejpam-4977	2	37	faculty	faculty	NOUN
ejpam-4977	2	38	of	of	ADP
ejpam-4977	2	39	science	science	NOUN
ejpam-4977	2	40	,	,	PUNCT
ejpam-4977	2	41	mahasarakham	mahasarakham	PROPN
ejpam-4977	2	42	university	university	PROPN
ejpam-4977	2	43	,	,	PUNCT
ejpam-4977	2	44	maha	maha	PROPN
ejpam-4977	2	45	sarakham	sarakham	PROPN
ejpam-4977	2	46	44150	44150	NUM
ejpam-4977	2	47	,	,	PUNCT
ejpam-4977	2	48	thailand	thailand	PROPN
ejpam-4977	2	49	abstract	abstract	PROPN
ejpam-4977	2	50	.	.	PUNCT
ejpam-4977	3	1	this	this	DET
ejpam-4977	3	2	paper	paper	NOUN
ejpam-4977	3	3	introduces	introduce	VERB
ejpam-4977	3	4	the	the	DET
ejpam-4977	3	5	notion	notion	NOUN
ejpam-4977	3	6	of	of	ADP
ejpam-4977	3	7	a	a	DET
ejpam-4977	3	8	b	b	NOUN
ejpam-4977	3	9	-	-	NOUN
ejpam-4977	3	10	filter	filter	NOUN
ejpam-4977	3	11	in	in	ADP
ejpam-4977	3	12	a	a	DET
ejpam-4977	3	13	b	b	NOUN
ejpam-4977	3	14	-	-	PUNCT
ejpam-4977	3	15	algebra	algebra	NOUN
ejpam-4977	3	16	(	(	PUNCT
ejpam-4977	3	17	x	x	X
ejpam-4977	3	18	,	,	PUNCT
ejpam-4977	3	19	∗	∗	NOUN
ejpam-4977	3	20	,	,	PUNCT
ejpam-4977	3	21	0	0	NUM
ejpam-4977	3	22	)	)	PUNCT
ejpam-4977	3	23	and	and	CCONJ
ejpam-4977	3	24	presents	present	VERB
ejpam-4977	3	25	characteristics	characteristic	NOUN
ejpam-4977	3	26	of	of	ADP
ejpam-4977	3	27	its	its	PRON
ejpam-4977	3	28	properties	property	NOUN
ejpam-4977	3	29	:	:	PUNCT
ejpam-4977	3	30	for	for	ADP
ejpam-4977	3	31	any	any	DET
ejpam-4977	3	32	a	a	DET
ejpam-4977	3	33	∈	∈	PROPN
ejpam-4977	3	34	x	x	NOUN
ejpam-4977	3	35	,	,	PUNCT
ejpam-4977	3	36	the	the	DET
ejpam-4977	3	37	set	set	NOUN
ejpam-4977	3	38	⟨a⟩b	⟨a⟩b	X
ejpam-4977	3	39	=	=	SYM
ejpam-4977	3	40	{	{	PUNCT
ejpam-4977	3	41	ak	ak	PROPN
ejpam-4977	3	42	:	:	PUNCT
ejpam-4977	3	43	k	k	PROPN
ejpam-4977	3	44	∈	∈	PROPN
ejpam-4977	3	45	z	z	PROPN
ejpam-4977	3	46	}	}	PUNCT
ejpam-4977	3	47	forms	form	VERB
ejpam-4977	3	48	a	a	DET
ejpam-4977	3	49	b	b	NOUN
ejpam-4977	3	50	-	-	PUNCT
ejpam-4977	3	51	ideal	ideal	ADJ
ejpam-4977	3	52	and	and	CCONJ
ejpam-4977	3	53	b	b	NOUN
ejpam-4977	3	54	-	-	NOUN
ejpam-4977	3	55	filter	filter	NOUN
ejpam-4977	3	56	of	of	ADP
ejpam-4977	3	57	x.	x.	NOUN
ejpam-4977	3	58	moreover	moreover	ADV
ejpam-4977	3	59	,	,	PUNCT
ejpam-4977	3	60	this	this	DET
ejpam-4977	3	61	paper	paper	NOUN
ejpam-4977	3	62	showns	shown	VERB
ejpam-4977	3	63	some	some	DET
ejpam-4977	3	64	properties	property	NOUN
ejpam-4977	3	65	of	of	ADP
ejpam-4977	3	66	exponents	exponent	NOUN
ejpam-4977	3	67	on	on	ADP
ejpam-4977	3	68	b	b	NOUN
ejpam-4977	3	69	-	-	PUNCT
ejpam-4977	3	70	algebra	algebra	NOUN
ejpam-4977	3	71	.	.	PUNCT
ejpam-4977	4	1	2020	2020	NUM
ejpam-4977	4	2	mathematics	mathematic	NOUN
ejpam-4977	4	3	subject	subject	NOUN
ejpam-4977	4	4	classifications	classification	NOUN
ejpam-4977	4	5	:	:	PUNCT
ejpam-4977	4	6	06f35	06f35	NUM
ejpam-4977	4	7	,	,	PUNCT
ejpam-4977	4	8	03g25	03g25	NOUN
ejpam-4977	4	9	key	key	ADJ
ejpam-4977	4	10	words	word	NOUN
ejpam-4977	4	11	and	and	CCONJ
ejpam-4977	4	12	phrases	phrase	NOUN
ejpam-4977	4	13	:	:	PUNCT
ejpam-4977	4	14	b	b	X
ejpam-4977	4	15	-	-	PUNCT
ejpam-4977	4	16	algebras	algebras	X
ejpam-4977	4	17	;	;	PUNCT
ejpam-4977	4	18	cyclic	cyclic	PROPN
ejpam-4977	4	19	b	b	X
ejpam-4977	4	20	-	-	PUNCT
ejpam-4977	4	21	algebras	algebras	X
ejpam-4977	4	22	;	;	PUNCT
ejpam-4977	4	23	b	b	X
ejpam-4977	4	24	-	-	PUNCT
ejpam-4977	4	25	ideal	ideal	NOUN
ejpam-4977	4	26	;	;	PUNCT
ejpam-4977	4	27	b	b	X
ejpam-4977	4	28	-	-	PUNCT
ejpam-4977	4	29	filter	filter	NOUN
ejpam-4977	4	30	1	1	NUM
ejpam-4977	4	31	.	.	PUNCT
ejpam-4977	5	1	introduction	introduction	NOUN
ejpam-4977	5	2	j.	j.	PROPN
ejpam-4977	5	3	neggers	neggers	PROPN
ejpam-4977	5	4	and	and	CCONJ
ejpam-4977	5	5	h.	h.	PROPN
ejpam-4977	5	6	s.	s.	PROPN
ejpam-4977	5	7	kim	kim	PROPN
ejpam-4977	5	8	introduced	introduce	VERB
ejpam-4977	5	9	in	in	ADP
ejpam-4977	5	10	[	[	X
ejpam-4977	5	11	1	1	X
ejpam-4977	5	12	]	]	PUNCT
ejpam-4977	5	13	the	the	DET
ejpam-4977	5	14	notion	notion	NOUN
ejpam-4977	5	15	of	of	ADP
ejpam-4977	5	16	b	b	NOUN
ejpam-4977	5	17	-	-	PUNCT
ejpam-4977	5	18	algebras	algebra	NOUN
ejpam-4977	5	19	and	and	CCONJ
ejpam-4977	5	20	some	some	DET
ejpam-4977	5	21	properties	property	NOUN
ejpam-4977	5	22	of	of	ADP
ejpam-4977	5	23	exponents	exponent	NOUN
ejpam-4977	5	24	on	on	ADP
ejpam-4977	5	25	its	its	PRON
ejpam-4977	5	26	.	.	PUNCT
ejpam-4977	6	1	furthermore	furthermore	ADV
ejpam-4977	6	2	,	,	PUNCT
ejpam-4977	6	3	they	they	PRON
ejpam-4977	6	4	investigated	investigate	VERB
ejpam-4977	6	5	the	the	DET
ejpam-4977	6	6	relationship	relationship	NOUN
ejpam-4977	6	7	between	between	ADP
ejpam-4977	6	8	b	b	NOUN
ejpam-4977	6	9	-	-	PUNCT
ejpam-4977	6	10	algebras	algebra	NOUN
ejpam-4977	6	11	and	and	CCONJ
ejpam-4977	6	12	groups	group	NOUN
ejpam-4977	6	13	and	and	CCONJ
ejpam-4977	6	14	asked	ask	VERB
ejpam-4977	6	15	whether	whether	SCONJ
ejpam-4977	6	16	a	a	DET
ejpam-4977	6	17	group	group	NOUN
ejpam-4977	6	18	determines	determine	VERB
ejpam-4977	6	19	a	a	DET
ejpam-4977	6	20	b	b	NOUN
ejpam-4977	6	21	-	-	PUNCT
ejpam-4977	6	22	algebra	algebra	NOUN
ejpam-4977	6	23	,	,	PUNCT
ejpam-4977	6	24	and	and	CCONJ
ejpam-4977	6	25	conversely	conversely	ADV
ejpam-4977	6	26	.	.	PUNCT
ejpam-4977	7	1	in	in	ADP
ejpam-4977	7	2	[	[	X
ejpam-4977	7	3	5	5	NUM
ejpam-4977	7	4	]	]	PUNCT
ejpam-4977	7	5	,	,	PUNCT
ejpam-4977	7	6	d.	d.	PROPN
ejpam-4977	7	7	al	al	PROPN
ejpam-4977	7	8	-	-	PUNCT
ejpam-4977	7	9	kadi	kadi	PROPN
ejpam-4977	7	10	introduced	introduce	VERB
ejpam-4977	7	11	the	the	DET
ejpam-4977	7	12	notion	notion	NOUN
ejpam-4977	7	13	of	of	ADP
ejpam-4977	7	14	b	b	NOUN
ejpam-4977	7	15	-	-	PUNCT
ejpam-4977	7	16	ideal	ideal	NOUN
ejpam-4977	7	17	and	and	CCONJ
ejpam-4977	7	18	then	then	ADV
ejpam-4977	7	19	k.	k.	PROPN
ejpam-4977	7	20	e.	e.	PROPN
ejpam-4977	7	21	belleza	belleza	PROPN
ejpam-4977	7	22	and	and	CCONJ
ejpam-4977	7	23	j.	j.	PROPN
ejpam-4977	7	24	p.	p.	PROPN
ejpam-4977	7	25	vilela	vilela	NOUN
ejpam-4977	7	26	in	in	ADP
ejpam-4977	7	27	[	[	X
ejpam-4977	7	28	7	7	NUM
ejpam-4977	7	29	]	]	PUNCT
ejpam-4977	7	30	presents	present	VERB
ejpam-4977	7	31	the	the	DET
ejpam-4977	7	32	characterizations	characterization	NOUN
ejpam-4977	7	33	and	and	CCONJ
ejpam-4977	7	34	properties	property	NOUN
ejpam-4977	7	35	of	of	ADP
ejpam-4977	7	36	b	b	NOUN
ejpam-4977	7	37	-	-	PUNCT
ejpam-4977	7	38	ideals	ideal	NOUN
ejpam-4977	7	39	in	in	ADP
ejpam-4977	7	40	a	a	DET
ejpam-4977	7	41	topological	topological	ADJ
ejpam-4977	7	42	b	b	NOUN
ejpam-4977	7	43	-	-	PUNCT
ejpam-4977	7	44	algebra	algebra	NOUN
ejpam-4977	7	45	and	and	CCONJ
ejpam-4977	7	46	introduces	introduce	VERB
ejpam-4977	7	47	the	the	DET
ejpam-4977	7	48	uniform	uniform	ADJ
ejpam-4977	7	49	topology	topology	NOUN
ejpam-4977	7	50	on	on	ADP
ejpam-4977	7	51	a	a	DET
ejpam-4977	7	52	b	b	NOUN
ejpam-4977	7	53	-	-	PUNCT
ejpam-4977	7	54	algebra	algebra	NOUN
ejpam-4977	7	55	in	in	ADP
ejpam-4977	7	56	terms	term	NOUN
ejpam-4977	7	57	of	of	ADP
ejpam-4977	7	58	its	its	PRON
ejpam-4977	7	59	b	b	NOUN
ejpam-4977	7	60	-	-	PUNCT
ejpam-4977	7	61	ideals	ideal	NOUN
ejpam-4977	7	62	.	.	PUNCT
ejpam-4977	8	1	moreover	moreover	ADV
ejpam-4977	8	2	,	,	PUNCT
ejpam-4977	8	3	they	they	PRON
ejpam-4977	8	4	have	have	AUX
ejpam-4977	8	5	shown	show	VERB
ejpam-4977	8	6	that	that	SCONJ
ejpam-4977	8	7	a	a	DET
ejpam-4977	8	8	uniform	uniform	ADJ
ejpam-4977	8	9	b	b	X
ejpam-4977	8	10	-	-	PUNCT
ejpam-4977	8	11	topological	topological	ADJ
ejpam-4977	8	12	space	space	NOUN
ejpam-4977	8	13	is	be	AUX
ejpam-4977	8	14	a	a	DET
ejpam-4977	8	15	topological	topological	ADJ
ejpam-4977	8	16	b	b	NOUN
ejpam-4977	8	17	-	-	PUNCT
ejpam-4977	8	18	algebra	algebra	NOUN
ejpam-4977	8	19	.	.	PUNCT
ejpam-4977	9	1	k.	k.	PROPN
ejpam-4977	9	2	e.	e.	PROPN
ejpam-4977	9	3	belleza	belleza	PROPN
ejpam-4977	9	4	and	and	CCONJ
ejpam-4977	9	5	j.	j.	PROPN
ejpam-4977	9	6	r.	r.	PROPN
ejpam-4977	9	7	albaracin	albaracin	PROPN
ejpam-4977	9	8	introduces	introduce	NOUN
ejpam-4977	9	9	and	and	CCONJ
ejpam-4977	9	10	characterized	characterize	VERB
ejpam-4977	9	11	the	the	DET
ejpam-4977	9	12	notion	notion	NOUN
ejpam-4977	9	13	of	of	ADP
ejpam-4977	9	14	a	a	DET
ejpam-4977	9	15	dual	dual	ADJ
ejpam-4977	9	16	b	b	NOUN
ejpam-4977	9	17	-	-	PUNCT
ejpam-4977	9	18	algebra	algebra	NOUN
ejpam-4977	9	19	,	,	PUNCT
ejpam-4977	9	20	in	in	ADP
ejpam-4977	9	21	[	[	PUNCT
ejpam-4977	9	22	6	6	NUM
ejpam-4977	9	23	]	]	PUNCT
ejpam-4977	9	24	.	.	PUNCT
ejpam-4977	10	1	moreover	moreover	ADV
ejpam-4977	10	2	in	in	ADP
ejpam-4977	10	3	the	the	DET
ejpam-4977	10	4	year	year	NOUN
ejpam-4977	10	5	2022	2022	NUM
ejpam-4977	10	6	,	,	PUNCT
ejpam-4977	10	7	k.	k.	PROPN
ejpam-4977	10	8	e.	e.	PROPN
ejpam-4977	10	9	belleza	belleza	PROPN
ejpam-4977	10	10	introduces	introduce	VERB
ejpam-4977	10	11	the	the	DET
ejpam-4977	10	12	dual	dual	ADJ
ejpam-4977	10	13	b	b	NOUN
ejpam-4977	10	14	-	-	PUNCT
ejpam-4977	10	15	topological	topological	ADJ
ejpam-4977	10	16	space	space	NOUN
ejpam-4977	10	17	,	,	PUNCT
ejpam-4977	10	18	dual	dual	ADJ
ejpam-4977	10	19	b	b	NOUN
ejpam-4977	10	20	-	-	PUNCT
ejpam-4977	10	21	ideals	ideal	NOUN
ejpam-4977	10	22	and	and	CCONJ
ejpam-4977	10	23	dual	dual	ADJ
ejpam-4977	10	24	b	b	NOUN
ejpam-4977	10	25	-	-	PUNCT
ejpam-4977	10	26	subalgebras	subalgebras	X
ejpam-4977	10	27	.	.	PUNCT
ejpam-4977	11	1	also	also	ADV
ejpam-4977	11	2	,	,	PUNCT
ejpam-4977	11	3	some	some	DET
ejpam-4977	11	4	properties	property	NOUN
ejpam-4977	11	5	of	of	ADP
ejpam-4977	11	6	a	a	DET
ejpam-4977	11	7	filterbase	filterbase	NOUN
ejpam-4977	11	8	on	on	ADP
ejpam-4977	11	9	a	a	DET
ejpam-4977	11	10	dual	dual	ADJ
ejpam-4977	11	11	b	b	NOUN
ejpam-4977	11	12	-	-	PUNCT
ejpam-4977	11	13	topological	topological	ADJ
ejpam-4977	11	14	space	space	NOUN
ejpam-4977	11	15	are	be	AUX
ejpam-4977	11	16	provided	provide	VERB
ejpam-4977	11	17	.	.	PUNCT
ejpam-4977	12	1	in	in	ADP
ejpam-4977	12	2	[	[	X
ejpam-4977	12	3	2	2	NUM
ejpam-4977	12	4	]	]	PUNCT
ejpam-4977	12	5	,	,	PUNCT
ejpam-4977	12	6	n.	n.	PROPN
ejpam-4977	12	7	c.	c.	PROPN
ejpam-4977	12	8	gonzaga	gonzaga	PROPN
ejpam-4977	12	9	,	,	PUNCT
ejpam-4977	12	10	jr	jr	PROPN
ejpam-4977	12	11	and	and	CCONJ
ejpam-4977	12	12	j.	j.	PROPN
ejpam-4977	12	13	p.	p.	PROPN
ejpam-4977	12	14	vilela	vilela	PROPN
ejpam-4977	12	15	introduced	introduce	VERB
ejpam-4977	12	16	the	the	DET
ejpam-4977	12	17	notion	notion	NOUN
ejpam-4977	12	18	of	of	ADP
ejpam-4977	12	19	cyclic	cyclic	PROPN
ejpam-4977	12	20	b	b	NOUN
ejpam-4977	12	21	-	-	PUNCT
ejpam-4977	12	22	algebras	algebras	PROPN
ejpam-4977	12	23	and	and	CCONJ
ejpam-4977	12	24	some	some	PRON
ejpam-4977	12	25	of	of	ADP
ejpam-4977	12	26	its	its	PRON
ejpam-4977	12	27	properties	property	NOUN
ejpam-4977	12	28	.	.	PUNCT
ejpam-4977	13	1	moreover	moreover	ADV
ejpam-4977	13	2	the	the	DET
ejpam-4977	13	3	authors	author	NOUN
ejpam-4977	13	4	had	have	AUX
ejpam-4977	13	5	investigated	investigate	VERB
ejpam-4977	13	6	the	the	DET
ejpam-4977	13	7	relationship	relationship	NOUN
ejpam-4977	13	8	between	between	ADP
ejpam-4977	13	9	the	the	DET
ejpam-4977	13	10	class	class	NOUN
ejpam-4977	13	11	of	of	ADP
ejpam-4977	13	12	cyclic	cyclic	ADJ
ejpam-4977	13	13	b	b	NOUN
ejpam-4977	13	14	-	-	PUNCT
ejpam-4977	13	15	algebras	algebras	PROPN
ejpam-4977	13	16	and	and	CCONJ
ejpam-4977	13	17	the	the	DET
ejpam-4977	13	18	class	class	NOUN
ejpam-4977	13	19	of	of	ADP
ejpam-4977	13	20	cyclic	cyclic	ADJ
ejpam-4977	13	21	groups	group	NOUN
ejpam-4977	13	22	coincide	coincide	NOUN
ejpam-4977	13	23	.	.	PUNCT
ejpam-4977	14	1	in	in	ADP
ejpam-4977	14	2	[	[	X
ejpam-4977	14	3	8	8	NUM
ejpam-4977	14	4	]	]	PUNCT
ejpam-4977	14	5	,	,	PUNCT
ejpam-4977	14	6	k.	k.	PROPN
ejpam-4977	14	7	e.	e.	PROPN
ejpam-4977	14	8	belleza	belleza	PROPN
ejpam-4977	14	9	and	and	CCONJ
ejpam-4977	14	10	j.	j.	PROPN
ejpam-4977	14	11	r.	r.	PROPN
ejpam-4977	14	12	albaracin	albaracin	PROPN
ejpam-4977	14	13	introduced	introduce	VERB
ejpam-4977	14	14	the	the	DET
ejpam-4977	14	15	notion	notion	NOUN
ejpam-4977	14	16	of	of	ADP
ejpam-4977	14	17	tdbalgebra	tdbalgebra	NOUN
ejpam-4977	14	18	,	,	PUNCT
ejpam-4977	14	19	presents	present	VERB
ejpam-4977	14	20	characteristics	characteristic	NOUN
ejpam-4977	14	21	and	and	CCONJ
ejpam-4977	14	22	properties	property	NOUN
ejpam-4977	14	23	of	of	ADP
ejpam-4977	14	24	dual	dual	ADJ
ejpam-4977	14	25	b	b	NOUN
ejpam-4977	14	26	-	-	PUNCT
ejpam-4977	14	27	filters	filter	NOUN
ejpam-4977	14	28	and	and	CCONJ
ejpam-4977	14	29	dual	dual	ADJ
ejpam-4977	14	30	b	b	NOUN
ejpam-4977	14	31	-	-	PUNCT
ejpam-4977	14	32	subalgebras	subalgebras	PROPN
ejpam-4977	14	33	in	in	ADP
ejpam-4977	14	34	a	a	DET
ejpam-4977	14	35	tdb	tdb	NOUN
ejpam-4977	14	36	-	-	NOUN
ejpam-4977	14	37	algebra	algebra	NOUN
ejpam-4977	14	38	,	,	PUNCT
ejpam-4977	14	39	and	and	CCONJ
ejpam-4977	14	40	introduces	introduce	VERB
ejpam-4977	14	41	the	the	DET
ejpam-4977	14	42	uniform	uniform	ADJ
ejpam-4977	14	43	topology	topology	NOUN
ejpam-4977	14	44	on	on	ADP
ejpam-4977	14	45	a	a	DET
ejpam-4977	14	46	dual	dual	ADJ
ejpam-4977	14	47	b	b	NOUN
ejpam-4977	14	48	-	-	PUNCT
ejpam-4977	14	49	algebra	algebra	NOUN
ejpam-4977	14	50	in	in	ADP
ejpam-4977	14	51	terms	term	NOUN
ejpam-4977	14	52	of	of	ADP
ejpam-4977	14	53	its	its	PRON
ejpam-4977	14	54	dual	dual	ADJ
ejpam-4977	14	55	b	b	NOUN
ejpam-4977	14	56	-	-	PUNCT
ejpam-4977	14	57	subalgebras	subalgebras	X
ejpam-4977	14	58	.	.	PUNCT
ejpam-4977	15	1	specifically	specifically	ADV
ejpam-4977	15	2	,	,	PUNCT
ejpam-4977	15	3	this	this	DET
ejpam-4977	15	4	paper	paper	NOUN
ejpam-4977	15	5	introduces	introduce	VERB
ejpam-4977	15	6	the	the	DET
ejpam-4977	15	7	notion	notion	NOUN
ejpam-4977	15	8	of	of	ADP
ejpam-4977	15	9	the	the	DET
ejpam-4977	15	10	b	b	NOUN
ejpam-4977	15	11	-	-	NOUN
ejpam-4977	15	12	filter	filter	NOUN
ejpam-4977	15	13	in	in	ADP
ejpam-4977	15	14	a	a	DET
ejpam-4977	15	15	b	b	NOUN
ejpam-4977	15	16	-	-	PUNCT
ejpam-4977	15	17	algebra	algebra	NOUN
ejpam-4977	15	18	and	and	CCONJ
ejpam-4977	15	19	we	we	PRON
ejpam-4977	15	20	have	have	AUX
ejpam-4977	15	21	show	show	VERB
ejpam-4977	15	22	that	that	SCONJ
ejpam-4977	15	23	for	for	ADP
ejpam-4977	15	24	any	any	DET
ejpam-4977	15	25	a	a	DET
ejpam-4977	15	26	∈	∈	PROPN
ejpam-4977	15	27	x	x	NOUN
ejpam-4977	15	28	,	,	PUNCT
ejpam-4977	15	29	the	the	DET
ejpam-4977	15	30	set	set	NOUN
ejpam-4977	15	31	⟨a⟩b	⟨a⟩b	X
ejpam-4977	15	32	=	=	SYM
ejpam-4977	15	33	{	{	PUNCT
ejpam-4977	15	34	ak	ak	PROPN
ejpam-4977	15	35	:	:	PUNCT
ejpam-4977	15	36	k	k	PROPN
ejpam-4977	15	37	∈	∈	PROPN
ejpam-4977	15	38	z	z	AUX
ejpam-4977	15	39	}	}	PUNCT
ejpam-4977	15	40	form	form	VERB
ejpam-4977	15	41	a	a	DET
ejpam-4977	15	42	b	b	NOUN
ejpam-4977	15	43	-	-	PUNCT
ejpam-4977	15	44	ideal	ideal	ADJ
ejpam-4977	15	45	and	and	CCONJ
ejpam-4977	15	46	b	b	NOUN
ejpam-4977	15	47	-	-	NOUN
ejpam-4977	15	48	filter	filter	NOUN
ejpam-4977	15	49	of	of	ADP
ejpam-4977	15	50	x.	x.	PROPN
ejpam-4977	15	51	doi	doi	PROPN
ejpam-4977	15	52	:	:	PUNCT
ejpam-4977	15	53	https://doi.org/10.29020/nybg.ejpam.v17i1.4977	https://doi.org/10.29020/nybg.ejpam.v17i1.4977	NUM
ejpam-4977	15	54	email	email	NOUN
ejpam-4977	15	55	address	address	NOUN
ejpam-4977	15	56	:	:	PUNCT
ejpam-4977	15	57	maliwan.t@msu.ac.th	maliwan.t@msu.ac.th	PROPN
ejpam-4977	15	58	(	(	PUNCT
ejpam-4977	15	59	m.	m.	NOUN
ejpam-4977	15	60	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	15	61	)	)	PUNCT
ejpam-4977	15	62	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4977	16	1	116	116	NUM
ejpam-4977	16	2	©	©	ADP
ejpam-4977	16	3	2024	2024	NUM
ejpam-4977	16	4	ejpam	ejpam	NOUN
ejpam-4977	16	5	all	all	DET
ejpam-4977	16	6	rights	right	NOUN
ejpam-4977	16	7	reserved	reserve	VERB
ejpam-4977	16	8	.	.	PUNCT
ejpam-4977	17	1	m.	m.	PROPN
ejpam-4977	17	2	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	17	3	/	/	SYM
ejpam-4977	17	4	eur	eur	PROPN
ejpam-4977	17	5	.	.	PUNCT
ejpam-4977	18	1	j.	j.	PROPN
ejpam-4977	18	2	pure	pure	PROPN
ejpam-4977	18	3	appl	appl	PROPN
ejpam-4977	18	4	.	.	PROPN
ejpam-4977	18	5	math	math	PROPN
ejpam-4977	18	6	,	,	PUNCT
ejpam-4977	18	7	17	17	NUM
ejpam-4977	18	8	(	(	PUNCT
ejpam-4977	18	9	1	1	NUM
ejpam-4977	18	10	)	)	PUNCT
ejpam-4977	18	11	(	(	PUNCT
ejpam-4977	18	12	2024	2024	NUM
ejpam-4977	18	13	)	)	PUNCT
ejpam-4977	18	14	,	,	PUNCT
ejpam-4977	18	15	116	116	NUM
ejpam-4977	18	16	-	-	SYM
ejpam-4977	18	17	123	123	NUM
ejpam-4977	18	18	117	117	NUM
ejpam-4977	18	19	2	2	NUM
ejpam-4977	18	20	.	.	PUNCT
ejpam-4977	18	21	preliminaries	preliminary	NOUN
ejpam-4977	18	22	first	first	ADV
ejpam-4977	18	23	,	,	PUNCT
ejpam-4977	18	24	we	we	PRON
ejpam-4977	18	25	will	will	AUX
ejpam-4977	18	26	review	review	VERB
ejpam-4977	18	27	some	some	DET
ejpam-4977	18	28	essential	essential	ADJ
ejpam-4977	18	29	notations	notation	NOUN
ejpam-4977	18	30	and	and	CCONJ
ejpam-4977	18	31	definitions	definition	NOUN
ejpam-4977	18	32	ofb	ofb	PROPN
ejpam-4977	18	33	-	-	PUNCT
ejpam-4977	18	34	algebras	algebras	PROPN
ejpam-4977	18	35	and	and	CCONJ
ejpam-4977	18	36	ordinary	ordinary	ADJ
ejpam-4977	18	37	senses	sense	NOUN
ejpam-4977	18	38	that	that	PRON
ejpam-4977	18	39	are	be	AUX
ejpam-4977	18	40	needed	need	VERB
ejpam-4977	18	41	for	for	ADP
ejpam-4977	18	42	this	this	DET
ejpam-4977	18	43	study	study	NOUN
ejpam-4977	18	44	in	in	ADP
ejpam-4977	18	45	this	this	DET
ejpam-4977	18	46	section	section	NOUN
ejpam-4977	18	47	.	.	PUNCT
ejpam-4977	19	1	throughout	throughout	ADP
ejpam-4977	19	2	this	this	DET
ejpam-4977	19	3	paper	paper	NOUN
ejpam-4977	19	4	,	,	PUNCT
ejpam-4977	19	5	x	x	PRON
ejpam-4977	19	6	will	will	AUX
ejpam-4977	19	7	denote	denote	VERB
ejpam-4977	19	8	the	the	DET
ejpam-4977	19	9	b	b	NOUN
ejpam-4977	19	10	-	-	PUNCT
ejpam-4977	19	11	algebra	algebra	NOUN
ejpam-4977	19	12	(	(	PUNCT
ejpam-4977	19	13	x	x	X
ejpam-4977	19	14	,	,	PUNCT
ejpam-4977	19	15	∗	∗	NOUN
ejpam-4977	19	16	,	,	PUNCT
ejpam-4977	19	17	0	0	NUM
ejpam-4977	19	18	)	)	PUNCT
ejpam-4977	19	19	unless	unless	SCONJ
ejpam-4977	19	20	otherwise	otherwise	ADV
ejpam-4977	19	21	specified	specify	VERB
ejpam-4977	19	22	.	.	PUNCT
ejpam-4977	20	1	definition	definition	NOUN
ejpam-4977	20	2	1	1	NUM
ejpam-4977	20	3	.	.	PUNCT
ejpam-4977	21	1	[	[	X
ejpam-4977	21	2	4	4	X
ejpam-4977	21	3	]	]	PUNCT
ejpam-4977	21	4	a	a	DET
ejpam-4977	21	5	b	b	X
ejpam-4977	21	6	-	-	PUNCT
ejpam-4977	21	7	algebra	algebra	NOUN
ejpam-4977	21	8	is	be	AUX
ejpam-4977	21	9	a	a	DET
ejpam-4977	21	10	non	non	ADJ
ejpam-4977	21	11	-	-	ADJ
ejpam-4977	21	12	empty	empty	ADJ
ejpam-4977	21	13	set	set	NOUN
ejpam-4977	21	14	x	x	PUNCT
ejpam-4977	21	15	with	with	ADP
ejpam-4977	21	16	a	a	DET
ejpam-4977	21	17	constant	constant	ADJ
ejpam-4977	21	18	0	0	NUM
ejpam-4977	21	19	and	and	CCONJ
ejpam-4977	21	20	a	a	DET
ejpam-4977	21	21	binary	binary	ADJ
ejpam-4977	21	22	operation	operation	NOUN
ejpam-4977	21	23	∗	∗	NOUN
ejpam-4977	21	24	satisfying	satisfy	VERB
ejpam-4977	21	25	the	the	DET
ejpam-4977	21	26	following	follow	VERB
ejpam-4977	21	27	axioms	axiom	NOUN
ejpam-4977	21	28	:	:	PUNCT
ejpam-4977	21	29	(	(	PUNCT
ejpam-4977	21	30	b1	b1	NOUN
ejpam-4977	21	31	)	)	PUNCT
ejpam-4977	21	32	x	x	SYM
ejpam-4977	21	33	∗	∗	NOUN
ejpam-4977	21	34	x	x	SYM
ejpam-4977	21	35	=	=	SYM
ejpam-4977	21	36	0	0	NUM
ejpam-4977	21	37	,	,	PUNCT
ejpam-4977	21	38	(	(	PUNCT
ejpam-4977	21	39	b2	b2	NOUN
ejpam-4977	21	40	)	)	PUNCT
ejpam-4977	21	41	x	x	SYM
ejpam-4977	21	42	∗	∗	NOUN
ejpam-4977	21	43	0	0	NUM
ejpam-4977	22	1	=	=	SYM
ejpam-4977	22	2	x	x	NOUN
ejpam-4977	22	3	,	,	PUNCT
ejpam-4977	22	4	(	(	PUNCT
ejpam-4977	22	5	b3	b3	PROPN
ejpam-4977	22	6	)	)	PUNCT
ejpam-4977	22	7	(	(	PUNCT
ejpam-4977	22	8	x	x	SYM
ejpam-4977	22	9	∗	∗	PROPN
ejpam-4977	22	10	y	y	NOUN
ejpam-4977	22	11	)	)	PUNCT
ejpam-4977	22	12	∗	∗	NOUN
ejpam-4977	22	13	z	z	NOUN
ejpam-4977	23	1	=	=	SYM
ejpam-4977	23	2	x	x	X
ejpam-4977	23	3	∗	∗	NOUN
ejpam-4977	23	4	(	(	PUNCT
ejpam-4977	23	5	z	z	NOUN
ejpam-4977	23	6	∗	∗	NOUN
ejpam-4977	23	7	(	(	PUNCT
ejpam-4977	23	8	0	0	NUM
ejpam-4977	23	9	∗	∗	PROPN
ejpam-4977	23	10	y	y	PROPN
ejpam-4977	23	11	)	)	PUNCT
ejpam-4977	23	12	)	)	PUNCT
ejpam-4977	23	13	for	for	ADP
ejpam-4977	23	14	all	all	DET
ejpam-4977	23	15	x	x	PROPN
ejpam-4977	23	16	,	,	PUNCT
ejpam-4977	23	17	y	y	PROPN
ejpam-4977	23	18	,	,	PUNCT
ejpam-4977	23	19	z	z	PROPN
ejpam-4977	23	20	∈	∈	PROPN
ejpam-4977	23	21	x.	x.	NOUN
ejpam-4977	24	1	a	a	DET
ejpam-4977	24	2	b	b	X
ejpam-4977	24	3	-	-	PUNCT
ejpam-4977	24	4	algebra	algebra	NOUN
ejpam-4977	24	5	(	(	PUNCT
ejpam-4977	24	6	x	x	X
ejpam-4977	24	7	,	,	PUNCT
ejpam-4977	24	8	∗	∗	NOUN
ejpam-4977	24	9	,	,	PUNCT
ejpam-4977	24	10	0	0	NUM
ejpam-4977	24	11	)	)	PUNCT
ejpam-4977	24	12	is	be	AUX
ejpam-4977	24	13	said	say	VERB
ejpam-4977	24	14	to	to	PART
ejpam-4977	24	15	be	be	AUX
ejpam-4977	24	16	commutative	commutative	ADJ
ejpam-4977	24	17	if	if	SCONJ
ejpam-4977	24	18	x	x	ADP
ejpam-4977	24	19	∗	∗	NOUN
ejpam-4977	24	20	(	(	PUNCT
ejpam-4977	24	21	0	0	NUM
ejpam-4977	24	22	∗	∗	NUM
ejpam-4977	24	23	y	y	NOUN
ejpam-4977	24	24	)	)	PUNCT
ejpam-4977	25	1	=	=	SYM
ejpam-4977	25	2	y	y	PROPN
ejpam-4977	25	3	∗	∗	NOUN
ejpam-4977	25	4	(	(	PUNCT
ejpam-4977	25	5	0	0	NUM
ejpam-4977	25	6	∗	∗	NOUN
ejpam-4977	25	7	x	x	NOUN
ejpam-4977	25	8	)	)	PUNCT
ejpam-4977	25	9	for	for	ADP
ejpam-4977	25	10	any	any	DET
ejpam-4977	25	11	x	x	NOUN
ejpam-4977	25	12	,	,	PUNCT
ejpam-4977	25	13	y	y	PROPN
ejpam-4977	25	14	∈	∈	PROPN
ejpam-4977	25	15	x	x	X
ejpam-4977	25	16	and	and	CCONJ
ejpam-4977	25	17	a	a	DET
ejpam-4977	25	18	nonempty	nonempty	NOUN
ejpam-4977	25	19	subset	subset	VERB
ejpam-4977	25	20	s	s	NOUN
ejpam-4977	25	21	of	of	ADP
ejpam-4977	25	22	a	a	PRON
ejpam-4977	25	23	x	x	PRON
ejpam-4977	25	24	is	be	AUX
ejpam-4977	25	25	called	call	VERB
ejpam-4977	25	26	a	a	DET
ejpam-4977	25	27	sub	sub	NOUN
ejpam-4977	25	28	-	-	NOUN
ejpam-4977	25	29	algebra	algebra	NOUN
ejpam-4977	25	30	of	of	ADP
ejpam-4977	25	31	x	x	SYM
ejpam-4977	25	32	if	if	SCONJ
ejpam-4977	25	33	x	x	PROPN
ejpam-4977	25	34	∗	∗	VERB
ejpam-4977	25	35	y	y	PROPN
ejpam-4977	25	36	∈	∈	PROPN
ejpam-4977	25	37	s	s	X
ejpam-4977	25	38	for	for	ADP
ejpam-4977	25	39	any	any	DET
ejpam-4977	25	40	x	x	NOUN
ejpam-4977	25	41	,	,	PUNCT
ejpam-4977	25	42	y	y	PROPN
ejpam-4977	25	43	∈	∈	PROPN
ejpam-4977	25	44	s.	s.	PROPN
ejpam-4977	25	45	also	also	ADV
ejpam-4977	25	46	,	,	PUNCT
ejpam-4977	25	47	the	the	DET
ejpam-4977	25	48	authors	author	NOUN
ejpam-4977	25	49	of	of	ADP
ejpam-4977	25	50	[	[	X
ejpam-4977	25	51	1	1	X
ejpam-4977	25	52	]	]	PUNCT
ejpam-4977	25	53	have	have	AUX
ejpam-4977	25	54	proved	prove	VERB
ejpam-4977	25	55	that	that	SCONJ
ejpam-4977	25	56	a	a	DET
ejpam-4977	25	57	b	b	NOUN
ejpam-4977	25	58	-	-	PUNCT
ejpam-4977	25	59	algebra	algebra	NOUN
ejpam-4977	25	60	x	x	PUNCT
ejpam-4977	25	61	is	be	AUX
ejpam-4977	25	62	commutative	commutative	ADJ
ejpam-4977	25	63	if	if	SCONJ
ejpam-4977	25	64	and	and	CCONJ
ejpam-4977	25	65	only	only	ADV
ejpam-4977	25	66	if	if	SCONJ
ejpam-4977	25	67	the	the	DET
ejpam-4977	25	68	equality	equality	NOUN
ejpam-4977	25	69	x	x	X
ejpam-4977	25	70	∗	∗	NOUN
ejpam-4977	25	71	(	(	PUNCT
ejpam-4977	25	72	x	x	X
ejpam-4977	25	73	∗	∗	NOUN
ejpam-4977	25	74	y	y	NOUN
ejpam-4977	25	75	)	)	PUNCT
ejpam-4977	25	76	=	=	SYM
ejpam-4977	26	1	y	y	PROPN
ejpam-4977	26	2	holds	hold	VERB
ejpam-4977	26	3	for	for	ADP
ejpam-4977	26	4	all	all	DET
ejpam-4977	26	5	x	x	NOUN
ejpam-4977	26	6	,	,	PUNCT
ejpam-4977	26	7	y	y	PROPN
ejpam-4977	26	8	∈	∈	PROPN
ejpam-4977	26	9	x.	x.	NOUN
ejpam-4977	26	10	example	example	NOUN
ejpam-4977	27	1	1	1	NUM
ejpam-4977	27	2	.	.	PUNCT
ejpam-4977	28	1	(	(	PUNCT
ejpam-4977	28	2	[	[	X
ejpam-4977	28	3	4],[5	4],[5	NUM
ejpam-4977	28	4	]	]	PUNCT
ejpam-4977	28	5	)	)	PUNCT
ejpam-4977	28	6	:	:	PUNCT
ejpam-4977	28	7	let	let	VERB
ejpam-4977	28	8	x	x	PUNCT
ejpam-4977	28	9	=	=	PUNCT
ejpam-4977	28	10	{	{	PUNCT
ejpam-4977	28	11	0	0	NUM
ejpam-4977	28	12	,	,	PUNCT
ejpam-4977	28	13	1	1	NUM
ejpam-4977	28	14	,	,	PUNCT
ejpam-4977	28	15	2	2	NUM
ejpam-4977	28	16	,	,	PUNCT
ejpam-4977	28	17	3	3	NUM
ejpam-4977	28	18	}	}	PUNCT
ejpam-4977	28	19	and	and	CCONJ
ejpam-4977	28	20	y	y	PROPN
ejpam-4977	28	21	=	=	PUNCT
ejpam-4977	28	22	{	{	PUNCT
ejpam-4977	28	23	0	0	NUM
ejpam-4977	28	24	,	,	PUNCT
ejpam-4977	28	25	1	1	NUM
ejpam-4977	28	26	,	,	PUNCT
ejpam-4977	28	27	2	2	NUM
ejpam-4977	28	28	,	,	PUNCT
ejpam-4977	28	29	3	3	NUM
ejpam-4977	28	30	,	,	PUNCT
ejpam-4977	28	31	4	4	NUM
ejpam-4977	28	32	,	,	PUNCT
ejpam-4977	28	33	5	5	NUM
ejpam-4977	28	34	}	}	PUNCT
ejpam-4977	28	35	.	.	PUNCT
ejpam-4977	29	1	define	define	VERB
ejpam-4977	29	2	binary	binary	ADJ
ejpam-4977	29	3	operations	operation	NOUN
ejpam-4977	29	4	∗	∗	NOUN
ejpam-4977	29	5	on	on	ADP
ejpam-4977	29	6	x	x	PUNCT
ejpam-4977	29	7	and	and	CCONJ
ejpam-4977	29	8	⊙	⊙	NOUN
ejpam-4977	29	9	on	on	ADP
ejpam-4977	29	10	y	y	PROPN
ejpam-4977	29	11	defined	define	VERB
ejpam-4977	29	12	by	by	ADP
ejpam-4977	29	13	the	the	DET
ejpam-4977	29	14	following	follow	VERB
ejpam-4977	29	15	two	two	NUM
ejpam-4977	29	16	tables	table	NOUN
ejpam-4977	29	17	respectively	respectively	ADV
ejpam-4977	29	18	:	:	PUNCT
ejpam-4977	29	19	∗	∗	NOUN
ejpam-4977	29	20	0	0	NUM
ejpam-4977	29	21	1	1	NUM
ejpam-4977	29	22	2	2	NUM
ejpam-4977	29	23	3	3	NUM
ejpam-4977	29	24	0	0	NUM
ejpam-4977	29	25	0	0	NUM
ejpam-4977	29	26	3	3	NUM
ejpam-4977	29	27	2	2	NUM
ejpam-4977	29	28	1	1	NUM
ejpam-4977	29	29	1	1	NUM
ejpam-4977	29	30	1	1	NUM
ejpam-4977	29	31	0	0	NUM
ejpam-4977	29	32	3	3	NUM
ejpam-4977	29	33	2	2	NUM
ejpam-4977	29	34	2	2	NUM
ejpam-4977	29	35	2	2	NUM
ejpam-4977	29	36	1	1	NUM
ejpam-4977	29	37	0	0	NUM
ejpam-4977	29	38	3	3	NUM
ejpam-4977	29	39	3	3	NUM
ejpam-4977	29	40	3	3	NUM
ejpam-4977	29	41	2	2	NUM
ejpam-4977	29	42	1	1	NUM
ejpam-4977	29	43	0	0	NUM
ejpam-4977	29	44	⊙	⊙	NOUN
ejpam-4977	29	45	0	0	NUM
ejpam-4977	29	46	1	1	NUM
ejpam-4977	29	47	2	2	NUM
ejpam-4977	29	48	3	3	NUM
ejpam-4977	29	49	4	4	NUM
ejpam-4977	29	50	5	5	NUM
ejpam-4977	29	51	0	0	NUM
ejpam-4977	29	52	0	0	NUM
ejpam-4977	29	53	2	2	NUM
ejpam-4977	29	54	1	1	NUM
ejpam-4977	29	55	3	3	NUM
ejpam-4977	29	56	4	4	NUM
ejpam-4977	29	57	5	5	NUM
ejpam-4977	29	58	1	1	NUM
ejpam-4977	29	59	1	1	NUM
ejpam-4977	29	60	0	0	NUM
ejpam-4977	29	61	2	2	NUM
ejpam-4977	29	62	4	4	NUM
ejpam-4977	29	63	5	5	NUM
ejpam-4977	29	64	3	3	NUM
ejpam-4977	29	65	2	2	NUM
ejpam-4977	29	66	2	2	NUM
ejpam-4977	29	67	1	1	NUM
ejpam-4977	29	68	0	0	NUM
ejpam-4977	29	69	5	5	NUM
ejpam-4977	29	70	3	3	NUM
ejpam-4977	29	71	4	4	NUM
ejpam-4977	29	72	3	3	NUM
ejpam-4977	29	73	3	3	NUM
ejpam-4977	29	74	4	4	NUM
ejpam-4977	29	75	5	5	NUM
ejpam-4977	29	76	0	0	NUM
ejpam-4977	29	77	2	2	NUM
ejpam-4977	29	78	1	1	NUM
ejpam-4977	29	79	4	4	NUM
ejpam-4977	29	80	4	4	NUM
ejpam-4977	29	81	5	5	NUM
ejpam-4977	29	82	3	3	NUM
ejpam-4977	29	83	1	1	NUM
ejpam-4977	29	84	0	0	NUM
ejpam-4977	29	85	2	2	NUM
ejpam-4977	29	86	5	5	NUM
ejpam-4977	29	87	5	5	NUM
ejpam-4977	29	88	3	3	NUM
ejpam-4977	29	89	4	4	NUM
ejpam-4977	29	90	2	2	NUM
ejpam-4977	29	91	1	1	NUM
ejpam-4977	29	92	0	0	NUM
ejpam-4977	29	93	then	then	ADV
ejpam-4977	29	94	(	(	PUNCT
ejpam-4977	29	95	x	x	X
ejpam-4977	29	96	,	,	PUNCT
ejpam-4977	29	97	∗	∗	NOUN
ejpam-4977	29	98	,	,	PUNCT
ejpam-4977	29	99	0	0	NUM
ejpam-4977	29	100	)	)	PUNCT
ejpam-4977	29	101	is	be	AUX
ejpam-4977	29	102	a	a	DET
ejpam-4977	29	103	commutative	commutative	ADJ
ejpam-4977	29	104	b	b	NOUN
ejpam-4977	29	105	-	-	PUNCT
ejpam-4977	29	106	algebra	algebra	NOUN
ejpam-4977	29	107	,	,	PUNCT
ejpam-4977	29	108	but	but	CCONJ
ejpam-4977	29	109	(	(	PUNCT
ejpam-4977	29	110	y,⊙	y,⊙	NOUN
ejpam-4977	29	111	,	,	PUNCT
ejpam-4977	29	112	0	0	NUM
ejpam-4977	29	113	)	)	PUNCT
ejpam-4977	29	114	is	be	AUX
ejpam-4977	29	115	a	a	DET
ejpam-4977	29	116	non	non	ADJ
ejpam-4977	29	117	commutative	commutative	ADJ
ejpam-4977	29	118	b	b	X
ejpam-4977	29	119	-	-	PUNCT
ejpam-4977	29	120	algebra	algebra	NOUN
ejpam-4977	29	121	,	,	PUNCT
ejpam-4977	29	122	since	since	SCONJ
ejpam-4977	29	123	2	2	NUM
ejpam-4977	29	124	∗	∗	NOUN
ejpam-4977	29	125	(	(	PUNCT
ejpam-4977	29	126	0	0	NUM
ejpam-4977	29	127	∗	∗	NOUN
ejpam-4977	29	128	5	5	NUM
ejpam-4977	29	129	)	)	PUNCT
ejpam-4977	29	130	=	=	SYM
ejpam-4977	30	1	2	2	NUM
ejpam-4977	30	2	∗	∗	NOUN
ejpam-4977	30	3	5	5	NUM
ejpam-4977	30	4	=	=	SYM
ejpam-4977	30	5	4	4	NUM
ejpam-4977	30	6	̸=	̸=	PROPN
ejpam-4977	30	7	3	3	NUM
ejpam-4977	30	8	=	=	SYM
ejpam-4977	30	9	5	5	NUM
ejpam-4977	30	10	∗	∗	NOUN
ejpam-4977	30	11	1	1	NUM
ejpam-4977	30	12	=	=	SYM
ejpam-4977	30	13	5	5	NUM
ejpam-4977	30	14	∗	∗	NOUN
ejpam-4977	30	15	(	(	PUNCT
ejpam-4977	30	16	0	0	NUM
ejpam-4977	30	17	∗	∗	NOUN
ejpam-4977	30	18	2	2	NUM
ejpam-4977	30	19	)	)	PUNCT
ejpam-4977	30	20	.	.	PUNCT
ejpam-4977	31	1	we	we	PRON
ejpam-4977	31	2	recall	recall	VERB
ejpam-4977	31	3	the	the	DET
ejpam-4977	31	4	following	follow	VERB
ejpam-4977	31	5	axioms	axiom	NOUN
ejpam-4977	31	6	for	for	ADP
ejpam-4977	31	7	the	the	DET
ejpam-4977	31	8	laws	law	NOUN
ejpam-4977	31	9	of	of	ADP
ejpam-4977	31	10	exponents	exponent	NOUN
ejpam-4977	31	11	for	for	ADP
ejpam-4977	31	12	b	b	NOUN
ejpam-4977	31	13	-	-	PUNCT
ejpam-4977	31	14	algebras	algebras	PROPN
ejpam-4977	31	15	.	.	PUNCT
ejpam-4977	32	1	theorem	theorem	NOUN
ejpam-4977	32	2	1	1	NUM
ejpam-4977	32	3	.	.	PUNCT
ejpam-4977	33	1	[	[	X
ejpam-4977	33	2	1	1	X
ejpam-4977	33	3	]	]	X
ejpam-4977	33	4	let	let	VERB
ejpam-4977	33	5	(	(	PUNCT
ejpam-4977	33	6	x	x	NOUN
ejpam-4977	33	7	,	,	PUNCT
ejpam-4977	33	8	∗	∗	NOUN
ejpam-4977	33	9	,	,	PUNCT
ejpam-4977	33	10	0	0	NUM
ejpam-4977	33	11	)	)	PUNCT
ejpam-4977	33	12	be	be	AUX
ejpam-4977	33	13	a	a	DET
ejpam-4977	33	14	b	b	NOUN
ejpam-4977	33	15	-	-	PUNCT
ejpam-4977	33	16	algebra	algebra	NOUN
ejpam-4977	33	17	.	.	PUNCT
ejpam-4977	34	1	then	then	ADV
ejpam-4977	34	2	the	the	DET
ejpam-4977	34	3	following	follow	VERB
ejpam-4977	34	4	conditions	condition	NOUN
ejpam-4977	34	5	hold	hold	VERB
ejpam-4977	34	6	for	for	ADP
ejpam-4977	34	7	any	any	DET
ejpam-4977	34	8	x	x	NOUN
ejpam-4977	34	9	,	,	PUNCT
ejpam-4977	34	10	y	y	PROPN
ejpam-4977	34	11	,	,	PUNCT
ejpam-4977	34	12	z	z	PROPN
ejpam-4977	34	13	∈	∈	PROPN
ejpam-4977	35	1	x	x	X
ejpam-4977	35	2	:	:	PUNCT
ejpam-4977	35	3	(	(	PUNCT
ejpam-4977	35	4	i	i	NOUN
ejpam-4977	35	5	)	)	PUNCT
ejpam-4977	35	6	x	x	X
ejpam-4977	35	7	=	=	PUNCT
ejpam-4977	35	8	(	(	PUNCT
ejpam-4977	35	9	x	x	X
ejpam-4977	35	10	∗	∗	PROPN
ejpam-4977	35	11	y	y	NOUN
ejpam-4977	35	12	)	)	PUNCT
ejpam-4977	35	13	∗	∗	NOUN
ejpam-4977	35	14	(	(	PUNCT
ejpam-4977	35	15	0	0	NUM
ejpam-4977	35	16	∗	∗	PROPN
ejpam-4977	35	17	y	y	PROPN
ejpam-4977	35	18	)	)	PUNCT
ejpam-4977	35	19	,	,	PUNCT
ejpam-4977	35	20	(	(	PUNCT
ejpam-4977	35	21	ii	ii	X
ejpam-4977	35	22	)	)	PUNCT
ejpam-4977	35	23	y	y	PROPN
ejpam-4977	35	24	∗	∗	NOUN
ejpam-4977	35	25	x	x	PUNCT
ejpam-4977	35	26	=	=	SYM
ejpam-4977	35	27	0	0	NUM
ejpam-4977	35	28	∗	∗	NOUN
ejpam-4977	35	29	(	(	PUNCT
ejpam-4977	35	30	x	x	X
ejpam-4977	35	31	∗	∗	PROPN
ejpam-4977	35	32	y	y	PROPN
ejpam-4977	35	33	)	)	PUNCT
ejpam-4977	35	34	,	,	PUNCT
ejpam-4977	35	35	(	(	PUNCT
ejpam-4977	35	36	iii	iii	NOUN
ejpam-4977	35	37	)	)	PUNCT
ejpam-4977	35	38	0	0	NUM
ejpam-4977	36	1	∗	∗	NOUN
ejpam-4977	36	2	(	(	PUNCT
ejpam-4977	36	3	0	0	NUM
ejpam-4977	36	4	∗	∗	NOUN
ejpam-4977	36	5	x	x	NOUN
ejpam-4977	36	6	)	)	PUNCT
ejpam-4977	36	7	=	=	SYM
ejpam-4977	37	1	x	x	NOUN
ejpam-4977	37	2	,	,	PUNCT
ejpam-4977	37	3	(	(	PUNCT
ejpam-4977	37	4	iv	iv	X
ejpam-4977	37	5	)	)	PUNCT
ejpam-4977	37	6	x	x	SYM
ejpam-4977	37	7	∗	∗	NOUN
ejpam-4977	37	8	(	(	PUNCT
ejpam-4977	37	9	y	y	PROPN
ejpam-4977	37	10	∗	∗	PROPN
ejpam-4977	37	11	z	z	NOUN
ejpam-4977	37	12	)	)	PUNCT
ejpam-4977	37	13	=	=	SYM
ejpam-4977	38	1	(	(	PUNCT
ejpam-4977	38	2	x	x	SYM
ejpam-4977	38	3	∗	∗	NOUN
ejpam-4977	38	4	(	(	PUNCT
ejpam-4977	38	5	0	0	NUM
ejpam-4977	38	6	∗	∗	NOUN
ejpam-4977	38	7	z	z	NOUN
ejpam-4977	38	8	)	)	PUNCT
ejpam-4977	38	9	)	)	PUNCT
ejpam-4977	38	10	∗	∗	PROPN
ejpam-4977	38	11	y	y	PROPN
ejpam-4977	38	12	,	,	PUNCT
ejpam-4977	38	13	m.	m.	NOUN
ejpam-4977	38	14	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	38	15	/	/	SYM
ejpam-4977	38	16	eur	eur	PROPN
ejpam-4977	38	17	.	.	PUNCT
ejpam-4977	39	1	j.	j.	PROPN
ejpam-4977	39	2	pure	pure	PROPN
ejpam-4977	39	3	appl	appl	PROPN
ejpam-4977	39	4	.	.	PROPN
ejpam-4977	39	5	math	math	PROPN
ejpam-4977	39	6	,	,	PUNCT
ejpam-4977	39	7	17	17	NUM
ejpam-4977	39	8	(	(	PUNCT
ejpam-4977	39	9	1	1	NUM
ejpam-4977	39	10	)	)	PUNCT
ejpam-4977	39	11	(	(	PUNCT
ejpam-4977	39	12	2024	2024	NUM
ejpam-4977	39	13	)	)	PUNCT
ejpam-4977	39	14	,	,	PUNCT
ejpam-4977	39	15	116	116	NUM
ejpam-4977	39	16	-	-	SYM
ejpam-4977	39	17	123	123	NUM
ejpam-4977	39	18	118	118	NUM
ejpam-4977	39	19	(	(	PUNCT
ejpam-4977	39	20	v	v	NOUN
ejpam-4977	39	21	)	)	PUNCT
ejpam-4977	39	22	x	x	SYM
ejpam-4977	39	23	∗	∗	NOUN
ejpam-4977	39	24	y	y	NOUN
ejpam-4977	39	25	=	=	PUNCT
ejpam-4977	39	26	x	x	SYM
ejpam-4977	39	27	∗	∗	NOUN
ejpam-4977	39	28	(	(	PUNCT
ejpam-4977	39	29	0	0	NUM
ejpam-4977	39	30	∗	∗	NOUN
ejpam-4977	39	31	(	(	PUNCT
ejpam-4977	39	32	0	0	NUM
ejpam-4977	39	33	∗	∗	PROPN
ejpam-4977	39	34	y	y	PROPN
ejpam-4977	39	35	)	)	PUNCT
ejpam-4977	39	36	,	,	PUNCT
ejpam-4977	39	37	(	(	PUNCT
ejpam-4977	39	38	vi	vi	NOUN
ejpam-4977	39	39	)	)	PUNCT
ejpam-4977	39	40	x	x	X
ejpam-4977	40	1	∗	∗	NOUN
ejpam-4977	40	2	y	y	NOUN
ejpam-4977	40	3	=	=	SYM
ejpam-4977	40	4	0	0	NUM
ejpam-4977	40	5	implies	imply	VERB
ejpam-4977	40	6	x	x	PUNCT
ejpam-4977	40	7	=	=	SYM
ejpam-4977	40	8	y	y	PROPN
ejpam-4977	40	9	,	,	PUNCT
ejpam-4977	40	10	(	(	PUNCT
ejpam-4977	40	11	vii	vii	PROPN
ejpam-4977	40	12	)	)	PUNCT
ejpam-4977	40	13	x	x	PROPN
ejpam-4977	40	14	∗	∗	NOUN
ejpam-4977	40	15	z	z	NOUN
ejpam-4977	40	16	=	=	SYM
ejpam-4977	40	17	y	y	PROPN
ejpam-4977	40	18	∗	∗	NOUN
ejpam-4977	40	19	z	z	PROPN
ejpam-4977	40	20	implies	imply	VERB
ejpam-4977	40	21	x	x	PUNCT
ejpam-4977	40	22	=	=	SYM
ejpam-4977	40	23	y	y	PROPN
ejpam-4977	40	24	and	and	CCONJ
ejpam-4977	40	25	(	(	PUNCT
ejpam-4977	40	26	viii	viii	NOUN
ejpam-4977	40	27	)	)	PUNCT
ejpam-4977	40	28	0	0	NUM
ejpam-4977	40	29	∗	∗	NOUN
ejpam-4977	40	30	x	x	X
ejpam-4977	41	1	=	=	SYM
ejpam-4977	41	2	0	0	NUM
ejpam-4977	41	3	∗	∗	NOUN
ejpam-4977	41	4	y	y	PROPN
ejpam-4977	41	5	,	,	PUNCT
ejpam-4977	41	6	implies	imply	VERB
ejpam-4977	41	7	x	x	PUNCT
ejpam-4977	41	8	=	=	PUNCT
ejpam-4977	41	9	y.	y.	NOUN
ejpam-4977	41	10	definition	definition	NOUN
ejpam-4977	41	11	2	2	NUM
ejpam-4977	41	12	.	.	PUNCT
ejpam-4977	42	1	[	[	X
ejpam-4977	42	2	5	5	NUM
ejpam-4977	42	3	]	]	X
ejpam-4977	42	4	let	let	VERB
ejpam-4977	42	5	(	(	PUNCT
ejpam-4977	42	6	x	x	NOUN
ejpam-4977	42	7	,	,	PUNCT
ejpam-4977	42	8	∗	∗	NOUN
ejpam-4977	42	9	,	,	PUNCT
ejpam-4977	42	10	0	0	NUM
ejpam-4977	42	11	)	)	PUNCT
ejpam-4977	42	12	be	be	AUX
ejpam-4977	42	13	a	a	DET
ejpam-4977	42	14	b	b	NOUN
ejpam-4977	42	15	-	-	PUNCT
ejpam-4977	42	16	algebra	algebra	NOUN
ejpam-4977	42	17	.	.	PUNCT
ejpam-4977	43	1	a	a	DET
ejpam-4977	43	2	nonempty	nonempty	NOUN
ejpam-4977	43	3	subset	subset	VERB
ejpam-4977	43	4	i	i	PRON
ejpam-4977	43	5	of	of	ADP
ejpam-4977	43	6	x	x	PRON
ejpam-4977	43	7	is	be	AUX
ejpam-4977	43	8	called	call	VERB
ejpam-4977	43	9	a	a	DET
ejpam-4977	43	10	b	b	NOUN
ejpam-4977	43	11	-	-	PUNCT
ejpam-4977	43	12	ideal	ideal	NOUN
ejpam-4977	43	13	of	of	ADP
ejpam-4977	43	14	x	x	PRON
ejpam-4977	43	15	if	if	SCONJ
ejpam-4977	43	16	it	it	PRON
ejpam-4977	43	17	satisfies	satisfy	VERB
ejpam-4977	43	18	the	the	DET
ejpam-4977	43	19	following	follow	VERB
ejpam-4977	43	20	conditions	condition	NOUN
ejpam-4977	43	21	for	for	ADP
ejpam-4977	43	22	any	any	DET
ejpam-4977	43	23	x	x	NOUN
ejpam-4977	43	24	,	,	PUNCT
ejpam-4977	43	25	y	y	PROPN
ejpam-4977	43	26	,	,	PUNCT
ejpam-4977	43	27	z	z	PROPN
ejpam-4977	43	28	∈	∈	PROPN
ejpam-4977	44	1	x	x	X
ejpam-4977	44	2	:	:	PUNCT
ejpam-4977	44	3	(	(	PUNCT
ejpam-4977	44	4	i	i	NOUN
ejpam-4977	44	5	)	)	PUNCT
ejpam-4977	44	6	0	0	PUNCT
ejpam-4977	45	1	∈	∈	PROPN
ejpam-4977	46	1	i	i	PRON
ejpam-4977	46	2	,	,	PUNCT
ejpam-4977	46	3	(	(	PUNCT
ejpam-4977	46	4	ii	ii	NOUN
ejpam-4977	46	5	)	)	PUNCT
ejpam-4977	46	6	if	if	SCONJ
ejpam-4977	46	7	x	x	PROPN
ejpam-4977	46	8	∗	∗	VERB
ejpam-4977	46	9	y	y	NOUN
ejpam-4977	46	10	∈	∈	PROPN
ejpam-4977	47	1	i	i	PRON
ejpam-4977	47	2	and	and	CCONJ
ejpam-4977	47	3	y	y	PROPN
ejpam-4977	47	4	∈	∈	PROPN
ejpam-4977	48	1	i	i	PRON
ejpam-4977	48	2	,	,	PUNCT
ejpam-4977	48	3	then	then	ADV
ejpam-4977	48	4	x	x	PART
ejpam-4977	48	5	∈	∈	PROPN
ejpam-4977	48	6	i.	i.	NOUN
ejpam-4977	48	7	definition	definition	NOUN
ejpam-4977	48	8	3	3	NUM
ejpam-4977	48	9	.	.	PUNCT
ejpam-4977	49	1	[	[	X
ejpam-4977	49	2	6	6	NUM
ejpam-4977	49	3	]	]	PUNCT
ejpam-4977	49	4	let	let	VERB
ejpam-4977	49	5	x	x	PRON
ejpam-4977	49	6	be	be	AUX
ejpam-4977	49	7	a	a	DET
ejpam-4977	49	8	non	non	ADJ
ejpam-4977	49	9	-	-	ADJ
ejpam-4977	49	10	empty	empty	ADJ
ejpam-4977	49	11	set	set	NOUN
ejpam-4977	49	12	with	with	ADP
ejpam-4977	49	13	a	a	DET
ejpam-4977	49	14	binary	binary	ADJ
ejpam-4977	49	15	operation	operation	NOUN
ejpam-4977	49	16	∗	∗	NOUN
ejpam-4977	49	17	and	and	CCONJ
ejpam-4977	49	18	a	a	DET
ejpam-4977	49	19	constant	constant	ADJ
ejpam-4977	49	20	0	0	NUM
ejpam-4977	49	21	.	.	PUNCT
ejpam-4977	50	1	then	then	ADV
ejpam-4977	50	2	the	the	DET
ejpam-4977	50	3	triple	triple	ADJ
ejpam-4977	50	4	(	(	PUNCT
ejpam-4977	50	5	x	x	NOUN
ejpam-4977	50	6	,	,	PUNCT
ejpam-4977	50	7	∗	∗	NOUN
ejpam-4977	50	8	,	,	PUNCT
ejpam-4977	50	9	0	0	NUM
ejpam-4977	50	10	)	)	PUNCT
ejpam-4977	50	11	is	be	AUX
ejpam-4977	50	12	a	a	DET
ejpam-4977	50	13	dual	dual	ADJ
ejpam-4977	50	14	b	b	NOUN
ejpam-4977	50	15	-	-	PUNCT
ejpam-4977	50	16	algebra	algebra	NOUN
ejpam-4977	50	17	if	if	SCONJ
ejpam-4977	50	18	its	its	PRON
ejpam-4977	50	19	satisfies	satisfie	NOUN
ejpam-4977	50	20	the	the	DET
ejpam-4977	50	21	following	following	ADJ
ejpam-4977	50	22	axioms	axiom	NOUN
ejpam-4977	50	23	for	for	ADP
ejpam-4977	50	24	all	all	DET
ejpam-4977	50	25	x	x	NOUN
ejpam-4977	50	26	,	,	PUNCT
ejpam-4977	50	27	y	y	PROPN
ejpam-4977	50	28	,	,	PUNCT
ejpam-4977	50	29	z	z	PROPN
ejpam-4977	50	30	∈	∈	PROPN
ejpam-4977	50	31	x	x	X
ejpam-4977	50	32	:	:	PUNCT
ejpam-4977	50	33	(	(	PUNCT
ejpam-4977	50	34	i	i	NOUN
ejpam-4977	50	35	)	)	PUNCT
ejpam-4977	51	1	x	x	SYM
ejpam-4977	51	2	∗	∗	NOUN
ejpam-4977	51	3	x	x	SYM
ejpam-4977	51	4	=	=	SYM
ejpam-4977	51	5	0	0	NUM
ejpam-4977	51	6	,	,	PUNCT
ejpam-4977	51	7	(	(	PUNCT
ejpam-4977	51	8	ii	ii	NOUN
ejpam-4977	51	9	)	)	PUNCT
ejpam-4977	51	10	0	0	NUM
ejpam-4977	52	1	∗	∗	NOUN
ejpam-4977	52	2	x	x	X
ejpam-4977	53	1	=	=	SYM
ejpam-4977	53	2	x	x	X
ejpam-4977	53	3	,	,	PUNCT
ejpam-4977	53	4	(	(	PUNCT
ejpam-4977	53	5	iii	iii	NOUN
ejpam-4977	53	6	)	)	PUNCT
ejpam-4977	53	7	x	x	SYM
ejpam-4977	53	8	∗	∗	NOUN
ejpam-4977	53	9	(	(	PUNCT
ejpam-4977	53	10	y	y	PROPN
ejpam-4977	53	11	∗	∗	PROPN
ejpam-4977	53	12	z	z	NOUN
ejpam-4977	53	13	)	)	PUNCT
ejpam-4977	53	14	=	=	SYM
ejpam-4977	53	15	(	(	PUNCT
ejpam-4977	53	16	(	(	PUNCT
ejpam-4977	53	17	y	y	NOUN
ejpam-4977	53	18	∗	∗	NOUN
ejpam-4977	53	19	0	0	NUM
ejpam-4977	53	20	)	)	PUNCT
ejpam-4977	53	21	∗	∗	NOUN
ejpam-4977	53	22	x	x	NOUN
ejpam-4977	53	23	)	)	PUNCT
ejpam-4977	53	24	∗	∗	NOUN
ejpam-4977	53	25	z.	z.	PROPN
ejpam-4977	53	26	definition	definition	NOUN
ejpam-4977	53	27	4	4	NUM
ejpam-4977	53	28	.	.	PUNCT
ejpam-4977	54	1	[	[	X
ejpam-4977	54	2	8	8	NUM
ejpam-4977	54	3	]	]	X
ejpam-4977	54	4	let	let	VERB
ejpam-4977	54	5	(	(	PUNCT
ejpam-4977	54	6	x	x	NOUN
ejpam-4977	54	7	,	,	PUNCT
ejpam-4977	54	8	∗	∗	NOUN
ejpam-4977	54	9	,	,	PUNCT
ejpam-4977	54	10	0	0	NUM
ejpam-4977	54	11	)	)	PUNCT
ejpam-4977	54	12	be	be	AUX
ejpam-4977	54	13	a	a	DET
ejpam-4977	54	14	dual	dual	ADJ
ejpam-4977	54	15	b	b	NOUN
ejpam-4977	54	16	-	-	PUNCT
ejpam-4977	54	17	algebra	algebra	NOUN
ejpam-4977	54	18	.	.	PUNCT
ejpam-4977	55	1	a	a	DET
ejpam-4977	55	2	nonempty	nonempty	NOUN
ejpam-4977	55	3	subset	subset	VERB
ejpam-4977	55	4	f	f	PROPN
ejpam-4977	55	5	of	of	ADP
ejpam-4977	55	6	x	x	PROPN
ejpam-4977	55	7	is	be	AUX
ejpam-4977	55	8	called	call	VERB
ejpam-4977	55	9	a	a	DET
ejpam-4977	55	10	dual	dual	ADJ
ejpam-4977	55	11	b	b	NOUN
ejpam-4977	55	12	-	-	NOUN
ejpam-4977	55	13	filter	filter	NOUN
ejpam-4977	55	14	of	of	ADP
ejpam-4977	55	15	x	x	PRON
ejpam-4977	55	16	if	if	SCONJ
ejpam-4977	55	17	it	it	PRON
ejpam-4977	55	18	satisfies	satisfy	VERB
ejpam-4977	55	19	the	the	DET
ejpam-4977	55	20	following	following	ADJ
ejpam-4977	55	21	axioms	axiom	NOUN
ejpam-4977	55	22	for	for	ADP
ejpam-4977	55	23	all	all	DET
ejpam-4977	55	24	x	x	NOUN
ejpam-4977	55	25	,	,	PUNCT
ejpam-4977	55	26	y	y	PROPN
ejpam-4977	55	27	,	,	PUNCT
ejpam-4977	55	28	z	z	PROPN
ejpam-4977	55	29	∈	∈	PROPN
ejpam-4977	56	1	x	x	X
ejpam-4977	56	2	:	:	PUNCT
ejpam-4977	56	3	(	(	PUNCT
ejpam-4977	56	4	i	i	NOUN
ejpam-4977	56	5	)	)	PUNCT
ejpam-4977	56	6	0	0	PUNCT
ejpam-4977	57	1	∈	∈	PROPN
ejpam-4977	57	2	f	f	X
ejpam-4977	57	3	,	,	PUNCT
ejpam-4977	57	4	(	(	PUNCT
ejpam-4977	57	5	ii	ii	NOUN
ejpam-4977	57	6	)	)	PUNCT
ejpam-4977	58	1	if	if	SCONJ
ejpam-4977	58	2	x	x	PROPN
ejpam-4977	58	3	∗	∗	VERB
ejpam-4977	58	4	y	y	PROPN
ejpam-4977	58	5	∈	∈	PROPN
ejpam-4977	58	6	f	f	PROPN
ejpam-4977	58	7	and	and	CCONJ
ejpam-4977	58	8	x	x	SYM
ejpam-4977	58	9	∈	∈	PROPN
ejpam-4977	58	10	f	f	PROPN
ejpam-4977	58	11	,	,	PUNCT
ejpam-4977	58	12	then	then	ADV
ejpam-4977	58	13	y	y	PROPN
ejpam-4977	58	14	∈	∈	PROPN
ejpam-4977	58	15	f	f	X
ejpam-4977	58	16	.	.	PUNCT
ejpam-4977	59	1	there	there	PRON
ejpam-4977	59	2	is	be	VERB
ejpam-4977	59	3	a	a	DET
ejpam-4977	59	4	b	b	NOUN
ejpam-4977	59	5	-	-	PUNCT
ejpam-4977	59	6	algebra	algebra	NOUN
ejpam-4977	59	7	that	that	PRON
ejpam-4977	59	8	is	be	AUX
ejpam-4977	59	9	also	also	ADV
ejpam-4977	59	10	a	a	DET
ejpam-4977	59	11	dual	dual	ADJ
ejpam-4977	59	12	b	b	NOUN
ejpam-4977	59	13	-	-	PUNCT
ejpam-4977	59	14	algebra	algebra	NOUN
ejpam-4977	59	15	in	in	ADP
ejpam-4977	59	16	the	the	DET
ejpam-4977	59	17	following	follow	VERB
ejpam-4977	59	18	example	example	NOUN
ejpam-4977	59	19	.	.	PUNCT
ejpam-4977	60	1	example	example	NOUN
ejpam-4977	61	1	2	2	NUM
ejpam-4977	61	2	.	.	PUNCT
ejpam-4977	62	1	[	[	X
ejpam-4977	62	2	6	6	NUM
ejpam-4977	62	3	]	]	PUNCT
ejpam-4977	62	4	let	let	VERB
ejpam-4977	62	5	x	x	PUNCT
ejpam-4977	62	6	=	=	PUNCT
ejpam-4977	62	7	{	{	PUNCT
ejpam-4977	62	8	0	0	NUM
ejpam-4977	62	9	,	,	PUNCT
ejpam-4977	62	10	a	a	DET
ejpam-4977	62	11	,	,	PUNCT
ejpam-4977	62	12	b	b	NOUN
ejpam-4977	62	13	,	,	PUNCT
ejpam-4977	62	14	c	c	NOUN
ejpam-4977	62	15	}	}	PUNCT
ejpam-4977	62	16	and	and	CCONJ
ejpam-4977	62	17	a	a	DET
ejpam-4977	62	18	binary	binary	ADJ
ejpam-4977	62	19	operations	operation	NOUN
ejpam-4977	62	20	∗	∗	NOUN
ejpam-4977	62	21	on	on	ADP
ejpam-4977	62	22	x	x	PUNCT
ejpam-4977	62	23	satisfying	satisfy	VERB
ejpam-4977	62	24	the	the	DET
ejpam-4977	62	25	following	follow	VERB
ejpam-4977	62	26	table	table	NOUN
ejpam-4977	62	27	:	:	PUNCT
ejpam-4977	62	28	∗	∗	NOUN
ejpam-4977	62	29	0	0	PUNCT
ejpam-4977	63	1	a	a	DET
ejpam-4977	63	2	b	b	NOUN
ejpam-4977	63	3	c	c	NOUN
ejpam-4977	63	4	0	0	NUM
ejpam-4977	63	5	0	0	NUM
ejpam-4977	63	6	a	a	DET
ejpam-4977	63	7	b	b	NOUN
ejpam-4977	63	8	c	c	NOUN
ejpam-4977	63	9	a	a	DET
ejpam-4977	63	10	a	a	DET
ejpam-4977	63	11	0	0	NUM
ejpam-4977	63	12	c	c	NOUN
ejpam-4977	63	13	b	b	PROPN
ejpam-4977	63	14	b	b	PROPN
ejpam-4977	63	15	b	b	PROPN
ejpam-4977	63	16	c	c	PROPN
ejpam-4977	63	17	0	0	NUM
ejpam-4977	64	1	a	a	DET
ejpam-4977	64	2	c	c	NOUN
ejpam-4977	64	3	c	c	NOUN
ejpam-4977	64	4	b	b	PROPN
ejpam-4977	64	5	a	a	DET
ejpam-4977	64	6	0	0	NUM
ejpam-4977	64	7	m.	m.	NOUN
ejpam-4977	64	8	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	64	9	/	/	SYM
ejpam-4977	64	10	eur	eur	PROPN
ejpam-4977	64	11	.	.	PUNCT
ejpam-4977	65	1	j.	j.	PROPN
ejpam-4977	65	2	pure	pure	PROPN
ejpam-4977	65	3	appl	appl	PROPN
ejpam-4977	65	4	.	.	PROPN
ejpam-4977	65	5	math	math	PROPN
ejpam-4977	65	6	,	,	PUNCT
ejpam-4977	65	7	17	17	NUM
ejpam-4977	65	8	(	(	PUNCT
ejpam-4977	65	9	1	1	NUM
ejpam-4977	65	10	)	)	PUNCT
ejpam-4977	65	11	(	(	PUNCT
ejpam-4977	65	12	2024	2024	NUM
ejpam-4977	65	13	)	)	PUNCT
ejpam-4977	65	14	,	,	PUNCT
ejpam-4977	65	15	116	116	NUM
ejpam-4977	65	16	-	-	SYM
ejpam-4977	65	17	123	123	NUM
ejpam-4977	65	18	119	119	NUM
ejpam-4977	65	19	example	example	NOUN
ejpam-4977	65	20	3	3	NUM
ejpam-4977	65	21	.	.	PUNCT
ejpam-4977	66	1	[	[	X
ejpam-4977	66	2	8	8	NUM
ejpam-4977	66	3	]	]	PUNCT
ejpam-4977	66	4	let	let	VERB
ejpam-4977	66	5	x	x	PUNCT
ejpam-4977	66	6	=	=	PUNCT
ejpam-4977	66	7	{	{	PUNCT
ejpam-4977	66	8	0	0	NUM
ejpam-4977	66	9	,	,	PUNCT
ejpam-4977	66	10	a	a	DET
ejpam-4977	66	11	,	,	PUNCT
ejpam-4977	66	12	b	b	NOUN
ejpam-4977	66	13	,	,	PUNCT
ejpam-4977	66	14	c	c	NOUN
ejpam-4977	66	15	,	,	PUNCT
ejpam-4977	66	16	d	d	NOUN
ejpam-4977	66	17	,	,	PUNCT
ejpam-4977	66	18	e	e	NOUN
ejpam-4977	66	19	}	}	PUNCT
ejpam-4977	66	20	and	and	CCONJ
ejpam-4977	66	21	a	a	DET
ejpam-4977	66	22	binary	binary	ADJ
ejpam-4977	66	23	operations	operation	NOUN
ejpam-4977	66	24	∗	∗	NOUN
ejpam-4977	66	25	on	on	ADP
ejpam-4977	66	26	x	x	PUNCT
ejpam-4977	66	27	satisfies	satisfie	NOUN
ejpam-4977	66	28	the	the	DET
ejpam-4977	66	29	following	follow	VERB
ejpam-4977	66	30	table	table	NOUN
ejpam-4977	66	31	:	:	PUNCT
ejpam-4977	66	32	∗	∗	NOUN
ejpam-4977	66	33	0	0	PUNCT
ejpam-4977	67	1	a	a	DET
ejpam-4977	67	2	b	b	NOUN
ejpam-4977	67	3	c	c	NOUN
ejpam-4977	67	4	d	d	X
ejpam-4977	67	5	e	e	X
ejpam-4977	67	6	0	0	NUM
ejpam-4977	67	7	0	0	NUM
ejpam-4977	67	8	a	a	DET
ejpam-4977	67	9	b	b	NOUN
ejpam-4977	67	10	c	c	NOUN
ejpam-4977	67	11	d	d	PROPN
ejpam-4977	67	12	e	e	PROPN
ejpam-4977	67	13	a	a	PRON
ejpam-4977	67	14	b	b	PROPN
ejpam-4977	67	15	0	0	NUM
ejpam-4977	67	16	a	a	PRON
ejpam-4977	67	17	d	d	X
ejpam-4977	67	18	e	e	NOUN
ejpam-4977	67	19	c	c	PROPN
ejpam-4977	67	20	b	b	PROPN
ejpam-4977	67	21	d	d	PROPN
ejpam-4977	67	22	b	b	PROPN
ejpam-4977	67	23	0	0	NUM
ejpam-4977	68	1	e	e	NOUN
ejpam-4977	68	2	c	c	NOUN
ejpam-4977	68	3	d	d	X
ejpam-4977	68	4	c	c	NOUN
ejpam-4977	68	5	c	c	NOUN
ejpam-4977	68	6	d	d	X
ejpam-4977	68	7	e	e	X
ejpam-4977	68	8	0	0	PUNCT
ejpam-4977	68	9	a	a	DET
ejpam-4977	68	10	b	b	NOUN
ejpam-4977	69	1	d	d	X
ejpam-4977	69	2	d	d	PROPN
ejpam-4977	69	3	e	e	PROPN
ejpam-4977	69	4	c	c	PROPN
ejpam-4977	69	5	b	b	PROPN
ejpam-4977	69	6	0	0	NUM
ejpam-4977	69	7	a	a	DET
ejpam-4977	69	8	e	e	NOUN
ejpam-4977	69	9	e	e	NOUN
ejpam-4977	69	10	c	c	PROPN
ejpam-4977	69	11	d	d	PROPN
ejpam-4977	69	12	a	a	DET
ejpam-4977	69	13	b	b	PROPN
ejpam-4977	69	14	0	0	NUM
ejpam-4977	69	15	then	then	ADV
ejpam-4977	69	16	(	(	PUNCT
ejpam-4977	69	17	x	x	X
ejpam-4977	69	18	,	,	PUNCT
ejpam-4977	69	19	∗	∗	NOUN
ejpam-4977	69	20	,	,	PUNCT
ejpam-4977	69	21	0	0	NUM
ejpam-4977	69	22	)	)	PUNCT
ejpam-4977	69	23	is	be	AUX
ejpam-4977	69	24	a	a	DET
ejpam-4977	69	25	dual	dual	ADJ
ejpam-4977	69	26	b	b	NOUN
ejpam-4977	69	27	-	-	PUNCT
ejpam-4977	69	28	algebra	algebra	NOUN
ejpam-4977	69	29	.	.	PUNCT
ejpam-4977	70	1	the	the	DET
ejpam-4977	70	2	sets	set	NOUN
ejpam-4977	70	3	f0	f0	PROPN
ejpam-4977	70	4	=	=	PUNCT
ejpam-4977	70	5	{	{	PUNCT
ejpam-4977	70	6	0	0	NUM
ejpam-4977	70	7	}	}	PUNCT
ejpam-4977	70	8	,	,	PUNCT
ejpam-4977	70	9	f2	f2	PROPN
ejpam-4977	70	10	=	=	PUNCT
ejpam-4977	70	11	{	{	PUNCT
ejpam-4977	70	12	0	0	NUM
ejpam-4977	70	13	,	,	PUNCT
ejpam-4977	70	14	c	c	NOUN
ejpam-4977	70	15	}	}	PUNCT
ejpam-4977	70	16	,	,	PUNCT
ejpam-4977	70	17	f3	f3	PROPN
ejpam-4977	70	18	=	=	SYM
ejpam-4977	70	19	{	{	PUNCT
ejpam-4977	70	20	0	0	NUM
ejpam-4977	70	21	,	,	PUNCT
ejpam-4977	70	22	d	d	NOUN
ejpam-4977	70	23	}	}	PUNCT
ejpam-4977	70	24	,	,	PUNCT
ejpam-4977	70	25	f4	f4	NOUN
ejpam-4977	70	26	=	=	SYM
ejpam-4977	70	27	{	{	PUNCT
ejpam-4977	70	28	0	0	NUM
ejpam-4977	70	29	,	,	PUNCT
ejpam-4977	70	30	e	e	NOUN
ejpam-4977	70	31	}	}	PUNCT
ejpam-4977	70	32	and	and	CCONJ
ejpam-4977	70	33	f5	f5	NOUN
ejpam-4977	70	34	=	=	SYM
ejpam-4977	70	35	{	{	PUNCT
ejpam-4977	70	36	0	0	NUM
ejpam-4977	70	37	,	,	PUNCT
ejpam-4977	70	38	a	a	PRON
ejpam-4977	70	39	,	,	PUNCT
ejpam-4977	70	40	b	b	X
ejpam-4977	70	41	}	}	PUNCT
ejpam-4977	70	42	are	be	AUX
ejpam-4977	70	43	dual	dual	ADJ
ejpam-4977	70	44	b	b	NOUN
ejpam-4977	70	45	-	-	PUNCT
ejpam-4977	70	46	filters	filter	NOUN
ejpam-4977	70	47	of	of	ADP
ejpam-4977	70	48	x	x	PRON
ejpam-4977	70	49	while	while	SCONJ
ejpam-4977	70	50	a	a	DET
ejpam-4977	70	51	=	=	X
ejpam-4977	70	52	{	{	PUNCT
ejpam-4977	70	53	0	0	NUM
ejpam-4977	70	54	,	,	PUNCT
ejpam-4977	70	55	a	a	PRON
ejpam-4977	70	56	,	,	PUNCT
ejpam-4977	70	57	e	e	NOUN
ejpam-4977	70	58	}	}	PUNCT
ejpam-4977	70	59	is	be	AUX
ejpam-4977	70	60	not	not	PART
ejpam-4977	70	61	a	a	DET
ejpam-4977	70	62	dual	dual	ADJ
ejpam-4977	70	63	b	b	NOUN
ejpam-4977	70	64	-	-	NOUN
ejpam-4977	70	65	filter	filter	NOUN
ejpam-4977	70	66	since	since	SCONJ
ejpam-4977	70	67	e	e	NOUN
ejpam-4977	70	68	∗	∗	NOUN
ejpam-4977	70	69	c	c	NOUN
ejpam-4977	71	1	=	=	PUNCT
ejpam-4977	71	2	a	a	DET
ejpam-4977	71	3	∈	∈	PROPN
ejpam-4977	71	4	a	a	PRON
ejpam-4977	71	5	where	where	SCONJ
ejpam-4977	71	6	e	e	X
ejpam-4977	71	7	∈	∈	PROPN
ejpam-4977	71	8	a	a	PRON
ejpam-4977	71	9	but	but	CCONJ
ejpam-4977	71	10	c	c	NOUN
ejpam-4977	71	11	/∈	/∈	PUNCT
ejpam-4977	72	1	a.	a.	NOUN
ejpam-4977	72	2	3	3	NUM
ejpam-4977	72	3	.	.	PUNCT
ejpam-4977	73	1	some	some	DET
ejpam-4977	73	2	axioms	axiom	NOUN
ejpam-4977	73	3	of	of	ADP
ejpam-4977	73	4	exponents	exponent	NOUN
ejpam-4977	73	5	for	for	ADP
ejpam-4977	73	6	b	b	NOUN
ejpam-4977	73	7	-	-	PUNCT
ejpam-4977	73	8	algebras	algebras	NOUN
ejpam-4977	73	9	in	in	ADP
ejpam-4977	73	10	this	this	DET
ejpam-4977	73	11	section	section	NOUN
ejpam-4977	73	12	,	,	PUNCT
ejpam-4977	73	13	we	we	PRON
ejpam-4977	73	14	recall	recall	VERB
ejpam-4977	73	15	the	the	DET
ejpam-4977	73	16	axioms	axiom	NOUN
ejpam-4977	73	17	for	for	ADP
ejpam-4977	73	18	a	a	DET
ejpam-4977	73	19	b	b	NOUN
ejpam-4977	73	20	-	-	PUNCT
ejpam-4977	73	21	algebra	algebra	NOUN
ejpam-4977	73	22	(	(	PUNCT
ejpam-4977	73	23	x	x	X
ejpam-4977	73	24	,	,	PUNCT
ejpam-4977	73	25	∗	∗	NOUN
ejpam-4977	73	26	,	,	PUNCT
ejpam-4977	73	27	0	0	NUM
ejpam-4977	73	28	)	)	PUNCT
ejpam-4977	73	29	.	.	PUNCT
ejpam-4977	74	1	the	the	DET
ejpam-4977	74	2	paper	paper	NOUN
ejpam-4977	75	1	[	[	X
ejpam-4977	75	2	1	1	X
ejpam-4977	75	3	]	]	PUNCT
ejpam-4977	75	4	and	and	CCONJ
ejpam-4977	75	5	[	[	X
ejpam-4977	75	6	2	2	NUM
ejpam-4977	75	7	]	]	PUNCT
ejpam-4977	75	8	introduced	introduce	VERB
ejpam-4977	75	9	the	the	DET
ejpam-4977	75	10	notions	notion	NOUN
ejpam-4977	75	11	of	of	ADP
ejpam-4977	75	12	exponents	exponent	NOUN
ejpam-4977	75	13	of	of	ADP
ejpam-4977	75	14	b	b	NOUN
ejpam-4977	75	15	-	-	PUNCT
ejpam-4977	75	16	algebra	algebra	NOUN
ejpam-4977	75	17	and	and	CCONJ
ejpam-4977	75	18	some	some	PRON
ejpam-4977	75	19	of	of	ADP
ejpam-4977	75	20	its	its	PRON
ejpam-4977	75	21	properties	property	NOUN
ejpam-4977	75	22	.	.	PUNCT
ejpam-4977	76	1	for	for	ADP
ejpam-4977	76	2	any	any	DET
ejpam-4977	76	3	x	x	NOUN
ejpam-4977	76	4	,	,	PUNCT
ejpam-4977	76	5	y	y	PROPN
ejpam-4977	76	6	∈	∈	PROPN
ejpam-4977	76	7	x	x	X
ejpam-4977	76	8	and	and	CCONJ
ejpam-4977	76	9	n	n	PRON
ejpam-4977	76	10	∈	∈	PROPN
ejpam-4977	76	11	z+	z+	NUM
ejpam-4977	76	12	,	,	PUNCT
ejpam-4977	76	13	defined	define	VERB
ejpam-4977	76	14	the	the	DET
ejpam-4977	76	15	relation	relation	NOUN
ejpam-4977	76	16	:	:	PUNCT
ejpam-4977	76	17	xn	xn	PROPN
ejpam-4977	77	1	=	=	SYM
ejpam-4977	77	2	xn−1	xn−1	PROPN
ejpam-4977	77	3	∗	∗	NOUN
ejpam-4977	77	4	(	(	PUNCT
ejpam-4977	77	5	0	0	NUM
ejpam-4977	77	6	∗	∗	NOUN
ejpam-4977	77	7	x	x	NOUN
ejpam-4977	77	8	)	)	PUNCT
ejpam-4977	77	9	and	and	CCONJ
ejpam-4977	77	10	−x	−x	NOUN
ejpam-4977	77	11	=	=	SYM
ejpam-4977	77	12	0	0	NUM
ejpam-4977	77	13	∗	∗	NOUN
ejpam-4977	77	14	x	x	SYM
ejpam-4977	77	15	where	where	SCONJ
ejpam-4977	77	16	x0	x0	PROPN
ejpam-4977	77	17	=	=	PUNCT
ejpam-4977	77	18	0	0	PUNCT
ejpam-4977	77	19	and	and	CCONJ
ejpam-4977	77	20	x1	x1	NUM
ejpam-4977	77	21	=	=	SYM
ejpam-4977	77	22	x0	x0	PROPN
ejpam-4977	77	23	∗	∗	NOUN
ejpam-4977	77	24	(	(	PUNCT
ejpam-4977	77	25	0	0	NUM
ejpam-4977	77	26	∗	∗	NOUN
ejpam-4977	77	27	x	x	NOUN
ejpam-4977	77	28	)	)	PUNCT
ejpam-4977	77	29	=	=	SYM
ejpam-4977	77	30	0	0	NUM
ejpam-4977	77	31	∗	∗	NOUN
ejpam-4977	77	32	(	(	PUNCT
ejpam-4977	77	33	0	0	NUM
ejpam-4977	77	34	∗	∗	NOUN
ejpam-4977	77	35	x	x	NOUN
ejpam-4977	77	36	)	)	PUNCT
ejpam-4977	77	37	=	=	SYM
ejpam-4977	77	38	x	x	PUNCT
ejpam-4977	77	39	and	and	CCONJ
ejpam-4977	77	40	denote	denote	VERB
ejpam-4977	77	41	that	that	DET
ejpam-4977	77	42	expression	expression	NOUN
ejpam-4977	77	43	x	x	PUNCT
ejpam-4977	77	44	∗	∗	PROPN
ejpam-4977	77	45	n∏	n∏	PROPN
ejpam-4977	77	46	y	y	PROPN
ejpam-4977	77	47	=	=	PRON
ejpam-4977	77	48	(	(	PUNCT
ejpam-4977	77	49	...	...	PUNCT
ejpam-4977	77	50	(	(	PUNCT
ejpam-4977	77	51	(	(	PUNCT
ejpam-4977	77	52	x	x	SYM
ejpam-4977	77	53	∗	∗	PROPN
ejpam-4977	77	54	y	y	NOUN
ejpam-4977	77	55	)	)	PUNCT
ejpam-4977	77	56	∗	∗	PROPN
ejpam-4977	77	57	y	y	PROPN
ejpam-4977	77	58	)	)	PUNCT
ejpam-4977	77	59	∗	∗	NOUN
ejpam-4977	77	60	...	...	PUNCT
ejpam-4977	77	61	)	)	PUNCT
ejpam-4977	78	1	∗	∗	PROPN
ejpam-4977	78	2	y	y	PROPN
ejpam-4977	78	3	,	,	PUNCT
ejpam-4977	78	4	where	where	SCONJ
ejpam-4977	78	5	y	y	PROPN
ejpam-4977	78	6	occurs	occur	VERB
ejpam-4977	78	7	n	n	PRON
ejpam-4977	78	8	times	time	NOUN
ejpam-4977	78	9	.	.	PUNCT
ejpam-4977	79	1	by	by	ADP
ejpam-4977	79	2	convention	convention	PROPN
ejpam-4977	79	3	,	,	PUNCT
ejpam-4977	79	4	x∗	x∗	PROPN
ejpam-4977	79	5	0∏	0∏	NUM
ejpam-4977	80	1	y	y	PROPN
ejpam-4977	80	2	means	mean	VERB
ejpam-4977	80	3	x∗0	x∗0	NOUN
ejpam-4977	80	4	=	=	SYM
ejpam-4977	80	5	x	x	NOUN
ejpam-4977	80	6	,	,	PUNCT
ejpam-4977	80	7	so	so	SCONJ
ejpam-4977	80	8	that	that	SCONJ
ejpam-4977	80	9	xn	xn	PUNCT
ejpam-4977	81	1	=	=	PUNCT
ejpam-4977	81	2	x∗(0∗	x∗(0∗	PROPN
ejpam-4977	81	3	n−1∏	n−1∏	PROPN
ejpam-4977	81	4	x	x	X
ejpam-4977	81	5	)	)	PUNCT
ejpam-4977	81	6	and	and	CCONJ
ejpam-4977	81	7	x−n	x−n	PROPN
ejpam-4977	81	8	=	=	PUNCT
ejpam-4977	81	9	(	(	PUNCT
ejpam-4977	81	10	−x)n	−x)n	NOUN
ejpam-4977	81	11	=	=	SYM
ejpam-4977	82	1	−(x)n	−(x)n	PROPN
ejpam-4977	82	2	=	=	SYM
ejpam-4977	82	3	0	0	NUM
ejpam-4977	82	4	∗	∗	NOUN
ejpam-4977	82	5	xn	xn	PROPN
ejpam-4977	82	6	implies	imply	VERB
ejpam-4977	82	7	that	that	SCONJ
ejpam-4977	82	8	(	(	PUNCT
ejpam-4977	82	9	x−1)−n	x−1)−n	PROPN
ejpam-4977	82	10	=	=	SYM
ejpam-4977	83	1	(	(	PUNCT
ejpam-4977	83	2	x−n)−1	x−n)−1	PROPN
ejpam-4977	83	3	=	=	SYM
ejpam-4977	83	4	xn	xn	PROPN
ejpam-4977	83	5	.	.	PUNCT
ejpam-4977	83	6	theorem	theorem	NOUN
ejpam-4977	83	7	2	2	NUM
ejpam-4977	83	8	.	.	PUNCT
ejpam-4977	84	1	[	[	X
ejpam-4977	84	2	1	1	X
ejpam-4977	84	3	]	]	X
ejpam-4977	84	4	let	let	VERB
ejpam-4977	84	5	(	(	PUNCT
ejpam-4977	84	6	x	x	NOUN
ejpam-4977	84	7	,	,	PUNCT
ejpam-4977	84	8	∗	∗	NOUN
ejpam-4977	84	9	,	,	PUNCT
ejpam-4977	84	10	0	0	NUM
ejpam-4977	84	11	)	)	PUNCT
ejpam-4977	84	12	be	be	AUX
ejpam-4977	84	13	a	a	DET
ejpam-4977	84	14	b	b	NOUN
ejpam-4977	84	15	-	-	PUNCT
ejpam-4977	84	16	algebra	algebra	NOUN
ejpam-4977	84	17	,	,	PUNCT
ejpam-4977	84	18	g	g	PROPN
ejpam-4977	84	19	∈	∈	PROPN
ejpam-4977	84	20	x	x	X
ejpam-4977	84	21	and	and	CCONJ
ejpam-4977	84	22	m	m	PROPN
ejpam-4977	84	23	,	,	PUNCT
ejpam-4977	84	24	n	n	PROPN
ejpam-4977	84	25	∈	∈	PROPN
ejpam-4977	84	26	z+	z+	PUNCT
ejpam-4977	84	27	.	.	PUNCT
ejpam-4977	85	1	then	then	ADV
ejpam-4977	85	2	gm	gm	PROPN
ejpam-4977	85	3	∗	∗	VERB
ejpam-4977	85	4	gn	gn	PROPN
ejpam-4977	86	1	=	=	PUNCT
ejpam-4977	86	2	{	{	PUNCT
ejpam-4977	86	3	gm−n	gm−n	NOUN
ejpam-4977	86	4	if	if	SCONJ
ejpam-4977	86	5	m	m	PROPN
ejpam-4977	86	6	≥	≥	VERB
ejpam-4977	86	7	n	n	CCONJ
ejpam-4977	86	8	0	0	NUM
ejpam-4977	86	9	∗	∗	NOUN
ejpam-4977	86	10	gn−m	gn−m	NOUN
ejpam-4977	86	11	if	if	SCONJ
ejpam-4977	86	12	m	m	VERB
ejpam-4977	86	13	<	<	X
ejpam-4977	86	14	n	n	PRON
ejpam-4977	86	15	corollary	corollary	ADJ
ejpam-4977	86	16	1	1	NUM
ejpam-4977	86	17	.	.	PUNCT
ejpam-4977	87	1	[	[	X
ejpam-4977	87	2	1	1	X
ejpam-4977	87	3	]	]	X
ejpam-4977	87	4	let	let	VERB
ejpam-4977	87	5	(	(	PUNCT
ejpam-4977	87	6	x	x	NOUN
ejpam-4977	87	7	,	,	PUNCT
ejpam-4977	87	8	∗	∗	NOUN
ejpam-4977	87	9	,	,	PUNCT
ejpam-4977	87	10	0	0	NUM
ejpam-4977	87	11	)	)	PUNCT
ejpam-4977	87	12	be	be	AUX
ejpam-4977	87	13	a	a	DET
ejpam-4977	87	14	b	b	NOUN
ejpam-4977	87	15	-	-	PUNCT
ejpam-4977	87	16	algebra	algebra	NOUN
ejpam-4977	87	17	.	.	PUNCT
ejpam-4977	88	1	then	then	ADV
ejpam-4977	88	2	the	the	DET
ejpam-4977	88	3	following	follow	VERB
ejpam-4977	88	4	equalities	equality	NOUN
ejpam-4977	88	5	hold	hold	VERB
ejpam-4977	88	6	for	for	ADP
ejpam-4977	88	7	all	all	DET
ejpam-4977	88	8	g	g	NOUN
ejpam-4977	88	9	∈	∈	PROPN
ejpam-4977	88	10	x	x	X
ejpam-4977	88	11	and	and	CCONJ
ejpam-4977	88	12	m	m	PROPN
ejpam-4977	88	13	,	,	PUNCT
ejpam-4977	88	14	n	n	PROPN
ejpam-4977	88	15	∈	∈	NOUN
ejpam-4977	88	16	z+	z+	NUM
ejpam-4977	88	17	:	:	PUNCT
ejpam-4977	88	18	(	(	PUNCT
ejpam-4977	88	19	i	i	NOUN
ejpam-4977	88	20	)	)	PUNCT
ejpam-4977	88	21	gm	gm	PROPN
ejpam-4977	88	22	∗	∗	NOUN
ejpam-4977	88	23	gn	gn	PROPN
ejpam-4977	89	1	=	=	PROPN
ejpam-4977	89	2	gm−n	gm−n	PROPN
ejpam-4977	89	3	,	,	PUNCT
ejpam-4977	89	4	(	(	PUNCT
ejpam-4977	89	5	ii	ii	NOUN
ejpam-4977	89	6	)	)	PUNCT
ejpam-4977	89	7	g	g	PROPN
ejpam-4977	89	8	∗	∗	NOUN
ejpam-4977	89	9	g−n	g−n	NOUN
ejpam-4977	89	10	=	=	SYM
ejpam-4977	89	11	gn+1	gn+1	ADJ
ejpam-4977	89	12	and	and	CCONJ
ejpam-4977	89	13	g−n	g−n	VERB
ejpam-4977	89	14	∗	∗	NOUN
ejpam-4977	89	15	g	g	NOUN
ejpam-4977	89	16	=	=	SYM
ejpam-4977	89	17	g−(n+1	g−(n+1	PROPN
ejpam-4977	89	18	)	)	PUNCT
ejpam-4977	89	19	,	,	PUNCT
ejpam-4977	89	20	m.	m.	NOUN
ejpam-4977	89	21	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	89	22	/	/	SYM
ejpam-4977	89	23	eur	eur	PROPN
ejpam-4977	89	24	.	.	PUNCT
ejpam-4977	90	1	j.	j.	PROPN
ejpam-4977	90	2	pure	pure	PROPN
ejpam-4977	90	3	appl	appl	PROPN
ejpam-4977	90	4	.	.	PROPN
ejpam-4977	90	5	math	math	PROPN
ejpam-4977	90	6	,	,	PUNCT
ejpam-4977	90	7	17	17	NUM
ejpam-4977	90	8	(	(	PUNCT
ejpam-4977	90	9	1	1	NUM
ejpam-4977	90	10	)	)	PUNCT
ejpam-4977	90	11	(	(	PUNCT
ejpam-4977	90	12	2024	2024	NUM
ejpam-4977	90	13	)	)	PUNCT
ejpam-4977	90	14	,	,	PUNCT
ejpam-4977	90	15	116	116	NUM
ejpam-4977	90	16	-	-	SYM
ejpam-4977	90	17	123	123	NUM
ejpam-4977	90	18	120	120	NUM
ejpam-4977	90	19	(	(	PUNCT
ejpam-4977	90	20	iii	iii	NOUN
ejpam-4977	90	21	)	)	PUNCT
ejpam-4977	90	22	−g	−g	NOUN
ejpam-4977	90	23	∗	∗	NOUN
ejpam-4977	90	24	gn	gn	PROPN
ejpam-4977	90	25	=	=	SYM
ejpam-4977	90	26	g−(n+1	g−(n+1	PROPN
ejpam-4977	90	27	)	)	PUNCT
ejpam-4977	90	28	,	,	PUNCT
ejpam-4977	90	29	(	(	PUNCT
ejpam-4977	90	30	iv	iv	X
ejpam-4977	90	31	)	)	PUNCT
ejpam-4977	90	32	gm	gm	PROPN
ejpam-4977	90	33	∗	∗	NOUN
ejpam-4977	90	34	(	(	PUNCT
ejpam-4977	90	35	−g	−g	NOUN
ejpam-4977	90	36	)	)	PUNCT
ejpam-4977	90	37	=	=	SYM
ejpam-4977	90	38	g(n+1	g(n+1	NOUN
ejpam-4977	90	39	)	)	PUNCT
ejpam-4977	90	40	,	,	PUNCT
ejpam-4977	90	41	(	(	PUNCT
ejpam-4977	90	42	v	v	NOUN
ejpam-4977	90	43	)	)	PUNCT
ejpam-4977	90	44	gm	gm	PROPN
ejpam-4977	90	45	∗	∗	NOUN
ejpam-4977	90	46	g−n	g−n	NOUN
ejpam-4977	90	47	=	=	SYM
ejpam-4977	90	48	g(m+n	g(m+n	NOUN
ejpam-4977	90	49	)	)	PUNCT
ejpam-4977	90	50	and	and	CCONJ
ejpam-4977	90	51	g−m	g−m	NOUN
ejpam-4977	90	52	∗	∗	NOUN
ejpam-4977	90	53	gn	gn	PROPN
ejpam-4977	91	1	=	=	SYM
ejpam-4977	91	2	g−(m+n	g−(m+n	PROPN
ejpam-4977	91	3	)	)	PUNCT
ejpam-4977	91	4	.	.	PUNCT
ejpam-4977	92	1	corollary	corollary	ADJ
ejpam-4977	92	2	2	2	NUM
ejpam-4977	92	3	.	.	PUNCT
ejpam-4977	93	1	[	[	X
ejpam-4977	93	2	2	2	NUM
ejpam-4977	93	3	]	]	X
ejpam-4977	93	4	let	let	VERB
ejpam-4977	93	5	(	(	PUNCT
ejpam-4977	93	6	x	x	NOUN
ejpam-4977	93	7	,	,	PUNCT
ejpam-4977	93	8	∗	∗	NOUN
ejpam-4977	93	9	,	,	PUNCT
ejpam-4977	93	10	0	0	NUM
ejpam-4977	93	11	)	)	PUNCT
ejpam-4977	93	12	be	be	AUX
ejpam-4977	93	13	a	a	DET
ejpam-4977	93	14	b	b	NOUN
ejpam-4977	93	15	-	-	PUNCT
ejpam-4977	93	16	algebra	algebra	NOUN
ejpam-4977	93	17	,	,	PUNCT
ejpam-4977	93	18	g	g	PROPN
ejpam-4977	93	19	∈	∈	PROPN
ejpam-4977	93	20	x	x	X
ejpam-4977	93	21	and	and	CCONJ
ejpam-4977	93	22	m	m	PROPN
ejpam-4977	93	23	,	,	PUNCT
ejpam-4977	93	24	n	n	PROPN
ejpam-4977	93	25	∈	∈	PROPN
ejpam-4977	93	26	z.	z.	NOUN
ejpam-4977	93	27	then	then	ADV
ejpam-4977	93	28	gm∗gn	gm∗gn	PROPN
ejpam-4977	93	29	=	=	SYM
ejpam-4977	93	30	gm−n	gm−n	NOUN
ejpam-4977	93	31	.	.	PUNCT
ejpam-4977	94	1	4	4	X
ejpam-4977	94	2	.	.	X
ejpam-4977	94	3	on	on	ADP
ejpam-4977	94	4	exponents	exponent	NOUN
ejpam-4977	94	5	and	and	CCONJ
ejpam-4977	94	6	cyclic	cyclic	ADJ
ejpam-4977	94	7	b	b	X
ejpam-4977	94	8	-	-	PUNCT
ejpam-4977	94	9	algebras	algebras	ADV
ejpam-4977	94	10	we	we	PRON
ejpam-4977	94	11	shall	shall	AUX
ejpam-4977	94	12	give	give	VERB
ejpam-4977	94	13	some	some	DET
ejpam-4977	94	14	elementary	elementary	ADJ
ejpam-4977	94	15	properties	property	NOUN
ejpam-4977	94	16	of	of	ADP
ejpam-4977	94	17	cyclic	cyclic	ADJ
ejpam-4977	94	18	b	b	NOUN
ejpam-4977	94	19	-	-	PUNCT
ejpam-4977	94	20	algebras	algebras	PROPN
ejpam-4977	94	21	.	.	PUNCT
ejpam-4977	95	1	recall	recall	VERB
ejpam-4977	95	2	that	that	PRON
ejpam-4977	95	3	for	for	ADP
ejpam-4977	95	4	a	a	DET
ejpam-4977	95	5	b	b	NOUN
ejpam-4977	95	6	-	-	PUNCT
ejpam-4977	95	7	algebra	algebra	NOUN
ejpam-4977	95	8	(	(	PUNCT
ejpam-4977	95	9	x	x	X
ejpam-4977	95	10	,	,	PUNCT
ejpam-4977	95	11	∗	∗	NOUN
ejpam-4977	95	12	,	,	PUNCT
ejpam-4977	95	13	0	0	NUM
ejpam-4977	95	14	)	)	PUNCT
ejpam-4977	95	15	(	(	PUNCT
ejpam-4977	95	16	see	see	VERB
ejpam-4977	95	17	[	[	X
ejpam-4977	95	18	2	2	NUM
ejpam-4977	95	19	]	]	PUNCT
ejpam-4977	95	20	)	)	PUNCT
ejpam-4977	95	21	if	if	SCONJ
ejpam-4977	95	22	there	there	PRON
ejpam-4977	95	23	is	be	VERB
ejpam-4977	95	24	an	an	DET
ejpam-4977	95	25	a	a	DET
ejpam-4977	95	26	∈	∈	NOUN
ejpam-4977	95	27	x	x	PUNCT
ejpam-4977	95	28	such	such	ADJ
ejpam-4977	95	29	that	that	SCONJ
ejpam-4977	95	30	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	95	31	=	=	SYM
ejpam-4977	95	32	{	{	PUNCT
ejpam-4977	95	33	ak	ak	PROPN
ejpam-4977	95	34	:	:	PUNCT
ejpam-4977	95	35	k	k	PROPN
ejpam-4977	95	36	∈	∈	PROPN
ejpam-4977	96	1	z	z	X
ejpam-4977	96	2	}	}	PUNCT
ejpam-4977	96	3	=	=	SYM
ejpam-4977	96	4	x	x	NOUN
ejpam-4977	96	5	,	,	PUNCT
ejpam-4977	96	6	then	then	ADV
ejpam-4977	96	7	x	x	PUNCT
ejpam-4977	96	8	is	be	AUX
ejpam-4977	96	9	called	call	VERB
ejpam-4977	96	10	a	a	DET
ejpam-4977	96	11	cyclic	cyclic	ADJ
ejpam-4977	96	12	b	b	X
ejpam-4977	96	13	-	-	PUNCT
ejpam-4977	96	14	algebra	algebra	NOUN
ejpam-4977	96	15	generated	generate	VERB
ejpam-4977	96	16	by	by	ADP
ejpam-4977	96	17	a.	a.	NOUN
ejpam-4977	96	18	also	also	ADV
ejpam-4977	96	19	,	,	PUNCT
ejpam-4977	96	20	the	the	DET
ejpam-4977	96	21	authors	author	NOUN
ejpam-4977	96	22	of	of	ADP
ejpam-4977	96	23	[	[	X
ejpam-4977	96	24	2	2	NUM
ejpam-4977	96	25	]	]	PUNCT
ejpam-4977	96	26	have	have	AUX
ejpam-4977	96	27	proved	prove	VERB
ejpam-4977	96	28	that	that	SCONJ
ejpam-4977	96	29	every	every	DET
ejpam-4977	96	30	cyclic	cyclic	ADJ
ejpam-4977	96	31	b	b	X
ejpam-4977	96	32	-	-	PUNCT
ejpam-4977	96	33	algebra	algebra	NOUN
ejpam-4977	96	34	is	be	AUX
ejpam-4977	96	35	commutative	commutative	ADJ
ejpam-4977	96	36	.	.	PUNCT
ejpam-4977	97	1	theorem	theorem	NOUN
ejpam-4977	97	2	3	3	X
ejpam-4977	97	3	.	.	PUNCT
ejpam-4977	98	1	let	let	AUX
ejpam-4977	98	2	(	(	PUNCT
ejpam-4977	98	3	x	x	X
ejpam-4977	98	4	,	,	PUNCT
ejpam-4977	98	5	∗	∗	NOUN
ejpam-4977	98	6	,	,	PUNCT
ejpam-4977	98	7	0	0	NUM
ejpam-4977	98	8	)	)	PUNCT
ejpam-4977	98	9	be	be	AUX
ejpam-4977	98	10	a	a	DET
ejpam-4977	98	11	b	b	NOUN
ejpam-4977	98	12	-	-	PUNCT
ejpam-4977	98	13	algebra	algebra	NOUN
ejpam-4977	98	14	and	and	CCONJ
ejpam-4977	98	15	x	x	NOUN
ejpam-4977	98	16	,	,	PUNCT
ejpam-4977	98	17	y	y	PROPN
ejpam-4977	98	18	∈	∈	PROPN
ejpam-4977	98	19	x	x	PUNCT
ejpam-4977	98	20	with	with	ADP
ejpam-4977	98	21	n	n	DET
ejpam-4977	98	22	∈	∈	PROPN
ejpam-4977	98	23	z+	z+	X
ejpam-4977	98	24	,	,	PUNCT
ejpam-4977	98	25	then	then	ADV
ejpam-4977	98	26	0	0	NUM
ejpam-4977	98	27	∗	∗	NOUN
ejpam-4977	98	28	(	(	PUNCT
ejpam-4977	98	29	x	x	X
ejpam-4977	98	30	∗	∗	NOUN
ejpam-4977	98	31	y)n	y)n	NUM
ejpam-4977	99	1	=	=	SYM
ejpam-4977	99	2	(	(	PUNCT
ejpam-4977	99	3	y	y	NOUN
ejpam-4977	99	4	∗	∗	NOUN
ejpam-4977	99	5	x)n	x)n	PUNCT
ejpam-4977	99	6	proof	proof	NOUN
ejpam-4977	99	7	.	.	PUNCT
ejpam-4977	100	1	cleary	cleary	PROPN
ejpam-4977	100	2	0	0	NUM
ejpam-4977	100	3	∗	∗	NOUN
ejpam-4977	100	4	(	(	PUNCT
ejpam-4977	100	5	x	x	X
ejpam-4977	100	6	∗	∗	PROPN
ejpam-4977	100	7	y	y	NOUN
ejpam-4977	100	8	)	)	PUNCT
ejpam-4977	100	9	=	=	SYM
ejpam-4977	100	10	(	(	PUNCT
ejpam-4977	100	11	0	0	NUM
ejpam-4977	100	12	∗	∗	NOUN
ejpam-4977	100	13	(	(	PUNCT
ejpam-4977	100	14	0	0	NUM
ejpam-4977	100	15	∗	∗	PROPN
ejpam-4977	100	16	y	y	NOUN
ejpam-4977	100	17	)	)	PUNCT
ejpam-4977	100	18	∗	∗	NOUN
ejpam-4977	100	19	x	x	NOUN
ejpam-4977	100	20	)	)	PUNCT
ejpam-4977	100	21	=	=	SYM
ejpam-4977	100	22	y	y	PROPN
ejpam-4977	100	23	∗	∗	NOUN
ejpam-4977	100	24	x.	x.	PUNCT
ejpam-4977	101	1	thus	thus	ADV
ejpam-4977	101	2	,	,	PUNCT
ejpam-4977	101	3	the	the	DET
ejpam-4977	101	4	equality	equality	NOUN
ejpam-4977	101	5	holds	hold	VERB
ejpam-4977	101	6	for	for	ADP
ejpam-4977	101	7	n	n	NOUN
ejpam-4977	101	8	=	=	SYM
ejpam-4977	101	9	1	1	NUM
ejpam-4977	101	10	next	next	ADV
ejpam-4977	101	11	,	,	PUNCT
ejpam-4977	101	12	suppose	suppose	VERB
ejpam-4977	101	13	that	that	SCONJ
ejpam-4977	101	14	0	0	NUM
ejpam-4977	101	15	∗	∗	NOUN
ejpam-4977	101	16	(	(	PUNCT
ejpam-4977	101	17	x	x	X
ejpam-4977	101	18	∗	∗	NOUN
ejpam-4977	101	19	y)n	y)n	NUM
ejpam-4977	101	20	=	=	SYM
ejpam-4977	101	21	(	(	PUNCT
ejpam-4977	101	22	y	y	NOUN
ejpam-4977	101	23	∗	∗	X
ejpam-4977	101	24	x)n	x)n	PUNCT
ejpam-4977	101	25	for	for	ADP
ejpam-4977	101	26	any	any	DET
ejpam-4977	101	27	n	n	NOUN
ejpam-4977	101	28	>	>	X
ejpam-4977	101	29	1	1	NUM
ejpam-4977	101	30	.	.	PUNCT
ejpam-4977	102	1	so	so	ADV
ejpam-4977	102	2	0	0	NUM
ejpam-4977	102	3	∗	∗	NOUN
ejpam-4977	102	4	(	(	PUNCT
ejpam-4977	102	5	x	x	SYM
ejpam-4977	102	6	∗	∗	X
ejpam-4977	102	7	y)n+1	y)n+1	NOUN
ejpam-4977	102	8	=	=	SYM
ejpam-4977	102	9	0	0	NUM
ejpam-4977	102	10	∗	∗	NOUN
ejpam-4977	102	11	{	{	PUNCT
ejpam-4977	102	12	(	(	PUNCT
ejpam-4977	102	13	x	x	NOUN
ejpam-4977	102	14	∗	∗	PROPN
ejpam-4977	102	15	y	y	NOUN
ejpam-4977	102	16	)	)	PUNCT
ejpam-4977	102	17	∗	∗	NOUN
ejpam-4977	102	18	(	(	PUNCT
ejpam-4977	102	19	0	0	NUM
ejpam-4977	102	20	∗	∗	NUM
ejpam-4977	102	21	n∏	n∏	PROPN
ejpam-4977	102	22	(	(	PUNCT
ejpam-4977	102	23	x	x	PROPN
ejpam-4977	102	24	∗	∗	PROPN
ejpam-4977	102	25	y	y	PROPN
ejpam-4977	102	26	)	)	PUNCT
ejpam-4977	102	27	)	)	PUNCT
ejpam-4977	102	28	}	}	PUNCT
ejpam-4977	103	1	=	=	SYM
ejpam-4977	103	2	0	0	NUM
ejpam-4977	103	3	∗	∗	NOUN
ejpam-4977	103	4	{	{	PUNCT
ejpam-4977	103	5	(	(	PUNCT
ejpam-4977	103	6	x	x	NOUN
ejpam-4977	103	7	∗	∗	PROPN
ejpam-4977	103	8	y	y	NOUN
ejpam-4977	103	9	)	)	PUNCT
ejpam-4977	103	10	∗	∗	NOUN
ejpam-4977	104	1	[	[	X
ejpam-4977	104	2	[	[	X
ejpam-4977	104	3	...	...	PUNCT
ejpam-4977	104	4	[	[	X
ejpam-4977	104	5	[	[	X
ejpam-4977	104	6	0	0	NUM
ejpam-4977	104	7	∗	∗	NOUN
ejpam-4977	104	8	(	(	PUNCT
ejpam-4977	104	9	x	x	X
ejpam-4977	104	10	∗	∗	PROPN
ejpam-4977	104	11	y	y	PROPN
ejpam-4977	104	12	)	)	PUNCT
ejpam-4977	104	13	]	]	PUNCT
ejpam-4977	105	1	∗	∗	NOUN
ejpam-4977	105	2	(	(	PUNCT
ejpam-4977	105	3	x	x	X
ejpam-4977	105	4	∗	∗	PROPN
ejpam-4977	105	5	y	y	PROPN
ejpam-4977	105	6	)	)	PUNCT
ejpam-4977	105	7	]	]	PUNCT
ejpam-4977	105	8	∗	∗	NOUN
ejpam-4977	105	9	...	...	PUNCT
ejpam-4977	105	10	]	]	PUNCT
ejpam-4977	105	11	∗	∗	NOUN
ejpam-4977	105	12	(	(	PUNCT
ejpam-4977	105	13	x	x	SYM
ejpam-4977	105	14	∗	∗	NOUN
ejpam-4977	105	15	y)]︸	y)]︸	PROPN
ejpam-4977	105	16	︷︷	︷︷	PROPN
ejpam-4977	105	17	︸	︸	X
ejpam-4977	105	18	(	(	PUNCT
ejpam-4977	105	19	x∗y	x∗y	X
ejpam-4977	105	20	)	)	PUNCT
ejpam-4977	105	21	occurs	occur	VERB
ejpam-4977	105	22	n	n	PROPN
ejpam-4977	105	23	times	time	NOUN
ejpam-4977	105	24	}	}	PUNCT
ejpam-4977	105	25	=	=	SYM
ejpam-4977	105	26	0	0	NUM
ejpam-4977	105	27	∗	∗	NOUN
ejpam-4977	105	28	{	{	PUNCT
ejpam-4977	106	1	[	[	X
ejpam-4977	106	2	(	(	PUNCT
ejpam-4977	106	3	x	x	X
ejpam-4977	106	4	∗	∗	PROPN
ejpam-4977	106	5	y	y	NOUN
ejpam-4977	106	6	)	)	PUNCT
ejpam-4977	106	7	∗	∗	NOUN
ejpam-4977	107	1	[	[	X
ejpam-4977	107	2	0	0	NUM
ejpam-4977	107	3	∗	∗	NOUN
ejpam-4977	107	4	(	(	PUNCT
ejpam-4977	107	5	x	x	X
ejpam-4977	107	6	∗	∗	PROPN
ejpam-4977	107	7	y	y	PROPN
ejpam-4977	107	8	)	)	PUNCT
ejpam-4977	107	9	]	]	PUNCT
ejpam-4977	107	10	]	]	X
ejpam-4977	107	11	∗	∗	NOUN
ejpam-4977	108	1	[	[	X
ejpam-4977	108	2	[	[	X
ejpam-4977	108	3	...	...	PUNCT
ejpam-4977	108	4	[	[	X
ejpam-4977	108	5	[	[	X
ejpam-4977	108	6	0	0	NUM
ejpam-4977	108	7	∗	∗	NOUN
ejpam-4977	108	8	{	{	PUNCT
ejpam-4977	108	9	(	(	PUNCT
ejpam-4977	108	10	x	x	NOUN
ejpam-4977	108	11	∗	∗	NOUN
ejpam-4977	108	12	y	y	PROPN
ejpam-4977	108	13	)	)	PUNCT
ejpam-4977	108	14	]	]	PUNCT
ejpam-4977	109	1	∗	∗	NOUN
ejpam-4977	109	2	(	(	PUNCT
ejpam-4977	109	3	x	x	X
ejpam-4977	109	4	∗	∗	PROPN
ejpam-4977	109	5	y	y	PROPN
ejpam-4977	109	6	)	)	PUNCT
ejpam-4977	109	7	]	]	PUNCT
ejpam-4977	109	8	∗	∗	NOUN
ejpam-4977	109	9	...	...	PUNCT
ejpam-4977	109	10	]	]	PUNCT
ejpam-4977	109	11	∗	∗	NOUN
ejpam-4977	109	12	(	(	PUNCT
ejpam-4977	109	13	x	x	SYM
ejpam-4977	109	14	∗	∗	NOUN
ejpam-4977	109	15	y)]︸	y)]︸	PROPN
ejpam-4977	109	16	︷︷	︷︷	PROPN
ejpam-4977	109	17	︸	︸	X
ejpam-4977	109	18	(	(	PUNCT
ejpam-4977	109	19	x∗y	x∗y	X
ejpam-4977	109	20	)	)	PUNCT
ejpam-4977	109	21	occurs	occur	VERB
ejpam-4977	109	22	n−1	n−1	PROPN
ejpam-4977	109	23	times	time	NOUN
ejpam-4977	109	24	}	}	PUNCT
ejpam-4977	109	25	=	=	SYM
ejpam-4977	109	26	0	0	NUM
ejpam-4977	109	27	∗	∗	NOUN
ejpam-4977	109	28	{	{	PUNCT
ejpam-4977	110	1	[	[	X
ejpam-4977	110	2	(	(	PUNCT
ejpam-4977	110	3	x	x	X
ejpam-4977	110	4	∗	∗	PROPN
ejpam-4977	110	5	y	y	NOUN
ejpam-4977	110	6	)	)	PUNCT
ejpam-4977	110	7	∗	∗	NOUN
ejpam-4977	110	8	(	(	PUNCT
ejpam-4977	110	9	y	y	PROPN
ejpam-4977	110	10	∗	∗	X
ejpam-4977	110	11	x	x	NOUN
ejpam-4977	110	12	)	)	PUNCT
ejpam-4977	110	13	∗	∗	NOUN
ejpam-4977	111	1	[	[	X
ejpam-4977	111	2	[	[	X
ejpam-4977	111	3	...	...	PUNCT
ejpam-4977	111	4	[	[	X
ejpam-4977	111	5	[	[	X
ejpam-4977	111	6	0	0	NUM
ejpam-4977	111	7	∗	∗	NOUN
ejpam-4977	111	8	(	(	PUNCT
ejpam-4977	111	9	x	x	X
ejpam-4977	111	10	∗	∗	PROPN
ejpam-4977	111	11	y	y	PROPN
ejpam-4977	111	12	)	)	PUNCT
ejpam-4977	111	13	]	]	PUNCT
ejpam-4977	112	1	∗	∗	NOUN
ejpam-4977	112	2	(	(	PUNCT
ejpam-4977	112	3	x	x	X
ejpam-4977	112	4	∗	∗	PROPN
ejpam-4977	112	5	y	y	PROPN
ejpam-4977	112	6	)	)	PUNCT
ejpam-4977	112	7	]	]	PUNCT
ejpam-4977	112	8	∗	∗	NOUN
ejpam-4977	112	9	...	...	PUNCT
ejpam-4977	112	10	]	]	PUNCT
ejpam-4977	112	11	∗	∗	NOUN
ejpam-4977	112	12	(	(	PUNCT
ejpam-4977	112	13	x	x	SYM
ejpam-4977	112	14	∗	∗	NOUN
ejpam-4977	112	15	y)]︸	y)]︸	PROPN
ejpam-4977	112	16	︷︷	︷︷	PROPN
ejpam-4977	112	17	︸	︸	X
ejpam-4977	112	18	(	(	PUNCT
ejpam-4977	112	19	x∗y	x∗y	X
ejpam-4977	112	20	)	)	PUNCT
ejpam-4977	112	21	occurs	occur	VERB
ejpam-4977	112	22	n−1	n−1	PROPN
ejpam-4977	112	23	times	time	NOUN
ejpam-4977	112	24	}	}	PUNCT
ejpam-4977	112	25	=	=	SYM
ejpam-4977	112	26	0	0	NUM
ejpam-4977	112	27	∗	∗	NOUN
ejpam-4977	112	28	{	{	PUNCT
ejpam-4977	113	1	[	[	X
ejpam-4977	113	2	[	[	X
ejpam-4977	113	3	...	...	PUNCT
ejpam-4977	113	4	[	[	X
ejpam-4977	113	5	(	(	PUNCT
ejpam-4977	113	6	x	x	SYM
ejpam-4977	113	7	∗	∗	PROPN
ejpam-4977	113	8	y	y	NOUN
ejpam-4977	113	9	)	)	PUNCT
ejpam-4977	113	10	∗	∗	NOUN
ejpam-4977	113	11	(	(	PUNCT
ejpam-4977	113	12	y	y	PROPN
ejpam-4977	113	13	∗	∗	NOUN
ejpam-4977	113	14	x	x	NOUN
ejpam-4977	113	15	)	)	PUNCT
ejpam-4977	113	16	]	]	PUNCT
ejpam-4977	113	17	∗	∗	NOUN
ejpam-4977	113	18	...	...	PUNCT
ejpam-4977	113	19	]	]	PUNCT
ejpam-4977	114	1	∗	∗	NOUN
ejpam-4977	114	2	(	(	PUNCT
ejpam-4977	114	3	y	y	PROPN
ejpam-4977	114	4	∗	∗	NOUN
ejpam-4977	114	5	x	x	NOUN
ejpam-4977	114	6	)	)	PUNCT
ejpam-4977	114	7	]	]	PUNCT
ejpam-4977	114	8	∗	∗	NOUN
ejpam-4977	114	9	(	(	PUNCT
ejpam-4977	114	10	y	y	PROPN
ejpam-4977	114	11	∗	∗	PROPN
ejpam-4977	114	12	x)︸	x)︸	PROPN
ejpam-4977	114	13	︷︷	︷︷	PROPN
ejpam-4977	114	14	︸	︸	X
ejpam-4977	114	15	(	(	PUNCT
ejpam-4977	114	16	y∗x	y∗x	NUM
ejpam-4977	114	17	)	)	PUNCT
ejpam-4977	114	18	occurs	occur	VERB
ejpam-4977	114	19	n−1	n−1	PROPN
ejpam-4977	114	20	times	time	NOUN
ejpam-4977	114	21	]	]	PUNCT
ejpam-4977	114	22	∗	∗	NOUN
ejpam-4977	115	1	[	[	X
ejpam-4977	115	2	0	0	NUM
ejpam-4977	115	3	∗	∗	NOUN
ejpam-4977	115	4	(	(	PUNCT
ejpam-4977	115	5	x	x	X
ejpam-4977	115	6	∗	∗	PROPN
ejpam-4977	115	7	y	y	PROPN
ejpam-4977	115	8	)	)	PUNCT
ejpam-4977	115	9	]	]	PUNCT
ejpam-4977	115	10	}	}	PUNCT
ejpam-4977	115	11	=	=	SYM
ejpam-4977	115	12	0	0	NUM
ejpam-4977	115	13	∗	∗	NOUN
ejpam-4977	115	14	{	{	PUNCT
ejpam-4977	115	15	[	[	X
ejpam-4977	115	16	[	[	X
ejpam-4977	115	17	[	[	X
ejpam-4977	115	18	...	...	PUNCT
ejpam-4977	116	1	[	[	X
ejpam-4977	116	2	[	[	X
ejpam-4977	116	3	0	0	NUM
ejpam-4977	116	4	∗	∗	NOUN
ejpam-4977	116	5	(	(	PUNCT
ejpam-4977	116	6	y	y	PROPN
ejpam-4977	116	7	∗	∗	NOUN
ejpam-4977	116	8	x	x	NOUN
ejpam-4977	116	9	)	)	PUNCT
ejpam-4977	116	10	]	]	PUNCT
ejpam-4977	116	11	∗	∗	NOUN
ejpam-4977	116	12	...	...	PUNCT
ejpam-4977	116	13	]	]	PUNCT
ejpam-4977	117	1	∗	∗	NOUN
ejpam-4977	117	2	(	(	PUNCT
ejpam-4977	117	3	y	y	PROPN
ejpam-4977	117	4	∗	∗	NOUN
ejpam-4977	117	5	x	x	NOUN
ejpam-4977	117	6	)	)	PUNCT
ejpam-4977	117	7	]	]	PUNCT
ejpam-4977	117	8	∗	∗	NOUN
ejpam-4977	117	9	(	(	PUNCT
ejpam-4977	117	10	y	y	PROPN
ejpam-4977	117	11	∗	∗	PROPN
ejpam-4977	117	12	x)︸	x)︸	PROPN
ejpam-4977	117	13	︷︷	︷︷	PROPN
ejpam-4977	117	14	︸	︸	X
ejpam-4977	117	15	(	(	PUNCT
ejpam-4977	117	16	y∗x	y∗x	NUM
ejpam-4977	117	17	)	)	PUNCT
ejpam-4977	117	18	occurs	occur	VERB
ejpam-4977	117	19	n−1	n−1	PROPN
ejpam-4977	117	20	times	time	NOUN
ejpam-4977	117	21	∗[0	∗[0	NUM
ejpam-4977	117	22	∗	∗	NOUN
ejpam-4977	117	23	(	(	PUNCT
ejpam-4977	117	24	x	x	X
ejpam-4977	117	25	∗	∗	PROPN
ejpam-4977	117	26	y	y	PROPN
ejpam-4977	117	27	)	)	PUNCT
ejpam-4977	117	28	]	]	PUNCT
ejpam-4977	117	29	∗	∗	NOUN
ejpam-4977	117	30	0	0	NUM
ejpam-4977	117	31	]	]	PUNCT
ejpam-4977	117	32	}	}	PUNCT
ejpam-4977	117	33	=	=	SYM
ejpam-4977	117	34	(	(	PUNCT
ejpam-4977	117	35	y	y	PROPN
ejpam-4977	117	36	∗	∗	X
ejpam-4977	117	37	x	x	NOUN
ejpam-4977	117	38	)	)	PUNCT
ejpam-4977	117	39	∗	∗	NOUN
ejpam-4977	118	1	[	[	X
ejpam-4977	118	2	[	[	X
ejpam-4977	118	3	...	...	PUNCT
ejpam-4977	118	4	[	[	X
ejpam-4977	118	5	[	[	X
ejpam-4977	118	6	0	0	NUM
ejpam-4977	118	7	∗	∗	NOUN
ejpam-4977	118	8	(	(	PUNCT
ejpam-4977	118	9	y	y	PROPN
ejpam-4977	118	10	∗	∗	NOUN
ejpam-4977	118	11	x	x	NOUN
ejpam-4977	118	12	)	)	PUNCT
ejpam-4977	118	13	]	]	PUNCT
ejpam-4977	119	1	∗	∗	NOUN
ejpam-4977	119	2	(	(	PUNCT
ejpam-4977	119	3	y	y	PROPN
ejpam-4977	119	4	∗	∗	NOUN
ejpam-4977	119	5	x	x	NOUN
ejpam-4977	119	6	)	)	PUNCT
ejpam-4977	119	7	]	]	PUNCT
ejpam-4977	119	8	]	]	X
ejpam-4977	119	9	∗	∗	NOUN
ejpam-4977	119	10	...	...	PUNCT
ejpam-4977	119	11	]	]	PUNCT
ejpam-4977	120	1	∗	∗	NOUN
ejpam-4977	120	2	(	(	PUNCT
ejpam-4977	120	3	y	y	PROPN
ejpam-4977	120	4	∗	∗	NOUN
ejpam-4977	120	5	x	x	NOUN
ejpam-4977	120	6	)	)	PUNCT
ejpam-4977	120	7	]	]	PUNCT
ejpam-4977	120	8	∗	∗	NOUN
ejpam-4977	120	9	(	(	PUNCT
ejpam-4977	120	10	y	y	PROPN
ejpam-4977	120	11	∗	∗	PROPN
ejpam-4977	120	12	x)︸	x)︸	PROPN
ejpam-4977	120	13	︷︷	︷︷	PROPN
ejpam-4977	120	14	︸	︸	X
ejpam-4977	120	15	(	(	PUNCT
ejpam-4977	120	16	y∗x	y∗x	NUM
ejpam-4977	120	17	)	)	PUNCT
ejpam-4977	120	18	occurs	occur	VERB
ejpam-4977	120	19	n	n	PRON
ejpam-4977	120	20	times	time	NOUN
ejpam-4977	120	21	=	=	SYM
ejpam-4977	120	22	(	(	PUNCT
ejpam-4977	120	23	y	y	PROPN
ejpam-4977	120	24	∗	∗	X
ejpam-4977	120	25	x	x	NOUN
ejpam-4977	120	26	)	)	PUNCT
ejpam-4977	120	27	∗	∗	NOUN
ejpam-4977	120	28	(	(	PUNCT
ejpam-4977	120	29	0	0	NUM
ejpam-4977	120	30	∗	∗	NUM
ejpam-4977	120	31	n∏	n∏	PROPN
ejpam-4977	120	32	(	(	PUNCT
ejpam-4977	120	33	y	y	PROPN
ejpam-4977	120	34	∗	∗	NOUN
ejpam-4977	120	35	x	x	NOUN
ejpam-4977	120	36	)	)	PUNCT
ejpam-4977	120	37	)	)	PUNCT
ejpam-4977	120	38	m.	m.	NOUN
ejpam-4977	120	39	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	120	40	/	/	SYM
ejpam-4977	120	41	eur	eur	PROPN
ejpam-4977	120	42	.	.	PUNCT
ejpam-4977	121	1	j.	j.	PROPN
ejpam-4977	121	2	pure	pure	PROPN
ejpam-4977	121	3	appl	appl	PROPN
ejpam-4977	121	4	.	.	PROPN
ejpam-4977	121	5	math	math	PROPN
ejpam-4977	121	6	,	,	PUNCT
ejpam-4977	121	7	17	17	NUM
ejpam-4977	121	8	(	(	PUNCT
ejpam-4977	121	9	1	1	NUM
ejpam-4977	121	10	)	)	PUNCT
ejpam-4977	121	11	(	(	PUNCT
ejpam-4977	121	12	2024	2024	NUM
ejpam-4977	121	13	)	)	PUNCT
ejpam-4977	121	14	,	,	PUNCT
ejpam-4977	121	15	116	116	NUM
ejpam-4977	121	16	-	-	SYM
ejpam-4977	121	17	123	123	NUM
ejpam-4977	121	18	121	121	NUM
ejpam-4977	121	19	=	=	SYM
ejpam-4977	121	20	(	(	PUNCT
ejpam-4977	121	21	y	y	PROPN
ejpam-4977	121	22	∗	∗	X
ejpam-4977	121	23	x)n+1	x)n+1	PROPN
ejpam-4977	122	1	therefore	therefore	ADV
ejpam-4977	122	2	,	,	PUNCT
ejpam-4977	122	3	0	0	NUM
ejpam-4977	122	4	∗	∗	NOUN
ejpam-4977	122	5	(	(	PUNCT
ejpam-4977	122	6	x	x	X
ejpam-4977	122	7	∗	∗	NOUN
ejpam-4977	122	8	y)n	y)n	NUM
ejpam-4977	123	1	=	=	SYM
ejpam-4977	123	2	(	(	PUNCT
ejpam-4977	123	3	y	y	NOUN
ejpam-4977	123	4	∗	∗	X
ejpam-4977	123	5	x)n	x)n	PUNCT
ejpam-4977	123	6	for	for	ADP
ejpam-4977	123	7	any	any	DET
ejpam-4977	123	8	n	n	PRON
ejpam-4977	123	9	∈	∈	PROPN
ejpam-4977	123	10	z+	z+	X
ejpam-4977	123	11	.	.	PUNCT
ejpam-4977	123	12	theorem	theorem	NOUN
ejpam-4977	123	13	4	4	NUM
ejpam-4977	123	14	.	.	PUNCT
ejpam-4977	124	1	let	let	AUX
ejpam-4977	124	2	(	(	PUNCT
ejpam-4977	124	3	x	x	X
ejpam-4977	124	4	,	,	PUNCT
ejpam-4977	124	5	∗	∗	NOUN
ejpam-4977	124	6	,	,	PUNCT
ejpam-4977	124	7	0	0	NUM
ejpam-4977	124	8	)	)	PUNCT
ejpam-4977	124	9	be	be	AUX
ejpam-4977	124	10	a	a	DET
ejpam-4977	124	11	b	b	NOUN
ejpam-4977	124	12	-	-	PUNCT
ejpam-4977	124	13	algebra	algebra	NOUN
ejpam-4977	124	14	and	and	CCONJ
ejpam-4977	124	15	x	x	NOUN
ejpam-4977	124	16	,	,	PUNCT
ejpam-4977	124	17	y	y	PROPN
ejpam-4977	124	18	∈	∈	PROPN
ejpam-4977	124	19	x	x	PUNCT
ejpam-4977	124	20	with	with	ADP
ejpam-4977	124	21	n	n	DET
ejpam-4977	124	22	∈	∈	PROPN
ejpam-4977	124	23	z+	z+	X
ejpam-4977	124	24	,	,	PUNCT
ejpam-4977	124	25	then	then	ADV
ejpam-4977	124	26	0	0	NUM
ejpam-4977	124	27	∗	∗	NOUN
ejpam-4977	124	28	(	(	PUNCT
ejpam-4977	124	29	0	0	NUM
ejpam-4977	124	30	∗	∗	NOUN
ejpam-4977	124	31	x)n	x)n	PUNCT
ejpam-4977	125	1	=	=	SYM
ejpam-4977	125	2	0	0	NUM
ejpam-4977	125	3	∗	∗	NOUN
ejpam-4977	125	4	(	(	PUNCT
ejpam-4977	125	5	0	0	NUM
ejpam-4977	125	6	∗	∗	NOUN
ejpam-4977	125	7	xn	xn	NOUN
ejpam-4977	125	8	)	)	PUNCT
ejpam-4977	125	9	proof	proof	NOUN
ejpam-4977	125	10	.	.	PUNCT
ejpam-4977	126	1	since	since	SCONJ
ejpam-4977	126	2	0	0	NUM
ejpam-4977	126	3	∗	∗	NOUN
ejpam-4977	126	4	xn	xn	PUNCT
ejpam-4977	126	5	=	=	PUNCT
ejpam-4977	126	6	x−n	x−n	PROPN
ejpam-4977	126	7	in	in	ADP
ejpam-4977	126	8	[	[	X
ejpam-4977	126	9	2	2	NUM
ejpam-4977	126	10	]	]	PUNCT
ejpam-4977	126	11	,	,	PUNCT
ejpam-4977	126	12	0	0	NUM
ejpam-4977	126	13	∗	∗	NOUN
ejpam-4977	126	14	(	(	PUNCT
ejpam-4977	126	15	0	0	NUM
ejpam-4977	126	16	∗	∗	NOUN
ejpam-4977	126	17	x)n	x)n	PUNCT
ejpam-4977	126	18	=	=	PRON
ejpam-4977	126	19	(	(	PUNCT
ejpam-4977	126	20	0	0	NUM
ejpam-4977	126	21	∗	∗	NOUN
ejpam-4977	126	22	x)−n	x)−n	PUNCT
ejpam-4977	127	1	=	=	PRON
ejpam-4977	128	1	(	(	PUNCT
ejpam-4977	128	2	x−1)−n	x−1)−n	PROPN
ejpam-4977	128	3	=	=	SYM
ejpam-4977	128	4	(	(	PUNCT
ejpam-4977	128	5	x−n)−1	x−n)−1	PROPN
ejpam-4977	128	6	=	=	SYM
ejpam-4977	128	7	0	0	NUM
ejpam-4977	128	8	∗	∗	NOUN
ejpam-4977	128	9	(	(	PUNCT
ejpam-4977	128	10	x−n	x−n	PROPN
ejpam-4977	128	11	)	)	PUNCT
ejpam-4977	128	12	=	=	SYM
ejpam-4977	128	13	0	0	NUM
ejpam-4977	128	14	∗	∗	NOUN
ejpam-4977	128	15	(	(	PUNCT
ejpam-4977	128	16	0	0	NUM
ejpam-4977	128	17	∗	∗	NOUN
ejpam-4977	128	18	xn	xn	NUM
ejpam-4977	128	19	)	)	PUNCT
ejpam-4977	128	20	.	.	PUNCT
ejpam-4977	129	1	theorem	theorem	NOUN
ejpam-4977	129	2	5	5	NUM
ejpam-4977	129	3	.	.	PUNCT
ejpam-4977	130	1	let	let	AUX
ejpam-4977	130	2	(	(	PUNCT
ejpam-4977	130	3	x	x	X
ejpam-4977	130	4	,	,	PUNCT
ejpam-4977	130	5	∗	∗	NOUN
ejpam-4977	130	6	,	,	PUNCT
ejpam-4977	130	7	0	0	NUM
ejpam-4977	130	8	)	)	PUNCT
ejpam-4977	130	9	be	be	AUX
ejpam-4977	130	10	a	a	DET
ejpam-4977	130	11	b	b	NOUN
ejpam-4977	130	12	-	-	PUNCT
ejpam-4977	130	13	algebra	algebra	NOUN
ejpam-4977	130	14	and	and	CCONJ
ejpam-4977	130	15	a	a	DET
ejpam-4977	130	16	∈	∈	ADJ
ejpam-4977	130	17	x	x	NOUN
ejpam-4977	130	18	,	,	PUNCT
ejpam-4977	130	19	then	then	ADV
ejpam-4977	130	20	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	130	21	is	be	AUX
ejpam-4977	130	22	a	a	DET
ejpam-4977	130	23	b	b	NOUN
ejpam-4977	130	24	-	-	PUNCT
ejpam-4977	130	25	ideal	ideal	NOUN
ejpam-4977	130	26	of	of	ADP
ejpam-4977	130	27	x.	x.	NOUN
ejpam-4977	130	28	proof	proof	NOUN
ejpam-4977	130	29	.	.	PUNCT
ejpam-4977	131	1	clearly	clearly	ADV
ejpam-4977	131	2	,	,	PUNCT
ejpam-4977	131	3	0	0	NUM
ejpam-4977	131	4	=	=	SYM
ejpam-4977	131	5	a0	a0	PROPN
ejpam-4977	131	6	∈	∈	PROPN
ejpam-4977	131	7	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	131	8	.	.	PUNCT
ejpam-4977	132	1	next	next	ADV
ejpam-4977	132	2	,	,	PUNCT
ejpam-4977	132	3	let	let	VERB
ejpam-4977	132	4	x	x	PRON
ejpam-4977	132	5	∗	∗	VERB
ejpam-4977	132	6	y	y	PROPN
ejpam-4977	132	7	∈	∈	PROPN
ejpam-4977	132	8	⟨a⟩b	⟨a⟩b	NOUN
ejpam-4977	132	9	and	and	CCONJ
ejpam-4977	132	10	y	y	PROPN
ejpam-4977	132	11	∈	∈	PROPN
ejpam-4977	132	12	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	132	13	,	,	PUNCT
ejpam-4977	132	14	then	then	ADV
ejpam-4977	132	15	x	x	X
ejpam-4977	132	16	∗	∗	NOUN
ejpam-4977	132	17	y	y	PROPN
ejpam-4977	132	18	=	=	SYM
ejpam-4977	132	19	ak	ak	PROPN
ejpam-4977	132	20	and	and	CCONJ
ejpam-4977	132	21	y	y	PROPN
ejpam-4977	132	22	=	=	PROPN
ejpam-4977	132	23	ar	ar	PROPN
ejpam-4977	132	24	for	for	ADP
ejpam-4977	132	25	some	some	DET
ejpam-4977	132	26	k	k	NOUN
ejpam-4977	132	27	,	,	PUNCT
ejpam-4977	133	1	r	r	NOUN
ejpam-4977	133	2	∈	∈	PROPN
ejpam-4977	133	3	z.	z.	PROPN
ejpam-4977	134	1	so	so	ADV
ejpam-4977	134	2	,	,	PUNCT
ejpam-4977	134	3	y∗x	y∗x	ADV
ejpam-4977	134	4	=	=	SYM
ejpam-4977	134	5	0∗(x∗y	0∗(x∗y	NUM
ejpam-4977	134	6	)	)	PUNCT
ejpam-4977	134	7	=	=	SYM
ejpam-4977	135	1	0∗ak	0∗ak	X
ejpam-4977	135	2	=	=	SYM
ejpam-4977	135	3	a0∗ak	a0∗ak	PROPN
ejpam-4977	135	4	=	=	SYM
ejpam-4977	135	5	a0−k	a0−k	PROPN
ejpam-4977	135	6	=	=	SYM
ejpam-4977	135	7	a−k	a−k	NOUN
ejpam-4977	135	8	∈	∈	PROPN
ejpam-4977	135	9	<	<	X
ejpam-4977	135	10	a	a	X
ejpam-4977	135	11	>	>	X
ejpam-4977	135	12	and	and	CCONJ
ejpam-4977	135	13	hance	hance	PROPN
ejpam-4977	135	14	by	by	ADP
ejpam-4977	135	15	[	[	X
ejpam-4977	135	16	1	1	NUM
ejpam-4977	135	17	]	]	PUNCT
ejpam-4977	135	18	,	,	PUNCT
ejpam-4977	135	19	x	x	SYM
ejpam-4977	135	20	=	=	SYM
ejpam-4977	135	21	y	y	PROPN
ejpam-4977	135	22	∗	∗	NOUN
ejpam-4977	135	23	(	(	PUNCT
ejpam-4977	135	24	y	y	PROPN
ejpam-4977	135	25	∗	∗	NOUN
ejpam-4977	135	26	x	x	NOUN
ejpam-4977	135	27	)	)	PUNCT
ejpam-4977	136	1	=	=	SYM
ejpam-4977	136	2	ar	ar	NOUN
ejpam-4977	136	3	∗	∗	NOUN
ejpam-4977	136	4	a−k	a−k	NOUN
ejpam-4977	136	5	=	=	SYM
ejpam-4977	136	6	ar−(−k	ar−(−k	NOUN
ejpam-4977	136	7	)	)	PUNCT
ejpam-4977	136	8	=	=	PUNCT
ejpam-4977	136	9	ar+k	ar+k	PROPN
ejpam-4977	136	10	∈	∈	PROPN
ejpam-4977	136	11	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	136	12	.	.	PUNCT
ejpam-4977	137	1	therefore	therefore	ADV
ejpam-4977	137	2	,	,	PUNCT
ejpam-4977	137	3	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	137	4	is	be	AUX
ejpam-4977	137	5	an	an	DET
ejpam-4977	137	6	ideal	ideal	NOUN
ejpam-4977	137	7	of	of	ADP
ejpam-4977	137	8	x.	x.	NOUN
ejpam-4977	137	9	in	in	ADP
ejpam-4977	137	10	2023	2023	NUM
ejpam-4977	137	11	,	,	PUNCT
ejpam-4977	137	12	k.	k.	PROPN
ejpam-4977	137	13	e.	e.	PROPN
ejpam-4977	137	14	belleza	belleza	PROPN
ejpam-4977	137	15	,	,	PUNCT
ejpam-4977	137	16	j.	j.	PROPN
ejpam-4977	137	17	r.	r.	PROPN
ejpam-4977	137	18	albaracin	albaracin	PROPN
ejpam-4977	138	1	[	[	X
ejpam-4977	138	2	8	8	NUM
ejpam-4977	138	3	]	]	PUNCT
ejpam-4977	138	4	introduced	introduce	VERB
ejpam-4977	138	5	the	the	DET
ejpam-4977	138	6	concept	concept	NOUN
ejpam-4977	138	7	of	of	ADP
ejpam-4977	138	8	dual	dual	ADJ
ejpam-4977	138	9	b	b	NOUN
ejpam-4977	138	10	-	-	PUNCT
ejpam-4977	138	11	filters	filter	NOUN
ejpam-4977	138	12	of	of	ADP
ejpam-4977	138	13	dual	dual	ADJ
ejpam-4977	138	14	b	b	NOUN
ejpam-4977	138	15	-	-	PUNCT
ejpam-4977	138	16	algebra	algebra	NOUN
ejpam-4977	138	17	.	.	PUNCT
ejpam-4977	139	1	thus	thus	ADV
ejpam-4977	139	2	,	,	PUNCT
ejpam-4977	139	3	we	we	PRON
ejpam-4977	139	4	can	can	AUX
ejpam-4977	139	5	have	have	VERB
ejpam-4977	139	6	the	the	DET
ejpam-4977	139	7	following	follow	VERB
ejpam-4977	139	8	definition	definition	NOUN
ejpam-4977	139	9	:	:	PUNCT
ejpam-4977	139	10	definition	definition	NOUN
ejpam-4977	139	11	5	5	NUM
ejpam-4977	139	12	.	.	PUNCT
ejpam-4977	140	1	let	let	VERB
ejpam-4977	140	2	(	(	PUNCT
ejpam-4977	140	3	x	x	X
ejpam-4977	140	4	,	,	PUNCT
ejpam-4977	140	5	∗	∗	NOUN
ejpam-4977	140	6	,	,	PUNCT
ejpam-4977	140	7	0	0	NUM
ejpam-4977	140	8	)	)	PUNCT
ejpam-4977	140	9	be	be	AUX
ejpam-4977	140	10	a	a	DET
ejpam-4977	140	11	b	b	NOUN
ejpam-4977	140	12	-	-	PUNCT
ejpam-4977	140	13	algebra	algebra	NOUN
ejpam-4977	140	14	.	.	PUNCT
ejpam-4977	141	1	a	a	DET
ejpam-4977	141	2	nonempty	nonempty	NOUN
ejpam-4977	141	3	subset	subset	VERB
ejpam-4977	141	4	f	f	PROPN
ejpam-4977	141	5	of	of	ADP
ejpam-4977	141	6	x	x	PROPN
ejpam-4977	141	7	is	be	AUX
ejpam-4977	141	8	called	call	VERB
ejpam-4977	141	9	a	a	DET
ejpam-4977	141	10	b	b	NOUN
ejpam-4977	141	11	-	-	NOUN
ejpam-4977	141	12	filter	filter	NOUN
ejpam-4977	141	13	of	of	ADP
ejpam-4977	141	14	x	x	PRON
ejpam-4977	141	15	if	if	SCONJ
ejpam-4977	141	16	it	it	PRON
ejpam-4977	141	17	satisfies	satisfy	VERB
ejpam-4977	141	18	the	the	DET
ejpam-4977	141	19	following	following	ADJ
ejpam-4977	141	20	axioms	axiom	NOUN
ejpam-4977	141	21	for	for	ADP
ejpam-4977	141	22	all	all	DET
ejpam-4977	141	23	x	x	NOUN
ejpam-4977	141	24	,	,	PUNCT
ejpam-4977	141	25	y	y	PROPN
ejpam-4977	141	26	∈	∈	PROPN
ejpam-4977	142	1	x	x	X
ejpam-4977	142	2	:	:	PUNCT
ejpam-4977	142	3	(	(	PUNCT
ejpam-4977	142	4	i	i	NOUN
ejpam-4977	142	5	)	)	PUNCT
ejpam-4977	142	6	0	0	PUNCT
ejpam-4977	143	1	∈	∈	PROPN
ejpam-4977	143	2	f	f	X
ejpam-4977	143	3	,	,	PUNCT
ejpam-4977	143	4	(	(	PUNCT
ejpam-4977	143	5	ii	ii	NOUN
ejpam-4977	143	6	)	)	PUNCT
ejpam-4977	144	1	if	if	SCONJ
ejpam-4977	144	2	x	x	PROPN
ejpam-4977	144	3	∗	∗	VERB
ejpam-4977	144	4	y	y	PROPN
ejpam-4977	144	5	∈	∈	PROPN
ejpam-4977	144	6	f	f	PROPN
ejpam-4977	144	7	and	and	CCONJ
ejpam-4977	144	8	x	x	SYM
ejpam-4977	144	9	∈	∈	PROPN
ejpam-4977	144	10	f	f	PROPN
ejpam-4977	144	11	,	,	PUNCT
ejpam-4977	144	12	then	then	ADV
ejpam-4977	144	13	y	y	PROPN
ejpam-4977	144	14	∈	∈	PROPN
ejpam-4977	144	15	f	f	AUX
ejpam-4977	144	16	.	.	PUNCT
ejpam-4977	144	17	example	example	NOUN
ejpam-4977	144	18	4	4	NUM
ejpam-4977	144	19	.	.	PUNCT
ejpam-4977	145	1	consider	consider	VERB
ejpam-4977	145	2	the	the	DET
ejpam-4977	145	3	b	b	NOUN
ejpam-4977	145	4	-	-	PUNCT
ejpam-4977	145	5	algebra	algebra	NOUN
ejpam-4977	145	6	x	x	X
ejpam-4977	145	7	=	=	SYM
ejpam-4977	145	8	{	{	PUNCT
ejpam-4977	145	9	0	0	NUM
ejpam-4977	145	10	,	,	PUNCT
ejpam-4977	145	11	1	1	NUM
ejpam-4977	145	12	,	,	PUNCT
ejpam-4977	145	13	2	2	NUM
ejpam-4977	145	14	,	,	PUNCT
ejpam-4977	145	15	3	3	NUM
ejpam-4977	145	16	}	}	PUNCT
ejpam-4977	145	17	from	from	ADP
ejpam-4977	145	18	example	example	NOUN
ejpam-4977	145	19	1	1	NUM
ejpam-4977	145	20	.	.	PUNCT
ejpam-4977	146	1	the	the	DET
ejpam-4977	146	2	sets	set	NOUN
ejpam-4977	146	3	f1	f1	NOUN
ejpam-4977	146	4	=	=	SYM
ejpam-4977	146	5	{	{	PUNCT
ejpam-4977	146	6	0	0	NUM
ejpam-4977	146	7	}	}	PUNCT
ejpam-4977	146	8	,	,	PUNCT
ejpam-4977	146	9	f2	f2	PROPN
ejpam-4977	146	10	=	=	SYM
ejpam-4977	146	11	{	{	PUNCT
ejpam-4977	146	12	0	0	NUM
ejpam-4977	146	13	,	,	PUNCT
ejpam-4977	146	14	2	2	NUM
ejpam-4977	146	15	}	}	PUNCT
ejpam-4977	146	16	are	be	AUX
ejpam-4977	146	17	b	b	NOUN
ejpam-4977	146	18	-	-	PUNCT
ejpam-4977	146	19	filters	filter	NOUN
ejpam-4977	146	20	of	of	ADP
ejpam-4977	146	21	x	x	PART
ejpam-4977	146	22	while	while	SCONJ
ejpam-4977	146	23	f3	f3	PROPN
ejpam-4977	146	24	=	=	SYM
ejpam-4977	146	25	{	{	PUNCT
ejpam-4977	146	26	0	0	NUM
ejpam-4977	146	27	,	,	PUNCT
ejpam-4977	146	28	3	3	NUM
ejpam-4977	146	29	}	}	PUNCT
ejpam-4977	146	30	is	be	AUX
ejpam-4977	146	31	not	not	PART
ejpam-4977	146	32	a	a	DET
ejpam-4977	146	33	b	b	NOUN
ejpam-4977	146	34	-	-	NOUN
ejpam-4977	146	35	filter	filter	NOUN
ejpam-4977	146	36	of	of	ADP
ejpam-4977	146	37	x	x	PRON
ejpam-4977	146	38	,	,	PUNCT
ejpam-4977	146	39	since	since	SCONJ
ejpam-4977	146	40	0	0	NUM
ejpam-4977	146	41	∗	∗	NOUN
ejpam-4977	146	42	1	1	NUM
ejpam-4977	146	43	=	=	SYM
ejpam-4977	146	44	3	3	NUM
ejpam-4977	146	45	∈	∈	NOUN
ejpam-4977	146	46	f3	f3	NOUN
ejpam-4977	146	47	where	where	SCONJ
ejpam-4977	146	48	0	0	NUM
ejpam-4977	146	49	∈	∈	NOUN
ejpam-4977	146	50	f3	f3	NOUN
ejpam-4977	146	51	but	but	CCONJ
ejpam-4977	146	52	1	1	NUM
ejpam-4977	146	53	/∈	/∈	ADV
ejpam-4977	146	54	f3	f3	PROPN
ejpam-4977	146	55	.	.	PUNCT
ejpam-4977	147	1	consider	consider	VERB
ejpam-4977	147	2	(	(	PUNCT
ejpam-4977	147	3	y,⊙	y,⊙	NOUN
ejpam-4977	147	4	,	,	PUNCT
ejpam-4977	147	5	0	0	NUM
ejpam-4977	147	6	)	)	PUNCT
ejpam-4977	147	7	,	,	PUNCT
ejpam-4977	147	8	the	the	DET
ejpam-4977	147	9	sets	set	VERB
ejpam-4977	147	10	f4	f4	NOUN
ejpam-4977	147	11	=	=	SYM
ejpam-4977	147	12	{	{	PUNCT
ejpam-4977	147	13	0	0	NUM
ejpam-4977	147	14	}	}	PUNCT
ejpam-4977	147	15	,	,	PUNCT
ejpam-4977	147	16	f5	f5	PROPN
ejpam-4977	147	17	=	=	SYM
ejpam-4977	147	18	{	{	PUNCT
ejpam-4977	147	19	0	0	NUM
ejpam-4977	147	20	,	,	PUNCT
ejpam-4977	147	21	3	3	NUM
ejpam-4977	147	22	}	}	PUNCT
ejpam-4977	147	23	,	,	PUNCT
ejpam-4977	147	24	f6	f6	X
ejpam-4977	147	25	=	=	PUNCT
ejpam-4977	147	26	{	{	PUNCT
ejpam-4977	147	27	0	0	NUM
ejpam-4977	147	28	,	,	PUNCT
ejpam-4977	147	29	4	4	NUM
ejpam-4977	147	30	}	}	PUNCT
ejpam-4977	147	31	,	,	PUNCT
ejpam-4977	147	32	f7	f7	PROPN
ejpam-4977	147	33	=	=	PUNCT
ejpam-4977	147	34	{	{	PUNCT
ejpam-4977	147	35	0	0	NUM
ejpam-4977	147	36	,	,	PUNCT
ejpam-4977	147	37	5	5	NUM
ejpam-4977	147	38	}	}	PUNCT
ejpam-4977	147	39	,	,	PUNCT
ejpam-4977	147	40	are	be	AUX
ejpam-4977	147	41	b	b	NOUN
ejpam-4977	147	42	-	-	PUNCT
ejpam-4977	147	43	filters	filter	NOUN
ejpam-4977	147	44	of	of	ADP
ejpam-4977	147	45	y	y	PROPN
ejpam-4977	147	46	.	.	PUNCT
ejpam-4977	148	1	theorem	theorem	ADJ
ejpam-4977	148	2	6	6	NUM
ejpam-4977	148	3	.	.	PUNCT
ejpam-4977	149	1	let	let	AUX
ejpam-4977	149	2	(	(	PUNCT
ejpam-4977	149	3	x	x	X
ejpam-4977	149	4	,	,	PUNCT
ejpam-4977	149	5	∗	∗	NOUN
ejpam-4977	149	6	,	,	PUNCT
ejpam-4977	149	7	0	0	NUM
ejpam-4977	149	8	)	)	PUNCT
ejpam-4977	149	9	be	be	AUX
ejpam-4977	149	10	a	a	DET
ejpam-4977	149	11	b	b	NOUN
ejpam-4977	149	12	-	-	PUNCT
ejpam-4977	149	13	algebra	algebra	NOUN
ejpam-4977	149	14	and	and	CCONJ
ejpam-4977	149	15	a	a	DET
ejpam-4977	149	16	∈	∈	ADJ
ejpam-4977	149	17	x	x	NOUN
ejpam-4977	149	18	,	,	PUNCT
ejpam-4977	149	19	then	then	ADV
ejpam-4977	149	20	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	149	21	is	be	AUX
ejpam-4977	149	22	a	a	DET
ejpam-4977	149	23	b	b	NOUN
ejpam-4977	149	24	-	-	NOUN
ejpam-4977	149	25	filter	filter	NOUN
ejpam-4977	149	26	of	of	ADP
ejpam-4977	149	27	x.	x.	NOUN
ejpam-4977	149	28	proof	proof	NOUN
ejpam-4977	149	29	.	.	PUNCT
ejpam-4977	150	1	clearly	clearly	ADV
ejpam-4977	150	2	0	0	X
ejpam-4977	150	3	=	=	SYM
ejpam-4977	150	4	a0	a0	PROPN
ejpam-4977	150	5	∈	∈	PROPN
ejpam-4977	150	6	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	150	7	.	.	PUNCT
ejpam-4977	151	1	next	next	ADV
ejpam-4977	151	2	,	,	PUNCT
ejpam-4977	151	3	let	let	VERB
ejpam-4977	151	4	x	x	PRON
ejpam-4977	151	5	∗	∗	VERB
ejpam-4977	151	6	y	y	PROPN
ejpam-4977	151	7	∈	∈	PROPN
ejpam-4977	151	8	⟨a⟩b	⟨a⟩b	NOUN
ejpam-4977	151	9	and	and	CCONJ
ejpam-4977	151	10	x	x	SYM
ejpam-4977	151	11	∈	∈	PROPN
ejpam-4977	151	12	⟨a⟩b	⟨a⟩b	NOUN
ejpam-4977	151	13	,	,	PUNCT
ejpam-4977	151	14	then	then	ADV
ejpam-4977	151	15	x	x	X
ejpam-4977	151	16	∗	∗	NOUN
ejpam-4977	151	17	y	y	PROPN
ejpam-4977	151	18	=	=	SYM
ejpam-4977	151	19	ak	ak	PROPN
ejpam-4977	151	20	and	and	CCONJ
ejpam-4977	151	21	x	x	X
ejpam-4977	151	22	=	=	NOUN
ejpam-4977	151	23	ar	ar	NOUN
ejpam-4977	151	24	for	for	ADP
ejpam-4977	151	25	some	some	DET
ejpam-4977	151	26	k	k	NOUN
ejpam-4977	151	27	,	,	PUNCT
ejpam-4977	151	28	r	r	NOUN
ejpam-4977	151	29	∈	∈	PROPN
ejpam-4977	151	30	z	z	NOUN
ejpam-4977	151	31	and	and	CCONJ
ejpam-4977	151	32	hence	hence	ADV
ejpam-4977	151	33	by	by	ADP
ejpam-4977	151	34	theory	theory	NOUN
ejpam-4977	151	35	3.2	3.2	NUM
ejpam-4977	151	36	in	in	ADP
ejpam-4977	151	37	[	[	X
ejpam-4977	151	38	1	1	NUM
ejpam-4977	151	39	]	]	PUNCT
ejpam-4977	151	40	,	,	PUNCT
ejpam-4977	151	41	y	y	PROPN
ejpam-4977	151	42	=	=	PUNCT
ejpam-4977	151	43	x	x	SYM
ejpam-4977	151	44	∗	∗	NOUN
ejpam-4977	151	45	(	(	PUNCT
ejpam-4977	151	46	x	x	X
ejpam-4977	151	47	∗	∗	PROPN
ejpam-4977	151	48	y	y	NOUN
ejpam-4977	151	49	)	)	PUNCT
ejpam-4977	151	50	=	=	SYM
ejpam-4977	151	51	ar	ar	PROPN
ejpam-4977	151	52	∗	∗	X
ejpam-4977	151	53	ak	ak	PROPN
ejpam-4977	151	54	=	=	PROPN
ejpam-4977	151	55	ar−k	ar−k	PROPN
ejpam-4977	151	56	∈	∈	PROPN
ejpam-4977	152	1	⟨a⟩b	⟨a⟩b	NOUN
ejpam-4977	152	2	therefore	therefore	ADV
ejpam-4977	152	3	,	,	PUNCT
ejpam-4977	152	4	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	152	5	is	be	AUX
ejpam-4977	152	6	a	a	DET
ejpam-4977	152	7	filter	filter	NOUN
ejpam-4977	152	8	of	of	ADP
ejpam-4977	152	9	x.	x.	PROPN
ejpam-4977	152	10	m.	m.	PROPN
ejpam-4977	152	11	phattarachaleekul	phattarachaleekul	PROPN
ejpam-4977	152	12	/	/	SYM
ejpam-4977	152	13	eur	eur	PROPN
ejpam-4977	152	14	.	.	PUNCT
ejpam-4977	153	1	j.	j.	PROPN
ejpam-4977	153	2	pure	pure	PROPN
ejpam-4977	153	3	appl	appl	PROPN
ejpam-4977	153	4	.	.	PROPN
ejpam-4977	153	5	math	math	PROPN
ejpam-4977	153	6	,	,	PUNCT
ejpam-4977	153	7	17	17	NUM
ejpam-4977	153	8	(	(	PUNCT
ejpam-4977	153	9	1	1	NUM
ejpam-4977	153	10	)	)	PUNCT
ejpam-4977	153	11	(	(	PUNCT
ejpam-4977	153	12	2024	2024	NUM
ejpam-4977	153	13	)	)	PUNCT
ejpam-4977	153	14	,	,	PUNCT
ejpam-4977	153	15	116	116	NUM
ejpam-4977	153	16	-	-	SYM
ejpam-4977	153	17	123	123	NUM
ejpam-4977	153	18	122	122	NUM
ejpam-4977	153	19	theorem	theorem	NOUN
ejpam-4977	153	20	7	7	NUM
ejpam-4977	153	21	.	.	PUNCT
ejpam-4977	154	1	let	let	AUX
ejpam-4977	154	2	(	(	PUNCT
ejpam-4977	154	3	x	x	X
ejpam-4977	154	4	,	,	PUNCT
ejpam-4977	154	5	∗	∗	NOUN
ejpam-4977	154	6	,	,	PUNCT
ejpam-4977	154	7	0	0	NUM
ejpam-4977	154	8	)	)	PUNCT
ejpam-4977	154	9	be	be	AUX
ejpam-4977	154	10	a	a	DET
ejpam-4977	154	11	b	b	NOUN
ejpam-4977	154	12	-	-	PUNCT
ejpam-4977	154	13	algebra	algebra	NOUN
ejpam-4977	154	14	and	and	CCONJ
ejpam-4977	154	15	a	a	DET
ejpam-4977	154	16	∈	∈	NOUN
ejpam-4977	154	17	x	x	PUNCT
ejpam-4977	154	18	with	with	ADP
ejpam-4977	154	19	0	0	NUM
ejpam-4977	154	20	∗	∗	NOUN
ejpam-4977	154	21	a	a	DET
ejpam-4977	154	22	=	=	NOUN
ejpam-4977	154	23	a	a	NOUN
ejpam-4977	154	24	,	,	PUNCT
ejpam-4977	154	25	then	then	ADV
ejpam-4977	154	26	⟨a⟩b	⟨a⟩b	NUM
ejpam-4977	154	27	=	=	SYM
ejpam-4977	154	28	{	{	PUNCT
ejpam-4977	154	29	0	0	NUM
ejpam-4977	154	30	,	,	PUNCT
ejpam-4977	154	31	a	a	DET
ejpam-4977	154	32	}	}	PUNCT
ejpam-4977	154	33	form	form	NOUN
ejpam-4977	154	34	a	a	DET
ejpam-4977	154	35	b	b	NOUN
ejpam-4977	154	36	-	-	NOUN
ejpam-4977	154	37	filter	filter	NOUN
ejpam-4977	154	38	of	of	ADP
ejpam-4977	154	39	x.	x.	NOUN
ejpam-4977	154	40	proof	proof	NOUN
ejpam-4977	154	41	.	.	PUNCT
ejpam-4977	155	1	let	let	VERB
ejpam-4977	155	2	a	a	DET
ejpam-4977	155	3	∈	∈	NOUN
ejpam-4977	155	4	x	x	PUNCT
ejpam-4977	155	5	with	with	ADP
ejpam-4977	155	6	0	0	NUM
ejpam-4977	155	7	∗	∗	NOUN
ejpam-4977	155	8	a	a	DET
ejpam-4977	155	9	=	=	NOUN
ejpam-4977	155	10	a.	a.	NOUN
ejpam-4977	155	11	consider	consider	VERB
ejpam-4977	155	12	a0	a0	NOUN
ejpam-4977	155	13	=	=	SYM
ejpam-4977	155	14	0	0	PROPN
ejpam-4977	155	15	,	,	PUNCT
ejpam-4977	155	16	a1	a1	NOUN
ejpam-4977	155	17	=	=	SYM
ejpam-4977	155	18	a	a	PROPN
ejpam-4977	155	19	,	,	PUNCT
ejpam-4977	155	20	a2	a2	PROPN
ejpam-4977	155	21	=	=	PUNCT
ejpam-4977	155	22	a	a	DET
ejpam-4977	155	23	∗	∗	NOUN
ejpam-4977	155	24	a	a	PRON
ejpam-4977	155	25	=	=	SYM
ejpam-4977	155	26	0	0	NUM
ejpam-4977	155	27	,	,	PUNCT
ejpam-4977	155	28	a3	a3	NOUN
ejpam-4977	155	29	=	=	SYM
ejpam-4977	155	30	a2	a2	PROPN
ejpam-4977	155	31	∗	∗	VERB
ejpam-4977	155	32	a	a	DET
ejpam-4977	155	33	=	=	SYM
ejpam-4977	155	34	0	0	NUM
ejpam-4977	155	35	∗	∗	NOUN
ejpam-4977	155	36	a	a	DET
ejpam-4977	155	37	=	=	NOUN
ejpam-4977	155	38	a	a	DET
ejpam-4977	155	39	,	,	PUNCT
ejpam-4977	155	40	a4	a4	NOUN
ejpam-4977	155	41	=	=	NOUN
ejpam-4977	155	42	a3	a3	NOUN
ejpam-4977	155	43	∗	∗	NOUN
ejpam-4977	156	1	a	a	PRON
ejpam-4977	156	2	=	=	NOUN
ejpam-4977	156	3	a	a	DET
ejpam-4977	156	4	∗	∗	NOUN
ejpam-4977	156	5	0	0	NUM
ejpam-4977	156	6	=	=	SYM
ejpam-4977	156	7	0	0	NUM
ejpam-4977	156	8	,	,	PUNCT
ejpam-4977	156	9	.	.	PUNCT
ejpam-4977	156	10	.	.	PUNCT
ejpam-4977	156	11	.	.	PUNCT
ejpam-4977	157	1	this	this	PRON
ejpam-4977	157	2	implies	imply	VERB
ejpam-4977	157	3	that	that	SCONJ
ejpam-4977	157	4	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	157	5	=	=	SYM
ejpam-4977	157	6	{	{	PUNCT
ejpam-4977	157	7	0	0	NUM
ejpam-4977	157	8	,	,	PUNCT
ejpam-4977	157	9	a	a	PRON
ejpam-4977	157	10	}	}	PUNCT
ejpam-4977	157	11	.	.	PUNCT
ejpam-4977	158	1	next	next	ADV
ejpam-4977	158	2	,	,	PUNCT
ejpam-4977	158	3	let	let	VERB
ejpam-4977	158	4	x	x	PRON
ejpam-4977	158	5	,	,	PUNCT
ejpam-4977	158	6	y	y	PROPN
ejpam-4977	158	7	∈	∈	PROPN
ejpam-4977	158	8	x	x	PROPN
ejpam-4977	158	9	,	,	PUNCT
ejpam-4977	158	10	f	f	PROPN
ejpam-4977	158	11	=	=	PUNCT
ejpam-4977	158	12	{	{	PUNCT
ejpam-4977	158	13	0	0	NUM
ejpam-4977	158	14	,	,	PUNCT
ejpam-4977	158	15	a	a	PRON
ejpam-4977	158	16	}	}	PUNCT
ejpam-4977	158	17	with	with	ADP
ejpam-4977	158	18	0	0	NUM
ejpam-4977	158	19	∗	∗	NOUN
ejpam-4977	158	20	a	a	DET
ejpam-4977	158	21	=	=	NOUN
ejpam-4977	158	22	a	a	NOUN
ejpam-4977	158	23	,	,	PUNCT
ejpam-4977	158	24	x	x	PUNCT
ejpam-4977	158	25	∗	∗	NOUN
ejpam-4977	158	26	y	y	PROPN
ejpam-4977	158	27	∈	∈	PROPN
ejpam-4977	158	28	f	f	PROPN
ejpam-4977	158	29	and	and	CCONJ
ejpam-4977	158	30	x	x	PROPN
ejpam-4977	158	31	∈	∈	PROPN
ejpam-4977	158	32	f.	f.	PROPN
ejpam-4977	158	33	case	case	NOUN
ejpam-4977	158	34	1	1	NUM
ejpam-4977	158	35	:	:	PUNCT
ejpam-4977	158	36	if	if	SCONJ
ejpam-4977	158	37	x	x	PROPN
ejpam-4977	158	38	∗	∗	VERB
ejpam-4977	158	39	y	y	NOUN
ejpam-4977	158	40	=	=	SYM
ejpam-4977	158	41	0	0	PUNCT
ejpam-4977	159	1	and	and	CCONJ
ejpam-4977	159	2	x	x	SYM
ejpam-4977	159	3	=	=	SYM
ejpam-4977	159	4	0	0	NUM
ejpam-4977	159	5	,	,	PUNCT
ejpam-4977	159	6	then	then	ADV
ejpam-4977	159	7	x	x	X
ejpam-4977	159	8	∗	∗	NOUN
ejpam-4977	159	9	y	y	NOUN
ejpam-4977	159	10	=	=	SYM
ejpam-4977	159	11	0	0	PROPN
ejpam-4977	159	12	∗	∗	NOUN
ejpam-4977	159	13	y	y	NOUN
ejpam-4977	159	14	=	=	SYM
ejpam-4977	159	15	0	0	PUNCT
ejpam-4977	159	16	=	=	SYM
ejpam-4977	159	17	y	y	PROPN
ejpam-4977	159	18	∗	∗	X
ejpam-4977	159	19	y	y	PROPN
ejpam-4977	159	20	by	by	ADP
ejpam-4977	159	21	theorem	theorem	NOUN
ejpam-4977	159	22	1	1	NUM
ejpam-4977	159	23	(	(	PUNCT
ejpam-4977	159	24	vii	vii	PROPN
ejpam-4977	159	25	)	)	PUNCT
ejpam-4977	159	26	,	,	PUNCT
ejpam-4977	159	27	y	y	PROPN
ejpam-4977	159	28	=	=	SYM
ejpam-4977	159	29	0	0	NUM
ejpam-4977	159	30	∈	∈	PROPN
ejpam-4977	159	31	f.	f.	NOUN
ejpam-4977	159	32	case	case	NOUN
ejpam-4977	159	33	2	2	X
ejpam-4977	159	34	:	:	PUNCT
ejpam-4977	159	35	if	if	SCONJ
ejpam-4977	159	36	x	x	PROPN
ejpam-4977	159	37	∗	∗	VERB
ejpam-4977	159	38	y	y	NOUN
ejpam-4977	159	39	=	=	SYM
ejpam-4977	159	40	0	0	PUNCT
ejpam-4977	160	1	and	and	CCONJ
ejpam-4977	160	2	x	x	X
ejpam-4977	160	3	=	=	SYM
ejpam-4977	160	4	a	a	NOUN
ejpam-4977	160	5	,	,	PUNCT
ejpam-4977	160	6	then	then	ADV
ejpam-4977	160	7	x	x	X
ejpam-4977	160	8	∗	∗	NOUN
ejpam-4977	160	9	y	y	NOUN
ejpam-4977	160	10	=	=	PUNCT
ejpam-4977	160	11	a	a	DET
ejpam-4977	160	12	∗	∗	X
ejpam-4977	160	13	y	y	NOUN
ejpam-4977	160	14	=	=	SYM
ejpam-4977	160	15	0	0	PUNCT
ejpam-4977	161	1	=	=	SYM
ejpam-4977	161	2	y	y	PROPN
ejpam-4977	161	3	∗	∗	X
ejpam-4977	161	4	y	y	PROPN
ejpam-4977	161	5	by	by	ADP
ejpam-4977	161	6	theorem	theorem	ADJ
ejpam-4977	161	7	1	1	NUM
ejpam-4977	161	8	(	(	PUNCT
ejpam-4977	161	9	vi	vi	NOUN
ejpam-4977	161	10	)	)	PUNCT
ejpam-4977	161	11	,	,	PUNCT
ejpam-4977	161	12	y	y	PROPN
ejpam-4977	161	13	=	=	PUNCT
ejpam-4977	161	14	a	a	DET
ejpam-4977	161	15	∈	∈	PROPN
ejpam-4977	161	16	f.	f.	PROPN
ejpam-4977	161	17	case	case	NOUN
ejpam-4977	161	18	3	3	X
ejpam-4977	161	19	:	:	PUNCT
ejpam-4977	161	20	if	if	SCONJ
ejpam-4977	161	21	x	x	PROPN
ejpam-4977	161	22	∗	∗	VERB
ejpam-4977	161	23	y	y	NOUN
ejpam-4977	161	24	=	=	PUNCT
ejpam-4977	161	25	a	a	PROPN
ejpam-4977	161	26	and	and	CCONJ
ejpam-4977	161	27	x	x	SYM
ejpam-4977	161	28	=	=	SYM
ejpam-4977	161	29	0	0	NUM
ejpam-4977	161	30	,	,	PUNCT
ejpam-4977	161	31	then	then	ADV
ejpam-4977	161	32	x	x	X
ejpam-4977	161	33	∗	∗	NOUN
ejpam-4977	161	34	y	y	NOUN
ejpam-4977	161	35	=	=	SYM
ejpam-4977	161	36	0	0	PROPN
ejpam-4977	161	37	∗	∗	NOUN
ejpam-4977	161	38	y	y	NOUN
ejpam-4977	161	39	=	=	PUNCT
ejpam-4977	161	40	a	a	PRON
ejpam-4977	161	41	by	by	ADP
ejpam-4977	161	42	assumption	assumption	NOUN
ejpam-4977	161	43	a	a	DET
ejpam-4977	161	44	=	=	SYM
ejpam-4977	161	45	0	0	NUM
ejpam-4977	161	46	∗	∗	NOUN
ejpam-4977	161	47	a	a	PRON
ejpam-4977	161	48	and	and	CCONJ
ejpam-4977	161	49	theorem	theorem	ADJ
ejpam-4977	161	50	1	1	NUM
ejpam-4977	161	51	(	(	PUNCT
ejpam-4977	161	52	viii	viii	NOUN
ejpam-4977	161	53	)	)	PUNCT
ejpam-4977	161	54	,	,	PUNCT
ejpam-4977	161	55	y	y	PROPN
ejpam-4977	161	56	=	=	PUNCT
ejpam-4977	161	57	a	a	DET
ejpam-4977	161	58	∈	∈	PROPN
ejpam-4977	161	59	f.	f.	PROPN
ejpam-4977	161	60	case	case	NOUN
ejpam-4977	161	61	4	4	NUM
ejpam-4977	161	62	:	:	PUNCT
ejpam-4977	161	63	if	if	SCONJ
ejpam-4977	161	64	x	x	PROPN
ejpam-4977	161	65	∗	∗	VERB
ejpam-4977	161	66	y	y	NOUN
ejpam-4977	161	67	=	=	PUNCT
ejpam-4977	161	68	a	a	PROPN
ejpam-4977	161	69	and	and	CCONJ
ejpam-4977	161	70	x	x	SYM
ejpam-4977	161	71	=	=	SYM
ejpam-4977	161	72	a	a	NOUN
ejpam-4977	161	73	,	,	PUNCT
ejpam-4977	161	74	then	then	ADV
ejpam-4977	161	75	x	x	X
ejpam-4977	161	76	∗	∗	NOUN
ejpam-4977	161	77	y	y	NOUN
ejpam-4977	161	78	=	=	PUNCT
ejpam-4977	161	79	a	a	DET
ejpam-4977	161	80	∗	∗	X
ejpam-4977	161	81	y	y	PROPN
ejpam-4977	161	82	=	=	PUNCT
ejpam-4977	161	83	a	a	PRON
ejpam-4977	161	84	by	by	ADP
ejpam-4977	161	85	theorem	theorem	NOUN
ejpam-4977	161	86	1	1	NUM
ejpam-4977	161	87	,	,	PUNCT
ejpam-4977	161	88	a	a	DET
ejpam-4977	161	89	=	=	X
ejpam-4977	161	90	a	a	DET
ejpam-4977	161	91	∗	∗	X
ejpam-4977	161	92	y	y	NOUN
ejpam-4977	161	93	=	=	SYM
ejpam-4977	161	94	0	0	NUM
ejpam-4977	161	95	∗	∗	NOUN
ejpam-4977	161	96	(	(	PUNCT
ejpam-4977	161	97	y	y	PROPN
ejpam-4977	161	98	∗	∗	X
ejpam-4977	161	99	a	a	X
ejpam-4977	161	100	)	)	PUNCT
ejpam-4977	161	101	=	=	SYM
ejpam-4977	161	102	0	0	NUM
ejpam-4977	161	103	∗	∗	NOUN
ejpam-4977	161	104	(	(	PUNCT
ejpam-4977	161	105	0	0	NUM
ejpam-4977	161	106	∗	∗	NOUN
ejpam-4977	161	107	a	a	NOUN
ejpam-4977	161	108	)	)	PUNCT
ejpam-4977	161	109	=	=	SYM
ejpam-4977	162	1	a	a	PRON
ejpam-4977	162	2	implies	imply	VERB
ejpam-4977	162	3	that	that	SCONJ
ejpam-4977	162	4	(	(	PUNCT
ejpam-4977	162	5	y	y	NOUN
ejpam-4977	162	6	∗	∗	X
ejpam-4977	162	7	a	a	X
ejpam-4977	162	8	)	)	PUNCT
ejpam-4977	162	9	=	=	SYM
ejpam-4977	162	10	(	(	PUNCT
ejpam-4977	162	11	0	0	NUM
ejpam-4977	162	12	∗	∗	NOUN
ejpam-4977	162	13	a	a	NOUN
ejpam-4977	162	14	)	)	PUNCT
ejpam-4977	162	15	and	and	CCONJ
ejpam-4977	162	16	hence	hence	ADV
ejpam-4977	162	17	y	y	NOUN
ejpam-4977	162	18	=	=	SYM
ejpam-4977	162	19	0	0	PUNCT
ejpam-4977	163	1	∈	∈	PROPN
ejpam-4977	163	2	f	f	PROPN
ejpam-4977	163	3	.	.	PUNCT
ejpam-4977	164	1	therefore	therefore	ADV
ejpam-4977	164	2	,	,	PUNCT
ejpam-4977	164	3	⟨a⟩b	⟨a⟩b	PROPN
ejpam-4977	164	4	=	=	SYM
ejpam-4977	164	5	f	f	PROPN
ejpam-4977	164	6	=	=	PUNCT
ejpam-4977	164	7	{	{	PUNCT
ejpam-4977	164	8	0	0	NUM
ejpam-4977	164	9	,	,	PUNCT
ejpam-4977	164	10	a	a	PRON
ejpam-4977	164	11	}	}	PUNCT
ejpam-4977	164	12	is	be	AUX
ejpam-4977	164	13	a	a	DET
ejpam-4977	164	14	b	b	NOUN
ejpam-4977	164	15	-	-	NOUN
ejpam-4977	164	16	filter	filter	NOUN
ejpam-4977	164	17	of	of	ADP
ejpam-4977	164	18	x.	x.	NOUN
ejpam-4977	164	19	5	5	NUM
ejpam-4977	164	20	.	.	PUNCT
ejpam-4977	164	21	conclusion	conclusion	NOUN
ejpam-4977	164	22	in	in	ADP
ejpam-4977	164	23	this	this	DET
ejpam-4977	164	24	paper	paper	NOUN
ejpam-4977	164	25	shown	show	VERB
ejpam-4977	164	26	some	some	DET
ejpam-4977	164	27	properties	property	NOUN
ejpam-4977	164	28	of	of	ADP
ejpam-4977	164	29	exponents	exponent	NOUN
ejpam-4977	164	30	on	on	ADP
ejpam-4977	164	31	b	b	NOUN
ejpam-4977	164	32	-	-	PUNCT
ejpam-4977	164	33	algebra	algebra	NOUN
ejpam-4977	164	34	and	and	CCONJ
ejpam-4977	164	35	we	we	PRON
ejpam-4977	164	36	introduces	introduce	VERB
ejpam-4977	164	37	the	the	DET
ejpam-4977	164	38	notion	notion	NOUN
ejpam-4977	164	39	of	of	ADP
ejpam-4977	164	40	b	b	NOUN
ejpam-4977	164	41	-	-	NOUN
ejpam-4977	164	42	filter	filter	NOUN
ejpam-4977	164	43	on	on	ADP
ejpam-4977	164	44	a	a	DET
ejpam-4977	164	45	b	b	NOUN
ejpam-4977	164	46	-	-	PUNCT
ejpam-4977	164	47	algebra	algebra	NOUN
ejpam-4977	164	48	and	and	CCONJ
ejpam-4977	164	49	presented	present	VERB
ejpam-4977	164	50	together	together	ADV
ejpam-4977	164	51	with	with	ADP
ejpam-4977	164	52	some	some	PRON
ejpam-4977	164	53	of	of	ADP
ejpam-4977	164	54	its	its	PRON
ejpam-4977	164	55	properties	property	NOUN
ejpam-4977	164	56	on	on	ADP
ejpam-4977	164	57	a	a	DET
ejpam-4977	164	58	cyclic	cyclic	ADJ
ejpam-4977	164	59	b	b	NOUN
ejpam-4977	164	60	-	-	PUNCT
ejpam-4977	164	61	algebra	algebra	NOUN
ejpam-4977	164	62	,	,	PUNCT
ejpam-4977	164	63	that	that	PRON
ejpam-4977	164	64	is	be	AUX
ejpam-4977	164	65	for	for	ADP
ejpam-4977	164	66	any	any	DET
ejpam-4977	164	67	an	an	DET
ejpam-4977	164	68	element	element	NOUN
ejpam-4977	164	69	a	a	PRON
ejpam-4977	164	70	in	in	ADP
ejpam-4977	164	71	a	a	DET
ejpam-4977	164	72	b	b	NOUN
ejpam-4977	164	73	-	-	PUNCT
ejpam-4977	164	74	algebra	algebra	NOUN
ejpam-4977	164	75	(	(	PUNCT
ejpam-4977	164	76	x	x	X
ejpam-4977	164	77	,	,	PUNCT
ejpam-4977	164	78	∗	∗	NOUN
ejpam-4977	164	79	,	,	PUNCT
ejpam-4977	164	80	0	0	NUM
ejpam-4977	164	81	)	)	PUNCT
ejpam-4977	164	82	,	,	PUNCT
ejpam-4977	164	83	we	we	PRON
ejpam-4977	164	84	show	show	VERB
ejpam-4977	164	85	that	that	SCONJ
ejpam-4977	164	86	the	the	DET
ejpam-4977	164	87	set	set	NOUN
ejpam-4977	164	88	⟨a⟩b	⟨a⟩b	X
ejpam-4977	164	89	=	=	SYM
ejpam-4977	164	90	{	{	PUNCT
ejpam-4977	164	91	ak	ak	PROPN
ejpam-4977	164	92	:	:	PUNCT
ejpam-4977	164	93	k	k	PROPN
ejpam-4977	164	94	∈	∈	PROPN
ejpam-4977	164	95	z	z	AUX
ejpam-4977	164	96	}	}	PUNCT
ejpam-4977	164	97	form	form	VERB
ejpam-4977	164	98	a	a	DET
ejpam-4977	164	99	b	b	NOUN
ejpam-4977	164	100	-	-	PUNCT
ejpam-4977	164	101	ideal	ideal	ADJ
ejpam-4977	164	102	and	and	CCONJ
ejpam-4977	164	103	b	b	NOUN
ejpam-4977	164	104	-	-	NOUN
ejpam-4977	164	105	filter	filter	NOUN
ejpam-4977	164	106	of	of	ADP
ejpam-4977	164	107	x.	x.	NOUN
ejpam-4977	164	108	moreover	moreover	ADV
ejpam-4977	164	109	,	,	PUNCT
ejpam-4977	164	110	if	if	SCONJ
ejpam-4977	164	111	0	0	NUM
ejpam-4977	164	112	∗	∗	NOUN
ejpam-4977	164	113	a	a	DET
ejpam-4977	164	114	=	=	NOUN
ejpam-4977	164	115	a	a	NOUN
ejpam-4977	164	116	,	,	PUNCT
ejpam-4977	164	117	we	we	PRON
ejpam-4977	164	118	obtain	obtain	VERB
ejpam-4977	164	119	⟨a⟩b	⟨a⟩b	NOUN
ejpam-4977	164	120	=	=	PUNCT
ejpam-4977	164	121	{	{	PUNCT
ejpam-4977	164	122	0	0	NUM
ejpam-4977	164	123	,	,	PUNCT
ejpam-4977	164	124	a	a	DET
ejpam-4977	164	125	}	}	PUNCT
ejpam-4977	164	126	form	form	NOUN
ejpam-4977	164	127	a	a	DET
ejpam-4977	164	128	b	b	NOUN
ejpam-4977	164	129	-	-	NOUN
ejpam-4977	164	130	filter	filter	NOUN
ejpam-4977	164	131	of	of	ADP
ejpam-4977	164	132	x.	x.	NOUN
ejpam-4977	164	133	acknowledgements	acknowledgement	NOUN
ejpam-4977	164	134	this	this	DET
ejpam-4977	164	135	research	research	NOUN
ejpam-4977	164	136	project	project	NOUN
ejpam-4977	164	137	was	be	AUX
ejpam-4977	164	138	financially	financially	ADV
ejpam-4977	164	139	supported	support	VERB
ejpam-4977	164	140	by	by	ADP
ejpam-4977	164	141	thailand	thailand	PROPN
ejpam-4977	164	142	science	science	PROPN
ejpam-4977	164	143	research	research	PROPN
ejpam-4977	164	144	and	and	CCONJ
ejpam-4977	164	145	innovation	innovation	NOUN
ejpam-4977	164	146	(	(	PUNCT
ejpam-4977	164	147	tsri	tsri	PROPN
ejpam-4977	164	148	)	)	PUNCT
ejpam-4977	164	149	.	.	PUNCT
ejpam-4977	165	1	references	reference	NOUN
ejpam-4977	165	2	123	123	NUM
ejpam-4977	165	3	references	reference	NOUN
ejpam-4977	165	4	[	[	X
ejpam-4977	165	5	1	1	NUM
ejpam-4977	165	6	]	]	PUNCT
ejpam-4977	165	7	j.	j.	PROPN
ejpam-4977	165	8	neggers	neggers	PROPN
ejpam-4977	165	9	,	,	PUNCT
ejpam-4977	165	10	and	and	CCONJ
ejpam-4977	165	11	h.	h.	PROPN
ejpam-4977	165	12	s.	s.	PROPN
ejpam-4977	165	13	kim	kim	PROPN
ejpam-4977	165	14	.	.	PUNCT
ejpam-4977	166	1	on	on	ADP
ejpam-4977	166	2	b	b	NOUN
ejpam-4977	166	3	-	-	PUNCT
ejpam-4977	166	4	algebras	algebras	X
ejpam-4977	166	5	.	.	PUNCT
ejpam-4977	167	1	matematički	matematički	PROPN
ejpam-4977	167	2	vesnik	vesnik	PROPN
ejpam-4977	167	3	,	,	PUNCT
ejpam-4977	167	4	54:21–29	54:21–29	NUM
ejpam-4977	167	5	,	,	PUNCT
ejpam-4977	167	6	2002	2002	NUM
ejpam-4977	167	7	.	.	PUNCT
ejpam-4977	168	1	[	[	X
ejpam-4977	168	2	2	2	NUM
ejpam-4977	168	3	]	]	X
ejpam-4977	168	4	n.	n.	PROPN
ejpam-4977	168	5	c.	c.	PROPN
ejpam-4977	168	6	gonzaga	gonzaga	PROPN
ejpam-4977	168	7	and	and	CCONJ
ejpam-4977	168	8	j	j	PROPN
ejpam-4977	168	9	p.	p.	PROPN
ejpam-4977	168	10	vilela	vilela	PROPN
ejpam-4977	168	11	.	.	PUNCT
ejpam-4977	169	1	on	on	ADP
ejpam-4977	169	2	cyclic	cyclic	PROPN
ejpam-4977	169	3	b	b	NOUN
ejpam-4977	169	4	-	-	PUNCT
ejpam-4977	169	5	algebras	algebras	PROPN
ejpam-4977	169	6	.	.	PUNCT
ejpam-4977	170	1	applied	apply	VERB
ejpam-4977	170	2	mathematical	mathematical	ADJ
ejpam-4977	170	3	sciences	science	NOUN
ejpam-4977	170	4	,	,	PUNCT
ejpam-4977	170	5	9:5507–5522	9:5507–5522	NUM
ejpam-4977	170	6	,	,	PUNCT
ejpam-4977	170	7	2015	2015	NUM
ejpam-4977	170	8	.	.	PUNCT
ejpam-4977	171	1	[	[	X
ejpam-4977	171	2	3	3	X
ejpam-4977	171	3	]	]	X
ejpam-4977	171	4	y.	y.	PROPN
ejpam-4977	171	5	b.	b.	PROPN
ejpam-4977	171	6	jun	jun	PROPN
ejpam-4977	171	7	,	,	PUNCT
ejpam-4977	171	8	e.	e.	PROPN
ejpam-4977	171	9	h.	h.	PROPN
ejpam-4977	171	10	roh	roh	PROPN
ejpam-4977	171	11	,	,	PUNCT
ejpam-4977	171	12	chinju	chinju	NOUN
ejpam-4977	171	13	and	and	CCONJ
ejpam-4977	171	14	h.	h.	PROPN
ejpam-4977	171	15	s.	s.	PROPN
ejpam-4977	171	16	kim	kim	PROPN
ejpam-4977	171	17	.	.	PUNCT
ejpam-4977	172	1	on	on	ADP
ejpam-4977	172	2	fuzzy	fuzzy	ADJ
ejpam-4977	172	3	b	b	NOUN
ejpam-4977	172	4	-	-	PUNCT
ejpam-4977	172	5	algebras	algebras	X
ejpam-4977	172	6	.	.	PUNCT
ejpam-4977	173	1	czechoslovak	czechoslovak	PROPN
ejpam-4977	173	2	mathematical	mathematical	PROPN
ejpam-4977	173	3	journal	journal	PROPN
ejpam-4977	173	4	,	,	PUNCT
ejpam-4977	173	5	52(127):375–384	52(127):375–384	PROPN
ejpam-4977	173	6	,	,	PUNCT
ejpam-4977	173	7	2002	2002	NUM
ejpam-4977	173	8	.	.	PUNCT
ejpam-4977	174	1	[	[	X
ejpam-4977	174	2	4	4	X
ejpam-4977	174	3	]	]	PUNCT
ejpam-4977	174	4	j.	j.	PROPN
ejpam-4977	174	5	neggers	neggers	PROPN
ejpam-4977	174	6	and	and	CCONJ
ejpam-4977	174	7	h.	h.	PROPN
ejpam-4977	174	8	s.	s.	PROPN
ejpam-4977	174	9	kim	kim	PROPN
ejpam-4977	174	10	.	.	PUNCT
ejpam-4977	175	1	a	a	DET
ejpam-4977	175	2	fundamental	fundamental	ADJ
ejpam-4977	175	3	theorem	theorem	NOUN
ejpam-4977	175	4	of	of	ADP
ejpam-4977	175	5	b	b	NOUN
ejpam-4977	175	6	-	-	PUNCT
ejpam-4977	175	7	homomorphism	homomorphism	NOUN
ejpam-4977	175	8	for	for	ADP
ejpam-4977	175	9	balgebras	balgebras	PROPN
ejpam-4977	175	10	.	.	PROPN
ejpam-4977	175	11	international	international	ADJ
ejpam-4977	175	12	mathematics	mathematics	PROPN
ejpam-4977	175	13	journal	journal	NOUN
ejpam-4977	175	14	,	,	PUNCT
ejpam-4977	175	15	2:207–214	2:207–214	NUM
ejpam-4977	175	16	,	,	PUNCT
ejpam-4977	175	17	2002	2002	NUM
ejpam-4977	175	18	.	.	PUNCT
ejpam-4977	176	1	[	[	X
ejpam-4977	176	2	5	5	X
ejpam-4977	176	3	]	]	PUNCT
ejpam-4977	176	4	d.	d.	PROPN
ejpam-4977	176	5	al	al	PROPN
ejpam-4977	176	6	-	-	PUNCT
ejpam-4977	176	7	kadi	kadi	PROPN
ejpam-4977	176	8	.	.	PUNCT
ejpam-4977	177	1	anti	anti	ADJ
ejpam-4977	177	2	fuzzy	fuzzy	ADJ
ejpam-4977	177	3	ideals	ideal	NOUN
ejpam-4977	177	4	of	of	ADP
ejpam-4977	177	5	b	b	NOUN
ejpam-4977	177	6	-	-	PUNCT
ejpam-4977	177	7	algebra	algebra	NOUN
ejpam-4977	177	8	.	.	PUNCT
ejpam-4977	178	1	international	international	ADJ
ejpam-4977	178	2	journal	journal	NOUN
ejpam-4977	178	3	of	of	ADP
ejpam-4977	178	4	pure	pure	ADJ
ejpam-4977	178	5	and	and	CCONJ
ejpam-4977	178	6	applied	applied	ADJ
ejpam-4977	178	7	mathematics	mathematic	NOUN
ejpam-4977	178	8	,	,	PUNCT
ejpam-4977	178	9	117(3):437–445	117(3):437–445	NUM
ejpam-4977	178	10	,	,	PUNCT
ejpam-4977	178	11	2017	2017	NUM
ejpam-4977	178	12	.	.	PUNCT
ejpam-4977	179	1	[	[	X
ejpam-4977	179	2	6	6	NUM
ejpam-4977	179	3	]	]	PUNCT
ejpam-4977	179	4	k.	k.	PROPN
ejpam-4977	179	5	e.	e.	PROPN
ejpam-4977	179	6	belleza	belleza	PROPN
ejpam-4977	179	7	and	and	CCONJ
ejpam-4977	179	8	j.	j.	PROPN
ejpam-4977	179	9	p.	p.	PROPN
ejpam-4977	179	10	vilela	vilela	PROPN
ejpam-4977	179	11	.	.	PUNCT
ejpam-4977	180	1	the	the	DET
ejpam-4977	180	2	dual	dual	ADJ
ejpam-4977	180	3	b	b	NOUN
ejpam-4977	180	4	-	-	PUNCT
ejpam-4977	180	5	algebra	algebra	NOUN
ejpam-4977	180	6	.	.	PUNCT
ejpam-4977	181	1	european	european	ADJ
ejpam-4977	181	2	journal	journal	PROPN
ejpam-4977	181	3	of	of	ADP
ejpam-4977	181	4	pure	pure	ADJ
ejpam-4977	181	5	and	and	CCONJ
ejpam-4977	181	6	applied	applied	ADJ
ejpam-4977	181	7	mathematics	mathematic	NOUN
ejpam-4977	181	8	,	,	PUNCT
ejpam-4977	181	9	12(4):1497–1507	12(4):1497–1507	NUM
ejpam-4977	181	10	,	,	PUNCT
ejpam-4977	181	11	2019	2019	NUM
ejpam-4977	181	12	.	.	PUNCT
ejpam-4977	182	1	[	[	X
ejpam-4977	182	2	7	7	X
ejpam-4977	182	3	]	]	PUNCT
ejpam-4977	182	4	k.	k.	PROPN
ejpam-4977	182	5	e.	e.	PROPN
ejpam-4977	182	6	belleza	belleza	PROPN
ejpam-4977	182	7	and	and	CCONJ
ejpam-4977	182	8	j.	j.	PROPN
ejpam-4977	182	9	p.	p.	PROPN
ejpam-4977	182	10	vilela	vilela	PROPN
ejpam-4977	182	11	.	.	PUNCT
ejpam-4977	183	1	on	on	ADP
ejpam-4977	183	2	b	b	NOUN
ejpam-4977	183	3	-	-	PUNCT
ejpam-4977	183	4	ideals	ideal	NOUN
ejpam-4977	183	5	in	in	ADP
ejpam-4977	183	6	a	a	DET
ejpam-4977	183	7	topological	topological	ADJ
ejpam-4977	183	8	b	b	NOUN
ejpam-4977	183	9	-	-	PUNCT
ejpam-4977	183	10	algebra	algebra	NOUN
ejpam-4977	183	11	and	and	CCONJ
ejpam-4977	183	12	the	the	DET
ejpam-4977	183	13	uniform	uniform	ADJ
ejpam-4977	183	14	b	b	X
ejpam-4977	183	15	-	-	PUNCT
ejpam-4977	183	16	topological	topological	ADJ
ejpam-4977	183	17	space	space	NOUN
ejpam-4977	183	18	.	.	PUNCT
ejpam-4977	184	1	european	european	ADJ
ejpam-4977	184	2	journal	journal	PROPN
ejpam-4977	184	3	of	of	ADP
ejpam-4977	184	4	pure	pure	ADJ
ejpam-4977	184	5	and	and	CCONJ
ejpam-4977	184	6	applied	applied	ADJ
ejpam-4977	184	7	mathematics	mathematic	NOUN
ejpam-4977	184	8	,	,	PUNCT
ejpam-4977	184	9	13(4):830	13(4):830	NUM
ejpam-4977	184	10	–	–	PUNCT
ejpam-4977	184	11	839	839	NUM
ejpam-4977	184	12	,	,	PUNCT
ejpam-4977	184	13	2020	2020	NUM
ejpam-4977	184	14	.	.	PUNCT
ejpam-4977	185	1	[	[	X
ejpam-4977	185	2	8	8	X
ejpam-4977	185	3	]	]	PUNCT
ejpam-4977	185	4	k.	k.	PROPN
ejpam-4977	185	5	e.	e.	PROPN
ejpam-4977	185	6	belleza	belleza	PROPN
ejpam-4977	185	7	and	and	CCONJ
ejpam-4977	185	8	j.	j.	PROPN
ejpam-4977	185	9	r.	r.	PROPN
ejpam-4977	185	10	albaracin	albaracin	PROPN
ejpam-4977	185	11	.	.	PUNCT
ejpam-4977	186	1	on	on	ADP
ejpam-4977	186	2	dual	dual	ADJ
ejpam-4977	186	3	b	b	NOUN
ejpam-4977	186	4	-	-	PUNCT
ejpam-4977	186	5	filters	filter	NOUN
ejpam-4977	186	6	and	and	CCONJ
ejpam-4977	186	7	dual	dual	ADJ
ejpam-4977	186	8	b	b	NOUN
ejpam-4977	186	9	-	-	PUNCT
ejpam-4977	186	10	subalgebras	subalgebras	PROPN
ejpam-4977	186	11	in	in	ADP
ejpam-4977	186	12	a	a	DET
ejpam-4977	186	13	topological	topological	ADJ
ejpam-4977	186	14	dual	dual	ADJ
ejpam-4977	186	15	b	b	NOUN
ejpam-4977	186	16	-	-	PUNCT
ejpam-4977	186	17	algebra	algebra	NOUN
ejpam-4977	186	18	.	.	PUNCT
ejpam-4977	187	1	journal	journal	NOUN
ejpam-4977	187	2	of	of	ADP
ejpam-4977	187	3	mathematics	mathematic	NOUN
ejpam-4977	187	4	and	and	CCONJ
ejpam-4977	187	5	computer	computer	NOUN
ejpam-4977	187	6	science	science	NOUN
ejpam-4977	187	7	,	,	PUNCT
ejpam-4977	187	8	28:1–10	28:1–10	NUM
ejpam-4977	187	9	,	,	PUNCT
ejpam-4977	187	10	2023	2023	NUM
ejpam-4977	187	11	.	.	PUNCT