id	sid	tid	token	lemma	pos
ejpam-5403	1	1	european	european	PROPN
ejpam-5403	1	2	journal	journal	PROPN
ejpam-5403	1	3	of	of	ADP
ejpam-5403	1	4	pure	pure	ADJ
ejpam-5403	1	5	and	and	CCONJ
ejpam-5403	1	6	applied	apply	VERB
ejpam-5403	1	7	mathematics	mathematic	NOUN
ejpam-5403	1	8	vol	vol	NOUN
ejpam-5403	1	9	.	.	PROPN
ejpam-5403	2	1	17	17	NUM
ejpam-5403	2	2	,	,	PUNCT
ejpam-5403	2	3	no	no	INTJ
ejpam-5403	2	4	.	.	NOUN
ejpam-5403	2	5	4	4	NUM
ejpam-5403	2	6	,	,	PUNCT
ejpam-5403	2	7	2024	2024	NUM
ejpam-5403	2	8	,	,	PUNCT
ejpam-5403	2	9	3268	3268	NUM
ejpam-5403	2	10	-	-	SYM
ejpam-5403	2	11	3276	3276	NUM
ejpam-5403	2	12	issn	issn	PROPN
ejpam-5403	2	13	1307	1307	NUM
ejpam-5403	2	14	-	-	SYM
ejpam-5403	2	15	5543	5543	NUM
ejpam-5403	2	16	–	–	PUNCT
ejpam-5403	2	17	ejpam.com	ejpam.com	X
ejpam-5403	2	18	published	publish	VERB
ejpam-5403	2	19	by	by	ADP
ejpam-5403	2	20	new	new	PROPN
ejpam-5403	2	21	york	york	PROPN
ejpam-5403	2	22	business	business	PROPN
ejpam-5403	2	23	global	global	ADJ
ejpam-5403	2	24	relations	relation	NOUN
ejpam-5403	2	25	between	between	ADP
ejpam-5403	2	26	g	g	NOUN
ejpam-5403	2	27	-	-	PUNCT
ejpam-5403	2	28	part	part	NOUN
ejpam-5403	2	29	and	and	CCONJ
ejpam-5403	2	30	atoms	atom	NOUN
ejpam-5403	2	31	in	in	ADP
ejpam-5403	2	32	q	q	ADJ
ejpam-5403	2	33	-	-	PUNCT
ejpam-5403	2	34	algebras	algebras	ADJ
ejpam-5403	2	35	ananya	ananya	PROPN
ejpam-5403	2	36	anantayasethi1,∗	anantayasethi1,∗	NOUN
ejpam-5403	2	37	,	,	PUNCT
ejpam-5403	2	38	tanabat	tanabat	PROPN
ejpam-5403	2	39	kunawat1	kunawat1	PROPN
ejpam-5403	2	40	,	,	PUNCT
ejpam-5403	2	41	panuwat	panuwat	VERB
ejpam-5403	2	42	moonnipa1	moonnipa1	NOUN
ejpam-5403	2	43	1	1	NUM
ejpam-5403	2	44	department	department	NOUN
ejpam-5403	2	45	of	of	ADP
ejpam-5403	2	46	mathematics	mathematic	NOUN
ejpam-5403	2	47	,	,	PUNCT
ejpam-5403	2	48	science	science	NOUN
ejpam-5403	2	49	faculty	faculty	NOUN
ejpam-5403	2	50	,	,	PUNCT
ejpam-5403	2	51	mahasarakham	mahasarakham	PROPN
ejpam-5403	2	52	university	university	PROPN
ejpam-5403	2	53	,	,	PUNCT
ejpam-5403	2	54	mahasarakham	mahasarakham	PROPN
ejpam-5403	2	55	,	,	PUNCT
ejpam-5403	2	56	44150	44150	NUM
ejpam-5403	2	57	,	,	PUNCT
ejpam-5403	2	58	thailand	thailand	PROPN
ejpam-5403	2	59	abstract	abstract	NOUN
ejpam-5403	2	60	.	.	PUNCT
ejpam-5403	3	1	in	in	ADP
ejpam-5403	3	2	this	this	DET
ejpam-5403	3	3	work	work	NOUN
ejpam-5403	3	4	the	the	DET
ejpam-5403	3	5	concepts	concept	NOUN
ejpam-5403	3	6	of	of	ADP
ejpam-5403	3	7	g	g	NOUN
ejpam-5403	3	8	-	-	PUNCT
ejpam-5403	3	9	part	part	NOUN
ejpam-5403	3	10	g(x	g(x	NOUN
ejpam-5403	3	11	)	)	PUNCT
ejpam-5403	3	12	,	,	PUNCT
ejpam-5403	3	13	atoms	atom	NOUN
ejpam-5403	3	14	and	and	CCONJ
ejpam-5403	3	15	strong	strong	ADJ
ejpam-5403	3	16	atoms	atom	NOUN
ejpam-5403	3	17	in	in	ADP
ejpam-5403	3	18	q	q	NOUN
ejpam-5403	3	19	-	-	PUNCT
ejpam-5403	3	20	algebras	algebra	NOUN
ejpam-5403	3	21	are	be	AUX
ejpam-5403	3	22	discussed	discuss	VERB
ejpam-5403	3	23	.	.	PUNCT
ejpam-5403	4	1	we	we	PRON
ejpam-5403	4	2	provide	provide	VERB
ejpam-5403	4	3	some	some	DET
ejpam-5403	4	4	connections	connection	NOUN
ejpam-5403	4	5	among	among	ADP
ejpam-5403	4	6	g(x	g(x	NOUN
ejpam-5403	4	7	)	)	PUNCT
ejpam-5403	4	8	,	,	PUNCT
ejpam-5403	4	9	set	set	VERB
ejpam-5403	4	10	of	of	ADP
ejpam-5403	4	11	all	all	DET
ejpam-5403	4	12	atoms	atom	NOUN
ejpam-5403	4	13	and	and	CCONJ
ejpam-5403	4	14	set	set	NOUN
ejpam-5403	4	15	of	of	ADP
ejpam-5403	4	16	all	all	DET
ejpam-5403	4	17	strong	strong	ADJ
ejpam-5403	4	18	atoms	atom	NOUN
ejpam-5403	4	19	of	of	ADP
ejpam-5403	4	20	x	x	PUNCT
ejpam-5403	4	21	which	which	PRON
ejpam-5403	4	22	related	relate	VERB
ejpam-5403	4	23	to	to	ADP
ejpam-5403	4	24	the	the	DET
ejpam-5403	4	25	concept	concept	NOUN
ejpam-5403	4	26	of	of	ADP
ejpam-5403	4	27	ideals	ideal	NOUN
ejpam-5403	4	28	.	.	PUNCT
ejpam-5403	5	1	we	we	PRON
ejpam-5403	5	2	prove	prove	VERB
ejpam-5403	5	3	that	that	SCONJ
ejpam-5403	5	4	a	a	DET
ejpam-5403	5	5	q	q	NOUN
ejpam-5403	5	6	-	-	NOUN
ejpam-5403	5	7	algebra	algebra	NOUN
ejpam-5403	5	8	x	x	AUX
ejpam-5403	5	9	does	do	AUX
ejpam-5403	5	10	not	not	PART
ejpam-5403	5	11	contain	contain	VERB
ejpam-5403	5	12	a	a	DET
ejpam-5403	5	13	strong	strong	ADJ
ejpam-5403	5	14	atom	atom	NOUN
ejpam-5403	5	15	whenever	whenever	SCONJ
ejpam-5403	5	16	it	it	PRON
ejpam-5403	5	17	contains	contain	VERB
ejpam-5403	5	18	a	a	DET
ejpam-5403	5	19	non	non	ADJ
ejpam-5403	5	20	-	-	ADJ
ejpam-5403	5	21	zero	zero	NUM
ejpam-5403	5	22	ideal	ideal	NOUN
ejpam-5403	5	23	g(x	g(x	NOUN
ejpam-5403	5	24	)	)	PUNCT
ejpam-5403	5	25	.	.	PUNCT
ejpam-5403	6	1	in	in	ADP
ejpam-5403	6	2	addition	addition	NOUN
ejpam-5403	6	3	,	,	PUNCT
ejpam-5403	6	4	we	we	PRON
ejpam-5403	6	5	provide	provide	VERB
ejpam-5403	6	6	some	some	DET
ejpam-5403	6	7	conditions	condition	NOUN
ejpam-5403	6	8	that	that	PRON
ejpam-5403	6	9	make	make	VERB
ejpam-5403	6	10	a	a	DET
ejpam-5403	6	11	set	set	NOUN
ejpam-5403	6	12	of	of	ADP
ejpam-5403	6	13	atoms	atom	NOUN
ejpam-5403	6	14	an	an	DET
ejpam-5403	6	15	abelian	abelian	ADJ
ejpam-5403	6	16	group	group	NOUN
ejpam-5403	6	17	.	.	PUNCT
ejpam-5403	7	1	2020	2020	NUM
ejpam-5403	7	2	mathematics	mathematics	PROPN
ejpam-5403	7	3	subject	subject	NOUN
ejpam-5403	7	4	classifications	classification	NOUN
ejpam-5403	7	5	:	:	PUNCT
ejpam-5403	7	6	03g25	03g25	NUM
ejpam-5403	7	7	,	,	PUNCT
ejpam-5403	7	8	03g27	03g27	NOUN
ejpam-5403	7	9	,	,	PUNCT
ejpam-5403	7	10	06f35	06f35	NUM
ejpam-5403	7	11	,	,	PUNCT
ejpam-5403	7	12	20k01	20k01	NUM
ejpam-5403	7	13	key	key	ADJ
ejpam-5403	7	14	words	word	NOUN
ejpam-5403	7	15	and	and	CCONJ
ejpam-5403	7	16	phrases	phrase	NOUN
ejpam-5403	7	17	:	:	PUNCT
ejpam-5403	7	18	q	q	X
ejpam-5403	7	19	-	-	PUNCT
ejpam-5403	7	20	algebra	algebra	NOUN
ejpam-5403	7	21	,	,	PUNCT
ejpam-5403	7	22	ideals	ideal	NOUN
ejpam-5403	7	23	,	,	PUNCT
ejpam-5403	7	24	g	g	NOUN
ejpam-5403	7	25	-	-	PUNCT
ejpam-5403	7	26	part	part	NOUN
ejpam-5403	7	27	,	,	PUNCT
ejpam-5403	7	28	g(x	g(x	NOUN
ejpam-5403	7	29	)	)	PUNCT
ejpam-5403	7	30	,	,	PUNCT
ejpam-5403	7	31	a(x	a(x	PROPN
ejpam-5403	7	32	)	)	PUNCT
ejpam-5403	7	33	,	,	PUNCT
ejpam-5403	7	34	atoms	atom	NOUN
ejpam-5403	7	35	,	,	PUNCT
ejpam-5403	7	36	strong	strong	ADJ
ejpam-5403	7	37	atom	atom	NOUN
ejpam-5403	7	38	,	,	PUNCT
ejpam-5403	7	39	abelian	abelian	PROPN
ejpam-5403	7	40	group	group	PROPN
ejpam-5403	7	41	1	1	NUM
ejpam-5403	7	42	.	.	PUNCT
ejpam-5403	7	43	introduction	introduction	NOUN
ejpam-5403	7	44	and	and	CCONJ
ejpam-5403	7	45	preliminaries	preliminary	NOUN
ejpam-5403	7	46	in	in	ADP
ejpam-5403	7	47	1996	1996	NUM
ejpam-5403	7	48	,	,	PUNCT
ejpam-5403	7	49	two	two	NUM
ejpam-5403	7	50	japanese	japanese	ADJ
ejpam-5403	7	51	mathematicians	mathematician	NOUN
ejpam-5403	7	52	y.	y.	PROPN
ejpam-5403	7	53	imai	imai	PROPN
ejpam-5403	7	54	and	and	CCONJ
ejpam-5403	7	55	k.	k.	PROPN
ejpam-5403	7	56	iseki	iseki	PROPN
ejpam-5403	8	1	[	[	X
ejpam-5403	8	2	6	6	NUM
ejpam-5403	8	3	]	]	PUNCT
ejpam-5403	8	4	introduced	introduce	VERB
ejpam-5403	8	5	a	a	DET
ejpam-5403	8	6	class	class	NOUN
ejpam-5403	8	7	of	of	ADP
ejpam-5403	8	8	logical	logical	ADJ
ejpam-5403	8	9	algebra	algebra	NOUN
ejpam-5403	8	10	which	which	PRON
ejpam-5403	8	11	is	be	AUX
ejpam-5403	8	12	called	call	VERB
ejpam-5403	8	13	a	a	DET
ejpam-5403	8	14	bck	bck	NOUN
ejpam-5403	8	15	-	-	PUNCT
ejpam-5403	8	16	algebra	algebra	NOUN
ejpam-5403	8	17	.	.	PUNCT
ejpam-5403	9	1	in	in	ADP
ejpam-5403	9	2	the	the	DET
ejpam-5403	9	3	same	same	ADJ
ejpam-5403	9	4	year	year	NOUN
ejpam-5403	9	5	the	the	DET
ejpam-5403	9	6	notion	notion	NOUN
ejpam-5403	9	7	of	of	ADP
ejpam-5403	9	8	bcialgebra	bcialgebra	NOUN
ejpam-5403	9	9	was	be	AUX
ejpam-5403	9	10	introduced	introduce	VERB
ejpam-5403	9	11	by	by	ADP
ejpam-5403	9	12	k.	k.	PROPN
ejpam-5403	9	13	iseki	iseki	PROPN
ejpam-5403	10	1	[	[	X
ejpam-5403	10	2	7	7	NUM
ejpam-5403	10	3	]	]	PUNCT
ejpam-5403	10	4	,	,	PUNCT
ejpam-5403	10	5	which	which	PRON
ejpam-5403	10	6	is	be	AUX
ejpam-5403	10	7	a	a	DET
ejpam-5403	10	8	super	super	ADJ
ejpam-5403	10	9	class	class	NOUN
ejpam-5403	10	10	of	of	ADP
ejpam-5403	10	11	bck	bck	NOUN
ejpam-5403	10	12	-	-	PUNCT
ejpam-5403	10	13	algebra	algebra	NOUN
ejpam-5403	10	14	.	.	PUNCT
ejpam-5403	11	1	for	for	ADP
ejpam-5403	11	2	more	more	ADJ
ejpam-5403	11	3	informations	information	NOUN
ejpam-5403	11	4	of	of	ADP
ejpam-5403	11	5	bck	bck	NOUN
ejpam-5403	11	6	-	-	PUNCT
ejpam-5403	11	7	algebra	algebra	PROPN
ejpam-5403	11	8	and	and	CCONJ
ejpam-5403	11	9	bci	bci	NOUN
ejpam-5403	11	10	-	-	NOUN
ejpam-5403	11	11	algebra	algebra	NOUN
ejpam-5403	11	12	see	see	VERB
ejpam-5403	11	13	also	also	ADV
ejpam-5403	11	14	[	[	PUNCT
ejpam-5403	11	15	[	[	X
ejpam-5403	11	16	16	16	NUM
ejpam-5403	11	17	]	]	PUNCT
ejpam-5403	11	18	,	,	PUNCT
ejpam-5403	11	19	[	[	X
ejpam-5403	11	20	8	8	NUM
ejpam-5403	11	21	]	]	SYM
ejpam-5403	11	22	]	]	PUNCT
ejpam-5403	11	23	.	.	PUNCT
ejpam-5403	12	1	it	it	PRON
ejpam-5403	12	2	is	be	AUX
ejpam-5403	12	3	natural	natural	ADJ
ejpam-5403	12	4	to	to	PART
ejpam-5403	12	5	study	study	VERB
ejpam-5403	12	6	a	a	DET
ejpam-5403	12	7	generalization	generalization	NOUN
ejpam-5403	12	8	of	of	ADP
ejpam-5403	12	9	these	these	DET
ejpam-5403	12	10	algebras	algebra	NOUN
ejpam-5403	12	11	.	.	PUNCT
ejpam-5403	13	1	later	later	ADV
ejpam-5403	13	2	on	on	ADV
ejpam-5403	13	3	there	there	PRON
ejpam-5403	13	4	is	be	VERB
ejpam-5403	13	5	a	a	DET
ejpam-5403	13	6	rich	rich	ADJ
ejpam-5403	13	7	literature	literature	NOUN
ejpam-5403	13	8	involved	involve	VERB
ejpam-5403	13	9	with	with	ADP
ejpam-5403	13	10	bckalgebra	bckalgebra	NOUN
ejpam-5403	13	11	and	and	CCONJ
ejpam-5403	13	12	bci	bci	NOUN
ejpam-5403	13	13	-	-	NOUN
ejpam-5403	13	14	algebra	algebra	NOUN
ejpam-5403	13	15	.	.	PUNCT
ejpam-5403	14	1	a	a	DET
ejpam-5403	14	2	bch	bch	PROPN
ejpam-5403	14	3	-	-	PUNCT
ejpam-5403	14	4	algebra	algebra	NOUN
ejpam-5403	14	5	was	be	AUX
ejpam-5403	14	6	emerged	emerge	VERB
ejpam-5403	14	7	in	in	ADP
ejpam-5403	14	8	1983	1983	NUM
ejpam-5403	14	9	by	by	ADP
ejpam-5403	14	10	q.	q.	PROPN
ejpam-5403	14	11	p.	p.	PROPN
ejpam-5403	14	12	hu	hu	PROPN
ejpam-5403	15	1	and	and	CCONJ
ejpam-5403	15	2	x.	x.	PROPN
ejpam-5403	15	3	li	li	PROPN
ejpam-5403	15	4	which	which	PRON
ejpam-5403	15	5	is	be	AUX
ejpam-5403	15	6	a	a	DET
ejpam-5403	15	7	generalization	generalization	NOUN
ejpam-5403	15	8	of	of	ADP
ejpam-5403	15	9	bck	bck	PROPN
ejpam-5403	15	10	,	,	PUNCT
ejpam-5403	15	11	bci	bci	NOUN
ejpam-5403	15	12	-	-	PUNCT
ejpam-5403	15	13	algebras	algebras	X
ejpam-5403	15	14	.	.	PUNCT
ejpam-5403	16	1	later	later	ADV
ejpam-5403	16	2	,	,	PUNCT
ejpam-5403	16	3	j.	j.	PROPN
ejpam-5403	16	4	neggers	neggers	PROPN
ejpam-5403	16	5	et	et	AUX
ejpam-5403	16	6	al	al	PROPN
ejpam-5403	16	7	.	.	PROPN
ejpam-5403	16	8	introduced	introduce	VERB
ejpam-5403	16	9	many	many	ADJ
ejpam-5403	16	10	algebras	algebra	NOUN
ejpam-5403	16	11	which	which	PRON
ejpam-5403	16	12	related	relate	VERB
ejpam-5403	16	13	to	to	PART
ejpam-5403	16	14	bck	bck	VERB
ejpam-5403	16	15	,	,	PUNCT
ejpam-5403	16	16	bci	bci	NOUN
ejpam-5403	16	17	-	-	PUNCT
ejpam-5403	16	18	algebras	algebra	NOUN
ejpam-5403	16	19	such	such	ADJ
ejpam-5403	16	20	as	as	ADP
ejpam-5403	16	21	d	d	NOUN
ejpam-5403	16	22	-	-	NOUN
ejpam-5403	16	23	algebra	algebra	ADJ
ejpam-5403	16	24	,	,	PUNCT
ejpam-5403	16	25	b	b	X
ejpam-5403	16	26	-	-	PUNCT
ejpam-5403	16	27	algebra	algebra	NOUN
ejpam-5403	16	28	and	and	CCONJ
ejpam-5403	16	29	qalgebra	qalgebra	NOUN
ejpam-5403	16	30	.	.	PUNCT
ejpam-5403	17	1	they	they	PRON
ejpam-5403	17	2	examined	examine	VERB
ejpam-5403	17	3	some	some	DET
ejpam-5403	17	4	relations	relation	NOUN
ejpam-5403	17	5	and	and	CCONJ
ejpam-5403	17	6	some	some	DET
ejpam-5403	17	7	properties	property	NOUN
ejpam-5403	17	8	of	of	ADP
ejpam-5403	17	9	theses	theses	PRON
ejpam-5403	17	10	algebras	algebra	NOUN
ejpam-5403	17	11	.	.	PUNCT
ejpam-5403	18	1	in	in	ADP
ejpam-5403	18	2	2001	2001	NUM
ejpam-5403	18	3	,	,	PUNCT
ejpam-5403	18	4	j.	j.	PROPN
ejpam-5403	18	5	neggers	neggers	PROPN
ejpam-5403	18	6	et	et	PROPN
ejpam-5403	18	7	al	al	PROPN
ejpam-5403	18	8	.	.	PUNCT
ejpam-5403	19	1	[	[	X
ejpam-5403	19	2	9	9	NUM
ejpam-5403	19	3	]	]	PUNCT
ejpam-5403	19	4	introduced	introduce	VERB
ejpam-5403	19	5	a	a	DET
ejpam-5403	19	6	new	new	ADJ
ejpam-5403	19	7	generalization	generalization	NOUN
ejpam-5403	19	8	of	of	ADP
ejpam-5403	19	9	bci	bci	NOUN
ejpam-5403	19	10	-	-	NOUN
ejpam-5403	19	11	algebra	algebra	NOUN
ejpam-5403	19	12	and	and	CCONJ
ejpam-5403	19	13	bck	bck	NOUN
ejpam-5403	19	14	-	-	PUNCT
ejpam-5403	19	15	algebra	algebra	NOUN
ejpam-5403	19	16	.	.	PUNCT
ejpam-5403	20	1	this	this	DET
ejpam-5403	20	2	new	new	ADJ
ejpam-5403	20	3	algebra	algebra	NOUN
ejpam-5403	20	4	was	be	AUX
ejpam-5403	20	5	known	know	VERB
ejpam-5403	20	6	as	as	ADP
ejpam-5403	20	7	q	q	NOUN
ejpam-5403	20	8	-	-	NOUN
ejpam-5403	20	9	algebra	algebra	NOUN
ejpam-5403	20	10	which	which	PRON
ejpam-5403	20	11	is	be	AUX
ejpam-5403	20	12	also	also	ADV
ejpam-5403	20	13	a	a	DET
ejpam-5403	20	14	generalization	generalization	NOUN
ejpam-5403	20	15	of	of	ADP
ejpam-5403	20	16	bch	bch	PROPN
ejpam-5403	20	17	-	-	PUNCT
ejpam-5403	20	18	algebra	algebra	NOUN
ejpam-5403	20	19	.	.	PUNCT
ejpam-5403	21	1	in	in	ADP
ejpam-5403	21	2	[	[	X
ejpam-5403	21	3	9	9	NUM
ejpam-5403	21	4	]	]	PUNCT
ejpam-5403	21	5	the	the	DET
ejpam-5403	21	6	authors	author	NOUN
ejpam-5403	21	7	generalized	generalize	VERB
ejpam-5403	21	8	some	some	DET
ejpam-5403	21	9	properties	property	NOUN
ejpam-5403	21	10	and	and	CCONJ
ejpam-5403	21	11	theorems	theorem	NOUN
ejpam-5403	21	12	discussed	discuss	VERB
ejpam-5403	21	13	in	in	ADP
ejpam-5403	21	14	bci	bci	NOUN
ejpam-5403	21	15	-	-	NOUN
ejpam-5403	21	16	algebra	algebra	NOUN
ejpam-5403	21	17	.	.	PUNCT
ejpam-5403	22	1	the	the	DET
ejpam-5403	22	2	concept	concept	NOUN
ejpam-5403	22	3	of	of	ADP
ejpam-5403	22	4	quadratic	quadratic	ADJ
ejpam-5403	22	5	q	q	NOUN
ejpam-5403	22	6	-	-	PUNCT
ejpam-5403	22	7	algebra	algebra	NOUN
ejpam-5403	22	8	is	be	AUX
ejpam-5403	22	9	also	also	ADV
ejpam-5403	22	10	offered	offer	VERB
ejpam-5403	22	11	in	in	ADP
ejpam-5403	22	12	[	[	X
ejpam-5403	22	13	9	9	NUM
ejpam-5403	22	14	]	]	PUNCT
ejpam-5403	22	15	.	.	PUNCT
ejpam-5403	23	1	a	a	DET
ejpam-5403	23	2	q	q	NOUN
ejpam-5403	23	3	-	-	PUNCT
ejpam-5403	23	4	algebra	algebra	NOUN
ejpam-5403	23	5	consists	consist	VERB
ejpam-5403	23	6	of	of	ADP
ejpam-5403	23	7	a	a	DET
ejpam-5403	23	8	nonempty	nonempty	ADV
ejpam-5403	23	9	set	set	VERB
ejpam-5403	23	10	x	x	PUNCT
ejpam-5403	23	11	and	and	CCONJ
ejpam-5403	23	12	a	a	DET
ejpam-5403	23	13	constant	constant	ADJ
ejpam-5403	23	14	0	0	NUM
ejpam-5403	23	15	∈	∈	NOUN
ejpam-5403	23	16	x	x	PUNCT
ejpam-5403	23	17	together	together	ADV
ejpam-5403	23	18	with	with	ADP
ejpam-5403	23	19	a	a	DET
ejpam-5403	23	20	binary	binary	ADJ
ejpam-5403	23	21	operation	operation	NOUN
ejpam-5403	23	22	∗	∗	NOUN
ejpam-5403	23	23	on	on	ADP
ejpam-5403	23	24	x	x	PUNCT
ejpam-5403	23	25	that	that	PRON
ejpam-5403	23	26	yields	yield	VERB
ejpam-5403	23	27	the	the	DET
ejpam-5403	23	28	following	following	NOUN
ejpam-5403	23	29	:	:	PUNCT
ejpam-5403	23	30	for	for	ADP
ejpam-5403	23	31	all	all	DET
ejpam-5403	23	32	x	x	NOUN
ejpam-5403	23	33	,	,	PUNCT
ejpam-5403	23	34	y	y	PROPN
ejpam-5403	23	35	,	,	PUNCT
ejpam-5403	23	36	z	z	NOUN
ejpam-5403	23	37	∈	∈	PROPN
ejpam-5403	23	38	x	x	SYM
ejpam-5403	23	39	(	(	PUNCT
ejpam-5403	23	40	q1	q1	PROPN
ejpam-5403	23	41	)	)	PUNCT
ejpam-5403	23	42	x	x	SYM
ejpam-5403	23	43	∗	∗	NOUN
ejpam-5403	23	44	x	x	SYM
ejpam-5403	24	1	=	=	SYM
ejpam-5403	24	2	0	0	NUM
ejpam-5403	24	3	,	,	PUNCT
ejpam-5403	24	4	∗corresponding	∗corresponde	VERB
ejpam-5403	24	5	author	author	NOUN
ejpam-5403	24	6	.	.	PUNCT
ejpam-5403	25	1	doi	doi	NOUN
ejpam-5403	25	2	:	:	PUNCT
ejpam-5403	25	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5403	https://doi.org/10.29020/nybg.ejpam.v17i4.5403	ADP
ejpam-5403	25	4	email	email	NOUN
ejpam-5403	25	5	addresses	address	VERB
ejpam-5403	25	6	:	:	PUNCT
ejpam-5403	25	7	ananya.a@msu.ac.th	ananya.a@msu.ac.th	ADP
ejpam-5403	25	8	(	(	PUNCT
ejpam-5403	25	9	a.	a.	NOUN
ejpam-5403	25	10	anantayasethi	anantayasethi	PROPN
ejpam-5403	25	11	)	)	PUNCT
ejpam-5403	25	12	,	,	PUNCT
ejpam-5403	25	13	64010213029@msu.ac.th	64010213029@msu.ac.th	INTJ
ejpam-5403	25	14	(	(	PUNCT
ejpam-5403	25	15	t.	t.	PROPN
ejpam-5403	25	16	kunawat	kunawat	PROPN
ejpam-5403	25	17	)	)	PUNCT
ejpam-5403	25	18	,	,	PUNCT
ejpam-5403	25	19	64010213047@msu.ac.th	64010213047@msu.ac.th	NUM
ejpam-5403	25	20	(	(	PUNCT
ejpam-5403	25	21	p.	p.	NOUN
ejpam-5403	25	22	moolnipa	moolnipa	NOUN
ejpam-5403	25	23	)	)	PUNCT
ejpam-5403	25	24	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5403	26	1	3268	3268	NUM
ejpam-5403	26	2	copyright	copyright	NOUN
ejpam-5403	26	3	:	:	PUNCT
ejpam-5403	26	4	©	©	PROPN
ejpam-5403	26	5	2024	2024	NUM
ejpam-5403	26	6	the	the	DET
ejpam-5403	26	7	author(s	author(s	NOUN
ejpam-5403	26	8	)	)	PUNCT
ejpam-5403	26	9	.	.	PUNCT
ejpam-5403	27	1	(	(	PUNCT
ejpam-5403	27	2	cc	cc	NOUN
ejpam-5403	27	3	by	by	ADP
ejpam-5403	27	4	-	-	PUNCT
ejpam-5403	27	5	nc	nc	PROPN
ejpam-5403	27	6	4.0	4.0	NUM
ejpam-5403	27	7	)	)	PUNCT
ejpam-5403	27	8	a.	a.	NOUN
ejpam-5403	27	9	anantayasethi	anantayasethi	PROPN
ejpam-5403	27	10	,	,	PUNCT
ejpam-5403	27	11	t.	t.	PROPN
ejpam-5403	27	12	kunawat	kunawat	PROPN
ejpam-5403	27	13	,	,	PUNCT
ejpam-5403	27	14	p.	p.	PROPN
ejpam-5403	27	15	moolnipa	moolnipa	PROPN
ejpam-5403	27	16	/	/	SYM
ejpam-5403	27	17	eur	eur	PROPN
ejpam-5403	27	18	.	.	PUNCT
ejpam-5403	28	1	j.	j.	PROPN
ejpam-5403	28	2	pure	pure	PROPN
ejpam-5403	28	3	appl	appl	PROPN
ejpam-5403	28	4	.	.	PROPN
ejpam-5403	28	5	math	math	PROPN
ejpam-5403	28	6	,	,	PUNCT
ejpam-5403	28	7	17	17	NUM
ejpam-5403	28	8	(	(	PUNCT
ejpam-5403	28	9	4	4	NUM
ejpam-5403	28	10	)	)	PUNCT
ejpam-5403	28	11	(	(	PUNCT
ejpam-5403	28	12	2024	2024	NUM
ejpam-5403	28	13	)	)	PUNCT
ejpam-5403	28	14	,	,	PUNCT
ejpam-5403	28	15	3268	3268	NUM
ejpam-5403	28	16	-	-	SYM
ejpam-5403	28	17	3276	3276	NUM
ejpam-5403	28	18	3269	3269	NUM
ejpam-5403	28	19	(	(	PUNCT
ejpam-5403	28	20	q2	q2	NOUN
ejpam-5403	28	21	)	)	PUNCT
ejpam-5403	28	22	x	x	SYM
ejpam-5403	29	1	∗	∗	NOUN
ejpam-5403	29	2	0	0	NUM
ejpam-5403	30	1	=	=	SYM
ejpam-5403	30	2	x	x	NOUN
ejpam-5403	30	3	,	,	PUNCT
ejpam-5403	30	4	(	(	PUNCT
ejpam-5403	30	5	q3	q3	PROPN
ejpam-5403	30	6	)	)	PUNCT
ejpam-5403	30	7	(	(	PUNCT
ejpam-5403	30	8	x	x	SYM
ejpam-5403	30	9	∗	∗	PROPN
ejpam-5403	30	10	y	y	NOUN
ejpam-5403	30	11	)	)	PUNCT
ejpam-5403	30	12	∗	∗	NOUN
ejpam-5403	30	13	z	z	NOUN
ejpam-5403	30	14	=	=	SYM
ejpam-5403	30	15	(	(	PUNCT
ejpam-5403	30	16	x	x	X
ejpam-5403	30	17	∗	∗	PROPN
ejpam-5403	30	18	z	z	NOUN
ejpam-5403	30	19	)	)	PUNCT
ejpam-5403	30	20	∗	∗	NOUN
ejpam-5403	30	21	y.	y.	NOUN
ejpam-5403	30	22	for	for	ADP
ejpam-5403	30	23	convenience	convenience	NOUN
ejpam-5403	30	24	,	,	PUNCT
ejpam-5403	30	25	we	we	PRON
ejpam-5403	30	26	write	write	VERB
ejpam-5403	30	27	xy	xy	PROPN
ejpam-5403	30	28	instead	instead	ADV
ejpam-5403	30	29	of	of	ADP
ejpam-5403	30	30	x	x	INTJ
ejpam-5403	30	31	∗	∗	NOUN
ejpam-5403	30	32	y.	y.	NOUN
ejpam-5403	30	33	since	since	SCONJ
ejpam-5403	30	34	then	then	ADV
ejpam-5403	30	35	,	,	PUNCT
ejpam-5403	30	36	there	there	PRON
ejpam-5403	30	37	are	be	VERB
ejpam-5403	30	38	many	many	ADJ
ejpam-5403	30	39	authors	author	NOUN
ejpam-5403	30	40	working	work	VERB
ejpam-5403	30	41	on	on	ADP
ejpam-5403	30	42	q	q	NOUN
ejpam-5403	30	43	-	-	PUNCT
ejpam-5403	30	44	algebras	algebras	X
ejpam-5403	30	45	[	[	PUNCT
ejpam-5403	30	46	see	see	VERB
ejpam-5403	30	47	[	[	X
ejpam-5403	30	48	2	2	NUM
ejpam-5403	30	49	]	]	PUNCT
ejpam-5403	30	50	,	,	PUNCT
ejpam-5403	31	1	[	[	X
ejpam-5403	31	2	1	1	NUM
ejpam-5403	31	3	]	]	PUNCT
ejpam-5403	31	4	,	,	PUNCT
ejpam-5403	31	5	[	[	X
ejpam-5403	31	6	3	3	NUM
ejpam-5403	31	7	]	]	PUNCT
ejpam-5403	31	8	,	,	PUNCT
ejpam-5403	31	9	[	[	X
ejpam-5403	31	10	14	14	NUM
ejpam-5403	31	11	]	]	PUNCT
ejpam-5403	31	12	,	,	PUNCT
ejpam-5403	31	13	[	[	X
ejpam-5403	31	14	12	12	NUM
ejpam-5403	31	15	]	]	PUNCT
ejpam-5403	31	16	,	,	PUNCT
ejpam-5403	31	17	[	[	X
ejpam-5403	31	18	10	10	NUM
ejpam-5403	31	19	]	]	PUNCT
ejpam-5403	31	20	,	,	PUNCT
ejpam-5403	32	1	[	[	X
ejpam-5403	32	2	11	11	NUM
ejpam-5403	32	3	]	]	PUNCT
ejpam-5403	32	4	,	,	PUNCT
ejpam-5403	32	5	[	[	X
ejpam-5403	32	6	13	13	NUM
ejpam-5403	32	7	]	]	SYM
ejpam-5403	32	8	]	]	PUNCT
ejpam-5403	32	9	.	.	PUNCT
ejpam-5403	33	1	in	in	ADP
ejpam-5403	33	2	[	[	X
ejpam-5403	33	3	9	9	NUM
ejpam-5403	33	4	]	]	PUNCT
ejpam-5403	33	5	,	,	PUNCT
ejpam-5403	33	6	the	the	DET
ejpam-5403	33	7	concepts	concept	NOUN
ejpam-5403	33	8	of	of	ADP
ejpam-5403	33	9	ideal	ideal	NOUN
ejpam-5403	33	10	and	and	CCONJ
ejpam-5403	33	11	g	g	NOUN
ejpam-5403	33	12	-	-	PUNCT
ejpam-5403	33	13	part	part	NOUN
ejpam-5403	33	14	were	be	AUX
ejpam-5403	33	15	established	establish	VERB
ejpam-5403	33	16	.	.	PUNCT
ejpam-5403	34	1	a	a	DET
ejpam-5403	34	2	non	non	ADJ
ejpam-5403	34	3	-	-	ADJ
ejpam-5403	34	4	empty	empty	ADJ
ejpam-5403	34	5	subset	subset	NOUN
ejpam-5403	34	6	i	i	PRON
ejpam-5403	34	7	of	of	ADP
ejpam-5403	34	8	a	a	DET
ejpam-5403	34	9	q	q	NOUN
ejpam-5403	34	10	-	-	NOUN
ejpam-5403	34	11	algebra	algebra	NOUN
ejpam-5403	34	12	x	x	PUNCT
ejpam-5403	34	13	is	be	AUX
ejpam-5403	34	14	an	an	DET
ejpam-5403	34	15	ideal	ideal	NOUN
ejpam-5403	34	16	if	if	SCONJ
ejpam-5403	34	17	the	the	DET
ejpam-5403	34	18	following	follow	VERB
ejpam-5403	34	19	conditions	condition	NOUN
ejpam-5403	34	20	were	be	AUX
ejpam-5403	34	21	fulfilled	fulfil	VERB
ejpam-5403	34	22	:	:	PUNCT
ejpam-5403	34	23	(	(	PUNCT
ejpam-5403	34	24	i1	i1	PROPN
ejpam-5403	34	25	)	)	PUNCT
ejpam-5403	34	26	0	0	PUNCT
ejpam-5403	35	1	∈	∈	PROPN
ejpam-5403	36	1	i	i	PRON
ejpam-5403	36	2	,	,	PUNCT
ejpam-5403	36	3	(	(	PUNCT
ejpam-5403	36	4	i2	i2	PROPN
ejpam-5403	36	5	)	)	PUNCT
ejpam-5403	36	6	xy	xy	PROPN
ejpam-5403	37	1	∈	∈	PROPN
ejpam-5403	38	1	i	i	PRON
ejpam-5403	38	2	and	and	CCONJ
ejpam-5403	38	3	y	y	PROPN
ejpam-5403	38	4	∈	∈	PROPN
ejpam-5403	39	1	i	i	PRON
ejpam-5403	39	2	imply	imply	VERB
ejpam-5403	39	3	x	x	X
ejpam-5403	39	4	∈	∈	PROPN
ejpam-5403	39	5	i.	i.	NOUN
ejpam-5403	39	6	a	a	DET
ejpam-5403	39	7	set	set	NOUN
ejpam-5403	39	8	{	{	PUNCT
ejpam-5403	39	9	0	0	NUM
ejpam-5403	39	10	}	}	PUNCT
ejpam-5403	39	11	and	and	CCONJ
ejpam-5403	39	12	x	x	PRON
ejpam-5403	39	13	are	be	AUX
ejpam-5403	39	14	always	always	ADV
ejpam-5403	39	15	ideals	ideal	NOUN
ejpam-5403	39	16	of	of	ADP
ejpam-5403	39	17	x.	x.	NOUN
ejpam-5403	39	18	a	a	DET
ejpam-5403	39	19	subset	subset	NOUN
ejpam-5403	39	20	g(x	g(x	NOUN
ejpam-5403	39	21	)	)	PUNCT
ejpam-5403	39	22	:	:	PUNCT
ejpam-5403	39	23	=	=	SYM
ejpam-5403	39	24	{	{	PUNCT
ejpam-5403	39	25	x	x	PUNCT
ejpam-5403	39	26	∈	∈	NOUN
ejpam-5403	39	27	x	x	PUNCT
ejpam-5403	39	28	|	|	ADV
ejpam-5403	39	29	0x	0x	NOUN
ejpam-5403	39	30	=	=	SYM
ejpam-5403	39	31	x	x	SYM
ejpam-5403	39	32	}	}	PUNCT
ejpam-5403	39	33	of	of	ADP
ejpam-5403	39	34	a	a	DET
ejpam-5403	39	35	qalgebra	qalgebra	NOUN
ejpam-5403	39	36	x	x	AUX
ejpam-5403	39	37	is	be	AUX
ejpam-5403	39	38	called	call	VERB
ejpam-5403	39	39	a	a	DET
ejpam-5403	39	40	g	g	NOUN
ejpam-5403	39	41	-	-	PUNCT
ejpam-5403	39	42	part	part	NOUN
ejpam-5403	39	43	of	of	ADP
ejpam-5403	39	44	x.	x.	NOUN
ejpam-5403	39	45	in	in	ADP
ejpam-5403	39	46	2004	2004	NUM
ejpam-5403	39	47	,	,	PUNCT
ejpam-5403	39	48	s.	s.	PROPN
ejpam-5403	39	49	s.	s.	PROPN
ejpam-5403	39	50	ahn	ahn	PROPN
ejpam-5403	39	51	et	et	PROPN
ejpam-5403	39	52	al	al	PROPN
ejpam-5403	39	53	.	.	PUNCT
ejpam-5403	40	1	[	[	X
ejpam-5403	40	2	14	14	NUM
ejpam-5403	40	3	]	]	PUNCT
ejpam-5403	40	4	introduced	introduce	VERB
ejpam-5403	40	5	the	the	DET
ejpam-5403	40	6	notion	notion	NOUN
ejpam-5403	40	7	of	of	ADP
ejpam-5403	40	8	implicative	implicative	ADJ
ejpam-5403	40	9	q	q	NOUN
ejpam-5403	40	10	-	-	PUNCT
ejpam-5403	40	11	algebra	algebra	NOUN
ejpam-5403	40	12	which	which	PRON
ejpam-5403	40	13	is	be	AUX
ejpam-5403	40	14	a	a	DET
ejpam-5403	40	15	q	q	NOUN
ejpam-5403	40	16	-	-	NOUN
ejpam-5403	40	17	algebra	algebra	NOUN
ejpam-5403	40	18	with	with	ADP
ejpam-5403	40	19	(	(	PUNCT
ejpam-5403	40	20	xy)(yz	xy)(yz	NOUN
ejpam-5403	40	21	)	)	PUNCT
ejpam-5403	40	22	=	=	SYM
ejpam-5403	40	23	(	(	PUNCT
ejpam-5403	40	24	xy)z	xy)z	NUM
ejpam-5403	40	25	for	for	ADP
ejpam-5403	40	26	all	all	DET
ejpam-5403	40	27	x	x	NOUN
ejpam-5403	40	28	,	,	PUNCT
ejpam-5403	40	29	y	y	PROPN
ejpam-5403	40	30	,	,	PUNCT
ejpam-5403	40	31	z	z	NOUN
ejpam-5403	40	32	∈	∈	NOUN
ejpam-5403	40	33	x.	x.	NOUN
ejpam-5403	41	1	many	many	ADJ
ejpam-5403	41	2	mathematicians	mathematician	NOUN
ejpam-5403	41	3	from	from	ADP
ejpam-5403	41	4	korea	korea	PROPN
ejpam-5403	41	5	and	and	CCONJ
ejpam-5403	41	6	egypt	egypt	PROPN
ejpam-5403	41	7	studied	study	VERB
ejpam-5403	41	8	on	on	ADP
ejpam-5403	41	9	mappings	mapping	NOUN
ejpam-5403	41	10	of	of	ADP
ejpam-5403	41	11	q	q	NOUN
ejpam-5403	41	12	-	-	PUNCT
ejpam-5403	41	13	algebras	algebras	ADJ
ejpam-5403	41	14	,	,	PUNCT
ejpam-5403	41	15	namely	namely	ADV
ejpam-5403	41	16	:	:	PUNCT
ejpam-5403	41	17	r	r	X
ejpam-5403	41	18	-	-	PUNCT
ejpam-5403	41	19	maps	map	NOUN
ejpam-5403	41	20	,	,	PUNCT
ejpam-5403	41	21	l	l	NOUN
ejpam-5403	41	22	-	-	NOUN
ejpam-5403	41	23	maps	map	NOUN
ejpam-5403	41	24	,	,	PUNCT
ejpam-5403	41	25	right	right	ADJ
ejpam-5403	41	26	fixed	fix	VERB
ejpam-5403	41	27	maps	map	NOUN
ejpam-5403	41	28	and	and	CCONJ
ejpam-5403	41	29	fuzzy	fuzzy	ADJ
ejpam-5403	41	30	set	set	NOUN
ejpam-5403	41	31	[	[	PUNCT
ejpam-5403	41	32	see	see	VERB
ejpam-5403	41	33	[	[	X
ejpam-5403	41	34	11	11	NUM
ejpam-5403	41	35	]	]	PUNCT
ejpam-5403	41	36	,	,	PUNCT
ejpam-5403	41	37	[	[	X
ejpam-5403	41	38	14	14	NUM
ejpam-5403	41	39	]	]	PUNCT
ejpam-5403	41	40	,	,	PUNCT
ejpam-5403	41	41	[	[	X
ejpam-5403	41	42	13	13	NUM
ejpam-5403	41	43	]	]	SYM
ejpam-5403	41	44	]	]	PUNCT
ejpam-5403	41	45	.	.	PUNCT
ejpam-5403	42	1	in	in	ADP
ejpam-5403	42	2	2010	2010	NUM
ejpam-5403	42	3	,	,	PUNCT
ejpam-5403	42	4	s.	s.	PROPN
ejpam-5403	42	5	s.	s.	PROPN
ejpam-5403	42	6	ahn	ahn	PROPN
ejpam-5403	42	7	and	and	CCONJ
ejpam-5403	42	8	k.	k.	PROPN
ejpam-5403	43	1	so	so	ADV
ejpam-5403	43	2	[	[	X
ejpam-5403	43	3	4	4	X
ejpam-5403	43	4	]	]	PUNCT
ejpam-5403	43	5	considered	consider	VERB
ejpam-5403	43	6	homomorphims	homomorphim	NOUN
ejpam-5403	43	7	and	and	CCONJ
ejpam-5403	43	8	congruence	congruence	VERB
ejpam-5403	43	9	in	in	ADP
ejpam-5403	43	10	q	q	NOUN
ejpam-5403	43	11	-	-	PUNCT
ejpam-5403	43	12	algebras	algebras	X
ejpam-5403	43	13	.	.	PUNCT
ejpam-5403	44	1	the	the	DET
ejpam-5403	44	2	authors	author	NOUN
ejpam-5403	44	3	in	in	ADP
ejpam-5403	44	4	[	[	X
ejpam-5403	44	5	4	4	NUM
ejpam-5403	44	6	]	]	PUNCT
ejpam-5403	44	7	provided	provide	VERB
ejpam-5403	44	8	some	some	DET
ejpam-5403	44	9	decompositions	decomposition	NOUN
ejpam-5403	44	10	of	of	ADP
ejpam-5403	44	11	ideals	ideal	NOUN
ejpam-5403	44	12	in	in	ADP
ejpam-5403	44	13	q	q	NOUN
ejpam-5403	44	14	-	-	PUNCT
ejpam-5403	44	15	algebras	algebras	X
ejpam-5403	44	16	.	.	PUNCT
ejpam-5403	45	1	recently	recently	ADV
ejpam-5403	45	2	,	,	PUNCT
ejpam-5403	45	3	the	the	DET
ejpam-5403	45	4	concept	concept	NOUN
ejpam-5403	45	5	of	of	ADP
ejpam-5403	45	6	ideal	ideal	NOUN
ejpam-5403	45	7	is	be	AUX
ejpam-5403	45	8	again	again	ADV
ejpam-5403	45	9	in	in	ADP
ejpam-5403	45	10	a	a	DET
ejpam-5403	45	11	spotlight	spotlight	NOUN
ejpam-5403	45	12	.	.	PUNCT
ejpam-5403	46	1	various	various	ADJ
ejpam-5403	46	2	kinds	kind	NOUN
ejpam-5403	46	3	of	of	ADP
ejpam-5403	46	4	ideals	ideal	NOUN
ejpam-5403	46	5	were	be	AUX
ejpam-5403	46	6	discussed	discuss	VERB
ejpam-5403	46	7	.	.	PUNCT
ejpam-5403	47	1	q	q	X
ejpam-5403	47	2	-	-	PUNCT
ejpam-5403	47	3	ideal	ideal	ADJ
ejpam-5403	47	4	,	,	PUNCT
ejpam-5403	47	5	prime	prime	ADJ
ejpam-5403	47	6	ideal	ideal	NOUN
ejpam-5403	47	7	,	,	PUNCT
ejpam-5403	47	8	fuzzy	fuzzy	ADJ
ejpam-5403	47	9	ideal	ideal	NOUN
ejpam-5403	47	10	,	,	PUNCT
ejpam-5403	47	11	intuitionistic	intuitionistic	ADJ
ejpam-5403	47	12	fuzzy	fuzzy	ADJ
ejpam-5403	47	13	prime	prime	ADJ
ejpam-5403	47	14	ideal	ideal	NOUN
ejpam-5403	47	15	,	,	PUNCT
ejpam-5403	47	16	g	g	NOUN
ejpam-5403	47	17	-	-	PUNCT
ejpam-5403	47	18	part	part	NOUN
ejpam-5403	47	19	ideal	ideal	NOUN
ejpam-5403	47	20	were	be	AUX
ejpam-5403	47	21	studied	study	VERB
ejpam-5403	47	22	in	in	ADP
ejpam-5403	47	23	[	[	X
ejpam-5403	47	24	2	2	NUM
ejpam-5403	47	25	]	]	PUNCT
ejpam-5403	47	26	,	,	PUNCT
ejpam-5403	48	1	[	[	X
ejpam-5403	48	2	12	12	NUM
ejpam-5403	48	3	]	]	PUNCT
ejpam-5403	48	4	,	,	PUNCT
ejpam-5403	49	1	[	[	X
ejpam-5403	49	2	10	10	NUM
ejpam-5403	49	3	]	]	PUNCT
ejpam-5403	49	4	,	,	PUNCT
ejpam-5403	49	5	[	[	X
ejpam-5403	49	6	13	13	NUM
ejpam-5403	49	7	]	]	PUNCT
ejpam-5403	49	8	.	.	PUNCT
ejpam-5403	50	1	in	in	ADP
ejpam-5403	50	2	the	the	DET
ejpam-5403	50	3	year	year	NOUN
ejpam-5403	50	4	2001	2001	NUM
ejpam-5403	50	5	,	,	PUNCT
ejpam-5403	50	6	d.	d.	PROPN
ejpam-5403	50	7	sun	sun	PROPN
ejpam-5403	50	8	[	[	X
ejpam-5403	50	9	15	15	NUM
ejpam-5403	50	10	]	]	PUNCT
ejpam-5403	50	11	introduced	introduce	VERB
ejpam-5403	50	12	the	the	DET
ejpam-5403	50	13	concept	concept	NOUN
ejpam-5403	50	14	of	of	ADP
ejpam-5403	50	15	atom	atom	NOUN
ejpam-5403	50	16	and	and	CCONJ
ejpam-5403	50	17	strong	strong	ADJ
ejpam-5403	50	18	atom	atom	NOUN
ejpam-5403	50	19	in	in	ADP
ejpam-5403	50	20	bckalgebra	bckalgebra	NOUN
ejpam-5403	50	21	.	.	PUNCT
ejpam-5403	51	1	he	he	PRON
ejpam-5403	51	2	proved	prove	VERB
ejpam-5403	51	3	that	that	SCONJ
ejpam-5403	51	4	a	a	DET
ejpam-5403	51	5	set	set	NOUN
ejpam-5403	51	6	of	of	ADP
ejpam-5403	51	7	all	all	DET
ejpam-5403	51	8	strong	strong	ADJ
ejpam-5403	51	9	atoms	atom	NOUN
ejpam-5403	51	10	and	and	CCONJ
ejpam-5403	51	11	together	together	ADV
ejpam-5403	51	12	with	with	ADP
ejpam-5403	51	13	zero	zero	NUM
ejpam-5403	51	14	element	element	NOUN
ejpam-5403	51	15	is	be	AUX
ejpam-5403	51	16	an	an	DET
ejpam-5403	51	17	ideal	ideal	NOUN
ejpam-5403	51	18	of	of	ADP
ejpam-5403	51	19	bck	bck	NOUN
ejpam-5403	51	20	-	-	PUNCT
ejpam-5403	51	21	algebra	algebra	NOUN
ejpam-5403	51	22	x.	x.	NOUN
ejpam-5403	51	23	in	in	ADP
ejpam-5403	51	24	2010	2010	NUM
ejpam-5403	51	25	,	,	PUNCT
ejpam-5403	52	1	s.	s.	PROPN
ejpam-5403	52	2	s.	s.	PROPN
ejpam-5403	52	3	ahn	ahn	PROPN
ejpam-5403	52	4	and	and	CCONJ
ejpam-5403	52	5	s.	s.	PROPN
ejpam-5403	52	6	e.	e.	PROPN
ejpam-5403	52	7	kang	kang	PROPN
ejpam-5403	53	1	[	[	X
ejpam-5403	53	2	3	3	X
ejpam-5403	53	3	]	]	PUNCT
ejpam-5403	53	4	introduced	introduce	VERB
ejpam-5403	53	5	the	the	DET
ejpam-5403	53	6	concepts	concept	NOUN
ejpam-5403	53	7	of	of	ADP
ejpam-5403	53	8	atoms	atom	NOUN
ejpam-5403	53	9	in	in	ADP
ejpam-5403	53	10	q	q	NOUN
ejpam-5403	53	11	-	-	NOUN
ejpam-5403	53	12	algebra	algebra	NOUN
ejpam-5403	53	13	.	.	PUNCT
ejpam-5403	54	1	an	an	DET
ejpam-5403	54	2	atom	atom	NOUN
ejpam-5403	54	3	of	of	ADP
ejpam-5403	54	4	x	x	NOUN
ejpam-5403	54	5	is	be	AUX
ejpam-5403	54	6	an	an	DET
ejpam-5403	54	7	element	element	NOUN
ejpam-5403	54	8	a	a	DET
ejpam-5403	54	9	∈	∈	NOUN
ejpam-5403	54	10	x	x	PUNCT
ejpam-5403	54	11	satisfying	satisfy	VERB
ejpam-5403	54	12	:	:	PUNCT
ejpam-5403	54	13	for	for	ADP
ejpam-5403	54	14	x	x	SYM
ejpam-5403	54	15	∈	∈	PROPN
ejpam-5403	54	16	x	x	X
ejpam-5403	54	17	,	,	PUNCT
ejpam-5403	54	18	xa	xa	PROPN
ejpam-5403	54	19	=	=	SYM
ejpam-5403	54	20	0	0	PROPN
ejpam-5403	54	21	implies	imply	VERB
ejpam-5403	54	22	x	x	X
ejpam-5403	55	1	=	=	PUNCT
ejpam-5403	55	2	a.	a.	NOUN
ejpam-5403	55	3	a	a	DET
ejpam-5403	55	4	set	set	NOUN
ejpam-5403	55	5	of	of	ADP
ejpam-5403	55	6	all	all	DET
ejpam-5403	55	7	atoms	atom	NOUN
ejpam-5403	55	8	of	of	ADP
ejpam-5403	55	9	x	x	PUNCT
ejpam-5403	55	10	is	be	AUX
ejpam-5403	55	11	denoted	denote	VERB
ejpam-5403	55	12	by	by	ADP
ejpam-5403	55	13	a(x	a(x	NOUN
ejpam-5403	55	14	)	)	PUNCT
ejpam-5403	55	15	.	.	PUNCT
ejpam-5403	56	1	some	some	DET
ejpam-5403	56	2	properties	property	NOUN
ejpam-5403	56	3	of	of	ADP
ejpam-5403	56	4	atoms	atom	NOUN
ejpam-5403	56	5	are	be	AUX
ejpam-5403	56	6	provided	provide	VERB
ejpam-5403	56	7	in	in	ADP
ejpam-5403	56	8	[	[	X
ejpam-5403	56	9	3	3	NUM
ejpam-5403	56	10	]	]	PUNCT
ejpam-5403	56	11	.	.	PUNCT
ejpam-5403	57	1	the	the	DET
ejpam-5403	57	2	authors	author	NOUN
ejpam-5403	57	3	showed	show	VERB
ejpam-5403	57	4	that	that	SCONJ
ejpam-5403	57	5	if	if	SCONJ
ejpam-5403	57	6	every	every	DET
ejpam-5403	57	7	non	non	ADJ
ejpam-5403	57	8	-	-	ADJ
ejpam-5403	57	9	zero	zero	NUM
ejpam-5403	57	10	element	element	NOUN
ejpam-5403	57	11	of	of	ADP
ejpam-5403	57	12	x	x	PUNCT
ejpam-5403	57	13	is	be	AUX
ejpam-5403	57	14	an	an	DET
ejpam-5403	57	15	atom	atom	NOUN
ejpam-5403	57	16	,	,	PUNCT
ejpam-5403	57	17	then	then	ADV
ejpam-5403	57	18	any	any	DET
ejpam-5403	57	19	subalgebra	subalgebra	NOUN
ejpam-5403	57	20	of	of	ADP
ejpam-5403	57	21	x	x	PUNCT
ejpam-5403	57	22	is	be	AUX
ejpam-5403	57	23	an	an	DET
ejpam-5403	57	24	ideal	ideal	NOUN
ejpam-5403	57	25	.	.	PUNCT
ejpam-5403	58	1	a	a	DET
ejpam-5403	58	2	subalgebra	subalgebra	NOUN
ejpam-5403	58	3	of	of	ADP
ejpam-5403	58	4	a	a	DET
ejpam-5403	58	5	q	q	NOUN
ejpam-5403	58	6	-	-	NOUN
ejpam-5403	58	7	algebra	algebra	NOUN
ejpam-5403	58	8	x	x	PUNCT
ejpam-5403	58	9	is	be	AUX
ejpam-5403	58	10	a	a	DET
ejpam-5403	58	11	non	non	ADJ
ejpam-5403	58	12	-	-	ADJ
ejpam-5403	58	13	empty	empty	ADJ
ejpam-5403	58	14	subset	subset	NOUN
ejpam-5403	58	15	i	i	PRON
ejpam-5403	58	16	of	of	ADP
ejpam-5403	58	17	x	x	PUNCT
ejpam-5403	58	18	with	with	ADP
ejpam-5403	58	19	ab	ab	PROPN
ejpam-5403	58	20	∈	∈	PROPN
ejpam-5403	59	1	i	i	PRON
ejpam-5403	59	2	for	for	ADP
ejpam-5403	59	3	all	all	DET
ejpam-5403	59	4	a	a	DET
ejpam-5403	59	5	,	,	PUNCT
ejpam-5403	59	6	b	b	X
ejpam-5403	59	7	∈	∈	PROPN
ejpam-5403	59	8	i.	i.	NOUN
ejpam-5403	59	9	moreover	moreover	ADV
ejpam-5403	59	10	,	,	PUNCT
ejpam-5403	59	11	the	the	DET
ejpam-5403	59	12	authors	author	NOUN
ejpam-5403	59	13	in	in	ADP
ejpam-5403	59	14	[	[	X
ejpam-5403	59	15	3	3	NUM
ejpam-5403	59	16	]	]	PUNCT
ejpam-5403	59	17	proved	prove	VERB
ejpam-5403	59	18	that	that	SCONJ
ejpam-5403	59	19	if	if	SCONJ
ejpam-5403	59	20	every	every	DET
ejpam-5403	59	21	non	non	ADJ
ejpam-5403	59	22	-	-	ADJ
ejpam-5403	59	23	zero	zero	NUM
ejpam-5403	59	24	element	element	NOUN
ejpam-5403	59	25	of	of	ADP
ejpam-5403	59	26	x	x	PUNCT
ejpam-5403	59	27	is	be	AUX
ejpam-5403	59	28	an	an	DET
ejpam-5403	59	29	atom	atom	NOUN
ejpam-5403	59	30	,	,	PUNCT
ejpam-5403	59	31	then	then	ADV
ejpam-5403	59	32	any	any	DET
ejpam-5403	59	33	subalgebra	subalgebra	NOUN
ejpam-5403	59	34	of	of	ADP
ejpam-5403	59	35	x	x	PUNCT
ejpam-5403	59	36	is	be	AUX
ejpam-5403	59	37	an	an	DET
ejpam-5403	59	38	ideal	ideal	NOUN
ejpam-5403	59	39	of	of	ADP
ejpam-5403	59	40	x.	x.	PROPN
ejpam-5403	59	41	example	example	NOUN
ejpam-5403	60	1	1	1	X
ejpam-5403	60	2	.	.	PUNCT
ejpam-5403	61	1	let	let	VERB
ejpam-5403	61	2	x	x	PUNCT
ejpam-5403	61	3	=	=	PUNCT
ejpam-5403	61	4	{	{	PUNCT
ejpam-5403	61	5	0	0	NUM
ejpam-5403	61	6	,	,	PUNCT
ejpam-5403	61	7	a	a	DET
ejpam-5403	61	8	,	,	PUNCT
ejpam-5403	61	9	b	b	NOUN
ejpam-5403	61	10	,	,	PUNCT
ejpam-5403	61	11	c	c	NOUN
ejpam-5403	61	12	,	,	PUNCT
ejpam-5403	61	13	d	d	NOUN
ejpam-5403	61	14	}	}	PUNCT
ejpam-5403	61	15	and	and	CCONJ
ejpam-5403	61	16	y	y	PROPN
ejpam-5403	61	17	=	=	PUNCT
ejpam-5403	61	18	{	{	PUNCT
ejpam-5403	61	19	0	0	NUM
ejpam-5403	61	20	,	,	PUNCT
ejpam-5403	61	21	a	a	DET
ejpam-5403	61	22	,	,	PUNCT
ejpam-5403	61	23	b	b	NOUN
ejpam-5403	61	24	}	}	PUNCT
ejpam-5403	61	25	.	.	PUNCT
ejpam-5403	62	1	the	the	DET
ejpam-5403	62	2	binary	binary	PROPN
ejpam-5403	62	3	operations	operation	NOUN
ejpam-5403	62	4	∗	∗	NOUN
ejpam-5403	62	5	and	and	CCONJ
ejpam-5403	62	6	•	•	NOUN
ejpam-5403	62	7	be	be	AUX
ejpam-5403	62	8	defined	define	VERB
ejpam-5403	62	9	on	on	ADP
ejpam-5403	62	10	x	x	PUNCT
ejpam-5403	62	11	and	and	CCONJ
ejpam-5403	62	12	y	y	PROPN
ejpam-5403	62	13	as	as	ADP
ejpam-5403	62	14	the	the	DET
ejpam-5403	62	15	following	following	ADJ
ejpam-5403	62	16	tables	table	NOUN
ejpam-5403	62	17	:	:	PUNCT
ejpam-5403	62	18	∗	∗	NOUN
ejpam-5403	62	19	0	0	NUM
ejpam-5403	63	1	a	a	DET
ejpam-5403	63	2	b	b	NOUN
ejpam-5403	63	3	c	c	NOUN
ejpam-5403	63	4	d	d	NOUN
ejpam-5403	63	5	0	0	NUM
ejpam-5403	63	6	0	0	NUM
ejpam-5403	63	7	a	a	DET
ejpam-5403	63	8	c	c	PROPN
ejpam-5403	63	9	b	b	PROPN
ejpam-5403	63	10	b	b	PROPN
ejpam-5403	63	11	a	a	PRON
ejpam-5403	63	12	a	a	DET
ejpam-5403	63	13	0	0	NUM
ejpam-5403	63	14	b	b	NOUN
ejpam-5403	63	15	c	c	NOUN
ejpam-5403	63	16	c	c	PROPN
ejpam-5403	63	17	b	b	PROPN
ejpam-5403	63	18	b	b	PROPN
ejpam-5403	63	19	c	c	PROPN
ejpam-5403	63	20	0	0	NUM
ejpam-5403	63	21	a	a	DET
ejpam-5403	63	22	a	a	DET
ejpam-5403	63	23	c	c	NOUN
ejpam-5403	63	24	c	c	NOUN
ejpam-5403	63	25	b	b	PROPN
ejpam-5403	63	26	a	a	PRON
ejpam-5403	63	27	0	0	NUM
ejpam-5403	63	28	0	0	NUM
ejpam-5403	63	29	d	d	PROPN
ejpam-5403	63	30	d	d	PROPN
ejpam-5403	63	31	b	b	PROPN
ejpam-5403	63	32	a	a	DET
ejpam-5403	63	33	0	0	NUM
ejpam-5403	63	34	0	0	NUM
ejpam-5403	63	35	•	•	NOUN
ejpam-5403	63	36	0	0	NUM
ejpam-5403	63	37	a	a	DET
ejpam-5403	63	38	b	b	NOUN
ejpam-5403	63	39	0	0	NUM
ejpam-5403	63	40	0	0	NUM
ejpam-5403	64	1	a	a	DET
ejpam-5403	64	2	a	a	DET
ejpam-5403	64	3	a	a	DET
ejpam-5403	64	4	a	a	DET
ejpam-5403	64	5	0	0	NUM
ejpam-5403	64	6	0	0	NUM
ejpam-5403	64	7	b	b	PROPN
ejpam-5403	64	8	b	b	NOUN
ejpam-5403	64	9	0	0	NUM
ejpam-5403	64	10	0	0	NUM
ejpam-5403	65	1	it	it	PRON
ejpam-5403	65	2	is	be	AUX
ejpam-5403	65	3	a	a	DET
ejpam-5403	65	4	routine	routine	NOUN
ejpam-5403	65	5	to	to	PART
ejpam-5403	65	6	check	check	VERB
ejpam-5403	65	7	that	that	PRON
ejpam-5403	65	8	(	(	PUNCT
ejpam-5403	65	9	x	x	X
ejpam-5403	65	10	;	;	PUNCT
ejpam-5403	65	11	∗	∗	NOUN
ejpam-5403	65	12	,	,	PUNCT
ejpam-5403	65	13	0	0	NUM
ejpam-5403	65	14	)	)	PUNCT
ejpam-5403	65	15	and	and	CCONJ
ejpam-5403	65	16	(	(	PUNCT
ejpam-5403	65	17	y	y	PROPN
ejpam-5403	65	18	;	;	PUNCT
ejpam-5403	65	19	•	•	X
ejpam-5403	65	20	,	,	PUNCT
ejpam-5403	65	21	0	0	NUM
ejpam-5403	65	22	)	)	PUNCT
ejpam-5403	65	23	are	be	AUX
ejpam-5403	65	24	q	q	NOUN
ejpam-5403	65	25	-	-	PUNCT
ejpam-5403	65	26	algebras	algebras	X
ejpam-5403	65	27	.	.	PUNCT
ejpam-5403	66	1	it	it	PRON
ejpam-5403	66	2	is	be	AUX
ejpam-5403	66	3	easy	easy	ADJ
ejpam-5403	66	4	to	to	PART
ejpam-5403	66	5	see	see	VERB
ejpam-5403	66	6	that	that	DET
ejpam-5403	66	7	g(x	g(x	NOUN
ejpam-5403	66	8	)	)	PUNCT
ejpam-5403	66	9	=	=	PRON
ejpam-5403	67	1	{	{	PUNCT
ejpam-5403	67	2	0	0	NUM
ejpam-5403	67	3	,	,	PUNCT
ejpam-5403	67	4	a	a	PRON
ejpam-5403	67	5	}	}	PUNCT
ejpam-5403	67	6	,	,	PUNCT
ejpam-5403	67	7	a(x	a(x	PROPN
ejpam-5403	67	8	)	)	PUNCT
ejpam-5403	67	9	=	=	PUNCT
ejpam-5403	67	10	{	{	PUNCT
ejpam-5403	67	11	0	0	NUM
ejpam-5403	67	12	,	,	PUNCT
ejpam-5403	67	13	a	a	PRON
ejpam-5403	67	14	,	,	PUNCT
ejpam-5403	67	15	b	b	NOUN
ejpam-5403	67	16	}	}	PUNCT
ejpam-5403	67	17	and	and	CCONJ
ejpam-5403	67	18	g(y	g(y	PROPN
ejpam-5403	67	19	)	)	PUNCT
ejpam-5403	68	1	=	=	PUNCT
ejpam-5403	68	2	{	{	PUNCT
ejpam-5403	68	3	0	0	NUM
ejpam-5403	68	4	,	,	PUNCT
ejpam-5403	68	5	a	a	PRON
ejpam-5403	68	6	}	}	PUNCT
ejpam-5403	68	7	,	,	PUNCT
ejpam-5403	68	8	a(y	a(y	PROPN
ejpam-5403	68	9	)	)	PUNCT
ejpam-5403	68	10	=	=	PUNCT
ejpam-5403	68	11	{	{	PUNCT
ejpam-5403	68	12	0	0	NUM
ejpam-5403	68	13	}	}	PUNCT
ejpam-5403	68	14	.	.	PUNCT
ejpam-5403	69	1	moreover	moreover	ADV
ejpam-5403	69	2	,	,	PUNCT
ejpam-5403	69	3	we	we	PRON
ejpam-5403	69	4	get	get	VERB
ejpam-5403	69	5	that	that	DET
ejpam-5403	69	6	g(x	g(x	NOUN
ejpam-5403	69	7	)	)	PUNCT
ejpam-5403	69	8	is	be	AUX
ejpam-5403	69	9	an	an	DET
ejpam-5403	69	10	ideal	ideal	NOUN
ejpam-5403	69	11	and	and	CCONJ
ejpam-5403	69	12	a	a	DET
ejpam-5403	69	13	subalgebra	subalgebra	NOUN
ejpam-5403	69	14	of	of	ADP
ejpam-5403	69	15	x.	x.	NOUN
ejpam-5403	69	16	in	in	ADP
ejpam-5403	69	17	this	this	DET
ejpam-5403	69	18	paper	paper	NOUN
ejpam-5403	69	19	,	,	PUNCT
ejpam-5403	69	20	we	we	PRON
ejpam-5403	69	21	examine	examine	VERB
ejpam-5403	69	22	the	the	DET
ejpam-5403	69	23	properties	property	NOUN
ejpam-5403	69	24	of	of	ADP
ejpam-5403	69	25	atoms	atom	NOUN
ejpam-5403	69	26	and	and	CCONJ
ejpam-5403	69	27	strong	strong	ADJ
ejpam-5403	69	28	atoms	atom	NOUN
ejpam-5403	69	29	.	.	PUNCT
ejpam-5403	70	1	we	we	PRON
ejpam-5403	70	2	also	also	ADV
ejpam-5403	70	3	show	show	VERB
ejpam-5403	70	4	some	some	DET
ejpam-5403	70	5	relations	relation	NOUN
ejpam-5403	70	6	between	between	ADP
ejpam-5403	70	7	a	a	DET
ejpam-5403	70	8	set	set	VERB
ejpam-5403	70	9	g	g	NOUN
ejpam-5403	70	10	-	-	PUNCT
ejpam-5403	70	11	part	part	NOUN
ejpam-5403	70	12	g(x	g(x	NOUN
ejpam-5403	70	13	)	)	PUNCT
ejpam-5403	70	14	,	,	PUNCT
ejpam-5403	70	15	a	a	DET
ejpam-5403	70	16	set	set	NOUN
ejpam-5403	70	17	of	of	ADP
ejpam-5403	70	18	all	all	DET
ejpam-5403	70	19	atoms	atom	NOUN
ejpam-5403	70	20	a(x	a(x	NOUN
ejpam-5403	70	21	)	)	PUNCT
ejpam-5403	70	22	and	and	CCONJ
ejpam-5403	70	23	a	a	DET
ejpam-5403	70	24	set	set	NOUN
ejpam-5403	70	25	of	of	ADP
ejpam-5403	70	26	all	all	DET
ejpam-5403	70	27	strong	strong	ADJ
ejpam-5403	70	28	atoms	atom	NOUN
ejpam-5403	70	29	of	of	ADP
ejpam-5403	70	30	a	a	DET
ejpam-5403	70	31	q	q	NOUN
ejpam-5403	70	32	-	-	NOUN
ejpam-5403	70	33	algebra	algebra	NOUN
ejpam-5403	70	34	x	x	PUNCT
ejpam-5403	70	35	that	that	PRON
ejpam-5403	70	36	involve	involve	VERB
ejpam-5403	70	37	with	with	ADP
ejpam-5403	70	38	ideal	ideal	ADJ
ejpam-5403	70	39	property	property	NOUN
ejpam-5403	70	40	.	.	PUNCT
ejpam-5403	71	1	now	now	ADV
ejpam-5403	71	2	we	we	PRON
ejpam-5403	71	3	will	will	AUX
ejpam-5403	71	4	review	review	VERB
ejpam-5403	71	5	some	some	DET
ejpam-5403	71	6	properties	property	NOUN
ejpam-5403	71	7	and	and	CCONJ
ejpam-5403	71	8	theorems	theorem	NOUN
ejpam-5403	71	9	that	that	SCONJ
ejpam-5403	71	10	we	we	PRON
ejpam-5403	71	11	will	will	AUX
ejpam-5403	71	12	use	use	VERB
ejpam-5403	71	13	later	later	ADV
ejpam-5403	71	14	.	.	PUNCT
ejpam-5403	72	1	in	in	ADP
ejpam-5403	72	2	[	[	X
ejpam-5403	72	3	9	9	NUM
ejpam-5403	72	4	]	]	PUNCT
ejpam-5403	72	5	and	and	CCONJ
ejpam-5403	72	6	[	[	X
ejpam-5403	72	7	3	3	X
ejpam-5403	72	8	]	]	PUNCT
ejpam-5403	72	9	gave	give	VERB
ejpam-5403	72	10	us	we	PRON
ejpam-5403	72	11	some	some	DET
ejpam-5403	72	12	calculations	calculation	NOUN
ejpam-5403	72	13	and	and	CCONJ
ejpam-5403	72	14	showed	show	VERB
ejpam-5403	72	15	a	a	DET
ejpam-5403	72	16	left	left	ADJ
ejpam-5403	72	17	cancellation	cancellation	NOUN
ejpam-5403	72	18	law	law	NOUN
ejpam-5403	72	19	in	in	ADP
ejpam-5403	72	20	a	a	DET
ejpam-5403	72	21	q	q	NOUN
ejpam-5403	72	22	-	-	PUNCT
ejpam-5403	72	23	algebra	algebra	NOUN
ejpam-5403	72	24	x.	x.	NOUN
ejpam-5403	72	25	lemma	lemma	PROPN
ejpam-5403	72	26	1	1	NUM
ejpam-5403	72	27	.	.	PUNCT
ejpam-5403	73	1	[	[	X
ejpam-5403	73	2	9	9	NUM
ejpam-5403	73	3	]	]	PUNCT
ejpam-5403	73	4	let	let	VERB
ejpam-5403	73	5	x	x	PRON
ejpam-5403	73	6	be	be	AUX
ejpam-5403	73	7	a	a	DET
ejpam-5403	73	8	q	q	NOUN
ejpam-5403	73	9	-	-	PUNCT
ejpam-5403	73	10	algebra	algebra	NOUN
ejpam-5403	73	11	and	and	CCONJ
ejpam-5403	73	12	a	a	DET
ejpam-5403	73	13	,	,	PUNCT
ejpam-5403	73	14	b	b	NOUN
ejpam-5403	73	15	,	,	PUNCT
ejpam-5403	73	16	c	c	PROPN
ejpam-5403	73	17	∈	∈	PROPN
ejpam-5403	73	18	x.	x.	NOUN
ejpam-5403	74	1	if	if	SCONJ
ejpam-5403	74	2	ab	ab	PROPN
ejpam-5403	74	3	=	=	SYM
ejpam-5403	74	4	ac	ac	PROPN
ejpam-5403	74	5	,	,	PUNCT
ejpam-5403	74	6	then	then	ADV
ejpam-5403	74	7	0b	0b	NOUN
ejpam-5403	74	8	=	=	SYM
ejpam-5403	74	9	0c	0c	PROPN
ejpam-5403	74	10	.	.	PUNCT
ejpam-5403	74	11	a.	a.	PROPN
ejpam-5403	74	12	anantayasethi	anantayasethi	PROPN
ejpam-5403	74	13	,	,	PUNCT
ejpam-5403	74	14	t.	t.	PROPN
ejpam-5403	74	15	kunawat	kunawat	PROPN
ejpam-5403	74	16	,	,	PUNCT
ejpam-5403	74	17	p.	p.	PROPN
ejpam-5403	74	18	moolnipa	moolnipa	PROPN
ejpam-5403	74	19	/	/	SYM
ejpam-5403	74	20	eur	eur	PROPN
ejpam-5403	74	21	.	.	PUNCT
ejpam-5403	75	1	j.	j.	PROPN
ejpam-5403	75	2	pure	pure	PROPN
ejpam-5403	75	3	appl	appl	PROPN
ejpam-5403	75	4	.	.	PROPN
ejpam-5403	75	5	math	math	PROPN
ejpam-5403	75	6	,	,	PUNCT
ejpam-5403	75	7	17	17	NUM
ejpam-5403	75	8	(	(	PUNCT
ejpam-5403	75	9	4	4	NUM
ejpam-5403	75	10	)	)	PUNCT
ejpam-5403	75	11	(	(	PUNCT
ejpam-5403	75	12	2024	2024	NUM
ejpam-5403	75	13	)	)	PUNCT
ejpam-5403	75	14	,	,	PUNCT
ejpam-5403	75	15	3268	3268	NUM
ejpam-5403	75	16	-	-	SYM
ejpam-5403	75	17	3276	3276	NUM
ejpam-5403	75	18	3270	3270	NUM
ejpam-5403	75	19	corollary	corollary	NOUN
ejpam-5403	75	20	1	1	NUM
ejpam-5403	75	21	.	.	PUNCT
ejpam-5403	76	1	[	[	X
ejpam-5403	76	2	9	9	NUM
ejpam-5403	76	3	]	]	PUNCT
ejpam-5403	76	4	a	a	DET
ejpam-5403	76	5	left	left	ADJ
ejpam-5403	76	6	cancellation	cancellation	NOUN
ejpam-5403	76	7	law	law	NOUN
ejpam-5403	76	8	holds	hold	VERB
ejpam-5403	76	9	in	in	ADP
ejpam-5403	76	10	g(x	g(x	NOUN
ejpam-5403	76	11	)	)	PUNCT
ejpam-5403	76	12	,	,	PUNCT
ejpam-5403	76	13	i.	i.	PROPN
ejpam-5403	76	14	e.	e.	PROPN
ejpam-5403	76	15	for	for	ADP
ejpam-5403	76	16	all	all	DET
ejpam-5403	76	17	a	a	DET
ejpam-5403	76	18	,	,	PUNCT
ejpam-5403	76	19	b	b	NOUN
ejpam-5403	76	20	,	,	PUNCT
ejpam-5403	76	21	c	c	PROPN
ejpam-5403	76	22	∈	∈	PROPN
ejpam-5403	76	23	g(x	g(x	PROPN
ejpam-5403	76	24	)	)	PUNCT
ejpam-5403	76	25	,	,	PUNCT
ejpam-5403	76	26	ab	ab	PROPN
ejpam-5403	76	27	=	=	PUNCT
ejpam-5403	76	28	ac	ac	PROPN
ejpam-5403	76	29	implies	imply	VERB
ejpam-5403	76	30	b	b	PROPN
ejpam-5403	76	31	=	=	SYM
ejpam-5403	76	32	c.	c.	PROPN
ejpam-5403	76	33	lemma	lemma	PROPN
ejpam-5403	76	34	2	2	X
ejpam-5403	76	35	.	.	PUNCT
ejpam-5403	77	1	[	[	X
ejpam-5403	77	2	3	3	X
ejpam-5403	77	3	]	]	PUNCT
ejpam-5403	77	4	every	every	DET
ejpam-5403	77	5	q	q	NOUN
ejpam-5403	77	6	-	-	NOUN
ejpam-5403	77	7	algebra	algebra	NOUN
ejpam-5403	77	8	x	x	SYM
ejpam-5403	77	9	satisfies	satisfy	VERB
ejpam-5403	77	10	the	the	DET
ejpam-5403	77	11	following	follow	VERB
ejpam-5403	77	12	property	property	NOUN
ejpam-5403	77	13	:	:	PUNCT
ejpam-5403	77	14	0(xy	0(xy	NUM
ejpam-5403	77	15	)	)	PUNCT
ejpam-5403	78	1	=	=	PRON
ejpam-5403	78	2	(	(	PUNCT
ejpam-5403	78	3	0x)(0y	0x)(0y	NOUN
ejpam-5403	78	4	)	)	PUNCT
ejpam-5403	78	5	for	for	ADP
ejpam-5403	78	6	all	all	DET
ejpam-5403	78	7	x	x	NOUN
ejpam-5403	78	8	,	,	PUNCT
ejpam-5403	78	9	y	y	PROPN
ejpam-5403	78	10	∈	∈	PROPN
ejpam-5403	78	11	x.	x.	NOUN
ejpam-5403	78	12	in	in	ADP
ejpam-5403	78	13	[	[	X
ejpam-5403	78	14	10	10	NUM
ejpam-5403	78	15	]	]	PUNCT
ejpam-5403	78	16	,	,	PUNCT
ejpam-5403	78	17	some	some	DET
ejpam-5403	78	18	informations	information	NOUN
ejpam-5403	78	19	and	and	CCONJ
ejpam-5403	78	20	calculations	calculation	NOUN
ejpam-5403	78	21	in	in	ADP
ejpam-5403	78	22	g(x	g(x	NOUN
ejpam-5403	78	23	)	)	PUNCT
ejpam-5403	78	24	are	be	AUX
ejpam-5403	78	25	presented	present	VERB
ejpam-5403	78	26	.	.	PUNCT
ejpam-5403	79	1	proposition	proposition	NOUN
ejpam-5403	79	2	1	1	NUM
ejpam-5403	79	3	.	.	PUNCT
ejpam-5403	80	1	[	[	X
ejpam-5403	80	2	10	10	NUM
ejpam-5403	80	3	]	]	PUNCT
ejpam-5403	80	4	let	let	VERB
ejpam-5403	80	5	x	x	PRON
ejpam-5403	80	6	be	be	AUX
ejpam-5403	80	7	a	a	DET
ejpam-5403	80	8	q	q	NOUN
ejpam-5403	80	9	-	-	PUNCT
ejpam-5403	80	10	algebra	algebra	NOUN
ejpam-5403	80	11	and	and	CCONJ
ejpam-5403	80	12	x	x	PART
ejpam-5403	80	13	∈	∈	PROPN
ejpam-5403	80	14	x.	x.	NOUN
ejpam-5403	80	15	then	then	ADV
ejpam-5403	80	16	0x	0x	NOUN
ejpam-5403	80	17	∈	∈	PROPN
ejpam-5403	80	18	g(x	g(x	NOUN
ejpam-5403	80	19	)	)	PUNCT
ejpam-5403	81	1	if	if	SCONJ
ejpam-5403	81	2	and	and	CCONJ
ejpam-5403	81	3	only	only	ADV
ejpam-5403	81	4	if	if	SCONJ
ejpam-5403	81	5	(	(	PUNCT
ejpam-5403	81	6	0x)x	0x)x	NUM
ejpam-5403	81	7	=	=	SYM
ejpam-5403	81	8	0	0	X
ejpam-5403	81	9	.	.	PUNCT
ejpam-5403	81	10	proposition	proposition	NOUN
ejpam-5403	81	11	2	2	NUM
ejpam-5403	81	12	.	.	PUNCT
ejpam-5403	82	1	[	[	X
ejpam-5403	82	2	10	10	NUM
ejpam-5403	82	3	]	]	PUNCT
ejpam-5403	82	4	let	let	VERB
ejpam-5403	82	5	x	x	PRON
ejpam-5403	82	6	be	be	AUX
ejpam-5403	82	7	a	a	DET
ejpam-5403	82	8	q	q	NOUN
ejpam-5403	82	9	-	-	NOUN
ejpam-5403	82	10	algebra	algebra	NOUN
ejpam-5403	82	11	.	.	PUNCT
ejpam-5403	83	1	if	if	SCONJ
ejpam-5403	83	2	a	a	PRON
ejpam-5403	83	3	,	,	PUNCT
ejpam-5403	83	4	b	b	PROPN
ejpam-5403	83	5	∈	∈	PROPN
ejpam-5403	83	6	g(x	g(x	PROPN
ejpam-5403	83	7	)	)	PUNCT
ejpam-5403	83	8	,	,	PUNCT
ejpam-5403	83	9	then	then	ADV
ejpam-5403	83	10	ab	ab	PROPN
ejpam-5403	83	11	=	=	PROPN
ejpam-5403	83	12	ba	ba	PROPN
ejpam-5403	83	13	.	.	PUNCT
ejpam-5403	83	14	proposition	proposition	NOUN
ejpam-5403	83	15	3	3	NUM
ejpam-5403	83	16	.	.	PUNCT
ejpam-5403	84	1	[	[	X
ejpam-5403	84	2	10	10	NUM
ejpam-5403	84	3	]	]	PUNCT
ejpam-5403	84	4	let	let	VERB
ejpam-5403	84	5	x	x	PRON
ejpam-5403	84	6	be	be	AUX
ejpam-5403	84	7	a	a	DET
ejpam-5403	84	8	q	q	NOUN
ejpam-5403	84	9	-	-	PUNCT
ejpam-5403	84	10	algebra	algebra	NOUN
ejpam-5403	84	11	and	and	CCONJ
ejpam-5403	84	12	a	a	DET
ejpam-5403	84	13	,	,	PUNCT
ejpam-5403	84	14	b	b	NOUN
ejpam-5403	84	15	,	,	PUNCT
ejpam-5403	84	16	c	c	PROPN
ejpam-5403	84	17	∈	∈	PROPN
ejpam-5403	84	18	g(x	g(x	PROPN
ejpam-5403	84	19	)	)	PUNCT
ejpam-5403	84	20	.	.	PUNCT
ejpam-5403	85	1	then	then	ADV
ejpam-5403	85	2	the	the	DET
ejpam-5403	85	3	following	follow	VERB
ejpam-5403	85	4	three	three	NUM
ejpam-5403	85	5	properties	property	NOUN
ejpam-5403	85	6	hold	hold	VERB
ejpam-5403	85	7	:	:	PUNCT
ejpam-5403	85	8	(	(	PUNCT
ejpam-5403	85	9	i	i	NOUN
ejpam-5403	85	10	)	)	PUNCT
ejpam-5403	85	11	if	if	SCONJ
ejpam-5403	85	12	a	a	DET
ejpam-5403	85	13	̸=	̸=	PROPN
ejpam-5403	85	14	b	b	PROPN
ejpam-5403	85	15	,	,	PUNCT
ejpam-5403	85	16	then	then	ADV
ejpam-5403	85	17	ab	ab	PROPN
ejpam-5403	85	18	/∈	/∈	PUNCT
ejpam-5403	86	1	{	{	PUNCT
ejpam-5403	86	2	0	0	NUM
ejpam-5403	86	3	,	,	PUNCT
ejpam-5403	86	4	a	a	DET
ejpam-5403	86	5	,	,	PUNCT
ejpam-5403	86	6	b	b	NOUN
ejpam-5403	86	7	}	}	PUNCT
ejpam-5403	86	8	for	for	ADP
ejpam-5403	86	9	a	a	DET
ejpam-5403	86	10	̸=	̸=	PROPN
ejpam-5403	86	11	0	0	NUM
ejpam-5403	86	12	and	and	CCONJ
ejpam-5403	86	13	b	b	X
ejpam-5403	86	14	̸=	̸=	PROPN
ejpam-5403	86	15	0	0	NUM
ejpam-5403	86	16	.	.	PUNCT
ejpam-5403	87	1	(	(	PUNCT
ejpam-5403	87	2	ii	ii	NOUN
ejpam-5403	87	3	)	)	PUNCT
ejpam-5403	88	1	if	if	SCONJ
ejpam-5403	88	2	ab	ab	PROPN
ejpam-5403	88	3	=	=	SYM
ejpam-5403	88	4	c	c	PROPN
ejpam-5403	88	5	,	,	PUNCT
ejpam-5403	88	6	then	then	ADV
ejpam-5403	88	7	ac	ac	PROPN
ejpam-5403	88	8	=	=	PROPN
ejpam-5403	88	9	b	b	PROPN
ejpam-5403	88	10	and	and	CCONJ
ejpam-5403	88	11	bc	bc	PROPN
ejpam-5403	88	12	=	=	SYM
ejpam-5403	88	13	a.	a.	PROPN
ejpam-5403	88	14	(	(	PUNCT
ejpam-5403	88	15	iii	iii	NOUN
ejpam-5403	88	16	)	)	PUNCT
ejpam-5403	88	17	xa	xa	PROPN
ejpam-5403	89	1	̸=	̸=	PROPN
ejpam-5403	89	2	x	x	PUNCT
ejpam-5403	89	3	for	for	ADP
ejpam-5403	89	4	all	all	DET
ejpam-5403	89	5	0	0	NUM
ejpam-5403	89	6	̸=	̸=	NOUN
ejpam-5403	89	7	x	x	SYM
ejpam-5403	89	8	∈	∈	PROPN
ejpam-5403	89	9	x	x	X
ejpam-5403	89	10	and	and	CCONJ
ejpam-5403	89	11	a	a	DET
ejpam-5403	89	12	̸=	̸=	PROPN
ejpam-5403	89	13	0	0	NUM
ejpam-5403	89	14	.	.	NOUN
ejpam-5403	89	15	2	2	NUM
ejpam-5403	89	16	.	.	X
ejpam-5403	89	17	g	g	NOUN
ejpam-5403	89	18	-	-	PUNCT
ejpam-5403	89	19	part	part	NOUN
ejpam-5403	89	20	and	and	CCONJ
ejpam-5403	89	21	atoms	atom	NOUN
ejpam-5403	89	22	in	in	ADP
ejpam-5403	89	23	this	this	DET
ejpam-5403	89	24	section	section	NOUN
ejpam-5403	89	25	,	,	PUNCT
ejpam-5403	89	26	we	we	PRON
ejpam-5403	89	27	investigate	investigate	VERB
ejpam-5403	89	28	some	some	DET
ejpam-5403	89	29	properties	property	NOUN
ejpam-5403	89	30	of	of	ADP
ejpam-5403	89	31	a	a	DET
ejpam-5403	89	32	set	set	VERB
ejpam-5403	89	33	g	g	NOUN
ejpam-5403	89	34	-	-	PUNCT
ejpam-5403	89	35	part	part	NOUN
ejpam-5403	89	36	g(x	g(x	NOUN
ejpam-5403	89	37	)	)	PUNCT
ejpam-5403	89	38	,	,	PUNCT
ejpam-5403	89	39	atoms	atom	NOUN
ejpam-5403	89	40	and	and	CCONJ
ejpam-5403	89	41	strong	strong	ADJ
ejpam-5403	89	42	atoms	atom	NOUN
ejpam-5403	89	43	of	of	ADP
ejpam-5403	89	44	a	a	DET
ejpam-5403	89	45	q	q	NOUN
ejpam-5403	89	46	-	-	PUNCT
ejpam-5403	89	47	algebra	algebra	NOUN
ejpam-5403	89	48	x.	x.	NOUN
ejpam-5403	89	49	we	we	PRON
ejpam-5403	89	50	also	also	ADV
ejpam-5403	89	51	present	present	VERB
ejpam-5403	89	52	some	some	DET
ejpam-5403	89	53	connections	connection	NOUN
ejpam-5403	89	54	among	among	ADP
ejpam-5403	89	55	them	they	PRON
ejpam-5403	89	56	.	.	PUNCT
ejpam-5403	90	1	first	first	ADV
ejpam-5403	90	2	,	,	PUNCT
ejpam-5403	90	3	we	we	PRON
ejpam-5403	90	4	will	will	AUX
ejpam-5403	90	5	mention	mention	VERB
ejpam-5403	90	6	some	some	DET
ejpam-5403	90	7	results	result	NOUN
ejpam-5403	90	8	of	of	ADP
ejpam-5403	90	9	atoms	atom	NOUN
ejpam-5403	90	10	in	in	ADP
ejpam-5403	90	11	[	[	X
ejpam-5403	90	12	3	3	NUM
ejpam-5403	90	13	]	]	PUNCT
ejpam-5403	90	14	.	.	PUNCT
ejpam-5403	91	1	theorem	theorem	NOUN
ejpam-5403	91	2	1	1	NUM
ejpam-5403	91	3	.	.	PUNCT
ejpam-5403	92	1	[	[	X
ejpam-5403	92	2	3	3	X
ejpam-5403	92	3	]	]	X
ejpam-5403	92	4	let	let	VERB
ejpam-5403	92	5	x	x	PRON
ejpam-5403	92	6	be	be	AUX
ejpam-5403	92	7	a	a	DET
ejpam-5403	92	8	q	q	NOUN
ejpam-5403	92	9	-	-	NOUN
ejpam-5403	92	10	algebra	algebra	NOUN
ejpam-5403	92	11	.	.	PUNCT
ejpam-5403	93	1	then	then	ADV
ejpam-5403	93	2	for	for	ADP
ejpam-5403	93	3	all	all	DET
ejpam-5403	93	4	x	x	NOUN
ejpam-5403	93	5	,	,	PUNCT
ejpam-5403	93	6	z	z	PROPN
ejpam-5403	93	7	,	,	PUNCT
ejpam-5403	93	8	u	u	NOUN
ejpam-5403	93	9	of	of	ADP
ejpam-5403	93	10	x	x	PRON
ejpam-5403	93	11	,	,	PUNCT
ejpam-5403	93	12	the	the	DET
ejpam-5403	93	13	following	follow	VERB
ejpam-5403	93	14	conditions	condition	NOUN
ejpam-5403	93	15	are	be	AUX
ejpam-5403	93	16	equivalent	equivalent	ADJ
ejpam-5403	93	17	:	:	PUNCT
ejpam-5403	93	18	(	(	PUNCT
ejpam-5403	93	19	i	i	NOUN
ejpam-5403	93	20	)	)	PUNCT
ejpam-5403	93	21	x	x	X
ejpam-5403	93	22	is	be	AUX
ejpam-5403	93	23	atom	atom	ADJ
ejpam-5403	93	24	;	;	PUNCT
ejpam-5403	93	25	(	(	PUNCT
ejpam-5403	93	26	ii	ii	NOUN
ejpam-5403	93	27	)	)	PUNCT
ejpam-5403	93	28	x	x	X
ejpam-5403	94	1	=	=	SYM
ejpam-5403	94	2	z(zx	z(zx	PROPN
ejpam-5403	94	3	)	)	PUNCT
ejpam-5403	94	4	;	;	PUNCT
ejpam-5403	94	5	(	(	PUNCT
ejpam-5403	94	6	iii	iii	X
ejpam-5403	94	7	)	)	PUNCT
ejpam-5403	94	8	(	(	PUNCT
ejpam-5403	94	9	zu)(zx	zu)(zx	PROPN
ejpam-5403	94	10	)	)	PUNCT
ejpam-5403	94	11	=	=	SYM
ejpam-5403	94	12	xu	xu	PROPN
ejpam-5403	94	13	.	.	PUNCT
ejpam-5403	94	14	theorem	theorem	VERB
ejpam-5403	94	15	2	2	NUM
ejpam-5403	94	16	.	.	PUNCT
ejpam-5403	95	1	[	[	X
ejpam-5403	95	2	3	3	X
ejpam-5403	95	3	]	]	X
ejpam-5403	95	4	let	let	VERB
ejpam-5403	95	5	x	x	PRON
ejpam-5403	95	6	be	be	AUX
ejpam-5403	95	7	a	a	DET
ejpam-5403	95	8	q	q	NOUN
ejpam-5403	95	9	-	-	PUNCT
ejpam-5403	95	10	algebra	algebra	NOUN
ejpam-5403	95	11	and	and	CCONJ
ejpam-5403	95	12	x	x	SYM
ejpam-5403	95	13	∈	∈	PROPN
ejpam-5403	95	14	x.	x.	NOUN
ejpam-5403	95	15	if	if	SCONJ
ejpam-5403	95	16	x	x	PRON
ejpam-5403	95	17	is	be	AUX
ejpam-5403	95	18	an	an	DET
ejpam-5403	95	19	atom	atom	NOUN
ejpam-5403	95	20	of	of	ADP
ejpam-5403	95	21	x	x	NOUN
ejpam-5403	95	22	,	,	PUNCT
ejpam-5403	95	23	then	then	ADV
ejpam-5403	95	24	the	the	DET
ejpam-5403	95	25	following	follow	VERB
ejpam-5403	95	26	properties	property	NOUN
ejpam-5403	95	27	are	be	AUX
ejpam-5403	95	28	satisfied	satisfied	ADJ
ejpam-5403	95	29	:	:	PUNCT
ejpam-5403	95	30	(	(	PUNCT
ejpam-5403	95	31	iv	iv	X
ejpam-5403	95	32	)	)	PUNCT
ejpam-5403	95	33	0(zx	0(zx	NOUN
ejpam-5403	95	34	)	)	PUNCT
ejpam-5403	96	1	=	=	SYM
ejpam-5403	96	2	xz	xz	PROPN
ejpam-5403	96	3	for	for	ADP
ejpam-5403	96	4	all	all	DET
ejpam-5403	96	5	z	z	NOUN
ejpam-5403	96	6	∈	∈	NOUN
ejpam-5403	96	7	x.	x.	NOUN
ejpam-5403	96	8	(	(	PUNCT
ejpam-5403	96	9	v	v	NOUN
ejpam-5403	96	10	)	)	PUNCT
ejpam-5403	96	11	0(0x	0(0x	NUM
ejpam-5403	96	12	)	)	PUNCT
ejpam-5403	97	1	=	=	PUNCT
ejpam-5403	97	2	x.	x.	NOUN
ejpam-5403	97	3	the	the	DET
ejpam-5403	97	4	converse	converse	NOUN
ejpam-5403	97	5	of	of	ADP
ejpam-5403	97	6	theorem	theorem	ADJ
ejpam-5403	97	7	2	2	NUM
ejpam-5403	97	8	is	be	AUX
ejpam-5403	97	9	not	not	PART
ejpam-5403	97	10	true	true	ADJ
ejpam-5403	97	11	.	.	PUNCT
ejpam-5403	98	1	the	the	DET
ejpam-5403	98	2	following	following	ADJ
ejpam-5403	98	3	example	example	NOUN
ejpam-5403	98	4	is	be	AUX
ejpam-5403	98	5	a	a	DET
ejpam-5403	98	6	counterexample	counterexample	NOUN
ejpam-5403	98	7	.	.	PUNCT
ejpam-5403	98	8	example	example	NOUN
ejpam-5403	99	1	2	2	NUM
ejpam-5403	99	2	.	.	X
ejpam-5403	99	3	consider	consider	VERB
ejpam-5403	99	4	a	a	DET
ejpam-5403	99	5	q	q	NOUN
ejpam-5403	99	6	-	-	NOUN
ejpam-5403	99	7	algebra	algebra	NOUN
ejpam-5403	99	8	x	x	VERB
ejpam-5403	99	9	from	from	ADP
ejpam-5403	99	10	example	example	NOUN
ejpam-5403	99	11	1	1	NUM
ejpam-5403	99	12	.	.	PUNCT
ejpam-5403	100	1	for	for	ADP
ejpam-5403	100	2	all	all	DET
ejpam-5403	100	3	z	z	NOUN
ejpam-5403	100	4	∈	∈	PROPN
ejpam-5403	100	5	x	x	NOUN
ejpam-5403	100	6	,	,	PUNCT
ejpam-5403	100	7	we	we	PRON
ejpam-5403	100	8	get	get	VERB
ejpam-5403	100	9	that	that	PRON
ejpam-5403	100	10	0(zc	0(zc	NOUN
ejpam-5403	100	11	)	)	PUNCT
ejpam-5403	101	1	=	=	SYM
ejpam-5403	101	2	cz	cz	NOUN
ejpam-5403	101	3	but	but	CCONJ
ejpam-5403	101	4	an	an	DET
ejpam-5403	101	5	element	element	NOUN
ejpam-5403	101	6	c	c	NOUN
ejpam-5403	101	7	is	be	AUX
ejpam-5403	101	8	not	not	PART
ejpam-5403	101	9	an	an	DET
ejpam-5403	101	10	atom	atom	NOUN
ejpam-5403	101	11	of	of	ADP
ejpam-5403	101	12	x.	x.	NOUN
ejpam-5403	101	13	hence	hence	ADV
ejpam-5403	101	14	,	,	PUNCT
ejpam-5403	101	15	the	the	DET
ejpam-5403	101	16	converse	converse	NOUN
ejpam-5403	101	17	of	of	ADP
ejpam-5403	101	18	theorem	theorem	NOUN
ejpam-5403	101	19	2(iv	2(iv	NUM
ejpam-5403	101	20	)	)	PUNCT
ejpam-5403	101	21	is	be	AUX
ejpam-5403	101	22	not	not	PART
ejpam-5403	101	23	ture	ture	ADJ
ejpam-5403	101	24	.	.	PUNCT
ejpam-5403	102	1	beside	beside	ADP
ejpam-5403	102	2	that	that	PRON
ejpam-5403	102	3	,	,	PUNCT
ejpam-5403	102	4	the	the	DET
ejpam-5403	102	5	converse	converse	NOUN
ejpam-5403	102	6	of	of	ADP
ejpam-5403	102	7	(	(	PUNCT
ejpam-5403	102	8	v	v	NOUN
ejpam-5403	102	9	)	)	PUNCT
ejpam-5403	102	10	is	be	AUX
ejpam-5403	102	11	also	also	ADV
ejpam-5403	102	12	not	not	PART
ejpam-5403	102	13	ture	ture	ADJ
ejpam-5403	102	14	since	since	SCONJ
ejpam-5403	102	15	0(0c	0(0c	NUM
ejpam-5403	102	16	)	)	PUNCT
ejpam-5403	102	17	=	=	NOUN
ejpam-5403	102	18	0b	0b	NOUN
ejpam-5403	102	19	=	=	SYM
ejpam-5403	102	20	c	c	NOUN
ejpam-5403	103	1	but	but	CCONJ
ejpam-5403	103	2	c	c	PROPN
ejpam-5403	103	3	is	be	AUX
ejpam-5403	103	4	not	not	PART
ejpam-5403	103	5	an	an	DET
ejpam-5403	103	6	atom	atom	NOUN
ejpam-5403	103	7	.	.	PUNCT
ejpam-5403	104	1	from	from	ADP
ejpam-5403	104	2	theorem	theorem	NOUN
ejpam-5403	104	3	2	2	NUM
ejpam-5403	104	4	we	we	PRON
ejpam-5403	104	5	get	get	VERB
ejpam-5403	104	6	that	that	SCONJ
ejpam-5403	104	7	every	every	DET
ejpam-5403	104	8	atom	atom	NOUN
ejpam-5403	104	9	of	of	ADP
ejpam-5403	104	10	q	q	NOUN
ejpam-5403	104	11	-	-	PUNCT
ejpam-5403	104	12	algebra	algebra	NOUN
ejpam-5403	104	13	x	x	PUNCT
ejpam-5403	104	14	is	be	AUX
ejpam-5403	104	15	a	a	DET
ejpam-5403	104	16	product	product	NOUN
ejpam-5403	104	17	of	of	ADP
ejpam-5403	104	18	0	0	NUM
ejpam-5403	104	19	and	and	CCONJ
ejpam-5403	104	20	some	some	DET
ejpam-5403	104	21	element	element	NOUN
ejpam-5403	104	22	of	of	ADP
ejpam-5403	104	23	x	x	PUNCT
ejpam-5403	104	24	as	as	ADP
ejpam-5403	104	25	the	the	DET
ejpam-5403	104	26	following	following	NOUN
ejpam-5403	104	27	:	:	PUNCT
ejpam-5403	104	28	a.	a.	NOUN
ejpam-5403	104	29	anantayasethi	anantayasethi	PROPN
ejpam-5403	104	30	,	,	PUNCT
ejpam-5403	104	31	t.	t.	PROPN
ejpam-5403	104	32	kunawat	kunawat	PROPN
ejpam-5403	104	33	,	,	PUNCT
ejpam-5403	104	34	p.	p.	PROPN
ejpam-5403	104	35	moolnipa	moolnipa	PROPN
ejpam-5403	104	36	/	/	SYM
ejpam-5403	104	37	eur	eur	PROPN
ejpam-5403	104	38	.	.	PUNCT
ejpam-5403	105	1	j.	j.	PROPN
ejpam-5403	105	2	pure	pure	PROPN
ejpam-5403	105	3	appl	appl	PROPN
ejpam-5403	105	4	.	.	PROPN
ejpam-5403	105	5	math	math	PROPN
ejpam-5403	105	6	,	,	PUNCT
ejpam-5403	105	7	17	17	NUM
ejpam-5403	105	8	(	(	PUNCT
ejpam-5403	105	9	4	4	NUM
ejpam-5403	105	10	)	)	PUNCT
ejpam-5403	105	11	(	(	PUNCT
ejpam-5403	105	12	2024	2024	NUM
ejpam-5403	105	13	)	)	PUNCT
ejpam-5403	105	14	,	,	PUNCT
ejpam-5403	105	15	3268	3268	NUM
ejpam-5403	105	16	-	-	SYM
ejpam-5403	105	17	3276	3276	NUM
ejpam-5403	105	18	3271	3271	NUM
ejpam-5403	105	19	corollary	corollary	ADJ
ejpam-5403	105	20	2	2	NUM
ejpam-5403	105	21	.	.	PUNCT
ejpam-5403	106	1	let	let	VERB
ejpam-5403	106	2	x	x	PRON
ejpam-5403	106	3	be	be	AUX
ejpam-5403	106	4	a	a	DET
ejpam-5403	106	5	q	q	NOUN
ejpam-5403	106	6	-	-	NOUN
ejpam-5403	106	7	algebra	algebra	NOUN
ejpam-5403	106	8	.	.	PUNCT
ejpam-5403	107	1	if	if	SCONJ
ejpam-5403	107	2	a	a	PRON
ejpam-5403	107	3	is	be	AUX
ejpam-5403	107	4	an	an	DET
ejpam-5403	107	5	atom	atom	NOUN
ejpam-5403	107	6	of	of	ADP
ejpam-5403	107	7	x	x	NOUN
ejpam-5403	107	8	,	,	PUNCT
ejpam-5403	107	9	then	then	ADV
ejpam-5403	107	10	a	a	DET
ejpam-5403	107	11	=	=	ADJ
ejpam-5403	107	12	0x	0x	NOUN
ejpam-5403	107	13	for	for	ADP
ejpam-5403	107	14	some	some	DET
ejpam-5403	107	15	x	x	SYM
ejpam-5403	107	16	∈	∈	PROPN
ejpam-5403	107	17	x.	x.	NOUN
ejpam-5403	108	1	the	the	DET
ejpam-5403	108	2	following	follow	VERB
ejpam-5403	108	3	proposition	proposition	NOUN
ejpam-5403	108	4	shows	show	VERB
ejpam-5403	108	5	some	some	DET
ejpam-5403	108	6	more	more	ADJ
ejpam-5403	108	7	properties	property	NOUN
ejpam-5403	108	8	of	of	ADP
ejpam-5403	108	9	atoms	atom	NOUN
ejpam-5403	108	10	in	in	ADP
ejpam-5403	108	11	q	q	NOUN
ejpam-5403	108	12	-	-	PUNCT
ejpam-5403	108	13	algebras	algebras	X
ejpam-5403	108	14	.	.	PUNCT
ejpam-5403	109	1	proposition	proposition	NOUN
ejpam-5403	109	2	4	4	NUM
ejpam-5403	109	3	.	.	PUNCT
ejpam-5403	110	1	let	let	VERB
ejpam-5403	110	2	x	x	PRON
ejpam-5403	110	3	be	be	AUX
ejpam-5403	110	4	a	a	DET
ejpam-5403	110	5	q	q	NOUN
ejpam-5403	110	6	-	-	NOUN
ejpam-5403	110	7	algebra	algebra	NOUN
ejpam-5403	110	8	and	and	CCONJ
ejpam-5403	110	9	let	let	VERB
ejpam-5403	110	10	a	a	DET
ejpam-5403	110	11	,	,	PUNCT
ejpam-5403	110	12	b	b	NOUN
ejpam-5403	110	13	be	be	AUX
ejpam-5403	110	14	atoms	atom	NOUN
ejpam-5403	110	15	of	of	ADP
ejpam-5403	110	16	x.	x.	NOUN
ejpam-5403	110	17	then	then	ADV
ejpam-5403	110	18	the	the	DET
ejpam-5403	110	19	following	follow	VERB
ejpam-5403	110	20	properties	property	NOUN
ejpam-5403	110	21	hold	hold	VERB
ejpam-5403	110	22	:	:	PUNCT
ejpam-5403	110	23	(	(	PUNCT
ejpam-5403	110	24	i	i	NOUN
ejpam-5403	110	25	)	)	PUNCT
ejpam-5403	110	26	a(xb	a(xb	PROPN
ejpam-5403	110	27	)	)	PUNCT
ejpam-5403	110	28	=	=	SYM
ejpam-5403	111	1	b(xa	b(xa	PROPN
ejpam-5403	111	2	)	)	PUNCT
ejpam-5403	111	3	for	for	ADP
ejpam-5403	111	4	all	all	PRON
ejpam-5403	111	5	x	x	SYM
ejpam-5403	111	6	∈	∈	NOUN
ejpam-5403	111	7	x.	x.	NOUN
ejpam-5403	111	8	(	(	PUNCT
ejpam-5403	111	9	ii	ii	PROPN
ejpam-5403	111	10	)	)	PUNCT
ejpam-5403	111	11	(	(	PUNCT
ejpam-5403	111	12	ax)(yb	ax)(yb	PROPN
ejpam-5403	111	13	)	)	PUNCT
ejpam-5403	111	14	=	=	SYM
ejpam-5403	111	15	(	(	PUNCT
ejpam-5403	111	16	bx)(ya	bx)(ya	PROPN
ejpam-5403	111	17	)	)	PUNCT
ejpam-5403	111	18	for	for	ADP
ejpam-5403	111	19	all	all	DET
ejpam-5403	111	20	x	x	NOUN
ejpam-5403	111	21	,	,	PUNCT
ejpam-5403	111	22	y	y	PROPN
ejpam-5403	111	23	∈	∈	PROPN
ejpam-5403	111	24	x.	x.	NOUN
ejpam-5403	111	25	proof	proof	NOUN
ejpam-5403	111	26	.	.	PUNCT
ejpam-5403	112	1	assume	assume	VERB
ejpam-5403	112	2	that	that	SCONJ
ejpam-5403	112	3	a	a	PRON
ejpam-5403	112	4	and	and	CCONJ
ejpam-5403	112	5	b	b	NOUN
ejpam-5403	112	6	are	be	AUX
ejpam-5403	112	7	atoms	atom	NOUN
ejpam-5403	112	8	of	of	ADP
ejpam-5403	112	9	x.	x.	NOUN
ejpam-5403	112	10	(	(	PUNCT
ejpam-5403	112	11	i	i	NOUN
ejpam-5403	112	12	):	):	PUNCT
ejpam-5403	112	13	let	let	VERB
ejpam-5403	112	14	x	x	PUNCT
ejpam-5403	112	15	∈	∈	PROPN
ejpam-5403	112	16	x.	x.	NOUN
ejpam-5403	112	17	then	then	ADV
ejpam-5403	112	18	by	by	ADP
ejpam-5403	112	19	theorem	theorem	NOUN
ejpam-5403	112	20	1(ii	1(ii	NUM
ejpam-5403	112	21	)	)	PUNCT
ejpam-5403	112	22	,	,	PUNCT
ejpam-5403	112	23	we	we	PRON
ejpam-5403	112	24	get	get	VERB
ejpam-5403	112	25	that	that	PRON
ejpam-5403	112	26	a	a	DET
ejpam-5403	112	27	=	=	SYM
ejpam-5403	112	28	x(xa	x(xa	NOUN
ejpam-5403	112	29	)	)	PUNCT
ejpam-5403	112	30	and	and	CCONJ
ejpam-5403	112	31	b	b	X
ejpam-5403	112	32	=	=	SYM
ejpam-5403	112	33	x(xb	x(xb	PROPN
ejpam-5403	112	34	)	)	PUNCT
ejpam-5403	112	35	.	.	PUNCT
ejpam-5403	113	1	then	then	ADV
ejpam-5403	113	2	by	by	ADP
ejpam-5403	113	3	(	(	PUNCT
ejpam-5403	113	4	q3	q3	PROPN
ejpam-5403	113	5	)	)	PUNCT
ejpam-5403	113	6	there	there	PRON
ejpam-5403	113	7	follows	follow	VERB
ejpam-5403	113	8	that	that	SCONJ
ejpam-5403	113	9	a(xb	a(xb	NOUN
ejpam-5403	113	10	)	)	PUNCT
ejpam-5403	113	11	=	=	SYM
ejpam-5403	113	12	(	(	PUNCT
ejpam-5403	113	13	x(xa))(xb	x(xa))(xb	PROPN
ejpam-5403	113	14	)	)	PUNCT
ejpam-5403	114	1	=	=	SYM
ejpam-5403	114	2	(	(	PUNCT
ejpam-5403	114	3	x(xb))(xa	x(xb))(xa	PROPN
ejpam-5403	114	4	)	)	PUNCT
ejpam-5403	114	5	=	=	SYM
ejpam-5403	114	6	b(xa	b(xa	PROPN
ejpam-5403	114	7	)	)	PUNCT
ejpam-5403	114	8	.	.	PUNCT
ejpam-5403	115	1	(	(	PUNCT
ejpam-5403	115	2	ii	ii	NOUN
ejpam-5403	115	3	):	):	PUNCT
ejpam-5403	115	4	let	let	VERB
ejpam-5403	115	5	x	x	PRON
ejpam-5403	115	6	,	,	PUNCT
ejpam-5403	115	7	y	y	PROPN
ejpam-5403	115	8	∈	∈	PROPN
ejpam-5403	115	9	x.	x.	NOUN
ejpam-5403	115	10	then	then	ADV
ejpam-5403	115	11	by	by	ADP
ejpam-5403	115	12	(	(	PUNCT
ejpam-5403	115	13	q3	q3	PROPN
ejpam-5403	115	14	)	)	PUNCT
ejpam-5403	115	15	and	and	CCONJ
ejpam-5403	115	16	(	(	PUNCT
ejpam-5403	115	17	i	i	NOUN
ejpam-5403	115	18	)	)	PUNCT
ejpam-5403	115	19	we	we	PRON
ejpam-5403	115	20	get	get	VERB
ejpam-5403	115	21	(	(	PUNCT
ejpam-5403	115	22	ax)(yb	ax)(yb	NOUN
ejpam-5403	115	23	)	)	PUNCT
ejpam-5403	115	24	=	=	SYM
ejpam-5403	116	1	(	(	PUNCT
ejpam-5403	116	2	a(yb))x	a(yb))x	NOUN
ejpam-5403	116	3	=	=	SYM
ejpam-5403	116	4	(	(	PUNCT
ejpam-5403	116	5	b(ya))x	b(ya))x	NOUN
ejpam-5403	116	6	=	=	SYM
ejpam-5403	116	7	(	(	PUNCT
ejpam-5403	116	8	bx)(ya	bx)(ya	PROPN
ejpam-5403	116	9	)	)	PUNCT
ejpam-5403	116	10	.	.	PUNCT
ejpam-5403	117	1	the	the	DET
ejpam-5403	117	2	converse	converse	NOUN
ejpam-5403	117	3	of	of	ADP
ejpam-5403	117	4	proposition	proposition	NOUN
ejpam-5403	117	5	4	4	NUM
ejpam-5403	117	6	is	be	AUX
ejpam-5403	117	7	not	not	PART
ejpam-5403	117	8	true	true	ADJ
ejpam-5403	117	9	as	as	SCONJ
ejpam-5403	117	10	seen	see	VERB
ejpam-5403	117	11	in	in	ADP
ejpam-5403	117	12	the	the	DET
ejpam-5403	117	13	following	follow	VERB
ejpam-5403	117	14	example	example	NOUN
ejpam-5403	117	15	.	.	PUNCT
ejpam-5403	118	1	example	example	NOUN
ejpam-5403	119	1	3	3	X
ejpam-5403	119	2	.	.	X
ejpam-5403	119	3	consider	consider	VERB
ejpam-5403	119	4	a	a	DET
ejpam-5403	119	5	q	q	NOUN
ejpam-5403	119	6	-	-	NOUN
ejpam-5403	119	7	algebra	algebra	NOUN
ejpam-5403	119	8	x	x	VERB
ejpam-5403	119	9	from	from	ADP
ejpam-5403	119	10	example	example	NOUN
ejpam-5403	119	11	1	1	NUM
ejpam-5403	119	12	.	.	PUNCT
ejpam-5403	120	1	let	let	VERB
ejpam-5403	120	2	us	we	PRON
ejpam-5403	120	3	focus	focus	VERB
ejpam-5403	120	4	on	on	ADP
ejpam-5403	120	5	elements	element	NOUN
ejpam-5403	120	6	c	c	NOUN
ejpam-5403	120	7	and	and	CCONJ
ejpam-5403	120	8	b	b	PROPN
ejpam-5403	120	9	of	of	ADP
ejpam-5403	120	10	x.	x.	NOUN
ejpam-5403	120	11	we	we	PRON
ejpam-5403	120	12	get	get	VERB
ejpam-5403	120	13	that	that	DET
ejpam-5403	120	14	b(xc	b(xc	NUM
ejpam-5403	121	1	)	)	PUNCT
ejpam-5403	121	2	=	=	SYM
ejpam-5403	121	3	c(xb	c(xb	PROPN
ejpam-5403	121	4	)	)	PUNCT
ejpam-5403	121	5	for	for	ADP
ejpam-5403	121	6	all	all	DET
ejpam-5403	121	7	x	x	SYM
ejpam-5403	121	8	∈	∈	NOUN
ejpam-5403	121	9	x	x	PUNCT
ejpam-5403	121	10	but	but	CCONJ
ejpam-5403	121	11	c	c	NOUN
ejpam-5403	121	12	is	be	AUX
ejpam-5403	121	13	not	not	PART
ejpam-5403	121	14	an	an	DET
ejpam-5403	121	15	atom	atom	NOUN
ejpam-5403	121	16	of	of	ADP
ejpam-5403	121	17	x.	x.	NOUN
ejpam-5403	121	18	hence	hence	ADV
ejpam-5403	121	19	,	,	PUNCT
ejpam-5403	121	20	the	the	DET
ejpam-5403	121	21	converse	converse	NOUN
ejpam-5403	121	22	of	of	ADP
ejpam-5403	121	23	proposition	proposition	NOUN
ejpam-5403	121	24	4(i	4(i	NUM
ejpam-5403	121	25	)	)	PUNCT
ejpam-5403	121	26	is	be	AUX
ejpam-5403	121	27	not	not	PART
ejpam-5403	121	28	true	true	ADJ
ejpam-5403	121	29	.	.	PUNCT
ejpam-5403	122	1	proposition	proposition	NOUN
ejpam-5403	122	2	5	5	NUM
ejpam-5403	122	3	.	.	PUNCT
ejpam-5403	123	1	let	let	VERB
ejpam-5403	123	2	x	x	PRON
ejpam-5403	123	3	be	be	AUX
ejpam-5403	123	4	a	a	DET
ejpam-5403	123	5	q	q	NOUN
ejpam-5403	123	6	-	-	NOUN
ejpam-5403	123	7	algebra	algebra	NOUN
ejpam-5403	123	8	.	.	PUNCT
ejpam-5403	124	1	every	every	DET
ejpam-5403	124	2	element	element	NOUN
ejpam-5403	124	3	of	of	ADP
ejpam-5403	124	4	x	x	PUNCT
ejpam-5403	124	5	is	be	AUX
ejpam-5403	124	6	an	an	DET
ejpam-5403	124	7	atom	atom	NOUN
ejpam-5403	124	8	if	if	SCONJ
ejpam-5403	124	9	and	and	CCONJ
ejpam-5403	124	10	only	only	ADV
ejpam-5403	124	11	if	if	SCONJ
ejpam-5403	124	12	a(xb	a(xb	PROPN
ejpam-5403	124	13	)	)	PUNCT
ejpam-5403	124	14	=	=	SYM
ejpam-5403	124	15	b(xa	b(xa	PROPN
ejpam-5403	124	16	)	)	PUNCT
ejpam-5403	124	17	for	for	ADP
ejpam-5403	124	18	all	all	DET
ejpam-5403	124	19	a	a	DET
ejpam-5403	124	20	,	,	PUNCT
ejpam-5403	124	21	b	b	NOUN
ejpam-5403	124	22	,	,	PUNCT
ejpam-5403	124	23	x	x	SYM
ejpam-5403	124	24	∈	∈	NOUN
ejpam-5403	124	25	x.	x.	NOUN
ejpam-5403	124	26	proof	proof	NOUN
ejpam-5403	124	27	.	.	PUNCT
ejpam-5403	125	1	(	(	PUNCT
ejpam-5403	125	2	⇒	⇒	PROPN
ejpam-5403	125	3	)	)	PUNCT
ejpam-5403	125	4	follows	follow	VERB
ejpam-5403	125	5	from	from	ADP
ejpam-5403	125	6	proposition	proposition	NOUN
ejpam-5403	125	7	4(i	4(i	NUM
ejpam-5403	125	8	)	)	PUNCT
ejpam-5403	125	9	.	.	PUNCT
ejpam-5403	126	1	(	(	PUNCT
ejpam-5403	126	2	⇐	⇐	NOUN
ejpam-5403	126	3	)	)	PUNCT
ejpam-5403	126	4	let	let	VERB
ejpam-5403	126	5	z	z	NOUN
ejpam-5403	126	6	∈	∈	PROPN
ejpam-5403	126	7	x.	x.	NOUN
ejpam-5403	126	8	then	then	ADV
ejpam-5403	126	9	by	by	ADP
ejpam-5403	126	10	assumption	assumption	NOUN
ejpam-5403	126	11	we	we	PRON
ejpam-5403	126	12	get	get	VERB
ejpam-5403	126	13	that	that	DET
ejpam-5403	126	14	z(xx	z(xx	NOUN
ejpam-5403	126	15	)	)	PUNCT
ejpam-5403	126	16	=	=	SYM
ejpam-5403	127	1	x(xz	x(xz	X
ejpam-5403	127	2	)	)	PUNCT
ejpam-5403	127	3	for	for	ADP
ejpam-5403	127	4	all	all	DET
ejpam-5403	127	5	x	x	SYM
ejpam-5403	127	6	∈	∈	ADJ
ejpam-5403	127	7	x.	x.	NOUN
ejpam-5403	127	8	there	there	PRON
ejpam-5403	127	9	follows	follow	VERB
ejpam-5403	127	10	that	that	PRON
ejpam-5403	127	11	z	z	NOUN
ejpam-5403	127	12	=	=	SYM
ejpam-5403	127	13	z0	z0	PROPN
ejpam-5403	127	14	=	=	SYM
ejpam-5403	127	15	z(xx	z(xx	PROPN
ejpam-5403	127	16	)	)	PUNCT
ejpam-5403	127	17	=	=	SYM
ejpam-5403	128	1	x(xz	x(xz	X
ejpam-5403	128	2	)	)	PUNCT
ejpam-5403	128	3	for	for	ADP
ejpam-5403	128	4	all	all	PRON
ejpam-5403	128	5	x	x	SYM
ejpam-5403	128	6	∈	∈	ADJ
ejpam-5403	128	7	x.	x.	NOUN
ejpam-5403	128	8	then	then	ADV
ejpam-5403	128	9	by	by	ADP
ejpam-5403	128	10	theorem	theorem	NOUN
ejpam-5403	128	11	1(ii	1(ii	NUM
ejpam-5403	128	12	)	)	PUNCT
ejpam-5403	128	13	,	,	PUNCT
ejpam-5403	128	14	z	z	PROPN
ejpam-5403	128	15	is	be	AUX
ejpam-5403	128	16	an	an	DET
ejpam-5403	128	17	atom	atom	NOUN
ejpam-5403	128	18	of	of	ADP
ejpam-5403	128	19	x.	x.	NOUN
ejpam-5403	128	20	in	in	ADP
ejpam-5403	128	21	2001	2001	NUM
ejpam-5403	128	22	,	,	PUNCT
ejpam-5403	128	23	d.	d.	PROPN
ejpam-5403	128	24	sun	sun	PROPN
ejpam-5403	129	1	[	[	X
ejpam-5403	129	2	15	15	NUM
ejpam-5403	129	3	]	]	PUNCT
ejpam-5403	129	4	introduced	introduce	VERB
ejpam-5403	129	5	the	the	DET
ejpam-5403	129	6	concept	concept	NOUN
ejpam-5403	129	7	of	of	ADP
ejpam-5403	129	8	strong	strong	ADJ
ejpam-5403	129	9	atoms	atom	NOUN
ejpam-5403	129	10	in	in	ADP
ejpam-5403	129	11	bck	bck	NOUN
ejpam-5403	129	12	-	-	PUNCT
ejpam-5403	129	13	algebra	algebra	NOUN
ejpam-5403	129	14	.	.	PUNCT
ejpam-5403	130	1	we	we	PRON
ejpam-5403	130	2	will	will	AUX
ejpam-5403	130	3	apply	apply	VERB
ejpam-5403	130	4	a	a	DET
ejpam-5403	130	5	concept	concept	NOUN
ejpam-5403	130	6	of	of	ADP
ejpam-5403	130	7	strong	strong	ADJ
ejpam-5403	130	8	atom	atom	NOUN
ejpam-5403	130	9	to	to	ADP
ejpam-5403	130	10	q	q	NOUN
ejpam-5403	130	11	-	-	PUNCT
ejpam-5403	130	12	algebras	algebras	ADJ
ejpam-5403	130	13	in	in	ADP
ejpam-5403	130	14	a	a	DET
ejpam-5403	130	15	similar	similar	ADJ
ejpam-5403	130	16	way	way	NOUN
ejpam-5403	130	17	.	.	PUNCT
ejpam-5403	131	1	let	let	VERB
ejpam-5403	131	2	a	a	PRON
ejpam-5403	131	3	be	be	AUX
ejpam-5403	131	4	an	an	DET
ejpam-5403	131	5	atom	atom	NOUN
ejpam-5403	131	6	of	of	ADP
ejpam-5403	131	7	a	a	DET
ejpam-5403	131	8	q	q	NOUN
ejpam-5403	131	9	-	-	PUNCT
ejpam-5403	131	10	algebra	algebra	NOUN
ejpam-5403	131	11	x.	x.	NOUN
ejpam-5403	131	12	an	an	DET
ejpam-5403	131	13	element	element	NOUN
ejpam-5403	131	14	a	a	PRON
ejpam-5403	131	15	is	be	AUX
ejpam-5403	131	16	called	call	VERB
ejpam-5403	131	17	a	a	DET
ejpam-5403	131	18	strong	strong	ADJ
ejpam-5403	131	19	atom	atom	NOUN
ejpam-5403	131	20	if	if	SCONJ
ejpam-5403	131	21	a	a	DET
ejpam-5403	131	22	̸=	̸=	PROPN
ejpam-5403	131	23	0	0	NUM
ejpam-5403	131	24	and	and	CCONJ
ejpam-5403	131	25	ax	ax	NOUN
ejpam-5403	131	26	=	=	PUNCT
ejpam-5403	131	27	a	a	PRON
ejpam-5403	131	28	for	for	ADP
ejpam-5403	131	29	all	all	PRON
ejpam-5403	131	30	x	x	SYM
ejpam-5403	131	31	∈	∈	ADJ
ejpam-5403	131	32	x	x	X
ejpam-5403	131	33	and	and	CCONJ
ejpam-5403	131	34	x	x	SYM
ejpam-5403	131	35	̸=	̸=	PROPN
ejpam-5403	131	36	a.	a.	NOUN
ejpam-5403	131	37	we	we	PRON
ejpam-5403	131	38	denote	denote	VERB
ejpam-5403	131	39	a	a	DET
ejpam-5403	131	40	set	set	NOUN
ejpam-5403	131	41	sa(x	sa(x	NOUN
ejpam-5403	131	42	)	)	PUNCT
ejpam-5403	131	43	as	as	SCONJ
ejpam-5403	131	44	follows	follow	VERB
ejpam-5403	131	45	:	:	PUNCT
ejpam-5403	131	46	sa(x	sa(x	NOUN
ejpam-5403	131	47	)	)	PUNCT
ejpam-5403	131	48	=	=	PRON
ejpam-5403	131	49	{	{	PUNCT
ejpam-5403	131	50	a	a	DET
ejpam-5403	131	51	∈	∈	PROPN
ejpam-5403	131	52	a(x	a(x	PROPN
ejpam-5403	131	53	)	)	PUNCT
ejpam-5403	131	54	|	|	ADV
ejpam-5403	131	55	a	a	PRON
ejpam-5403	131	56	is	be	AUX
ejpam-5403	131	57	a	a	DET
ejpam-5403	131	58	strong	strong	ADJ
ejpam-5403	131	59	atom	atom	NOUN
ejpam-5403	131	60	of	of	ADP
ejpam-5403	131	61	x	x	PUNCT
ejpam-5403	131	62	}	}	PUNCT
ejpam-5403	131	63	∪	∪	X
ejpam-5403	131	64	{	{	PUNCT
ejpam-5403	131	65	0	0	NUM
ejpam-5403	131	66	}	}	PUNCT
ejpam-5403	131	67	.	.	PUNCT
ejpam-5403	132	1	there	there	PRON
ejpam-5403	132	2	is	be	VERB
ejpam-5403	132	3	a	a	DET
ejpam-5403	132	4	connection	connection	NOUN
ejpam-5403	132	5	between	between	ADP
ejpam-5403	132	6	strong	strong	ADJ
ejpam-5403	132	7	atoms	atom	NOUN
ejpam-5403	132	8	and	and	CCONJ
ejpam-5403	132	9	g	g	NOUN
ejpam-5403	132	10	-	-	PUNCT
ejpam-5403	132	11	part	part	NOUN
ejpam-5403	132	12	of	of	ADP
ejpam-5403	132	13	x.	x.	NOUN
ejpam-5403	132	14	the	the	DET
ejpam-5403	132	15	following	follow	VERB
ejpam-5403	132	16	properties	property	NOUN
ejpam-5403	132	17	show	show	VERB
ejpam-5403	132	18	that	that	SCONJ
ejpam-5403	132	19	x	x	PRON
ejpam-5403	132	20	does	do	AUX
ejpam-5403	132	21	not	not	PART
ejpam-5403	132	22	contain	contain	VERB
ejpam-5403	132	23	any	any	DET
ejpam-5403	132	24	strong	strong	ADJ
ejpam-5403	132	25	storm	storm	NOUN
ejpam-5403	132	26	whenever	whenever	SCONJ
ejpam-5403	132	27	x	x	PRON
ejpam-5403	132	28	contains	contain	VERB
ejpam-5403	132	29	g	g	NOUN
ejpam-5403	132	30	-	-	PUNCT
ejpam-5403	132	31	part	part	NOUN
ejpam-5403	132	32	which	which	PRON
ejpam-5403	132	33	is	be	AUX
ejpam-5403	132	34	an	an	DET
ejpam-5403	132	35	ideal	ideal	NOUN
ejpam-5403	132	36	with	with	ADP
ejpam-5403	132	37	the	the	DET
ejpam-5403	132	38	cardinality	cardinality	NOUN
ejpam-5403	132	39	greater	great	ADJ
ejpam-5403	132	40	or	or	CCONJ
ejpam-5403	132	41	equal	equal	ADJ
ejpam-5403	132	42	to	to	ADP
ejpam-5403	132	43	2	2	NUM
ejpam-5403	132	44	.	.	PUNCT
ejpam-5403	133	1	first	first	ADV
ejpam-5403	133	2	,	,	PUNCT
ejpam-5403	133	3	we	we	PRON
ejpam-5403	133	4	need	need	VERB
ejpam-5403	133	5	the	the	DET
ejpam-5403	133	6	following	follow	VERB
ejpam-5403	133	7	proposition	proposition	NOUN
ejpam-5403	133	8	:	:	PUNCT
ejpam-5403	133	9	proposition	proposition	NOUN
ejpam-5403	133	10	6	6	NUM
ejpam-5403	133	11	.	.	PUNCT
ejpam-5403	134	1	[	[	X
ejpam-5403	134	2	5	5	X
ejpam-5403	134	3	]	]	PUNCT
ejpam-5403	134	4	let	let	VERB
ejpam-5403	134	5	x	x	PRON
ejpam-5403	134	6	be	be	AUX
ejpam-5403	134	7	a	a	DET
ejpam-5403	134	8	q	q	NOUN
ejpam-5403	134	9	-	-	NOUN
ejpam-5403	134	10	algebra	algebra	NOUN
ejpam-5403	134	11	with	with	ADP
ejpam-5403	134	12	|x|	|x|	PROPN
ejpam-5403	134	13	=	=	SYM
ejpam-5403	134	14	n	n	PROPN
ejpam-5403	134	15	and	and	CCONJ
ejpam-5403	134	16	g(x	g(x	NOUN
ejpam-5403	134	17	)	)	PUNCT
ejpam-5403	134	18	̸=	̸=	PROPN
ejpam-5403	134	19	x.	x.	NOUN
ejpam-5403	134	20	if	if	SCONJ
ejpam-5403	134	21	g(x	g(x	NOUN
ejpam-5403	134	22	)	)	PUNCT
ejpam-5403	134	23	is	be	AUX
ejpam-5403	134	24	an	an	DET
ejpam-5403	134	25	ideal	ideal	NOUN
ejpam-5403	134	26	of	of	ADP
ejpam-5403	134	27	x	x	NOUN
ejpam-5403	134	28	,	,	PUNCT
ejpam-5403	134	29	then	then	ADV
ejpam-5403	134	30	|g(x)|	|g(x)|	PROPN
ejpam-5403	134	31	≤	≤	NUM
ejpam-5403	134	32	n	n	PRON
ejpam-5403	134	33	2	2	NUM
ejpam-5403	134	34	.	.	PUNCT
ejpam-5403	135	1	proposition	proposition	NOUN
ejpam-5403	135	2	7	7	NUM
ejpam-5403	135	3	.	.	PUNCT
ejpam-5403	136	1	let	let	VERB
ejpam-5403	136	2	x	x	PRON
ejpam-5403	136	3	be	be	AUX
ejpam-5403	136	4	a	a	DET
ejpam-5403	136	5	q	q	NOUN
ejpam-5403	136	6	-	-	NOUN
ejpam-5403	136	7	algebra	algebra	NOUN
ejpam-5403	136	8	.	.	PUNCT
ejpam-5403	137	1	if	if	SCONJ
ejpam-5403	137	2	g(x	g(x	NOUN
ejpam-5403	137	3	)	)	PUNCT
ejpam-5403	137	4	is	be	AUX
ejpam-5403	137	5	an	an	DET
ejpam-5403	137	6	ideal	ideal	ADJ
ejpam-5403	137	7	and	and	CCONJ
ejpam-5403	137	8	|g(x)|	|g(x)|	PROPN
ejpam-5403	137	9	=	=	SYM
ejpam-5403	137	10	2	2	NUM
ejpam-5403	137	11	,	,	PUNCT
ejpam-5403	137	12	then	then	ADV
ejpam-5403	137	13	sa(x	sa(x	NOUN
ejpam-5403	137	14	)	)	PUNCT
ejpam-5403	137	15	=	=	PUNCT
ejpam-5403	137	16	{	{	PUNCT
ejpam-5403	137	17	0	0	NUM
ejpam-5403	137	18	}	}	PUNCT
ejpam-5403	137	19	.	.	PUNCT
ejpam-5403	138	1	proof	proof	NOUN
ejpam-5403	138	2	.	.	PUNCT
ejpam-5403	139	1	assume	assume	VERB
ejpam-5403	139	2	that	that	SCONJ
ejpam-5403	139	3	g(x	g(x	NOUN
ejpam-5403	139	4	)	)	PUNCT
ejpam-5403	139	5	is	be	AUX
ejpam-5403	139	6	an	an	DET
ejpam-5403	139	7	ideal	ideal	NOUN
ejpam-5403	139	8	of	of	ADP
ejpam-5403	139	9	x	x	X
ejpam-5403	139	10	and	and	CCONJ
ejpam-5403	139	11	|g(x)|	|g(x)|	PROPN
ejpam-5403	139	12	=	=	SYM
ejpam-5403	140	1	2	2	X
ejpam-5403	140	2	.	.	X
ejpam-5403	140	3	we	we	PRON
ejpam-5403	140	4	assume	assume	VERB
ejpam-5403	140	5	that	that	SCONJ
ejpam-5403	140	6	g(x	g(x	NOUN
ejpam-5403	140	7	)	)	PUNCT
ejpam-5403	141	1	=	=	PRON
ejpam-5403	141	2	{	{	PUNCT
ejpam-5403	141	3	0	0	NUM
ejpam-5403	141	4	,	,	PUNCT
ejpam-5403	141	5	a	a	PRON
ejpam-5403	141	6	}	}	PUNCT
ejpam-5403	141	7	.	.	PUNCT
ejpam-5403	141	8	suppose	suppose	VERB
ejpam-5403	141	9	that	that	SCONJ
ejpam-5403	141	10	a	a	PRON
ejpam-5403	141	11	is	be	AUX
ejpam-5403	141	12	a	a	DET
ejpam-5403	141	13	strong	strong	ADJ
ejpam-5403	141	14	atom	atom	NOUN
ejpam-5403	141	15	ofx	ofx	NOUN
ejpam-5403	141	16	.	.	PUNCT
ejpam-5403	142	1	sinceg(x	sinceg(x	ADJ
ejpam-5403	142	2	)	)	PUNCT
ejpam-5403	142	3	is	be	AUX
ejpam-5403	142	4	an	an	DET
ejpam-5403	142	5	ideal	ideal	NOUN
ejpam-5403	142	6	,	,	PUNCT
ejpam-5403	142	7	then	then	ADV
ejpam-5403	142	8	by	by	ADP
ejpam-5403	142	9	proposition	proposition	NOUN
ejpam-5403	142	10	6	6	NUM
ejpam-5403	142	11	,	,	PUNCT
ejpam-5403	142	12	a.	a.	NOUN
ejpam-5403	142	13	anantayasethi	anantayasethi	PROPN
ejpam-5403	142	14	,	,	PUNCT
ejpam-5403	142	15	t.	t.	PROPN
ejpam-5403	142	16	kunawat	kunawat	PROPN
ejpam-5403	142	17	,	,	PUNCT
ejpam-5403	142	18	p.	p.	PROPN
ejpam-5403	142	19	moolnipa	moolnipa	PROPN
ejpam-5403	142	20	/	/	SYM
ejpam-5403	142	21	eur	eur	PROPN
ejpam-5403	142	22	.	.	PUNCT
ejpam-5403	143	1	j.	j.	PROPN
ejpam-5403	143	2	pure	pure	PROPN
ejpam-5403	143	3	appl	appl	PROPN
ejpam-5403	143	4	.	.	PROPN
ejpam-5403	143	5	math	math	PROPN
ejpam-5403	143	6	,	,	PUNCT
ejpam-5403	143	7	17	17	NUM
ejpam-5403	143	8	(	(	PUNCT
ejpam-5403	143	9	4	4	NUM
ejpam-5403	143	10	)	)	PUNCT
ejpam-5403	143	11	(	(	PUNCT
ejpam-5403	143	12	2024	2024	NUM
ejpam-5403	143	13	)	)	PUNCT
ejpam-5403	143	14	,	,	PUNCT
ejpam-5403	143	15	3268	3268	NUM
ejpam-5403	143	16	-	-	SYM
ejpam-5403	143	17	3276	3276	NUM
ejpam-5403	143	18	3272	3272	NUM
ejpam-5403	143	19	we	we	PRON
ejpam-5403	143	20	get	get	VERB
ejpam-5403	143	21	|g(x)|	|g(x)|	NOUN
ejpam-5403	143	22	≤	≤	NUM
ejpam-5403	143	23	|x|	|x|	PROPN
ejpam-5403	143	24	2	2	NUM
ejpam-5403	143	25	.	.	PUNCT
ejpam-5403	144	1	therefore	therefore	ADV
ejpam-5403	144	2	,	,	PUNCT
ejpam-5403	144	3	|x|	|x|	PROPN
ejpam-5403	144	4	≥	≥	NUM
ejpam-5403	144	5	4	4	NUM
ejpam-5403	144	6	.	.	PUNCT
ejpam-5403	144	7	let	let	VERB
ejpam-5403	144	8	b	b	NOUN
ejpam-5403	144	9	∈	∈	PROPN
ejpam-5403	144	10	x	x	PUNCT
ejpam-5403	144	11	such	such	ADJ
ejpam-5403	144	12	that	that	DET
ejpam-5403	144	13	b	b	NOUN
ejpam-5403	144	14	/∈	/∈	PUNCT
ejpam-5403	144	15	g(x	g(x	NOUN
ejpam-5403	144	16	)	)	PUNCT
ejpam-5403	144	17	.	.	PUNCT
ejpam-5403	145	1	since	since	SCONJ
ejpam-5403	145	2	a	a	PRON
ejpam-5403	145	3	is	be	AUX
ejpam-5403	145	4	a	a	DET
ejpam-5403	145	5	strong	strong	ADJ
ejpam-5403	145	6	atom	atom	NOUN
ejpam-5403	145	7	,	,	PUNCT
ejpam-5403	145	8	then	then	ADV
ejpam-5403	145	9	ab	ab	PROPN
ejpam-5403	145	10	=	=	PUNCT
ejpam-5403	145	11	a.	a.	NOUN
ejpam-5403	146	1	it	it	PRON
ejpam-5403	146	2	follows	follow	VERB
ejpam-5403	146	3	that	that	DET
ejpam-5403	146	4	0b	0b	NOUN
ejpam-5403	146	5	=	=	SYM
ejpam-5403	146	6	(	(	PUNCT
ejpam-5403	146	7	aa)b	aa)b	PROPN
ejpam-5403	146	8	=	=	X
ejpam-5403	146	9	(	(	PUNCT
ejpam-5403	146	10	ab)a	ab)a	PROPN
ejpam-5403	146	11	=	=	PUNCT
ejpam-5403	146	12	aa	aa	NOUN
ejpam-5403	146	13	=	=	NOUN
ejpam-5403	146	14	0	0	X
ejpam-5403	146	15	.	.	PUNCT
ejpam-5403	147	1	since	since	SCONJ
ejpam-5403	147	2	b	b	PROPN
ejpam-5403	147	3	/∈	/∈	PUNCT
ejpam-5403	147	4	g(x	g(x	NOUN
ejpam-5403	147	5	)	)	PUNCT
ejpam-5403	147	6	,	,	PUNCT
ejpam-5403	147	7	a	a	DET
ejpam-5403	147	8	∈	∈	PROPN
ejpam-5403	147	9	g(x	g(x	NOUN
ejpam-5403	147	10	)	)	PUNCT
ejpam-5403	147	11	and	and	CCONJ
ejpam-5403	147	12	g(x	g(x	NOUN
ejpam-5403	147	13	)	)	PUNCT
ejpam-5403	147	14	is	be	AUX
ejpam-5403	147	15	an	an	DET
ejpam-5403	147	16	ideal	ideal	NOUN
ejpam-5403	147	17	,	,	PUNCT
ejpam-5403	147	18	then	then	ADV
ejpam-5403	147	19	ba	ba	PROPN
ejpam-5403	147	20	/∈	/∈	PUNCT
ejpam-5403	148	1	g(x	g(x	NOUN
ejpam-5403	148	2	)	)	PUNCT
ejpam-5403	148	3	.	.	PUNCT
ejpam-5403	149	1	moreover	moreover	ADV
ejpam-5403	149	2	,	,	PUNCT
ejpam-5403	149	3	by	by	ADP
ejpam-5403	149	4	proposition	proposition	NOUN
ejpam-5403	149	5	3(iii	3(iii	NUM
ejpam-5403	149	6	)	)	PUNCT
ejpam-5403	149	7	,	,	PUNCT
ejpam-5403	149	8	we	we	PRON
ejpam-5403	149	9	get	get	VERB
ejpam-5403	149	10	that	that	DET
ejpam-5403	149	11	ba	ba	PROPN
ejpam-5403	149	12	̸=	̸=	PROPN
ejpam-5403	149	13	b.	b.	PROPN
ejpam-5403	150	1	therefore	therefore	ADV
ejpam-5403	150	2	,	,	PUNCT
ejpam-5403	150	3	ba	ba	PROPN
ejpam-5403	150	4	/∈	/∈	PUNCT
ejpam-5403	150	5	{	{	PUNCT
ejpam-5403	150	6	0	0	NUM
ejpam-5403	150	7	,	,	PUNCT
ejpam-5403	150	8	a	a	DET
ejpam-5403	150	9	,	,	PUNCT
ejpam-5403	150	10	b	b	NOUN
ejpam-5403	150	11	}	}	PUNCT
ejpam-5403	150	12	.	.	PUNCT
ejpam-5403	151	1	we	we	PRON
ejpam-5403	151	2	may	may	AUX
ejpam-5403	151	3	assume	assume	VERB
ejpam-5403	151	4	that	that	SCONJ
ejpam-5403	151	5	ba	ba	PROPN
ejpam-5403	151	6	=	=	PUNCT
ejpam-5403	151	7	c	c	PROPN
ejpam-5403	151	8	for	for	ADP
ejpam-5403	151	9	some	some	DET
ejpam-5403	151	10	c	c	PROPN
ejpam-5403	151	11	∈	∈	PROPN
ejpam-5403	151	12	x\{0	x\{0	PROPN
ejpam-5403	151	13	,	,	PUNCT
ejpam-5403	151	14	a	a	DET
ejpam-5403	151	15	,	,	PUNCT
ejpam-5403	151	16	b	b	NOUN
ejpam-5403	151	17	}	}	PUNCT
ejpam-5403	151	18	.	.	PUNCT
ejpam-5403	152	1	similarly	similarly	ADV
ejpam-5403	152	2	,	,	PUNCT
ejpam-5403	152	3	we	we	PRON
ejpam-5403	152	4	get	get	VERB
ejpam-5403	152	5	ac	ac	ADP
ejpam-5403	153	1	=	=	PUNCT
ejpam-5403	153	2	a	a	PROPN
ejpam-5403	153	3	and	and	CCONJ
ejpam-5403	153	4	ca	ca	NOUN
ejpam-5403	153	5	/∈	/∈	PUNCT
ejpam-5403	153	6	{	{	PUNCT
ejpam-5403	153	7	0	0	NUM
ejpam-5403	153	8	,	,	PUNCT
ejpam-5403	153	9	a	a	PRON
ejpam-5403	153	10	,	,	PUNCT
ejpam-5403	153	11	c	c	NOUN
ejpam-5403	153	12	}	}	PUNCT
ejpam-5403	153	13	.	.	PUNCT
ejpam-5403	154	1	then	then	ADV
ejpam-5403	154	2	we	we	PRON
ejpam-5403	154	3	get	get	VERB
ejpam-5403	154	4	cb	cb	X
ejpam-5403	154	5	=	=	PRON
ejpam-5403	154	6	(	(	PUNCT
ejpam-5403	154	7	ba)b	ba)b	PROPN
ejpam-5403	154	8	=	=	SYM
ejpam-5403	154	9	(	(	PUNCT
ejpam-5403	154	10	bb)a	bb)a	PROPN
ejpam-5403	154	11	=	=	SYM
ejpam-5403	154	12	0a	0a	PROPN
ejpam-5403	154	13	=	=	PUNCT
ejpam-5403	155	1	a.	a.	NOUN
ejpam-5403	155	2	it	it	PRON
ejpam-5403	155	3	follows	follow	VERB
ejpam-5403	155	4	that	that	SCONJ
ejpam-5403	155	5	a	a	DET
ejpam-5403	155	6	=	=	X
ejpam-5403	155	7	ac	ac	PROPN
ejpam-5403	155	8	=	=	PUNCT
ejpam-5403	155	9	(	(	PUNCT
ejpam-5403	155	10	cb)c	cb)c	NOUN
ejpam-5403	155	11	=	=	PUNCT
ejpam-5403	155	12	(	(	PUNCT
ejpam-5403	155	13	cc)b	cc)b	PROPN
ejpam-5403	155	14	=	=	SYM
ejpam-5403	155	15	0b	0b	NOUN
ejpam-5403	155	16	=	=	SYM
ejpam-5403	155	17	0	0	PROPN
ejpam-5403	155	18	,	,	PUNCT
ejpam-5403	155	19	a	a	DET
ejpam-5403	155	20	contradiction	contradiction	NOUN
ejpam-5403	155	21	.	.	PUNCT
ejpam-5403	156	1	hence	hence	ADV
ejpam-5403	156	2	,	,	PUNCT
ejpam-5403	156	3	a	a	DET
ejpam-5403	156	4	/∈	/∈	NOUN
ejpam-5403	156	5	sa(x	sa(x	NOUN
ejpam-5403	156	6	)	)	PUNCT
ejpam-5403	156	7	.	.	PUNCT
ejpam-5403	157	1	let	let	VERB
ejpam-5403	157	2	x	x	SYM
ejpam-5403	157	3	∈	∈	PROPN
ejpam-5403	157	4	x	x	X
ejpam-5403	157	5	\g(x	\g(x	NOUN
ejpam-5403	157	6	)	)	PUNCT
ejpam-5403	157	7	.	.	PUNCT
ejpam-5403	158	1	since	since	SCONJ
ejpam-5403	158	2	a	a	DET
ejpam-5403	158	3	∈	∈	PROPN
ejpam-5403	158	4	g(x	g(x	NOUN
ejpam-5403	158	5	)	)	PUNCT
ejpam-5403	158	6	and	and	CCONJ
ejpam-5403	158	7	x	x	SYM
ejpam-5403	158	8	̸=	̸=	PROPN
ejpam-5403	158	9	0	0	NUM
ejpam-5403	158	10	,	,	PUNCT
ejpam-5403	158	11	then	then	ADV
ejpam-5403	158	12	xa	xa	PROPN
ejpam-5403	158	13	̸=	̸=	PROPN
ejpam-5403	158	14	x	x	PUNCT
ejpam-5403	158	15	by	by	ADP
ejpam-5403	158	16	by	by	ADP
ejpam-5403	158	17	proposition	proposition	NOUN
ejpam-5403	158	18	3(iii	3(iii	NUM
ejpam-5403	158	19	)	)	PUNCT
ejpam-5403	158	20	.	.	PUNCT
ejpam-5403	159	1	it	it	PRON
ejpam-5403	159	2	follows	follow	VERB
ejpam-5403	159	3	that	that	SCONJ
ejpam-5403	159	4	x	x	SYM
ejpam-5403	159	5	/∈	/∈	PUNCT
ejpam-5403	159	6	sa(x	sa(x	NOUN
ejpam-5403	159	7	)	)	PUNCT
ejpam-5403	159	8	.	.	PUNCT
ejpam-5403	160	1	altogether	altogether	ADV
ejpam-5403	160	2	,	,	PUNCT
ejpam-5403	160	3	sa(x	sa(x	NOUN
ejpam-5403	160	4	)	)	PUNCT
ejpam-5403	160	5	=	=	PUNCT
ejpam-5403	160	6	{	{	PUNCT
ejpam-5403	160	7	0	0	NUM
ejpam-5403	160	8	}	}	PUNCT
ejpam-5403	160	9	.	.	PUNCT
ejpam-5403	161	1	proposition	proposition	NOUN
ejpam-5403	161	2	8	8	NUM
ejpam-5403	161	3	.	.	PUNCT
ejpam-5403	162	1	let	let	VERB
ejpam-5403	162	2	x	x	PRON
ejpam-5403	162	3	be	be	AUX
ejpam-5403	162	4	a	a	DET
ejpam-5403	162	5	q	q	NOUN
ejpam-5403	162	6	-	-	NOUN
ejpam-5403	162	7	algebra	algebra	NOUN
ejpam-5403	162	8	.	.	PUNCT
ejpam-5403	163	1	if	if	SCONJ
ejpam-5403	163	2	|g(x)|	|g(x)|	NOUN
ejpam-5403	163	3	≥	≥	NUM
ejpam-5403	163	4	3	3	NUM
ejpam-5403	163	5	,	,	PUNCT
ejpam-5403	163	6	then	then	ADV
ejpam-5403	163	7	sa(x	sa(x	NOUN
ejpam-5403	163	8	)	)	PUNCT
ejpam-5403	163	9	=	=	PUNCT
ejpam-5403	163	10	{	{	PUNCT
ejpam-5403	163	11	0	0	NUM
ejpam-5403	163	12	}	}	PUNCT
ejpam-5403	163	13	.	.	PUNCT
ejpam-5403	164	1	proof	proof	NOUN
ejpam-5403	164	2	.	.	PUNCT
ejpam-5403	165	1	assume	assume	VERB
ejpam-5403	165	2	that	that	SCONJ
ejpam-5403	165	3	|g(x)|	|g(x)|	NOUN
ejpam-5403	165	4	≥	≥	NUM
ejpam-5403	165	5	3	3	NUM
ejpam-5403	165	6	.	.	PUNCT
ejpam-5403	166	1	then	then	ADV
ejpam-5403	166	2	there	there	PRON
ejpam-5403	166	3	are	be	VERB
ejpam-5403	166	4	a	a	DET
ejpam-5403	166	5	,	,	PUNCT
ejpam-5403	166	6	b	b	PROPN
ejpam-5403	166	7	∈	∈	PROPN
ejpam-5403	166	8	g(x	g(x	NOUN
ejpam-5403	166	9	)	)	PUNCT
ejpam-5403	166	10	such	such	ADJ
ejpam-5403	166	11	that	that	SCONJ
ejpam-5403	166	12	a	a	DET
ejpam-5403	166	13	,	,	PUNCT
ejpam-5403	166	14	b	b	NOUN
ejpam-5403	166	15	/∈	/∈	PUNCT
ejpam-5403	166	16	{	{	PUNCT
ejpam-5403	166	17	0	0	NUM
ejpam-5403	166	18	}	}	PUNCT
ejpam-5403	166	19	and	and	CCONJ
ejpam-5403	166	20	a	a	DET
ejpam-5403	166	21	̸=	̸=	PROPN
ejpam-5403	166	22	b.	b.	NOUN
ejpam-5403	166	23	let	let	VERB
ejpam-5403	166	24	x	x	PUNCT
ejpam-5403	166	25	∈	∈	PROPN
ejpam-5403	166	26	x	x	X
ejpam-5403	166	27	and	and	CCONJ
ejpam-5403	166	28	x	x	SYM
ejpam-5403	166	29	̸=	̸=	PROPN
ejpam-5403	166	30	0	0	NUM
ejpam-5403	166	31	.	.	PUNCT
ejpam-5403	167	1	if	if	SCONJ
ejpam-5403	167	2	x	x	X
ejpam-5403	167	3	=	=	SYM
ejpam-5403	167	4	a	a	X
ejpam-5403	167	5	,	,	PUNCT
ejpam-5403	167	6	then	then	ADV
ejpam-5403	167	7	xb	xb	PROPN
ejpam-5403	167	8	=	=	PROPN
ejpam-5403	167	9	ab	ab	PROPN
ejpam-5403	167	10	/∈	/∈	PUNCT
ejpam-5403	167	11	{	{	PUNCT
ejpam-5403	167	12	0	0	NUM
ejpam-5403	167	13	,	,	PUNCT
ejpam-5403	167	14	a	a	DET
ejpam-5403	167	15	,	,	PUNCT
ejpam-5403	167	16	b	b	NOUN
ejpam-5403	167	17	}	}	PUNCT
ejpam-5403	167	18	by	by	ADP
ejpam-5403	167	19	proposition	proposition	NOUN
ejpam-5403	167	20	3(i	3(i	NUM
ejpam-5403	167	21	)	)	PUNCT
ejpam-5403	167	22	.	.	PUNCT
ejpam-5403	168	1	therefore	therefore	ADV
ejpam-5403	168	2	,	,	PUNCT
ejpam-5403	168	3	xb	xb	PROPN
ejpam-5403	168	4	̸=	̸=	PROPN
ejpam-5403	168	5	x	x	PUNCT
ejpam-5403	168	6	there	there	PRON
ejpam-5403	168	7	follows	follow	VERB
ejpam-5403	168	8	that	that	SCONJ
ejpam-5403	168	9	x	x	SYM
ejpam-5403	168	10	/∈	/∈	PUNCT
ejpam-5403	168	11	sa(x	sa(x	NOUN
ejpam-5403	168	12	)	)	PUNCT
ejpam-5403	168	13	.	.	PUNCT
ejpam-5403	169	1	if	if	SCONJ
ejpam-5403	169	2	x	x	PROPN
ejpam-5403	169	3	̸=	̸=	PROPN
ejpam-5403	169	4	a	a	PRON
ejpam-5403	169	5	,	,	PUNCT
ejpam-5403	169	6	then	then	ADV
ejpam-5403	169	7	by	by	ADP
ejpam-5403	169	8	proposition	proposition	NOUN
ejpam-5403	169	9	3(i	3(i	NUM
ejpam-5403	169	10	)	)	PUNCT
ejpam-5403	169	11	,	,	PUNCT
ejpam-5403	169	12	xa	xa	PROPN
ejpam-5403	169	13	̸=	̸=	PROPN
ejpam-5403	169	14	x.	x.	PUNCT
ejpam-5403	169	15	thus	thus	ADV
ejpam-5403	169	16	,	,	PUNCT
ejpam-5403	169	17	x	x	X
ejpam-5403	169	18	/∈	/∈	PUNCT
ejpam-5403	169	19	sa(x	sa(x	NOUN
ejpam-5403	169	20	)	)	PUNCT
ejpam-5403	169	21	.	.	PUNCT
ejpam-5403	170	1	altogether	altogether	ADV
ejpam-5403	170	2	,	,	PUNCT
ejpam-5403	170	3	we	we	PRON
ejpam-5403	170	4	get	get	VERB
ejpam-5403	170	5	sa(x	sa(x	NOUN
ejpam-5403	170	6	)	)	PUNCT
ejpam-5403	170	7	=	=	PUNCT
ejpam-5403	170	8	{	{	PUNCT
ejpam-5403	170	9	0	0	NUM
ejpam-5403	170	10	}	}	PUNCT
ejpam-5403	170	11	.	.	PUNCT
ejpam-5403	171	1	proposition	proposition	NOUN
ejpam-5403	171	2	7	7	NUM
ejpam-5403	171	3	and	and	CCONJ
ejpam-5403	171	4	proposition	proposition	NOUN
ejpam-5403	171	5	8	8	NUM
ejpam-5403	171	6	give	give	VERB
ejpam-5403	171	7	the	the	DET
ejpam-5403	171	8	following	following	NOUN
ejpam-5403	171	9	theorem	theorem	NOUN
ejpam-5403	171	10	:	:	PUNCT
ejpam-5403	171	11	theorem	theorem	NOUN
ejpam-5403	171	12	3	3	X
ejpam-5403	171	13	.	.	PUNCT
ejpam-5403	172	1	let	let	VERB
ejpam-5403	172	2	x	x	PRON
ejpam-5403	172	3	be	be	AUX
ejpam-5403	172	4	a	a	DET
ejpam-5403	172	5	q	q	NOUN
ejpam-5403	172	6	-	-	PUNCT
ejpam-5403	172	7	algebra	algebra	NOUN
ejpam-5403	172	8	and	and	CCONJ
ejpam-5403	172	9	g(x	g(x	NOUN
ejpam-5403	172	10	)	)	PUNCT
ejpam-5403	172	11	̸=	̸=	PROPN
ejpam-5403	172	12	{	{	PUNCT
ejpam-5403	172	13	0	0	NUM
ejpam-5403	172	14	}	}	PUNCT
ejpam-5403	172	15	.	.	PUNCT
ejpam-5403	173	1	if	if	SCONJ
ejpam-5403	173	2	g(x	g(x	NOUN
ejpam-5403	173	3	)	)	PUNCT
ejpam-5403	173	4	is	be	AUX
ejpam-5403	173	5	an	an	DET
ejpam-5403	173	6	ideal	ideal	ADJ
ejpam-5403	173	7	,	,	PUNCT
ejpam-5403	173	8	then	then	ADV
ejpam-5403	173	9	sa(x	sa(x	NOUN
ejpam-5403	173	10	)	)	PUNCT
ejpam-5403	173	11	=	=	PUNCT
ejpam-5403	173	12	{	{	PUNCT
ejpam-5403	173	13	0	0	NUM
ejpam-5403	173	14	}	}	PUNCT
ejpam-5403	173	15	.	.	PUNCT
ejpam-5403	174	1	it	it	PRON
ejpam-5403	174	2	is	be	AUX
ejpam-5403	174	3	clear	clear	ADJ
ejpam-5403	174	4	that	that	SCONJ
ejpam-5403	174	5	a	a	DET
ejpam-5403	174	6	set	set	NOUN
ejpam-5403	174	7	of	of	ADP
ejpam-5403	174	8	all	all	DET
ejpam-5403	174	9	atoms	atom	NOUN
ejpam-5403	174	10	of	of	ADP
ejpam-5403	174	11	a	a	DET
ejpam-5403	174	12	q	q	NOUN
ejpam-5403	174	13	-	-	NOUN
ejpam-5403	174	14	algebra	algebra	NOUN
ejpam-5403	174	15	x	x	PUNCT
ejpam-5403	174	16	is	be	AUX
ejpam-5403	174	17	not	not	PART
ejpam-5403	174	18	closed	closed	ADJ
ejpam-5403	174	19	.	.	PUNCT
ejpam-5403	175	1	but	but	CCONJ
ejpam-5403	175	2	if	if	SCONJ
ejpam-5403	175	3	we	we	PRON
ejpam-5403	175	4	focus	focus	VERB
ejpam-5403	175	5	on	on	ADP
ejpam-5403	175	6	a	a	DET
ejpam-5403	175	7	set	set	NOUN
ejpam-5403	175	8	of	of	ADP
ejpam-5403	175	9	strong	strong	ADJ
ejpam-5403	175	10	atoms	atom	NOUN
ejpam-5403	175	11	,	,	PUNCT
ejpam-5403	175	12	we	we	PRON
ejpam-5403	175	13	get	get	VERB
ejpam-5403	175	14	that	that	SCONJ
ejpam-5403	175	15	the	the	DET
ejpam-5403	175	16	product	product	NOUN
ejpam-5403	175	17	of	of	ADP
ejpam-5403	175	18	strong	strong	ADJ
ejpam-5403	175	19	atoms	atom	NOUN
ejpam-5403	175	20	is	be	AUX
ejpam-5403	175	21	again	again	ADV
ejpam-5403	175	22	a	a	DET
ejpam-5403	175	23	strong	strong	ADJ
ejpam-5403	175	24	atom	atom	NOUN
ejpam-5403	175	25	.	.	PUNCT
ejpam-5403	176	1	it	it	PRON
ejpam-5403	176	2	follows	follow	VERB
ejpam-5403	176	3	that	that	SCONJ
ejpam-5403	176	4	sa(x	sa(x	NOUN
ejpam-5403	176	5	)	)	PUNCT
ejpam-5403	176	6	is	be	AUX
ejpam-5403	176	7	a	a	DET
ejpam-5403	176	8	subalgebra	subalgebra	NOUN
ejpam-5403	176	9	of	of	ADP
ejpam-5403	176	10	x.	x.	NOUN
ejpam-5403	176	11	proposition	proposition	NOUN
ejpam-5403	176	12	9	9	NUM
ejpam-5403	176	13	.	.	PUNCT
ejpam-5403	177	1	let	let	VERB
ejpam-5403	177	2	x	x	PRON
ejpam-5403	177	3	be	be	AUX
ejpam-5403	177	4	a	a	DET
ejpam-5403	177	5	q	q	NOUN
ejpam-5403	177	6	-	-	NOUN
ejpam-5403	177	7	algebra	algebra	NOUN
ejpam-5403	177	8	.	.	PUNCT
ejpam-5403	178	1	then	then	ADV
ejpam-5403	178	2	sa(x	sa(x	NOUN
ejpam-5403	178	3	)	)	PUNCT
ejpam-5403	178	4	is	be	AUX
ejpam-5403	178	5	a	a	DET
ejpam-5403	178	6	subalgebra	subalgebra	NOUN
ejpam-5403	178	7	of	of	ADP
ejpam-5403	178	8	x.	x.	NOUN
ejpam-5403	178	9	proof	proof	NOUN
ejpam-5403	178	10	.	.	PUNCT
ejpam-5403	179	1	if	if	SCONJ
ejpam-5403	179	2	|sa(x)|	|sa(x)|	NOUN
ejpam-5403	179	3	≤	≤	ADV
ejpam-5403	179	4	2	2	NUM
ejpam-5403	179	5	,	,	PUNCT
ejpam-5403	179	6	then	then	ADV
ejpam-5403	179	7	it	it	PRON
ejpam-5403	179	8	is	be	AUX
ejpam-5403	179	9	clear	clear	ADJ
ejpam-5403	179	10	that	that	SCONJ
ejpam-5403	179	11	sa(x	sa(x	NOUN
ejpam-5403	179	12	)	)	PUNCT
ejpam-5403	179	13	is	be	AUX
ejpam-5403	179	14	a	a	DET
ejpam-5403	179	15	subalgebra	subalgebra	NOUN
ejpam-5403	179	16	.	.	PUNCT
ejpam-5403	180	1	assume	assume	VERB
ejpam-5403	180	2	now	now	ADV
ejpam-5403	180	3	that	that	PRON
ejpam-5403	180	4	|sa(x)|	|sa(x)|	PROPN
ejpam-5403	180	5	≥	≥	NOUN
ejpam-5403	180	6	3	3	X
ejpam-5403	180	7	.	.	PUNCT
ejpam-5403	181	1	let	let	VERB
ejpam-5403	181	2	a	a	DET
ejpam-5403	181	3	,	,	PUNCT
ejpam-5403	181	4	b	b	NOUN
ejpam-5403	181	5	∈	∈	PROPN
ejpam-5403	181	6	sa(x	sa(x	NOUN
ejpam-5403	181	7	)	)	PUNCT
ejpam-5403	181	8	.	.	PUNCT
ejpam-5403	182	1	if	if	SCONJ
ejpam-5403	182	2	b	b	X
ejpam-5403	182	3	=	=	SYM
ejpam-5403	182	4	0	0	NUM
ejpam-5403	182	5	,	,	PUNCT
ejpam-5403	182	6	then	then	ADV
ejpam-5403	182	7	ab	ab	PROPN
ejpam-5403	182	8	=	=	PROPN
ejpam-5403	182	9	a0	a0	PROPN
ejpam-5403	182	10	=	=	PUNCT
ejpam-5403	182	11	a	a	DET
ejpam-5403	182	12	∈	∈	NOUN
ejpam-5403	182	13	sa(x	sa(x	NOUN
ejpam-5403	182	14	)	)	PUNCT
ejpam-5403	182	15	.	.	PUNCT
ejpam-5403	183	1	if	if	SCONJ
ejpam-5403	183	2	a	a	DET
ejpam-5403	183	3	̸=	̸=	PROPN
ejpam-5403	183	4	0	0	NUM
ejpam-5403	183	5	and	and	CCONJ
ejpam-5403	183	6	b	b	PROPN
ejpam-5403	183	7	̸=	̸=	PROPN
ejpam-5403	183	8	0	0	NUM
ejpam-5403	183	9	,	,	PUNCT
ejpam-5403	183	10	then	then	ADV
ejpam-5403	183	11	ab	ab	PROPN
ejpam-5403	183	12	=	=	PUNCT
ejpam-5403	183	13	a	a	PRON
ejpam-5403	183	14	since	since	SCONJ
ejpam-5403	183	15	a	a	PRON
ejpam-5403	183	16	is	be	AUX
ejpam-5403	183	17	a	a	DET
ejpam-5403	183	18	strong	strong	ADJ
ejpam-5403	183	19	atom	atom	NOUN
ejpam-5403	183	20	.	.	PUNCT
ejpam-5403	184	1	therefore	therefore	ADV
ejpam-5403	184	2	,	,	PUNCT
ejpam-5403	184	3	ab	ab	PROPN
ejpam-5403	184	4	=	=	PUNCT
ejpam-5403	184	5	a	a	DET
ejpam-5403	184	6	∈	∈	NOUN
ejpam-5403	184	7	sa(x	sa(x	NOUN
ejpam-5403	184	8	)	)	PUNCT
ejpam-5403	184	9	.	.	PUNCT
ejpam-5403	185	1	if	if	SCONJ
ejpam-5403	185	2	a	a	DET
ejpam-5403	185	3	=	=	NOUN
ejpam-5403	185	4	0	0	NUM
ejpam-5403	185	5	and	and	CCONJ
ejpam-5403	185	6	b	b	PROPN
ejpam-5403	185	7	̸=	̸=	PROPN
ejpam-5403	185	8	0	0	NUM
ejpam-5403	185	9	,	,	PUNCT
ejpam-5403	185	10	then	then	ADV
ejpam-5403	185	11	ab	ab	PROPN
ejpam-5403	185	12	=	=	PUNCT
ejpam-5403	185	13	0b	0b	PROPN
ejpam-5403	185	14	.	.	PUNCT
ejpam-5403	186	1	since	since	SCONJ
ejpam-5403	186	2	|sa(x)|	|sa(x)|	PROPN
ejpam-5403	186	3	≥	≥	NOUN
ejpam-5403	186	4	3	3	NUM
ejpam-5403	186	5	,	,	PUNCT
ejpam-5403	186	6	then	then	ADV
ejpam-5403	186	7	there	there	PRON
ejpam-5403	186	8	is	be	VERB
ejpam-5403	186	9	a	a	DET
ejpam-5403	186	10	strong	strong	ADJ
ejpam-5403	186	11	atom	atom	NOUN
ejpam-5403	186	12	c	c	NOUN
ejpam-5403	186	13	such	such	ADJ
ejpam-5403	186	14	that	that	PRON
ejpam-5403	186	15	c	c	NOUN
ejpam-5403	186	16	/∈	/∈	PUNCT
ejpam-5403	187	1	{	{	PUNCT
ejpam-5403	187	2	0	0	NUM
ejpam-5403	187	3	,	,	PUNCT
ejpam-5403	187	4	b	b	NOUN
ejpam-5403	187	5	}	}	PUNCT
ejpam-5403	187	6	.	.	PUNCT
ejpam-5403	188	1	then	then	ADV
ejpam-5403	188	2	cb	cb	PROPN
ejpam-5403	188	3	=	=	PROPN
ejpam-5403	188	4	c.	c.	PROPN
ejpam-5403	188	5	by	by	ADP
ejpam-5403	188	6	proposition	proposition	NOUN
ejpam-5403	188	7	4(i	4(i	NUM
ejpam-5403	188	8	)	)	PUNCT
ejpam-5403	188	9	we	we	PRON
ejpam-5403	188	10	get	get	VERB
ejpam-5403	188	11	that	that	PRON
ejpam-5403	188	12	ab	ab	PROPN
ejpam-5403	188	13	=	=	SYM
ejpam-5403	188	14	0b	0b	PROPN
ejpam-5403	188	15	=	=	SYM
ejpam-5403	188	16	0(bc	0(bc	NUM
ejpam-5403	188	17	)	)	PUNCT
ejpam-5403	188	18	=	=	SYM
ejpam-5403	188	19	c(b0	c(b0	NOUN
ejpam-5403	188	20	)	)	PUNCT
ejpam-5403	188	21	=	=	SYM
ejpam-5403	188	22	cb	cb	PROPN
ejpam-5403	188	23	=	=	PROPN
ejpam-5403	188	24	c.	c.	PROPN
ejpam-5403	188	25	therefore	therefore	ADV
ejpam-5403	188	26	,	,	PUNCT
ejpam-5403	188	27	ab	ab	PROPN
ejpam-5403	188	28	∈	∈	PROPN
ejpam-5403	188	29	sa(x	sa(x	NOUN
ejpam-5403	188	30	)	)	PUNCT
ejpam-5403	188	31	.	.	PUNCT
ejpam-5403	189	1	altogether	altogether	ADV
ejpam-5403	189	2	,	,	PUNCT
ejpam-5403	189	3	we	we	PRON
ejpam-5403	189	4	get	get	VERB
ejpam-5403	189	5	sa(x	sa(x	NOUN
ejpam-5403	189	6	)	)	PUNCT
ejpam-5403	189	7	is	be	AUX
ejpam-5403	189	8	a	a	DET
ejpam-5403	189	9	subalgebra	subalgebra	NOUN
ejpam-5403	189	10	of	of	ADP
ejpam-5403	189	11	x.	x.	NOUN
ejpam-5403	189	12	next	next	ADV
ejpam-5403	189	13	we	we	PRON
ejpam-5403	189	14	will	will	AUX
ejpam-5403	189	15	examine	examine	VERB
ejpam-5403	189	16	some	some	DET
ejpam-5403	189	17	properties	property	NOUN
ejpam-5403	189	18	of	of	ADP
ejpam-5403	189	19	a	a	DET
ejpam-5403	189	20	set	set	NOUN
ejpam-5403	189	21	of	of	ADP
ejpam-5403	189	22	all	all	DET
ejpam-5403	189	23	atoms	atom	NOUN
ejpam-5403	189	24	a(x	a(x	NOUN
ejpam-5403	189	25	)	)	PUNCT
ejpam-5403	189	26	of	of	ADP
ejpam-5403	189	27	any	any	DET
ejpam-5403	189	28	qalgebra	qalgebra	PROPN
ejpam-5403	189	29	x.	x.	NOUN
ejpam-5403	189	30	in	in	ADP
ejpam-5403	189	31	general	general	ADJ
ejpam-5403	189	32	,	,	PUNCT
ejpam-5403	189	33	a	a	DET
ejpam-5403	189	34	set	set	NOUN
ejpam-5403	189	35	a(x	a(x	NOUN
ejpam-5403	189	36	)	)	PUNCT
ejpam-5403	189	37	need	need	VERB
ejpam-5403	189	38	not	not	PART
ejpam-5403	189	39	to	to	PART
ejpam-5403	189	40	be	be	AUX
ejpam-5403	189	41	closed	close	VERB
ejpam-5403	189	42	and	and	CCONJ
ejpam-5403	189	43	also	also	ADV
ejpam-5403	189	44	need	need	VERB
ejpam-5403	189	45	not	not	PART
ejpam-5403	189	46	to	to	PART
ejpam-5403	189	47	be	be	AUX
ejpam-5403	189	48	an	an	DET
ejpam-5403	189	49	ideal	ideal	NOUN
ejpam-5403	189	50	of	of	ADP
ejpam-5403	189	51	x	x	PUNCT
ejpam-5403	189	52	as	as	ADP
ejpam-5403	189	53	the	the	DET
ejpam-5403	189	54	following	follow	VERB
ejpam-5403	189	55	example	example	NOUN
ejpam-5403	189	56	.	.	PUNCT
ejpam-5403	190	1	example	example	NOUN
ejpam-5403	191	1	4	4	NUM
ejpam-5403	191	2	.	.	PUNCT
ejpam-5403	191	3	let	let	VERB
ejpam-5403	191	4	x	x	PUNCT
ejpam-5403	191	5	=	=	PUNCT
ejpam-5403	191	6	{	{	PUNCT
ejpam-5403	191	7	0	0	NUM
ejpam-5403	191	8	,	,	PUNCT
ejpam-5403	191	9	a	a	DET
ejpam-5403	191	10	,	,	PUNCT
ejpam-5403	191	11	b	b	NOUN
ejpam-5403	191	12	,	,	PUNCT
ejpam-5403	191	13	c	c	NOUN
ejpam-5403	191	14	,	,	PUNCT
ejpam-5403	191	15	d	d	NOUN
ejpam-5403	191	16	,	,	PUNCT
ejpam-5403	191	17	f	f	NOUN
ejpam-5403	191	18	}	}	PUNCT
ejpam-5403	191	19	and	and	CCONJ
ejpam-5403	191	20	let	let	VERB
ejpam-5403	191	21	a	a	DET
ejpam-5403	191	22	binary	binary	ADJ
ejpam-5403	191	23	operation	operation	NOUN
ejpam-5403	191	24	∗	∗	NOUN
ejpam-5403	191	25	be	be	AUX
ejpam-5403	191	26	defined	define	VERB
ejpam-5403	191	27	on	on	ADP
ejpam-5403	191	28	x	x	PUNCT
ejpam-5403	191	29	as	as	ADP
ejpam-5403	191	30	the	the	DET
ejpam-5403	191	31	following	following	NOUN
ejpam-5403	191	32	:	:	PUNCT
ejpam-5403	191	33	∗	∗	NOUN
ejpam-5403	191	34	0	0	NUM
ejpam-5403	192	1	a	a	DET
ejpam-5403	192	2	b	b	NOUN
ejpam-5403	192	3	c	c	NOUN
ejpam-5403	192	4	d	d	X
ejpam-5403	192	5	f	f	PROPN
ejpam-5403	192	6	0	0	NUM
ejpam-5403	192	7	0	0	NUM
ejpam-5403	192	8	a	a	DET
ejpam-5403	192	9	c	c	PROPN
ejpam-5403	192	10	b	b	PROPN
ejpam-5403	192	11	b	b	PROPN
ejpam-5403	192	12	b	b	PROPN
ejpam-5403	192	13	a	a	DET
ejpam-5403	192	14	a	a	DET
ejpam-5403	192	15	0	0	NUM
ejpam-5403	192	16	b	b	NOUN
ejpam-5403	192	17	c	c	NOUN
ejpam-5403	192	18	c	c	NOUN
ejpam-5403	192	19	c	c	PROPN
ejpam-5403	192	20	b	b	PROPN
ejpam-5403	192	21	b	b	PROPN
ejpam-5403	192	22	c	c	PROPN
ejpam-5403	192	23	0	0	NUM
ejpam-5403	192	24	a	a	DET
ejpam-5403	192	25	a	a	DET
ejpam-5403	192	26	a	a	DET
ejpam-5403	192	27	c	c	NOUN
ejpam-5403	192	28	c	c	NOUN
ejpam-5403	192	29	b	b	PROPN
ejpam-5403	192	30	a	a	PRON
ejpam-5403	192	31	0	0	NUM
ejpam-5403	192	32	0	0	NUM
ejpam-5403	192	33	0	0	NUM
ejpam-5403	193	1	d	d	PROPN
ejpam-5403	193	2	d	d	PROPN
ejpam-5403	193	3	b	b	PROPN
ejpam-5403	193	4	a	a	PRON
ejpam-5403	193	5	0	0	NUM
ejpam-5403	193	6	0	0	NUM
ejpam-5403	193	7	0	0	NUM
ejpam-5403	194	1	f	f	PROPN
ejpam-5403	194	2	f	f	PROPN
ejpam-5403	194	3	b	b	PROPN
ejpam-5403	194	4	a	a	DET
ejpam-5403	194	5	0	0	NUM
ejpam-5403	194	6	0	0	NUM
ejpam-5403	194	7	0	0	NUM
ejpam-5403	194	8	a.	a.	NOUN
ejpam-5403	194	9	anantayasethi	anantayasethi	PROPN
ejpam-5403	194	10	,	,	PUNCT
ejpam-5403	194	11	t.	t.	PROPN
ejpam-5403	194	12	kunawat	kunawat	PROPN
ejpam-5403	194	13	,	,	PUNCT
ejpam-5403	194	14	p.	p.	PROPN
ejpam-5403	194	15	moolnipa	moolnipa	PROPN
ejpam-5403	194	16	/	/	SYM
ejpam-5403	194	17	eur	eur	PROPN
ejpam-5403	194	18	.	.	PUNCT
ejpam-5403	195	1	j.	j.	PROPN
ejpam-5403	195	2	pure	pure	PROPN
ejpam-5403	195	3	appl	appl	PROPN
ejpam-5403	195	4	.	.	PROPN
ejpam-5403	195	5	math	math	PROPN
ejpam-5403	195	6	,	,	PUNCT
ejpam-5403	195	7	17	17	NUM
ejpam-5403	195	8	(	(	PUNCT
ejpam-5403	195	9	4	4	NUM
ejpam-5403	195	10	)	)	PUNCT
ejpam-5403	195	11	(	(	PUNCT
ejpam-5403	195	12	2024	2024	NUM
ejpam-5403	195	13	)	)	PUNCT
ejpam-5403	195	14	,	,	PUNCT
ejpam-5403	195	15	3268	3268	NUM
ejpam-5403	195	16	-	-	SYM
ejpam-5403	195	17	3276	3276	NUM
ejpam-5403	195	18	3273	3273	NUM
ejpam-5403	195	19	it	it	PRON
ejpam-5403	195	20	is	be	AUX
ejpam-5403	195	21	a	a	DET
ejpam-5403	195	22	routine	routine	NOUN
ejpam-5403	195	23	to	to	PART
ejpam-5403	195	24	check	check	VERB
ejpam-5403	195	25	that	that	PRON
ejpam-5403	195	26	(	(	PUNCT
ejpam-5403	195	27	x	x	X
ejpam-5403	195	28	;	;	PUNCT
ejpam-5403	195	29	∗	∗	NOUN
ejpam-5403	195	30	,	,	PUNCT
ejpam-5403	195	31	0	0	NUM
ejpam-5403	195	32	)	)	PUNCT
ejpam-5403	195	33	is	be	AUX
ejpam-5403	195	34	a	a	DET
ejpam-5403	195	35	q	q	NOUN
ejpam-5403	195	36	-	-	NOUN
ejpam-5403	195	37	algebra	algebra	NOUN
ejpam-5403	195	38	.	.	PUNCT
ejpam-5403	196	1	it	it	PRON
ejpam-5403	196	2	is	be	AUX
ejpam-5403	196	3	easy	easy	ADJ
ejpam-5403	196	4	to	to	PART
ejpam-5403	196	5	see	see	VERB
ejpam-5403	196	6	that	that	SCONJ
ejpam-5403	196	7	a(x	a(x	NOUN
ejpam-5403	196	8	)	)	PUNCT
ejpam-5403	196	9	=	=	PUNCT
ejpam-5403	196	10	{	{	PUNCT
ejpam-5403	196	11	0	0	NUM
ejpam-5403	196	12	,	,	PUNCT
ejpam-5403	196	13	a	a	DET
ejpam-5403	196	14	,	,	PUNCT
ejpam-5403	196	15	b	b	NOUN
ejpam-5403	196	16	}	}	PUNCT
ejpam-5403	196	17	.	.	PUNCT
ejpam-5403	197	1	we	we	PRON
ejpam-5403	197	2	get	get	VERB
ejpam-5403	197	3	that	that	DET
ejpam-5403	197	4	a(x	a(x	NOUN
ejpam-5403	197	5	)	)	PUNCT
ejpam-5403	197	6	is	be	AUX
ejpam-5403	197	7	not	not	PART
ejpam-5403	197	8	a	a	DET
ejpam-5403	197	9	subalgebra	subalgebra	NOUN
ejpam-5403	197	10	since	since	SCONJ
ejpam-5403	197	11	0	0	NUM
ejpam-5403	197	12	,	,	PUNCT
ejpam-5403	197	13	b	b	X
ejpam-5403	197	14	∈	∈	PROPN
ejpam-5403	197	15	a(x	a(x	PROPN
ejpam-5403	197	16	)	)	PUNCT
ejpam-5403	197	17	but	but	CCONJ
ejpam-5403	197	18	0b	0b	NOUN
ejpam-5403	197	19	=	=	SYM
ejpam-5403	197	20	c	c	PROPN
ejpam-5403	197	21	/∈	/∈	PUNCT
ejpam-5403	197	22	a(x	a(x	NOUN
ejpam-5403	197	23	)	)	PUNCT
ejpam-5403	197	24	.	.	PUNCT
ejpam-5403	198	1	moreover	moreover	ADV
ejpam-5403	198	2	,	,	PUNCT
ejpam-5403	198	3	a(x	a(x	NOUN
ejpam-5403	198	4	)	)	PUNCT
ejpam-5403	198	5	is	be	AUX
ejpam-5403	198	6	not	not	PART
ejpam-5403	198	7	an	an	DET
ejpam-5403	198	8	ideal	ideal	NOUN
ejpam-5403	198	9	of	of	ADP
ejpam-5403	198	10	x.	x.	NOUN
ejpam-5403	198	11	indeed	indeed	ADV
ejpam-5403	198	12	,	,	PUNCT
ejpam-5403	198	13	db	db	PROPN
ejpam-5403	198	14	=	=	PUNCT
ejpam-5403	198	15	a	a	DET
ejpam-5403	198	16	∈	∈	PROPN
ejpam-5403	198	17	a(x	a(x	PROPN
ejpam-5403	198	18	)	)	PUNCT
ejpam-5403	198	19	and	and	CCONJ
ejpam-5403	198	20	b	b	X
ejpam-5403	198	21	∈	∈	NOUN
ejpam-5403	198	22	a(x	a(x	PROPN
ejpam-5403	198	23	)	)	PUNCT
ejpam-5403	198	24	but	but	CCONJ
ejpam-5403	198	25	d	d	NOUN
ejpam-5403	198	26	/∈	/∈	PUNCT
ejpam-5403	198	27	a(x	a(x	NOUN
ejpam-5403	198	28	)	)	PUNCT
ejpam-5403	198	29	.	.	PUNCT
ejpam-5403	199	1	from	from	ADP
ejpam-5403	199	2	example	example	NOUN
ejpam-5403	199	3	4	4	NUM
ejpam-5403	199	4	,	,	PUNCT
ejpam-5403	199	5	let	let	VERB
ejpam-5403	199	6	we	we	PRON
ejpam-5403	199	7	mention	mention	VERB
ejpam-5403	199	8	some	some	DET
ejpam-5403	199	9	errors	error	NOUN
ejpam-5403	199	10	in	in	ADP
ejpam-5403	199	11	[	[	X
ejpam-5403	199	12	3	3	NUM
ejpam-5403	199	13	]	]	PUNCT
ejpam-5403	199	14	,	,	PUNCT
ejpam-5403	199	15	namely	namely	ADV
ejpam-5403	199	16	[	[	PUNCT
ejpam-5403	199	17	corollary	corollary	ADJ
ejpam-5403	199	18	3.6	3.6	NUM
ejpam-5403	199	19	]	]	PUNCT
ejpam-5403	199	20	:	:	PUNCT
ejpam-5403	199	21	”	"	PUNCT
ejpam-5403	199	22	let	let	VERB
ejpam-5403	199	23	x	x	PRON
ejpam-5403	199	24	be	be	AUX
ejpam-5403	199	25	a	a	DET
ejpam-5403	199	26	q	q	NOUN
ejpam-5403	199	27	-	-	NOUN
ejpam-5403	199	28	algebra	algebra	NOUN
ejpam-5403	199	29	.	.	PUNCT
ejpam-5403	200	1	if	if	SCONJ
ejpam-5403	200	2	a	a	PRON
ejpam-5403	200	3	is	be	AUX
ejpam-5403	200	4	an	an	DET
ejpam-5403	200	5	atom	atom	NOUN
ejpam-5403	200	6	of	of	ADP
ejpam-5403	200	7	x	x	NOUN
ejpam-5403	200	8	,	,	PUNCT
ejpam-5403	200	9	then	then	ADV
ejpam-5403	200	10	for	for	ADP
ejpam-5403	200	11	all	all	DET
ejpam-5403	200	12	x	x	NOUN
ejpam-5403	200	13	in	in	ADP
ejpam-5403	200	14	x	x	X
ejpam-5403	200	15	,	,	PUNCT
ejpam-5403	200	16	ax	ax	NOUN
ejpam-5403	200	17	is	be	AUX
ejpam-5403	200	18	an	an	DET
ejpam-5403	200	19	atom	atom	NOUN
ejpam-5403	200	20	.	.	PUNCT
ejpam-5403	201	1	hence	hence	ADV
ejpam-5403	201	2	,	,	PUNCT
ejpam-5403	201	3	a(x	a(x	PROPN
ejpam-5403	201	4	)	)	PUNCT
ejpam-5403	201	5	is	be	AUX
ejpam-5403	201	6	a	a	DET
ejpam-5403	201	7	subalgebra	subalgebra	NOUN
ejpam-5403	201	8	of	of	ADP
ejpam-5403	201	9	x.	x.	NOUN
ejpam-5403	201	10	for	for	ADP
ejpam-5403	201	11	every	every	DET
ejpam-5403	201	12	x	x	PROPN
ejpam-5403	201	13	of	of	ADP
ejpam-5403	201	14	x	x	PRON
ejpam-5403	201	15	,	,	PUNCT
ejpam-5403	201	16	there	there	PRON
ejpam-5403	201	17	is	be	VERB
ejpam-5403	201	18	an	an	DET
ejpam-5403	201	19	atom	atom	NOUN
ejpam-5403	201	20	a	a	DET
ejpam-5403	201	21	such	such	ADJ
ejpam-5403	201	22	that	that	DET
ejpam-5403	201	23	ax	ax	NOUN
ejpam-5403	201	24	=	=	SYM
ejpam-5403	201	25	0	0	NUM
ejpam-5403	201	26	,	,	PUNCT
ejpam-5403	201	27	i.e.	i.e.	X
ejpam-5403	201	28	every	every	DET
ejpam-5403	201	29	q	q	NOUN
ejpam-5403	201	30	-	-	PUNCT
ejpam-5403	201	31	algebra	algebra	NOUN
ejpam-5403	201	32	is	be	AUX
ejpam-5403	201	33	generated	generate	VERB
ejpam-5403	201	34	by	by	ADP
ejpam-5403	201	35	atoms	atom	NOUN
ejpam-5403	201	36	.	.	PUNCT
ejpam-5403	201	37	”	"	PUNCT
ejpam-5403	202	1	is	be	AUX
ejpam-5403	202	2	invalid	invalid	ADJ
ejpam-5403	202	3	.	.	PUNCT
ejpam-5403	203	1	the	the	DET
ejpam-5403	203	2	mistakes	mistake	NOUN
ejpam-5403	203	3	show	show	VERB
ejpam-5403	203	4	in	in	ADP
ejpam-5403	203	5	example	example	NOUN
ejpam-5403	203	6	4	4	NUM
ejpam-5403	203	7	.	.	PUNCT
ejpam-5403	204	1	next	next	ADV
ejpam-5403	204	2	,	,	PUNCT
ejpam-5403	204	3	we	we	PRON
ejpam-5403	204	4	investigate	investigate	VERB
ejpam-5403	204	5	some	some	DET
ejpam-5403	204	6	relations	relation	NOUN
ejpam-5403	204	7	between	between	ADP
ejpam-5403	204	8	g	g	NOUN
ejpam-5403	204	9	-	-	PUNCT
ejpam-5403	204	10	part	part	NOUN
ejpam-5403	204	11	g(x	g(x	NOUN
ejpam-5403	204	12	)	)	PUNCT
ejpam-5403	204	13	and	and	CCONJ
ejpam-5403	204	14	set	set	VERB
ejpam-5403	204	15	of	of	ADP
ejpam-5403	204	16	all	all	DET
ejpam-5403	204	17	atoms	atom	NOUN
ejpam-5403	204	18	a(x	a(x	NOUN
ejpam-5403	204	19	)	)	PUNCT
ejpam-5403	204	20	.	.	PUNCT
ejpam-5403	205	1	we	we	PRON
ejpam-5403	205	2	know	know	VERB
ejpam-5403	205	3	that	that	SCONJ
ejpam-5403	205	4	g(x)∩a(x	g(x)∩a(x	NOUN
ejpam-5403	205	5	)	)	PUNCT
ejpam-5403	205	6	̸=	̸=	PROPN
ejpam-5403	205	7	∅	∅	NOUN
ejpam-5403	205	8	since	since	SCONJ
ejpam-5403	205	9	an	an	DET
ejpam-5403	205	10	element	element	NOUN
ejpam-5403	205	11	0	0	NUM
ejpam-5403	205	12	is	be	AUX
ejpam-5403	205	13	an	an	DET
ejpam-5403	205	14	atom	atom	NOUN
ejpam-5403	205	15	and	and	CCONJ
ejpam-5403	205	16	00	00	NUM
ejpam-5403	205	17	=	=	SYM
ejpam-5403	205	18	0	0	NUM
ejpam-5403	205	19	∈	∈	PROPN
ejpam-5403	205	20	g(x	g(x	PROPN
ejpam-5403	205	21	)	)	PUNCT
ejpam-5403	205	22	.	.	PUNCT
ejpam-5403	206	1	the	the	DET
ejpam-5403	206	2	set	set	ADJ
ejpam-5403	206	3	g(x	g(x	NOUN
ejpam-5403	206	4	)	)	PUNCT
ejpam-5403	206	5	need	need	VERB
ejpam-5403	206	6	not	not	PART
ejpam-5403	206	7	to	to	PART
ejpam-5403	206	8	be	be	AUX
ejpam-5403	206	9	a	a	DET
ejpam-5403	206	10	subset	subset	NOUN
ejpam-5403	206	11	of	of	ADP
ejpam-5403	206	12	a(x	a(x	NOUN
ejpam-5403	206	13	)	)	PUNCT
ejpam-5403	206	14	and	and	CCONJ
ejpam-5403	206	15	vice	vice	ADV
ejpam-5403	206	16	versa	versa	ADV
ejpam-5403	206	17	.	.	PUNCT
ejpam-5403	207	1	from	from	ADP
ejpam-5403	207	2	example	example	NOUN
ejpam-5403	207	3	1	1	NUM
ejpam-5403	207	4	we	we	PRON
ejpam-5403	207	5	have	have	VERB
ejpam-5403	207	6	that	that	DET
ejpam-5403	207	7	g(x	g(x	NOUN
ejpam-5403	207	8	)	)	PUNCT
ejpam-5403	207	9	⊆	⊆	NUM
ejpam-5403	207	10	a(x	a(x	NOUN
ejpam-5403	207	11	)	)	PUNCT
ejpam-5403	207	12	and	and	CCONJ
ejpam-5403	207	13	a(y	a(y	PROPN
ejpam-5403	207	14	)	)	PUNCT
ejpam-5403	208	1	⊆	⊆	NUM
ejpam-5403	208	2	g(y	g(y	PROPN
ejpam-5403	208	3	)	)	PUNCT
ejpam-5403	208	4	.	.	PUNCT
ejpam-5403	209	1	our	our	PRON
ejpam-5403	209	2	aim	aim	NOUN
ejpam-5403	209	3	is	be	AUX
ejpam-5403	209	4	to	to	PART
ejpam-5403	209	5	find	find	VERB
ejpam-5403	209	6	some	some	DET
ejpam-5403	209	7	conditions	condition	NOUN
ejpam-5403	209	8	that	that	PRON
ejpam-5403	209	9	yield	yield	VERB
ejpam-5403	209	10	previous	previous	ADJ
ejpam-5403	209	11	inclusions	inclusion	NOUN
ejpam-5403	209	12	.	.	PUNCT
ejpam-5403	210	1	next	next	ADJ
ejpam-5403	210	2	proprosition	proprosition	NOUN
ejpam-5403	210	3	shows	show	VERB
ejpam-5403	210	4	a	a	DET
ejpam-5403	210	5	sufficient	sufficient	ADJ
ejpam-5403	210	6	condition	condition	NOUN
ejpam-5403	210	7	of	of	ADP
ejpam-5403	210	8	an	an	DET
ejpam-5403	210	9	element	element	NOUN
ejpam-5403	210	10	of	of	ADP
ejpam-5403	210	11	g(x	g(x	NOUN
ejpam-5403	210	12	)	)	PUNCT
ejpam-5403	210	13	to	to	PART
ejpam-5403	210	14	be	be	AUX
ejpam-5403	210	15	an	an	DET
ejpam-5403	210	16	atom	atom	NOUN
ejpam-5403	210	17	of	of	ADP
ejpam-5403	210	18	x.	x.	NOUN
ejpam-5403	210	19	proposition	proposition	PROPN
ejpam-5403	210	20	10	10	NUM
ejpam-5403	210	21	.	.	PUNCT
ejpam-5403	211	1	let	let	VERB
ejpam-5403	211	2	x	x	PRON
ejpam-5403	211	3	be	be	AUX
ejpam-5403	211	4	a	a	DET
ejpam-5403	211	5	q	q	NOUN
ejpam-5403	211	6	-	-	NOUN
ejpam-5403	211	7	algebra	algebra	NOUN
ejpam-5403	211	8	.	.	PUNCT
ejpam-5403	212	1	if	if	SCONJ
ejpam-5403	212	2	g(x	g(x	NOUN
ejpam-5403	212	3	)	)	PUNCT
ejpam-5403	212	4	is	be	AUX
ejpam-5403	212	5	an	an	DET
ejpam-5403	212	6	ideal	ideal	NOUN
ejpam-5403	212	7	of	of	ADP
ejpam-5403	212	8	x	x	NOUN
ejpam-5403	212	9	,	,	PUNCT
ejpam-5403	212	10	then	then	ADV
ejpam-5403	212	11	g(x	g(x	NOUN
ejpam-5403	212	12	)	)	PUNCT
ejpam-5403	212	13	⊆	⊆	NUM
ejpam-5403	212	14	a(x	a(x	NOUN
ejpam-5403	212	15	)	)	PUNCT
ejpam-5403	212	16	.	.	PUNCT
ejpam-5403	213	1	proof	proof	NOUN
ejpam-5403	213	2	.	.	PUNCT
ejpam-5403	214	1	assume	assume	VERB
ejpam-5403	214	2	that	that	SCONJ
ejpam-5403	214	3	g(x	g(x	NOUN
ejpam-5403	214	4	)	)	PUNCT
ejpam-5403	214	5	is	be	AUX
ejpam-5403	214	6	an	an	DET
ejpam-5403	214	7	ideal	ideal	NOUN
ejpam-5403	214	8	of	of	ADP
ejpam-5403	214	9	x.	x.	NOUN
ejpam-5403	214	10	let	let	VERB
ejpam-5403	214	11	a	a	DET
ejpam-5403	214	12	∈	∈	PROPN
ejpam-5403	214	13	g(x	g(x	NOUN
ejpam-5403	214	14	)	)	PUNCT
ejpam-5403	214	15	.	.	PUNCT
ejpam-5403	215	1	if	if	SCONJ
ejpam-5403	215	2	a	a	DET
ejpam-5403	215	3	=	=	SYM
ejpam-5403	215	4	0	0	NUM
ejpam-5403	215	5	,	,	PUNCT
ejpam-5403	215	6	then	then	ADV
ejpam-5403	215	7	a	a	DET
ejpam-5403	215	8	∈	∈	PROPN
ejpam-5403	215	9	a(x	a(x	NOUN
ejpam-5403	215	10	)	)	PUNCT
ejpam-5403	215	11	.	.	PUNCT
ejpam-5403	216	1	now	now	ADV
ejpam-5403	216	2	we	we	PRON
ejpam-5403	216	3	assume	assume	VERB
ejpam-5403	216	4	that	that	SCONJ
ejpam-5403	216	5	a	a	DET
ejpam-5403	216	6	̸=	̸=	PROPN
ejpam-5403	216	7	0	0	NUM
ejpam-5403	216	8	.	.	PUNCT
ejpam-5403	216	9	suppose	suppose	VERB
ejpam-5403	216	10	that	that	SCONJ
ejpam-5403	216	11	there	there	PRON
ejpam-5403	216	12	is	be	VERB
ejpam-5403	216	13	an	an	DET
ejpam-5403	216	14	element	element	NOUN
ejpam-5403	216	15	w	w	PROPN
ejpam-5403	216	16	∈	∈	PROPN
ejpam-5403	216	17	x	x	X
ejpam-5403	216	18	,	,	PUNCT
ejpam-5403	216	19	w	w	PROPN
ejpam-5403	216	20	̸=	̸=	PROPN
ejpam-5403	216	21	a	a	DET
ejpam-5403	216	22	such	such	ADJ
ejpam-5403	216	23	that	that	DET
ejpam-5403	216	24	wa	wa	NOUN
ejpam-5403	216	25	=	=	NOUN
ejpam-5403	216	26	0	0	PROPN
ejpam-5403	216	27	.	.	PUNCT
ejpam-5403	217	1	since	since	SCONJ
ejpam-5403	217	2	a	a	DET
ejpam-5403	217	3	∈	∈	PROPN
ejpam-5403	217	4	g(x	g(x	NOUN
ejpam-5403	217	5	)	)	PUNCT
ejpam-5403	217	6	,	,	PUNCT
ejpam-5403	217	7	then	then	ADV
ejpam-5403	217	8	0a	0a	PROPN
ejpam-5403	217	9	=	=	PUNCT
ejpam-5403	217	10	a	a	DET
ejpam-5403	217	11	there	there	PRON
ejpam-5403	217	12	follows	follow	VERB
ejpam-5403	217	13	that	that	SCONJ
ejpam-5403	217	14	w	w	PROPN
ejpam-5403	217	15	̸=	̸=	PROPN
ejpam-5403	217	16	0	0	NUM
ejpam-5403	217	17	.	.	PUNCT
ejpam-5403	218	1	since	since	SCONJ
ejpam-5403	218	2	wa	wa	NOUN
ejpam-5403	218	3	=	=	SYM
ejpam-5403	218	4	0	0	NUM
ejpam-5403	218	5	∈	∈	PROPN
ejpam-5403	218	6	g(x	g(x	PROPN
ejpam-5403	218	7	)	)	PUNCT
ejpam-5403	218	8	,	,	PUNCT
ejpam-5403	218	9	a	a	DET
ejpam-5403	218	10	∈	∈	PROPN
ejpam-5403	218	11	g(x	g(x	NOUN
ejpam-5403	218	12	)	)	PUNCT
ejpam-5403	218	13	and	and	CCONJ
ejpam-5403	218	14	g(x	g(x	NOUN
ejpam-5403	218	15	)	)	PUNCT
ejpam-5403	218	16	is	be	AUX
ejpam-5403	218	17	ideal	ideal	ADJ
ejpam-5403	218	18	,	,	PUNCT
ejpam-5403	218	19	then	then	ADV
ejpam-5403	218	20	w	w	PROPN
ejpam-5403	218	21	∈	∈	PROPN
ejpam-5403	218	22	g(x	g(x	PROPN
ejpam-5403	218	23	)	)	PUNCT
ejpam-5403	218	24	.	.	PUNCT
ejpam-5403	219	1	now	now	ADV
ejpam-5403	219	2	,	,	PUNCT
ejpam-5403	219	3	there	there	PRON
ejpam-5403	219	4	are	be	VERB
ejpam-5403	219	5	0	0	NUM
ejpam-5403	219	6	,	,	PUNCT
ejpam-5403	219	7	a	a	PRON
ejpam-5403	219	8	and	and	CCONJ
ejpam-5403	219	9	w	w	NOUN
ejpam-5403	219	10	belong	belong	VERB
ejpam-5403	219	11	to	to	ADP
ejpam-5403	219	12	g(x	g(x	NOUN
ejpam-5403	219	13	)	)	PUNCT
ejpam-5403	219	14	and	and	CCONJ
ejpam-5403	219	15	wa	wa	NOUN
ejpam-5403	219	16	=	=	NOUN
ejpam-5403	219	17	0	0	PROPN
ejpam-5403	219	18	,	,	PUNCT
ejpam-5403	219	19	then	then	ADV
ejpam-5403	219	20	by	by	ADP
ejpam-5403	219	21	proposition	proposition	NOUN
ejpam-5403	219	22	3(ii	3(ii	NUM
ejpam-5403	219	23	)	)	PUNCT
ejpam-5403	219	24	we	we	PRON
ejpam-5403	219	25	get	get	VERB
ejpam-5403	219	26	that	that	DET
ejpam-5403	219	27	a0	a0	PROPN
ejpam-5403	219	28	=	=	SYM
ejpam-5403	219	29	w.	w.	PROPN
ejpam-5403	219	30	thus	thus	ADV
ejpam-5403	219	31	,	,	PUNCT
ejpam-5403	219	32	by	by	ADP
ejpam-5403	219	33	(	(	PUNCT
ejpam-5403	219	34	q2	q2	NOUN
ejpam-5403	219	35	)	)	PUNCT
ejpam-5403	219	36	we	we	PRON
ejpam-5403	219	37	get	get	VERB
ejpam-5403	219	38	that	that	DET
ejpam-5403	219	39	w	w	PROPN
ejpam-5403	219	40	=	=	SYM
ejpam-5403	219	41	a0	a0	PROPN
ejpam-5403	219	42	=	=	PUNCT
ejpam-5403	219	43	a	a	PROPN
ejpam-5403	219	44	,	,	PUNCT
ejpam-5403	219	45	a	a	DET
ejpam-5403	219	46	contradiction	contradiction	NOUN
ejpam-5403	219	47	.	.	PUNCT
ejpam-5403	220	1	hence	hence	ADV
ejpam-5403	220	2	,	,	PUNCT
ejpam-5403	220	3	wa	wa	NOUN
ejpam-5403	220	4	=	=	SYM
ejpam-5403	220	5	0	0	PROPN
ejpam-5403	220	6	implies	imply	VERB
ejpam-5403	220	7	w	w	NOUN
ejpam-5403	220	8	=	=	VERB
ejpam-5403	220	9	a.	a.	NOUN
ejpam-5403	220	10	this	this	PRON
ejpam-5403	220	11	gives	give	VERB
ejpam-5403	220	12	a	a	PRON
ejpam-5403	220	13	is	be	AUX
ejpam-5403	220	14	an	an	DET
ejpam-5403	220	15	atom	atom	NOUN
ejpam-5403	220	16	of	of	ADP
ejpam-5403	220	17	x.	x.	NOUN
ejpam-5403	220	18	altogether	altogether	ADV
ejpam-5403	220	19	,	,	PUNCT
ejpam-5403	220	20	we	we	PRON
ejpam-5403	220	21	get	get	VERB
ejpam-5403	220	22	g(x	g(x	NOUN
ejpam-5403	220	23	)	)	PUNCT
ejpam-5403	220	24	⊆	⊆	NUM
ejpam-5403	220	25	a(x	a(x	NOUN
ejpam-5403	220	26	)	)	PUNCT
ejpam-5403	220	27	.	.	PUNCT
ejpam-5403	221	1	the	the	DET
ejpam-5403	221	2	converse	converse	NOUN
ejpam-5403	221	3	of	of	ADP
ejpam-5403	221	4	proposition	proposition	NOUN
ejpam-5403	221	5	10	10	NUM
ejpam-5403	221	6	is	be	AUX
ejpam-5403	221	7	not	not	PART
ejpam-5403	221	8	true	true	ADJ
ejpam-5403	221	9	,	,	PUNCT
ejpam-5403	221	10	i.e.	i.e.	X
ejpam-5403	221	11	if	if	SCONJ
ejpam-5403	221	12	all	all	DET
ejpam-5403	221	13	members	member	NOUN
ejpam-5403	221	14	of	of	ADP
ejpam-5403	221	15	g(x	g(x	NOUN
ejpam-5403	221	16	)	)	PUNCT
ejpam-5403	221	17	are	be	AUX
ejpam-5403	221	18	atoms	atom	NOUN
ejpam-5403	221	19	of	of	ADP
ejpam-5403	221	20	x	x	NOUN
ejpam-5403	221	21	,	,	PUNCT
ejpam-5403	221	22	then	then	ADV
ejpam-5403	221	23	g(x	g(x	NOUN
ejpam-5403	221	24	)	)	PUNCT
ejpam-5403	221	25	need	need	VERB
ejpam-5403	221	26	not	not	PART
ejpam-5403	221	27	to	to	PART
ejpam-5403	221	28	be	be	AUX
ejpam-5403	221	29	an	an	DET
ejpam-5403	221	30	ideal	ideal	NOUN
ejpam-5403	221	31	of	of	ADP
ejpam-5403	221	32	x.	x.	NOUN
ejpam-5403	221	33	the	the	DET
ejpam-5403	221	34	following	follow	VERB
ejpam-5403	221	35	example	example	NOUN
ejpam-5403	221	36	is	be	AUX
ejpam-5403	221	37	the	the	DET
ejpam-5403	221	38	counterexample	counterexample	NOUN
ejpam-5403	221	39	of	of	ADP
ejpam-5403	221	40	the	the	DET
ejpam-5403	221	41	converse	converse	NOUN
ejpam-5403	221	42	.	.	PUNCT
ejpam-5403	221	43	example	example	NOUN
ejpam-5403	222	1	5	5	NUM
ejpam-5403	222	2	.	.	PUNCT
ejpam-5403	223	1	[	[	X
ejpam-5403	223	2	13	13	NUM
ejpam-5403	223	3	]	]	PUNCT
ejpam-5403	223	4	let	let	AUX
ejpam-5403	223	5	consider	consider	VERB
ejpam-5403	223	6	a	a	DET
ejpam-5403	223	7	q	q	NOUN
ejpam-5403	223	8	-	-	PUNCT
ejpam-5403	223	9	algebra	algebra	NOUN
ejpam-5403	223	10	x	x	NOUN
ejpam-5403	223	11	,	,	PUNCT
ejpam-5403	223	12	defined	define	VERB
ejpam-5403	223	13	as	as	ADP
ejpam-5403	223	14	the	the	DET
ejpam-5403	223	15	following	follow	VERB
ejpam-5403	223	16	table	table	NOUN
ejpam-5403	223	17	:	:	PUNCT
ejpam-5403	223	18	∗	∗	NOUN
ejpam-5403	223	19	0	0	NUM
ejpam-5403	224	1	1	1	NUM
ejpam-5403	224	2	2	2	NUM
ejpam-5403	224	3	3	3	NUM
ejpam-5403	224	4	4	4	NUM
ejpam-5403	224	5	0	0	NUM
ejpam-5403	224	6	0	0	NUM
ejpam-5403	224	7	0	0	NUM
ejpam-5403	224	8	0	0	NUM
ejpam-5403	224	9	0	0	NUM
ejpam-5403	224	10	4	4	NUM
ejpam-5403	224	11	1	1	NUM
ejpam-5403	224	12	1	1	NUM
ejpam-5403	224	13	0	0	NUM
ejpam-5403	224	14	0	0	NUM
ejpam-5403	224	15	1	1	NUM
ejpam-5403	224	16	4	4	NUM
ejpam-5403	224	17	2	2	NUM
ejpam-5403	224	18	2	2	NUM
ejpam-5403	224	19	2	2	NUM
ejpam-5403	224	20	0	0	NUM
ejpam-5403	224	21	0	0	NUM
ejpam-5403	224	22	4	4	NUM
ejpam-5403	224	23	3	3	NUM
ejpam-5403	224	24	3	3	NUM
ejpam-5403	224	25	0	0	NUM
ejpam-5403	224	26	3	3	NUM
ejpam-5403	224	27	0	0	NUM
ejpam-5403	224	28	4	4	NUM
ejpam-5403	224	29	4	4	NUM
ejpam-5403	224	30	4	4	NUM
ejpam-5403	224	31	4	4	NUM
ejpam-5403	224	32	4	4	NUM
ejpam-5403	224	33	4	4	NUM
ejpam-5403	224	34	0	0	NUM
ejpam-5403	224	35	it	it	PRON
ejpam-5403	224	36	is	be	AUX
ejpam-5403	224	37	not	not	PART
ejpam-5403	224	38	difficult	difficult	ADJ
ejpam-5403	224	39	to	to	PART
ejpam-5403	224	40	verify	verify	VERB
ejpam-5403	224	41	that	that	DET
ejpam-5403	224	42	g(x	g(x	NOUN
ejpam-5403	224	43	)	)	PUNCT
ejpam-5403	225	1	=	=	PRON
ejpam-5403	225	2	{	{	PUNCT
ejpam-5403	225	3	0	0	NUM
ejpam-5403	225	4	,	,	PUNCT
ejpam-5403	225	5	4	4	NUM
ejpam-5403	225	6	}	}	PUNCT
ejpam-5403	225	7	and	and	CCONJ
ejpam-5403	225	8	a(x	a(x	PROPN
ejpam-5403	225	9	)	)	PUNCT
ejpam-5403	225	10	=	=	PUNCT
ejpam-5403	225	11	{	{	PUNCT
ejpam-5403	225	12	0	0	NUM
ejpam-5403	225	13	,	,	PUNCT
ejpam-5403	225	14	4	4	NUM
ejpam-5403	225	15	}	}	PUNCT
ejpam-5403	225	16	.	.	PUNCT
ejpam-5403	226	1	then	then	ADV
ejpam-5403	226	2	g(x	g(x	NOUN
ejpam-5403	226	3	)	)	PUNCT
ejpam-5403	226	4	⊆	⊆	NUM
ejpam-5403	226	5	a(x	a(x	NOUN
ejpam-5403	226	6	)	)	PUNCT
ejpam-5403	226	7	.	.	PUNCT
ejpam-5403	227	1	but	but	CCONJ
ejpam-5403	227	2	g(x	g(x	NOUN
ejpam-5403	227	3	)	)	PUNCT
ejpam-5403	227	4	is	be	AUX
ejpam-5403	227	5	not	not	PART
ejpam-5403	227	6	an	an	DET
ejpam-5403	227	7	ideal	ideal	NOUN
ejpam-5403	227	8	of	of	ADP
ejpam-5403	227	9	x.	x.	NOUN
ejpam-5403	227	10	indeed	indeed	ADV
ejpam-5403	227	11	,	,	PUNCT
ejpam-5403	227	12	2(4	2(4	NUM
ejpam-5403	227	13	)	)	PUNCT
ejpam-5403	227	14	=	=	SYM
ejpam-5403	227	15	2	2	NUM
ejpam-5403	227	16	∈	∈	NOUN
ejpam-5403	227	17	g(x	g(x	NOUN
ejpam-5403	227	18	)	)	PUNCT
ejpam-5403	227	19	and	and	CCONJ
ejpam-5403	227	20	4	4	NUM
ejpam-5403	227	21	∈	∈	NOUN
ejpam-5403	227	22	g(x	g(x	NOUN
ejpam-5403	227	23	)	)	PUNCT
ejpam-5403	227	24	but	but	CCONJ
ejpam-5403	227	25	2	2	NUM
ejpam-5403	227	26	/∈	/∈	INTJ
ejpam-5403	227	27	g(x	g(x	NOUN
ejpam-5403	227	28	)	)	PUNCT
ejpam-5403	227	29	.	.	PUNCT
ejpam-5403	228	1	let	let	VERB
ejpam-5403	228	2	a	a	PRON
ejpam-5403	228	3	and	and	CCONJ
ejpam-5403	228	4	b	b	NOUN
ejpam-5403	228	5	be	be	AUX
ejpam-5403	228	6	non	non	ADJ
ejpam-5403	228	7	-	-	ADJ
ejpam-5403	228	8	empty	empty	ADJ
ejpam-5403	228	9	subset	subset	NOUN
ejpam-5403	228	10	of	of	ADP
ejpam-5403	228	11	a	a	DET
ejpam-5403	228	12	q	q	NOUN
ejpam-5403	228	13	-	-	PUNCT
ejpam-5403	228	14	algebra	algebra	NOUN
ejpam-5403	228	15	x.	x.	NOUN
ejpam-5403	228	16	we	we	PRON
ejpam-5403	228	17	define	define	VERB
ejpam-5403	228	18	ab	ab	NOUN
ejpam-5403	228	19	as	as	ADP
ejpam-5403	228	20	following	follow	VERB
ejpam-5403	228	21	:	:	PUNCT
ejpam-5403	228	22	ab	ab	PROPN
ejpam-5403	228	23	=	=	PUNCT
ejpam-5403	228	24	{	{	PUNCT
ejpam-5403	228	25	ab	ab	PROPN
ejpam-5403	228	26	|	|	ADV
ejpam-5403	228	27	a	a	DET
ejpam-5403	228	28	∈	∈	PROPN
ejpam-5403	228	29	a	a	PRON
ejpam-5403	228	30	,	,	PUNCT
ejpam-5403	228	31	b	b	PROPN
ejpam-5403	228	32	∈	∈	PROPN
ejpam-5403	228	33	b	b	NOUN
ejpam-5403	228	34	}	}	PUNCT
ejpam-5403	228	35	.	.	PUNCT
ejpam-5403	229	1	then	then	ADV
ejpam-5403	229	2	we	we	PRON
ejpam-5403	229	3	get	get	VERB
ejpam-5403	229	4	some	some	DET
ejpam-5403	229	5	important	important	ADJ
ejpam-5403	229	6	informations	information	NOUN
ejpam-5403	229	7	of	of	ADP
ejpam-5403	229	8	g(x	g(x	NOUN
ejpam-5403	229	9	)	)	PUNCT
ejpam-5403	229	10	and	and	CCONJ
ejpam-5403	229	11	a(x	a(x	PROPN
ejpam-5403	229	12	):	):	PUNCT
ejpam-5403	229	13	remark	remark	NOUN
ejpam-5403	229	14	1	1	NUM
ejpam-5403	229	15	.	.	PUNCT
ejpam-5403	230	1	let	let	VERB
ejpam-5403	230	2	x	x	PRON
ejpam-5403	230	3	be	be	AUX
ejpam-5403	230	4	a	a	DET
ejpam-5403	230	5	q	q	NOUN
ejpam-5403	230	6	-	-	NOUN
ejpam-5403	230	7	algebra	algebra	NOUN
ejpam-5403	230	8	.	.	PUNCT
ejpam-5403	231	1	then	then	ADV
ejpam-5403	231	2	we	we	PRON
ejpam-5403	231	3	get	get	VERB
ejpam-5403	231	4	:	:	PUNCT
ejpam-5403	231	5	(	(	PUNCT
ejpam-5403	231	6	i	i	NOUN
ejpam-5403	231	7	)	)	PUNCT
ejpam-5403	231	8	g(x	g(x	NOUN
ejpam-5403	231	9	)	)	PUNCT
ejpam-5403	231	10	⊆	⊆	NUM
ejpam-5403	231	11	g(x)g(x	g(x)g(x	NOUN
ejpam-5403	231	12	)	)	PUNCT
ejpam-5403	231	13	,	,	PUNCT
ejpam-5403	231	14	a(x	a(x	PROPN
ejpam-5403	231	15	)	)	PUNCT
ejpam-5403	231	16	⊆	⊆	NUM
ejpam-5403	231	17	a(x)a(x	a(x)a(x	NOUN
ejpam-5403	231	18	)	)	PUNCT
ejpam-5403	231	19	and	and	CCONJ
ejpam-5403	231	20	g(x	g(x	NOUN
ejpam-5403	231	21	)	)	PUNCT
ejpam-5403	231	22	⊆	⊆	NUM
ejpam-5403	231	23	g(x)a(x	g(x)a(x	NOUN
ejpam-5403	231	24	)	)	PUNCT
ejpam-5403	231	25	.	.	PUNCT
ejpam-5403	232	1	a.	a.	NOUN
ejpam-5403	232	2	anantayasethi	anantayasethi	PROPN
ejpam-5403	232	3	,	,	PUNCT
ejpam-5403	232	4	t.	t.	PROPN
ejpam-5403	232	5	kunawat	kunawat	PROPN
ejpam-5403	232	6	,	,	PUNCT
ejpam-5403	232	7	p.	p.	PROPN
ejpam-5403	232	8	moolnipa	moolnipa	PROPN
ejpam-5403	232	9	/	/	SYM
ejpam-5403	232	10	eur	eur	PROPN
ejpam-5403	232	11	.	.	PUNCT
ejpam-5403	233	1	j.	j.	PROPN
ejpam-5403	233	2	pure	pure	PROPN
ejpam-5403	233	3	appl	appl	PROPN
ejpam-5403	233	4	.	.	PROPN
ejpam-5403	233	5	math	math	PROPN
ejpam-5403	233	6	,	,	PUNCT
ejpam-5403	233	7	17	17	NUM
ejpam-5403	233	8	(	(	PUNCT
ejpam-5403	233	9	4	4	NUM
ejpam-5403	233	10	)	)	PUNCT
ejpam-5403	233	11	(	(	PUNCT
ejpam-5403	233	12	2024	2024	NUM
ejpam-5403	233	13	)	)	PUNCT
ejpam-5403	233	14	,	,	PUNCT
ejpam-5403	233	15	3268	3268	NUM
ejpam-5403	233	16	-	-	SYM
ejpam-5403	233	17	3276	3276	NUM
ejpam-5403	233	18	3274	3274	NUM
ejpam-5403	233	19	(	(	PUNCT
ejpam-5403	233	20	ii	ii	NOUN
ejpam-5403	233	21	)	)	PUNCT
ejpam-5403	233	22	g(x	g(x	NOUN
ejpam-5403	233	23	)	)	PUNCT
ejpam-5403	234	1	⊆	⊆	NUM
ejpam-5403	234	2	a(x)g(x	a(x)g(x	NUM
ejpam-5403	234	3	)	)	PUNCT
ejpam-5403	234	4	,	,	PUNCT
ejpam-5403	234	5	a(x	a(x	PROPN
ejpam-5403	234	6	)	)	PUNCT
ejpam-5403	234	7	⊆	⊆	NUM
ejpam-5403	234	8	a(x)g(x	a(x)g(x	NUM
ejpam-5403	234	9	)	)	PUNCT
ejpam-5403	234	10	and	and	CCONJ
ejpam-5403	234	11	g(x	g(x	NOUN
ejpam-5403	234	12	)	)	PUNCT
ejpam-5403	234	13	∪a(x	∪a(x	NOUN
ejpam-5403	234	14	)	)	PUNCT
ejpam-5403	234	15	⊆	⊆	NUM
ejpam-5403	234	16	a(x)g(x	a(x)g(x	NUM
ejpam-5403	234	17	)	)	PUNCT
ejpam-5403	234	18	.	.	PUNCT
ejpam-5403	235	1	(	(	PUNCT
ejpam-5403	235	2	iii	iii	X
ejpam-5403	235	3	)	)	PUNCT
ejpam-5403	235	4	g(x	g(x	NOUN
ejpam-5403	235	5	)	)	PUNCT
ejpam-5403	235	6	⊆	⊆	NUM
ejpam-5403	235	7	g(x)a(x	g(x)a(x	NOUN
ejpam-5403	235	8	)	)	PUNCT
ejpam-5403	235	9	∩a(x)g(x	∩a(x)g(x	NOUN
ejpam-5403	235	10	)	)	PUNCT
ejpam-5403	235	11	.	.	PUNCT
ejpam-5403	236	1	(	(	PUNCT
ejpam-5403	236	2	iv	iv	X
ejpam-5403	236	3	)	)	PUNCT
ejpam-5403	236	4	a(x	a(x	PROPN
ejpam-5403	236	5	)	)	PUNCT
ejpam-5403	236	6	⊆	⊆	NUM
ejpam-5403	236	7	a(x)a(x	a(x)a(x	NOUN
ejpam-5403	236	8	)	)	PUNCT
ejpam-5403	236	9	∩a(x)g(x	∩a(x)g(x	ADJ
ejpam-5403	236	10	)	)	PUNCT
ejpam-5403	237	1	=	=	SYM
ejpam-5403	237	2	a(x	a(x	PROPN
ejpam-5403	237	3	)	)	PUNCT
ejpam-5403	237	4	(	(	PUNCT
ejpam-5403	237	5	a(x	a(x	PROPN
ejpam-5403	237	6	)	)	PUNCT
ejpam-5403	237	7	∩g(x	∩g(x	ADJ
ejpam-5403	237	8	)	)	PUNCT
ejpam-5403	237	9	)	)	PUNCT
ejpam-5403	237	10	.	.	PUNCT
ejpam-5403	238	1	a	a	DET
ejpam-5403	238	2	set	set	NOUN
ejpam-5403	238	3	of	of	ADP
ejpam-5403	238	4	all	all	DET
ejpam-5403	238	5	atoms	atom	NOUN
ejpam-5403	238	6	a(x	a(x	NOUN
ejpam-5403	238	7	)	)	PUNCT
ejpam-5403	238	8	need	need	VERB
ejpam-5403	238	9	not	not	PART
ejpam-5403	238	10	to	to	PART
ejpam-5403	238	11	be	be	AUX
ejpam-5403	238	12	closed	close	VERB
ejpam-5403	238	13	,	,	PUNCT
ejpam-5403	238	14	i.e.	i.e.	X
ejpam-5403	238	15	in	in	ADP
ejpam-5403	238	16	general	general	ADJ
ejpam-5403	238	17	a(x	a(x	PROPN
ejpam-5403	238	18	)	)	PUNCT
ejpam-5403	238	19	̸=	̸=	PROPN
ejpam-5403	238	20	a(x)a(x	a(x)a(x	NOUN
ejpam-5403	238	21	)	)	PUNCT
ejpam-5403	238	22	.	.	PUNCT
ejpam-5403	239	1	from	from	ADP
ejpam-5403	239	2	remark	remark	NOUN
ejpam-5403	239	3	1(iv	1(iv	NUM
ejpam-5403	239	4	)	)	PUNCT
ejpam-5403	239	5	,	,	PUNCT
ejpam-5403	239	6	we	we	PRON
ejpam-5403	239	7	have	have	VERB
ejpam-5403	239	8	that	that	PRON
ejpam-5403	239	9	a(x	a(x	NOUN
ejpam-5403	239	10	)	)	PUNCT
ejpam-5403	239	11	⊆	⊆	NUM
ejpam-5403	239	12	a(x	a(x	NOUN
ejpam-5403	239	13	)	)	PUNCT
ejpam-5403	239	14	(	(	PUNCT
ejpam-5403	239	15	a(x	a(x	PROPN
ejpam-5403	239	16	)	)	PUNCT
ejpam-5403	239	17	∩	∩	NOUN
ejpam-5403	239	18	g(x	g(x	NOUN
ejpam-5403	239	19	)	)	PUNCT
ejpam-5403	239	20	)	)	PUNCT
ejpam-5403	239	21	.	.	PUNCT
ejpam-5403	240	1	it	it	PRON
ejpam-5403	240	2	follows	follow	VERB
ejpam-5403	240	3	that	that	SCONJ
ejpam-5403	240	4	every	every	DET
ejpam-5403	240	5	atom	atom	NOUN
ejpam-5403	240	6	of	of	ADP
ejpam-5403	240	7	x	x	PRON
ejpam-5403	240	8	can	can	AUX
ejpam-5403	240	9	be	be	AUX
ejpam-5403	240	10	written	write	VERB
ejpam-5403	240	11	in	in	ADP
ejpam-5403	240	12	the	the	DET
ejpam-5403	240	13	form	form	NOUN
ejpam-5403	240	14	of	of	ADP
ejpam-5403	240	15	products	product	NOUN
ejpam-5403	240	16	of	of	ADP
ejpam-5403	240	17	atoms	atom	NOUN
ejpam-5403	240	18	.	.	PUNCT
ejpam-5403	241	1	but	but	CCONJ
ejpam-5403	241	2	the	the	DET
ejpam-5403	241	3	product	product	NOUN
ejpam-5403	241	4	za	za	PROPN
ejpam-5403	241	5	of	of	ADP
ejpam-5403	241	6	atom	atom	PROPN
ejpam-5403	241	7	z	z	PROPN
ejpam-5403	241	8	and	and	CCONJ
ejpam-5403	241	9	atom	atom	NOUN
ejpam-5403	241	10	a	a	DET
ejpam-5403	241	11	need	need	NOUN
ejpam-5403	241	12	not	not	PART
ejpam-5403	241	13	to	to	PART
ejpam-5403	241	14	be	be	AUX
ejpam-5403	241	15	atom	atom	NOUN
ejpam-5403	241	16	as	as	SCONJ
ejpam-5403	241	17	seen	see	VERB
ejpam-5403	241	18	from	from	ADP
ejpam-5403	241	19	example	example	NOUN
ejpam-5403	241	20	4	4	NUM
ejpam-5403	241	21	.	.	PUNCT
ejpam-5403	242	1	next	next	PROPN
ejpam-5403	242	2	lemma	lemma	PROPN
ejpam-5403	242	3	shows	show	VERB
ejpam-5403	242	4	the	the	DET
ejpam-5403	242	5	condition	condition	NOUN
ejpam-5403	242	6	that	that	PRON
ejpam-5403	242	7	gives	give	VERB
ejpam-5403	242	8	equality	equality	NOUN
ejpam-5403	242	9	of	of	ADP
ejpam-5403	242	10	remark	remark	NOUN
ejpam-5403	242	11	1(iv	1(iv	NUM
ejpam-5403	242	12	)	)	PUNCT
ejpam-5403	242	13	.	.	PUNCT
ejpam-5403	243	1	proposition	proposition	NOUN
ejpam-5403	243	2	11	11	NUM
ejpam-5403	243	3	.	.	PUNCT
ejpam-5403	244	1	let	let	VERB
ejpam-5403	244	2	x	x	PRON
ejpam-5403	244	3	be	be	AUX
ejpam-5403	244	4	a	a	DET
ejpam-5403	244	5	q	q	NOUN
ejpam-5403	244	6	-	-	NOUN
ejpam-5403	244	7	algebra	algebra	NOUN
ejpam-5403	244	8	.	.	PUNCT
ejpam-5403	245	1	if	if	SCONJ
ejpam-5403	245	2	a(x	a(x	NOUN
ejpam-5403	245	3	)	)	PUNCT
ejpam-5403	245	4	is	be	AUX
ejpam-5403	245	5	an	an	DET
ejpam-5403	245	6	ideal	ideal	NOUN
ejpam-5403	245	7	of	of	ADP
ejpam-5403	245	8	x	x	NOUN
ejpam-5403	245	9	,	,	PUNCT
ejpam-5403	245	10	then	then	ADV
ejpam-5403	245	11	a(x	a(x	PROPN
ejpam-5403	245	12	)	)	PUNCT
ejpam-5403	245	13	=	=	SYM
ejpam-5403	245	14	a(x	a(x	PROPN
ejpam-5403	245	15	)	)	PUNCT
ejpam-5403	245	16	(	(	PUNCT
ejpam-5403	245	17	a(x	a(x	PROPN
ejpam-5403	245	18	)	)	PUNCT
ejpam-5403	245	19	∩g(x	∩g(x	ADJ
ejpam-5403	245	20	)	)	PUNCT
ejpam-5403	245	21	)	)	PUNCT
ejpam-5403	245	22	.	.	PUNCT
ejpam-5403	246	1	proof	proof	NOUN
ejpam-5403	246	2	.	.	PUNCT
ejpam-5403	247	1	assume	assume	VERB
ejpam-5403	247	2	that	that	SCONJ
ejpam-5403	247	3	a(x	a(x	NOUN
ejpam-5403	247	4	)	)	PUNCT
ejpam-5403	247	5	is	be	AUX
ejpam-5403	247	6	an	an	DET
ejpam-5403	247	7	ideal	ideal	NOUN
ejpam-5403	247	8	of	of	ADP
ejpam-5403	247	9	x.	x.	NOUN
ejpam-5403	247	10	let	let	VERB
ejpam-5403	247	11	z	z	PROPN
ejpam-5403	247	12	∈	∈	PROPN
ejpam-5403	247	13	a(x	a(x	PROPN
ejpam-5403	247	14	)	)	PUNCT
ejpam-5403	247	15	and	and	CCONJ
ejpam-5403	247	16	a	a	DET
ejpam-5403	247	17	∈	∈	PROPN
ejpam-5403	247	18	a(x	a(x	PROPN
ejpam-5403	247	19	)	)	PUNCT
ejpam-5403	247	20	∩	∩	NOUN
ejpam-5403	247	21	g(x	g(x	NOUN
ejpam-5403	247	22	)	)	PUNCT
ejpam-5403	247	23	.	.	PUNCT
ejpam-5403	248	1	then	then	ADV
ejpam-5403	248	2	we	we	PRON
ejpam-5403	248	3	get	get	VERB
ejpam-5403	248	4	a	a	DET
ejpam-5403	248	5	=	=	SYM
ejpam-5403	248	6	0a	0a	X
ejpam-5403	248	7	=	=	SYM
ejpam-5403	248	8	(	(	PUNCT
ejpam-5403	248	9	zz)a	zz)a	PROPN
ejpam-5403	248	10	=	=	SYM
ejpam-5403	248	11	(	(	PUNCT
ejpam-5403	248	12	za)z	za)z	PROPN
ejpam-5403	248	13	.	.	PUNCT
ejpam-5403	249	1	it	it	PRON
ejpam-5403	249	2	follows	follow	VERB
ejpam-5403	249	3	that	that	SCONJ
ejpam-5403	249	4	(	(	PUNCT
ejpam-5403	249	5	za)z	za)z	PROPN
ejpam-5403	249	6	=	=	SYM
ejpam-5403	249	7	a	a	DET
ejpam-5403	249	8	∈	∈	PROPN
ejpam-5403	249	9	g(x	g(x	NOUN
ejpam-5403	249	10	)	)	PUNCT
ejpam-5403	249	11	⊆	⊆	NUM
ejpam-5403	249	12	a(x	a(x	NOUN
ejpam-5403	249	13	)	)	PUNCT
ejpam-5403	249	14	.	.	PUNCT
ejpam-5403	250	1	since	since	SCONJ
ejpam-5403	250	2	(	(	PUNCT
ejpam-5403	250	3	za)z	za)z	PROPN
ejpam-5403	250	4	∈	∈	NOUN
ejpam-5403	250	5	a(x	a(x	NOUN
ejpam-5403	250	6	)	)	PUNCT
ejpam-5403	250	7	,	,	PUNCT
ejpam-5403	250	8	a	a	DET
ejpam-5403	250	9	∈	∈	PROPN
ejpam-5403	250	10	a(x	a(x	NOUN
ejpam-5403	250	11	)	)	PUNCT
ejpam-5403	250	12	and	and	CCONJ
ejpam-5403	250	13	a(x	a(x	PROPN
ejpam-5403	250	14	)	)	PUNCT
ejpam-5403	250	15	is	be	AUX
ejpam-5403	250	16	an	an	DET
ejpam-5403	250	17	ideal	ideal	NOUN
ejpam-5403	250	18	,	,	PUNCT
ejpam-5403	250	19	then	then	ADV
ejpam-5403	250	20	za	za	PROPN
ejpam-5403	250	21	∈	∈	PROPN
ejpam-5403	250	22	a(x	a(x	PROPN
ejpam-5403	250	23	)	)	PUNCT
ejpam-5403	250	24	.	.	PUNCT
ejpam-5403	251	1	therefore	therefore	ADV
ejpam-5403	251	2	,	,	PUNCT
ejpam-5403	251	3	a(x	a(x	PROPN
ejpam-5403	251	4	)	)	PUNCT
ejpam-5403	251	5	(	(	PUNCT
ejpam-5403	251	6	a(x	a(x	PROPN
ejpam-5403	251	7	)	)	PUNCT
ejpam-5403	251	8	∩g(x	∩g(x	ADJ
ejpam-5403	251	9	)	)	PUNCT
ejpam-5403	251	10	)	)	PUNCT
ejpam-5403	252	1	⊆	⊆	NUM
ejpam-5403	252	2	a(x	a(x	NOUN
ejpam-5403	252	3	)	)	PUNCT
ejpam-5403	252	4	.	.	PUNCT
ejpam-5403	253	1	the	the	DET
ejpam-5403	253	2	inclusion	inclusion	NOUN
ejpam-5403	253	3	a(x	a(x	NOUN
ejpam-5403	253	4	)	)	PUNCT
ejpam-5403	253	5	⊆	⊆	NUM
ejpam-5403	253	6	a(x	a(x	NOUN
ejpam-5403	253	7	)	)	PUNCT
ejpam-5403	253	8	(	(	PUNCT
ejpam-5403	253	9	a(x	a(x	PROPN
ejpam-5403	253	10	)	)	PUNCT
ejpam-5403	253	11	∩g(x	∩g(x	ADJ
ejpam-5403	253	12	)	)	PUNCT
ejpam-5403	253	13	)	)	PUNCT
ejpam-5403	253	14	follows	follow	VERB
ejpam-5403	253	15	from	from	ADP
ejpam-5403	253	16	remark	remark	NOUN
ejpam-5403	253	17	1(iv	1(iv	NUM
ejpam-5403	253	18	)	)	PUNCT
ejpam-5403	253	19	.	.	PUNCT
ejpam-5403	254	1	hence	hence	ADV
ejpam-5403	254	2	,	,	PUNCT
ejpam-5403	254	3	a(x	a(x	PROPN
ejpam-5403	254	4	)	)	PUNCT
ejpam-5403	254	5	=	=	SYM
ejpam-5403	254	6	a(x	a(x	PROPN
ejpam-5403	254	7	)	)	PUNCT
ejpam-5403	254	8	(	(	PUNCT
ejpam-5403	254	9	a(x	a(x	PROPN
ejpam-5403	254	10	)	)	PUNCT
ejpam-5403	254	11	∩g(x	∩g(x	ADJ
ejpam-5403	254	12	)	)	PUNCT
ejpam-5403	254	13	)	)	PUNCT
ejpam-5403	254	14	.	.	PUNCT
ejpam-5403	255	1	as	as	ADP
ejpam-5403	255	2	a	a	DET
ejpam-5403	255	3	consequence	consequence	NOUN
ejpam-5403	255	4	of	of	ADP
ejpam-5403	255	5	proposition	proposition	NOUN
ejpam-5403	255	6	11	11	NUM
ejpam-5403	255	7	,	,	PUNCT
ejpam-5403	255	8	the	the	DET
ejpam-5403	255	9	product	product	NOUN
ejpam-5403	255	10	ab	ab	PROPN
ejpam-5403	255	11	of	of	ADP
ejpam-5403	255	12	atom	atom	PROPN
ejpam-5403	255	13	a	a	PRON
ejpam-5403	255	14	and	and	CCONJ
ejpam-5403	255	15	atom	atom	PROPN
ejpam-5403	255	16	b	b	PROPN
ejpam-5403	255	17	with	with	ADP
ejpam-5403	255	18	0b	0b	NOUN
ejpam-5403	255	19	=	=	SYM
ejpam-5403	255	20	b	b	PROPN
ejpam-5403	255	21	is	be	AUX
ejpam-5403	255	22	again	again	ADV
ejpam-5403	255	23	an	an	DET
ejpam-5403	255	24	atom	atom	NOUN
ejpam-5403	255	25	of	of	ADP
ejpam-5403	255	26	x.	x.	NOUN
ejpam-5403	255	27	proposition	proposition	PROPN
ejpam-5403	255	28	12	12	NUM
ejpam-5403	255	29	.	.	PUNCT
ejpam-5403	256	1	let	let	VERB
ejpam-5403	256	2	x	x	PRON
ejpam-5403	256	3	be	be	AUX
ejpam-5403	256	4	a	a	DET
ejpam-5403	256	5	q	q	NOUN
ejpam-5403	256	6	-	-	NOUN
ejpam-5403	256	7	algebra	algebra	NOUN
ejpam-5403	256	8	.	.	PUNCT
ejpam-5403	257	1	if	if	SCONJ
ejpam-5403	257	2	a(x	a(x	NOUN
ejpam-5403	257	3	)	)	PUNCT
ejpam-5403	257	4	is	be	AUX
ejpam-5403	257	5	an	an	DET
ejpam-5403	257	6	ideal	ideal	NOUN
ejpam-5403	257	7	of	of	ADP
ejpam-5403	257	8	x	x	NOUN
ejpam-5403	257	9	,	,	PUNCT
ejpam-5403	257	10	then	then	ADV
ejpam-5403	257	11	a(x	a(x	PROPN
ejpam-5403	257	12	)	)	PUNCT
ejpam-5403	257	13	∩g(x	∩g(x	ADV
ejpam-5403	257	14	)	)	PUNCT
ejpam-5403	257	15	is	be	AUX
ejpam-5403	257	16	an	an	DET
ejpam-5403	257	17	abelian	abelian	ADJ
ejpam-5403	257	18	group	group	NOUN
ejpam-5403	257	19	.	.	PUNCT
ejpam-5403	258	1	proof	proof	NOUN
ejpam-5403	258	2	.	.	PUNCT
ejpam-5403	259	1	let	let	VERB
ejpam-5403	259	2	x	x	PRON
ejpam-5403	259	3	,	,	PUNCT
ejpam-5403	259	4	y	y	PROPN
ejpam-5403	259	5	,	,	PUNCT
ejpam-5403	259	6	z	z	PROPN
ejpam-5403	259	7	∈	∈	PROPN
ejpam-5403	259	8	a(x	a(x	PROPN
ejpam-5403	259	9	)	)	PUNCT
ejpam-5403	259	10	∩g(x	∩g(x	ADJ
ejpam-5403	259	11	)	)	PUNCT
ejpam-5403	259	12	.	.	PUNCT
ejpam-5403	260	1	then	then	ADV
ejpam-5403	260	2	by	by	ADP
ejpam-5403	260	3	lemma	lemma	PROPN
ejpam-5403	260	4	2	2	NUM
ejpam-5403	260	5	we	we	PRON
ejpam-5403	260	6	get	get	VERB
ejpam-5403	260	7	that	that	DET
ejpam-5403	260	8	0(xy	0(xy	NOUN
ejpam-5403	260	9	)	)	PUNCT
ejpam-5403	261	1	=	=	PRON
ejpam-5403	261	2	(	(	PUNCT
ejpam-5403	261	3	0x)(0y	0x)(0y	NOUN
ejpam-5403	261	4	)	)	PUNCT
ejpam-5403	261	5	=	=	VERB
ejpam-5403	262	1	xy	xy	PROPN
ejpam-5403	262	2	.	.	PUNCT
ejpam-5403	263	1	therefore	therefore	ADV
ejpam-5403	263	2	,	,	PUNCT
ejpam-5403	263	3	xy	xy	PROPN
ejpam-5403	263	4	∈	∈	PROPN
ejpam-5403	263	5	g(x	g(x	PROPN
ejpam-5403	263	6	)	)	PUNCT
ejpam-5403	263	7	.	.	PUNCT
ejpam-5403	264	1	since	since	SCONJ
ejpam-5403	264	2	x	x	PROPN
ejpam-5403	264	3	∈	∈	PROPN
ejpam-5403	264	4	a(x	a(x	PROPN
ejpam-5403	264	5	)	)	PUNCT
ejpam-5403	264	6	and	and	CCONJ
ejpam-5403	264	7	y	y	PROPN
ejpam-5403	264	8	∈	∈	PROPN
ejpam-5403	264	9	a(x)∩g(x	a(x)∩g(x	NUM
ejpam-5403	264	10	)	)	PUNCT
ejpam-5403	264	11	,	,	PUNCT
ejpam-5403	264	12	then	then	ADV
ejpam-5403	264	13	by	by	ADP
ejpam-5403	264	14	proposition	proposition	NOUN
ejpam-5403	264	15	11	11	NUM
ejpam-5403	264	16	we	we	PRON
ejpam-5403	264	17	get	get	VERB
ejpam-5403	264	18	xy	xy	PROPN
ejpam-5403	264	19	∈	∈	PROPN
ejpam-5403	264	20	a(x	a(x	NOUN
ejpam-5403	264	21	)	)	PUNCT
ejpam-5403	264	22	.	.	PUNCT
ejpam-5403	265	1	thus	thus	ADV
ejpam-5403	265	2	,	,	PUNCT
ejpam-5403	265	3	xy	xy	PROPN
ejpam-5403	265	4	∈	∈	PROPN
ejpam-5403	265	5	a(x	a(x	PROPN
ejpam-5403	265	6	)	)	PUNCT
ejpam-5403	265	7	∩	∩	NOUN
ejpam-5403	265	8	g(x	g(x	NOUN
ejpam-5403	265	9	)	)	PUNCT
ejpam-5403	265	10	.	.	PUNCT
ejpam-5403	266	1	the	the	DET
ejpam-5403	266	2	commutative	commutative	ADJ
ejpam-5403	266	3	property	property	NOUN
ejpam-5403	266	4	follows	follow	VERB
ejpam-5403	266	5	from	from	ADP
ejpam-5403	266	6	proposition	proposition	NOUN
ejpam-5403	266	7	2	2	NUM
ejpam-5403	266	8	.	.	PUNCT
ejpam-5403	267	1	since	since	SCONJ
ejpam-5403	267	2	the	the	DET
ejpam-5403	267	3	commutative	commutative	ADJ
ejpam-5403	267	4	property	property	NOUN
ejpam-5403	267	5	is	be	AUX
ejpam-5403	267	6	hold	hold	NOUN
ejpam-5403	267	7	,	,	PUNCT
ejpam-5403	267	8	then	then	ADV
ejpam-5403	267	9	we	we	PRON
ejpam-5403	267	10	get	get	VERB
ejpam-5403	267	11	(	(	PUNCT
ejpam-5403	267	12	xy)z	xy)z	PUNCT
ejpam-5403	267	13	=	=	SYM
ejpam-5403	267	14	(	(	PUNCT
ejpam-5403	267	15	yx)z	yx)z	PROPN
ejpam-5403	267	16	=	=	SYM
ejpam-5403	267	17	(	(	PUNCT
ejpam-5403	267	18	yz)x	yz)x	PROPN
ejpam-5403	267	19	=	=	SYM
ejpam-5403	267	20	x(yz	x(yz	PROPN
ejpam-5403	267	21	)	)	PUNCT
ejpam-5403	267	22	.	.	PUNCT
ejpam-5403	268	1	hence	hence	ADV
ejpam-5403	268	2	,	,	PUNCT
ejpam-5403	268	3	an	an	DET
ejpam-5403	268	4	associative	associative	ADJ
ejpam-5403	268	5	law	law	NOUN
ejpam-5403	268	6	is	be	AUX
ejpam-5403	268	7	hold	hold	NOUN
ejpam-5403	268	8	.	.	PUNCT
ejpam-5403	269	1	moreover	moreover	ADV
ejpam-5403	269	2	,	,	PUNCT
ejpam-5403	269	3	0	0	NUM
ejpam-5403	269	4	∈	∈	PROPN
ejpam-5403	269	5	a(x	a(x	PROPN
ejpam-5403	269	6	)	)	PUNCT
ejpam-5403	269	7	∩	∩	NOUN
ejpam-5403	269	8	g(x	g(x	NOUN
ejpam-5403	269	9	)	)	PUNCT
ejpam-5403	269	10	,	,	PUNCT
ejpam-5403	269	11	by	by	ADP
ejpam-5403	269	12	(	(	PUNCT
ejpam-5403	269	13	q1	q1	PROPN
ejpam-5403	269	14	)	)	PUNCT
ejpam-5403	269	15	and	and	CCONJ
ejpam-5403	269	16	x	x	PUNCT
ejpam-5403	269	17	∈	∈	PROPN
ejpam-5403	269	18	g(x	g(x	NOUN
ejpam-5403	269	19	)	)	PUNCT
ejpam-5403	269	20	we	we	PRON
ejpam-5403	269	21	get	get	VERB
ejpam-5403	269	22	x0	x0	PROPN
ejpam-5403	270	1	=	=	PUNCT
ejpam-5403	271	1	x	x	PUNCT
ejpam-5403	272	1	=	=	SYM
ejpam-5403	273	1	0x	0x	NOUN
ejpam-5403	273	2	.	.	PUNCT
ejpam-5403	274	1	therefore	therefore	ADV
ejpam-5403	274	2	,	,	PUNCT
ejpam-5403	274	3	0	0	NUM
ejpam-5403	274	4	is	be	AUX
ejpam-5403	274	5	an	an	DET
ejpam-5403	274	6	identity	identity	NOUN
ejpam-5403	274	7	of	of	ADP
ejpam-5403	274	8	a(x	a(x	NOUN
ejpam-5403	274	9	)	)	PUNCT
ejpam-5403	274	10	∩	∩	NOUN
ejpam-5403	274	11	g(x	g(x	NOUN
ejpam-5403	274	12	)	)	PUNCT
ejpam-5403	274	13	.	.	PUNCT
ejpam-5403	275	1	an	an	DET
ejpam-5403	275	2	inverse	inverse	NOUN
ejpam-5403	275	3	property	property	NOUN
ejpam-5403	275	4	follows	follow	VERB
ejpam-5403	275	5	from	from	ADP
ejpam-5403	275	6	(	(	PUNCT
ejpam-5403	275	7	q2	q2	NOUN
ejpam-5403	275	8	)	)	PUNCT
ejpam-5403	275	9	.	.	PUNCT
ejpam-5403	276	1	altogether	altogether	ADV
ejpam-5403	276	2	,	,	PUNCT
ejpam-5403	276	3	we	we	PRON
ejpam-5403	276	4	get	get	VERB
ejpam-5403	276	5	that	that	DET
ejpam-5403	276	6	a(x	a(x	NOUN
ejpam-5403	276	7	)	)	PUNCT
ejpam-5403	276	8	∩	∩	NOUN
ejpam-5403	276	9	g(x	g(x	NOUN
ejpam-5403	276	10	)	)	PUNCT
ejpam-5403	276	11	is	be	AUX
ejpam-5403	276	12	an	an	DET
ejpam-5403	276	13	abelian	abelian	ADJ
ejpam-5403	276	14	group	group	NOUN
ejpam-5403	276	15	.	.	PUNCT
ejpam-5403	277	1	proposition	proposition	NOUN
ejpam-5403	277	2	13	13	NUM
ejpam-5403	277	3	.	.	PUNCT
ejpam-5403	278	1	let	let	VERB
ejpam-5403	278	2	x	x	PRON
ejpam-5403	278	3	be	be	AUX
ejpam-5403	278	4	a	a	DET
ejpam-5403	278	5	q	q	NOUN
ejpam-5403	278	6	-	-	NOUN
ejpam-5403	278	7	algebra	algebra	NOUN
ejpam-5403	278	8	.	.	PUNCT
ejpam-5403	279	1	if	if	SCONJ
ejpam-5403	279	2	a(x	a(x	NOUN
ejpam-5403	279	3	)	)	PUNCT
ejpam-5403	279	4	⊆	⊆	NUM
ejpam-5403	279	5	g(x	g(x	NOUN
ejpam-5403	279	6	)	)	PUNCT
ejpam-5403	279	7	and	and	CCONJ
ejpam-5403	279	8	a(x	a(x	NOUN
ejpam-5403	279	9	)	)	PUNCT
ejpam-5403	279	10	is	be	AUX
ejpam-5403	279	11	an	an	DET
ejpam-5403	279	12	ideal	ideal	NOUN
ejpam-5403	279	13	of	of	ADP
ejpam-5403	279	14	x	x	PRON
ejpam-5403	279	15	,	,	PUNCT
ejpam-5403	279	16	then	then	ADV
ejpam-5403	279	17	(	(	PUNCT
ejpam-5403	279	18	i	i	NOUN
ejpam-5403	279	19	)	)	PUNCT
ejpam-5403	279	20	a(x	a(x	PROPN
ejpam-5403	279	21	)	)	PUNCT
ejpam-5403	279	22	is	be	AUX
ejpam-5403	279	23	a	a	DET
ejpam-5403	279	24	subalgebra	subalgebra	NOUN
ejpam-5403	279	25	of	of	ADP
ejpam-5403	279	26	x	x	SYM
ejpam-5403	279	27	(	(	PUNCT
ejpam-5403	279	28	ii	ii	NOUN
ejpam-5403	279	29	)	)	PUNCT
ejpam-5403	279	30	a(x	a(x	PROPN
ejpam-5403	279	31	)	)	PUNCT
ejpam-5403	279	32	is	be	AUX
ejpam-5403	279	33	an	an	DET
ejpam-5403	279	34	abelian	abelian	ADJ
ejpam-5403	279	35	group	group	NOUN
ejpam-5403	279	36	proof	proof	NOUN
ejpam-5403	279	37	.	.	PUNCT
ejpam-5403	280	1	(	(	PUNCT
ejpam-5403	280	2	i	i	NOUN
ejpam-5403	280	3	)	)	PUNCT
ejpam-5403	280	4	since	since	SCONJ
ejpam-5403	280	5	a(x	a(x	NOUN
ejpam-5403	280	6	)	)	PUNCT
ejpam-5403	280	7	⊆	⊆	NUM
ejpam-5403	280	8	g(x	g(x	NOUN
ejpam-5403	280	9	)	)	PUNCT
ejpam-5403	280	10	,	,	PUNCT
ejpam-5403	280	11	then	then	ADV
ejpam-5403	280	12	a(x	a(x	PROPN
ejpam-5403	280	13	)	)	PUNCT
ejpam-5403	280	14	∩g(x	∩g(x	ADV
ejpam-5403	280	15	)	)	PUNCT
ejpam-5403	280	16	=	=	SYM
ejpam-5403	280	17	a(x	a(x	NOUN
ejpam-5403	280	18	)	)	PUNCT
ejpam-5403	280	19	.	.	PUNCT
ejpam-5403	281	1	then	then	ADV
ejpam-5403	281	2	by	by	ADP
ejpam-5403	281	3	proposition	proposition	NOUN
ejpam-5403	281	4	11	11	NUM
ejpam-5403	281	5	we	we	PRON
ejpam-5403	281	6	get	get	VERB
ejpam-5403	281	7	that	that	PRON
ejpam-5403	281	8	a(x	a(x	NOUN
ejpam-5403	281	9	)	)	PUNCT
ejpam-5403	281	10	=	=	SYM
ejpam-5403	281	11	a(x	a(x	PROPN
ejpam-5403	281	12	)	)	PUNCT
ejpam-5403	281	13	(	(	PUNCT
ejpam-5403	281	14	a(x	a(x	PROPN
ejpam-5403	281	15	)	)	PUNCT
ejpam-5403	281	16	∩g(x	∩g(x	ADJ
ejpam-5403	281	17	)	)	PUNCT
ejpam-5403	281	18	)	)	PUNCT
ejpam-5403	282	1	=	=	SYM
ejpam-5403	282	2	a(x)a(x	a(x)a(x	NOUN
ejpam-5403	282	3	)	)	PUNCT
ejpam-5403	282	4	.	.	PUNCT
ejpam-5403	283	1	hence	hence	ADV
ejpam-5403	283	2	,	,	PUNCT
ejpam-5403	283	3	a(x	a(x	PROPN
ejpam-5403	283	4	)	)	PUNCT
ejpam-5403	283	5	is	be	AUX
ejpam-5403	283	6	a	a	DET
ejpam-5403	283	7	subalgebra	subalgebra	NOUN
ejpam-5403	283	8	of	of	ADP
ejpam-5403	283	9	x.	x.	PROPN
ejpam-5403	283	10	(	(	PUNCT
ejpam-5403	283	11	ii	ii	PROPN
ejpam-5403	283	12	)	)	PUNCT
ejpam-5403	283	13	since	since	SCONJ
ejpam-5403	283	14	a(x	a(x	NOUN
ejpam-5403	283	15	)	)	PUNCT
ejpam-5403	283	16	∩	∩	NOUN
ejpam-5403	283	17	g(x	g(x	NOUN
ejpam-5403	283	18	)	)	PUNCT
ejpam-5403	283	19	=	=	SYM
ejpam-5403	283	20	a(x	a(x	NOUN
ejpam-5403	283	21	)	)	PUNCT
ejpam-5403	283	22	,	,	PUNCT
ejpam-5403	283	23	then	then	ADV
ejpam-5403	283	24	by	by	ADP
ejpam-5403	283	25	proposition	proposition	NOUN
ejpam-5403	283	26	12	12	NUM
ejpam-5403	283	27	we	we	PRON
ejpam-5403	283	28	get	get	VERB
ejpam-5403	283	29	that	that	DET
ejpam-5403	283	30	a(x	a(x	NOUN
ejpam-5403	283	31	)	)	PUNCT
ejpam-5403	283	32	is	be	AUX
ejpam-5403	283	33	an	an	DET
ejpam-5403	283	34	abelian	abelian	ADJ
ejpam-5403	283	35	group	group	NOUN
ejpam-5403	283	36	.	.	PUNCT
ejpam-5403	284	1	references	reference	NOUN
ejpam-5403	284	2	3275	3275	NUM
ejpam-5403	284	3	proposition	proposition	NOUN
ejpam-5403	284	4	14	14	NUM
ejpam-5403	284	5	.	.	PUNCT
ejpam-5403	285	1	let	let	VERB
ejpam-5403	285	2	x	x	PRON
ejpam-5403	285	3	be	be	AUX
ejpam-5403	285	4	a	a	DET
ejpam-5403	285	5	q	q	NOUN
ejpam-5403	285	6	-	-	NOUN
ejpam-5403	285	7	algebra	algebra	NOUN
ejpam-5403	285	8	.	.	PUNCT
ejpam-5403	286	1	if	if	SCONJ
ejpam-5403	286	2	g(x	g(x	NOUN
ejpam-5403	286	3	)	)	PUNCT
ejpam-5403	286	4	and	and	CCONJ
ejpam-5403	286	5	a(x	a(x	NOUN
ejpam-5403	286	6	)	)	PUNCT
ejpam-5403	286	7	are	be	AUX
ejpam-5403	286	8	ideals	ideal	NOUN
ejpam-5403	286	9	of	of	ADP
ejpam-5403	286	10	x	x	NOUN
ejpam-5403	286	11	,	,	PUNCT
ejpam-5403	286	12	then	then	ADV
ejpam-5403	286	13	a(x)g(x	a(x)g(x	NUM
ejpam-5403	286	14	)	)	PUNCT
ejpam-5403	286	15	is	be	AUX
ejpam-5403	286	16	an	an	DET
ejpam-5403	286	17	ideal	ideal	NOUN
ejpam-5403	286	18	of	of	ADP
ejpam-5403	286	19	x.	x.	NOUN
ejpam-5403	286	20	proof	proof	PROPN
ejpam-5403	286	21	.	.	PUNCT
ejpam-5403	287	1	assume	assume	VERB
ejpam-5403	287	2	that	that	SCONJ
ejpam-5403	287	3	g(x	g(x	NOUN
ejpam-5403	287	4	)	)	PUNCT
ejpam-5403	287	5	and	and	CCONJ
ejpam-5403	287	6	a(x	a(x	NOUN
ejpam-5403	287	7	)	)	PUNCT
ejpam-5403	287	8	are	be	AUX
ejpam-5403	287	9	ideals	ideal	NOUN
ejpam-5403	287	10	of	of	ADP
ejpam-5403	287	11	x.	x.	NOUN
ejpam-5403	287	12	then	then	ADV
ejpam-5403	287	13	by	by	ADP
ejpam-5403	287	14	proposition	proposition	NOUN
ejpam-5403	287	15	10	10	NUM
ejpam-5403	287	16	we	we	PRON
ejpam-5403	287	17	get	get	VERB
ejpam-5403	287	18	that	that	DET
ejpam-5403	287	19	g(x	g(x	NOUN
ejpam-5403	287	20	)	)	PUNCT
ejpam-5403	287	21	⊆	⊆	NUM
ejpam-5403	287	22	a(x	a(x	NOUN
ejpam-5403	287	23	)	)	PUNCT
ejpam-5403	287	24	.	.	PUNCT
ejpam-5403	288	1	since	since	SCONJ
ejpam-5403	288	2	a(x	a(x	NOUN
ejpam-5403	288	3	)	)	PUNCT
ejpam-5403	288	4	is	be	AUX
ejpam-5403	288	5	an	an	DET
ejpam-5403	288	6	ideal	ideal	NOUN
ejpam-5403	288	7	,	,	PUNCT
ejpam-5403	288	8	then	then	ADV
ejpam-5403	288	9	a(x	a(x	PROPN
ejpam-5403	288	10	)	)	PUNCT
ejpam-5403	288	11	=	=	SYM
ejpam-5403	288	12	a(x	a(x	PROPN
ejpam-5403	288	13	)	)	PUNCT
ejpam-5403	288	14	(	(	PUNCT
ejpam-5403	288	15	a(x	a(x	PROPN
ejpam-5403	288	16	)	)	PUNCT
ejpam-5403	288	17	∩	∩	NOUN
ejpam-5403	288	18	g(x	g(x	NOUN
ejpam-5403	288	19	)	)	PUNCT
ejpam-5403	288	20	)	)	PUNCT
ejpam-5403	288	21	.	.	PUNCT
ejpam-5403	289	1	there	there	PRON
ejpam-5403	289	2	follows	follow	VERB
ejpam-5403	289	3	that	that	SCONJ
ejpam-5403	289	4	a(x	a(x	NOUN
ejpam-5403	289	5	)	)	PUNCT
ejpam-5403	289	6	=	=	PUNCT
ejpam-5403	289	7	a(x)g(x	a(x)g(x	NOUN
ejpam-5403	289	8	)	)	PUNCT
ejpam-5403	289	9	.	.	PUNCT
ejpam-5403	290	1	hence	hence	ADV
ejpam-5403	290	2	,	,	PUNCT
ejpam-5403	290	3	a(x)g(x	a(x)g(x	NUM
ejpam-5403	290	4	)	)	PUNCT
ejpam-5403	290	5	is	be	AUX
ejpam-5403	290	6	an	an	DET
ejpam-5403	290	7	ideal	ideal	NOUN
ejpam-5403	290	8	of	of	ADP
ejpam-5403	290	9	x.	x.	NOUN
ejpam-5403	290	10	3	3	NUM
ejpam-5403	290	11	.	.	X
ejpam-5403	290	12	conclusion	conclusion	NOUN
ejpam-5403	290	13	the	the	DET
ejpam-5403	290	14	concept	concept	NOUN
ejpam-5403	290	15	of	of	ADP
ejpam-5403	290	16	ideal	ideal	NOUN
ejpam-5403	290	17	plays	play	VERB
ejpam-5403	290	18	an	an	DET
ejpam-5403	290	19	important	important	ADJ
ejpam-5403	290	20	role	role	NOUN
ejpam-5403	290	21	in	in	ADP
ejpam-5403	290	22	studying	study	VERB
ejpam-5403	290	23	q	q	ADJ
ejpam-5403	290	24	-	-	PUNCT
ejpam-5403	290	25	algebra	algebra	NOUN
ejpam-5403	290	26	structures	structure	NOUN
ejpam-5403	290	27	.	.	PUNCT
ejpam-5403	291	1	many	many	ADJ
ejpam-5403	291	2	mathematicians	mathematician	NOUN
ejpam-5403	291	3	examine	examine	VERB
ejpam-5403	291	4	various	various	ADJ
ejpam-5403	291	5	subsets	subset	NOUN
ejpam-5403	291	6	of	of	ADP
ejpam-5403	291	7	a	a	DET
ejpam-5403	291	8	q	q	NOUN
ejpam-5403	291	9	-	-	PUNCT
ejpam-5403	291	10	algebra	algebra	NOUN
ejpam-5403	291	11	which	which	PRON
ejpam-5403	291	12	are	be	AUX
ejpam-5403	291	13	ideals	ideal	NOUN
ejpam-5403	291	14	.	.	PUNCT
ejpam-5403	292	1	in	in	ADP
ejpam-5403	292	2	this	this	DET
ejpam-5403	292	3	work	work	NOUN
ejpam-5403	292	4	,	,	PUNCT
ejpam-5403	292	5	we	we	PRON
ejpam-5403	292	6	obtain	obtain	VERB
ejpam-5403	292	7	information	information	NOUN
ejpam-5403	292	8	that	that	SCONJ
ejpam-5403	292	9	all	all	DET
ejpam-5403	292	10	elements	element	NOUN
ejpam-5403	292	11	of	of	ADP
ejpam-5403	292	12	g(x	g(x	NOUN
ejpam-5403	292	13	)	)	PUNCT
ejpam-5403	292	14	are	be	AUX
ejpam-5403	292	15	atoms	atom	NOUN
ejpam-5403	292	16	whenever	whenever	SCONJ
ejpam-5403	292	17	g(x	g(x	NOUN
ejpam-5403	292	18	)	)	PUNCT
ejpam-5403	292	19	is	be	AUX
ejpam-5403	292	20	an	an	DET
ejpam-5403	292	21	ideal	ideal	NOUN
ejpam-5403	292	22	.	.	PUNCT
ejpam-5403	293	1	moreover	moreover	ADV
ejpam-5403	293	2	,	,	PUNCT
ejpam-5403	293	3	we	we	PRON
ejpam-5403	293	4	get	get	VERB
ejpam-5403	293	5	that	that	SCONJ
ejpam-5403	293	6	a	a	DET
ejpam-5403	293	7	q	q	NOUN
ejpam-5403	293	8	-	-	NOUN
ejpam-5403	293	9	algebra	algebra	NOUN
ejpam-5403	293	10	x	x	PUNCT
ejpam-5403	293	11	such	such	ADJ
ejpam-5403	293	12	that	that	SCONJ
ejpam-5403	293	13	g(x	g(x	NOUN
ejpam-5403	293	14	)	)	PUNCT
ejpam-5403	293	15	is	be	AUX
ejpam-5403	293	16	an	an	DET
ejpam-5403	293	17	ideal	ideal	ADJ
ejpam-5403	293	18	and	and	CCONJ
ejpam-5403	293	19	g(x	g(x	NOUN
ejpam-5403	293	20	)	)	PUNCT
ejpam-5403	293	21	̸=	̸=	PROPN
ejpam-5403	293	22	{	{	PUNCT
ejpam-5403	293	23	0	0	NUM
ejpam-5403	293	24	}	}	PUNCT
ejpam-5403	293	25	,	,	PUNCT
ejpam-5403	293	26	does	do	AUX
ejpam-5403	293	27	not	not	PART
ejpam-5403	293	28	contain	contain	VERB
ejpam-5403	293	29	a	a	DET
ejpam-5403	293	30	strong	strong	ADJ
ejpam-5403	293	31	atom	atom	NOUN
ejpam-5403	293	32	.	.	PUNCT
ejpam-5403	294	1	for	for	ADP
ejpam-5403	294	2	future	future	ADJ
ejpam-5403	294	3	study	study	NOUN
ejpam-5403	294	4	one	one	PRON
ejpam-5403	294	5	can	can	AUX
ejpam-5403	294	6	investigate	investigate	VERB
ejpam-5403	294	7	when	when	SCONJ
ejpam-5403	294	8	a	a	DET
ejpam-5403	294	9	set	set	NOUN
ejpam-5403	294	10	of	of	ADP
ejpam-5403	294	11	all	all	DET
ejpam-5403	294	12	atoms	atom	NOUN
ejpam-5403	294	13	a(x	a(x	NOUN
ejpam-5403	294	14	)	)	PUNCT
ejpam-5403	294	15	is	be	AUX
ejpam-5403	294	16	an	an	DET
ejpam-5403	294	17	ideal	ideal	NOUN
ejpam-5403	294	18	of	of	ADP
ejpam-5403	294	19	x	x	PUNCT
ejpam-5403	294	20	and	and	CCONJ
ejpam-5403	294	21	which	which	DET
ejpam-5403	294	22	conditions	condition	NOUN
ejpam-5403	294	23	that	that	PRON
ejpam-5403	294	24	make	make	VERB
ejpam-5403	294	25	x	x	PUNCT
ejpam-5403	294	26	contains	contain	VERB
ejpam-5403	294	27	both	both	DET
ejpam-5403	294	28	non	non	ADJ
ejpam-5403	294	29	-	-	ADJ
ejpam-5403	294	30	zero	zero	NUM
ejpam-5403	294	31	atoms	atom	NOUN
ejpam-5403	294	32	and	and	CCONJ
ejpam-5403	294	33	strong	strong	ADJ
ejpam-5403	294	34	atoms	atom	NOUN
ejpam-5403	294	35	.	.	PUNCT
ejpam-5403	295	1	also	also	ADV
ejpam-5403	295	2	,	,	PUNCT
ejpam-5403	295	3	for	for	ADP
ejpam-5403	295	4	any	any	DET
ejpam-5403	295	5	q	q	NOUN
ejpam-5403	295	6	-	-	NOUN
ejpam-5403	295	7	algebra	algebra	NOUN
ejpam-5403	295	8	x	x	PUNCT
ejpam-5403	295	9	one	one	PRON
ejpam-5403	295	10	can	can	AUX
ejpam-5403	295	11	find	find	VERB
ejpam-5403	295	12	the	the	DET
ejpam-5403	295	13	sufficient	sufficient	ADJ
ejpam-5403	295	14	condition	condition	NOUN
ejpam-5403	295	15	of	of	ADP
ejpam-5403	295	16	a(x	a(x	NOUN
ejpam-5403	295	17	)	)	PUNCT
ejpam-5403	295	18	to	to	PART
ejpam-5403	295	19	be	be	AUX
ejpam-5403	295	20	an	an	DET
ejpam-5403	295	21	ideal	ideal	NOUN
ejpam-5403	295	22	of	of	ADP
ejpam-5403	295	23	x.	x.	NOUN
ejpam-5403	295	24	acknowledgements	acknowledgement	NOUN
ejpam-5403	295	25	this	this	DET
ejpam-5403	295	26	research	research	NOUN
ejpam-5403	295	27	project	project	NOUN
ejpam-5403	295	28	was	be	AUX
ejpam-5403	295	29	financially	financially	ADV
ejpam-5403	295	30	supported	support	VERB
ejpam-5403	295	31	by	by	ADP
ejpam-5403	295	32	mahasarakham	mahasarakham	PROPN
ejpam-5403	295	33	university	university	PROPN
ejpam-5403	295	34	,	,	PUNCT
ejpam-5403	295	35	thailand	thailand	PROPN
ejpam-5403	295	36	.	.	PUNCT
ejpam-5403	296	1	references	reference	NOUN
ejpam-5403	296	2	[	[	X
ejpam-5403	296	3	1	1	NUM
ejpam-5403	296	4	]	]	PUNCT
ejpam-5403	296	5	h.	h.	PROPN
ejpam-5403	296	6	k.	k.	PROPN
ejpam-5403	296	7	abdullah	abdullah	PROPN
ejpam-5403	296	8	and	and	CCONJ
ejpam-5403	296	9	m.	m.	NOUN
ejpam-5403	296	10	tach	tach	PROPN
ejpam-5403	296	11	.	.	PUNCT
ejpam-5403	297	1	intuitionistic	intuitionistic	ADJ
ejpam-5403	297	2	fuzzy	fuzzy	ADJ
ejpam-5403	297	3	prime	prime	ADJ
ejpam-5403	297	4	ideal	ideal	NOUN
ejpam-5403	297	5	on	on	ADP
ejpam-5403	297	6	q	q	NOUN
ejpam-5403	297	7	-	-	PUNCT
ejpam-5403	297	8	algebras	algebra	NOUN
ejpam-5403	297	9	.	.	PUNCT
ejpam-5403	298	1	international	international	ADJ
ejpam-5403	298	2	journal	journal	PROPN
ejpam-5403	298	3	of	of	ADP
ejpam-5403	298	4	academic	academic	ADJ
ejpam-5403	298	5	and	and	CCONJ
ejpam-5403	298	6	applied	applied	ADJ
ejpam-5403	298	7	research	research	NOUN
ejpam-5403	298	8	,	,	PUNCT
ejpam-5403	298	9	4(10):66–78	4(10):66–78	NOUN
ejpam-5403	298	10	,	,	PUNCT
ejpam-5403	298	11	2020	2020	NUM
ejpam-5403	298	12	.	.	PUNCT
ejpam-5403	299	1	[	[	X
ejpam-5403	299	2	2	2	X
ejpam-5403	299	3	]	]	PUNCT
ejpam-5403	299	4	h.	h.	PROPN
ejpam-5403	299	5	k.	k.	PROPN
ejpam-5403	299	6	abdullah	abdullah	PROPN
ejpam-5403	299	7	and	and	CCONJ
ejpam-5403	299	8	m.	m.	NOUN
ejpam-5403	299	9	tach	tach	PROPN
ejpam-5403	299	10	.	.	PUNCT
ejpam-5403	300	1	prime	prime	PROPN
ejpam-5403	300	2	ideal	ideal	NOUN
ejpam-5403	300	3	in	in	ADP
ejpam-5403	300	4	q	q	NOUN
ejpam-5403	300	5	-	-	NOUN
ejpam-5403	300	6	algebra	algebra	NOUN
ejpam-5403	300	7	.	.	PUNCT
ejpam-5403	301	1	international	international	ADJ
ejpam-5403	301	2	journal	journal	NOUN
ejpam-5403	301	3	of	of	ADP
ejpam-5403	301	4	academic	academic	ADJ
ejpam-5403	301	5	and	and	CCONJ
ejpam-5403	301	6	applied	applied	ADJ
ejpam-5403	301	7	research	research	NOUN
ejpam-5403	301	8	,	,	PUNCT
ejpam-5403	301	9	4(10):79–87	4(10):79–87	NUM
ejpam-5403	301	10	,	,	PUNCT
ejpam-5403	301	11	2020	2020	NUM
ejpam-5403	301	12	.	.	PUNCT
ejpam-5403	302	1	[	[	X
ejpam-5403	302	2	3	3	X
ejpam-5403	302	3	]	]	PUNCT
ejpam-5403	302	4	s.	s.	PROPN
ejpam-5403	302	5	ahn	ahn	PROPN
ejpam-5403	302	6	and	and	CCONJ
ejpam-5403	302	7	s.	s.	PROPN
ejpam-5403	302	8	e.	e.	PROPN
ejpam-5403	302	9	kang	kang	PROPN
ejpam-5403	302	10	.	.	PUNCT
ejpam-5403	303	1	the	the	DET
ejpam-5403	303	2	role	role	NOUN
ejpam-5403	303	3	of	of	ADP
ejpam-5403	303	4	t(x	t(x	PROPN
ejpam-5403	303	5	)	)	PUNCT
ejpam-5403	303	6	in	in	ADP
ejpam-5403	303	7	the	the	DET
ejpam-5403	303	8	ideal	ideal	ADJ
ejpam-5403	303	9	theory	theory	NOUN
ejpam-5403	303	10	of	of	ADP
ejpam-5403	303	11	q	q	NOUN
ejpam-5403	303	12	-	-	PUNCT
ejpam-5403	303	13	algebras	algebras	X
ejpam-5403	303	14	.	.	PUNCT
ejpam-5403	304	1	honam	honam	PROPN
ejpam-5403	304	2	mathematical	mathematical	PROPN
ejpam-5403	304	3	journal	journal	PROPN
ejpam-5403	304	4	,	,	PUNCT
ejpam-5403	304	5	32(3):515–523	32(3):515–523	PROPN
ejpam-5403	304	6	,	,	PUNCT
ejpam-5403	304	7	2010	2010	NUM
ejpam-5403	304	8	.	.	PUNCT
ejpam-5403	305	1	[	[	X
ejpam-5403	305	2	4	4	X
ejpam-5403	305	3	]	]	PUNCT
ejpam-5403	305	4	s.	s.	PROPN
ejpam-5403	305	5	s.	s.	PROPN
ejpam-5403	305	6	ahn	ahn	PROPN
ejpam-5403	305	7	and	and	CCONJ
ejpam-5403	305	8	k.	k.	PROPN
ejpam-5403	305	9	so	so	ADV
ejpam-5403	305	10	.	.	PUNCT
ejpam-5403	306	1	on	on	ADP
ejpam-5403	306	2	medial	medial	ADJ
ejpam-5403	306	3	q	q	NOUN
ejpam-5403	306	4	-	-	PUNCT
ejpam-5403	306	5	algebras	algebra	NOUN
ejpam-5403	306	6	.	.	PUNCT
ejpam-5403	307	1	communications	communication	NOUN
ejpam-5403	307	2	of	of	ADP
ejpam-5403	307	3	the	the	DET
ejpam-5403	307	4	korean	korean	ADJ
ejpam-5403	307	5	mathematical	mathematical	ADJ
ejpam-5403	307	6	society	society	NOUN
ejpam-5403	307	7	,	,	PUNCT
ejpam-5403	307	8	25(3):365–372	25(3):365–372	NUM
ejpam-5403	307	9	,	,	PUNCT
ejpam-5403	307	10	2010	2010	NUM
ejpam-5403	307	11	.	.	PUNCT
ejpam-5403	308	1	[	[	X
ejpam-5403	308	2	5	5	NUM
ejpam-5403	308	3	]	]	PUNCT
ejpam-5403	308	4	a.	a.	NOUN
ejpam-5403	308	5	anantayasethi	anantayasethi	PROPN
ejpam-5403	308	6	and	and	CCONJ
ejpam-5403	308	7	j.	j.	PROPN
ejpam-5403	308	8	koppitz	koppitz	PROPN
ejpam-5403	308	9	.	.	PUNCT
ejpam-5403	309	1	all	all	DET
ejpam-5403	309	2	the	the	DET
ejpam-5403	309	3	cardinal	cardinal	ADJ
ejpam-5403	309	4	numbers	number	NOUN
ejpam-5403	309	5	of	of	ADP
ejpam-5403	309	6	ideals	ideal	NOUN
ejpam-5403	309	7	g	g	NOUN
ejpam-5403	309	8	-	-	PUNCT
ejpam-5403	309	9	part	part	NOUN
ejpam-5403	309	10	g(x	g(x	NOUN
ejpam-5403	309	11	)	)	PUNCT
ejpam-5403	309	12	of	of	ADP
ejpam-5403	309	13	q	q	NOUN
ejpam-5403	309	14	-	-	PUNCT
ejpam-5403	309	15	algebras	algebra	NOUN
ejpam-5403	309	16	,	,	PUNCT
ejpam-5403	309	17	2024	2024	NUM
ejpam-5403	309	18	.	.	PUNCT
ejpam-5403	310	1	preprint	preprint	NOUN
ejpam-5403	310	2	.	.	PUNCT
ejpam-5403	311	1	[	[	X
ejpam-5403	311	2	6	6	NUM
ejpam-5403	311	3	]	]	X
ejpam-5403	311	4	y.	y.	PROPN
ejpam-5403	311	5	imai	imai	PROPN
ejpam-5403	311	6	and	and	CCONJ
ejpam-5403	311	7	k.	k.	PROPN
ejpam-5403	311	8	iseki	iseki	PROPN
ejpam-5403	311	9	.	.	PUNCT
ejpam-5403	312	1	on	on	ADP
ejpam-5403	312	2	axiom	axiom	NOUN
ejpam-5403	312	3	system	system	NOUN
ejpam-5403	312	4	of	of	ADP
ejpam-5403	312	5	propositional	propositional	ADJ
ejpam-5403	312	6	calculi	calculi	PROPN
ejpam-5403	312	7	.	.	PUNCT
ejpam-5403	313	1	xiv	xiv	PROPN
ejpam-5403	313	2	.	.	PUNCT
ejpam-5403	314	1	proceedings	proceeding	NOUN
ejpam-5403	314	2	of	of	ADP
ejpam-5403	314	3	the	the	DET
ejpam-5403	314	4	japan	japan	PROPN
ejpam-5403	314	5	academy	academy	PROPN
ejpam-5403	314	6	,	,	PUNCT
ejpam-5403	314	7	42:19–22	42:19–22	NUM
ejpam-5403	314	8	,	,	PUNCT
ejpam-5403	314	9	1966	1966	NUM
ejpam-5403	314	10	.	.	PUNCT
ejpam-5403	315	1	[	[	X
ejpam-5403	315	2	7	7	X
ejpam-5403	315	3	]	]	PUNCT
ejpam-5403	315	4	k.	k.	PROPN
ejpam-5403	315	5	iseki	iseki	PROPN
ejpam-5403	315	6	.	.	PUNCT
ejpam-5403	316	1	an	an	DET
ejpam-5403	316	2	algebra	algebra	NOUN
ejpam-5403	316	3	related	relate	VERB
ejpam-5403	316	4	with	with	ADP
ejpam-5403	316	5	a	a	DET
ejpam-5403	316	6	propositional	propositional	ADJ
ejpam-5403	316	7	calculus	calculus	NOUN
ejpam-5403	316	8	calculi	calculi	PROPN
ejpam-5403	316	9	.	.	PUNCT
ejpam-5403	317	1	proceedings	proceeding	NOUN
ejpam-5403	317	2	of	of	ADP
ejpam-5403	317	3	the	the	DET
ejpam-5403	317	4	japan	japan	PROPN
ejpam-5403	317	5	academy	academy	PROPN
ejpam-5403	317	6	,	,	PUNCT
ejpam-5403	317	7	42:26–29	42:26–29	PROPN
ejpam-5403	317	8	,	,	PUNCT
ejpam-5403	317	9	1966	1966	NUM
ejpam-5403	317	10	.	.	PUNCT
ejpam-5403	318	1	references	reference	NOUN
ejpam-5403	318	2	3276	3276	NUM
ejpam-5403	318	3	[	[	X
ejpam-5403	318	4	8	8	NUM
ejpam-5403	318	5	]	]	PUNCT
ejpam-5403	318	6	k.	k.	PROPN
ejpam-5403	318	7	iseki	iseki	PROPN
ejpam-5403	318	8	and	and	CCONJ
ejpam-5403	318	9	s.	s.	PROPN
ejpam-5403	318	10	tanaka	tanaka	PROPN
ejpam-5403	318	11	.	.	PUNCT
ejpam-5403	319	1	an	an	DET
ejpam-5403	319	2	introduction	introduction	NOUN
ejpam-5403	319	3	to	to	ADP
ejpam-5403	319	4	theory	theory	NOUN
ejpam-5403	319	5	of	of	ADP
ejpam-5403	319	6	bck	bck	NOUN
ejpam-5403	319	7	-	-	PUNCT
ejpam-5403	319	8	algebra	algebra	NOUN
ejpam-5403	319	9	.	.	PUNCT
ejpam-5403	320	1	mathematica	mathematica	PROPN
ejpam-5403	320	2	japonica	japonica	PROPN
ejpam-5403	320	3	,	,	PUNCT
ejpam-5403	320	4	23:1–26	23:1–26	NUM
ejpam-5403	320	5	,	,	PUNCT
ejpam-5403	320	6	1978	1978	NUM
ejpam-5403	320	7	.	.	PUNCT
ejpam-5403	321	1	[	[	X
ejpam-5403	321	2	9	9	NUM
ejpam-5403	321	3	]	]	PUNCT
ejpam-5403	321	4	s.	s.	PROPN
ejpam-5403	321	5	ahn	ahn	PROPN
ejpam-5403	321	6	j.	j.	PROPN
ejpam-5403	321	7	neggers	neggers	PROPN
ejpam-5403	321	8	and	and	CCONJ
ejpam-5403	321	9	h.	h.	PROPN
ejpam-5403	321	10	s.	s.	PROPN
ejpam-5403	321	11	kim	kim	PROPN
ejpam-5403	321	12	.	.	PUNCT
ejpam-5403	322	1	on	on	ADP
ejpam-5403	322	2	q	q	NOUN
ejpam-5403	322	3	-	-	PUNCT
ejpam-5403	322	4	algebras	algebra	NOUN
ejpam-5403	322	5	.	.	PUNCT
ejpam-5403	323	1	international	international	ADJ
ejpam-5403	323	2	journal	journal	PROPN
ejpam-5403	323	3	of	of	ADP
ejpam-5403	323	4	mathematics	mathematics	PROPN
ejpam-5403	323	5	and	and	CCONJ
ejpam-5403	323	6	mathematical	mathematical	ADJ
ejpam-5403	323	7	sciences	science	NOUN
ejpam-5403	323	8	,	,	PUNCT
ejpam-5403	323	9	27(12):749–757	27(12):749–757	NOUN
ejpam-5403	323	10	,	,	PUNCT
ejpam-5403	323	11	2001	2001	NUM
ejpam-5403	323	12	.	.	PUNCT
ejpam-5403	324	1	[	[	X
ejpam-5403	324	2	10	10	NUM
ejpam-5403	324	3	]	]	X
ejpam-5403	324	4	j.	j.	PROPN
ejpam-5403	324	5	koppitzs	koppitzs	PROPN
ejpam-5403	324	6	and	and	CCONJ
ejpam-5403	324	7	a.	a.	NOUN
ejpam-5403	324	8	anantayasethi	anantayasethi	PROPN
ejpam-5403	324	9	.	.	PUNCT
ejpam-5403	325	1	characterization	characterization	NOUN
ejpam-5403	325	2	of	of	ADP
ejpam-5403	325	3	ideals	ideal	NOUN
ejpam-5403	325	4	of	of	ADP
ejpam-5403	325	5	q	q	NOUN
ejpam-5403	325	6	-	-	PUNCT
ejpam-5403	325	7	algebras	algebras	ADV
ejpam-5403	325	8	related	relate	VERB
ejpam-5403	325	9	to	to	ADP
ejpam-5403	325	10	its	its	PRON
ejpam-5403	325	11	g	g	NOUN
ejpam-5403	325	12	-	-	PUNCT
ejpam-5403	325	13	part	part	NOUN
ejpam-5403	325	14	,	,	PUNCT
ejpam-5403	325	15	2024	2024	NUM
ejpam-5403	325	16	.	.	PUNCT
ejpam-5403	326	1	preprint	preprint	NOUN
ejpam-5403	326	2	.	.	PUNCT
ejpam-5403	327	1	[	[	X
ejpam-5403	327	2	11	11	NUM
ejpam-5403	327	3	]	]	PUNCT
ejpam-5403	327	4	s.	s.	PROPN
ejpam-5403	327	5	m.	m.	PROPN
ejpam-5403	327	6	lee	lee	PROPN
ejpam-5403	327	7	and	and	CCONJ
ejpam-5403	327	8	k.	k.	PROPN
ejpam-5403	327	9	h.	h.	PROPN
ejpam-5403	327	10	kim	kim	PROPN
ejpam-5403	327	11	.	.	PUNCT
ejpam-5403	328	1	on	on	ADP
ejpam-5403	328	2	right	right	ADJ
ejpam-5403	328	3	fixed	fix	VERB
ejpam-5403	328	4	maps	map	NOUN
ejpam-5403	328	5	of	of	ADP
ejpam-5403	328	6	q	q	NOUN
ejpam-5403	328	7	-	-	PUNCT
ejpam-5403	328	8	algebras	algebra	NOUN
ejpam-5403	328	9	.	.	PUNCT
ejpam-5403	329	1	international	international	PROPN
ejpam-5403	329	2	mathematical	mathematical	PROPN
ejpam-5403	329	3	forum	forum	PROPN
ejpam-5403	329	4	,	,	PUNCT
ejpam-5403	329	5	6(1):31–37	6(1):31–37	NUM
ejpam-5403	329	6	,	,	PUNCT
ejpam-5403	329	7	2011	2011	NUM
ejpam-5403	329	8	.	.	PUNCT
ejpam-5403	330	1	[	[	X
ejpam-5403	330	2	12	12	NUM
ejpam-5403	330	3	]	]	X
ejpam-5403	330	4	c.	c.	PROPN
ejpam-5403	330	5	granados	granados	PROPN
ejpam-5403	330	6	s.	s.	PROPN
ejpam-5403	330	7	das	das	PROPN
ejpam-5403	330	8	,	,	PUNCT
ejpam-5403	330	9	r.	r.	PROPN
ejpam-5403	330	10	das	das	PROPN
ejpam-5403	330	11	and	and	CCONJ
ejpam-5403	330	12	a.	a.	NOUN
ejpam-5403	330	13	mukherjee	mukherjee	PROPN
ejpam-5403	330	14	.	.	PUNCT
ejpam-5403	331	1	pentapartitioned	pentapartitione	VERB
ejpam-5403	331	2	neutrosophic	neutrosophic	ADJ
ejpam-5403	331	3	qideals	qideal	NOUN
ejpam-5403	331	4	of	of	ADP
ejpam-5403	331	5	q	q	NOUN
ejpam-5403	331	6	-	-	PUNCT
ejpam-5403	331	7	algebra	algebra	NOUN
ejpam-5403	331	8	.	.	PUNCT
ejpam-5403	332	1	neutrosophic	neutrosophic	ADJ
ejpam-5403	332	2	sets	set	NOUN
ejpam-5403	332	3	and	and	CCONJ
ejpam-5403	332	4	systems	system	NOUN
ejpam-5403	332	5	,	,	PUNCT
ejpam-5403	332	6	41:52–63	41:52–63	NUM
ejpam-5403	332	7	,	,	PUNCT
ejpam-5403	332	8	2021	2021	NUM
ejpam-5403	332	9	.	.	PUNCT
ejpam-5403	333	1	[	[	X
ejpam-5403	333	2	13	13	NUM
ejpam-5403	333	3	]	]	PUNCT
ejpam-5403	333	4	m.	m.	NOUN
ejpam-5403	333	5	a.	a.	PROPN
ejpam-5403	333	6	naby	naby	PROPN
ejpam-5403	333	7	s.	s.	PROPN
ejpam-5403	333	8	m.	m.	PROPN
ejpam-5403	333	9	mostafa	mostafa	PROPN
ejpam-5403	333	10	and	and	CCONJ
ejpam-5403	333	11	o.	o.	PROPN
ejpam-5403	333	12	r.	r.	PROPN
ejpam-5403	333	13	elgendy	elgendy	PROPN
ejpam-5403	333	14	.	.	PUNCT
ejpam-5403	334	1	fuzzy	fuzzy	ADJ
ejpam-5403	334	2	q	q	NOUN
ejpam-5403	334	3	-	-	NOUN
ejpam-5403	334	4	ideals	ideal	NOUN
ejpam-5403	334	5	in	in	ADP
ejpam-5403	334	6	q	q	NOUN
ejpam-5403	334	7	-	-	PUNCT
ejpam-5403	334	8	algebras	algebra	NOUN
ejpam-5403	334	9	.	.	PUNCT
ejpam-5403	335	1	world	world	NOUN
ejpam-5403	335	2	applied	apply	VERB
ejpam-5403	335	3	programming	programming	NOUN
ejpam-5403	335	4	,	,	PUNCT
ejpam-5403	335	5	2(2):69–80	2(2):69–80	NUM
ejpam-5403	335	6	,	,	PUNCT
ejpam-5403	335	7	2012	2012	NUM
ejpam-5403	335	8	.	.	PUNCT
ejpam-5403	336	1	[	[	X
ejpam-5403	336	2	14	14	NUM
ejpam-5403	336	3	]	]	PUNCT
ejpam-5403	336	4	h.	h.	PROPN
ejpam-5403	336	5	s.	s.	PROPN
ejpam-5403	336	6	kim	kim	PROPN
ejpam-5403	336	7	s.	s.	PROPN
ejpam-5403	336	8	s.	s.	PROPN
ejpam-5403	336	9	ahn	ahn	PROPN
ejpam-5403	336	10	and	and	CCONJ
ejpam-5403	336	11	h.	h.	PROPN
ejpam-5403	336	12	d.	d.	PROPN
ejpam-5403	336	13	lee	lee	PROPN
ejpam-5403	336	14	.	.	PUNCT
ejpam-5403	337	1	r	r	X
ejpam-5403	337	2	-	-	PUNCT
ejpam-5403	337	3	maps	map	NOUN
ejpam-5403	337	4	and	and	CCONJ
ejpam-5403	337	5	l	l	NOUN
ejpam-5403	337	6	-	-	NOUN
ejpam-5403	337	7	map	map	NOUN
ejpam-5403	337	8	in	in	ADP
ejpam-5403	337	9	q	q	NOUN
ejpam-5403	337	10	-	-	PUNCT
ejpam-5403	337	11	algebras	algebra	NOUN
ejpam-5403	337	12	.	.	PUNCT
ejpam-5403	338	1	international	international	ADJ
ejpam-5403	338	2	journal	journal	PROPN
ejpam-5403	338	3	of	of	ADP
ejpam-5403	338	4	pure	pure	ADJ
ejpam-5403	338	5	and	and	CCONJ
ejpam-5403	338	6	applied	applied	ADJ
ejpam-5403	338	7	mathematics	mathematic	NOUN
ejpam-5403	338	8	,	,	PUNCT
ejpam-5403	338	9	12(4):419–425	12(4):419–425	NOUN
ejpam-5403	338	10	,	,	PUNCT
ejpam-5403	338	11	2004	2004	NUM
ejpam-5403	338	12	.	.	PUNCT
ejpam-5403	339	1	[	[	X
ejpam-5403	339	2	15	15	NUM
ejpam-5403	339	3	]	]	X
ejpam-5403	339	4	d.	d.	PROPN
ejpam-5403	339	5	sun	sun	PROPN
ejpam-5403	339	6	.	.	PUNCT
ejpam-5403	340	1	on	on	ADP
ejpam-5403	340	2	atoms	atom	NOUN
ejpam-5403	340	3	of	of	ADP
ejpam-5403	340	4	bck	bck	NOUN
ejpam-5403	340	5	-	-	PUNCT
ejpam-5403	340	6	algebras	algebras	PROPN
ejpam-5403	340	7	.	.	PUNCT
ejpam-5403	341	1	scientiae	scientiae	PROPN
ejpam-5403	341	2	mathematicae	mathematicae	VERB
ejpam-5403	341	3	japonicae	japonicae	PROPN
ejpam-5403	341	4	online	online	PROPN
ejpam-5403	341	5	,	,	PUNCT
ejpam-5403	341	6	2(4):115–124	2(4):115–124	NUM
ejpam-5403	341	7	,	,	PUNCT
ejpam-5403	341	8	2001	2001	NUM
ejpam-5403	341	9	.	.	PUNCT
ejpam-5403	342	1	[	[	X
ejpam-5403	342	2	16	16	NUM
ejpam-5403	342	3	]	]	PUNCT
ejpam-5403	342	4	k.	k.	PROPN
ejpam-5403	342	5	iseki	iseki	PROPN
ejpam-5403	342	6	y.	y.	PROPN
ejpam-5403	342	7	arai	arai	PROPN
ejpam-5403	342	8	and	and	CCONJ
ejpam-5403	342	9	s.	s.	PROPN
ejpam-5403	342	10	tanaka	tanaka	PROPN
ejpam-5403	342	11	.	.	PUNCT
ejpam-5403	343	1	characterizations	characterization	NOUN
ejpam-5403	343	2	of	of	ADP
ejpam-5403	343	3	bci	bci	PROPN
ejpam-5403	343	4	,	,	PUNCT
ejpam-5403	343	5	bck	bck	NOUN
ejpam-5403	343	6	-	-	PUNCT
ejpam-5403	343	7	algebra	algebra	NOUN
ejpam-5403	343	8	.	.	PUNCT
ejpam-5403	344	1	proceedings	proceeding	NOUN
ejpam-5403	344	2	of	of	ADP
ejpam-5403	344	3	the	the	DET
ejpam-5403	344	4	japan	japan	PROPN
ejpam-5403	344	5	academy	academy	PROPN
ejpam-5403	344	6	,	,	PUNCT
ejpam-5403	344	7	42:105–107	42:105–107	PROPN
ejpam-5403	344	8	,	,	PUNCT
ejpam-5403	344	9	1966	1966	NUM
ejpam-5403	344	10	.	.	PUNCT