id	sid	tid	token	lemma	pos
ejpam-3283	1	1	european	european	PROPN
ejpam-3283	1	2	journal	journal	PROPN
ejpam-3283	1	3	of	of	ADP
ejpam-3283	1	4	pure	pure	ADJ
ejpam-3283	1	5	and	and	CCONJ
ejpam-3283	1	6	applied	apply	VERB
ejpam-3283	1	7	mathematics	mathematic	NOUN
ejpam-3283	1	8	vol	vol	NOUN
ejpam-3283	1	9	.	.	PUNCT
ejpam-3283	2	1	11	11	NUM
ejpam-3283	2	2	,	,	PUNCT
ejpam-3283	2	3	no	no	INTJ
ejpam-3283	2	4	.	.	NOUN
ejpam-3283	2	5	3	3	NUM
ejpam-3283	2	6	,	,	PUNCT
ejpam-3283	2	7	2018	2018	NUM
ejpam-3283	2	8	,	,	PUNCT
ejpam-3283	2	9	762	762	NUM
ejpam-3283	2	10	-	-	SYM
ejpam-3283	2	11	773	773	NUM
ejpam-3283	2	12	issn	issn	PROPN
ejpam-3283	2	13	1307	1307	NUM
ejpam-3283	2	14	-	-	SYM
ejpam-3283	2	15	5543	5543	NUM
ejpam-3283	2	16	–	–	PUNCT
ejpam-3283	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3283	2	18	published	publish	VERB
ejpam-3283	2	19	by	by	ADP
ejpam-3283	2	20	new	new	PROPN
ejpam-3283	2	21	york	york	PROPN
ejpam-3283	2	22	business	business	PROPN
ejpam-3283	2	23	global	global	PROPN
ejpam-3283	2	24	the	the	DET
ejpam-3283	2	25	minimality	minimality	NOUN
ejpam-3283	2	26	and	and	CCONJ
ejpam-3283	2	27	maximality	maximality	PROPN
ejpam-3283	2	28	of	of	ADP
ejpam-3283	2	29	n	n	CCONJ
ejpam-3283	2	30	-	-	PUNCT
ejpam-3283	2	31	ideals	ideal	NOUN
ejpam-3283	2	32	in	in	ADP
ejpam-3283	2	33	n	n	CCONJ
ejpam-3283	2	34	-	-	PUNCT
ejpam-3283	2	35	ary	ary	PROPN
ejpam-3283	2	36	semigroups	semigroups	PROPN
ejpam-3283	2	37	pattarawan	pattarawan	PROPN
ejpam-3283	2	38	petchkaew1	petchkaew1	PROPN
ejpam-3283	2	39	,	,	PUNCT
ejpam-3283	2	40	ronnason	ronnason	NOUN
ejpam-3283	2	41	chinram2,3,∗	chinram2,3,∗	PROPN
ejpam-3283	2	42	1	1	NUM
ejpam-3283	2	43	mathematics	mathematic	NOUN
ejpam-3283	2	44	and	and	CCONJ
ejpam-3283	2	45	statistics	statistic	NOUN
ejpam-3283	2	46	program	program	NOUN
ejpam-3283	2	47	,	,	PUNCT
ejpam-3283	2	48	faculty	faculty	NOUN
ejpam-3283	2	49	of	of	ADP
ejpam-3283	2	50	science	science	NOUN
ejpam-3283	2	51	and	and	CCONJ
ejpam-3283	2	52	technology	technology	NOUN
ejpam-3283	2	53	,	,	PUNCT
ejpam-3283	2	54	songkhla	songkhla	VERB
ejpam-3283	2	55	rajabhat	rajabhat	ADJ
ejpam-3283	2	56	university	university	NOUN
ejpam-3283	2	57	,	,	PUNCT
ejpam-3283	2	58	songkhla	songkhla	ADJ
ejpam-3283	2	59	,	,	PUNCT
ejpam-3283	2	60	90000	90000	NUM
ejpam-3283	2	61	,	,	PUNCT
ejpam-3283	2	62	thailand	thailand	PROPN
ejpam-3283	2	63	2	2	NUM
ejpam-3283	2	64	algebra	algebra	NOUN
ejpam-3283	2	65	and	and	CCONJ
ejpam-3283	2	66	applications	application	NOUN
ejpam-3283	2	67	research	research	NOUN
ejpam-3283	2	68	unit	unit	NOUN
ejpam-3283	2	69	,	,	PUNCT
ejpam-3283	2	70	department	department	NOUN
ejpam-3283	2	71	of	of	ADP
ejpam-3283	2	72	mathematics	mathematics	PROPN
ejpam-3283	2	73	and	and	CCONJ
ejpam-3283	2	74	statistics	statistic	NOUN
ejpam-3283	2	75	,	,	PUNCT
ejpam-3283	2	76	faculty	faculty	NOUN
ejpam-3283	2	77	of	of	ADP
ejpam-3283	2	78	science	science	NOUN
ejpam-3283	2	79	,	,	PUNCT
ejpam-3283	2	80	prince	prince	NOUN
ejpam-3283	2	81	of	of	ADP
ejpam-3283	2	82	songkla	songkla	PROPN
ejpam-3283	2	83	university	university	PROPN
ejpam-3283	2	84	,	,	PUNCT
ejpam-3283	2	85	hat	hat	PROPN
ejpam-3283	2	86	yai	yai	PROPN
ejpam-3283	2	87	,	,	PUNCT
ejpam-3283	2	88	songkhla	songkhla	ADJ
ejpam-3283	2	89	,	,	PUNCT
ejpam-3283	2	90	90110	90110	NUM
ejpam-3283	2	91	,	,	PUNCT
ejpam-3283	2	92	thailand	thailand	PROPN
ejpam-3283	2	93	3	3	NUM
ejpam-3283	2	94	centre	centre	NOUN
ejpam-3283	2	95	of	of	ADP
ejpam-3283	2	96	excellence	excellence	NOUN
ejpam-3283	2	97	in	in	ADP
ejpam-3283	2	98	mathematics	mathematics	PROPN
ejpam-3283	2	99	,	,	PUNCT
ejpam-3283	2	100	che	che	PROPN
ejpam-3283	2	101	,	,	PUNCT
ejpam-3283	2	102	si	si	PROPN
ejpam-3283	2	103	ayuthaya	ayuthaya	PROPN
ejpam-3283	2	104	road	road	PROPN
ejpam-3283	2	105	,	,	PUNCT
ejpam-3283	2	106	bangkok	bangkok	PROPN
ejpam-3283	2	107	10400	10400	NUM
ejpam-3283	2	108	,	,	PUNCT
ejpam-3283	2	109	thailand	thailand	PROPN
ejpam-3283	2	110	abstract	abstract	PROPN
ejpam-3283	2	111	.	.	PUNCT
ejpam-3283	3	1	one	one	NUM
ejpam-3283	3	2	knows	know	VERB
ejpam-3283	3	3	that	that	SCONJ
ejpam-3283	3	4	the	the	DET
ejpam-3283	3	5	concept	concept	NOUN
ejpam-3283	3	6	of	of	ADP
ejpam-3283	3	7	minimality	minimality	NOUN
ejpam-3283	3	8	and	and	CCONJ
ejpam-3283	3	9	maximality	maximality	NOUN
ejpam-3283	3	10	of	of	ADP
ejpam-3283	3	11	left	left	ADJ
ejpam-3283	3	12	ideals	ideal	NOUN
ejpam-3283	3	13	and	and	CCONJ
ejpam-3283	3	14	right	right	ADJ
ejpam-3283	3	15	ideals	ideal	NOUN
ejpam-3283	3	16	play	play	VERB
ejpam-3283	3	17	an	an	DET
ejpam-3283	3	18	important	important	ADJ
ejpam-3283	3	19	role	role	NOUN
ejpam-3283	3	20	in	in	ADP
ejpam-3283	3	21	semigroups	semigroup	NOUN
ejpam-3283	3	22	.	.	PUNCT
ejpam-3283	4	1	in	in	ADP
ejpam-3283	4	2	this	this	DET
ejpam-3283	4	3	paper	paper	NOUN
ejpam-3283	4	4	,	,	PUNCT
ejpam-3283	4	5	we	we	PRON
ejpam-3283	4	6	extend	extend	VERB
ejpam-3283	4	7	this	this	DET
ejpam-3283	4	8	concept	concept	NOUN
ejpam-3283	4	9	to	to	PART
ejpam-3283	4	10	consider	consider	VERB
ejpam-3283	4	11	in	in	ADP
ejpam-3283	4	12	n	n	CCONJ
ejpam-3283	4	13	-	-	PUNCT
ejpam-3283	4	14	ary	ary	NOUN
ejpam-3283	4	15	semigroups	semigroup	NOUN
ejpam-3283	4	16	.	.	PUNCT
ejpam-3283	5	1	a	a	DET
ejpam-3283	5	2	number	number	NOUN
ejpam-3283	5	3	of	of	ADP
ejpam-3283	5	4	results	result	NOUN
ejpam-3283	5	5	concerning	concern	VERB
ejpam-3283	5	6	relationships	relationship	NOUN
ejpam-3283	5	7	between	between	ADP
ejpam-3283	5	8	minimality	minimality	NOUN
ejpam-3283	5	9	and	and	CCONJ
ejpam-3283	5	10	maximality	maximality	PROPN
ejpam-3283	5	11	of	of	ADP
ejpam-3283	5	12	n	n	CCONJ
ejpam-3283	5	13	-	-	PUNCT
ejpam-3283	5	14	ideals	ideal	NOUN
ejpam-3283	5	15	of	of	ADP
ejpam-3283	5	16	n	n	CCONJ
ejpam-3283	5	17	-	-	PUNCT
ejpam-3283	5	18	ary	ary	NOUN
ejpam-3283	5	19	semigroups	semigroup	NOUN
ejpam-3283	5	20	and	and	CCONJ
ejpam-3283	5	21	n	n	CCONJ
ejpam-3283	5	22	-	-	PUNCT
ejpam-3283	5	23	simple	simple	ADJ
ejpam-3283	5	24	(	(	PUNCT
ejpam-3283	5	25	0	0	NUM
ejpam-3283	5	26	-	-	PUNCT
ejpam-3283	5	27	n	n	CCONJ
ejpam-3283	5	28	-	-	PUNCT
ejpam-3283	5	29	simple	simple	ADJ
ejpam-3283	5	30	)	)	PUNCT
ejpam-3283	5	31	n	n	CCONJ
ejpam-3283	5	32	-	-	PUNCT
ejpam-3283	5	33	ary	ary	PROPN
ejpam-3283	5	34	semigroups	semigroup	NOUN
ejpam-3283	5	35	as	as	ADV
ejpam-3283	5	36	well	well	ADV
ejpam-3283	5	37	as	as	ADP
ejpam-3283	5	38	some	some	DET
ejpam-3283	5	39	characterizations	characterization	NOUN
ejpam-3283	5	40	of	of	ADP
ejpam-3283	5	41	minimality	minimality	NOUN
ejpam-3283	5	42	and	and	CCONJ
ejpam-3283	5	43	maximality	maximality	NOUN
ejpam-3283	5	44	of	of	ADP
ejpam-3283	5	45	n	n	CCONJ
ejpam-3283	5	46	-	-	PUNCT
ejpam-3283	5	47	ideals	ideal	NOUN
ejpam-3283	5	48	of	of	ADP
ejpam-3283	5	49	n	n	CCONJ
ejpam-3283	5	50	-	-	PUNCT
ejpam-3283	5	51	ary	ary	NOUN
ejpam-3283	5	52	semigroups	semigroup	NOUN
ejpam-3283	5	53	are	be	AUX
ejpam-3283	5	54	given	give	VERB
ejpam-3283	5	55	.	.	PUNCT
ejpam-3283	6	1	2010	2010	NUM
ejpam-3283	6	2	mathematics	mathematic	NOUN
ejpam-3283	6	3	subject	subject	NOUN
ejpam-3283	6	4	classifications	classification	NOUN
ejpam-3283	6	5	:	:	PUNCT
ejpam-3283	6	6	20n15	20n15	NUM
ejpam-3283	6	7	,	,	PUNCT
ejpam-3283	6	8	20m12	20m12	NUM
ejpam-3283	6	9	key	key	ADJ
ejpam-3283	6	10	words	word	NOUN
ejpam-3283	6	11	and	and	CCONJ
ejpam-3283	6	12	phrases	phrase	NOUN
ejpam-3283	6	13	:	:	PUNCT
ejpam-3283	6	14	n	n	NUM
ejpam-3283	6	15	-	-	PUNCT
ejpam-3283	6	16	ary	ary	PROPN
ejpam-3283	6	17	semigroups	semigroup	NOUN
ejpam-3283	6	18	,	,	PUNCT
ejpam-3283	6	19	i	i	PRON
ejpam-3283	6	20	-	-	PUNCT
ejpam-3283	6	21	ideals	ideal	NOUN
ejpam-3283	6	22	,	,	PUNCT
ejpam-3283	6	23	n	n	CCONJ
ejpam-3283	6	24	-	-	PUNCT
ejpam-3283	6	25	ideals	ideal	NOUN
ejpam-3283	6	26	,	,	PUNCT
ejpam-3283	6	27	minimal	minimal	ADJ
ejpam-3283	6	28	ideals	ideal	NOUN
ejpam-3283	6	29	,	,	PUNCT
ejpam-3283	6	30	maximal	maximal	ADJ
ejpam-3283	6	31	ideals	ideal	NOUN
ejpam-3283	6	32	,	,	PUNCT
ejpam-3283	6	33	n	n	CCONJ
ejpam-3283	6	34	-	-	PUNCT
ejpam-3283	6	35	simple	simple	ADJ
ejpam-3283	6	36	,	,	PUNCT
ejpam-3283	6	37	0	0	NUM
ejpam-3283	6	38	-	-	PUNCT
ejpam-3283	6	39	n	n	CCONJ
ejpam-3283	6	40	-	-	PUNCT
ejpam-3283	6	41	simple	simple	ADJ
ejpam-3283	6	42	.	.	PUNCT
ejpam-3283	7	1	1	1	X
ejpam-3283	7	2	.	.	X
ejpam-3283	7	3	introduction	introduction	NOUN
ejpam-3283	7	4	the	the	DET
ejpam-3283	7	5	generalization	generalization	NOUN
ejpam-3283	7	6	of	of	ADP
ejpam-3283	7	7	classical	classical	ADJ
ejpam-3283	7	8	algebraic	algebraic	ADJ
ejpam-3283	7	9	structures	structure	NOUN
ejpam-3283	7	10	to	to	ADP
ejpam-3283	7	11	n	n	CCONJ
ejpam-3283	7	12	-	-	PUNCT
ejpam-3283	7	13	ary	ary	PROPN
ejpam-3283	7	14	structures	structure	NOUN
ejpam-3283	7	15	was	be	AUX
ejpam-3283	7	16	first	first	ADV
ejpam-3283	7	17	introduced	introduce	VERB
ejpam-3283	7	18	by	by	ADP
ejpam-3283	7	19	kasner	kasner	NOUN
ejpam-3283	7	20	[	[	X
ejpam-3283	7	21	10	10	NUM
ejpam-3283	7	22	]	]	PUNCT
ejpam-3283	7	23	in	in	ADP
ejpam-3283	7	24	1904	1904	NUM
ejpam-3283	7	25	.	.	PUNCT
ejpam-3283	8	1	in	in	ADP
ejpam-3283	8	2	[	[	X
ejpam-3283	8	3	12	12	NUM
ejpam-3283	8	4	]	]	PUNCT
ejpam-3283	8	5	,	,	PUNCT
ejpam-3283	8	6	sioson	sioson	PROPN
ejpam-3283	8	7	introduced	introduce	VERB
ejpam-3283	8	8	regular	regular	ADJ
ejpam-3283	8	9	n	n	CCONJ
ejpam-3283	8	10	-	-	PUNCT
ejpam-3283	8	11	ary	ary	NOUN
ejpam-3283	8	12	semigroups	semigroup	NOUN
ejpam-3283	8	13	and	and	CCONJ
ejpam-3283	8	14	verified	verify	VERB
ejpam-3283	8	15	their	their	PRON
ejpam-3283	8	16	properties	property	NOUN
ejpam-3283	8	17	.	.	PUNCT
ejpam-3283	9	1	in	in	ADP
ejpam-3283	9	2	[	[	X
ejpam-3283	9	3	3	3	NUM
ejpam-3283	9	4	]	]	PUNCT
ejpam-3283	9	5	,	,	PUNCT
ejpam-3283	9	6	dudek	dudek	PROPN
ejpam-3283	9	7	and	and	CCONJ
ejpam-3283	9	8	grozdinska	grozdinska	NOUN
ejpam-3283	9	9	investigated	investigate	VERB
ejpam-3283	9	10	the	the	DET
ejpam-3283	9	11	nature	nature	NOUN
ejpam-3283	9	12	of	of	ADP
ejpam-3283	9	13	regular	regular	ADJ
ejpam-3283	9	14	n	n	CCONJ
ejpam-3283	9	15	-	-	PUNCT
ejpam-3283	9	16	ary	ary	PROPN
ejpam-3283	9	17	semigroups	semigroup	NOUN
ejpam-3283	9	18	in	in	ADP
ejpam-3283	9	19	detail	detail	NOUN
ejpam-3283	9	20	;	;	PUNCT
ejpam-3283	9	21	moreover	moreover	ADV
ejpam-3283	9	22	,	,	PUNCT
ejpam-3283	9	23	dudek	dudek	PROPN
ejpam-3283	9	24	proved	prove	VERB
ejpam-3283	9	25	several	several	ADJ
ejpam-3283	9	26	results	result	NOUN
ejpam-3283	9	27	and	and	CCONJ
ejpam-3283	9	28	gave	give	VERB
ejpam-3283	9	29	many	many	ADJ
ejpam-3283	9	30	examples	example	NOUN
ejpam-3283	9	31	of	of	ADP
ejpam-3283	9	32	n	n	CCONJ
ejpam-3283	9	33	-	-	PUNCT
ejpam-3283	9	34	ary	ary	NOUN
ejpam-3283	9	35	groups	group	NOUN
ejpam-3283	9	36	in	in	ADP
ejpam-3283	9	37	[	[	X
ejpam-3283	9	38	4	4	NUM
ejpam-3283	9	39	]	]	PUNCT
ejpam-3283	9	40	,	,	PUNCT
ejpam-3283	9	41	[	[	X
ejpam-3283	9	42	5	5	NUM
ejpam-3283	9	43	]	]	PUNCT
ejpam-3283	9	44	and	and	CCONJ
ejpam-3283	9	45	[	[	X
ejpam-3283	9	46	6	6	NUM
ejpam-3283	9	47	]	]	PUNCT
ejpam-3283	9	48	.	.	PUNCT
ejpam-3283	10	1	furthermore	furthermore	ADV
ejpam-3283	10	2	,	,	PUNCT
ejpam-3283	10	3	dudek	dudek	PROPN
ejpam-3283	10	4	also	also	ADV
ejpam-3283	10	5	investigated	investigate	VERB
ejpam-3283	10	6	the	the	DET
ejpam-3283	10	7	properties	property	NOUN
ejpam-3283	10	8	of	of	ADP
ejpam-3283	10	9	ideals	ideal	NOUN
ejpam-3283	10	10	of	of	ADP
ejpam-3283	10	11	some	some	DET
ejpam-3283	10	12	elements	element	NOUN
ejpam-3283	10	13	of	of	ADP
ejpam-3283	10	14	n	n	CCONJ
ejpam-3283	10	15	-	-	PUNCT
ejpam-3283	10	16	ary	ary	PROPN
ejpam-3283	10	17	(	(	PUNCT
ejpam-3283	10	18	n	n	CCONJ
ejpam-3283	10	19	≥	≥	NOUN
ejpam-3283	10	20	3	3	NUM
ejpam-3283	10	21	)	)	PUNCT
ejpam-3283	10	22	semigroups	semigroup	NOUN
ejpam-3283	10	23	containing	contain	VERB
ejpam-3283	10	24	an	an	DET
ejpam-3283	10	25	idempotent	idempotent	NOUN
ejpam-3283	10	26	in	in	ADP
ejpam-3283	10	27	[	[	X
ejpam-3283	10	28	7	7	NUM
ejpam-3283	10	29	]	]	PUNCT
ejpam-3283	10	30	.	.	PUNCT
ejpam-3283	11	1	in	in	ADP
ejpam-3283	11	2	[	[	X
ejpam-3283	11	3	15	15	NUM
ejpam-3283	11	4	]	]	PUNCT
ejpam-3283	11	5	,	,	PUNCT
ejpam-3283	11	6	the	the	DET
ejpam-3283	11	7	relation	relation	NOUN
ejpam-3283	11	8	between	between	ADP
ejpam-3283	11	9	soft	soft	ADJ
ejpam-3283	11	10	regular	regular	ADJ
ejpam-3283	11	11	n	n	CCONJ
ejpam-3283	11	12	-	-	PUNCT
ejpam-3283	11	13	ary	ary	NOUN
ejpam-3283	11	14	semigroups	semigroup	NOUN
ejpam-3283	11	15	and	and	CCONJ
ejpam-3283	11	16	regular	regular	ADJ
ejpam-3283	11	17	n	n	CCONJ
ejpam-3283	11	18	-	-	PUNCT
ejpam-3283	11	19	ary	ary	NOUN
ejpam-3283	11	20	semigroups	semigroup	NOUN
ejpam-3283	11	21	was	be	AUX
ejpam-3283	11	22	discussed	discuss	VERB
ejpam-3283	11	23	by	by	ADP
ejpam-3283	11	24	wang	wang	PROPN
ejpam-3283	11	25	,	,	PUNCT
ejpam-3283	11	26	zhou	zhou	PROPN
ejpam-3283	11	27	and	and	CCONJ
ejpam-3283	11	28	zhan	zhan	PROPN
ejpam-3283	11	29	.	.	PUNCT
ejpam-3283	12	1	nowadays	nowadays	ADV
ejpam-3283	12	2	,	,	PUNCT
ejpam-3283	12	3	the	the	DET
ejpam-3283	12	4	theory	theory	NOUN
ejpam-3283	12	5	of	of	ADP
ejpam-3283	12	6	n	n	CCONJ
ejpam-3283	12	7	-	-	PUNCT
ejpam-3283	12	8	ary	ary	PROPN
ejpam-3283	12	9	systems	system	NOUN
ejpam-3283	12	10	has	have	VERB
ejpam-3283	12	11	many	many	ADJ
ejpam-3283	12	12	applications	application	NOUN
ejpam-3283	12	13	,	,	PUNCT
ejpam-3283	12	14	for	for	ADP
ejpam-3283	12	15	instance	instance	NOUN
ejpam-3283	12	16	,	,	PUNCT
ejpam-3283	12	17	application	application	NOUN
ejpam-3283	12	18	in	in	ADP
ejpam-3283	12	19	physics	physics	NOUN
ejpam-3283	12	20	(	(	PUNCT
ejpam-3283	12	21	[	[	X
ejpam-3283	12	22	11	11	NUM
ejpam-3283	12	23	]	]	PUNCT
ejpam-3283	12	24	and	and	CCONJ
ejpam-3283	12	25	[	[	X
ejpam-3283	12	26	14	14	NUM
ejpam-3283	12	27	]	]	PUNCT
ejpam-3283	12	28	)	)	PUNCT
ejpam-3283	12	29	and	and	CCONJ
ejpam-3283	12	30	application	application	NOUN
ejpam-3283	12	31	in	in	ADP
ejpam-3283	12	32	automata	automata	NOUN
ejpam-3283	12	33	theory	theory	NOUN
ejpam-3283	12	34	[	[	X
ejpam-3283	12	35	8	8	NUM
ejpam-3283	12	36	]	]	PUNCT
ejpam-3283	12	37	.	.	PUNCT
ejpam-3283	13	1	recently	recently	ADV
ejpam-3283	13	2	,	,	PUNCT
ejpam-3283	13	3	solano	solano	PROPN
ejpam-3283	13	4	,	,	PUNCT
ejpam-3283	13	5	suebsung	suebsung	PROPN
ejpam-3283	13	6	and	and	CCONJ
ejpam-3283	13	7	chinram	chinram	PROPN
ejpam-3283	13	8	studied	study	VERB
ejpam-3283	13	9	ideals	ideal	NOUN
ejpam-3283	13	10	of	of	ADP
ejpam-3283	13	11	fuzzy	fuzzy	ADJ
ejpam-3283	13	12	points	point	NOUN
ejpam-3283	13	13	n	n	CCONJ
ejpam-3283	13	14	-	-	PUNCT
ejpam-3283	13	15	ary	ary	NOUN
ejpam-3283	13	16	semigroups	semigroup	NOUN
ejpam-3283	13	17	in	in	ADP
ejpam-3283	13	18	[	[	X
ejpam-3283	13	19	13	13	NUM
ejpam-3283	13	20	]	]	PUNCT
ejpam-3283	13	21	.	.	PUNCT
ejpam-3283	14	1	∗corresponding	∗corresponde	VERB
ejpam-3283	14	2	author	author	NOUN
ejpam-3283	14	3	.	.	PUNCT
ejpam-3283	15	1	doi	doi	NOUN
ejpam-3283	15	2	:	:	PUNCT
ejpam-3283	15	3	https://doi.org/10.29020/nybg.ejpam.v11i3.3283	https://doi.org/10.29020/nybg.ejpam.v11i3.3283	NUM
ejpam-3283	15	4	email	email	NOUN
ejpam-3283	15	5	addresses	address	NOUN
ejpam-3283	15	6	:	:	PUNCT
ejpam-3283	15	7	pattarawan.pe@gmail.com	pattarawan.pe@gmail.com	PROPN
ejpam-3283	15	8	(	(	PUNCT
ejpam-3283	15	9	p.	p.	NOUN
ejpam-3283	15	10	petchkaew	petchkaew	NOUN
ejpam-3283	15	11	)	)	PUNCT
ejpam-3283	15	12	,	,	PUNCT
ejpam-3283	15	13	ronnason.c@psu.ac.th	ronnason.c@psu.ac.th	PROPN
ejpam-3283	15	14	(	(	PUNCT
ejpam-3283	15	15	r.	r.	PROPN
ejpam-3283	15	16	chinram	chinram	PROPN
ejpam-3283	15	17	)	)	PUNCT
ejpam-3283	15	18	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3283	16	1	762	762	NUM
ejpam-3283	16	2	c	c	NOUN
ejpam-3283	16	3	©	©	PROPN
ejpam-3283	16	4	2018	2018	NUM
ejpam-3283	16	5	ejpam	ejpam	VERB
ejpam-3283	16	6	all	all	DET
ejpam-3283	16	7	rights	right	NOUN
ejpam-3283	16	8	reserved	reserve	VERB
ejpam-3283	16	9	.	.	PUNCT
ejpam-3283	17	1	p.	p.	NOUN
ejpam-3283	17	2	petchkaew	petchkaew	PROPN
ejpam-3283	17	3	,	,	PUNCT
ejpam-3283	17	4	r.	r.	PROPN
ejpam-3283	17	5	chinram	chinram	PROPN
ejpam-3283	17	6	/	/	SYM
ejpam-3283	17	7	eur	eur	PROPN
ejpam-3283	17	8	.	.	PUNCT
ejpam-3283	18	1	j.	j.	PROPN
ejpam-3283	18	2	pure	pure	PROPN
ejpam-3283	18	3	appl	appl	PROPN
ejpam-3283	18	4	.	.	PROPN
ejpam-3283	18	5	math	math	PROPN
ejpam-3283	18	6	,	,	PUNCT
ejpam-3283	18	7	11	11	NUM
ejpam-3283	18	8	(	(	PUNCT
ejpam-3283	18	9	3	3	NUM
ejpam-3283	18	10	)	)	PUNCT
ejpam-3283	18	11	(	(	PUNCT
ejpam-3283	18	12	2018	2018	NUM
ejpam-3283	18	13	)	)	PUNCT
ejpam-3283	18	14	,	,	PUNCT
ejpam-3283	18	15	762	762	NUM
ejpam-3283	18	16	-	-	SYM
ejpam-3283	18	17	773	773	NUM
ejpam-3283	18	18	763	763	NUM
ejpam-3283	18	19	in	in	ADP
ejpam-3283	18	20	2000	2000	NUM
ejpam-3283	18	21	,	,	PUNCT
ejpam-3283	18	22	cao	cao	PROPN
ejpam-3283	18	23	and	and	CCONJ
ejpam-3283	18	24	xu	xu	PROPN
ejpam-3283	18	25	studied	study	VERB
ejpam-3283	18	26	about	about	ADP
ejpam-3283	18	27	the	the	DET
ejpam-3283	18	28	minimal	minimal	ADJ
ejpam-3283	18	29	and	and	CCONJ
ejpam-3283	18	30	maximal	maximal	ADJ
ejpam-3283	18	31	left	leave	VERB
ejpam-3283	18	32	ideals	ideal	NOUN
ejpam-3283	18	33	in	in	ADP
ejpam-3283	18	34	ordered	order	VERB
ejpam-3283	18	35	semigroups	semigroup	NOUN
ejpam-3283	18	36	and	and	CCONJ
ejpam-3283	18	37	gave	give	VERB
ejpam-3283	18	38	some	some	DET
ejpam-3283	18	39	characterizations	characterization	NOUN
ejpam-3283	18	40	of	of	ADP
ejpam-3283	18	41	them	they	PRON
ejpam-3283	18	42	in	in	ADP
ejpam-3283	18	43	[	[	X
ejpam-3283	18	44	2	2	NUM
ejpam-3283	18	45	]	]	PUNCT
ejpam-3283	18	46	.	.	PUNCT
ejpam-3283	19	1	after	after	ADP
ejpam-3283	19	2	that	that	PRON
ejpam-3283	19	3	,	,	PUNCT
ejpam-3283	19	4	in	in	ADP
ejpam-3283	19	5	[	[	X
ejpam-3283	19	6	1	1	NUM
ejpam-3283	19	7	]	]	PUNCT
ejpam-3283	19	8	,	,	PUNCT
ejpam-3283	19	9	arslanov	arslanov	NOUN
ejpam-3283	19	10	and	and	CCONJ
ejpam-3283	19	11	kehayopulu	kehayopulu	VERB
ejpam-3283	19	12	characterized	characterize	VERB
ejpam-3283	19	13	the	the	DET
ejpam-3283	19	14	minimal	minimal	ADJ
ejpam-3283	19	15	and	and	CCONJ
ejpam-3283	19	16	maximal	maximal	ADJ
ejpam-3283	19	17	ideals	ideal	NOUN
ejpam-3283	19	18	in	in	ADP
ejpam-3283	19	19	ordered	order	VERB
ejpam-3283	19	20	semigroups	semigroup	NOUN
ejpam-3283	19	21	.	.	PUNCT
ejpam-3283	20	1	in	in	ADP
ejpam-3283	20	2	2010	2010	NUM
ejpam-3283	20	3	,	,	PUNCT
ejpam-3283	20	4	iampan	iampan	NOUN
ejpam-3283	20	5	gave	give	VERB
ejpam-3283	20	6	some	some	DET
ejpam-3283	20	7	characterization	characterization	NOUN
ejpam-3283	20	8	of	of	ADP
ejpam-3283	20	9	minimality	minimality	NOUN
ejpam-3283	20	10	and	and	CCONJ
ejpam-3283	20	11	maximality	maximality	NOUN
ejpam-3283	20	12	of	of	ADP
ejpam-3283	20	13	left	left	ADJ
ejpam-3283	20	14	ideals	ideal	NOUN
ejpam-3283	20	15	and	and	CCONJ
ejpam-3283	20	16	right	right	ADJ
ejpam-3283	20	17	ideals	ideal	NOUN
ejpam-3283	20	18	in	in	ADP
ejpam-3283	20	19	ternary	ternary	ADJ
ejpam-3283	20	20	semigroups	semigroup	NOUN
ejpam-3283	20	21	in	in	ADP
ejpam-3283	20	22	[	[	X
ejpam-3283	20	23	9	9	NUM
ejpam-3283	20	24	]	]	PUNCT
ejpam-3283	20	25	and	and	CCONJ
ejpam-3283	20	26	this	this	PRON
ejpam-3283	20	27	is	be	AUX
ejpam-3283	20	28	an	an	DET
ejpam-3283	20	29	our	our	PRON
ejpam-3283	20	30	motivation	motivation	NOUN
ejpam-3283	20	31	to	to	PART
ejpam-3283	20	32	do	do	AUX
ejpam-3283	20	33	this	this	DET
ejpam-3283	20	34	paper	paper	NOUN
ejpam-3283	20	35	.	.	PUNCT
ejpam-3283	21	1	in	in	ADP
ejpam-3283	21	2	this	this	DET
ejpam-3283	21	3	paper	paper	NOUN
ejpam-3283	21	4	,	,	PUNCT
ejpam-3283	21	5	we	we	PRON
ejpam-3283	21	6	extend	extend	VERB
ejpam-3283	21	7	those	those	DET
ejpam-3283	21	8	results	result	NOUN
ejpam-3283	21	9	in	in	ADP
ejpam-3283	21	10	[	[	X
ejpam-3283	21	11	9	9	NUM
ejpam-3283	21	12	]	]	PUNCT
ejpam-3283	21	13	to	to	ADP
ejpam-3283	21	14	n	n	CCONJ
ejpam-3283	21	15	-	-	PUNCT
ejpam-3283	21	16	ary	ary	PROPN
ejpam-3283	21	17	semigroups	semigroup	NOUN
ejpam-3283	21	18	.	.	PUNCT
ejpam-3283	22	1	we	we	PRON
ejpam-3283	22	2	investigate	investigate	VERB
ejpam-3283	22	3	the	the	DET
ejpam-3283	22	4	minimality	minimality	NOUN
ejpam-3283	22	5	and	and	CCONJ
ejpam-3283	22	6	maximality	maximality	NOUN
ejpam-3283	22	7	of	of	ADP
ejpam-3283	22	8	n	n	CCONJ
ejpam-3283	22	9	-	-	PUNCT
ejpam-3283	22	10	ideals	ideal	NOUN
ejpam-3283	22	11	in	in	ADP
ejpam-3283	22	12	n	n	CCONJ
ejpam-3283	22	13	-	-	PUNCT
ejpam-3283	22	14	ary	ary	NOUN
ejpam-3283	22	15	semigroups	semigroup	NOUN
ejpam-3283	22	16	and	and	CCONJ
ejpam-3283	22	17	give	give	VERB
ejpam-3283	22	18	some	some	DET
ejpam-3283	22	19	characterizations	characterization	NOUN
ejpam-3283	22	20	of	of	ADP
ejpam-3283	22	21	minimality	minimality	NOUN
ejpam-3283	22	22	and	and	CCONJ
ejpam-3283	22	23	maximality	maximality	NOUN
ejpam-3283	22	24	of	of	ADP
ejpam-3283	22	25	n	n	CCONJ
ejpam-3283	22	26	-	-	PUNCT
ejpam-3283	22	27	ideals	ideal	NOUN
ejpam-3283	22	28	in	in	ADP
ejpam-3283	22	29	n	n	CCONJ
ejpam-3283	22	30	-	-	PUNCT
ejpam-3283	22	31	ary	ary	NOUN
ejpam-3283	22	32	semigroups	semigroup	NOUN
ejpam-3283	22	33	.	.	PUNCT
ejpam-3283	23	1	2	2	X
ejpam-3283	23	2	.	.	NUM
ejpam-3283	23	3	preliminaries	preliminary	NOUN
ejpam-3283	23	4	for	for	ADP
ejpam-3283	23	5	the	the	DET
ejpam-3283	23	6	sake	sake	NOUN
ejpam-3283	23	7	of	of	ADP
ejpam-3283	23	8	completeness	completeness	NOUN
ejpam-3283	23	9	,	,	PUNCT
ejpam-3283	23	10	we	we	PRON
ejpam-3283	23	11	state	state	VERB
ejpam-3283	23	12	some	some	DET
ejpam-3283	23	13	definitions	definition	NOUN
ejpam-3283	23	14	in	in	ADP
ejpam-3283	23	15	the	the	DET
ejpam-3283	23	16	same	same	ADJ
ejpam-3283	23	17	fashion	fashion	NOUN
ejpam-3283	23	18	as	as	SCONJ
ejpam-3283	23	19	found	find	VERB
ejpam-3283	23	20	in	in	ADP
ejpam-3283	23	21	[	[	X
ejpam-3283	23	22	15	15	NUM
ejpam-3283	23	23	]	]	PUNCT
ejpam-3283	23	24	and	and	CCONJ
ejpam-3283	23	25	[	[	X
ejpam-3283	23	26	9	9	NUM
ejpam-3283	23	27	]	]	PUNCT
ejpam-3283	23	28	which	which	PRON
ejpam-3283	23	29	are	be	AUX
ejpam-3283	23	30	used	use	VERB
ejpam-3283	23	31	throughout	throughout	ADP
ejpam-3283	23	32	this	this	DET
ejpam-3283	23	33	paper	paper	NOUN
ejpam-3283	23	34	.	.	PUNCT
ejpam-3283	24	1	first	first	ADV
ejpam-3283	24	2	,	,	PUNCT
ejpam-3283	24	3	we	we	PRON
ejpam-3283	24	4	would	would	AUX
ejpam-3283	24	5	like	like	VERB
ejpam-3283	24	6	to	to	PART
ejpam-3283	24	7	introduce	introduce	VERB
ejpam-3283	24	8	the	the	DET
ejpam-3283	24	9	definition	definition	NOUN
ejpam-3283	24	10	of	of	ADP
ejpam-3283	24	11	n	n	CCONJ
ejpam-3283	24	12	-	-	PUNCT
ejpam-3283	24	13	ary	ary	NOUN
ejpam-3283	24	14	semigroup	semigroup	NOUN
ejpam-3283	24	15	which	which	PRON
ejpam-3283	24	16	was	be	AUX
ejpam-3283	24	17	stated	state	VERB
ejpam-3283	24	18	in	in	ADP
ejpam-3283	24	19	[	[	X
ejpam-3283	24	20	15	15	NUM
ejpam-3283	24	21	]	]	X
ejpam-3283	24	22	,	,	PUNCT
ejpam-3283	24	23	a	a	DET
ejpam-3283	24	24	nonempty	nonempty	ADJ
ejpam-3283	24	25	set	set	VERB
ejpam-3283	24	26	s	s	NOUN
ejpam-3283	24	27	together	together	ADV
ejpam-3283	24	28	with	with	ADP
ejpam-3283	24	29	an	an	DET
ejpam-3283	24	30	n	n	CCONJ
ejpam-3283	24	31	-	-	PUNCT
ejpam-3283	24	32	ary	ary	NOUN
ejpam-3283	24	33	operation	operation	NOUN
ejpam-3283	24	34	given	give	VERB
ejpam-3283	24	35	by	by	ADP
ejpam-3283	24	36	f	f	PROPN
ejpam-3283	24	37	:	:	PUNCT
ejpam-3283	24	38	sn	sn	PROPN
ejpam-3283	24	39	→	→	SYM
ejpam-3283	24	40	s	s	PROPN
ejpam-3283	24	41	,	,	PUNCT
ejpam-3283	24	42	where	where	SCONJ
ejpam-3283	24	43	n	n	PRON
ejpam-3283	24	44	≥	≥	NOUN
ejpam-3283	24	45	2	2	NUM
ejpam-3283	24	46	,	,	PUNCT
ejpam-3283	24	47	is	be	AUX
ejpam-3283	24	48	called	call	VERB
ejpam-3283	24	49	an	an	DET
ejpam-3283	24	50	n	n	CCONJ
ejpam-3283	24	51	-	-	PUNCT
ejpam-3283	24	52	ary	ary	NOUN
ejpam-3283	24	53	groupoid	groupoid	PROPN
ejpam-3283	24	54	and	and	CCONJ
ejpam-3283	24	55	is	be	AUX
ejpam-3283	24	56	denoted	denote	VERB
ejpam-3283	24	57	by	by	ADP
ejpam-3283	24	58	(	(	PUNCT
ejpam-3283	24	59	s	s	PROPN
ejpam-3283	24	60	,	,	PUNCT
ejpam-3283	24	61	f	f	NOUN
ejpam-3283	24	62	)	)	PUNCT
ejpam-3283	24	63	.	.	PUNCT
ejpam-3283	25	1	according	accord	VERB
ejpam-3283	25	2	to	to	ADP
ejpam-3283	25	3	the	the	DET
ejpam-3283	25	4	general	general	ADJ
ejpam-3283	25	5	convention	convention	NOUN
ejpam-3283	25	6	used	use	VERB
ejpam-3283	25	7	in	in	ADP
ejpam-3283	25	8	the	the	DET
ejpam-3283	25	9	theory	theory	NOUN
ejpam-3283	25	10	of	of	ADP
ejpam-3283	25	11	n	n	CCONJ
ejpam-3283	25	12	-	-	PUNCT
ejpam-3283	25	13	ary	ary	PROPN
ejpam-3283	25	14	groupoids	groupoid	NOUN
ejpam-3283	25	15	,	,	PUNCT
ejpam-3283	25	16	the	the	DET
ejpam-3283	25	17	sequence	sequence	NOUN
ejpam-3283	25	18	of	of	ADP
ejpam-3283	25	19	elements	element	NOUN
ejpam-3283	25	20	xi	xi	PROPN
ejpam-3283	25	21	,	,	PUNCT
ejpam-3283	25	22	xi+1	xi+1	PROPN
ejpam-3283	25	23	,	,	PUNCT
ejpam-3283	25	24	.	.	PUNCT
ejpam-3283	25	25	.	.	PUNCT
ejpam-3283	25	26	.	.	PUNCT
ejpam-3283	26	1	,	,	PUNCT
ejpam-3283	26	2	xj	xj	PROPN
ejpam-3283	26	3	is	be	AUX
ejpam-3283	26	4	denoted	denote	VERB
ejpam-3283	26	5	by	by	ADP
ejpam-3283	26	6	xji	xji	PROPN
ejpam-3283	26	7	.	.	PUNCT
ejpam-3283	27	1	in	in	ADP
ejpam-3283	27	2	the	the	DET
ejpam-3283	27	3	case	case	NOUN
ejpam-3283	27	4	j	j	X
ejpam-3283	27	5	<	<	X
ejpam-3283	27	6	i	i	X
ejpam-3283	27	7	,	,	PUNCT
ejpam-3283	27	8	it	it	PRON
ejpam-3283	27	9	is	be	AUX
ejpam-3283	27	10	the	the	DET
ejpam-3283	27	11	empty	empty	ADJ
ejpam-3283	27	12	symbol	symbol	NOUN
ejpam-3283	27	13	.	.	PUNCT
ejpam-3283	28	1	if	if	SCONJ
ejpam-3283	28	2	xi+1	xi+1	NUM
ejpam-3283	28	3	=	=	SYM
ejpam-3283	28	4	xi+2	xi+2	PUNCT
ejpam-3283	28	5	=	=	SYM
ejpam-3283	28	6	·	·	PUNCT
ejpam-3283	28	7	·	·	PUNCT
ejpam-3283	28	8	·	·	PUNCT
ejpam-3283	29	1	=	=	PUNCT
ejpam-3283	29	2	xi+t	xi+t	PUNCT
ejpam-3283	30	1	=	=	PUNCT
ejpam-3283	31	1	x	x	X
ejpam-3283	31	2	,	,	PUNCT
ejpam-3283	31	3	then	then	ADV
ejpam-3283	31	4	we	we	PRON
ejpam-3283	31	5	write	write	VERB
ejpam-3283	31	6	xt	xt	PROPN
ejpam-3283	31	7	instead	instead	ADV
ejpam-3283	31	8	of	of	ADP
ejpam-3283	31	9	xi+ti+1	xi+ti+1	PROPN
ejpam-3283	31	10	.	.	PROPN
ejpam-3283	32	1	in	in	ADP
ejpam-3283	32	2	this	this	DET
ejpam-3283	32	3	convention	convention	NOUN
ejpam-3283	32	4	,	,	PUNCT
ejpam-3283	32	5	f(x1	f(x1	NOUN
ejpam-3283	32	6	,	,	PUNCT
ejpam-3283	32	7	x2	x2	PROPN
ejpam-3283	32	8	,	,	PUNCT
ejpam-3283	32	9	.	.	PUNCT
ejpam-3283	32	10	.	.	PUNCT
ejpam-3283	32	11	.	.	PUNCT
ejpam-3283	33	1	,	,	PUNCT
ejpam-3283	33	2	xn	xn	X
ejpam-3283	33	3	)	)	PUNCT
ejpam-3283	33	4	=	=	SYM
ejpam-3283	33	5	f(xn1	f(xn1	PROPN
ejpam-3283	33	6	)	)	PUNCT
ejpam-3283	33	7	and	and	CCONJ
ejpam-3283	33	8	f(x1	f(x1	NOUN
ejpam-3283	33	9	,	,	PUNCT
ejpam-3283	33	10	.	.	PUNCT
ejpam-3283	33	11	.	.	PUNCT
ejpam-3283	33	12	.	.	PUNCT
ejpam-3283	34	1	,	,	PUNCT
ejpam-3283	34	2	xi	xi	X
ejpam-3283	34	3	,	,	PUNCT
ejpam-3283	34	4	x	x	X
ejpam-3283	34	5	.	.	PUNCT
ejpam-3283	34	6	.	.	PUNCT
ejpam-3283	34	7	.	.	PUNCT
ejpam-3283	35	1	,	,	PUNCT
ejpam-3283	35	2	x︸	x︸	VERB
ejpam-3283	35	3	︷︷	︷︷	PROPN
ejpam-3283	35	4	︸	︸	PRON
ejpam-3283	35	5	t	t	PROPN
ejpam-3283	35	6	,	,	PUNCT
ejpam-3283	35	7	xi+t+1	xi+t+1	X
ejpam-3283	35	8	,	,	PUNCT
ejpam-3283	35	9	.	.	PUNCT
ejpam-3283	35	10	.	.	PUNCT
ejpam-3283	36	1	.	.	PUNCT
ejpam-3283	37	1	,	,	PUNCT
ejpam-3283	37	2	xn	xn	X
ejpam-3283	37	3	)	)	PUNCT
ejpam-3283	37	4	=	=	SYM
ejpam-3283	37	5	f(xi1	f(xi1	NOUN
ejpam-3283	37	6	,	,	PUNCT
ejpam-3283	37	7	x	x	PROPN
ejpam-3283	37	8	t	t	PROPN
ejpam-3283	37	9	,	,	PUNCT
ejpam-3283	37	10	xni+t+1	xni+t+1	PROPN
ejpam-3283	37	11	)	)	PUNCT
ejpam-3283	37	12	.	.	PUNCT
ejpam-3283	38	1	an	an	PRON
ejpam-3283	38	2	n	n	NUM
ejpam-3283	38	3	-	-	PUNCT
ejpam-3283	38	4	ary	ary	NOUN
ejpam-3283	38	5	groupoid	groupoid	PROPN
ejpam-3283	38	6	(	(	PUNCT
ejpam-3283	38	7	s	s	PROPN
ejpam-3283	38	8	,	,	PUNCT
ejpam-3283	38	9	f	f	X
ejpam-3283	38	10	)	)	PUNCT
ejpam-3283	38	11	is	be	AUX
ejpam-3283	38	12	called	call	VERB
ejpam-3283	38	13	(	(	PUNCT
ejpam-3283	38	14	i	i	PROPN
ejpam-3283	38	15	,	,	PUNCT
ejpam-3283	38	16	j)-associative	j)-associative	ADJ
ejpam-3283	38	17	if	if	SCONJ
ejpam-3283	38	18	f(xi−1	f(xi−1	PROPN
ejpam-3283	38	19	1	1	NUM
ejpam-3283	38	20	,	,	PUNCT
ejpam-3283	38	21	f(xn+i−1	f(xn+i−1	PROPN
ejpam-3283	38	22	i	i	PROPN
ejpam-3283	38	23	)	)	PUNCT
ejpam-3283	38	24	,	,	PUNCT
ejpam-3283	38	25	x2n−1	x2n−1	PROPN
ejpam-3283	38	26	n+i	n+i	NUM
ejpam-3283	38	27	)	)	PUNCT
ejpam-3283	39	1	=	=	NOUN
ejpam-3283	39	2	f(xj−1	f(xj−1	NUM
ejpam-3283	39	3	1	1	NUM
ejpam-3283	39	4	,	,	PUNCT
ejpam-3283	39	5	f(xn+j−1	f(xn+j−1	NOUN
ejpam-3283	39	6	j	j	PROPN
ejpam-3283	39	7	)	)	PUNCT
ejpam-3283	39	8	,	,	PUNCT
ejpam-3283	39	9	x2n−1	x2n−1	PROPN
ejpam-3283	39	10	n+j	n+j	PROPN
ejpam-3283	39	11	)	)	PUNCT
ejpam-3283	39	12	hold	hold	VERB
ejpam-3283	39	13	for	for	ADP
ejpam-3283	39	14	all	all	DET
ejpam-3283	39	15	x1	x1	PROPN
ejpam-3283	39	16	,	,	PUNCT
ejpam-3283	39	17	x2	x2	PROPN
ejpam-3283	39	18	,	,	PUNCT
ejpam-3283	39	19	.	.	PUNCT
ejpam-3283	39	20	.	.	PUNCT
ejpam-3283	40	1	.	.	PUNCT
ejpam-3283	41	1	,	,	PUNCT
ejpam-3283	41	2	x2n−1	x2n−1	PROPN
ejpam-3283	41	3	∈	∈	PROPN
ejpam-3283	41	4	s.	s.	PROPN
ejpam-3283	42	1	the	the	DET
ejpam-3283	42	2	operation	operation	NOUN
ejpam-3283	42	3	f	f	PROPN
ejpam-3283	42	4	is	be	AUX
ejpam-3283	42	5	associative	associative	ADJ
ejpam-3283	42	6	if	if	SCONJ
ejpam-3283	42	7	the	the	DET
ejpam-3283	42	8	above	above	ADJ
ejpam-3283	42	9	identity	identity	NOUN
ejpam-3283	42	10	holds	hold	VERB
ejpam-3283	42	11	for	for	ADP
ejpam-3283	42	12	every	every	DET
ejpam-3283	42	13	1	1	NUM
ejpam-3283	42	14	≤	≤	NUM
ejpam-3283	42	15	i	i	PRON
ejpam-3283	42	16	≤	≤	NUM
ejpam-3283	42	17	j	j	PROPN
ejpam-3283	42	18	≤	≤	NUM
ejpam-3283	42	19	n	n	CCONJ
ejpam-3283	42	20	,	,	PUNCT
ejpam-3283	42	21	and	and	CCONJ
ejpam-3283	42	22	(	(	PUNCT
ejpam-3283	42	23	s	s	X
ejpam-3283	42	24	,	,	PUNCT
ejpam-3283	42	25	f	f	X
ejpam-3283	42	26	)	)	PUNCT
ejpam-3283	42	27	is	be	AUX
ejpam-3283	42	28	called	call	VERB
ejpam-3283	42	29	an	an	DET
ejpam-3283	42	30	n	n	CCONJ
ejpam-3283	42	31	-	-	PUNCT
ejpam-3283	42	32	ary	ary	NOUN
ejpam-3283	42	33	semigroup	semigroup	PROPN
ejpam-3283	42	34	.	.	PUNCT
ejpam-3283	43	1	a	a	DET
ejpam-3283	43	2	nonempty	nonempty	ADV
ejpam-3283	43	3	subset	subset	VERB
ejpam-3283	43	4	h	h	NOUN
ejpam-3283	43	5	of	of	ADP
ejpam-3283	43	6	an	an	DET
ejpam-3283	43	7	n	n	CCONJ
ejpam-3283	43	8	-	-	PUNCT
ejpam-3283	43	9	ary	ary	NOUN
ejpam-3283	43	10	semigroup	semigroup	NOUN
ejpam-3283	43	11	(	(	PUNCT
ejpam-3283	43	12	s	s	PROPN
ejpam-3283	43	13	,	,	PUNCT
ejpam-3283	43	14	f	f	X
ejpam-3283	43	15	)	)	PUNCT
ejpam-3283	43	16	is	be	AUX
ejpam-3283	43	17	called	call	VERB
ejpam-3283	43	18	an	an	DET
ejpam-3283	43	19	n	n	CCONJ
ejpam-3283	43	20	-	-	PUNCT
ejpam-3283	43	21	ary	ary	NOUN
ejpam-3283	43	22	subsemigroup	subsemigroup	NOUN
ejpam-3283	43	23	of	of	ADP
ejpam-3283	43	24	s	s	PRON
ejpam-3283	43	25	if	if	SCONJ
ejpam-3283	43	26	f(an1	f(an1	PROPN
ejpam-3283	43	27	)	)	PUNCT
ejpam-3283	43	28	∈	∈	PROPN
ejpam-3283	43	29	h	h	NOUN
ejpam-3283	43	30	for	for	ADP
ejpam-3283	43	31	all	all	DET
ejpam-3283	43	32	a1	a1	NOUN
ejpam-3283	43	33	,	,	PUNCT
ejpam-3283	43	34	a2	a2	PROPN
ejpam-3283	43	35	,	,	PUNCT
ejpam-3283	43	36	.	.	PUNCT
ejpam-3283	43	37	.	.	PUNCT
ejpam-3283	44	1	.	.	PUNCT
ejpam-3283	45	1	,	,	PUNCT
ejpam-3283	45	2	an	an	DET
ejpam-3283	45	3	∈	∈	PROPN
ejpam-3283	45	4	h.	h.	NOUN
ejpam-3283	45	5	a	a	DET
ejpam-3283	45	6	nonempty	nonempty	NOUN
ejpam-3283	45	7	subset	subset	VERB
ejpam-3283	45	8	i	i	PRON
ejpam-3283	45	9	of	of	ADP
ejpam-3283	45	10	s	s	PROPN
ejpam-3283	45	11	is	be	AUX
ejpam-3283	45	12	called	call	VERB
ejpam-3283	45	13	an	an	DET
ejpam-3283	45	14	i	i	NOUN
ejpam-3283	45	15	-	-	PUNCT
ejpam-3283	45	16	ideal	ideal	NOUN
ejpam-3283	45	17	of	of	ADP
ejpam-3283	45	18	s	s	PRON
ejpam-3283	45	19	if	if	SCONJ
ejpam-3283	45	20	for	for	ADP
ejpam-3283	45	21	every	every	DET
ejpam-3283	45	22	x1	x1	PROPN
ejpam-3283	45	23	,	,	PUNCT
ejpam-3283	45	24	...	...	PUNCT
ejpam-3283	45	25	,	,	PUNCT
ejpam-3283	45	26	xi−1	xi−1	PROPN
ejpam-3283	45	27	,	,	PUNCT
ejpam-3283	45	28	xi+1	xi+1	PROPN
ejpam-3283	45	29	,	,	PUNCT
ejpam-3283	45	30	...	...	PUNCT
ejpam-3283	45	31	,	,	PUNCT
ejpam-3283	45	32	xn	xn	PROPN
ejpam-3283	45	33	∈	∈	PROPN
ejpam-3283	45	34	s	s	VERB
ejpam-3283	45	35	with	with	ADP
ejpam-3283	45	36	a	a	DET
ejpam-3283	45	37	∈	∈	PROPN
ejpam-3283	46	1	i	i	PRON
ejpam-3283	46	2	,	,	PUNCT
ejpam-3283	46	3	then	then	ADV
ejpam-3283	46	4	f(xi−1	f(xi−1	PROPN
ejpam-3283	46	5	1	1	NUM
ejpam-3283	46	6	,	,	PUNCT
ejpam-3283	46	7	a	a	DET
ejpam-3283	46	8	,	,	PUNCT
ejpam-3283	46	9	xni+1	xni+1	PROPN
ejpam-3283	46	10	)	)	PUNCT
ejpam-3283	46	11	∈	∈	PROPN
ejpam-3283	46	12	i.	i.	NOUN
ejpam-3283	46	13	a	a	DET
ejpam-3283	46	14	nonempty	nonempty	NOUN
ejpam-3283	46	15	subset	subset	VERB
ejpam-3283	46	16	i	i	PRON
ejpam-3283	46	17	of	of	ADP
ejpam-3283	46	18	s	s	PROPN
ejpam-3283	46	19	is	be	AUX
ejpam-3283	46	20	called	call	VERB
ejpam-3283	46	21	an	an	DET
ejpam-3283	46	22	ideal	ideal	NOUN
ejpam-3283	46	23	of	of	ADP
ejpam-3283	46	24	s	s	PRON
ejpam-3283	46	25	if	if	SCONJ
ejpam-3283	46	26	i	i	PRON
ejpam-3283	46	27	is	be	AUX
ejpam-3283	46	28	an	an	DET
ejpam-3283	46	29	i	i	NOUN
ejpam-3283	46	30	-	-	PUNCT
ejpam-3283	46	31	ideal	ideal	NOUN
ejpam-3283	46	32	for	for	ADP
ejpam-3283	46	33	every	every	DET
ejpam-3283	46	34	1	1	NUM
ejpam-3283	46	35	≤	≤	NUM
ejpam-3283	46	36	i	i	PRON
ejpam-3283	46	37	≤	≤	ADJ
ejpam-3283	46	38	n.	n.	NOUN
ejpam-3283	46	39	for	for	ADP
ejpam-3283	46	40	nonempty	nonempty	ADJ
ejpam-3283	46	41	subset	subset	VERB
ejpam-3283	46	42	a1	a1	NOUN
ejpam-3283	46	43	,	,	PUNCT
ejpam-3283	46	44	a2	a2	PROPN
ejpam-3283	46	45	,	,	PUNCT
ejpam-3283	46	46	.	.	PUNCT
ejpam-3283	46	47	.	.	PUNCT
ejpam-3283	47	1	.	.	PUNCT
ejpam-3283	48	1	,	,	PUNCT
ejpam-3283	48	2	an	an	PRON
ejpam-3283	48	3	of	of	ADP
ejpam-3283	48	4	s	s	NOUN
ejpam-3283	48	5	,	,	PUNCT
ejpam-3283	48	6	let	let	VERB
ejpam-3283	48	7	f(an1	f(an1	PROPN
ejpam-3283	48	8	)	)	PUNCT
ejpam-3283	48	9	:	:	PUNCT
ejpam-3283	49	1	=	=	SYM
ejpam-3283	49	2	{	{	PUNCT
ejpam-3283	49	3	f(an1	f(an1	PROPN
ejpam-3283	49	4	)	)	PUNCT
ejpam-3283	49	5	|	|	ADV
ejpam-3283	49	6	ai	ai	INTJ
ejpam-3283	49	7	∈	∈	PROPN
ejpam-3283	49	8	ai	ai	VERB
ejpam-3283	49	9	for	for	ADP
ejpam-3283	49	10	all	all	PRON
ejpam-3283	49	11	i	i	PRON
ejpam-3283	49	12	∈	∈	PROPN
ejpam-3283	49	13	{	{	PUNCT
ejpam-3283	49	14	1	1	NUM
ejpam-3283	49	15	,	,	PUNCT
ejpam-3283	49	16	2	2	NUM
ejpam-3283	49	17	,	,	PUNCT
ejpam-3283	49	18	.	.	PUNCT
ejpam-3283	49	19	.	.	PUNCT
ejpam-3283	50	1	.	.	PUNCT
ejpam-3283	50	2	,	,	PUNCT
ejpam-3283	51	1	n	n	CCONJ
ejpam-3283	51	2	}	}	PUNCT
ejpam-3283	51	3	}	}	PUNCT
ejpam-3283	51	4	.	.	PUNCT
ejpam-3283	52	1	if	if	SCONJ
ejpam-3283	52	2	a1	a1	NOUN
ejpam-3283	52	3	=	=	SYM
ejpam-3283	52	4	{	{	PUNCT
ejpam-3283	52	5	a1	a1	NOUN
ejpam-3283	52	6	}	}	PUNCT
ejpam-3283	52	7	,	,	PUNCT
ejpam-3283	52	8	then	then	ADV
ejpam-3283	52	9	we	we	PRON
ejpam-3283	52	10	write	write	VERB
ejpam-3283	52	11	f({a1	f({a1	NOUN
ejpam-3283	52	12	}	}	PUNCT
ejpam-3283	52	13	,	,	PUNCT
ejpam-3283	52	14	an2	an2	PROPN
ejpam-3283	52	15	)	)	PUNCT
ejpam-3283	52	16	as	as	ADP
ejpam-3283	52	17	f(a1	f(a1	NOUN
ejpam-3283	52	18	,	,	PUNCT
ejpam-3283	52	19	a	a	DET
ejpam-3283	52	20	n	n	NOUN
ejpam-3283	52	21	2	2	NUM
ejpam-3283	52	22	)	)	PUNCT
ejpam-3283	52	23	,	,	PUNCT
ejpam-3283	52	24	and	and	CCONJ
ejpam-3283	52	25	similarly	similarly	ADV
ejpam-3283	52	26	in	in	ADP
ejpam-3283	52	27	another	another	DET
ejpam-3283	52	28	case	case	NOUN
ejpam-3283	52	29	such	such	ADJ
ejpam-3283	52	30	as	as	SCONJ
ejpam-3283	52	31	we	we	PRON
ejpam-3283	52	32	write	write	VERB
ejpam-3283	52	33	f({a1	f({a1	NOUN
ejpam-3283	52	34	}	}	PUNCT
ejpam-3283	52	35	,	,	PUNCT
ejpam-3283	52	36	an−1	an−1	PROPN
ejpam-3283	52	37	2	2	NUM
ejpam-3283	52	38	,	,	PUNCT
ejpam-3283	52	39	{	{	PUNCT
ejpam-3283	52	40	an	an	NOUN
ejpam-3283	52	41	}	}	PUNCT
ejpam-3283	52	42	)	)	PUNCT
ejpam-3283	52	43	as	as	ADP
ejpam-3283	52	44	f(a1	f(a1	NOUN
ejpam-3283	52	45	,	,	PUNCT
ejpam-3283	52	46	a	a	DET
ejpam-3283	52	47	n	n	DET
ejpam-3283	52	48	2	2	NUM
ejpam-3283	52	49	,	,	PUNCT
ejpam-3283	52	50	an	an	PRON
ejpam-3283	52	51	)	)	PUNCT
ejpam-3283	52	52	and	and	CCONJ
ejpam-3283	52	53	so	so	ADV
ejpam-3283	52	54	on	on	ADV
ejpam-3283	52	55	.	.	PUNCT
ejpam-3283	53	1	the	the	DET
ejpam-3283	53	2	intersection	intersection	NOUN
ejpam-3283	53	3	of	of	ADP
ejpam-3283	53	4	all	all	DET
ejpam-3283	53	5	n	n	CCONJ
ejpam-3283	53	6	-	-	PUNCT
ejpam-3283	53	7	ideals	ideal	NOUN
ejpam-3283	53	8	of	of	ADP
ejpam-3283	53	9	an	an	DET
ejpam-3283	53	10	n	n	CCONJ
ejpam-3283	53	11	-	-	PUNCT
ejpam-3283	53	12	ary	ary	NOUN
ejpam-3283	53	13	subsemigroup	subsemigroup	PROPN
ejpam-3283	53	14	h	h	PROPN
ejpam-3283	53	15	of	of	ADP
ejpam-3283	53	16	an	an	DET
ejpam-3283	53	17	n	n	CCONJ
ejpam-3283	53	18	-	-	PUNCT
ejpam-3283	53	19	ary	ary	NOUN
ejpam-3283	53	20	semigroup	semigroup	PROPN
ejpam-3283	53	21	s	s	AUX
ejpam-3283	53	22	containing	contain	VERB
ejpam-3283	53	23	a	a	DET
ejpam-3283	53	24	nonempty	nonempty	NOUN
ejpam-3283	53	25	subset	subset	VERB
ejpam-3283	53	26	a	a	PRON
ejpam-3283	53	27	of	of	ADP
ejpam-3283	53	28	h	h	NOUN
ejpam-3283	53	29	is	be	AUX
ejpam-3283	53	30	the	the	DET
ejpam-3283	53	31	n	n	CCONJ
ejpam-3283	53	32	-	-	PUNCT
ejpam-3283	53	33	ideal	ideal	NOUN
ejpam-3283	53	34	of	of	ADP
ejpam-3283	53	35	h	h	NOUN
ejpam-3283	53	36	generated	generate	VERB
ejpam-3283	53	37	by	by	ADP
ejpam-3283	53	38	a.	a.	NOUN
ejpam-3283	53	39	for	for	ADP
ejpam-3283	53	40	a	a	DET
ejpam-3283	53	41	=	=	X
ejpam-3283	53	42	{	{	PUNCT
ejpam-3283	53	43	a	a	NOUN
ejpam-3283	53	44	}	}	PUNCT
ejpam-3283	53	45	,	,	PUNCT
ejpam-3283	53	46	we	we	PRON
ejpam-3283	53	47	p.	p.	NOUN
ejpam-3283	53	48	petchkaew	petchkaew	VERB
ejpam-3283	53	49	,	,	PUNCT
ejpam-3283	53	50	r.	r.	PROPN
ejpam-3283	53	51	chinram	chinram	PROPN
ejpam-3283	53	52	/	/	SYM
ejpam-3283	53	53	eur	eur	PROPN
ejpam-3283	53	54	.	.	PUNCT
ejpam-3283	54	1	j.	j.	PROPN
ejpam-3283	54	2	pure	pure	PROPN
ejpam-3283	54	3	appl	appl	PROPN
ejpam-3283	54	4	.	.	PROPN
ejpam-3283	54	5	math	math	PROPN
ejpam-3283	54	6	,	,	PUNCT
ejpam-3283	54	7	11	11	NUM
ejpam-3283	54	8	(	(	PUNCT
ejpam-3283	54	9	3	3	NUM
ejpam-3283	54	10	)	)	PUNCT
ejpam-3283	54	11	(	(	PUNCT
ejpam-3283	54	12	2018	2018	NUM
ejpam-3283	54	13	)	)	PUNCT
ejpam-3283	54	14	,	,	PUNCT
ejpam-3283	54	15	762	762	NUM
ejpam-3283	54	16	-	-	SYM
ejpam-3283	54	17	773	773	NUM
ejpam-3283	54	18	764	764	NUM
ejpam-3283	54	19	donote	donote	NOUN
ejpam-3283	54	20	in	in	ADP
ejpam-3283	54	21	,	,	PUNCT
ejpam-3283	54	22	h(a	h(a	PROPN
ejpam-3283	54	23	)	)	PUNCT
ejpam-3283	54	24	to	to	PART
ejpam-3283	54	25	be	be	AUX
ejpam-3283	54	26	the	the	DET
ejpam-3283	54	27	n	n	CCONJ
ejpam-3283	54	28	-	-	PUNCT
ejpam-3283	54	29	ideal	ideal	NOUN
ejpam-3283	54	30	of	of	ADP
ejpam-3283	54	31	h	h	NOUN
ejpam-3283	54	32	generate	generate	NOUN
ejpam-3283	54	33	by	by	ADP
ejpam-3283	54	34	{	{	PUNCT
ejpam-3283	54	35	a	a	X
ejpam-3283	54	36	}	}	PUNCT
ejpam-3283	54	37	.	.	PUNCT
ejpam-3283	55	1	if	if	SCONJ
ejpam-3283	55	2	h	h	NOUN
ejpam-3283	55	3	=	=	SYM
ejpam-3283	55	4	s	s	PROPN
ejpam-3283	55	5	,	,	PUNCT
ejpam-3283	55	6	then	then	ADV
ejpam-3283	55	7	we	we	PRON
ejpam-3283	55	8	write	write	VERB
ejpam-3283	55	9	in	in	ADP
ejpam-3283	55	10	,	,	PUNCT
ejpam-3283	55	11	s(a	s(a	PROPN
ejpam-3283	55	12	)	)	PUNCT
ejpam-3283	55	13	as	as	ADP
ejpam-3283	55	14	in(a	in(a	NUM
ejpam-3283	55	15	)	)	PUNCT
ejpam-3283	55	16	.	.	PUNCT
ejpam-3283	56	1	an	an	DET
ejpam-3283	56	2	element	element	NOUN
ejpam-3283	56	3	a	a	PRON
ejpam-3283	56	4	of	of	ADP
ejpam-3283	56	5	an	an	DET
ejpam-3283	56	6	n	n	CCONJ
ejpam-3283	56	7	-	-	PUNCT
ejpam-3283	56	8	ary	ary	NOUN
ejpam-3283	56	9	semigroup	semigroup	PROPN
ejpam-3283	56	10	s	s	PROPN
ejpam-3283	56	11	with	with	ADP
ejpam-3283	56	12	at	at	ADV
ejpam-3283	56	13	least	least	ADV
ejpam-3283	56	14	two	two	NUM
ejpam-3283	56	15	elements	element	NOUN
ejpam-3283	56	16	is	be	AUX
ejpam-3283	56	17	called	call	VERB
ejpam-3283	56	18	zero	zero	NUM
ejpam-3283	56	19	element	element	NOUN
ejpam-3283	56	20	of	of	ADP
ejpam-3283	56	21	s	s	PRON
ejpam-3283	56	22	if	if	SCONJ
ejpam-3283	56	23	f(xi−1	f(xi−1	PROPN
ejpam-3283	56	24	1	1	NUM
ejpam-3283	56	25	,	,	PUNCT
ejpam-3283	56	26	a	a	DET
ejpam-3283	56	27	,	,	PUNCT
ejpam-3283	56	28	xni+1	xni+1	PROPN
ejpam-3283	56	29	)	)	PUNCT
ejpam-3283	56	30	=	=	SYM
ejpam-3283	56	31	a	a	PRON
ejpam-3283	56	32	for	for	ADP
ejpam-3283	56	33	all	all	DET
ejpam-3283	56	34	x1	x1	PROPN
ejpam-3283	56	35	,	,	PUNCT
ejpam-3283	56	36	x2	x2	PROPN
ejpam-3283	56	37	,	,	PUNCT
ejpam-3283	56	38	.	.	PUNCT
ejpam-3283	56	39	.	.	PUNCT
ejpam-3283	57	1	.	.	PUNCT
ejpam-3283	58	1	,	,	PUNCT
ejpam-3283	58	2	xn−1	xn−1	PROPN
ejpam-3283	58	3	,	,	PUNCT
ejpam-3283	58	4	xn+1	xn+1	PROPN
ejpam-3283	58	5	,	,	PUNCT
ejpam-3283	58	6	.	.	PUNCT
ejpam-3283	58	7	.	.	PUNCT
ejpam-3283	58	8	.	.	PUNCT
ejpam-3283	59	1	,	,	PUNCT
ejpam-3283	59	2	xn	xn	PUNCT
ejpam-3283	59	3	∈	∈	PROPN
ejpam-3283	59	4	s	s	PART
ejpam-3283	59	5	and	and	CCONJ
ejpam-3283	59	6	denote	denote	VERB
ejpam-3283	59	7	it	it	PRON
ejpam-3283	59	8	by	by	ADP
ejpam-3283	59	9	0	0	PROPN
ejpam-3283	59	10	.	.	PUNCT
ejpam-3283	60	1	if	if	SCONJ
ejpam-3283	60	2	an	an	DET
ejpam-3283	60	3	n	n	CCONJ
ejpam-3283	60	4	-	-	PUNCT
ejpam-3283	60	5	ary	ary	NOUN
ejpam-3283	60	6	semigroup	semigroup	PROPN
ejpam-3283	60	7	s	s	PROPN
ejpam-3283	60	8	contains	contain	VERB
ejpam-3283	60	9	a	a	DET
ejpam-3283	60	10	zero	zero	NUM
ejpam-3283	60	11	element	element	NOUN
ejpam-3283	60	12	,	,	PUNCT
ejpam-3283	60	13	then	then	ADV
ejpam-3283	60	14	every	every	DET
ejpam-3283	60	15	n	n	CCONJ
ejpam-3283	60	16	-	-	PUNCT
ejpam-3283	60	17	ideal	ideal	NOUN
ejpam-3283	60	18	of	of	ADP
ejpam-3283	60	19	s	s	PRON
ejpam-3283	60	20	also	also	ADV
ejpam-3283	60	21	contains	contain	VERB
ejpam-3283	60	22	a	a	DET
ejpam-3283	60	23	zero	zero	NUM
ejpam-3283	60	24	element	element	NOUN
ejpam-3283	60	25	.	.	PUNCT
ejpam-3283	61	1	an	an	DET
ejpam-3283	61	2	n	n	NUM
ejpam-3283	61	3	-	-	PUNCT
ejpam-3283	61	4	ary	ary	NOUN
ejpam-3283	61	5	semigroup	semigroup	PROPN
ejpam-3283	61	6	s	s	PROPN
ejpam-3283	61	7	without	without	ADP
ejpam-3283	61	8	zero	zero	NUM
ejpam-3283	61	9	is	be	AUX
ejpam-3283	61	10	called	call	VERB
ejpam-3283	61	11	n	n	CCONJ
ejpam-3283	61	12	-	-	PUNCT
ejpam-3283	61	13	simple	simple	ADJ
ejpam-3283	61	14	if	if	SCONJ
ejpam-3283	61	15	it	it	PRON
ejpam-3283	61	16	has	have	VERB
ejpam-3283	61	17	no	no	DET
ejpam-3283	61	18	proper	proper	ADJ
ejpam-3283	61	19	n	n	CCONJ
ejpam-3283	61	20	-	-	PUNCT
ejpam-3283	61	21	ideals	ideal	NOUN
ejpam-3283	61	22	.	.	PUNCT
ejpam-3283	62	1	an	an	DET
ejpam-3283	62	2	n	n	NUM
ejpam-3283	62	3	-	-	PUNCT
ejpam-3283	62	4	ary	ary	NOUN
ejpam-3283	62	5	semigroup	semigroup	PROPN
ejpam-3283	62	6	s	s	PROPN
ejpam-3283	62	7	with	with	ADP
ejpam-3283	62	8	zero	zero	NUM
ejpam-3283	62	9	is	be	AUX
ejpam-3283	62	10	called	call	VERB
ejpam-3283	62	11	0	0	NUM
ejpam-3283	62	12	-	-	PUNCT
ejpam-3283	62	13	n	n	CCONJ
ejpam-3283	62	14	-	-	PUNCT
ejpam-3283	62	15	simple	simple	ADJ
ejpam-3283	62	16	if	if	SCONJ
ejpam-3283	62	17	it	it	PRON
ejpam-3283	62	18	has	have	VERB
ejpam-3283	62	19	no	no	DET
ejpam-3283	62	20	nonzero	nonzero	NOUN
ejpam-3283	62	21	proper	proper	ADJ
ejpam-3283	62	22	n	n	CCONJ
ejpam-3283	62	23	-	-	PUNCT
ejpam-3283	62	24	ideals	ideal	NOUN
ejpam-3283	62	25	and	and	CCONJ
ejpam-3283	62	26	f(sn	f(sn	NOUN
ejpam-3283	62	27	)	)	PUNCT
ejpam-3283	62	28	6=	6=	PUNCT
ejpam-3283	62	29	{	{	PUNCT
ejpam-3283	62	30	0	0	NUM
ejpam-3283	62	31	}	}	PUNCT
ejpam-3283	62	32	.	.	PUNCT
ejpam-3283	63	1	an	an	DET
ejpam-3283	63	2	n	n	CCONJ
ejpam-3283	63	3	-	-	PUNCT
ejpam-3283	63	4	ideal	ideal	NOUN
ejpam-3283	63	5	i	i	PRON
ejpam-3283	63	6	of	of	ADP
ejpam-3283	63	7	an	an	DET
ejpam-3283	63	8	n	n	CCONJ
ejpam-3283	63	9	-	-	PUNCT
ejpam-3283	63	10	ary	ary	NOUN
ejpam-3283	63	11	semigroup	semigroup	PROPN
ejpam-3283	63	12	s	s	PROPN
ejpam-3283	63	13	without	without	ADP
ejpam-3283	63	14	zero	zero	NUM
ejpam-3283	63	15	is	be	AUX
ejpam-3283	63	16	called	call	VERB
ejpam-3283	63	17	a	a	DET
ejpam-3283	63	18	minimal	minimal	ADJ
ejpam-3283	63	19	n	n	CCONJ
ejpam-3283	63	20	-	-	PUNCT
ejpam-3283	63	21	ideal	ideal	NOUN
ejpam-3283	63	22	of	of	ADP
ejpam-3283	63	23	s	s	PRON
ejpam-3283	63	24	if	if	SCONJ
ejpam-3283	63	25	there	there	PRON
ejpam-3283	63	26	is	be	VERB
ejpam-3283	63	27	no	no	DET
ejpam-3283	63	28	n	n	CCONJ
ejpam-3283	63	29	-	-	PUNCT
ejpam-3283	63	30	ideal	ideal	NOUN
ejpam-3283	63	31	j	j	PROPN
ejpam-3283	63	32	of	of	ADP
ejpam-3283	63	33	s	s	PRON
ejpam-3283	63	34	such	such	ADJ
ejpam-3283	63	35	that	that	SCONJ
ejpam-3283	63	36	j	j	PROPN
ejpam-3283	63	37	(	(	PUNCT
ejpam-3283	63	38	i.	i.	PROPN
ejpam-3283	63	39	this	this	PRON
ejpam-3283	63	40	implies	imply	VERB
ejpam-3283	63	41	that	that	SCONJ
ejpam-3283	63	42	if	if	SCONJ
ejpam-3283	63	43	there	there	PRON
ejpam-3283	63	44	is	be	VERB
ejpam-3283	63	45	an	an	DET
ejpam-3283	63	46	n	n	CCONJ
ejpam-3283	63	47	-	-	PUNCT
ejpam-3283	63	48	ideal	ideal	NOUN
ejpam-3283	63	49	j	j	PROPN
ejpam-3283	63	50	of	of	ADP
ejpam-3283	63	51	s	s	PRON
ejpam-3283	63	52	such	such	ADJ
ejpam-3283	63	53	that	that	SCONJ
ejpam-3283	63	54	j	j	PROPN
ejpam-3283	63	55	⊆	⊆	NUM
ejpam-3283	63	56	i	i	PROPN
ejpam-3283	63	57	,	,	PUNCT
ejpam-3283	63	58	we	we	PRON
ejpam-3283	63	59	obtain	obtain	VERB
ejpam-3283	63	60	that	that	DET
ejpam-3283	63	61	j	j	PROPN
ejpam-3283	63	62	=	=	PROPN
ejpam-3283	63	63	i.	i.	PROPN
ejpam-3283	63	64	a	a	DET
ejpam-3283	63	65	nonzero	nonzero	PROPN
ejpam-3283	63	66	n	n	CCONJ
ejpam-3283	63	67	-	-	PUNCT
ejpam-3283	63	68	ideal	ideal	NOUN
ejpam-3283	63	69	i	i	PRON
ejpam-3283	63	70	of	of	ADP
ejpam-3283	63	71	an	an	DET
ejpam-3283	63	72	n	n	CCONJ
ejpam-3283	63	73	-	-	PUNCT
ejpam-3283	63	74	ary	ary	NOUN
ejpam-3283	63	75	semigroup	semigroup	PROPN
ejpam-3283	63	76	s	s	PROPN
ejpam-3283	63	77	with	with	ADP
ejpam-3283	63	78	zero	zero	NUM
ejpam-3283	63	79	is	be	AUX
ejpam-3283	63	80	called	call	VERB
ejpam-3283	63	81	a	a	DET
ejpam-3283	63	82	0	0	NUM
ejpam-3283	63	83	-	-	PUNCT
ejpam-3283	63	84	minimal	minimal	ADJ
ejpam-3283	63	85	n	n	CCONJ
ejpam-3283	63	86	-	-	PUNCT
ejpam-3283	63	87	ideal	ideal	NOUN
ejpam-3283	63	88	of	of	ADP
ejpam-3283	63	89	s	s	PRON
ejpam-3283	63	90	if	if	SCONJ
ejpam-3283	63	91	there	there	PRON
ejpam-3283	63	92	is	be	VERB
ejpam-3283	63	93	no	no	DET
ejpam-3283	63	94	nonzero	nonzero	ADJ
ejpam-3283	63	95	n	n	CCONJ
ejpam-3283	63	96	-	-	PUNCT
ejpam-3283	63	97	ideal	ideal	NOUN
ejpam-3283	63	98	j	j	PROPN
ejpam-3283	63	99	of	of	ADP
ejpam-3283	63	100	s	s	PRON
ejpam-3283	63	101	such	such	ADJ
ejpam-3283	63	102	that	that	SCONJ
ejpam-3283	63	103	j	j	PROPN
ejpam-3283	63	104	(	(	PUNCT
ejpam-3283	63	105	i.	i.	PROPN
ejpam-3283	63	106	equivalently	equivalently	ADV
ejpam-3283	63	107	,	,	PUNCT
ejpam-3283	63	108	if	if	SCONJ
ejpam-3283	63	109	s	s	PROPN
ejpam-3283	63	110	has	have	VERB
ejpam-3283	63	111	an	an	DET
ejpam-3283	63	112	n	n	CCONJ
ejpam-3283	63	113	-	-	PUNCT
ejpam-3283	63	114	ideal	ideal	NOUN
ejpam-3283	63	115	j	j	PROPN
ejpam-3283	63	116	such	such	ADJ
ejpam-3283	63	117	that	that	SCONJ
ejpam-3283	63	118	j	j	PROPN
ejpam-3283	63	119	(	(	PUNCT
ejpam-3283	63	120	i	i	PRON
ejpam-3283	63	121	,	,	PUNCT
ejpam-3283	63	122	we	we	PRON
ejpam-3283	63	123	acquire	acquire	VERB
ejpam-3283	63	124	that	that	DET
ejpam-3283	63	125	j	j	PROPN
ejpam-3283	64	1	=	=	PUNCT
ejpam-3283	64	2	{	{	PUNCT
ejpam-3283	64	3	0	0	NUM
ejpam-3283	64	4	}	}	PUNCT
ejpam-3283	64	5	.	.	PUNCT
ejpam-3283	65	1	a	a	DET
ejpam-3283	65	2	proper	proper	ADJ
ejpam-3283	65	3	n	n	CCONJ
ejpam-3283	65	4	-	-	PUNCT
ejpam-3283	65	5	ideal	ideal	NOUN
ejpam-3283	65	6	i	i	PRON
ejpam-3283	65	7	of	of	ADP
ejpam-3283	65	8	an	an	DET
ejpam-3283	65	9	n	n	CCONJ
ejpam-3283	65	10	-	-	PUNCT
ejpam-3283	65	11	ary	ary	NOUN
ejpam-3283	65	12	semigroup	semigroup	PROPN
ejpam-3283	65	13	s	s	PART
ejpam-3283	65	14	is	be	AUX
ejpam-3283	65	15	called	call	VERB
ejpam-3283	65	16	a	a	DET
ejpam-3283	65	17	maximal	maximal	ADJ
ejpam-3283	65	18	n	n	CCONJ
ejpam-3283	65	19	-	-	PUNCT
ejpam-3283	65	20	ideal	ideal	NOUN
ejpam-3283	65	21	of	of	ADP
ejpam-3283	65	22	s	s	PRON
ejpam-3283	65	23	if	if	SCONJ
ejpam-3283	65	24	for	for	ADP
ejpam-3283	65	25	any	any	DET
ejpam-3283	65	26	n	n	CCONJ
ejpam-3283	65	27	-	-	PUNCT
ejpam-3283	65	28	ideal	ideal	NOUN
ejpam-3283	65	29	j	j	PROPN
ejpam-3283	65	30	of	of	ADP
ejpam-3283	65	31	s	s	PRON
ejpam-3283	65	32	such	such	ADJ
ejpam-3283	65	33	that	that	SCONJ
ejpam-3283	65	34	i	i	PRON
ejpam-3283	65	35	(	(	PUNCT
ejpam-3283	65	36	j	j	PROPN
ejpam-3283	65	37	,	,	PUNCT
ejpam-3283	65	38	we	we	PRON
ejpam-3283	65	39	have	have	VERB
ejpam-3283	65	40	j	j	PROPN
ejpam-3283	65	41	=	=	PROPN
ejpam-3283	65	42	s.	s.	PROPN
ejpam-3283	65	43	equivalently	equivalently	PROPN
ejpam-3283	65	44	,	,	PUNCT
ejpam-3283	65	45	if	if	SCONJ
ejpam-3283	65	46	j	j	PROPN
ejpam-3283	65	47	is	be	AUX
ejpam-3283	65	48	a	a	DET
ejpam-3283	65	49	proper	proper	ADJ
ejpam-3283	65	50	n	n	CCONJ
ejpam-3283	65	51	-	-	PUNCT
ejpam-3283	65	52	ideal	ideal	NOUN
ejpam-3283	65	53	of	of	ADP
ejpam-3283	65	54	s	s	PRON
ejpam-3283	65	55	such	such	ADJ
ejpam-3283	65	56	that	that	SCONJ
ejpam-3283	65	57	i	i	PRON
ejpam-3283	65	58	⊆	⊆	NUM
ejpam-3283	65	59	j	j	PROPN
ejpam-3283	65	60	,	,	PUNCT
ejpam-3283	65	61	we	we	PRON
ejpam-3283	65	62	gain	gain	VERB
ejpam-3283	65	63	that	that	PRON
ejpam-3283	65	64	j	j	PROPN
ejpam-3283	65	65	=	=	PROPN
ejpam-3283	65	66	i.	i.	PROPN
ejpam-3283	65	67	3	3	NUM
ejpam-3283	65	68	.	.	PUNCT
ejpam-3283	65	69	main	main	ADJ
ejpam-3283	65	70	results	result	NOUN
ejpam-3283	65	71	throughout	throughout	ADP
ejpam-3283	65	72	this	this	DET
ejpam-3283	65	73	paper	paper	NOUN
ejpam-3283	65	74	,	,	PUNCT
ejpam-3283	65	75	s	s	VERB
ejpam-3283	65	76	is	be	AUX
ejpam-3283	65	77	assumed	assume	VERB
ejpam-3283	65	78	to	to	PART
ejpam-3283	65	79	be	be	AUX
ejpam-3283	65	80	an	an	DET
ejpam-3283	65	81	n	n	CCONJ
ejpam-3283	65	82	-	-	PUNCT
ejpam-3283	65	83	ary	ary	NOUN
ejpam-3283	65	84	semigroup	semigroup	NOUN
ejpam-3283	65	85	.	.	PUNCT
ejpam-3283	66	1	in	in	ADP
ejpam-3283	66	2	this	this	DET
ejpam-3283	66	3	section	section	NOUN
ejpam-3283	66	4	,	,	PUNCT
ejpam-3283	66	5	we	we	PRON
ejpam-3283	66	6	provide	provide	VERB
ejpam-3283	66	7	some	some	DET
ejpam-3283	66	8	idea	idea	NOUN
ejpam-3283	66	9	,	,	PUNCT
ejpam-3283	66	10	elementary	elementary	ADJ
ejpam-3283	66	11	properties	property	NOUN
ejpam-3283	66	12	and	and	CCONJ
ejpam-3283	66	13	some	some	DET
ejpam-3283	66	14	our	our	PRON
ejpam-3283	66	15	fundamental	fundamental	ADJ
ejpam-3283	66	16	results	result	NOUN
ejpam-3283	66	17	which	which	PRON
ejpam-3283	66	18	relate	relate	VERB
ejpam-3283	66	19	to	to	ADP
ejpam-3283	66	20	n	n	NOUN
ejpam-3283	66	21	-	-	PUNCT
ejpam-3283	66	22	ideals	ideal	NOUN
ejpam-3283	66	23	,	,	PUNCT
ejpam-3283	66	24	n	n	CCONJ
ejpam-3283	66	25	-	-	PUNCT
ejpam-3283	66	26	simples	simple	NOUN
ejpam-3283	66	27	,	,	PUNCT
ejpam-3283	66	28	and	and	CCONJ
ejpam-3283	66	29	0	0	NUM
ejpam-3283	66	30	-	-	PUNCT
ejpam-3283	66	31	n	n	CCONJ
ejpam-3283	66	32	-	-	PUNCT
ejpam-3283	66	33	simples	simple	NOUN
ejpam-3283	66	34	.	.	PUNCT
ejpam-3283	67	1	lemma	lemma	PROPN
ejpam-3283	67	2	1	1	X
ejpam-3283	67	3	.	.	PUNCT
ejpam-3283	68	1	let	let	VERB
ejpam-3283	68	2	a	a	DET
ejpam-3283	68	3	be	be	AUX
ejpam-3283	68	4	any	any	DET
ejpam-3283	68	5	nonempty	nonempty	NOUN
ejpam-3283	68	6	subset	subset	NOUN
ejpam-3283	68	7	of	of	ADP
ejpam-3283	68	8	s.	s.	PROPN
ejpam-3283	68	9	then	then	ADV
ejpam-3283	68	10	f(sn−1	f(sn−1	PROPN
ejpam-3283	68	11	,	,	PUNCT
ejpam-3283	68	12	a	a	DET
ejpam-3283	68	13	)	)	PUNCT
ejpam-3283	68	14	∪	∪	ADP
ejpam-3283	68	15	a	a	PRON
ejpam-3283	68	16	is	be	AUX
ejpam-3283	68	17	the	the	DET
ejpam-3283	68	18	smallest	small	ADJ
ejpam-3283	68	19	n	n	CCONJ
ejpam-3283	68	20	-	-	PUNCT
ejpam-3283	68	21	ideal	ideal	NOUN
ejpam-3283	68	22	of	of	ADP
ejpam-3283	68	23	s	s	AUX
ejpam-3283	68	24	containing	contain	VERB
ejpam-3283	68	25	a.	a.	NOUN
ejpam-3283	68	26	proof	proof	NOUN
ejpam-3283	68	27	.	.	PUNCT
ejpam-3283	69	1	first	first	ADV
ejpam-3283	69	2	,	,	PUNCT
ejpam-3283	69	3	we	we	PRON
ejpam-3283	69	4	show	show	VERB
ejpam-3283	69	5	that	that	SCONJ
ejpam-3283	69	6	f(sn−1	f(sn−1	ADP
ejpam-3283	69	7	,	,	PUNCT
ejpam-3283	69	8	a	a	PRON
ejpam-3283	69	9	)	)	PUNCT
ejpam-3283	69	10	∪a	∪a	NUM
ejpam-3283	69	11	is	be	AUX
ejpam-3283	69	12	an	an	DET
ejpam-3283	69	13	n	n	CCONJ
ejpam-3283	69	14	-	-	PUNCT
ejpam-3283	69	15	ideal	ideal	NOUN
ejpam-3283	69	16	of	of	ADP
ejpam-3283	69	17	s.	s.	PROPN
ejpam-3283	69	18	let	let	VERB
ejpam-3283	69	19	x1	x1	PROPN
ejpam-3283	69	20	,	,	PUNCT
ejpam-3283	69	21	x2	x2	PROPN
ejpam-3283	69	22	,	,	PUNCT
ejpam-3283	69	23	.	.	PUNCT
ejpam-3283	69	24	.	.	PUNCT
ejpam-3283	69	25	.	.	PUNCT
ejpam-3283	70	1	,	,	PUNCT
ejpam-3283	70	2	xn−1	xn−1	PROPN
ejpam-3283	70	3	∈	∈	PROPN
ejpam-3283	70	4	s	s	PART
ejpam-3283	70	5	and	and	CCONJ
ejpam-3283	70	6	y	y	PROPN
ejpam-3283	70	7	∈	∈	PROPN
ejpam-3283	70	8	f(sn−1	f(sn−1	PROPN
ejpam-3283	70	9	,	,	PUNCT
ejpam-3283	70	10	a	a	PRON
ejpam-3283	70	11	)	)	PUNCT
ejpam-3283	70	12	∪a	∪a	NUM
ejpam-3283	70	13	.	.	PUNCT
ejpam-3283	71	1	we	we	PRON
ejpam-3283	71	2	divide	divide	VERB
ejpam-3283	71	3	into	into	ADP
ejpam-3283	71	4	two	two	NUM
ejpam-3283	71	5	cases	case	NOUN
ejpam-3283	71	6	.	.	PUNCT
ejpam-3283	72	1	case	case	NOUN
ejpam-3283	72	2	1	1	NUM
ejpam-3283	72	3	:	:	PUNCT
ejpam-3283	72	4	if	if	SCONJ
ejpam-3283	72	5	y	y	PROPN
ejpam-3283	72	6	∈	∈	PROPN
ejpam-3283	72	7	f(sn−1	f(sn−1	PROPN
ejpam-3283	72	8	,	,	PUNCT
ejpam-3283	72	9	a	a	PRON
ejpam-3283	72	10	)	)	PUNCT
ejpam-3283	72	11	,	,	PUNCT
ejpam-3283	72	12	then	then	ADV
ejpam-3283	72	13	y	y	PROPN
ejpam-3283	72	14	=	=	PUNCT
ejpam-3283	72	15	f(sn−1	f(sn−1	PROPN
ejpam-3283	72	16	1	1	NUM
ejpam-3283	72	17	,	,	PUNCT
ejpam-3283	72	18	a	a	PRON
ejpam-3283	72	19	)	)	PUNCT
ejpam-3283	72	20	for	for	ADP
ejpam-3283	72	21	some	some	DET
ejpam-3283	72	22	s1	s1	NOUN
ejpam-3283	72	23	,	,	PUNCT
ejpam-3283	72	24	s2	s2	NOUN
ejpam-3283	72	25	,	,	PUNCT
ejpam-3283	72	26	.	.	PUNCT
ejpam-3283	72	27	.	.	PUNCT
ejpam-3283	72	28	.	.	PUNCT
ejpam-3283	73	1	,	,	PUNCT
ejpam-3283	73	2	sn−1	sn−1	PROPN
ejpam-3283	73	3	∈	∈	PROPN
ejpam-3283	73	4	s	s	X
ejpam-3283	73	5	and	and	CCONJ
ejpam-3283	73	6	for	for	ADP
ejpam-3283	73	7	some	some	PRON
ejpam-3283	73	8	a	a	DET
ejpam-3283	73	9	∈	∈	NOUN
ejpam-3283	73	10	a.	a.	NOUN
ejpam-3283	73	11	then	then	ADV
ejpam-3283	73	12	f(xn−1	f(xn−1	ADJ
ejpam-3283	73	13	1	1	NUM
ejpam-3283	73	14	,	,	PUNCT
ejpam-3283	73	15	y	y	PROPN
ejpam-3283	73	16	)	)	PUNCT
ejpam-3283	73	17	=	=	SYM
ejpam-3283	73	18	f(xn−1	f(xn−1	NOUN
ejpam-3283	73	19	1	1	NUM
ejpam-3283	73	20	,	,	PUNCT
ejpam-3283	73	21	f(sn−1	f(sn−1	ADV
ejpam-3283	73	22	1	1	NUM
ejpam-3283	73	23	,	,	PUNCT
ejpam-3283	73	24	a	a	NOUN
ejpam-3283	73	25	)	)	PUNCT
ejpam-3283	73	26	)	)	PUNCT
ejpam-3283	73	27	=	=	SYM
ejpam-3283	73	28	f(f(xn−1	f(f(xn−1	NOUN
ejpam-3283	73	29	1	1	NUM
ejpam-3283	73	30	,	,	PUNCT
ejpam-3283	73	31	s1	s1	NOUN
ejpam-3283	73	32	)	)	PUNCT
ejpam-3283	73	33	,	,	PUNCT
ejpam-3283	73	34	sn−1	sn−1	PROPN
ejpam-3283	73	35	2	2	NUM
ejpam-3283	73	36	,	,	PUNCT
ejpam-3283	73	37	a	a	PRON
ejpam-3283	73	38	)	)	PUNCT
ejpam-3283	73	39	∈	∈	PROPN
ejpam-3283	73	40	f(sn−1	f(sn−1	PROPN
ejpam-3283	73	41	,	,	PUNCT
ejpam-3283	73	42	a	a	PRON
ejpam-3283	73	43	)	)	PUNCT
ejpam-3283	73	44	⊆	⊆	NUM
ejpam-3283	73	45	f(sn−1	f(sn−1	PROPN
ejpam-3283	73	46	,	,	PUNCT
ejpam-3283	73	47	a	a	PRON
ejpam-3283	73	48	)	)	PUNCT
ejpam-3283	73	49	∪a	∪a	NUM
ejpam-3283	73	50	.	.	PUNCT
ejpam-3283	74	1	case	case	NOUN
ejpam-3283	74	2	2	2	NUM
ejpam-3283	74	3	:	:	PUNCT
ejpam-3283	75	1	if	if	SCONJ
ejpam-3283	75	2	y	y	PROPN
ejpam-3283	75	3	∈	∈	PROPN
ejpam-3283	75	4	a	a	PRON
ejpam-3283	75	5	,	,	PUNCT
ejpam-3283	75	6	then	then	ADV
ejpam-3283	75	7	f(xn−1	f(xn−1	ADJ
ejpam-3283	75	8	1	1	NUM
ejpam-3283	75	9	,	,	PUNCT
ejpam-3283	75	10	y	y	PROPN
ejpam-3283	75	11	)	)	PUNCT
ejpam-3283	75	12	∈	∈	PROPN
ejpam-3283	75	13	f(sn−1	f(sn−1	PROPN
ejpam-3283	75	14	,	,	PUNCT
ejpam-3283	75	15	a	a	PRON
ejpam-3283	75	16	)	)	PUNCT
ejpam-3283	75	17	⊆	⊆	NUM
ejpam-3283	75	18	f(sn−1	f(sn−1	PROPN
ejpam-3283	75	19	,	,	PUNCT
ejpam-3283	75	20	a	a	PRON
ejpam-3283	75	21	)	)	PUNCT
ejpam-3283	75	22	∪a	∪a	NUM
ejpam-3283	75	23	.	.	PUNCT
ejpam-3283	76	1	from	from	ADP
ejpam-3283	76	2	case	case	NOUN
ejpam-3283	76	3	1	1	NUM
ejpam-3283	76	4	and	and	CCONJ
ejpam-3283	76	5	case	case	NOUN
ejpam-3283	76	6	2	2	NUM
ejpam-3283	76	7	,	,	PUNCT
ejpam-3283	76	8	we	we	PRON
ejpam-3283	76	9	can	can	AUX
ejpam-3283	76	10	conclude	conclude	VERB
ejpam-3283	76	11	that	that	PRON
ejpam-3283	76	12	f(sn−1	f(sn−1	ADP
ejpam-3283	76	13	,	,	PUNCT
ejpam-3283	76	14	a	a	PRON
ejpam-3283	76	15	)	)	PUNCT
ejpam-3283	76	16	∪a	∪a	NUM
ejpam-3283	76	17	is	be	AUX
ejpam-3283	76	18	a	a	DET
ejpam-3283	76	19	n	n	CCONJ
ejpam-3283	76	20	-	-	PUNCT
ejpam-3283	76	21	ideal	ideal	NOUN
ejpam-3283	76	22	of	of	ADP
ejpam-3283	76	23	s.	s.	PROPN
ejpam-3283	76	24	next	next	ADV
ejpam-3283	76	25	,	,	PUNCT
ejpam-3283	76	26	we	we	PRON
ejpam-3283	76	27	show	show	VERB
ejpam-3283	76	28	that	that	SCONJ
ejpam-3283	76	29	f(sn−1	f(sn−1	ADP
ejpam-3283	76	30	,	,	PUNCT
ejpam-3283	76	31	a	a	PRON
ejpam-3283	76	32	)	)	PUNCT
ejpam-3283	76	33	∪	∪	ADP
ejpam-3283	76	34	a	a	PRON
ejpam-3283	76	35	is	be	AUX
ejpam-3283	76	36	a	a	DET
ejpam-3283	76	37	smallest	small	ADJ
ejpam-3283	76	38	n	n	CCONJ
ejpam-3283	76	39	-	-	PUNCT
ejpam-3283	76	40	ideal	ideal	NOUN
ejpam-3283	76	41	of	of	ADP
ejpam-3283	76	42	s	s	AUX
ejpam-3283	76	43	containing	contain	VERB
ejpam-3283	76	44	a.	a.	NOUN
ejpam-3283	76	45	let	let	VERB
ejpam-3283	76	46	i	i	PRON
ejpam-3283	76	47	be	be	AUX
ejpam-3283	76	48	any	any	DET
ejpam-3283	76	49	n	n	CCONJ
ejpam-3283	76	50	-	-	PUNCT
ejpam-3283	76	51	ideal	ideal	NOUN
ejpam-3283	76	52	of	of	ADP
ejpam-3283	76	53	s	s	AUX
ejpam-3283	76	54	containing	contain	VERB
ejpam-3283	76	55	a.	a.	NOUN
ejpam-3283	76	56	let	let	VERB
ejpam-3283	76	57	y	y	PROPN
ejpam-3283	76	58	∈	∈	PROPN
ejpam-3283	76	59	f(sn−1	f(sn−1	PROPN
ejpam-3283	76	60	,	,	PUNCT
ejpam-3283	76	61	a	a	PRON
ejpam-3283	76	62	)	)	PUNCT
ejpam-3283	76	63	∪	∪	ADP
ejpam-3283	76	64	a.	a.	NOUN
ejpam-3283	76	65	if	if	SCONJ
ejpam-3283	76	66	y	y	PROPN
ejpam-3283	76	67	∈	∈	PROPN
ejpam-3283	76	68	a	a	PRON
ejpam-3283	76	69	,	,	PUNCT
ejpam-3283	76	70	then	then	ADV
ejpam-3283	76	71	y	y	PROPN
ejpam-3283	76	72	∈	∈	PROPN
ejpam-3283	77	1	i	i	PRON
ejpam-3283	77	2	because	because	SCONJ
ejpam-3283	77	3	a	a	DET
ejpam-3283	77	4	⊆	⊆	NUM
ejpam-3283	77	5	i.	i.	NOUN
ejpam-3283	77	6	if	if	SCONJ
ejpam-3283	77	7	y	y	PROPN
ejpam-3283	77	8	∈	∈	PROPN
ejpam-3283	77	9	f(sn−1	f(sn−1	PROPN
ejpam-3283	77	10	,	,	PUNCT
ejpam-3283	77	11	a	a	PRON
ejpam-3283	77	12	)	)	PUNCT
ejpam-3283	77	13	,	,	PUNCT
ejpam-3283	77	14	then	then	ADV
ejpam-3283	77	15	y	y	PROPN
ejpam-3283	77	16	=	=	PUNCT
ejpam-3283	77	17	f(sn−1	f(sn−1	PROPN
ejpam-3283	77	18	1	1	NUM
ejpam-3283	77	19	,	,	PUNCT
ejpam-3283	77	20	a	a	PRON
ejpam-3283	77	21	)	)	PUNCT
ejpam-3283	77	22	for	for	ADP
ejpam-3283	77	23	some	some	DET
ejpam-3283	77	24	s1	s1	NOUN
ejpam-3283	77	25	,	,	PUNCT
ejpam-3283	77	26	s2	s2	NOUN
ejpam-3283	77	27	,	,	PUNCT
ejpam-3283	77	28	.	.	PUNCT
ejpam-3283	77	29	.	.	PUNCT
ejpam-3283	77	30	.	.	PUNCT
ejpam-3283	78	1	,	,	PUNCT
ejpam-3283	78	2	sn−1	sn−1	PROPN
ejpam-3283	78	3	∈	∈	PROPN
ejpam-3283	78	4	s	s	X
ejpam-3283	78	5	and	and	CCONJ
ejpam-3283	78	6	for	for	ADP
ejpam-3283	78	7	some	some	PRON
ejpam-3283	78	8	a	a	DET
ejpam-3283	78	9	∈	∈	NOUN
ejpam-3283	78	10	a.	a.	NOUN
ejpam-3283	78	11	thus	thus	ADV
ejpam-3283	78	12	a	a	DET
ejpam-3283	78	13	∈	∈	NOUN
ejpam-3283	78	14	i	i	PRON
ejpam-3283	78	15	because	because	SCONJ
ejpam-3283	78	16	a	a	DET
ejpam-3283	78	17	⊆	⊆	NUM
ejpam-3283	78	18	i.	i.	NOUN
ejpam-3283	78	19	hence	hence	ADV
ejpam-3283	78	20	y	y	PROPN
ejpam-3283	78	21	=	=	PUNCT
ejpam-3283	78	22	f(sn−1	f(sn−1	PROPN
ejpam-3283	78	23	1	1	NUM
ejpam-3283	78	24	,	,	PUNCT
ejpam-3283	78	25	a	a	PRON
ejpam-3283	78	26	)	)	PUNCT
ejpam-3283	78	27	∈	∈	NOUN
ejpam-3283	79	1	i	i	PRON
ejpam-3283	79	2	since	since	SCONJ
ejpam-3283	79	3	i	i	PRON
ejpam-3283	79	4	is	be	AUX
ejpam-3283	79	5	an	an	DET
ejpam-3283	79	6	n	n	CCONJ
ejpam-3283	79	7	-	-	PUNCT
ejpam-3283	79	8	ideal	ideal	NOUN
ejpam-3283	79	9	of	of	ADP
ejpam-3283	79	10	s.	s.	PROPN
ejpam-3283	79	11	therefore	therefore	ADV
ejpam-3283	79	12	,	,	PUNCT
ejpam-3283	79	13	we	we	PRON
ejpam-3283	79	14	obtain	obtain	VERB
ejpam-3283	79	15	f(sn−1	f(sn−1	ADP
ejpam-3283	79	16	,	,	PUNCT
ejpam-3283	79	17	a	a	PRON
ejpam-3283	79	18	)	)	PUNCT
ejpam-3283	79	19	∪	∪	ADP
ejpam-3283	79	20	a	a	DET
ejpam-3283	79	21	⊆	⊆	NUM
ejpam-3283	79	22	i.	i.	NOUN
ejpam-3283	79	23	since	since	SCONJ
ejpam-3283	79	24	i	i	PRON
ejpam-3283	79	25	is	be	AUX
ejpam-3283	79	26	an	an	DET
ejpam-3283	79	27	arbitrary	arbitrary	ADJ
ejpam-3283	79	28	n	n	CCONJ
ejpam-3283	79	29	-	-	PUNCT
ejpam-3283	79	30	ideal	ideal	NOUN
ejpam-3283	79	31	of	of	ADP
ejpam-3283	79	32	s	s	AUX
ejpam-3283	79	33	containing	contain	VERB
ejpam-3283	79	34	a	a	PRON
ejpam-3283	79	35	,	,	PUNCT
ejpam-3283	79	36	we	we	PRON
ejpam-3283	79	37	obtain	obtain	VERB
ejpam-3283	79	38	that	that	PRON
ejpam-3283	79	39	f(sn−1	f(sn−1	ADP
ejpam-3283	79	40	,	,	PUNCT
ejpam-3283	79	41	a	a	DET
ejpam-3283	79	42	)	)	PUNCT
ejpam-3283	79	43	∪a	∪a	NUM
ejpam-3283	79	44	is	be	AUX
ejpam-3283	79	45	a	a	DET
ejpam-3283	79	46	smallest	small	ADJ
ejpam-3283	79	47	n	n	CCONJ
ejpam-3283	79	48	-	-	PUNCT
ejpam-3283	79	49	ideal	ideal	NOUN
ejpam-3283	79	50	of	of	ADP
ejpam-3283	79	51	s	s	AUX
ejpam-3283	79	52	containing	contain	VERB
ejpam-3283	79	53	a.	a.	NOUN
ejpam-3283	79	54	corollary	corollary	NOUN
ejpam-3283	79	55	1	1	PROPN
ejpam-3283	79	56	.	.	PUNCT
ejpam-3283	80	1	for	for	ADP
ejpam-3283	80	2	any	any	DET
ejpam-3283	80	3	an	an	DET
ejpam-3283	80	4	element	element	NOUN
ejpam-3283	80	5	a	a	PRON
ejpam-3283	80	6	of	of	ADP
ejpam-3283	80	7	s	s	NOUN
ejpam-3283	80	8	,	,	PUNCT
ejpam-3283	80	9	in(a	in(a	NUM
ejpam-3283	80	10	)	)	PUNCT
ejpam-3283	80	11	=	=	SYM
ejpam-3283	80	12	f(sn−1	f(sn−1	PROPN
ejpam-3283	80	13	,	,	PUNCT
ejpam-3283	80	14	a	a	PRON
ejpam-3283	80	15	)	)	PUNCT
ejpam-3283	80	16	∪	∪	NOUN
ejpam-3283	80	17	{	{	PUNCT
ejpam-3283	80	18	a	a	NOUN
ejpam-3283	80	19	}	}	PUNCT
ejpam-3283	80	20	.	.	PUNCT
ejpam-3283	81	1	proof	proof	NOUN
ejpam-3283	81	2	.	.	PUNCT
ejpam-3283	82	1	this	this	PRON
ejpam-3283	82	2	follows	follow	VERB
ejpam-3283	82	3	from	from	ADP
ejpam-3283	82	4	lemma	lemma	PROPN
ejpam-3283	82	5	1	1	NUM
ejpam-3283	82	6	.	.	PUNCT
ejpam-3283	83	1	p.	p.	NOUN
ejpam-3283	83	2	petchkaew	petchkaew	NOUN
ejpam-3283	83	3	,	,	PUNCT
ejpam-3283	83	4	r.	r.	PROPN
ejpam-3283	83	5	chinram	chinram	PROPN
ejpam-3283	83	6	/	/	SYM
ejpam-3283	83	7	eur	eur	PROPN
ejpam-3283	83	8	.	.	PUNCT
ejpam-3283	84	1	j.	j.	PROPN
ejpam-3283	84	2	pure	pure	PROPN
ejpam-3283	84	3	appl	appl	PROPN
ejpam-3283	84	4	.	.	PROPN
ejpam-3283	84	5	math	math	PROPN
ejpam-3283	84	6	,	,	PUNCT
ejpam-3283	84	7	11	11	NUM
ejpam-3283	84	8	(	(	PUNCT
ejpam-3283	84	9	3	3	NUM
ejpam-3283	84	10	)	)	PUNCT
ejpam-3283	84	11	(	(	PUNCT
ejpam-3283	84	12	2018	2018	NUM
ejpam-3283	84	13	)	)	PUNCT
ejpam-3283	84	14	,	,	PUNCT
ejpam-3283	84	15	762	762	NUM
ejpam-3283	84	16	-	-	SYM
ejpam-3283	84	17	773	773	NUM
ejpam-3283	84	18	765	765	NUM
ejpam-3283	84	19	lemma	lemma	PROPN
ejpam-3283	84	20	2	2	NUM
ejpam-3283	84	21	.	.	PUNCT
ejpam-3283	84	22	let	let	VERB
ejpam-3283	84	23	a	a	DET
ejpam-3283	84	24	be	be	AUX
ejpam-3283	84	25	any	any	DET
ejpam-3283	84	26	nonempty	nonempty	NOUN
ejpam-3283	84	27	subset	subset	NOUN
ejpam-3283	84	28	of	of	ADP
ejpam-3283	84	29	s.	s.	PROPN
ejpam-3283	84	30	then	then	ADV
ejpam-3283	84	31	f(sn−1	f(sn−1	PROPN
ejpam-3283	84	32	,	,	PUNCT
ejpam-3283	84	33	a	a	DET
ejpam-3283	84	34	)	)	PUNCT
ejpam-3283	84	35	is	be	AUX
ejpam-3283	84	36	an	an	DET
ejpam-3283	84	37	n	n	CCONJ
ejpam-3283	84	38	-	-	PUNCT
ejpam-3283	84	39	ideal	ideal	NOUN
ejpam-3283	84	40	of	of	ADP
ejpam-3283	84	41	s.	s.	PROPN
ejpam-3283	84	42	proof	proof	PROPN
ejpam-3283	84	43	.	.	PUNCT
ejpam-3283	85	1	this	this	PRON
ejpam-3283	85	2	follows	follow	VERB
ejpam-3283	85	3	from	from	ADP
ejpam-3283	85	4	one	one	NUM
ejpam-3283	85	5	of	of	ADP
ejpam-3283	85	6	the	the	DET
ejpam-3283	85	7	proof	proof	NOUN
ejpam-3283	85	8	of	of	ADP
ejpam-3283	85	9	lemma	lemma	PROPN
ejpam-3283	85	10	1	1	NUM
ejpam-3283	85	11	.	.	PUNCT
ejpam-3283	86	1	lemma	lemma	PROPN
ejpam-3283	86	2	3	3	X
ejpam-3283	86	3	.	.	PUNCT
ejpam-3283	87	1	if	if	SCONJ
ejpam-3283	87	2	s	s	PROPN
ejpam-3283	87	3	has	have	VERB
ejpam-3283	87	4	no	no	DET
ejpam-3283	87	5	zero	zero	NUM
ejpam-3283	87	6	element	element	NOUN
ejpam-3283	87	7	,	,	PUNCT
ejpam-3283	87	8	then	then	ADV
ejpam-3283	87	9	the	the	DET
ejpam-3283	87	10	following	following	ADJ
ejpam-3283	87	11	statements	statement	NOUN
ejpam-3283	87	12	are	be	AUX
ejpam-3283	87	13	equivalent	equivalent	ADJ
ejpam-3283	87	14	:	:	PUNCT
ejpam-3283	87	15	(	(	PUNCT
ejpam-3283	87	16	1	1	X
ejpam-3283	87	17	)	)	PUNCT
ejpam-3283	87	18	s	s	VERB
ejpam-3283	87	19	is	be	AUX
ejpam-3283	87	20	n	n	PRON
ejpam-3283	87	21	-	-	PUNCT
ejpam-3283	87	22	simple	simple	NOUN
ejpam-3283	87	23	.	.	PUNCT
ejpam-3283	88	1	(	(	PUNCT
ejpam-3283	88	2	2	2	NUM
ejpam-3283	88	3	)	)	PUNCT
ejpam-3283	88	4	f(sn−1	f(sn−1	NOUN
ejpam-3283	88	5	,	,	PUNCT
ejpam-3283	88	6	a	a	PRON
ejpam-3283	88	7	)	)	PUNCT
ejpam-3283	88	8	=	=	SYM
ejpam-3283	88	9	s	s	PROPN
ejpam-3283	88	10	for	for	ADP
ejpam-3283	88	11	all	all	DET
ejpam-3283	88	12	a	a	DET
ejpam-3283	88	13	∈	∈	NOUN
ejpam-3283	88	14	s.	s.	PROPN
ejpam-3283	88	15	(	(	PUNCT
ejpam-3283	88	16	3	3	NUM
ejpam-3283	88	17	)	)	PUNCT
ejpam-3283	88	18	in(a	in(a	PUNCT
ejpam-3283	88	19	)	)	PUNCT
ejpam-3283	89	1	=	=	SYM
ejpam-3283	89	2	s	s	PROPN
ejpam-3283	89	3	for	for	ADP
ejpam-3283	89	4	all	all	DET
ejpam-3283	89	5	a	a	DET
ejpam-3283	89	6	∈	∈	PROPN
ejpam-3283	89	7	s.	s.	PROPN
ejpam-3283	89	8	proof	proof	NOUN
ejpam-3283	89	9	.	.	PUNCT
ejpam-3283	90	1	first	first	ADV
ejpam-3283	90	2	,	,	PUNCT
ejpam-3283	90	3	we	we	PRON
ejpam-3283	90	4	show	show	VERB
ejpam-3283	90	5	(	(	PUNCT
ejpam-3283	90	6	1)⇒	1)⇒	NUM
ejpam-3283	90	7	(	(	PUNCT
ejpam-3283	90	8	2	2	NUM
ejpam-3283	90	9	)	)	PUNCT
ejpam-3283	90	10	.	.	PUNCT
ejpam-3283	91	1	assume	assume	VERB
ejpam-3283	91	2	that	that	SCONJ
ejpam-3283	91	3	s	s	VERB
ejpam-3283	91	4	is	be	AUX
ejpam-3283	91	5	n	n	PRON
ejpam-3283	91	6	-	-	PUNCT
ejpam-3283	91	7	simple	simple	NOUN
ejpam-3283	91	8	.	.	PUNCT
ejpam-3283	92	1	by	by	ADP
ejpam-3283	92	2	lemma	lemma	PROPN
ejpam-3283	92	3	2	2	NUM
ejpam-3283	92	4	,	,	PUNCT
ejpam-3283	92	5	f(sn−1	f(sn−1	PROPN
ejpam-3283	92	6	,	,	PUNCT
ejpam-3283	92	7	a	a	PRON
ejpam-3283	92	8	)	)	PUNCT
ejpam-3283	92	9	is	be	AUX
ejpam-3283	92	10	an	an	DET
ejpam-3283	92	11	n	n	CCONJ
ejpam-3283	92	12	-	-	PUNCT
ejpam-3283	92	13	ideal	ideal	NOUN
ejpam-3283	92	14	of	of	ADP
ejpam-3283	92	15	s	s	PRON
ejpam-3283	92	16	for	for	ADP
ejpam-3283	92	17	all	all	DET
ejpam-3283	92	18	a	a	DET
ejpam-3283	92	19	∈	∈	NOUN
ejpam-3283	92	20	s.	s.	PROPN
ejpam-3283	92	21	hence	hence	ADV
ejpam-3283	92	22	f(sn−1	f(sn−1	PROPN
ejpam-3283	92	23	,	,	PUNCT
ejpam-3283	92	24	a	a	PRON
ejpam-3283	92	25	)	)	PUNCT
ejpam-3283	92	26	=	=	SYM
ejpam-3283	92	27	s	s	PROPN
ejpam-3283	92	28	for	for	ADP
ejpam-3283	92	29	all	all	DET
ejpam-3283	92	30	a	a	DET
ejpam-3283	92	31	∈	∈	NOUN
ejpam-3283	92	32	s	s	NOUN
ejpam-3283	92	33	because	because	SCONJ
ejpam-3283	92	34	s	s	PROPN
ejpam-3283	92	35	is	be	AUX
ejpam-3283	92	36	n	n	PRON
ejpam-3283	92	37	-	-	PUNCT
ejpam-3283	92	38	simple	simple	NOUN
ejpam-3283	92	39	.	.	PUNCT
ejpam-3283	93	1	next	next	ADV
ejpam-3283	93	2	,	,	PUNCT
ejpam-3283	93	3	we	we	PRON
ejpam-3283	93	4	show	show	VERB
ejpam-3283	93	5	(	(	PUNCT
ejpam-3283	93	6	2)⇒	2)⇒	NUM
ejpam-3283	93	7	(	(	PUNCT
ejpam-3283	93	8	3	3	NUM
ejpam-3283	93	9	)	)	PUNCT
ejpam-3283	93	10	.	.	PUNCT
ejpam-3283	94	1	suppose	suppose	VERB
ejpam-3283	94	2	that	that	SCONJ
ejpam-3283	94	3	f(sn−1	f(sn−1	PROPN
ejpam-3283	94	4	,	,	PUNCT
ejpam-3283	94	5	a	a	PRON
ejpam-3283	94	6	)	)	PUNCT
ejpam-3283	94	7	=	=	SYM
ejpam-3283	94	8	s	s	PROPN
ejpam-3283	94	9	for	for	ADP
ejpam-3283	94	10	all	all	DET
ejpam-3283	94	11	a	a	DET
ejpam-3283	94	12	∈	∈	NOUN
ejpam-3283	94	13	s.	s.	PROPN
ejpam-3283	94	14	by	by	ADP
ejpam-3283	94	15	corollary	corollary	ADJ
ejpam-3283	94	16	1	1	NUM
ejpam-3283	94	17	,	,	PUNCT
ejpam-3283	94	18	we	we	PRON
ejpam-3283	94	19	gain	gain	VERB
ejpam-3283	94	20	in(a	in(a	PUNCT
ejpam-3283	94	21	)	)	PUNCT
ejpam-3283	94	22	=	=	SYM
ejpam-3283	94	23	f(sn−1	f(sn−1	PROPN
ejpam-3283	94	24	,	,	PUNCT
ejpam-3283	94	25	a	a	PRON
ejpam-3283	94	26	)	)	PUNCT
ejpam-3283	94	27	∪	∪	NOUN
ejpam-3283	94	28	{	{	PUNCT
ejpam-3283	94	29	a	a	NOUN
ejpam-3283	94	30	}	}	PUNCT
ejpam-3283	94	31	=	=	SYM
ejpam-3283	94	32	s	s	NOUN
ejpam-3283	94	33	∪	∪	X
ejpam-3283	94	34	{	{	PUNCT
ejpam-3283	94	35	a	a	PRON
ejpam-3283	94	36	}	}	PUNCT
ejpam-3283	94	37	=	=	SYM
ejpam-3283	94	38	s.	s.	PROPN
ejpam-3283	94	39	therefore	therefore	ADV
ejpam-3283	94	40	,	,	PUNCT
ejpam-3283	94	41	in(a	in(a	PUNCT
ejpam-3283	94	42	)	)	PUNCT
ejpam-3283	94	43	=	=	SYM
ejpam-3283	94	44	s	s	PROPN
ejpam-3283	94	45	for	for	ADP
ejpam-3283	94	46	all	all	DET
ejpam-3283	94	47	a	a	DET
ejpam-3283	94	48	∈	∈	NOUN
ejpam-3283	94	49	s.	s.	PROPN
ejpam-3283	94	50	finally	finally	ADV
ejpam-3283	94	51	,	,	PUNCT
ejpam-3283	94	52	we	we	PRON
ejpam-3283	94	53	show	show	VERB
ejpam-3283	94	54	(	(	PUNCT
ejpam-3283	94	55	3	3	NUM
ejpam-3283	94	56	)	)	PUNCT
ejpam-3283	94	57	⇒	⇒	NOUN
ejpam-3283	94	58	(	(	PUNCT
ejpam-3283	94	59	1	1	NUM
ejpam-3283	94	60	)	)	PUNCT
ejpam-3283	94	61	.	.	PUNCT
ejpam-3283	95	1	assume	assume	VERB
ejpam-3283	95	2	the	the	DET
ejpam-3283	95	3	statement	statement	NOUN
ejpam-3283	95	4	(	(	PUNCT
ejpam-3283	95	5	3	3	X
ejpam-3283	95	6	)	)	PUNCT
ejpam-3283	95	7	holds	hold	VERB
ejpam-3283	95	8	.	.	PUNCT
ejpam-3283	96	1	let	let	VERB
ejpam-3283	96	2	i	i	PRON
ejpam-3283	96	3	be	be	AUX
ejpam-3283	96	4	any	any	DET
ejpam-3283	96	5	n	n	CCONJ
ejpam-3283	96	6	-	-	PUNCT
ejpam-3283	96	7	ideal	ideal	NOUN
ejpam-3283	96	8	of	of	ADP
ejpam-3283	96	9	s.	s.	PROPN
ejpam-3283	96	10	since	since	SCONJ
ejpam-3283	96	11	i	i	PRON
ejpam-3283	96	12	is	be	AUX
ejpam-3283	96	13	a	a	DET
ejpam-3283	96	14	nonempty	nonempty	ADJ
ejpam-3283	96	15	set	set	NOUN
ejpam-3283	96	16	,	,	PUNCT
ejpam-3283	96	17	there	there	PRON
ejpam-3283	96	18	exists	exist	VERB
ejpam-3283	96	19	x	x	X
ejpam-3283	96	20	∈	∈	PROPN
ejpam-3283	96	21	i.	i.	NOUN
ejpam-3283	96	22	then	then	ADV
ejpam-3283	96	23	s	s	VERB
ejpam-3283	96	24	=	=	PUNCT
ejpam-3283	96	25	in(x	in(x	X
ejpam-3283	96	26	)	)	PUNCT
ejpam-3283	96	27	⊆	⊆	NUM
ejpam-3283	96	28	i	i	NOUN
ejpam-3283	96	29	⊆	⊆	NUM
ejpam-3283	96	30	s.	s.	PROPN
ejpam-3283	96	31	this	this	PRON
ejpam-3283	96	32	implies	imply	VERB
ejpam-3283	96	33	that	that	SCONJ
ejpam-3283	96	34	i	i	PRON
ejpam-3283	96	35	=	=	PROPN
ejpam-3283	96	36	s.	s.	PROPN
ejpam-3283	96	37	therefore	therefore	ADV
ejpam-3283	96	38	,	,	PUNCT
ejpam-3283	96	39	s	s	X
ejpam-3283	96	40	is	be	AUX
ejpam-3283	96	41	n	n	PRON
ejpam-3283	96	42	-	-	PUNCT
ejpam-3283	96	43	simple	simple	NOUN
ejpam-3283	96	44	.	.	PUNCT
ejpam-3283	97	1	example	example	NOUN
ejpam-3283	97	2	1	1	NUM
ejpam-3283	97	3	.	.	X
ejpam-3283	98	1	consider	consider	VERB
ejpam-3283	98	2	z30	z30	ADJ
ejpam-3283	98	3	,	,	PUNCT
ejpam-3283	98	4	let	let	VERB
ejpam-3283	98	5	s	s	PRON
ejpam-3283	98	6	=	=	PUNCT
ejpam-3283	98	7	{	{	PUNCT
ejpam-3283	98	8	5	5	NUM
ejpam-3283	98	9	,	,	PUNCT
ejpam-3283	98	10	25	25	NUM
ejpam-3283	98	11	}	}	PUNCT
ejpam-3283	98	12	.	.	PUNCT
ejpam-3283	99	1	define	define	VERB
ejpam-3283	99	2	f	f	PROPN
ejpam-3283	99	3	:	:	PUNCT
ejpam-3283	99	4	sn	sn	PROPN
ejpam-3283	99	5	→	→	SYM
ejpam-3283	99	6	s	s	X
ejpam-3283	99	7	by	by	ADP
ejpam-3283	99	8	f(xn1	f(xn1	NOUN
ejpam-3283	99	9	)	)	PUNCT
ejpam-3283	99	10	=	=	PUNCT
ejpam-3283	100	1	x1	x1	PROPN
ejpam-3283	100	2	·	·	PUNCT
ejpam-3283	100	3	x2	x2	X
ejpam-3283	100	4	·	·	PUNCT
ejpam-3283	100	5	.	.	PUNCT
ejpam-3283	100	6	.	.	PUNCT
ejpam-3283	100	7	.	.	PUNCT
ejpam-3283	101	1	·	·	PUNCT
ejpam-3283	101	2	xn	xn	PUNCT
ejpam-3283	102	1	for	for	ADP
ejpam-3283	102	2	all	all	DET
ejpam-3283	102	3	x1	x1	PROPN
ejpam-3283	102	4	,	,	PUNCT
ejpam-3283	102	5	x2	x2	PROPN
ejpam-3283	102	6	,	,	PUNCT
ejpam-3283	102	7	.	.	PUNCT
ejpam-3283	102	8	.	.	PUNCT
ejpam-3283	102	9	.	.	PUNCT
ejpam-3283	103	1	,	,	PUNCT
ejpam-3283	103	2	xn	xn	PUNCT
ejpam-3283	103	3	∈	∈	PROPN
ejpam-3283	103	4	s	s	VERB
ejpam-3283	103	5	where	where	SCONJ
ejpam-3283	103	6	·	·	PUNCT
ejpam-3283	103	7	is	be	AUX
ejpam-3283	103	8	the	the	DET
ejpam-3283	103	9	multiplication	multiplication	NOUN
ejpam-3283	103	10	of	of	ADP
ejpam-3283	103	11	z30	z30	NOUN
ejpam-3283	103	12	.	.	PUNCT
ejpam-3283	104	1	it	it	PRON
ejpam-3283	104	2	is	be	AUX
ejpam-3283	104	3	easy	easy	ADJ
ejpam-3283	104	4	to	to	PART
ejpam-3283	104	5	see	see	VERB
ejpam-3283	104	6	that	that	PRON
ejpam-3283	104	7	s	s	VERB
ejpam-3283	104	8	is	be	AUX
ejpam-3283	104	9	n	n	PRON
ejpam-3283	104	10	-	-	PUNCT
ejpam-3283	104	11	simple	simple	NOUN
ejpam-3283	104	12	.	.	PUNCT
ejpam-3283	105	1	lemma	lemma	PROPN
ejpam-3283	105	2	4	4	X
ejpam-3283	105	3	.	.	PUNCT
ejpam-3283	106	1	if	if	SCONJ
ejpam-3283	106	2	s	s	PROPN
ejpam-3283	106	3	has	have	VERB
ejpam-3283	106	4	a	a	DET
ejpam-3283	106	5	zero	zero	NUM
ejpam-3283	106	6	element	element	NOUN
ejpam-3283	106	7	,	,	PUNCT
ejpam-3283	106	8	then	then	ADV
ejpam-3283	106	9	the	the	DET
ejpam-3283	106	10	following	following	ADJ
ejpam-3283	106	11	statements	statement	NOUN
ejpam-3283	106	12	hold	hold	VERB
ejpam-3283	106	13	:	:	PUNCT
ejpam-3283	106	14	(	(	PUNCT
ejpam-3283	106	15	1	1	X
ejpam-3283	106	16	)	)	PUNCT
ejpam-3283	106	17	if	if	SCONJ
ejpam-3283	106	18	s	s	NOUN
ejpam-3283	106	19	is	be	AUX
ejpam-3283	106	20	0	0	NUM
ejpam-3283	106	21	-	-	PUNCT
ejpam-3283	106	22	n	n	CCONJ
ejpam-3283	106	23	-	-	PUNCT
ejpam-3283	106	24	simple	simple	ADJ
ejpam-3283	106	25	,	,	PUNCT
ejpam-3283	106	26	then	then	ADV
ejpam-3283	106	27	in(a	in(a	PUNCT
ejpam-3283	106	28	)	)	PUNCT
ejpam-3283	107	1	=	=	SYM
ejpam-3283	107	2	s	s	PROPN
ejpam-3283	107	3	for	for	ADP
ejpam-3283	107	4	all	all	DET
ejpam-3283	107	5	a	a	DET
ejpam-3283	107	6	∈	∈	NOUN
ejpam-3283	107	7	s	s	PART
ejpam-3283	107	8	r	r	NOUN
ejpam-3283	107	9	{	{	PUNCT
ejpam-3283	107	10	0	0	NUM
ejpam-3283	107	11	}	}	PUNCT
ejpam-3283	107	12	.	.	PUNCT
ejpam-3283	108	1	(	(	PUNCT
ejpam-3283	108	2	2	2	X
ejpam-3283	108	3	)	)	PUNCT
ejpam-3283	108	4	if	if	SCONJ
ejpam-3283	108	5	in(a	in(a	NOUN
ejpam-3283	108	6	)	)	PUNCT
ejpam-3283	108	7	=	=	SYM
ejpam-3283	108	8	s	s	PROPN
ejpam-3283	108	9	for	for	ADP
ejpam-3283	108	10	all	all	DET
ejpam-3283	108	11	a	a	DET
ejpam-3283	108	12	∈	∈	NOUN
ejpam-3283	108	13	s	s	PART
ejpam-3283	108	14	r	r	NOUN
ejpam-3283	108	15	{	{	PUNCT
ejpam-3283	108	16	0	0	NUM
ejpam-3283	108	17	}	}	PUNCT
ejpam-3283	108	18	,	,	PUNCT
ejpam-3283	108	19	then	then	ADV
ejpam-3283	108	20	either	either	CCONJ
ejpam-3283	108	21	f(sn	f(sn	NUM
ejpam-3283	108	22	)	)	PUNCT
ejpam-3283	108	23	=	=	SYM
ejpam-3283	108	24	{	{	PUNCT
ejpam-3283	108	25	0	0	NUM
ejpam-3283	108	26	}	}	PUNCT
ejpam-3283	108	27	or	or	CCONJ
ejpam-3283	108	28	s	s	NOUN
ejpam-3283	108	29	is	be	AUX
ejpam-3283	108	30	0	0	NUM
ejpam-3283	108	31	-	-	PUNCT
ejpam-3283	108	32	n	n	CCONJ
ejpam-3283	108	33	-	-	PUNCT
ejpam-3283	108	34	simple	simple	ADJ
ejpam-3283	108	35	.	.	PUNCT
ejpam-3283	109	1	proof	proof	NOUN
ejpam-3283	109	2	.	.	PUNCT
ejpam-3283	110	1	(	(	PUNCT
ejpam-3283	110	2	1	1	X
ejpam-3283	110	3	)	)	PUNCT
ejpam-3283	110	4	suppose	suppose	VERB
ejpam-3283	110	5	that	that	SCONJ
ejpam-3283	110	6	s	s	VERB
ejpam-3283	110	7	is	be	AUX
ejpam-3283	110	8	0	0	NUM
ejpam-3283	110	9	-	-	PUNCT
ejpam-3283	110	10	n	n	CCONJ
ejpam-3283	110	11	-	-	PUNCT
ejpam-3283	110	12	simple	simple	NOUN
ejpam-3283	110	13	.	.	PUNCT
ejpam-3283	111	1	since	since	SCONJ
ejpam-3283	111	2	in(a	in(a	NUM
ejpam-3283	111	3	)	)	PUNCT
ejpam-3283	111	4	is	be	AUX
ejpam-3283	111	5	a	a	DET
ejpam-3283	111	6	nonzero	nonzero	ADJ
ejpam-3283	111	7	n	n	CCONJ
ejpam-3283	111	8	-	-	PUNCT
ejpam-3283	111	9	ideal	ideal	NOUN
ejpam-3283	111	10	of	of	ADP
ejpam-3283	111	11	s	s	PRON
ejpam-3283	111	12	for	for	ADP
ejpam-3283	111	13	all	all	DET
ejpam-3283	111	14	a	a	DET
ejpam-3283	111	15	∈	∈	NOUN
ejpam-3283	111	16	s	s	PART
ejpam-3283	111	17	r	r	NOUN
ejpam-3283	111	18	{	{	PUNCT
ejpam-3283	111	19	0	0	NUM
ejpam-3283	111	20	}	}	PUNCT
ejpam-3283	111	21	,	,	PUNCT
ejpam-3283	111	22	we	we	PRON
ejpam-3283	111	23	obtain	obtain	VERB
ejpam-3283	111	24	that	that	PRON
ejpam-3283	111	25	in(a	in(a	PUNCT
ejpam-3283	111	26	)	)	PUNCT
ejpam-3283	111	27	=	=	SYM
ejpam-3283	111	28	s	s	PROPN
ejpam-3283	111	29	for	for	ADP
ejpam-3283	111	30	all	all	DET
ejpam-3283	111	31	a	a	DET
ejpam-3283	111	32	∈	∈	NOUN
ejpam-3283	111	33	s	s	PART
ejpam-3283	111	34	r	r	NOUN
ejpam-3283	111	35	{	{	PUNCT
ejpam-3283	111	36	0	0	NUM
ejpam-3283	111	37	}	}	PUNCT
ejpam-3283	111	38	.	.	PUNCT
ejpam-3283	112	1	(	(	PUNCT
ejpam-3283	112	2	2	2	X
ejpam-3283	112	3	)	)	PUNCT
ejpam-3283	112	4	assume	assume	VERB
ejpam-3283	112	5	that	that	SCONJ
ejpam-3283	112	6	in(a	in(a	X
ejpam-3283	112	7	)	)	PUNCT
ejpam-3283	112	8	=	=	SYM
ejpam-3283	112	9	s	s	PROPN
ejpam-3283	112	10	for	for	ADP
ejpam-3283	112	11	all	all	DET
ejpam-3283	112	12	a	a	DET
ejpam-3283	112	13	∈	∈	NOUN
ejpam-3283	112	14	s	s	PART
ejpam-3283	112	15	r	r	NOUN
ejpam-3283	112	16	{	{	PUNCT
ejpam-3283	112	17	0	0	NUM
ejpam-3283	112	18	}	}	PUNCT
ejpam-3283	112	19	and	and	CCONJ
ejpam-3283	112	20	suppose	suppose	VERB
ejpam-3283	112	21	that	that	SCONJ
ejpam-3283	112	22	f(sn	f(sn	NOUN
ejpam-3283	112	23	)	)	PUNCT
ejpam-3283	112	24	6=	6=	PUNCT
ejpam-3283	112	25	{	{	PUNCT
ejpam-3283	112	26	0	0	NUM
ejpam-3283	112	27	}	}	PUNCT
ejpam-3283	112	28	.	.	PUNCT
ejpam-3283	113	1	let	let	VERB
ejpam-3283	113	2	i	i	PRON
ejpam-3283	113	3	be	be	AUX
ejpam-3283	113	4	a	a	DET
ejpam-3283	113	5	nonzero	nonzero	ADJ
ejpam-3283	113	6	n	n	CCONJ
ejpam-3283	113	7	-	-	PUNCT
ejpam-3283	113	8	ideal	ideal	NOUN
ejpam-3283	113	9	of	of	ADP
ejpam-3283	113	10	s.	s.	PROPN
ejpam-3283	113	11	then	then	ADV
ejpam-3283	113	12	there	there	PRON
ejpam-3283	113	13	exists	exist	VERB
ejpam-3283	113	14	x	x	X
ejpam-3283	113	15	∈	∈	NOUN
ejpam-3283	114	1	i	i	PRON
ejpam-3283	114	2	r	r	NOUN
ejpam-3283	114	3	{	{	PUNCT
ejpam-3283	114	4	0	0	NUM
ejpam-3283	114	5	}	}	PUNCT
ejpam-3283	114	6	.	.	PUNCT
ejpam-3283	115	1	hence	hence	ADV
ejpam-3283	115	2	s	s	X
ejpam-3283	115	3	=	=	PUNCT
ejpam-3283	115	4	in(x	in(x	X
ejpam-3283	115	5	)	)	PUNCT
ejpam-3283	115	6	⊆	⊆	NUM
ejpam-3283	115	7	i	i	NOUN
ejpam-3283	115	8	⊆	⊆	NUM
ejpam-3283	115	9	s	s	NOUN
ejpam-3283	115	10	,	,	PUNCT
ejpam-3283	115	11	and	and	CCONJ
ejpam-3283	115	12	so	so	ADV
ejpam-3283	115	13	i	i	PRON
ejpam-3283	115	14	=	=	PUNCT
ejpam-3283	115	15	s.	s.	PROPN
ejpam-3283	115	16	therefore	therefore	ADV
ejpam-3283	115	17	,	,	PUNCT
ejpam-3283	115	18	s	s	X
ejpam-3283	115	19	is	be	AUX
ejpam-3283	115	20	0	0	NUM
ejpam-3283	115	21	-	-	PUNCT
ejpam-3283	115	22	n	n	CCONJ
ejpam-3283	115	23	-	-	PUNCT
ejpam-3283	115	24	simple	simple	NOUN
ejpam-3283	115	25	.	.	PUNCT
ejpam-3283	116	1	example	example	NOUN
ejpam-3283	116	2	2	2	NUM
ejpam-3283	116	3	.	.	X
ejpam-3283	116	4	consider	consider	VERB
ejpam-3283	116	5	z30	z30	ADJ
ejpam-3283	116	6	,	,	PUNCT
ejpam-3283	116	7	let	let	VERB
ejpam-3283	116	8	s	s	PRON
ejpam-3283	116	9	=	=	VERB
ejpam-3283	116	10	{	{	PUNCT
ejpam-3283	116	11	0	0	NUM
ejpam-3283	116	12	,	,	PUNCT
ejpam-3283	116	13	5	5	NUM
ejpam-3283	116	14	,	,	PUNCT
ejpam-3283	116	15	25	25	NUM
ejpam-3283	116	16	}	}	PUNCT
ejpam-3283	116	17	.	.	PUNCT
ejpam-3283	117	1	define	define	VERB
ejpam-3283	117	2	f	f	PROPN
ejpam-3283	117	3	:	:	PUNCT
ejpam-3283	117	4	sn	sn	PROPN
ejpam-3283	117	5	→	→	SYM
ejpam-3283	117	6	s	s	X
ejpam-3283	117	7	by	by	ADP
ejpam-3283	117	8	f(xn1	f(xn1	NOUN
ejpam-3283	117	9	)	)	PUNCT
ejpam-3283	117	10	=	=	PUNCT
ejpam-3283	118	1	x1	x1	PROPN
ejpam-3283	118	2	·	·	PUNCT
ejpam-3283	118	3	x2	x2	X
ejpam-3283	118	4	·	·	PUNCT
ejpam-3283	118	5	.	.	PUNCT
ejpam-3283	118	6	.	.	PUNCT
ejpam-3283	118	7	.	.	PUNCT
ejpam-3283	119	1	·	·	PUNCT
ejpam-3283	119	2	xn	xn	PUNCT
ejpam-3283	120	1	for	for	ADP
ejpam-3283	120	2	all	all	DET
ejpam-3283	120	3	x1	x1	PROPN
ejpam-3283	120	4	,	,	PUNCT
ejpam-3283	120	5	x2	x2	PROPN
ejpam-3283	120	6	,	,	PUNCT
ejpam-3283	120	7	.	.	PUNCT
ejpam-3283	120	8	.	.	PUNCT
ejpam-3283	120	9	.	.	PUNCT
ejpam-3283	121	1	,	,	PUNCT
ejpam-3283	121	2	xn	xn	PUNCT
ejpam-3283	121	3	∈	∈	PROPN
ejpam-3283	121	4	s	s	VERB
ejpam-3283	121	5	where	where	SCONJ
ejpam-3283	121	6	·	·	PUNCT
ejpam-3283	121	7	is	be	AUX
ejpam-3283	121	8	the	the	DET
ejpam-3283	121	9	multiplication	multiplication	NOUN
ejpam-3283	121	10	of	of	ADP
ejpam-3283	121	11	z30	z30	NOUN
ejpam-3283	121	12	.	.	PUNCT
ejpam-3283	122	1	it	it	PRON
ejpam-3283	122	2	is	be	AUX
ejpam-3283	122	3	easy	easy	ADJ
ejpam-3283	122	4	to	to	PART
ejpam-3283	122	5	see	see	VERB
ejpam-3283	122	6	that	that	PRON
ejpam-3283	122	7	s	s	VERB
ejpam-3283	122	8	is	be	AUX
ejpam-3283	122	9	0	0	NUM
ejpam-3283	122	10	-	-	PUNCT
ejpam-3283	122	11	n	n	CCONJ
ejpam-3283	122	12	-	-	PUNCT
ejpam-3283	122	13	simple	simple	NOUN
ejpam-3283	122	14	.	.	PUNCT
ejpam-3283	123	1	lemma	lemma	PROPN
ejpam-3283	123	2	5	5	X
ejpam-3283	123	3	.	.	PUNCT
ejpam-3283	124	1	let	let	AUX
ejpam-3283	124	2	{	{	PUNCT
ejpam-3283	124	3	iγ	iγ	VERB
ejpam-3283	124	4	|	|	ADV
ejpam-3283	124	5	γ	γ	X
ejpam-3283	124	6	∈	∈	PROPN
ejpam-3283	124	7	γ	γ	AUX
ejpam-3283	124	8	}	}	PUNCT
ejpam-3283	124	9	be	be	AUX
ejpam-3283	124	10	a	a	DET
ejpam-3283	124	11	family	family	NOUN
ejpam-3283	124	12	of	of	ADP
ejpam-3283	124	13	n	n	CCONJ
ejpam-3283	124	14	-	-	PUNCT
ejpam-3283	124	15	ideals	ideal	NOUN
ejpam-3283	124	16	of	of	ADP
ejpam-3283	124	17	s.	s.	PROPN
ejpam-3283	124	18	then	then	ADV
ejpam-3283	124	19	⋃	⋃	PROPN
ejpam-3283	124	20	γ∈γ	γ∈γ	ADJ
ejpam-3283	124	21	iγ	iγ	NOUN
ejpam-3283	124	22	is	be	AUX
ejpam-3283	124	23	an	an	DET
ejpam-3283	124	24	n	n	CCONJ
ejpam-3283	124	25	-	-	PUNCT
ejpam-3283	124	26	ideal	ideal	NOUN
ejpam-3283	124	27	of	of	ADP
ejpam-3283	124	28	s	s	PRON
ejpam-3283	124	29	and	and	CCONJ
ejpam-3283	124	30	⋂	⋂	PROPN
ejpam-3283	124	31	γ∈γ	γ∈γ	ADJ
ejpam-3283	124	32	iγ	iγ	NOUN
ejpam-3283	124	33	is	be	AUX
ejpam-3283	124	34	also	also	ADV
ejpam-3283	124	35	an	an	DET
ejpam-3283	124	36	n	n	CCONJ
ejpam-3283	124	37	-	-	PUNCT
ejpam-3283	124	38	ideal	ideal	NOUN
ejpam-3283	124	39	of	of	ADP
ejpam-3283	124	40	s	s	PRON
ejpam-3283	124	41	if	if	SCONJ
ejpam-3283	124	42	it	it	PRON
ejpam-3283	124	43	’s	’	VERB
ejpam-3283	124	44	not	not	PART
ejpam-3283	124	45	empty	empty	ADJ
ejpam-3283	124	46	.	.	PUNCT
ejpam-3283	125	1	p.	p.	NOUN
ejpam-3283	125	2	petchkaew	petchkaew	NOUN
ejpam-3283	125	3	,	,	PUNCT
ejpam-3283	125	4	r.	r.	PROPN
ejpam-3283	125	5	chinram	chinram	PROPN
ejpam-3283	125	6	/	/	SYM
ejpam-3283	125	7	eur	eur	PROPN
ejpam-3283	125	8	.	.	PUNCT
ejpam-3283	126	1	j.	j.	PROPN
ejpam-3283	126	2	pure	pure	PROPN
ejpam-3283	126	3	appl	appl	PROPN
ejpam-3283	126	4	.	.	PROPN
ejpam-3283	126	5	math	math	PROPN
ejpam-3283	126	6	,	,	PUNCT
ejpam-3283	126	7	11	11	NUM
ejpam-3283	126	8	(	(	PUNCT
ejpam-3283	126	9	3	3	NUM
ejpam-3283	126	10	)	)	PUNCT
ejpam-3283	126	11	(	(	PUNCT
ejpam-3283	126	12	2018	2018	NUM
ejpam-3283	126	13	)	)	PUNCT
ejpam-3283	126	14	,	,	PUNCT
ejpam-3283	126	15	762	762	NUM
ejpam-3283	126	16	-	-	SYM
ejpam-3283	126	17	773	773	NUM
ejpam-3283	126	18	766	766	NUM
ejpam-3283	126	19	proof	proof	NOUN
ejpam-3283	126	20	.	.	PUNCT
ejpam-3283	127	1	the	the	DET
ejpam-3283	127	2	proof	proof	NOUN
ejpam-3283	127	3	is	be	AUX
ejpam-3283	127	4	straightforward	straightforward	ADJ
ejpam-3283	127	5	.	.	PUNCT
ejpam-3283	128	1	lemma	lemma	PROPN
ejpam-3283	128	2	6	6	NUM
ejpam-3283	128	3	.	.	PUNCT
ejpam-3283	129	1	let	let	VERB
ejpam-3283	129	2	i	i	PRON
ejpam-3283	129	3	be	be	AUX
ejpam-3283	129	4	an	an	DET
ejpam-3283	129	5	n	n	CCONJ
ejpam-3283	129	6	-	-	PUNCT
ejpam-3283	129	7	ideal	ideal	NOUN
ejpam-3283	129	8	of	of	ADP
ejpam-3283	129	9	s	s	PRON
ejpam-3283	129	10	and	and	CCONJ
ejpam-3283	129	11	h	h	NOUN
ejpam-3283	129	12	be	be	AUX
ejpam-3283	129	13	an	an	DET
ejpam-3283	129	14	n	n	CCONJ
ejpam-3283	129	15	-	-	PUNCT
ejpam-3283	129	16	ary	ary	NOUN
ejpam-3283	129	17	subsemigroup	subsemigroup	NOUN
ejpam-3283	129	18	of	of	ADP
ejpam-3283	129	19	s	s	PROPN
ejpam-3283	129	20	,	,	PUNCT
ejpam-3283	129	21	then	then	ADV
ejpam-3283	129	22	the	the	DET
ejpam-3283	129	23	following	following	ADJ
ejpam-3283	129	24	statements	statement	NOUN
ejpam-3283	129	25	hold	hold	VERB
ejpam-3283	129	26	:	:	PUNCT
ejpam-3283	129	27	(	(	PUNCT
ejpam-3283	129	28	1	1	X
ejpam-3283	129	29	)	)	PUNCT
ejpam-3283	129	30	if	if	SCONJ
ejpam-3283	129	31	h	h	NOUN
ejpam-3283	129	32	is	be	AUX
ejpam-3283	129	33	n	n	PRON
ejpam-3283	129	34	-	-	PUNCT
ejpam-3283	129	35	simple	simple	NOUN
ejpam-3283	129	36	such	such	ADJ
ejpam-3283	129	37	that	that	SCONJ
ejpam-3283	129	38	h	h	NOUN
ejpam-3283	129	39	∩	∩	NOUN
ejpam-3283	129	40	i	i	ADP
ejpam-3283	129	41	6=	6=	PROPN
ejpam-3283	129	42	∅	∅	NOUN
ejpam-3283	129	43	,	,	PUNCT
ejpam-3283	129	44	then	then	ADV
ejpam-3283	129	45	h	h	PROPN
ejpam-3283	129	46	⊆	⊆	NUM
ejpam-3283	129	47	i.	i.	NOUN
ejpam-3283	129	48	(	(	PUNCT
ejpam-3283	129	49	2	2	NUM
ejpam-3283	129	50	)	)	PUNCT
ejpam-3283	129	51	if	if	SCONJ
ejpam-3283	129	52	h	h	NOUN
ejpam-3283	129	53	is	be	AUX
ejpam-3283	129	54	0	0	NUM
ejpam-3283	129	55	-	-	PUNCT
ejpam-3283	129	56	n	n	CCONJ
ejpam-3283	129	57	-	-	PUNCT
ejpam-3283	129	58	simple	simple	NOUN
ejpam-3283	129	59	such	such	ADJ
ejpam-3283	129	60	that	that	PRON
ejpam-3283	129	61	(	(	PUNCT
ejpam-3283	129	62	h	h	NOUN
ejpam-3283	129	63	r	r	NOUN
ejpam-3283	129	64	{	{	PUNCT
ejpam-3283	129	65	0	0	NUM
ejpam-3283	129	66	}	}	PUNCT
ejpam-3283	129	67	)	)	PUNCT
ejpam-3283	129	68	∩	∩	NOUN
ejpam-3283	130	1	i	i	PRON
ejpam-3283	130	2	6=	6=	PROPN
ejpam-3283	130	3	∅	∅	NOUN
ejpam-3283	130	4	,	,	PUNCT
ejpam-3283	130	5	then	then	ADV
ejpam-3283	130	6	h	h	PROPN
ejpam-3283	130	7	⊆	⊆	NUM
ejpam-3283	130	8	i.	i.	NOUN
ejpam-3283	130	9	proof	proof	NOUN
ejpam-3283	130	10	.	.	PUNCT
ejpam-3283	131	1	(	(	PUNCT
ejpam-3283	131	2	1	1	X
ejpam-3283	131	3	)	)	PUNCT
ejpam-3283	131	4	assume	assume	VERB
ejpam-3283	131	5	that	that	SCONJ
ejpam-3283	131	6	h	h	NOUN
ejpam-3283	131	7	is	be	AUX
ejpam-3283	131	8	n	n	PRON
ejpam-3283	131	9	-	-	PUNCT
ejpam-3283	131	10	simple	simple	NOUN
ejpam-3283	131	11	such	such	ADJ
ejpam-3283	131	12	that	that	SCONJ
ejpam-3283	131	13	h∩i	h∩i	NOUN
ejpam-3283	132	1	6=	6=	PRON
ejpam-3283	132	2	∅.	∅.	VERB
ejpam-3283	132	3	then	then	ADV
ejpam-3283	132	4	there	there	PRON
ejpam-3283	132	5	exists	exist	VERB
ejpam-3283	132	6	a	a	DET
ejpam-3283	132	7	∈	∈	PROPN
ejpam-3283	132	8	h∩i	h∩i	NOUN
ejpam-3283	132	9	.	.	PUNCT
ejpam-3283	133	1	by	by	ADP
ejpam-3283	133	2	lemma	lemma	PROPN
ejpam-3283	133	3	2	2	NUM
ejpam-3283	133	4	,	,	PUNCT
ejpam-3283	133	5	we	we	PRON
ejpam-3283	133	6	obtain	obtain	VERB
ejpam-3283	133	7	that	that	SCONJ
ejpam-3283	133	8	f(hn−1	f(hn−1	PROPN
ejpam-3283	133	9	,	,	PUNCT
ejpam-3283	133	10	a	a	DET
ejpam-3283	133	11	)	)	PUNCT
ejpam-3283	133	12	∩h	∩h	NOUN
ejpam-3283	133	13	is	be	AUX
ejpam-3283	133	14	an	an	DET
ejpam-3283	133	15	n	n	CCONJ
ejpam-3283	133	16	-	-	PUNCT
ejpam-3283	133	17	ideal	ideal	NOUN
ejpam-3283	133	18	of	of	ADP
ejpam-3283	133	19	h.	h.	PROPN
ejpam-3283	133	20	since	since	SCONJ
ejpam-3283	133	21	h	h	PROPN
ejpam-3283	133	22	is	be	AUX
ejpam-3283	133	23	n	n	PRON
ejpam-3283	133	24	-	-	PUNCT
ejpam-3283	133	25	simple	simple	ADJ
ejpam-3283	133	26	,	,	PUNCT
ejpam-3283	133	27	we	we	PRON
ejpam-3283	133	28	gain	gain	VERB
ejpam-3283	133	29	f(hn−1	f(hn−1	PROPN
ejpam-3283	133	30	,	,	PUNCT
ejpam-3283	133	31	a	a	PRON
ejpam-3283	133	32	)	)	PUNCT
ejpam-3283	133	33	∩	∩	ADJ
ejpam-3283	133	34	h	h	NOUN
ejpam-3283	133	35	=	=	PUNCT
ejpam-3283	133	36	h.	h.	PROPN
ejpam-3283	133	37	this	this	PRON
ejpam-3283	133	38	implies	imply	VERB
ejpam-3283	133	39	h	h	NOUN
ejpam-3283	133	40	⊆	⊆	NUM
ejpam-3283	133	41	f(hn−1	f(hn−1	PROPN
ejpam-3283	133	42	,	,	PUNCT
ejpam-3283	133	43	a	a	PRON
ejpam-3283	133	44	)	)	PUNCT
ejpam-3283	134	1	⊆	⊆	NUM
ejpam-3283	134	2	f(sn−1	f(sn−1	NOUN
ejpam-3283	134	3	,	,	PUNCT
ejpam-3283	134	4	i	i	PROPN
ejpam-3283	134	5	)	)	PUNCT
ejpam-3283	134	6	⊆	⊆	NUM
ejpam-3283	134	7	i.	i.	NOUN
ejpam-3283	134	8	therefore	therefore	ADV
ejpam-3283	134	9	,	,	PUNCT
ejpam-3283	134	10	h	h	PROPN
ejpam-3283	134	11	⊆	⊆	NUM
ejpam-3283	134	12	i.	i.	NOUN
ejpam-3283	134	13	(	(	PUNCT
ejpam-3283	134	14	2	2	X
ejpam-3283	134	15	)	)	PUNCT
ejpam-3283	134	16	suppose	suppose	VERB
ejpam-3283	134	17	that	that	SCONJ
ejpam-3283	134	18	h	h	PROPN
ejpam-3283	134	19	is	be	AUX
ejpam-3283	134	20	0	0	NUM
ejpam-3283	134	21	-	-	PUNCT
ejpam-3283	134	22	n	n	CCONJ
ejpam-3283	134	23	-	-	PUNCT
ejpam-3283	134	24	simple	simple	NOUN
ejpam-3283	134	25	such	such	ADJ
ejpam-3283	134	26	that	that	PRON
ejpam-3283	134	27	(	(	PUNCT
ejpam-3283	134	28	h	h	NOUN
ejpam-3283	134	29	r	r	NOUN
ejpam-3283	134	30	{	{	PUNCT
ejpam-3283	134	31	0	0	NUM
ejpam-3283	134	32	}	}	PUNCT
ejpam-3283	134	33	)	)	PUNCT
ejpam-3283	134	34	∩	∩	NOUN
ejpam-3283	135	1	i	i	PRON
ejpam-3283	135	2	6=	6=	NOUN
ejpam-3283	135	3	∅.	∅.	VERB
ejpam-3283	135	4	then	then	ADV
ejpam-3283	135	5	there	there	PRON
ejpam-3283	135	6	exists	exist	VERB
ejpam-3283	135	7	a	a	DET
ejpam-3283	135	8	∈	∈	NOUN
ejpam-3283	135	9	hr{0}∩i	hr{0}∩i	NOUN
ejpam-3283	135	10	.	.	PUNCT
ejpam-3283	136	1	by	by	ADP
ejpam-3283	136	2	lemma	lemma	PROPN
ejpam-3283	136	3	4(1	4(1	PROPN
ejpam-3283	136	4	)	)	PUNCT
ejpam-3283	136	5	and	and	CCONJ
ejpam-3283	136	6	corollary	corollary	ADJ
ejpam-3283	136	7	1	1	NUM
ejpam-3283	136	8	,	,	PUNCT
ejpam-3283	136	9	we	we	PRON
ejpam-3283	136	10	obtain	obtain	VERB
ejpam-3283	136	11	h	h	NOUN
ejpam-3283	136	12	=	=	PUNCT
ejpam-3283	136	13	in	in	ADP
ejpam-3283	136	14	,	,	PUNCT
ejpam-3283	136	15	h(a	h(a	PROPN
ejpam-3283	136	16	)	)	PUNCT
ejpam-3283	137	1	=	=	PRON
ejpam-3283	137	2	(	(	PUNCT
ejpam-3283	137	3	f(hn−1	f(hn−1	PROPN
ejpam-3283	137	4	,	,	PUNCT
ejpam-3283	137	5	a)∪	a)∪	ADV
ejpam-3283	137	6	{	{	PUNCT
ejpam-3283	137	7	a	a	NOUN
ejpam-3283	137	8	}	}	PUNCT
ejpam-3283	137	9	)	)	PUNCT
ejpam-3283	137	10	∩h	∩h	PROPN
ejpam-3283	137	11	⊆	⊆	NUM
ejpam-3283	137	12	f(sn−1	f(sn−1	PROPN
ejpam-3283	137	13	,	,	PUNCT
ejpam-3283	137	14	a	a	PRON
ejpam-3283	137	15	)	)	PUNCT
ejpam-3283	137	16	∪	∪	NOUN
ejpam-3283	137	17	{	{	PUNCT
ejpam-3283	137	18	a	a	PRON
ejpam-3283	137	19	}	}	PUNCT
ejpam-3283	137	20	=	=	SYM
ejpam-3283	137	21	in(a	in(a	X
ejpam-3283	137	22	)	)	PUNCT
ejpam-3283	137	23	⊆	⊆	NUM
ejpam-3283	137	24	i.	i.	NOUN
ejpam-3283	137	25	therefore	therefore	ADV
ejpam-3283	137	26	,	,	PUNCT
ejpam-3283	137	27	h	h	NOUN
ejpam-3283	137	28	⊆	⊆	NUM
ejpam-3283	137	29	i	i	PRON
ejpam-3283	137	30	as	as	ADP
ejpam-3283	137	31	desire	desire	NOUN
ejpam-3283	137	32	.	.	PUNCT
ejpam-3283	138	1	lemma	lemma	PROPN
ejpam-3283	138	2	7	7	X
ejpam-3283	138	3	.	.	PUNCT
ejpam-3283	139	1	let	let	VERB
ejpam-3283	139	2	a	a	DET
ejpam-3283	139	3	be	be	AUX
ejpam-3283	139	4	a	a	DET
ejpam-3283	139	5	nonempty	nonempty	ADJ
ejpam-3283	139	6	subset	subset	NOUN
ejpam-3283	139	7	of	of	ADP
ejpam-3283	139	8	an	an	DET
ejpam-3283	139	9	n	n	CCONJ
ejpam-3283	139	10	-	-	PUNCT
ejpam-3283	139	11	ideal	ideal	NOUN
ejpam-3283	139	12	i	i	PRON
ejpam-3283	139	13	of	of	ADP
ejpam-3283	139	14	s.	s.	PROPN
ejpam-3283	139	15	then	then	ADV
ejpam-3283	139	16	f(in−1	f(in−1	ADP
ejpam-3283	139	17	,	,	PUNCT
ejpam-3283	139	18	a	a	PRON
ejpam-3283	139	19	)	)	PUNCT
ejpam-3283	139	20	is	be	AUX
ejpam-3283	139	21	an	an	DET
ejpam-3283	139	22	n	n	CCONJ
ejpam-3283	139	23	-	-	PUNCT
ejpam-3283	139	24	ideal	ideal	NOUN
ejpam-3283	139	25	of	of	ADP
ejpam-3283	139	26	s.	s.	PROPN
ejpam-3283	139	27	proof	proof	PROPN
ejpam-3283	139	28	.	.	PUNCT
ejpam-3283	140	1	let	let	VERB
ejpam-3283	140	2	s1	s1	NOUN
ejpam-3283	140	3	,	,	PUNCT
ejpam-3283	140	4	s2	s2	PROPN
ejpam-3283	140	5	,	,	PUNCT
ejpam-3283	140	6	.	.	PUNCT
ejpam-3283	140	7	.	.	PUNCT
ejpam-3283	141	1	.	.	PUNCT
ejpam-3283	142	1	,	,	PUNCT
ejpam-3283	142	2	sn−1	sn−1	PROPN
ejpam-3283	142	3	∈	∈	PROPN
ejpam-3283	142	4	s	s	PART
ejpam-3283	142	5	and	and	CCONJ
ejpam-3283	142	6	let	let	VERB
ejpam-3283	142	7	y	y	PROPN
ejpam-3283	142	8	∈	∈	PROPN
ejpam-3283	142	9	f(in−1	f(in−1	PROPN
ejpam-3283	142	10	,	,	PUNCT
ejpam-3283	142	11	a	a	PRON
ejpam-3283	142	12	)	)	PUNCT
ejpam-3283	142	13	.	.	PUNCT
ejpam-3283	143	1	then	then	ADV
ejpam-3283	143	2	y	y	PROPN
ejpam-3283	143	3	=	=	PROPN
ejpam-3283	143	4	f(xn−1	f(xn−1	PROPN
ejpam-3283	143	5	1	1	NUM
ejpam-3283	143	6	,	,	PUNCT
ejpam-3283	143	7	a	a	PRON
ejpam-3283	143	8	)	)	PUNCT
ejpam-3283	143	9	for	for	ADP
ejpam-3283	143	10	some	some	DET
ejpam-3283	143	11	x1	x1	PROPN
ejpam-3283	143	12	,	,	PUNCT
ejpam-3283	143	13	x2	x2	PROPN
ejpam-3283	143	14	,	,	PUNCT
ejpam-3283	143	15	.	.	PUNCT
ejpam-3283	143	16	.	.	PUNCT
ejpam-3283	143	17	.	.	PUNCT
ejpam-3283	144	1	,	,	PUNCT
ejpam-3283	144	2	xn−1	xn−1	PROPN
ejpam-3283	144	3	∈	∈	PROPN
ejpam-3283	145	1	i	i	PRON
ejpam-3283	145	2	and	and	CCONJ
ejpam-3283	145	3	for	for	ADP
ejpam-3283	145	4	some	some	PRON
ejpam-3283	145	5	a	a	DET
ejpam-3283	145	6	∈	∈	NOUN
ejpam-3283	145	7	a.	a.	NOUN
ejpam-3283	145	8	then	then	ADV
ejpam-3283	145	9	f(sn−1	f(sn−1	ADP
ejpam-3283	145	10	1	1	NUM
ejpam-3283	145	11	,	,	PUNCT
ejpam-3283	145	12	y	y	NOUN
ejpam-3283	145	13	)	)	PUNCT
ejpam-3283	145	14	=	=	SYM
ejpam-3283	145	15	f(sn−1	f(sn−1	ADJ
ejpam-3283	145	16	1	1	NUM
ejpam-3283	145	17	,	,	PUNCT
ejpam-3283	145	18	f(xn−1	f(xn−1	PROPN
ejpam-3283	145	19	,	,	PUNCT
ejpam-3283	145	20	a	a	NOUN
ejpam-3283	145	21	)	)	PUNCT
ejpam-3283	145	22	)	)	PUNCT
ejpam-3283	146	1	=	=	PUNCT
ejpam-3283	147	1	f(f(sn−1	f(f(sn−1	ADJ
ejpam-3283	147	2	1	1	NUM
ejpam-3283	147	3	,	,	PUNCT
ejpam-3283	147	4	x1	x1	PROPN
ejpam-3283	147	5	)	)	PUNCT
ejpam-3283	147	6	,	,	PUNCT
ejpam-3283	147	7	xn−1	xn−1	PROPN
ejpam-3283	147	8	2	2	NUM
ejpam-3283	147	9	,	,	PUNCT
ejpam-3283	147	10	a	a	PRON
ejpam-3283	147	11	)	)	PUNCT
ejpam-3283	147	12	∈	∈	NOUN
ejpam-3283	147	13	f(in−1	f(in−1	X
ejpam-3283	147	14	,	,	PUNCT
ejpam-3283	147	15	a	a	PRON
ejpam-3283	147	16	)	)	PUNCT
ejpam-3283	147	17	because	because	SCONJ
ejpam-3283	147	18	i	i	PRON
ejpam-3283	147	19	is	be	AUX
ejpam-3283	147	20	an	an	DET
ejpam-3283	147	21	n	n	CCONJ
ejpam-3283	147	22	-	-	PUNCT
ejpam-3283	147	23	ideal	ideal	NOUN
ejpam-3283	147	24	of	of	ADP
ejpam-3283	147	25	s	s	PRON
ejpam-3283	147	26	and	and	CCONJ
ejpam-3283	147	27	xi	xi	ADP
ejpam-3283	147	28	∈	∈	PROPN
ejpam-3283	148	1	i	i	PRON
ejpam-3283	148	2	for	for	ADP
ejpam-3283	148	3	all	all	PRON
ejpam-3283	148	4	i	i	PRON
ejpam-3283	148	5	∈	∈	PROPN
ejpam-3283	148	6	{	{	PUNCT
ejpam-3283	148	7	1	1	NUM
ejpam-3283	148	8	,	,	PUNCT
ejpam-3283	148	9	2	2	NUM
ejpam-3283	148	10	,	,	PUNCT
ejpam-3283	148	11	.	.	PUNCT
ejpam-3283	148	12	.	.	PUNCT
ejpam-3283	148	13	.	.	PUNCT
ejpam-3283	149	1	,	,	PUNCT
ejpam-3283	149	2	n−	n−	NOUN
ejpam-3283	149	3	1	1	NUM
ejpam-3283	149	4	}	}	PUNCT
ejpam-3283	149	5	.	.	PUNCT
ejpam-3283	150	1	this	this	PRON
ejpam-3283	150	2	implies	imply	VERB
ejpam-3283	150	3	that	that	SCONJ
ejpam-3283	150	4	f(in−1	f(in−1	ADP
ejpam-3283	150	5	,	,	PUNCT
ejpam-3283	150	6	a	a	PRON
ejpam-3283	150	7	)	)	PUNCT
ejpam-3283	150	8	is	be	AUX
ejpam-3283	150	9	an	an	DET
ejpam-3283	150	10	n	n	CCONJ
ejpam-3283	150	11	-	-	PUNCT
ejpam-3283	150	12	ideal	ideal	NOUN
ejpam-3283	150	13	of	of	ADP
ejpam-3283	150	14	s.	s.	PROPN
ejpam-3283	150	15	4	4	NUM
ejpam-3283	150	16	.	.	PUNCT
ejpam-3283	150	17	minimality	minimality	NOUN
ejpam-3283	150	18	of	of	ADP
ejpam-3283	150	19	n	n	CCONJ
ejpam-3283	150	20	-	-	PUNCT
ejpam-3283	150	21	ideals	ideal	NOUN
ejpam-3283	150	22	in	in	ADP
ejpam-3283	150	23	this	this	DET
ejpam-3283	150	24	section	section	NOUN
ejpam-3283	150	25	,	,	PUNCT
ejpam-3283	150	26	we	we	PRON
ejpam-3283	150	27	investigate	investigate	VERB
ejpam-3283	150	28	the	the	DET
ejpam-3283	150	29	relationship	relationship	NOUN
ejpam-3283	150	30	between	between	ADP
ejpam-3283	150	31	the	the	DET
ejpam-3283	150	32	minimality	minimality	NOUN
ejpam-3283	150	33	of	of	ADP
ejpam-3283	150	34	n	n	CCONJ
ejpam-3283	150	35	-	-	PUNCT
ejpam-3283	150	36	ideals	ideal	NOUN
ejpam-3283	150	37	and	and	CCONJ
ejpam-3283	150	38	n	n	CCONJ
ejpam-3283	150	39	-	-	PUNCT
ejpam-3283	150	40	simple	simple	ADJ
ejpam-3283	150	41	(	(	PUNCT
ejpam-3283	150	42	0	0	NUM
ejpam-3283	150	43	-	-	PUNCT
ejpam-3283	150	44	n	n	CCONJ
ejpam-3283	150	45	-	-	PUNCT
ejpam-3283	150	46	simple	simple	ADJ
ejpam-3283	150	47	)	)	PUNCT
ejpam-3283	150	48	n	n	CCONJ
ejpam-3283	150	49	-	-	PUNCT
ejpam-3283	150	50	ary	ary	PROPN
ejpam-3283	150	51	semigroups	semigroup	NOUN
ejpam-3283	150	52	.	.	PUNCT
ejpam-3283	151	1	theorem	theorem	NOUN
ejpam-3283	151	2	1	1	NUM
ejpam-3283	151	3	.	.	PUNCT
ejpam-3283	152	1	let	let	VERB
ejpam-3283	152	2	s	s	PRON
ejpam-3283	152	3	be	be	AUX
ejpam-3283	152	4	an	an	DET
ejpam-3283	152	5	n	n	CCONJ
ejpam-3283	152	6	-	-	PUNCT
ejpam-3283	152	7	ary	ary	NOUN
ejpam-3283	152	8	semigroup	semigroup	NOUN
ejpam-3283	152	9	without	without	ADP
ejpam-3283	152	10	zero	zero	NUM
ejpam-3283	153	1	and	and	CCONJ
ejpam-3283	153	2	i	i	PRON
ejpam-3283	153	3	be	be	VERB
ejpam-3283	153	4	an	an	DET
ejpam-3283	153	5	n	n	CCONJ
ejpam-3283	153	6	-	-	PUNCT
ejpam-3283	153	7	ideal	ideal	NOUN
ejpam-3283	153	8	of	of	ADP
ejpam-3283	153	9	s.	s.	PROPN
ejpam-3283	154	1	then	then	ADV
ejpam-3283	154	2	i	i	PRON
ejpam-3283	154	3	is	be	AUX
ejpam-3283	154	4	a	a	DET
ejpam-3283	154	5	minimal	minimal	ADJ
ejpam-3283	154	6	n	n	CCONJ
ejpam-3283	154	7	-	-	PUNCT
ejpam-3283	154	8	ideal	ideal	NOUN
ejpam-3283	154	9	of	of	ADP
ejpam-3283	154	10	s	s	PRON
ejpam-3283	154	11	if	if	SCONJ
ejpam-3283	155	1	and	and	CCONJ
ejpam-3283	155	2	only	only	ADV
ejpam-3283	155	3	if	if	SCONJ
ejpam-3283	155	4	i	i	PRON
ejpam-3283	155	5	is	be	AUX
ejpam-3283	155	6	n	n	PRON
ejpam-3283	155	7	-	-	PUNCT
ejpam-3283	155	8	simple	simple	ADJ
ejpam-3283	155	9	.	.	PUNCT
ejpam-3283	156	1	proof	proof	NOUN
ejpam-3283	156	2	.	.	PUNCT
ejpam-3283	157	1	(	(	PUNCT
ejpam-3283	157	2	1	1	X
ejpam-3283	157	3	)	)	PUNCT
ejpam-3283	157	4	assume	assume	VERB
ejpam-3283	157	5	that	that	SCONJ
ejpam-3283	157	6	i	i	PRON
ejpam-3283	157	7	is	be	AUX
ejpam-3283	157	8	a	a	DET
ejpam-3283	157	9	minimal	minimal	ADJ
ejpam-3283	157	10	n	n	CCONJ
ejpam-3283	157	11	-	-	PUNCT
ejpam-3283	157	12	ideal	ideal	NOUN
ejpam-3283	157	13	of	of	ADP
ejpam-3283	157	14	s.	s.	PROPN
ejpam-3283	157	15	let	let	VERB
ejpam-3283	157	16	j	j	PROPN
ejpam-3283	157	17	be	be	AUX
ejpam-3283	157	18	any	any	DET
ejpam-3283	157	19	n	n	CCONJ
ejpam-3283	157	20	-	-	PUNCT
ejpam-3283	157	21	ideal	ideal	NOUN
ejpam-3283	157	22	of	of	ADP
ejpam-3283	157	23	i.	i.	NOUN
ejpam-3283	157	24	thus	thus	ADV
ejpam-3283	157	25	f(in−1	f(in−1	X
ejpam-3283	157	26	,	,	PUNCT
ejpam-3283	157	27	j	j	NOUN
ejpam-3283	157	28	)	)	PUNCT
ejpam-3283	158	1	⊆	⊆	NUM
ejpam-3283	158	2	j	j	NOUN
ejpam-3283	158	3	⊆	⊆	NUM
ejpam-3283	158	4	i.	i.	NOUN
ejpam-3283	158	5	by	by	ADP
ejpam-3283	158	6	lemma	lemma	PROPN
ejpam-3283	158	7	7	7	NUM
ejpam-3283	158	8	,	,	PUNCT
ejpam-3283	158	9	f(in−1	f(in−1	ADJ
ejpam-3283	158	10	,	,	PUNCT
ejpam-3283	158	11	j	j	NOUN
ejpam-3283	158	12	)	)	PUNCT
ejpam-3283	158	13	is	be	AUX
ejpam-3283	158	14	an	an	DET
ejpam-3283	158	15	n	n	CCONJ
ejpam-3283	158	16	-	-	PUNCT
ejpam-3283	158	17	ideal	ideal	NOUN
ejpam-3283	158	18	of	of	ADP
ejpam-3283	158	19	s.	s.	PROPN
ejpam-3283	158	20	since	since	SCONJ
ejpam-3283	158	21	i	i	PRON
ejpam-3283	158	22	is	be	AUX
ejpam-3283	158	23	a	a	DET
ejpam-3283	158	24	minimal	minimal	ADJ
ejpam-3283	158	25	n	n	CCONJ
ejpam-3283	158	26	-	-	PUNCT
ejpam-3283	158	27	ideal	ideal	NOUN
ejpam-3283	158	28	,	,	PUNCT
ejpam-3283	158	29	i	i	PRON
ejpam-3283	158	30	⊆	⊆	NUM
ejpam-3283	158	31	f(in−1	f(in−1	X
ejpam-3283	158	32	,	,	PUNCT
ejpam-3283	158	33	j	j	NOUN
ejpam-3283	158	34	)	)	PUNCT
ejpam-3283	158	35	and	and	CCONJ
ejpam-3283	158	36	then	then	ADV
ejpam-3283	158	37	f(in−1	f(in−1	X
ejpam-3283	158	38	,	,	PUNCT
ejpam-3283	158	39	j	j	NOUN
ejpam-3283	158	40	)	)	PUNCT
ejpam-3283	158	41	=	=	SYM
ejpam-3283	158	42	i.	i.	PROPN
ejpam-3283	158	43	therefore	therefore	ADV
ejpam-3283	158	44	,	,	PUNCT
ejpam-3283	158	45	i	i	PRON
ejpam-3283	158	46	is	be	AUX
ejpam-3283	158	47	n	n	PRON
ejpam-3283	158	48	-	-	PUNCT
ejpam-3283	158	49	simple	simple	ADJ
ejpam-3283	158	50	.	.	PUNCT
ejpam-3283	159	1	conversely	conversely	ADV
ejpam-3283	159	2	,	,	PUNCT
ejpam-3283	159	3	suppose	suppose	VERB
ejpam-3283	159	4	that	that	SCONJ
ejpam-3283	159	5	i	i	PRON
ejpam-3283	159	6	is	be	AUX
ejpam-3283	159	7	n	n	PRON
ejpam-3283	159	8	-	-	PUNCT
ejpam-3283	159	9	simple	simple	ADJ
ejpam-3283	159	10	.	.	PUNCT
ejpam-3283	160	1	let	let	VERB
ejpam-3283	160	2	j	j	PROPN
ejpam-3283	160	3	be	be	AUX
ejpam-3283	160	4	an	an	DET
ejpam-3283	160	5	n	n	CCONJ
ejpam-3283	160	6	-	-	PUNCT
ejpam-3283	160	7	ideal	ideal	NOUN
ejpam-3283	160	8	of	of	ADP
ejpam-3283	160	9	s	s	PRON
ejpam-3283	160	10	such	such	ADJ
ejpam-3283	160	11	that	that	SCONJ
ejpam-3283	160	12	j	j	PROPN
ejpam-3283	160	13	⊆	⊆	NUM
ejpam-3283	160	14	i.	i.	NOUN
ejpam-3283	161	1	so	so	SCONJ
ejpam-3283	161	2	i	i	PRON
ejpam-3283	161	3	∩	∩	VERB
ejpam-3283	161	4	i	i	ADP
ejpam-3283	161	5	6=	6=	NOUN
ejpam-3283	161	6	∅	∅	NOUN
ejpam-3283	161	7	,	,	PUNCT
ejpam-3283	161	8	and	and	CCONJ
ejpam-3283	161	9	hence	hence	ADV
ejpam-3283	161	10	i	i	PRON
ejpam-3283	161	11	⊆	⊆	NUM
ejpam-3283	161	12	j	j	PROPN
ejpam-3283	161	13	by	by	ADP
ejpam-3283	161	14	lemma	lemma	PROPN
ejpam-3283	161	15	6(1	6(1	NUM
ejpam-3283	161	16	)	)	PUNCT
ejpam-3283	161	17	.	.	PUNCT
ejpam-3283	162	1	this	this	PRON
ejpam-3283	162	2	implies	imply	VERB
ejpam-3283	162	3	that	that	SCONJ
ejpam-3283	162	4	j	j	PROPN
ejpam-3283	162	5	=	=	PROPN
ejpam-3283	162	6	i.	i.	PROPN
ejpam-3283	162	7	therefore	therefore	ADV
ejpam-3283	162	8	,	,	PUNCT
ejpam-3283	162	9	i	i	PRON
ejpam-3283	162	10	is	be	AUX
ejpam-3283	162	11	a	a	DET
ejpam-3283	162	12	minimal	minimal	ADJ
ejpam-3283	162	13	n	n	CCONJ
ejpam-3283	162	14	-	-	PUNCT
ejpam-3283	162	15	ideal	ideal	NOUN
ejpam-3283	162	16	of	of	ADP
ejpam-3283	162	17	s.	s.	PROPN
ejpam-3283	162	18	example	example	PROPN
ejpam-3283	163	1	3	3	X
ejpam-3283	163	2	.	.	X
ejpam-3283	163	3	consider	consider	VERB
ejpam-3283	163	4	z30	z30	ADJ
ejpam-3283	163	5	,	,	PUNCT
ejpam-3283	163	6	let	let	VERB
ejpam-3283	163	7	s	s	PRON
ejpam-3283	163	8	=	=	VERB
ejpam-3283	163	9	{	{	PUNCT
ejpam-3283	163	10	1	1	NUM
ejpam-3283	163	11	,	,	PUNCT
ejpam-3283	163	12	5	5	NUM
ejpam-3283	163	13	,	,	PUNCT
ejpam-3283	163	14	25	25	NUM
ejpam-3283	163	15	}	}	PUNCT
ejpam-3283	163	16	.	.	PUNCT
ejpam-3283	164	1	define	define	VERB
ejpam-3283	164	2	f	f	PROPN
ejpam-3283	164	3	:	:	PUNCT
ejpam-3283	164	4	sn	sn	PROPN
ejpam-3283	164	5	→	→	SYM
ejpam-3283	164	6	s	s	X
ejpam-3283	164	7	by	by	ADP
ejpam-3283	164	8	f(xn1	f(xn1	NOUN
ejpam-3283	164	9	)	)	PUNCT
ejpam-3283	164	10	=	=	PUNCT
ejpam-3283	165	1	x1	x1	PROPN
ejpam-3283	165	2	·	·	PUNCT
ejpam-3283	165	3	x2	x2	X
ejpam-3283	165	4	·	·	PUNCT
ejpam-3283	165	5	.	.	PUNCT
ejpam-3283	165	6	.	.	PUNCT
ejpam-3283	165	7	.	.	PUNCT
ejpam-3283	166	1	·	·	PUNCT
ejpam-3283	166	2	xn	xn	PUNCT
ejpam-3283	167	1	for	for	ADP
ejpam-3283	167	2	all	all	DET
ejpam-3283	167	3	x1	x1	PROPN
ejpam-3283	167	4	,	,	PUNCT
ejpam-3283	167	5	x2	x2	PROPN
ejpam-3283	167	6	,	,	PUNCT
ejpam-3283	167	7	.	.	PUNCT
ejpam-3283	167	8	.	.	PUNCT
ejpam-3283	167	9	.	.	PUNCT
ejpam-3283	168	1	,	,	PUNCT
ejpam-3283	168	2	xn	xn	PUNCT
ejpam-3283	168	3	∈	∈	PROPN
ejpam-3283	168	4	s	s	VERB
ejpam-3283	168	5	where	where	SCONJ
ejpam-3283	168	6	·	·	PUNCT
ejpam-3283	168	7	is	be	AUX
ejpam-3283	168	8	the	the	DET
ejpam-3283	168	9	multiplication	multiplication	NOUN
ejpam-3283	168	10	of	of	ADP
ejpam-3283	168	11	z30	z30	NOUN
ejpam-3283	168	12	.	.	PUNCT
ejpam-3283	169	1	it	it	PRON
ejpam-3283	169	2	is	be	AUX
ejpam-3283	169	3	easy	easy	ADJ
ejpam-3283	169	4	to	to	PART
ejpam-3283	169	5	see	see	VERB
ejpam-3283	169	6	that	that	PRON
ejpam-3283	169	7	i	i	PRON
ejpam-3283	169	8	=	=	PUNCT
ejpam-3283	169	9	{	{	PUNCT
ejpam-3283	169	10	5	5	NUM
ejpam-3283	169	11	,	,	PUNCT
ejpam-3283	169	12	25	25	NUM
ejpam-3283	169	13	}	}	PUNCT
ejpam-3283	169	14	is	be	AUX
ejpam-3283	169	15	a	a	DET
ejpam-3283	169	16	minimal	minimal	ADJ
ejpam-3283	169	17	n	n	CCONJ
ejpam-3283	169	18	-	-	PUNCT
ejpam-3283	169	19	ideal	ideal	NOUN
ejpam-3283	169	20	of	of	ADP
ejpam-3283	169	21	s.	s.	PROPN
ejpam-3283	169	22	p.	p.	PROPN
ejpam-3283	169	23	petchkaew	petchkaew	PROPN
ejpam-3283	169	24	,	,	PUNCT
ejpam-3283	169	25	r.	r.	PROPN
ejpam-3283	169	26	chinram	chinram	PROPN
ejpam-3283	169	27	/	/	SYM
ejpam-3283	169	28	eur	eur	PROPN
ejpam-3283	169	29	.	.	PUNCT
ejpam-3283	170	1	j.	j.	PROPN
ejpam-3283	170	2	pure	pure	PROPN
ejpam-3283	170	3	appl	appl	PROPN
ejpam-3283	170	4	.	.	PROPN
ejpam-3283	170	5	math	math	PROPN
ejpam-3283	170	6	,	,	PUNCT
ejpam-3283	170	7	11	11	NUM
ejpam-3283	170	8	(	(	PUNCT
ejpam-3283	170	9	3	3	NUM
ejpam-3283	170	10	)	)	PUNCT
ejpam-3283	170	11	(	(	PUNCT
ejpam-3283	170	12	2018	2018	NUM
ejpam-3283	170	13	)	)	PUNCT
ejpam-3283	170	14	,	,	PUNCT
ejpam-3283	170	15	762	762	NUM
ejpam-3283	170	16	-	-	SYM
ejpam-3283	170	17	773	773	NUM
ejpam-3283	170	18	767	767	NUM
ejpam-3283	170	19	theorem	theorem	NOUN
ejpam-3283	170	20	2	2	NUM
ejpam-3283	170	21	.	.	PUNCT
ejpam-3283	171	1	if	if	SCONJ
ejpam-3283	171	2	s	s	PROPN
ejpam-3283	171	3	has	have	VERB
ejpam-3283	171	4	a	a	DET
ejpam-3283	171	5	zero	zero	NUM
ejpam-3283	171	6	element	element	NOUN
ejpam-3283	171	7	and	and	CCONJ
ejpam-3283	171	8	i	i	PRON
ejpam-3283	171	9	is	be	AUX
ejpam-3283	171	10	a	a	DET
ejpam-3283	171	11	nonzero	nonzero	ADJ
ejpam-3283	171	12	n	n	CCONJ
ejpam-3283	171	13	-	-	PUNCT
ejpam-3283	171	14	ideal	ideal	NOUN
ejpam-3283	171	15	of	of	ADP
ejpam-3283	171	16	s	s	PROPN
ejpam-3283	171	17	,	,	PUNCT
ejpam-3283	171	18	then	then	ADV
ejpam-3283	171	19	the	the	DET
ejpam-3283	171	20	following	follow	VERB
ejpam-3283	171	21	statement	statement	NOUN
ejpam-3283	171	22	hold	hold	VERB
ejpam-3283	171	23	:	:	PUNCT
ejpam-3283	171	24	(	(	PUNCT
ejpam-3283	171	25	1	1	X
ejpam-3283	171	26	)	)	PUNCT
ejpam-3283	171	27	if	if	SCONJ
ejpam-3283	171	28	i	i	PRON
ejpam-3283	171	29	is	be	AUX
ejpam-3283	171	30	a	a	DET
ejpam-3283	171	31	0	0	NUM
ejpam-3283	171	32	-	-	PUNCT
ejpam-3283	171	33	minimal	minimal	ADJ
ejpam-3283	171	34	n	n	CCONJ
ejpam-3283	171	35	-	-	PUNCT
ejpam-3283	171	36	ideal	ideal	NOUN
ejpam-3283	171	37	of	of	ADP
ejpam-3283	171	38	s	s	PROPN
ejpam-3283	171	39	,	,	PUNCT
ejpam-3283	171	40	then	then	ADV
ejpam-3283	171	41	either	either	CCONJ
ejpam-3283	171	42	f(in−1	f(in−1	ADP
ejpam-3283	171	43	,	,	PUNCT
ejpam-3283	171	44	j	j	NOUN
ejpam-3283	171	45	)	)	PUNCT
ejpam-3283	171	46	=	=	PRON
ejpam-3283	171	47	{	{	PUNCT
ejpam-3283	171	48	0	0	NUM
ejpam-3283	171	49	}	}	PUNCT
ejpam-3283	171	50	for	for	ADP
ejpam-3283	171	51	some	some	DET
ejpam-3283	171	52	nonzero	nonzero	ADJ
ejpam-3283	171	53	n	n	CCONJ
ejpam-3283	171	54	-	-	PUNCT
ejpam-3283	171	55	ideal	ideal	NOUN
ejpam-3283	171	56	j	j	PROPN
ejpam-3283	171	57	of	of	ADP
ejpam-3283	171	58	i	i	PRON
ejpam-3283	171	59	or	or	CCONJ
ejpam-3283	171	60	i	i	PRON
ejpam-3283	171	61	is	be	AUX
ejpam-3283	171	62	0	0	NUM
ejpam-3283	171	63	-	-	PUNCT
ejpam-3283	171	64	n	n	CCONJ
ejpam-3283	171	65	-	-	PUNCT
ejpam-3283	171	66	simple	simple	NOUN
ejpam-3283	171	67	.	.	PUNCT
ejpam-3283	172	1	(	(	PUNCT
ejpam-3283	172	2	2	2	X
ejpam-3283	172	3	)	)	PUNCT
ejpam-3283	172	4	if	if	SCONJ
ejpam-3283	172	5	i	i	PRON
ejpam-3283	172	6	is	be	AUX
ejpam-3283	172	7	0	0	NUM
ejpam-3283	172	8	-	-	PUNCT
ejpam-3283	172	9	n	n	CCONJ
ejpam-3283	172	10	-	-	PUNCT
ejpam-3283	172	11	simple	simple	NOUN
ejpam-3283	172	12	,	,	PUNCT
ejpam-3283	172	13	then	then	ADV
ejpam-3283	172	14	i	i	PRON
ejpam-3283	172	15	is	be	AUX
ejpam-3283	172	16	a	a	DET
ejpam-3283	172	17	0	0	NUM
ejpam-3283	172	18	-	-	PUNCT
ejpam-3283	172	19	minimal	minimal	ADJ
ejpam-3283	172	20	n	n	CCONJ
ejpam-3283	172	21	-	-	PUNCT
ejpam-3283	172	22	ideal	ideal	NOUN
ejpam-3283	172	23	of	of	ADP
ejpam-3283	172	24	s.	s.	PROPN
ejpam-3283	172	25	proof	proof	PROPN
ejpam-3283	172	26	.	.	PUNCT
ejpam-3283	173	1	(	(	PUNCT
ejpam-3283	173	2	1	1	X
ejpam-3283	173	3	)	)	PUNCT
ejpam-3283	173	4	assume	assume	VERB
ejpam-3283	173	5	that	that	SCONJ
ejpam-3283	173	6	i	i	PRON
ejpam-3283	173	7	is	be	AUX
ejpam-3283	173	8	a	a	DET
ejpam-3283	173	9	0	0	NUM
ejpam-3283	173	10	-	-	PUNCT
ejpam-3283	173	11	minimal	minimal	ADJ
ejpam-3283	173	12	n	n	CCONJ
ejpam-3283	173	13	-	-	PUNCT
ejpam-3283	173	14	ideal	ideal	NOUN
ejpam-3283	173	15	of	of	ADP
ejpam-3283	173	16	s	s	PRON
ejpam-3283	173	17	and	and	CCONJ
ejpam-3283	173	18	assume	assume	VERB
ejpam-3283	173	19	that	that	SCONJ
ejpam-3283	173	20	f(in−1	f(in−1	ADP
ejpam-3283	173	21	,	,	PUNCT
ejpam-3283	173	22	j	j	NOUN
ejpam-3283	173	23	)	)	PUNCT
ejpam-3283	173	24	6=	6=	ADP
ejpam-3283	173	25	{	{	PUNCT
ejpam-3283	173	26	0	0	NUM
ejpam-3283	173	27	}	}	PUNCT
ejpam-3283	173	28	for	for	ADP
ejpam-3283	173	29	any	any	DET
ejpam-3283	173	30	nonzero	nonzero	NOUN
ejpam-3283	173	31	n	n	CCONJ
ejpam-3283	173	32	-	-	PUNCT
ejpam-3283	173	33	ideal	ideal	NOUN
ejpam-3283	173	34	j	j	PROPN
ejpam-3283	173	35	of	of	ADP
ejpam-3283	173	36	i.	i.	PROPN
ejpam-3283	173	37	let	let	VERB
ejpam-3283	173	38	j	j	PROPN
ejpam-3283	173	39	be	be	AUX
ejpam-3283	173	40	a	a	DET
ejpam-3283	173	41	nonzero	nonzero	ADJ
ejpam-3283	173	42	n	n	CCONJ
ejpam-3283	173	43	-	-	PUNCT
ejpam-3283	173	44	ideal	ideal	NOUN
ejpam-3283	173	45	of	of	ADP
ejpam-3283	173	46	i.	i.	NOUN
ejpam-3283	173	47	then	then	ADV
ejpam-3283	173	48	{	{	PUNCT
ejpam-3283	173	49	0	0	NUM
ejpam-3283	173	50	}	}	PUNCT
ejpam-3283	173	51	6=	6=	ADP
ejpam-3283	173	52	f(in−1	f(in−1	PROPN
ejpam-3283	173	53	,	,	PUNCT
ejpam-3283	173	54	j	j	NOUN
ejpam-3283	173	55	)	)	PUNCT
ejpam-3283	173	56	⊆	⊆	NUM
ejpam-3283	173	57	j	j	PROPN
ejpam-3283	173	58	⊆	⊆	NUM
ejpam-3283	173	59	i.	i.	NOUN
ejpam-3283	173	60	moreover	moreover	ADV
ejpam-3283	173	61	,	,	PUNCT
ejpam-3283	173	62	we	we	PRON
ejpam-3283	173	63	obtain	obtain	VERB
ejpam-3283	173	64	that	that	PRON
ejpam-3283	173	65	f(in−1	f(in−1	ADP
ejpam-3283	173	66	,	,	PUNCT
ejpam-3283	173	67	j	j	NOUN
ejpam-3283	173	68	)	)	PUNCT
ejpam-3283	173	69	is	be	AUX
ejpam-3283	173	70	an	an	DET
ejpam-3283	173	71	n	n	CCONJ
ejpam-3283	173	72	-	-	PUNCT
ejpam-3283	173	73	ideal	ideal	NOUN
ejpam-3283	173	74	of	of	ADP
ejpam-3283	173	75	s	s	PRON
ejpam-3283	173	76	by	by	ADP
ejpam-3283	173	77	lemma	lemma	PROPN
ejpam-3283	173	78	7	7	NUM
ejpam-3283	173	79	.	.	PUNCT
ejpam-3283	174	1	since	since	SCONJ
ejpam-3283	174	2	i	i	PRON
ejpam-3283	174	3	is	be	AUX
ejpam-3283	174	4	a	a	DET
ejpam-3283	174	5	0	0	NUM
ejpam-3283	174	6	-	-	PUNCT
ejpam-3283	174	7	minimal	minimal	ADJ
ejpam-3283	174	8	n	n	CCONJ
ejpam-3283	174	9	-	-	PUNCT
ejpam-3283	174	10	ideal	ideal	NOUN
ejpam-3283	174	11	of	of	ADP
ejpam-3283	174	12	s	s	PROPN
ejpam-3283	174	13	,	,	PUNCT
ejpam-3283	174	14	i	i	PRON
ejpam-3283	174	15	⊆	⊆	NUM
ejpam-3283	174	16	f(in−1	f(in−1	X
ejpam-3283	174	17	,	,	PUNCT
ejpam-3283	174	18	j	j	NOUN
ejpam-3283	174	19	)	)	PUNCT
ejpam-3283	174	20	.	.	PUNCT
ejpam-3283	175	1	this	this	PRON
ejpam-3283	175	2	implies	imply	VERB
ejpam-3283	175	3	that	that	SCONJ
ejpam-3283	175	4	f(in−1	f(in−1	ADP
ejpam-3283	175	5	,	,	PUNCT
ejpam-3283	175	6	j	j	NOUN
ejpam-3283	175	7	)	)	PUNCT
ejpam-3283	175	8	=	=	SYM
ejpam-3283	176	1	j	j	PROPN
ejpam-3283	176	2	=	=	PROPN
ejpam-3283	176	3	i.	i.	PROPN
ejpam-3283	176	4	therefore	therefore	ADV
ejpam-3283	176	5	,	,	PUNCT
ejpam-3283	176	6	i	i	PRON
ejpam-3283	176	7	is	be	AUX
ejpam-3283	176	8	an	an	DET
ejpam-3283	176	9	0	0	NUM
ejpam-3283	176	10	-	-	PUNCT
ejpam-3283	176	11	n	n	CCONJ
ejpam-3283	176	12	-	-	PUNCT
ejpam-3283	176	13	simple	simple	NOUN
ejpam-3283	176	14	.	.	PUNCT
ejpam-3283	177	1	(	(	PUNCT
ejpam-3283	177	2	2	2	X
ejpam-3283	177	3	)	)	PUNCT
ejpam-3283	177	4	assume	assume	VERB
ejpam-3283	177	5	that	that	SCONJ
ejpam-3283	177	6	i	i	PRON
ejpam-3283	177	7	is	be	AUX
ejpam-3283	177	8	0	0	NUM
ejpam-3283	177	9	-	-	PUNCT
ejpam-3283	177	10	n	n	CCONJ
ejpam-3283	177	11	-	-	PUNCT
ejpam-3283	177	12	simple	simple	NOUN
ejpam-3283	177	13	.	.	PUNCT
ejpam-3283	178	1	let	let	VERB
ejpam-3283	178	2	j	j	NOUN
ejpam-3283	178	3	be	be	AUX
ejpam-3283	178	4	a	a	DET
ejpam-3283	178	5	nonzero	nonzero	ADJ
ejpam-3283	178	6	n	n	CCONJ
ejpam-3283	178	7	-	-	PUNCT
ejpam-3283	178	8	ideal	ideal	NOUN
ejpam-3283	178	9	of	of	ADP
ejpam-3283	178	10	s	s	PRON
ejpam-3283	178	11	such	such	ADJ
ejpam-3283	178	12	that	that	SCONJ
ejpam-3283	178	13	j	j	PROPN
ejpam-3283	178	14	⊆	⊆	NUM
ejpam-3283	178	15	i.	i.	NOUN
ejpam-3283	178	16	this	this	PRON
ejpam-3283	178	17	implies	imply	VERB
ejpam-3283	178	18	that	that	SCONJ
ejpam-3283	178	19	i	i	PRON
ejpam-3283	178	20	r	r	NOUN
ejpam-3283	178	21	{	{	PUNCT
ejpam-3283	178	22	0	0	NUM
ejpam-3283	178	23	}	}	PUNCT
ejpam-3283	178	24	∩	∩	ADJ
ejpam-3283	178	25	j	j	PROPN
ejpam-3283	178	26	6=	6=	PROPN
ejpam-3283	178	27	∅	∅	NOUN
ejpam-3283	178	28	and	and	CCONJ
ejpam-3283	178	29	so	so	ADV
ejpam-3283	178	30	i	i	PRON
ejpam-3283	178	31	⊆	⊆	NUM
ejpam-3283	178	32	j	j	PROPN
ejpam-3283	178	33	by	by	ADP
ejpam-3283	178	34	lemma	lemma	PROPN
ejpam-3283	178	35	6(2	6(2	NUM
ejpam-3283	178	36	)	)	PUNCT
ejpam-3283	178	37	.	.	PUNCT
ejpam-3283	179	1	hence	hence	ADV
ejpam-3283	179	2	j	j	PROPN
ejpam-3283	179	3	=	=	PROPN
ejpam-3283	179	4	i.	i.	PROPN
ejpam-3283	179	5	therefore	therefore	ADV
ejpam-3283	179	6	,	,	PUNCT
ejpam-3283	179	7	i	i	PRON
ejpam-3283	179	8	is	be	AUX
ejpam-3283	179	9	a	a	DET
ejpam-3283	179	10	0	0	NUM
ejpam-3283	179	11	-	-	PUNCT
ejpam-3283	179	12	minimal	minimal	ADJ
ejpam-3283	179	13	n	n	CCONJ
ejpam-3283	179	14	-	-	PUNCT
ejpam-3283	179	15	ideal	ideal	NOUN
ejpam-3283	179	16	of	of	ADP
ejpam-3283	179	17	s.	s.	PROPN
ejpam-3283	179	18	example	example	PROPN
ejpam-3283	179	19	4	4	X
ejpam-3283	179	20	.	.	X
ejpam-3283	179	21	consider	consider	VERB
ejpam-3283	179	22	z30	z30	ADJ
ejpam-3283	179	23	,	,	PUNCT
ejpam-3283	179	24	let	let	VERB
ejpam-3283	179	25	s	s	PRON
ejpam-3283	179	26	=	=	VERB
ejpam-3283	179	27	{	{	PUNCT
ejpam-3283	179	28	0	0	NUM
ejpam-3283	179	29	,	,	PUNCT
ejpam-3283	179	30	1	1	NUM
ejpam-3283	179	31	,	,	PUNCT
ejpam-3283	179	32	5	5	NUM
ejpam-3283	179	33	,	,	PUNCT
ejpam-3283	179	34	25	25	NUM
ejpam-3283	179	35	}	}	PUNCT
ejpam-3283	179	36	.	.	PUNCT
ejpam-3283	180	1	define	define	VERB
ejpam-3283	180	2	f	f	PROPN
ejpam-3283	180	3	:	:	PUNCT
ejpam-3283	180	4	sn	sn	PROPN
ejpam-3283	180	5	→	→	SYM
ejpam-3283	180	6	s	s	X
ejpam-3283	180	7	by	by	ADP
ejpam-3283	180	8	f(xn1	f(xn1	NOUN
ejpam-3283	180	9	)	)	PUNCT
ejpam-3283	180	10	=	=	PUNCT
ejpam-3283	181	1	x1	x1	PROPN
ejpam-3283	181	2	·	·	PUNCT
ejpam-3283	181	3	x2	x2	X
ejpam-3283	181	4	·	·	PUNCT
ejpam-3283	181	5	.	.	PUNCT
ejpam-3283	181	6	.	.	PUNCT
ejpam-3283	181	7	.	.	PUNCT
ejpam-3283	182	1	·	·	PUNCT
ejpam-3283	182	2	xn	xn	PUNCT
ejpam-3283	183	1	for	for	ADP
ejpam-3283	183	2	all	all	DET
ejpam-3283	183	3	x1	x1	PROPN
ejpam-3283	183	4	,	,	PUNCT
ejpam-3283	183	5	x2	x2	PROPN
ejpam-3283	183	6	,	,	PUNCT
ejpam-3283	183	7	.	.	PUNCT
ejpam-3283	183	8	.	.	PUNCT
ejpam-3283	183	9	.	.	PUNCT
ejpam-3283	184	1	,	,	PUNCT
ejpam-3283	184	2	xn	xn	PUNCT
ejpam-3283	184	3	∈	∈	PROPN
ejpam-3283	184	4	s	s	VERB
ejpam-3283	184	5	where	where	SCONJ
ejpam-3283	184	6	·	·	PUNCT
ejpam-3283	184	7	is	be	AUX
ejpam-3283	184	8	the	the	DET
ejpam-3283	184	9	multiplication	multiplication	NOUN
ejpam-3283	184	10	of	of	ADP
ejpam-3283	184	11	z30	z30	NOUN
ejpam-3283	184	12	.	.	PUNCT
ejpam-3283	185	1	it	it	PRON
ejpam-3283	185	2	is	be	AUX
ejpam-3283	185	3	easy	easy	ADJ
ejpam-3283	185	4	to	to	PART
ejpam-3283	185	5	see	see	VERB
ejpam-3283	185	6	that	that	PRON
ejpam-3283	185	7	i	i	PRON
ejpam-3283	185	8	=	=	PUNCT
ejpam-3283	185	9	{	{	PUNCT
ejpam-3283	185	10	0	0	NUM
ejpam-3283	185	11	,	,	PUNCT
ejpam-3283	185	12	5	5	NUM
ejpam-3283	185	13	,	,	PUNCT
ejpam-3283	185	14	25	25	NUM
ejpam-3283	185	15	}	}	PUNCT
ejpam-3283	185	16	is	be	AUX
ejpam-3283	185	17	a	a	DET
ejpam-3283	185	18	0	0	NUM
ejpam-3283	185	19	-	-	PUNCT
ejpam-3283	185	20	minimal	minimal	ADJ
ejpam-3283	185	21	n	n	CCONJ
ejpam-3283	185	22	-	-	PUNCT
ejpam-3283	185	23	ideal	ideal	NOUN
ejpam-3283	185	24	of	of	ADP
ejpam-3283	185	25	s.	s.	PROPN
ejpam-3283	185	26	theorem	theorem	VERB
ejpam-3283	185	27	3	3	X
ejpam-3283	185	28	.	.	PUNCT
ejpam-3283	186	1	if	if	SCONJ
ejpam-3283	186	2	s	s	PROPN
ejpam-3283	186	3	has	have	VERB
ejpam-3283	186	4	no	no	DET
ejpam-3283	186	5	zero	zero	NUM
ejpam-3283	186	6	element	element	NOUN
ejpam-3283	186	7	but	but	CCONJ
ejpam-3283	186	8	it	it	PRON
ejpam-3283	186	9	has	have	VERB
ejpam-3283	186	10	proper	proper	ADJ
ejpam-3283	186	11	n	n	CCONJ
ejpam-3283	186	12	-	-	PUNCT
ejpam-3283	186	13	ideals	ideal	NOUN
ejpam-3283	186	14	,	,	PUNCT
ejpam-3283	186	15	then	then	ADV
ejpam-3283	186	16	every	every	DET
ejpam-3283	186	17	proper	proper	ADJ
ejpam-3283	186	18	n	n	CCONJ
ejpam-3283	186	19	-	-	PUNCT
ejpam-3283	186	20	ideal	ideal	NOUN
ejpam-3283	186	21	of	of	ADP
ejpam-3283	186	22	s	s	NOUN
ejpam-3283	186	23	is	be	AUX
ejpam-3283	186	24	minimal	minimal	ADJ
ejpam-3283	186	25	if	if	SCONJ
ejpam-3283	186	26	and	and	CCONJ
ejpam-3283	186	27	only	only	ADV
ejpam-3283	186	28	if	if	SCONJ
ejpam-3283	186	29	s	s	NOUN
ejpam-3283	186	30	contains	contain	VERB
ejpam-3283	186	31	exactly	exactly	ADV
ejpam-3283	186	32	one	one	NUM
ejpam-3283	186	33	proper	proper	ADJ
ejpam-3283	186	34	n	n	CCONJ
ejpam-3283	186	35	-	-	PUNCT
ejpam-3283	186	36	ideal	ideal	NOUN
ejpam-3283	186	37	or	or	CCONJ
ejpam-3283	186	38	s	s	NOUN
ejpam-3283	186	39	contains	contain	VERB
ejpam-3283	186	40	exactly	exactly	ADV
ejpam-3283	186	41	two	two	NUM
ejpam-3283	186	42	proper	proper	ADJ
ejpam-3283	186	43	n	n	CCONJ
ejpam-3283	186	44	-	-	PUNCT
ejpam-3283	186	45	ideals	ideal	NOUN
ejpam-3283	186	46	i1	i1	NOUN
ejpam-3283	186	47	and	and	CCONJ
ejpam-3283	186	48	i2	i2	PROPN
ejpam-3283	186	49	such	such	ADJ
ejpam-3283	186	50	that	that	SCONJ
ejpam-3283	186	51	i1	i1	PROPN
ejpam-3283	186	52	∪	∪	PROPN
ejpam-3283	186	53	i2	i2	PROPN
ejpam-3283	186	54	=	=	SYM
ejpam-3283	186	55	s	s	PROPN
ejpam-3283	186	56	and	and	CCONJ
ejpam-3283	186	57	i1	i1	PROPN
ejpam-3283	186	58	∩	∩	PROPN
ejpam-3283	186	59	i2	i2	PROPN
ejpam-3283	186	60	=	=	PUNCT
ejpam-3283	186	61	∅.	∅.	NOUN
ejpam-3283	186	62	proof	proof	NOUN
ejpam-3283	186	63	.	.	PUNCT
ejpam-3283	187	1	assume	assume	VERB
ejpam-3283	187	2	that	that	SCONJ
ejpam-3283	187	3	every	every	DET
ejpam-3283	187	4	proper	proper	ADJ
ejpam-3283	187	5	n	n	CCONJ
ejpam-3283	187	6	-	-	PUNCT
ejpam-3283	187	7	ideal	ideal	NOUN
ejpam-3283	187	8	of	of	ADP
ejpam-3283	187	9	s	s	NOUN
ejpam-3283	187	10	is	be	AUX
ejpam-3283	187	11	minimal	minimal	ADJ
ejpam-3283	187	12	.	.	PUNCT
ejpam-3283	188	1	let	let	VERB
ejpam-3283	188	2	i	i	PRON
ejpam-3283	188	3	be	be	AUX
ejpam-3283	188	4	a	a	DET
ejpam-3283	188	5	proper	proper	ADJ
ejpam-3283	188	6	n	n	CCONJ
ejpam-3283	188	7	-	-	PUNCT
ejpam-3283	188	8	ideal	ideal	NOUN
ejpam-3283	188	9	of	of	ADP
ejpam-3283	188	10	s.	s.	PROPN
ejpam-3283	189	1	then	then	ADV
ejpam-3283	189	2	i	i	PRON
ejpam-3283	189	3	is	be	AUX
ejpam-3283	189	4	a	a	DET
ejpam-3283	189	5	minimal	minimal	ADJ
ejpam-3283	189	6	n	n	CCONJ
ejpam-3283	189	7	-	-	PUNCT
ejpam-3283	189	8	ideal	ideal	NOUN
ejpam-3283	189	9	of	of	ADP
ejpam-3283	189	10	s.	s.	PROPN
ejpam-3283	189	11	we	we	PRON
ejpam-3283	189	12	divide	divide	VERB
ejpam-3283	189	13	into	into	ADP
ejpam-3283	189	14	two	two	NUM
ejpam-3283	189	15	cases	case	NOUN
ejpam-3283	189	16	:	:	PUNCT
ejpam-3283	189	17	case	case	NOUN
ejpam-3283	189	18	1	1	NUM
ejpam-3283	189	19	:	:	PUNCT
ejpam-3283	189	20	suppose	suppose	VERB
ejpam-3283	189	21	that	that	SCONJ
ejpam-3283	189	22	s	s	NOUN
ejpam-3283	189	23	=	=	NOUN
ejpam-3283	189	24	in(a	in(a	X
ejpam-3283	189	25	)	)	PUNCT
ejpam-3283	189	26	for	for	ADP
ejpam-3283	189	27	all	all	DET
ejpam-3283	189	28	a	a	DET
ejpam-3283	189	29	∈	∈	NOUN
ejpam-3283	189	30	s	s	PART
ejpam-3283	189	31	r	r	NOUN
ejpam-3283	189	32	i.	i.	NOUN
ejpam-3283	189	33	let	let	VERB
ejpam-3283	189	34	j	j	PROPN
ejpam-3283	189	35	be	be	AUX
ejpam-3283	189	36	a	a	DET
ejpam-3283	189	37	proper	proper	ADJ
ejpam-3283	189	38	n	n	CCONJ
ejpam-3283	189	39	-	-	PUNCT
ejpam-3283	189	40	ideal	ideal	NOUN
ejpam-3283	189	41	of	of	ADP
ejpam-3283	189	42	s	s	NOUN
ejpam-3283	189	43	and	and	CCONJ
ejpam-3283	189	44	j	j	PROPN
ejpam-3283	189	45	6=	6=	PROPN
ejpam-3283	190	1	i	i	PRON
ejpam-3283	190	2	,	,	PUNCT
ejpam-3283	190	3	then	then	ADV
ejpam-3283	190	4	j	j	PROPN
ejpam-3283	190	5	r	r	NOUN
ejpam-3283	190	6	i	i	PROPN
ejpam-3283	190	7	6=	6=	NOUN
ejpam-3283	190	8	∅	∅	NOUN
ejpam-3283	190	9	because	because	SCONJ
ejpam-3283	190	10	i	i	PRON
ejpam-3283	190	11	is	be	AUX
ejpam-3283	190	12	a	a	DET
ejpam-3283	190	13	minimal	minimal	ADJ
ejpam-3283	190	14	n	n	CCONJ
ejpam-3283	190	15	-	-	PUNCT
ejpam-3283	190	16	ideals	ideal	NOUN
ejpam-3283	190	17	of	of	ADP
ejpam-3283	190	18	s.	s.	PROPN
ejpam-3283	190	19	thus	thus	ADV
ejpam-3283	190	20	there	there	PRON
ejpam-3283	190	21	exists	exist	VERB
ejpam-3283	190	22	a	a	DET
ejpam-3283	190	23	∈	∈	PROPN
ejpam-3283	190	24	j	j	NOUN
ejpam-3283	190	25	r	r	NOUN
ejpam-3283	190	26	i	i	NOUN
ejpam-3283	191	1	⊆	⊆	NUM
ejpam-3283	191	2	s	s	NOUN
ejpam-3283	191	3	r	r	NOUN
ejpam-3283	191	4	i.	i.	NOUN
ejpam-3283	191	5	hence	hence	ADV
ejpam-3283	191	6	s	s	PART
ejpam-3283	191	7	=	=	NOUN
ejpam-3283	191	8	in(a	in(a	X
ejpam-3283	191	9	)	)	PUNCT
ejpam-3283	191	10	⊆	⊆	NUM
ejpam-3283	191	11	j	j	PROPN
ejpam-3283	191	12	⊆	⊆	NUM
ejpam-3283	191	13	s.	s.	PROPN
ejpam-3283	191	14	so	so	ADV
ejpam-3283	191	15	j	j	PROPN
ejpam-3283	191	16	=	=	SYM
ejpam-3283	191	17	s	s	PROPN
ejpam-3283	191	18	,	,	PUNCT
ejpam-3283	191	19	which	which	PRON
ejpam-3283	191	20	is	be	AUX
ejpam-3283	191	21	a	a	DET
ejpam-3283	191	22	contradiction	contradiction	NOUN
ejpam-3283	191	23	.	.	PUNCT
ejpam-3283	192	1	this	this	PRON
ejpam-3283	192	2	implies	imply	VERB
ejpam-3283	192	3	that	that	SCONJ
ejpam-3283	192	4	j	j	PROPN
ejpam-3283	192	5	=	=	PROPN
ejpam-3283	192	6	i.	i.	PROPN
ejpam-3283	192	7	in	in	ADP
ejpam-3283	192	8	this	this	DET
ejpam-3283	192	9	case	case	NOUN
ejpam-3283	192	10	,	,	PUNCT
ejpam-3283	192	11	we	we	PRON
ejpam-3283	192	12	can	can	AUX
ejpam-3283	192	13	conclude	conclude	VERB
ejpam-3283	192	14	that	that	PRON
ejpam-3283	192	15	s	s	NOUN
ejpam-3283	192	16	contains	contain	VERB
ejpam-3283	192	17	exactly	exactly	ADV
ejpam-3283	192	18	one	one	NUM
ejpam-3283	192	19	proper	proper	ADJ
ejpam-3283	192	20	n	n	CCONJ
ejpam-3283	192	21	-	-	PUNCT
ejpam-3283	192	22	ideal	ideal	NOUN
ejpam-3283	192	23	of	of	ADP
ejpam-3283	192	24	s.	s.	PROPN
ejpam-3283	192	25	case	case	PROPN
ejpam-3283	192	26	2	2	NUM
ejpam-3283	192	27	:	:	PUNCT
ejpam-3283	192	28	suppose	suppose	VERB
ejpam-3283	192	29	that	that	SCONJ
ejpam-3283	192	30	there	there	PRON
ejpam-3283	192	31	exists	exist	VERB
ejpam-3283	192	32	a	a	DET
ejpam-3283	192	33	∈	∈	NOUN
ejpam-3283	192	34	s	s	PART
ejpam-3283	192	35	r	r	NOUN
ejpam-3283	192	36	i	i	PRON
ejpam-3283	192	37	such	such	ADJ
ejpam-3283	192	38	that	that	DET
ejpam-3283	192	39	s	s	PROPN
ejpam-3283	192	40	6=	6=	NUM
ejpam-3283	192	41	in(a	in(a	NUM
ejpam-3283	192	42	)	)	PUNCT
ejpam-3283	192	43	.	.	PUNCT
ejpam-3283	193	1	this	this	PRON
ejpam-3283	193	2	implies	imply	VERB
ejpam-3283	193	3	that	that	SCONJ
ejpam-3283	193	4	in(a	in(a	NUM
ejpam-3283	193	5	)	)	PUNCT
ejpam-3283	193	6	6=	6=	ADP
ejpam-3283	194	1	i	i	PRON
ejpam-3283	194	2	and	and	CCONJ
ejpam-3283	194	3	in(a	in(a	NUM
ejpam-3283	194	4	)	)	PUNCT
ejpam-3283	194	5	is	be	AUX
ejpam-3283	194	6	a	a	DET
ejpam-3283	194	7	minimal	minimal	ADJ
ejpam-3283	194	8	n	n	CCONJ
ejpam-3283	194	9	-	-	PUNCT
ejpam-3283	194	10	ideal	ideal	NOUN
ejpam-3283	194	11	of	of	ADP
ejpam-3283	194	12	s	s	PRON
ejpam-3283	194	13	by	by	ADP
ejpam-3283	194	14	the	the	DET
ejpam-3283	194	15	fact	fact	NOUN
ejpam-3283	194	16	that	that	SCONJ
ejpam-3283	194	17	in(a	in(a	NUM
ejpam-3283	194	18	)	)	PUNCT
ejpam-3283	194	19	is	be	AUX
ejpam-3283	194	20	a	a	DET
ejpam-3283	194	21	proper	proper	ADJ
ejpam-3283	194	22	n	n	CCONJ
ejpam-3283	194	23	-	-	PUNCT
ejpam-3283	194	24	ideal	ideal	NOUN
ejpam-3283	194	25	of	of	ADP
ejpam-3283	194	26	s.	s.	PROPN
ejpam-3283	194	27	by	by	ADP
ejpam-3283	194	28	lemma	lemma	PROPN
ejpam-3283	194	29	5	5	NUM
ejpam-3283	194	30	,	,	PUNCT
ejpam-3283	194	31	we	we	PRON
ejpam-3283	194	32	gain	gain	VERB
ejpam-3283	194	33	that	that	SCONJ
ejpam-3283	194	34	in(a)∪	in(a)∪	PRON
ejpam-3283	195	1	i	i	PRON
ejpam-3283	195	2	is	be	AUX
ejpam-3283	195	3	an	an	DET
ejpam-3283	195	4	n	n	CCONJ
ejpam-3283	195	5	-	-	PUNCT
ejpam-3283	195	6	ideal	ideal	NOUN
ejpam-3283	195	7	of	of	ADP
ejpam-3283	195	8	s.	s.	PROPN
ejpam-3283	195	9	since	since	SCONJ
ejpam-3283	195	10	i	i	PRON
ejpam-3283	195	11	is	be	AUX
ejpam-3283	195	12	a	a	DET
ejpam-3283	195	13	minimal	minimal	ADJ
ejpam-3283	195	14	n	n	CCONJ
ejpam-3283	195	15	-	-	PUNCT
ejpam-3283	195	16	ideal	ideal	NOUN
ejpam-3283	195	17	of	of	ADP
ejpam-3283	195	18	s	s	PRON
ejpam-3283	196	1	and	and	CCONJ
ejpam-3283	196	2	i	i	PRON
ejpam-3283	196	3	(	(	PUNCT
ejpam-3283	196	4	in(a)∪	in(a)∪	INTJ
ejpam-3283	196	5	i	i	PRON
ejpam-3283	196	6	,	,	PUNCT
ejpam-3283	196	7	we	we	PRON
ejpam-3283	196	8	acquire	acquire	VERB
ejpam-3283	196	9	that	that	SCONJ
ejpam-3283	196	10	in(a)∪	in(a)∪	PRON
ejpam-3283	197	1	i	i	NOUN
ejpam-3283	197	2	=	=	SYM
ejpam-3283	198	1	s	s	VERB
ejpam-3283	198	2	otherwise	otherwise	ADV
ejpam-3283	199	1	in(a)∪	in(a)∪	INTJ
ejpam-3283	200	1	i	i	PRON
ejpam-3283	200	2	must	must	AUX
ejpam-3283	200	3	be	be	AUX
ejpam-3283	200	4	a	a	DET
ejpam-3283	200	5	minimal	minimal	ADJ
ejpam-3283	200	6	n	n	CCONJ
ejpam-3283	200	7	-	-	PUNCT
ejpam-3283	200	8	ideal	ideal	NOUN
ejpam-3283	200	9	of	of	ADP
ejpam-3283	200	10	s	s	PROPN
ejpam-3283	200	11	,	,	PUNCT
ejpam-3283	200	12	it	it	PRON
ejpam-3283	200	13	is	be	AUX
ejpam-3283	200	14	impossible	impossible	ADJ
ejpam-3283	200	15	.	.	PUNCT
ejpam-3283	201	1	by	by	ADP
ejpam-3283	201	2	the	the	DET
ejpam-3283	201	3	minimality	minimality	NOUN
ejpam-3283	201	4	of	of	ADP
ejpam-3283	201	5	an	an	DET
ejpam-3283	201	6	n	n	CCONJ
ejpam-3283	201	7	-	-	PUNCT
ejpam-3283	201	8	ideal	ideal	NOUN
ejpam-3283	201	9	in(a	in(a	NUM
ejpam-3283	201	10	)	)	PUNCT
ejpam-3283	201	11	and	and	CCONJ
ejpam-3283	201	12	by	by	ADP
ejpam-3283	201	13	the	the	DET
ejpam-3283	201	14	fact	fact	NOUN
ejpam-3283	201	15	that	that	SCONJ
ejpam-3283	201	16	in(a)∩i	in(a)∩i	PROPN
ejpam-3283	201	17	(	(	PUNCT
ejpam-3283	201	18	in(a	in(a	NUM
ejpam-3283	201	19	)	)	PUNCT
ejpam-3283	201	20	,	,	PUNCT
ejpam-3283	201	21	we	we	PRON
ejpam-3283	201	22	have	have	VERB
ejpam-3283	201	23	that	that	DET
ejpam-3283	201	24	in(a)∩i	in(a)∩i	PROPN
ejpam-3283	201	25	=	=	PUNCT
ejpam-3283	201	26	∅.	∅.	ADP
ejpam-3283	201	27	next	next	ADV
ejpam-3283	201	28	,	,	PUNCT
ejpam-3283	201	29	we	we	PRON
ejpam-3283	201	30	show	show	VERB
ejpam-3283	201	31	that	that	SCONJ
ejpam-3283	201	32	s	s	VERB
ejpam-3283	201	33	has	have	VERB
ejpam-3283	201	34	exactly	exactly	ADV
ejpam-3283	201	35	two	two	NUM
ejpam-3283	201	36	proper	proper	ADJ
ejpam-3283	201	37	n	n	CCONJ
ejpam-3283	201	38	-	-	PUNCT
ejpam-3283	201	39	ideals	ideal	NOUN
ejpam-3283	201	40	i	i	PRON
ejpam-3283	201	41	and	and	CCONJ
ejpam-3283	201	42	in(a	in(a	NUM
ejpam-3283	201	43	)	)	PUNCT
ejpam-3283	201	44	.	.	PUNCT
ejpam-3283	202	1	suppose	suppose	VERB
ejpam-3283	202	2	that	that	SCONJ
ejpam-3283	202	3	m	m	PROPN
ejpam-3283	202	4	is	be	AUX
ejpam-3283	202	5	a	a	DET
ejpam-3283	202	6	proper	proper	ADJ
ejpam-3283	202	7	n	n	CCONJ
ejpam-3283	202	8	-	-	PUNCT
ejpam-3283	202	9	ideal	ideal	NOUN
ejpam-3283	202	10	of	of	ADP
ejpam-3283	202	11	s.	s.	PROPN
ejpam-3283	202	12	then	then	ADV
ejpam-3283	202	13	m	m	PROPN
ejpam-3283	202	14	is	be	AUX
ejpam-3283	202	15	a	a	DET
ejpam-3283	202	16	minimal	minimal	ADJ
ejpam-3283	202	17	n	n	CCONJ
ejpam-3283	202	18	-	-	PUNCT
ejpam-3283	202	19	ideal	ideal	NOUN
ejpam-3283	202	20	of	of	ADP
ejpam-3283	202	21	s	s	PRON
ejpam-3283	202	22	by	by	ADP
ejpam-3283	202	23	the	the	DET
ejpam-3283	202	24	hypothesis	hypothesis	NOUN
ejpam-3283	202	25	.	.	PUNCT
ejpam-3283	203	1	thus	thus	ADV
ejpam-3283	203	2	m	m	VERB
ejpam-3283	203	3	=	=	SYM
ejpam-3283	203	4	m∩s	m∩s	PROPN
ejpam-3283	203	5	=	=	SYM
ejpam-3283	203	6	m∩(in(a)∪i	m∩(in(a)∪i	PROPN
ejpam-3283	203	7	)	)	PUNCT
ejpam-3283	203	8	=	=	SYM
ejpam-3283	203	9	(	(	PUNCT
ejpam-3283	203	10	m∩in(a))∪(m∩i	m∩in(a))∪(m∩i	NOUN
ejpam-3283	203	11	)	)	PUNCT
ejpam-3283	203	12	.	.	PUNCT
ejpam-3283	204	1	if	if	SCONJ
ejpam-3283	204	2	m	m	NOUN
ejpam-3283	204	3	∩	∩	VERB
ejpam-3283	204	4	i	i	ADP
ejpam-3283	204	5	6=	6=	PROPN
ejpam-3283	204	6	∅	∅	NOUN
ejpam-3283	204	7	,	,	PUNCT
ejpam-3283	204	8	then	then	ADV
ejpam-3283	204	9	m	m	VERB
ejpam-3283	204	10	=	=	ADJ
ejpam-3283	204	11	i	i	PRON
ejpam-3283	204	12	because	because	SCONJ
ejpam-3283	204	13	m	m	VERB
ejpam-3283	204	14	and	and	CCONJ
ejpam-3283	204	15	i	i	PRON
ejpam-3283	204	16	are	be	AUX
ejpam-3283	204	17	both	both	PRON
ejpam-3283	204	18	minimal	minimal	ADJ
ejpam-3283	204	19	n	n	CCONJ
ejpam-3283	204	20	-	-	PUNCT
ejpam-3283	204	21	ideals	ideal	NOUN
ejpam-3283	204	22	of	of	ADP
ejpam-3283	204	23	s.	s.	PROPN
ejpam-3283	204	24	if	if	SCONJ
ejpam-3283	204	25	m	m	VERB
ejpam-3283	204	26	∩	∩	ADJ
ejpam-3283	204	27	i	i	NOUN
ejpam-3283	204	28	=	=	SYM
ejpam-3283	204	29	∅	∅	NOUN
ejpam-3283	204	30	,	,	PUNCT
ejpam-3283	204	31	then	then	ADV
ejpam-3283	204	32	m	m	VERB
ejpam-3283	204	33	=	=	ADJ
ejpam-3283	204	34	m	m	NOUN
ejpam-3283	204	35	∩	∩	NOUN
ejpam-3283	204	36	in(a	in(a	NUM
ejpam-3283	204	37	)	)	PUNCT
ejpam-3283	204	38	.	.	PUNCT
ejpam-3283	205	1	hence	hence	ADV
ejpam-3283	205	2	m	m	VERB
ejpam-3283	205	3	∩	∩	NOUN
ejpam-3283	205	4	in(a	in(a	NUM
ejpam-3283	205	5	)	)	PUNCT
ejpam-3283	206	1	6=	6=	ADP
ejpam-3283	206	2	∅.	∅.	ADP
ejpam-3283	206	3	this	this	PRON
ejpam-3283	206	4	implies	imply	VERB
ejpam-3283	206	5	that	that	SCONJ
ejpam-3283	206	6	m	m	VERB
ejpam-3283	206	7	=	=	NOUN
ejpam-3283	206	8	in(a	in(a	NOUN
ejpam-3283	206	9	)	)	PUNCT
ejpam-3283	206	10	because	because	SCONJ
ejpam-3283	206	11	m	m	PROPN
ejpam-3283	206	12	and	and	CCONJ
ejpam-3283	206	13	in(a	in(a	NUM
ejpam-3283	206	14	)	)	PUNCT
ejpam-3283	206	15	are	be	AUX
ejpam-3283	206	16	both	both	PRON
ejpam-3283	206	17	minimal	minimal	ADJ
ejpam-3283	206	18	n	n	CCONJ
ejpam-3283	206	19	-	-	PUNCT
ejpam-3283	206	20	ideals	ideal	NOUN
ejpam-3283	206	21	of	of	ADP
ejpam-3283	206	22	s.	s.	PROPN
ejpam-3283	206	23	in	in	ADP
ejpam-3283	206	24	this	this	DET
ejpam-3283	206	25	case	case	NOUN
ejpam-3283	206	26	,	,	PUNCT
ejpam-3283	206	27	we	we	PRON
ejpam-3283	206	28	can	can	AUX
ejpam-3283	206	29	conclude	conclude	VERB
ejpam-3283	206	30	that	that	PRON
ejpam-3283	206	31	s	s	NOUN
ejpam-3283	206	32	contains	contain	VERB
ejpam-3283	206	33	exactly	exactly	ADV
ejpam-3283	206	34	two	two	NUM
ejpam-3283	206	35	proper	proper	ADJ
ejpam-3283	206	36	n	n	CCONJ
ejpam-3283	206	37	-	-	PUNCT
ejpam-3283	206	38	ideals	ideal	NOUN
ejpam-3283	206	39	i	i	PRON
ejpam-3283	206	40	and	and	CCONJ
ejpam-3283	206	41	in(a	in(a	NUM
ejpam-3283	206	42	)	)	PUNCT
ejpam-3283	206	43	,	,	PUNCT
ejpam-3283	206	44	moreover	moreover	ADV
ejpam-3283	206	45	,	,	PUNCT
ejpam-3283	206	46	we	we	PRON
ejpam-3283	206	47	obtain	obtain	VERB
ejpam-3283	206	48	in(a	in(a	NOUN
ejpam-3283	206	49	)	)	PUNCT
ejpam-3283	206	50	∪	∪	ADP
ejpam-3283	206	51	i	i	PRON
ejpam-3283	206	52	=	=	SYM
ejpam-3283	206	53	s	s	X
ejpam-3283	206	54	and	and	CCONJ
ejpam-3283	206	55	in(a	in(a	NUM
ejpam-3283	206	56	)	)	PUNCT
ejpam-3283	206	57	∩	∩	NOUN
ejpam-3283	207	1	i	i	PRON
ejpam-3283	207	2	=	=	PUNCT
ejpam-3283	207	3	∅.	∅.	PROPN
ejpam-3283	207	4	p.	p.	NOUN
ejpam-3283	207	5	petchkaew	petchkaew	NOUN
ejpam-3283	207	6	,	,	PUNCT
ejpam-3283	207	7	r.	r.	PROPN
ejpam-3283	207	8	chinram	chinram	PROPN
ejpam-3283	207	9	/	/	SYM
ejpam-3283	207	10	eur	eur	PROPN
ejpam-3283	207	11	.	.	PUNCT
ejpam-3283	208	1	j.	j.	PROPN
ejpam-3283	208	2	pure	pure	PROPN
ejpam-3283	208	3	appl	appl	PROPN
ejpam-3283	208	4	.	.	PROPN
ejpam-3283	208	5	math	math	PROPN
ejpam-3283	208	6	,	,	PUNCT
ejpam-3283	208	7	11	11	NUM
ejpam-3283	208	8	(	(	PUNCT
ejpam-3283	208	9	3	3	NUM
ejpam-3283	208	10	)	)	PUNCT
ejpam-3283	208	11	(	(	PUNCT
ejpam-3283	208	12	2018	2018	NUM
ejpam-3283	208	13	)	)	PUNCT
ejpam-3283	208	14	,	,	PUNCT
ejpam-3283	208	15	762	762	NUM
ejpam-3283	208	16	-	-	SYM
ejpam-3283	208	17	773	773	NUM
ejpam-3283	208	18	768	768	NUM
ejpam-3283	208	19	conversely	conversely	ADV
ejpam-3283	208	20	,	,	PUNCT
ejpam-3283	208	21	if	if	SCONJ
ejpam-3283	208	22	s	s	PROPN
ejpam-3283	208	23	has	have	VERB
ejpam-3283	208	24	exactly	exactly	ADV
ejpam-3283	208	25	one	one	NUM
ejpam-3283	208	26	proper	proper	ADJ
ejpam-3283	208	27	n	n	CCONJ
ejpam-3283	208	28	-	-	PUNCT
ejpam-3283	208	29	ideal	ideal	NOUN
ejpam-3283	208	30	,	,	PUNCT
ejpam-3283	208	31	then	then	ADV
ejpam-3283	208	32	it	it	PRON
ejpam-3283	208	33	is	be	AUX
ejpam-3283	208	34	clearly	clearly	ADV
ejpam-3283	208	35	that	that	SCONJ
ejpam-3283	208	36	it	it	PRON
ejpam-3283	208	37	is	be	AUX
ejpam-3283	208	38	just	just	ADV
ejpam-3283	208	39	a	a	DET
ejpam-3283	208	40	minimal	minimal	ADJ
ejpam-3283	208	41	n	n	CCONJ
ejpam-3283	208	42	-	-	PUNCT
ejpam-3283	208	43	ideal	ideal	NOUN
ejpam-3283	208	44	.	.	PUNCT
ejpam-3283	209	1	next	next	ADV
ejpam-3283	209	2	,	,	PUNCT
ejpam-3283	209	3	suppose	suppose	VERB
ejpam-3283	209	4	that	that	SCONJ
ejpam-3283	209	5	s	s	VERB
ejpam-3283	209	6	has	have	VERB
ejpam-3283	209	7	exactly	exactly	ADV
ejpam-3283	209	8	two	two	NUM
ejpam-3283	209	9	proper	proper	ADJ
ejpam-3283	209	10	n	n	CCONJ
ejpam-3283	209	11	-	-	PUNCT
ejpam-3283	209	12	ideals	ideal	NOUN
ejpam-3283	209	13	i1	i1	NOUN
ejpam-3283	209	14	and	and	CCONJ
ejpam-3283	209	15	i2	i2	PROPN
ejpam-3283	209	16	such	such	ADJ
ejpam-3283	209	17	that	that	SCONJ
ejpam-3283	209	18	i1	i1	PROPN
ejpam-3283	209	19	∪	∪	PROPN
ejpam-3283	209	20	i2	i2	PROPN
ejpam-3283	209	21	=	=	SYM
ejpam-3283	209	22	s	s	PROPN
ejpam-3283	209	23	and	and	CCONJ
ejpam-3283	209	24	i1	i1	PROPN
ejpam-3283	209	25	∩	∩	PROPN
ejpam-3283	209	26	i2	i2	PROPN
ejpam-3283	209	27	=	=	PUNCT
ejpam-3283	209	28	∅.	∅.	NOUN
ejpam-3283	209	29	since	since	SCONJ
ejpam-3283	209	30	i1	i1	PROPN
ejpam-3283	209	31	∩	∩	PROPN
ejpam-3283	209	32	i2	i2	PROPN
ejpam-3283	209	33	=	=	NOUN
ejpam-3283	209	34	∅	∅	NOUN
ejpam-3283	209	35	,	,	PUNCT
ejpam-3283	209	36	we	we	PRON
ejpam-3283	209	37	have	have	VERB
ejpam-3283	209	38	that	that	DET
ejpam-3283	209	39	i1	i1	PROPN
ejpam-3283	209	40	6⊆	6⊆	PROPN
ejpam-3283	209	41	i2	i2	PROPN
ejpam-3283	209	42	and	and	CCONJ
ejpam-3283	209	43	i2	i2	PROPN
ejpam-3283	209	44	6⊆	6⊆	PROPN
ejpam-3283	209	45	i1	i1	PROPN
ejpam-3283	209	46	.	.	PUNCT
ejpam-3283	210	1	hence	hence	ADV
ejpam-3283	210	2	i1	i1	PROPN
ejpam-3283	210	3	and	and	CCONJ
ejpam-3283	210	4	i2	i2	PROPN
ejpam-3283	210	5	are	be	AUX
ejpam-3283	210	6	both	both	PRON
ejpam-3283	210	7	minimal	minimal	ADJ
ejpam-3283	210	8	n	n	CCONJ
ejpam-3283	210	9	-	-	PUNCT
ejpam-3283	210	10	ideals	ideal	NOUN
ejpam-3283	210	11	of	of	ADP
ejpam-3283	210	12	s.	s.	PROPN
ejpam-3283	210	13	so	so	SCONJ
ejpam-3283	210	14	we	we	PRON
ejpam-3283	210	15	can	can	AUX
ejpam-3283	210	16	conclude	conclude	VERB
ejpam-3283	210	17	that	that	SCONJ
ejpam-3283	210	18	every	every	DET
ejpam-3283	210	19	proper	proper	ADJ
ejpam-3283	210	20	n	n	CCONJ
ejpam-3283	210	21	-	-	PUNCT
ejpam-3283	210	22	ideal	ideal	NOUN
ejpam-3283	210	23	of	of	ADP
ejpam-3283	210	24	s	s	NOUN
ejpam-3283	210	25	is	be	AUX
ejpam-3283	210	26	minimal	minimal	ADJ
ejpam-3283	210	27	.	.	PUNCT
ejpam-3283	211	1	therefore	therefore	ADV
ejpam-3283	211	2	,	,	PUNCT
ejpam-3283	211	3	the	the	DET
ejpam-3283	211	4	proof	proof	NOUN
ejpam-3283	211	5	is	be	AUX
ejpam-3283	211	6	completed	complete	VERB
ejpam-3283	211	7	.	.	PUNCT
ejpam-3283	212	1	theorem	theorem	VERB
ejpam-3283	212	2	4	4	NUM
ejpam-3283	212	3	.	.	PUNCT
ejpam-3283	213	1	if	if	SCONJ
ejpam-3283	213	2	s	s	PROPN
ejpam-3283	213	3	has	have	VERB
ejpam-3283	213	4	a	a	DET
ejpam-3283	213	5	zero	zero	NUM
ejpam-3283	213	6	element	element	NOUN
ejpam-3283	213	7	and	and	CCONJ
ejpam-3283	213	8	nonzero	nonzero	NOUN
ejpam-3283	213	9	proper	proper	ADJ
ejpam-3283	213	10	n	n	CCONJ
ejpam-3283	213	11	-	-	PUNCT
ejpam-3283	213	12	ideals	ideal	NOUN
ejpam-3283	213	13	,	,	PUNCT
ejpam-3283	213	14	then	then	ADV
ejpam-3283	213	15	every	every	DET
ejpam-3283	213	16	nonzero	nonzero	ADJ
ejpam-3283	213	17	proper	proper	ADJ
ejpam-3283	213	18	n	n	CCONJ
ejpam-3283	213	19	-	-	PUNCT
ejpam-3283	213	20	ideal	ideal	NOUN
ejpam-3283	213	21	of	of	ADP
ejpam-3283	213	22	s	s	PROPN
ejpam-3283	213	23	is	be	AUX
ejpam-3283	213	24	0	0	NUM
ejpam-3283	213	25	-	-	PUNCT
ejpam-3283	213	26	minimal	minimal	ADJ
ejpam-3283	213	27	if	if	SCONJ
ejpam-3283	213	28	and	and	CCONJ
ejpam-3283	213	29	only	only	ADV
ejpam-3283	213	30	if	if	SCONJ
ejpam-3283	213	31	s	s	NOUN
ejpam-3283	213	32	contains	contain	VERB
ejpam-3283	213	33	exactly	exactly	ADV
ejpam-3283	213	34	one	one	NUM
ejpam-3283	213	35	nonzero	nonzero	NOUN
ejpam-3283	213	36	proper	proper	ADJ
ejpam-3283	213	37	nideal	nideal	ADJ
ejpam-3283	213	38	or	or	CCONJ
ejpam-3283	213	39	s	s	NOUN
ejpam-3283	213	40	contains	contain	VERB
ejpam-3283	213	41	exactly	exactly	ADV
ejpam-3283	213	42	two	two	NUM
ejpam-3283	213	43	nonzero	nonzero	ADJ
ejpam-3283	213	44	proper	proper	ADJ
ejpam-3283	213	45	n	n	CCONJ
ejpam-3283	213	46	-	-	PUNCT
ejpam-3283	213	47	ideals	ideal	NOUN
ejpam-3283	213	48	i1	i1	NOUN
ejpam-3283	213	49	and	and	CCONJ
ejpam-3283	213	50	i2	i2	PROPN
ejpam-3283	213	51	such	such	ADJ
ejpam-3283	213	52	that	that	SCONJ
ejpam-3283	213	53	i1	i1	PROPN
ejpam-3283	213	54	∪	∪	PROPN
ejpam-3283	213	55	i2	i2	PROPN
ejpam-3283	213	56	=	=	SYM
ejpam-3283	213	57	s	s	PROPN
ejpam-3283	213	58	and	and	CCONJ
ejpam-3283	213	59	i1	i1	PROPN
ejpam-3283	213	60	∩	∩	PROPN
ejpam-3283	213	61	i2	i2	PROPN
ejpam-3283	213	62	=	=	PUNCT
ejpam-3283	213	63	{	{	PUNCT
ejpam-3283	213	64	0	0	NUM
ejpam-3283	213	65	}	}	PUNCT
ejpam-3283	213	66	.	.	PUNCT
ejpam-3283	214	1	proof	proof	NOUN
ejpam-3283	214	2	.	.	PUNCT
ejpam-3283	215	1	it	it	PRON
ejpam-3283	215	2	follows	follow	VERB
ejpam-3283	215	3	from	from	ADP
ejpam-3283	215	4	the	the	DET
ejpam-3283	215	5	proof	proof	NOUN
ejpam-3283	215	6	of	of	ADP
ejpam-3283	215	7	theorem	theorem	ADJ
ejpam-3283	215	8	3	3	NUM
ejpam-3283	215	9	and	and	CCONJ
ejpam-3283	215	10	use	use	VERB
ejpam-3283	215	11	the	the	DET
ejpam-3283	215	12	fact	fact	NOUN
ejpam-3283	215	13	that	that	SCONJ
ejpam-3283	215	14	every	every	DET
ejpam-3283	215	15	n	n	NOUN
ejpam-3283	215	16	-	-	PUNCT
ejpam-3283	215	17	ideal	ideal	NOUN
ejpam-3283	215	18	of	of	ADP
ejpam-3283	215	19	s	s	PROPN
ejpam-3283	215	20	contains	contain	VERB
ejpam-3283	215	21	a	a	DET
ejpam-3283	215	22	zero	zero	NUM
ejpam-3283	215	23	element	element	NOUN
ejpam-3283	215	24	.	.	PUNCT
ejpam-3283	216	1	5	5	X
ejpam-3283	216	2	.	.	X
ejpam-3283	216	3	maximality	maximality	NOUN
ejpam-3283	216	4	of	of	ADP
ejpam-3283	216	5	n	n	CCONJ
ejpam-3283	216	6	-	-	PUNCT
ejpam-3283	216	7	ideals	ideal	NOUN
ejpam-3283	216	8	as	as	ADP
ejpam-3283	216	9	a	a	DET
ejpam-3283	216	10	result	result	NOUN
ejpam-3283	216	11	of	of	ADP
ejpam-3283	216	12	this	this	DET
ejpam-3283	216	13	section	section	NOUN
ejpam-3283	216	14	,	,	PUNCT
ejpam-3283	216	15	we	we	PRON
ejpam-3283	216	16	give	give	VERB
ejpam-3283	216	17	some	some	DET
ejpam-3283	216	18	characterization	characterization	NOUN
ejpam-3283	216	19	of	of	ADP
ejpam-3283	216	20	the	the	DET
ejpam-3283	216	21	minimality	minimality	NOUN
ejpam-3283	216	22	of	of	ADP
ejpam-3283	216	23	n	n	CCONJ
ejpam-3283	216	24	-	-	PUNCT
ejpam-3283	216	25	ideals	ideal	NOUN
ejpam-3283	216	26	of	of	ADP
ejpam-3283	216	27	n	n	CCONJ
ejpam-3283	216	28	-	-	PUNCT
ejpam-3283	216	29	ary	ary	NOUN
ejpam-3283	216	30	semigroups	semigroup	NOUN
ejpam-3283	216	31	as	as	ADV
ejpam-3283	216	32	well	well	ADV
ejpam-3283	216	33	as	as	ADP
ejpam-3283	216	34	the	the	DET
ejpam-3283	216	35	relationship	relationship	NOUN
ejpam-3283	216	36	between	between	ADP
ejpam-3283	216	37	maximality	maximality	NOUN
ejpam-3283	216	38	of	of	ADP
ejpam-3283	216	39	n	n	CCONJ
ejpam-3283	216	40	-	-	PUNCT
ejpam-3283	216	41	ideals	ideal	NOUN
ejpam-3283	216	42	and	and	CCONJ
ejpam-3283	216	43	the	the	DET
ejpam-3283	216	44	union	union	NOUN
ejpam-3283	216	45	u	u	NOUN
ejpam-3283	216	46	of	of	ADP
ejpam-3283	216	47	all	all	DET
ejpam-3283	216	48	(	(	PUNCT
ejpam-3283	216	49	nonzero	nonzero	NOUN
ejpam-3283	216	50	)	)	PUNCT
ejpam-3283	216	51	proper	proper	ADJ
ejpam-3283	216	52	n	n	CCONJ
ejpam-3283	216	53	-	-	PUNCT
ejpam-3283	216	54	ideals	ideal	NOUN
ejpam-3283	216	55	of	of	ADP
ejpam-3283	216	56	n	n	CCONJ
ejpam-3283	216	57	-	-	PUNCT
ejpam-3283	216	58	ary	ary	NOUN
ejpam-3283	216	59	semigroups	semigroup	NOUN
ejpam-3283	216	60	are	be	AUX
ejpam-3283	216	61	characterized	characterize	VERB
ejpam-3283	216	62	.	.	PUNCT
ejpam-3283	217	1	theorem	theorem	ADJ
ejpam-3283	217	2	5	5	NUM
ejpam-3283	217	3	.	.	PUNCT
ejpam-3283	218	1	if	if	SCONJ
ejpam-3283	218	2	s	s	PROPN
ejpam-3283	218	3	has	have	VERB
ejpam-3283	218	4	no	no	DET
ejpam-3283	218	5	zero	zero	NUM
ejpam-3283	218	6	element	element	NOUN
ejpam-3283	218	7	but	but	CCONJ
ejpam-3283	218	8	it	it	PRON
ejpam-3283	218	9	has	have	VERB
ejpam-3283	218	10	proper	proper	ADJ
ejpam-3283	218	11	n	n	CCONJ
ejpam-3283	218	12	-	-	PUNCT
ejpam-3283	218	13	ideals	ideal	NOUN
ejpam-3283	218	14	,	,	PUNCT
ejpam-3283	218	15	then	then	ADV
ejpam-3283	218	16	every	every	DET
ejpam-3283	218	17	proper	proper	ADJ
ejpam-3283	218	18	n	n	CCONJ
ejpam-3283	218	19	-	-	PUNCT
ejpam-3283	218	20	ideal	ideal	NOUN
ejpam-3283	218	21	of	of	ADP
ejpam-3283	218	22	s	s	PRON
ejpam-3283	218	23	is	be	AUX
ejpam-3283	218	24	maximal	maximal	ADJ
ejpam-3283	218	25	if	if	SCONJ
ejpam-3283	218	26	and	and	CCONJ
ejpam-3283	218	27	only	only	ADV
ejpam-3283	218	28	if	if	SCONJ
ejpam-3283	218	29	s	s	NOUN
ejpam-3283	218	30	contains	contain	VERB
ejpam-3283	218	31	exactly	exactly	ADV
ejpam-3283	218	32	one	one	NUM
ejpam-3283	218	33	proper	proper	ADJ
ejpam-3283	218	34	n	n	CCONJ
ejpam-3283	218	35	-	-	PUNCT
ejpam-3283	218	36	ideal	ideal	NOUN
ejpam-3283	218	37	or	or	CCONJ
ejpam-3283	218	38	s	s	NOUN
ejpam-3283	218	39	contains	contain	VERB
ejpam-3283	218	40	exactly	exactly	ADV
ejpam-3283	218	41	two	two	NUM
ejpam-3283	218	42	proper	proper	ADJ
ejpam-3283	218	43	n	n	CCONJ
ejpam-3283	218	44	-	-	PUNCT
ejpam-3283	218	45	ideals	ideal	NOUN
ejpam-3283	218	46	i1	i1	NOUN
ejpam-3283	218	47	and	and	CCONJ
ejpam-3283	218	48	i2	i2	PROPN
ejpam-3283	218	49	such	such	ADJ
ejpam-3283	218	50	that	that	SCONJ
ejpam-3283	218	51	i1	i1	PROPN
ejpam-3283	218	52	∪	∪	PROPN
ejpam-3283	218	53	i2	i2	PROPN
ejpam-3283	218	54	=	=	SYM
ejpam-3283	218	55	s	s	PROPN
ejpam-3283	218	56	and	and	CCONJ
ejpam-3283	218	57	i1	i1	PROPN
ejpam-3283	218	58	∩	∩	PROPN
ejpam-3283	218	59	i2	i2	PROPN
ejpam-3283	218	60	=	=	PUNCT
ejpam-3283	218	61	∅.	∅.	NOUN
ejpam-3283	218	62	proof	proof	NOUN
ejpam-3283	218	63	.	.	PUNCT
ejpam-3283	219	1	assume	assume	VERB
ejpam-3283	219	2	that	that	SCONJ
ejpam-3283	219	3	every	every	DET
ejpam-3283	219	4	proper	proper	ADJ
ejpam-3283	219	5	n	n	CCONJ
ejpam-3283	219	6	-	-	PUNCT
ejpam-3283	219	7	ideal	ideal	NOUN
ejpam-3283	219	8	of	of	ADP
ejpam-3283	219	9	s	s	NOUN
ejpam-3283	219	10	is	be	AUX
ejpam-3283	219	11	maximal	maximal	ADJ
ejpam-3283	219	12	.	.	PUNCT
ejpam-3283	220	1	let	let	VERB
ejpam-3283	220	2	i	i	PRON
ejpam-3283	220	3	be	be	AUX
ejpam-3283	220	4	a	a	DET
ejpam-3283	220	5	proper	proper	ADJ
ejpam-3283	220	6	n	n	CCONJ
ejpam-3283	220	7	-	-	PUNCT
ejpam-3283	220	8	ideal	ideal	NOUN
ejpam-3283	220	9	of	of	ADP
ejpam-3283	220	10	s.	s.	PROPN
ejpam-3283	221	1	then	then	ADV
ejpam-3283	221	2	i	i	PRON
ejpam-3283	221	3	is	be	AUX
ejpam-3283	221	4	maximal	maximal	ADJ
ejpam-3283	221	5	n	n	CCONJ
ejpam-3283	221	6	-	-	PUNCT
ejpam-3283	221	7	ideal	ideal	NOUN
ejpam-3283	221	8	of	of	ADP
ejpam-3283	221	9	s.	s.	PROPN
ejpam-3283	221	10	we	we	PRON
ejpam-3283	221	11	divide	divide	VERB
ejpam-3283	221	12	into	into	ADP
ejpam-3283	221	13	two	two	NUM
ejpam-3283	221	14	cases	case	NOUN
ejpam-3283	221	15	:	:	PUNCT
ejpam-3283	221	16	case	case	NOUN
ejpam-3283	221	17	1	1	NUM
ejpam-3283	221	18	:	:	PUNCT
ejpam-3283	221	19	suppose	suppose	VERB
ejpam-3283	221	20	that	that	SCONJ
ejpam-3283	221	21	s	s	NOUN
ejpam-3283	221	22	=	=	NOUN
ejpam-3283	221	23	in(a	in(a	X
ejpam-3283	221	24	)	)	PUNCT
ejpam-3283	221	25	for	for	ADP
ejpam-3283	221	26	all	all	DET
ejpam-3283	221	27	a	a	DET
ejpam-3283	221	28	∈	∈	NOUN
ejpam-3283	221	29	s	s	PART
ejpam-3283	221	30	r	r	NOUN
ejpam-3283	221	31	i.	i.	NOUN
ejpam-3283	221	32	let	let	VERB
ejpam-3283	221	33	j	j	PROPN
ejpam-3283	221	34	be	be	AUX
ejpam-3283	221	35	also	also	ADV
ejpam-3283	221	36	a	a	DET
ejpam-3283	221	37	proper	proper	ADJ
ejpam-3283	221	38	n	n	CCONJ
ejpam-3283	221	39	-	-	PUNCT
ejpam-3283	221	40	ideal	ideal	NOUN
ejpam-3283	221	41	of	of	ADP
ejpam-3283	221	42	s	s	NOUN
ejpam-3283	221	43	and	and	CCONJ
ejpam-3283	221	44	j	j	PROPN
ejpam-3283	221	45	6=	6=	PROPN
ejpam-3283	221	46	i.	i.	PROPN
ejpam-3283	221	47	then	then	ADV
ejpam-3283	221	48	j	j	PROPN
ejpam-3283	221	49	is	be	AUX
ejpam-3283	221	50	a	a	DET
ejpam-3283	221	51	maximal	maximal	ADJ
ejpam-3283	221	52	n	n	CCONJ
ejpam-3283	221	53	-	-	PUNCT
ejpam-3283	221	54	ideal	ideal	NOUN
ejpam-3283	221	55	of	of	ADP
ejpam-3283	221	56	s	s	PROPN
ejpam-3283	221	57	,	,	PUNCT
ejpam-3283	221	58	and	and	CCONJ
ejpam-3283	222	1	so	so	ADV
ejpam-3283	222	2	j	j	PROPN
ejpam-3283	222	3	r	r	NOUN
ejpam-3283	222	4	i	i	PROPN
ejpam-3283	222	5	6=	6=	X
ejpam-3283	222	6	∅.	∅.	VERB
ejpam-3283	222	7	then	then	ADV
ejpam-3283	222	8	there	there	PRON
ejpam-3283	222	9	exists	exist	VERB
ejpam-3283	222	10	a	a	DET
ejpam-3283	222	11	∈	∈	PROPN
ejpam-3283	223	1	j	j	NOUN
ejpam-3283	223	2	r	r	NOUN
ejpam-3283	223	3	i	i	NOUN
ejpam-3283	224	1	⊆	⊆	NUM
ejpam-3283	224	2	s	s	NOUN
ejpam-3283	224	3	r	r	NOUN
ejpam-3283	224	4	i.	i.	NOUN
ejpam-3283	224	5	hence	hence	ADV
ejpam-3283	224	6	s	s	PART
ejpam-3283	224	7	=	=	NOUN
ejpam-3283	224	8	in(a	in(a	X
ejpam-3283	224	9	)	)	PUNCT
ejpam-3283	224	10	⊆	⊆	NUM
ejpam-3283	224	11	j	j	PROPN
ejpam-3283	224	12	⊆	⊆	NUM
ejpam-3283	224	13	s	s	NOUN
ejpam-3283	224	14	,	,	PUNCT
ejpam-3283	224	15	and	and	CCONJ
ejpam-3283	224	16	so	so	ADV
ejpam-3283	224	17	j	j	PROPN
ejpam-3283	224	18	=	=	SYM
ejpam-3283	224	19	s	s	PROPN
ejpam-3283	224	20	,	,	PUNCT
ejpam-3283	224	21	which	which	PRON
ejpam-3283	224	22	is	be	AUX
ejpam-3283	224	23	a	a	DET
ejpam-3283	224	24	contradiction	contradiction	NOUN
ejpam-3283	224	25	.	.	PUNCT
ejpam-3283	225	1	this	this	PRON
ejpam-3283	225	2	implies	imply	VERB
ejpam-3283	225	3	that	that	SCONJ
ejpam-3283	225	4	j	j	PROPN
ejpam-3283	225	5	=	=	PROPN
ejpam-3283	225	6	i.	i.	PROPN
ejpam-3283	225	7	in	in	ADP
ejpam-3283	225	8	this	this	DET
ejpam-3283	225	9	case	case	NOUN
ejpam-3283	225	10	,	,	PUNCT
ejpam-3283	225	11	we	we	PRON
ejpam-3283	225	12	can	can	AUX
ejpam-3283	225	13	conclude	conclude	VERB
ejpam-3283	225	14	that	that	SCONJ
ejpam-3283	225	15	i	i	PRON
ejpam-3283	225	16	is	be	AUX
ejpam-3283	225	17	the	the	DET
ejpam-3283	225	18	unique	unique	ADJ
ejpam-3283	225	19	n	n	CCONJ
ejpam-3283	225	20	-	-	PUNCT
ejpam-3283	225	21	ideal	ideal	NOUN
ejpam-3283	225	22	of	of	ADP
ejpam-3283	225	23	s	s	NOUN
ejpam-3283	225	24	case	case	NOUN
ejpam-3283	225	25	2	2	NUM
ejpam-3283	225	26	:	:	PUNCT
ejpam-3283	225	27	suppose	suppose	VERB
ejpam-3283	225	28	that	that	SCONJ
ejpam-3283	225	29	there	there	PRON
ejpam-3283	225	30	exists	exist	VERB
ejpam-3283	225	31	a	a	DET
ejpam-3283	225	32	∈	∈	NOUN
ejpam-3283	225	33	s	s	PART
ejpam-3283	225	34	r	r	NOUN
ejpam-3283	225	35	i	i	PRON
ejpam-3283	225	36	such	such	ADJ
ejpam-3283	225	37	that	that	DET
ejpam-3283	225	38	s	s	PROPN
ejpam-3283	225	39	6=	6=	NUM
ejpam-3283	225	40	in(a	in(a	NUM
ejpam-3283	225	41	)	)	PUNCT
ejpam-3283	225	42	.	.	PUNCT
ejpam-3283	226	1	this	this	PRON
ejpam-3283	226	2	implies	imply	VERB
ejpam-3283	226	3	that	that	SCONJ
ejpam-3283	226	4	in(a	in(a	NUM
ejpam-3283	226	5	)	)	PUNCT
ejpam-3283	226	6	6=	6=	ADP
ejpam-3283	227	1	i	i	PRON
ejpam-3283	227	2	and	and	CCONJ
ejpam-3283	227	3	in(a	in(a	NUM
ejpam-3283	227	4	)	)	PUNCT
ejpam-3283	227	5	is	be	AUX
ejpam-3283	227	6	a	a	DET
ejpam-3283	227	7	maximal	maximal	ADJ
ejpam-3283	227	8	n	n	CCONJ
ejpam-3283	227	9	-	-	PUNCT
ejpam-3283	227	10	ideal	ideal	NOUN
ejpam-3283	227	11	of	of	ADP
ejpam-3283	227	12	s	s	PRON
ejpam-3283	227	13	by	by	ADP
ejpam-3283	227	14	the	the	DET
ejpam-3283	227	15	fact	fact	NOUN
ejpam-3283	227	16	that	that	SCONJ
ejpam-3283	227	17	in(a	in(a	NUM
ejpam-3283	227	18	)	)	PUNCT
ejpam-3283	227	19	is	be	AUX
ejpam-3283	227	20	a	a	DET
ejpam-3283	227	21	proper	proper	ADJ
ejpam-3283	227	22	n	n	CCONJ
ejpam-3283	227	23	-	-	PUNCT
ejpam-3283	227	24	ideal	ideal	NOUN
ejpam-3283	227	25	of	of	ADP
ejpam-3283	227	26	s.	s.	PROPN
ejpam-3283	227	27	by	by	ADP
ejpam-3283	227	28	lemma	lemma	PROPN
ejpam-3283	227	29	5	5	NUM
ejpam-3283	227	30	,	,	PUNCT
ejpam-3283	227	31	we	we	PRON
ejpam-3283	227	32	have	have	VERB
ejpam-3283	227	33	that	that	PRON
ejpam-3283	227	34	in(a	in(a	X
ejpam-3283	227	35	)	)	PUNCT
ejpam-3283	227	36	∪	∪	ADP
ejpam-3283	227	37	i	i	PRON
ejpam-3283	227	38	is	be	AUX
ejpam-3283	227	39	an	an	DET
ejpam-3283	227	40	n	n	CCONJ
ejpam-3283	227	41	-	-	PUNCT
ejpam-3283	227	42	ideal	ideal	NOUN
ejpam-3283	227	43	of	of	ADP
ejpam-3283	227	44	s.	s.	PROPN
ejpam-3283	227	45	since	since	SCONJ
ejpam-3283	227	46	i	i	PRON
ejpam-3283	227	47	is	be	AUX
ejpam-3283	227	48	a	a	DET
ejpam-3283	227	49	maximal	maximal	ADJ
ejpam-3283	227	50	n	n	CCONJ
ejpam-3283	227	51	-	-	PUNCT
ejpam-3283	227	52	ideal	ideal	NOUN
ejpam-3283	227	53	of	of	ADP
ejpam-3283	227	54	s	s	PRON
ejpam-3283	227	55	and	and	CCONJ
ejpam-3283	227	56	i	i	PRON
ejpam-3283	227	57	(	(	PUNCT
ejpam-3283	227	58	in(a	in(a	ADV
ejpam-3283	227	59	)	)	PUNCT
ejpam-3283	227	60	∪	∪	ADP
ejpam-3283	228	1	i	i	PRON
ejpam-3283	228	2	,	,	PUNCT
ejpam-3283	228	3	we	we	PRON
ejpam-3283	228	4	obtain	obtain	VERB
ejpam-3283	228	5	that	that	PRON
ejpam-3283	228	6	in(a	in(a	X
ejpam-3283	228	7	)	)	PUNCT
ejpam-3283	228	8	∪	∪	ADP
ejpam-3283	228	9	i	i	PRON
ejpam-3283	228	10	=	=	PUNCT
ejpam-3283	228	11	s.	s.	PROPN
ejpam-3283	228	12	by	by	ADP
ejpam-3283	228	13	the	the	DET
ejpam-3283	228	14	maximality	maximality	NOUN
ejpam-3283	228	15	of	of	ADP
ejpam-3283	228	16	an	an	DET
ejpam-3283	228	17	n	n	CCONJ
ejpam-3283	228	18	-	-	PUNCT
ejpam-3283	228	19	ideal	ideal	NOUN
ejpam-3283	228	20	in(a	in(a	NUM
ejpam-3283	228	21	)	)	PUNCT
ejpam-3283	228	22	and	and	CCONJ
ejpam-3283	228	23	by	by	ADP
ejpam-3283	228	24	the	the	DET
ejpam-3283	228	25	fact	fact	NOUN
ejpam-3283	228	26	that	that	SCONJ
ejpam-3283	228	27	in(a	in(a	NOUN
ejpam-3283	228	28	)	)	PUNCT
ejpam-3283	228	29	∩	∩	NOUN
ejpam-3283	228	30	i	i	PRON
ejpam-3283	228	31	(	(	PUNCT
ejpam-3283	228	32	in(a	in(a	NUM
ejpam-3283	228	33	)	)	PUNCT
ejpam-3283	228	34	,	,	PUNCT
ejpam-3283	228	35	we	we	PRON
ejpam-3283	228	36	gain	gain	VERB
ejpam-3283	228	37	that	that	PRON
ejpam-3283	228	38	in(a	in(a	NOUN
ejpam-3283	228	39	)	)	PUNCT
ejpam-3283	228	40	∩	∩	NOUN
ejpam-3283	228	41	i	i	PRON
ejpam-3283	228	42	=	=	PUNCT
ejpam-3283	228	43	∅.	∅.	VERB
ejpam-3283	228	44	next	next	ADV
ejpam-3283	228	45	,	,	PUNCT
ejpam-3283	228	46	we	we	PRON
ejpam-3283	228	47	show	show	VERB
ejpam-3283	228	48	that	that	SCONJ
ejpam-3283	228	49	s	s	VERB
ejpam-3283	228	50	has	have	VERB
ejpam-3283	228	51	exactly	exactly	ADV
ejpam-3283	228	52	two	two	NUM
ejpam-3283	228	53	proper	proper	ADJ
ejpam-3283	228	54	n	n	CCONJ
ejpam-3283	228	55	-	-	PUNCT
ejpam-3283	228	56	ideal	ideal	NOUN
ejpam-3283	228	57	i	i	PRON
ejpam-3283	228	58	and	and	CCONJ
ejpam-3283	228	59	in(a	in(a	NUM
ejpam-3283	228	60	)	)	PUNCT
ejpam-3283	228	61	.	.	PUNCT
ejpam-3283	229	1	suppose	suppose	VERB
ejpam-3283	229	2	that	that	SCONJ
ejpam-3283	229	3	m	m	PROPN
ejpam-3283	229	4	is	be	AUX
ejpam-3283	229	5	a	a	DET
ejpam-3283	229	6	proper	proper	ADJ
ejpam-3283	229	7	n	n	CCONJ
ejpam-3283	229	8	-	-	PUNCT
ejpam-3283	229	9	ideal	ideal	NOUN
ejpam-3283	229	10	of	of	ADP
ejpam-3283	229	11	s.	s.	PROPN
ejpam-3283	229	12	then	then	ADV
ejpam-3283	229	13	m	m	VERB
ejpam-3283	229	14	is	be	AUX
ejpam-3283	229	15	a	a	DET
ejpam-3283	229	16	maximal	maximal	ADJ
ejpam-3283	229	17	n	n	CCONJ
ejpam-3283	229	18	-	-	PUNCT
ejpam-3283	229	19	ideal	ideal	NOUN
ejpam-3283	229	20	of	of	ADP
ejpam-3283	229	21	s	s	PRON
ejpam-3283	229	22	by	by	ADP
ejpam-3283	229	23	the	the	DET
ejpam-3283	229	24	hypothesis	hypothesis	NOUN
ejpam-3283	229	25	.	.	PUNCT
ejpam-3283	230	1	hence	hence	ADV
ejpam-3283	230	2	m	m	VERB
ejpam-3283	230	3	=	=	ADJ
ejpam-3283	230	4	m	m	NOUN
ejpam-3283	230	5	∩	∩	NOUN
ejpam-3283	230	6	s	s	PART
ejpam-3283	230	7	=	=	X
ejpam-3283	230	8	m	m	NOUN
ejpam-3283	230	9	∩	∩	NOUN
ejpam-3283	230	10	(	(	PUNCT
ejpam-3283	230	11	in(a	in(a	NUM
ejpam-3283	230	12	)	)	PUNCT
ejpam-3283	230	13	∪	∪	PROPN
ejpam-3283	230	14	i	i	PROPN
ejpam-3283	230	15	)	)	PUNCT
ejpam-3283	230	16	=	=	SYM
ejpam-3283	231	1	(	(	PUNCT
ejpam-3283	231	2	m	m	NOUN
ejpam-3283	231	3	∩	∩	NOUN
ejpam-3283	231	4	in(a	in(a	NUM
ejpam-3283	231	5	)	)	PUNCT
ejpam-3283	231	6	)	)	PUNCT
ejpam-3283	231	7	∪	∪	ADP
ejpam-3283	231	8	(	(	PUNCT
ejpam-3283	231	9	m	m	NOUN
ejpam-3283	231	10	∩	∩	ADJ
ejpam-3283	231	11	i	i	NOUN
ejpam-3283	231	12	)	)	PUNCT
ejpam-3283	231	13	.	.	PUNCT
ejpam-3283	232	1	if	if	SCONJ
ejpam-3283	232	2	m	m	NOUN
ejpam-3283	232	3	∩	∩	VERB
ejpam-3283	232	4	i	i	ADP
ejpam-3283	232	5	6=	6=	PROPN
ejpam-3283	232	6	∅	∅	NOUN
ejpam-3283	232	7	,	,	PUNCT
ejpam-3283	232	8	then	then	ADV
ejpam-3283	232	9	m	m	VERB
ejpam-3283	232	10	=	=	ADJ
ejpam-3283	232	11	i	i	PRON
ejpam-3283	232	12	because	because	SCONJ
ejpam-3283	232	13	m	m	VERB
ejpam-3283	232	14	and	and	CCONJ
ejpam-3283	232	15	i	i	PRON
ejpam-3283	232	16	are	be	AUX
ejpam-3283	232	17	both	both	ADV
ejpam-3283	232	18	maximal	maximal	ADJ
ejpam-3283	232	19	n	n	CCONJ
ejpam-3283	232	20	-	-	PUNCT
ejpam-3283	232	21	ideals	ideal	NOUN
ejpam-3283	232	22	of	of	ADP
ejpam-3283	232	23	s.	s.	PROPN
ejpam-3283	232	24	if	if	SCONJ
ejpam-3283	232	25	m	m	VERB
ejpam-3283	232	26	∩	∩	ADJ
ejpam-3283	232	27	i	i	NOUN
ejpam-3283	232	28	=	=	SYM
ejpam-3283	232	29	∅	∅	NOUN
ejpam-3283	232	30	,	,	PUNCT
ejpam-3283	232	31	then	then	ADV
ejpam-3283	232	32	m	m	VERB
ejpam-3283	232	33	=	=	ADJ
ejpam-3283	232	34	m	m	NOUN
ejpam-3283	232	35	∩	∩	NOUN
ejpam-3283	232	36	in(a	in(a	NUM
ejpam-3283	232	37	)	)	PUNCT
ejpam-3283	232	38	.	.	PUNCT
ejpam-3283	233	1	then	then	ADV
ejpam-3283	233	2	m	m	VERB
ejpam-3283	233	3	∩in(a	∩in(a	PROPN
ejpam-3283	233	4	)	)	PUNCT
ejpam-3283	233	5	6=	6=	ADP
ejpam-3283	233	6	∅.	∅.	ADP
ejpam-3283	233	7	this	this	PRON
ejpam-3283	233	8	implies	imply	VERB
ejpam-3283	233	9	that	that	SCONJ
ejpam-3283	233	10	m	m	VERB
ejpam-3283	233	11	=	=	NOUN
ejpam-3283	233	12	in(a	in(a	NOUN
ejpam-3283	233	13	)	)	PUNCT
ejpam-3283	233	14	because	because	SCONJ
ejpam-3283	233	15	m	m	PROPN
ejpam-3283	233	16	and	and	CCONJ
ejpam-3283	233	17	in(a	in(a	NUM
ejpam-3283	233	18	)	)	PUNCT
ejpam-3283	233	19	are	be	AUX
ejpam-3283	233	20	both	both	PRON
ejpam-3283	233	21	maximal	maximal	ADJ
ejpam-3283	233	22	n	n	CCONJ
ejpam-3283	233	23	-	-	PUNCT
ejpam-3283	233	24	ideals	ideal	NOUN
ejpam-3283	233	25	of	of	ADP
ejpam-3283	233	26	s.	s.	PROPN
ejpam-3283	233	27	in	in	ADP
ejpam-3283	233	28	this	this	DET
ejpam-3283	233	29	case	case	NOUN
ejpam-3283	233	30	,	,	PUNCT
ejpam-3283	233	31	we	we	PRON
ejpam-3283	233	32	can	can	AUX
ejpam-3283	233	33	conclude	conclude	VERB
ejpam-3283	233	34	that	that	PRON
ejpam-3283	233	35	s	s	NOUN
ejpam-3283	233	36	contains	contain	VERB
ejpam-3283	233	37	exactly	exactly	ADV
ejpam-3283	233	38	two	two	NUM
ejpam-3283	233	39	proper	proper	ADJ
ejpam-3283	233	40	n	n	CCONJ
ejpam-3283	233	41	-	-	PUNCT
ejpam-3283	233	42	ideal	ideal	NOUN
ejpam-3283	233	43	i	i	PRON
ejpam-3283	233	44	and	and	CCONJ
ejpam-3283	233	45	in(a	in(a	NUM
ejpam-3283	233	46	)	)	PUNCT
ejpam-3283	233	47	,	,	PUNCT
ejpam-3283	233	48	moreover	moreover	ADV
ejpam-3283	233	49	,	,	PUNCT
ejpam-3283	233	50	we	we	PRON
ejpam-3283	233	51	obtain	obtain	VERB
ejpam-3283	233	52	in(a	in(a	NOUN
ejpam-3283	233	53	)	)	PUNCT
ejpam-3283	233	54	∪	∪	ADP
ejpam-3283	233	55	i	i	PRON
ejpam-3283	233	56	=	=	SYM
ejpam-3283	233	57	s	s	X
ejpam-3283	233	58	and	and	CCONJ
ejpam-3283	233	59	in(a	in(a	NUM
ejpam-3283	233	60	)	)	PUNCT
ejpam-3283	233	61	∩	∩	NOUN
ejpam-3283	234	1	i	i	PRON
ejpam-3283	234	2	=	=	PUNCT
ejpam-3283	234	3	∅.	∅.	PROPN
ejpam-3283	234	4	p.	p.	NOUN
ejpam-3283	234	5	petchkaew	petchkaew	NOUN
ejpam-3283	234	6	,	,	PUNCT
ejpam-3283	234	7	r.	r.	PROPN
ejpam-3283	234	8	chinram	chinram	PROPN
ejpam-3283	234	9	/	/	SYM
ejpam-3283	234	10	eur	eur	PROPN
ejpam-3283	234	11	.	.	PUNCT
ejpam-3283	235	1	j.	j.	PROPN
ejpam-3283	235	2	pure	pure	PROPN
ejpam-3283	235	3	appl	appl	PROPN
ejpam-3283	235	4	.	.	PROPN
ejpam-3283	235	5	math	math	PROPN
ejpam-3283	235	6	,	,	PUNCT
ejpam-3283	235	7	11	11	NUM
ejpam-3283	235	8	(	(	PUNCT
ejpam-3283	235	9	3	3	NUM
ejpam-3283	235	10	)	)	PUNCT
ejpam-3283	235	11	(	(	PUNCT
ejpam-3283	235	12	2018	2018	NUM
ejpam-3283	235	13	)	)	PUNCT
ejpam-3283	235	14	,	,	PUNCT
ejpam-3283	235	15	762	762	NUM
ejpam-3283	235	16	-	-	SYM
ejpam-3283	235	17	773	773	NUM
ejpam-3283	235	18	769	769	NUM
ejpam-3283	235	19	conversely	conversely	ADV
ejpam-3283	235	20	,	,	PUNCT
ejpam-3283	235	21	if	if	SCONJ
ejpam-3283	235	22	s	s	NOUN
ejpam-3283	235	23	contains	contain	VERB
ejpam-3283	235	24	exactly	exactly	ADV
ejpam-3283	235	25	one	one	NUM
ejpam-3283	235	26	proper	proper	ADJ
ejpam-3283	235	27	n	n	CCONJ
ejpam-3283	235	28	-	-	PUNCT
ejpam-3283	235	29	ideal	ideal	NOUN
ejpam-3283	235	30	,	,	PUNCT
ejpam-3283	235	31	then	then	ADV
ejpam-3283	235	32	it	it	PRON
ejpam-3283	235	33	is	be	AUX
ejpam-3283	235	34	clearly	clearly	ADV
ejpam-3283	235	35	that	that	SCONJ
ejpam-3283	235	36	it	it	PRON
ejpam-3283	235	37	is	be	AUX
ejpam-3283	235	38	just	just	ADV
ejpam-3283	235	39	a	a	DET
ejpam-3283	235	40	maximal	maximal	ADJ
ejpam-3283	235	41	n	n	CCONJ
ejpam-3283	235	42	-	-	PUNCT
ejpam-3283	235	43	ideal	ideal	NOUN
ejpam-3283	235	44	.	.	PUNCT
ejpam-3283	236	1	next	next	ADV
ejpam-3283	236	2	,	,	PUNCT
ejpam-3283	236	3	assume	assume	VERB
ejpam-3283	236	4	that	that	SCONJ
ejpam-3283	236	5	s	s	VERB
ejpam-3283	236	6	contains	contain	VERB
ejpam-3283	236	7	exactly	exactly	ADV
ejpam-3283	236	8	two	two	NUM
ejpam-3283	236	9	proper	proper	ADJ
ejpam-3283	236	10	n	n	CCONJ
ejpam-3283	236	11	-	-	PUNCT
ejpam-3283	236	12	ideals	ideal	NOUN
ejpam-3283	236	13	i1	i1	NOUN
ejpam-3283	236	14	and	and	CCONJ
ejpam-3283	236	15	i2	i2	PROPN
ejpam-3283	236	16	such	such	ADJ
ejpam-3283	236	17	that	that	SCONJ
ejpam-3283	236	18	i1	i1	PROPN
ejpam-3283	236	19	∪	∪	PROPN
ejpam-3283	236	20	i2	i2	PROPN
ejpam-3283	236	21	=	=	SYM
ejpam-3283	236	22	s	s	PROPN
ejpam-3283	236	23	and	and	CCONJ
ejpam-3283	236	24	i1	i1	PROPN
ejpam-3283	236	25	∩	∩	PROPN
ejpam-3283	236	26	i2	i2	PROPN
ejpam-3283	236	27	=	=	PUNCT
ejpam-3283	236	28	∅.	∅.	NOUN
ejpam-3283	236	29	since	since	SCONJ
ejpam-3283	236	30	i1	i1	PROPN
ejpam-3283	236	31	∩	∩	PROPN
ejpam-3283	236	32	i2	i2	PROPN
ejpam-3283	236	33	=	=	NOUN
ejpam-3283	236	34	∅	∅	NOUN
ejpam-3283	236	35	,	,	PUNCT
ejpam-3283	236	36	we	we	PRON
ejpam-3283	236	37	obtain	obtain	VERB
ejpam-3283	236	38	that	that	DET
ejpam-3283	236	39	i1	i1	PROPN
ejpam-3283	236	40	6⊆	6⊆	PROPN
ejpam-3283	236	41	i2	i2	PROPN
ejpam-3283	236	42	and	and	CCONJ
ejpam-3283	236	43	i2	i2	PROPN
ejpam-3283	236	44	6⊆	6⊆	PROPN
ejpam-3283	236	45	i1	i1	PROPN
ejpam-3283	236	46	.	.	PUNCT
ejpam-3283	237	1	hence	hence	ADV
ejpam-3283	237	2	i1	i1	PROPN
ejpam-3283	237	3	and	and	CCONJ
ejpam-3283	237	4	i2	i2	PROPN
ejpam-3283	237	5	are	be	AUX
ejpam-3283	237	6	both	both	ADV
ejpam-3283	237	7	maximal	maximal	ADJ
ejpam-3283	237	8	n	n	CCONJ
ejpam-3283	237	9	-	-	PUNCT
ejpam-3283	237	10	ideals	ideal	NOUN
ejpam-3283	237	11	of	of	ADP
ejpam-3283	237	12	s.	s.	PROPN
ejpam-3283	237	13	therefore	therefore	ADV
ejpam-3283	237	14	,	,	PUNCT
ejpam-3283	237	15	the	the	DET
ejpam-3283	237	16	proof	proof	NOUN
ejpam-3283	237	17	is	be	AUX
ejpam-3283	237	18	completed	complete	VERB
ejpam-3283	237	19	.	.	PUNCT
ejpam-3283	238	1	theorem	theorem	VERB
ejpam-3283	238	2	6	6	NUM
ejpam-3283	238	3	.	.	PUNCT
ejpam-3283	239	1	if	if	SCONJ
ejpam-3283	239	2	s	s	PROPN
ejpam-3283	239	3	has	have	VERB
ejpam-3283	239	4	a	a	DET
ejpam-3283	239	5	zero	zero	NUM
ejpam-3283	239	6	element	element	NOUN
ejpam-3283	239	7	and	and	CCONJ
ejpam-3283	239	8	nonzero	nonzero	NOUN
ejpam-3283	239	9	proper	proper	ADJ
ejpam-3283	239	10	n	n	CCONJ
ejpam-3283	239	11	-	-	PUNCT
ejpam-3283	239	12	ideals	ideal	NOUN
ejpam-3283	239	13	,	,	PUNCT
ejpam-3283	239	14	then	then	ADV
ejpam-3283	239	15	every	every	DET
ejpam-3283	239	16	nonzero	nonzero	ADJ
ejpam-3283	239	17	proper	proper	ADJ
ejpam-3283	239	18	n	n	CCONJ
ejpam-3283	239	19	-	-	PUNCT
ejpam-3283	239	20	ideal	ideal	NOUN
ejpam-3283	239	21	of	of	ADP
ejpam-3283	239	22	s	s	PRON
ejpam-3283	239	23	is	be	AUX
ejpam-3283	239	24	maximal	maximal	ADJ
ejpam-3283	239	25	if	if	SCONJ
ejpam-3283	239	26	and	and	CCONJ
ejpam-3283	239	27	only	only	ADV
ejpam-3283	239	28	if	if	SCONJ
ejpam-3283	239	29	s	s	NOUN
ejpam-3283	239	30	contains	contain	VERB
ejpam-3283	239	31	exactly	exactly	ADV
ejpam-3283	239	32	one	one	NUM
ejpam-3283	239	33	nonzero	nonzero	NOUN
ejpam-3283	239	34	proper	proper	ADJ
ejpam-3283	239	35	nideal	nideal	ADJ
ejpam-3283	239	36	or	or	CCONJ
ejpam-3283	239	37	s	s	NOUN
ejpam-3283	239	38	contains	contain	VERB
ejpam-3283	239	39	exactly	exactly	ADV
ejpam-3283	239	40	two	two	NUM
ejpam-3283	239	41	nonzero	nonzero	ADJ
ejpam-3283	239	42	proper	proper	ADJ
ejpam-3283	239	43	n	n	CCONJ
ejpam-3283	239	44	-	-	PUNCT
ejpam-3283	239	45	ideals	ideal	NOUN
ejpam-3283	239	46	i1	i1	NOUN
ejpam-3283	239	47	and	and	CCONJ
ejpam-3283	239	48	i2	i2	PROPN
ejpam-3283	239	49	such	such	ADJ
ejpam-3283	239	50	that	that	SCONJ
ejpam-3283	239	51	i1	i1	PROPN
ejpam-3283	239	52	∪	∪	PROPN
ejpam-3283	239	53	i2	i2	PROPN
ejpam-3283	239	54	=	=	SYM
ejpam-3283	239	55	s	s	PROPN
ejpam-3283	239	56	and	and	CCONJ
ejpam-3283	239	57	i1	i1	PROPN
ejpam-3283	239	58	∩	∩	PROPN
ejpam-3283	239	59	i2	i2	PROPN
ejpam-3283	239	60	=	=	PUNCT
ejpam-3283	239	61	{	{	PUNCT
ejpam-3283	239	62	0	0	NUM
ejpam-3283	239	63	}	}	PUNCT
ejpam-3283	239	64	.	.	PUNCT
ejpam-3283	240	1	proof	proof	NOUN
ejpam-3283	240	2	.	.	PUNCT
ejpam-3283	241	1	the	the	DET
ejpam-3283	241	2	proof	proof	NOUN
ejpam-3283	241	3	of	of	ADP
ejpam-3283	241	4	this	this	DET
ejpam-3283	241	5	theorem	theorem	NOUN
ejpam-3283	241	6	follows	follow	VERB
ejpam-3283	241	7	from	from	ADP
ejpam-3283	241	8	the	the	DET
ejpam-3283	241	9	proof	proof	NOUN
ejpam-3283	241	10	of	of	ADP
ejpam-3283	241	11	theorem	theorem	NOUN
ejpam-3283	241	12	5	5	NUM
ejpam-3283	241	13	and	and	CCONJ
ejpam-3283	241	14	the	the	DET
ejpam-3283	241	15	fact	fact	NOUN
ejpam-3283	241	16	that	that	SCONJ
ejpam-3283	241	17	every	every	DET
ejpam-3283	241	18	n	n	NOUN
ejpam-3283	241	19	-	-	PUNCT
ejpam-3283	241	20	ideal	ideal	NOUN
ejpam-3283	241	21	of	of	ADP
ejpam-3283	241	22	s	s	PROPN
ejpam-3283	241	23	contains	contain	VERB
ejpam-3283	241	24	a	a	DET
ejpam-3283	241	25	zero	zero	NUM
ejpam-3283	241	26	element	element	NOUN
ejpam-3283	241	27	.	.	PUNCT
ejpam-3283	242	1	theorem	theorem	VERB
ejpam-3283	242	2	7	7	NUM
ejpam-3283	242	3	.	.	PUNCT
ejpam-3283	243	1	let	let	VERB
ejpam-3283	243	2	i	i	PRON
ejpam-3283	243	3	be	be	AUX
ejpam-3283	243	4	a	a	DET
ejpam-3283	243	5	proper	proper	ADJ
ejpam-3283	243	6	n	n	CCONJ
ejpam-3283	243	7	-	-	PUNCT
ejpam-3283	243	8	ideal	ideal	NOUN
ejpam-3283	243	9	of	of	ADP
ejpam-3283	243	10	s.	s.	PROPN
ejpam-3283	244	1	then	then	ADV
ejpam-3283	244	2	i	i	PRON
ejpam-3283	244	3	is	be	AUX
ejpam-3283	244	4	maximal	maximal	ADJ
ejpam-3283	244	5	n	n	CCONJ
ejpam-3283	244	6	-	-	PUNCT
ejpam-3283	244	7	ideal	ideal	NOUN
ejpam-3283	244	8	if	if	SCONJ
ejpam-3283	244	9	and	and	CCONJ
ejpam-3283	244	10	only	only	ADV
ejpam-3283	244	11	if	if	SCONJ
ejpam-3283	244	12	(	(	PUNCT
ejpam-3283	244	13	1	1	X
ejpam-3283	244	14	)	)	PUNCT
ejpam-3283	244	15	s	s	PART
ejpam-3283	244	16	r	r	NOUN
ejpam-3283	244	17	i	i	NOUN
ejpam-3283	244	18	=	=	PUNCT
ejpam-3283	244	19	{	{	PUNCT
ejpam-3283	244	20	a	a	NOUN
ejpam-3283	244	21	}	}	PUNCT
ejpam-3283	244	22	and	and	CCONJ
ejpam-3283	244	23	f(a	f(a	PROPN
ejpam-3283	244	24	,	,	PUNCT
ejpam-3283	244	25	sn−2	sn−2	PROPN
ejpam-3283	244	26	,	,	PUNCT
ejpam-3283	244	27	a	a	PRON
ejpam-3283	244	28	)	)	PUNCT
ejpam-3283	244	29	⊆	⊆	NUM
ejpam-3283	244	30	i	i	PRON
ejpam-3283	244	31	for	for	ADP
ejpam-3283	244	32	some	some	DET
ejpam-3283	244	33	a	a	DET
ejpam-3283	244	34	∈	∈	NOUN
ejpam-3283	244	35	s	s	PART
ejpam-3283	244	36	or	or	CCONJ
ejpam-3283	244	37	(	(	PUNCT
ejpam-3283	244	38	2	2	NUM
ejpam-3283	244	39	)	)	PUNCT
ejpam-3283	244	40	s	s	PART
ejpam-3283	244	41	r	r	NOUN
ejpam-3283	244	42	i	i	PROPN
ejpam-3283	244	43	⊆	⊆	PROPN
ejpam-3283	244	44	f(sn−1	f(sn−1	ADP
ejpam-3283	244	45	,	,	PUNCT
ejpam-3283	244	46	a	a	PRON
ejpam-3283	244	47	)	)	PUNCT
ejpam-3283	244	48	for	for	ADP
ejpam-3283	244	49	all	all	DET
ejpam-3283	244	50	a	a	DET
ejpam-3283	244	51	∈	∈	NOUN
ejpam-3283	244	52	s	s	PART
ejpam-3283	244	53	r	r	NOUN
ejpam-3283	244	54	i.	i.	NOUN
ejpam-3283	244	55	proof	proof	NOUN
ejpam-3283	244	56	.	.	PUNCT
ejpam-3283	245	1	assume	assume	VERB
ejpam-3283	245	2	that	that	SCONJ
ejpam-3283	245	3	i	i	PRON
ejpam-3283	245	4	is	be	AUX
ejpam-3283	245	5	maximal	maximal	ADJ
ejpam-3283	245	6	n	n	CCONJ
ejpam-3283	245	7	-	-	PUNCT
ejpam-3283	245	8	ideal	ideal	NOUN
ejpam-3283	245	9	of	of	ADP
ejpam-3283	245	10	s.	s.	PROPN
ejpam-3283	245	11	we	we	PRON
ejpam-3283	245	12	consider	consider	VERB
ejpam-3283	245	13	the	the	DET
ejpam-3283	245	14	following	follow	VERB
ejpam-3283	245	15	two	two	NUM
ejpam-3283	245	16	cases	case	NOUN
ejpam-3283	245	17	:	:	PUNCT
ejpam-3283	245	18	case	case	NOUN
ejpam-3283	245	19	1	1	NUM
ejpam-3283	245	20	:	:	PUNCT
ejpam-3283	245	21	suppose	suppose	VERB
ejpam-3283	245	22	that	that	SCONJ
ejpam-3283	245	23	there	there	PRON
ejpam-3283	245	24	exists	exist	VERB
ejpam-3283	245	25	a	a	DET
ejpam-3283	245	26	∈	∈	NOUN
ejpam-3283	245	27	s	s	PART
ejpam-3283	245	28	r	r	NOUN
ejpam-3283	245	29	i	i	PRON
ejpam-3283	245	30	such	such	ADJ
ejpam-3283	245	31	that	that	PRON
ejpam-3283	245	32	f(sn−1	f(sn−1	PROPN
ejpam-3283	245	33	,	,	PUNCT
ejpam-3283	245	34	a	a	PRON
ejpam-3283	245	35	)	)	PUNCT
ejpam-3283	245	36	⊆	⊆	NUM
ejpam-3283	245	37	i.	i.	NOUN
ejpam-3283	245	38	then	then	ADV
ejpam-3283	245	39	f(a	f(a	PROPN
ejpam-3283	245	40	,	,	PUNCT
ejpam-3283	245	41	sn−2	sn−2	PROPN
ejpam-3283	245	42	,	,	PUNCT
ejpam-3283	245	43	a	a	PRON
ejpam-3283	245	44	)	)	PUNCT
ejpam-3283	246	1	⊆	⊆	NUM
ejpam-3283	246	2	f(sn−1	f(sn−1	PROPN
ejpam-3283	246	3	,	,	PUNCT
ejpam-3283	246	4	a	a	PRON
ejpam-3283	246	5	)	)	PUNCT
ejpam-3283	246	6	⊆	⊆	NUM
ejpam-3283	246	7	i.	i.	NOUN
ejpam-3283	246	8	by	by	ADP
ejpam-3283	246	9	corollary	corollary	ADJ
ejpam-3283	246	10	1	1	NUM
ejpam-3283	246	11	,	,	PUNCT
ejpam-3283	246	12	we	we	PRON
ejpam-3283	246	13	obtain	obtain	VERB
ejpam-3283	246	14	i∪{a	i∪{a	NOUN
ejpam-3283	246	15	}	}	PUNCT
ejpam-3283	246	16	=	=	SYM
ejpam-3283	246	17	(	(	PUNCT
ejpam-3283	246	18	i∪f(sn−1	i∪f(sn−1	ADJ
ejpam-3283	246	19	,	,	PUNCT
ejpam-3283	246	20	a))∪{a	a))∪{a	NOUN
ejpam-3283	246	21	}	}	PUNCT
ejpam-3283	246	22	=	=	PUNCT
ejpam-3283	247	1	i	i	PRON
ejpam-3283	247	2	∪	∪	VERB
ejpam-3283	247	3	(	(	PUNCT
ejpam-3283	247	4	f(sn−1	f(sn−1	PROPN
ejpam-3283	247	5	,	,	PUNCT
ejpam-3283	247	6	a	a	PRON
ejpam-3283	247	7	)	)	PUNCT
ejpam-3283	247	8	∪	∪	NOUN
ejpam-3283	247	9	{	{	PUNCT
ejpam-3283	247	10	a	a	PRON
ejpam-3283	247	11	}	}	PUNCT
ejpam-3283	247	12	)	)	PUNCT
ejpam-3283	248	1	=	=	PUNCT
ejpam-3283	248	2	i	i	PRON
ejpam-3283	248	3	∪	∪	VERB
ejpam-3283	248	4	in(a	in(a	NOUN
ejpam-3283	248	5	)	)	PUNCT
ejpam-3283	248	6	.	.	PUNCT
ejpam-3283	249	1	this	this	PRON
ejpam-3283	249	2	implies	imply	VERB
ejpam-3283	249	3	that	that	SCONJ
ejpam-3283	249	4	i	i	PRON
ejpam-3283	249	5	∪	∪	VERB
ejpam-3283	249	6	{	{	PUNCT
ejpam-3283	249	7	a	a	PRON
ejpam-3283	249	8	}	}	PUNCT
ejpam-3283	249	9	is	be	AUX
ejpam-3283	249	10	an	an	DET
ejpam-3283	249	11	n	n	CCONJ
ejpam-3283	249	12	-	-	PUNCT
ejpam-3283	249	13	ideal	ideal	NOUN
ejpam-3283	249	14	of	of	ADP
ejpam-3283	249	15	s	s	PRON
ejpam-3283	249	16	because	because	SCONJ
ejpam-3283	249	17	i	i	PRON
ejpam-3283	249	18	∪	∪	VERB
ejpam-3283	249	19	in(a	in(a	PUNCT
ejpam-3283	249	20	)	)	PUNCT
ejpam-3283	249	21	is	be	AUX
ejpam-3283	249	22	an	an	DET
ejpam-3283	249	23	n	n	CCONJ
ejpam-3283	249	24	-	-	PUNCT
ejpam-3283	249	25	ideal	ideal	NOUN
ejpam-3283	249	26	of	of	ADP
ejpam-3283	249	27	s.	s.	PROPN
ejpam-3283	249	28	since	since	SCONJ
ejpam-3283	249	29	i	i	PRON
ejpam-3283	249	30	is	be	AUX
ejpam-3283	249	31	a	a	DET
ejpam-3283	249	32	maximal	maximal	ADJ
ejpam-3283	249	33	n	n	CCONJ
ejpam-3283	249	34	-	-	PUNCT
ejpam-3283	249	35	ideal	ideal	NOUN
ejpam-3283	249	36	of	of	ADP
ejpam-3283	249	37	s	s	PRON
ejpam-3283	249	38	and	and	CCONJ
ejpam-3283	249	39	i	i	PRON
ejpam-3283	249	40	(	(	PUNCT
ejpam-3283	249	41	i	i	PRON
ejpam-3283	249	42	∪	∪	VERB
ejpam-3283	249	43	{	{	PUNCT
ejpam-3283	249	44	a	a	X
ejpam-3283	249	45	}	}	PUNCT
ejpam-3283	249	46	,	,	PUNCT
ejpam-3283	249	47	we	we	PRON
ejpam-3283	249	48	obtain	obtain	VERB
ejpam-3283	249	49	that	that	SCONJ
ejpam-3283	249	50	i	i	PRON
ejpam-3283	249	51	∪	∪	VERB
ejpam-3283	249	52	{	{	PUNCT
ejpam-3283	249	53	a	a	PRON
ejpam-3283	249	54	}	}	PUNCT
ejpam-3283	249	55	=	=	PUNCT
ejpam-3283	249	56	s.	s.	PROPN
ejpam-3283	249	57	this	this	PRON
ejpam-3283	249	58	implies	imply	VERB
ejpam-3283	249	59	that	that	SCONJ
ejpam-3283	249	60	s	s	VERB
ejpam-3283	249	61	r	r	NOUN
ejpam-3283	249	62	i	i	NOUN
ejpam-3283	249	63	=	=	PUNCT
ejpam-3283	249	64	{	{	PUNCT
ejpam-3283	249	65	a	a	NOUN
ejpam-3283	249	66	}	}	PUNCT
ejpam-3283	249	67	.	.	PUNCT
ejpam-3283	250	1	hence	hence	ADV
ejpam-3283	250	2	we	we	PRON
ejpam-3283	250	3	have	have	VERB
ejpam-3283	250	4	that	that	PRON
ejpam-3283	250	5	s	s	ADP
ejpam-3283	250	6	r	r	NOUN
ejpam-3283	250	7	i	i	NOUN
ejpam-3283	250	8	=	=	PUNCT
ejpam-3283	250	9	{	{	PUNCT
ejpam-3283	250	10	a	a	NOUN
ejpam-3283	250	11	}	}	PUNCT
ejpam-3283	250	12	and	and	CCONJ
ejpam-3283	250	13	f(a	f(a	PROPN
ejpam-3283	250	14	,	,	PUNCT
ejpam-3283	250	15	sn−2	sn−2	PROPN
ejpam-3283	250	16	,	,	PUNCT
ejpam-3283	250	17	a	a	PRON
ejpam-3283	250	18	)	)	PUNCT
ejpam-3283	250	19	⊆	⊆	NUM
ejpam-3283	250	20	i	i	PRON
ejpam-3283	250	21	for	for	ADP
ejpam-3283	250	22	some	some	DET
ejpam-3283	250	23	a	a	DET
ejpam-3283	250	24	∈	∈	NOUN
ejpam-3283	250	25	s	s	NOUN
ejpam-3283	250	26	as	as	ADP
ejpam-3283	250	27	desire	desire	NOUN
ejpam-3283	250	28	.	.	PUNCT
ejpam-3283	251	1	in	in	ADP
ejpam-3283	251	2	this	this	DET
ejpam-3283	251	3	case	case	NOUN
ejpam-3283	251	4	,	,	PUNCT
ejpam-3283	251	5	the	the	DET
ejpam-3283	251	6	statement	statement	NOUN
ejpam-3283	251	7	(	(	PUNCT
ejpam-3283	251	8	1	1	X
ejpam-3283	251	9	)	)	PUNCT
ejpam-3283	251	10	is	be	AUX
ejpam-3283	251	11	satisfied	satisfied	ADJ
ejpam-3283	251	12	.	.	PUNCT
ejpam-3283	252	1	case	case	NOUN
ejpam-3283	252	2	2	2	NUM
ejpam-3283	252	3	:	:	PUNCT
ejpam-3283	252	4	suppose	suppose	VERB
ejpam-3283	252	5	that	that	SCONJ
ejpam-3283	252	6	f(sn−1	f(sn−1	PROPN
ejpam-3283	252	7	,	,	PUNCT
ejpam-3283	252	8	a	a	PRON
ejpam-3283	252	9	)	)	PUNCT
ejpam-3283	252	10	6⊆	6⊆	NUM
ejpam-3283	252	11	i	i	PRON
ejpam-3283	252	12	for	for	ADP
ejpam-3283	252	13	all	all	DET
ejpam-3283	252	14	a	a	DET
ejpam-3283	252	15	∈	∈	NOUN
ejpam-3283	252	16	s	s	PART
ejpam-3283	252	17	r	r	NOUN
ejpam-3283	252	18	i.	i.	NOUN
ejpam-3283	252	19	let	let	VERB
ejpam-3283	252	20	a	a	DET
ejpam-3283	252	21	∈	∈	NOUN
ejpam-3283	252	22	s	s	PART
ejpam-3283	252	23	r	r	NOUN
ejpam-3283	252	24	i.	i.	NOUN
ejpam-3283	252	25	then	then	ADV
ejpam-3283	252	26	f(sn−1	f(sn−1	PROPN
ejpam-3283	252	27	,	,	PUNCT
ejpam-3283	252	28	a	a	PRON
ejpam-3283	252	29	)	)	PUNCT
ejpam-3283	252	30	6⊆	6⊆	PROPN
ejpam-3283	252	31	i.	i.	PROPN
ejpam-3283	252	32	moreover	moreover	ADV
ejpam-3283	252	33	,	,	PUNCT
ejpam-3283	252	34	we	we	PRON
ejpam-3283	252	35	obtain	obtain	VERB
ejpam-3283	252	36	that	that	PRON
ejpam-3283	252	37	f(sn−1	f(sn−1	ADP
ejpam-3283	252	38	,	,	PUNCT
ejpam-3283	252	39	a	a	DET
ejpam-3283	252	40	)	)	PUNCT
ejpam-3283	252	41	is	be	AUX
ejpam-3283	252	42	an	an	DET
ejpam-3283	252	43	n	n	CCONJ
ejpam-3283	252	44	-	-	PUNCT
ejpam-3283	252	45	ideal	ideal	NOUN
ejpam-3283	252	46	of	of	ADP
ejpam-3283	252	47	s	s	PRON
ejpam-3283	252	48	by	by	ADP
ejpam-3283	252	49	lemma	lemma	PROPN
ejpam-3283	252	50	2	2	NUM
ejpam-3283	252	51	.	.	PUNCT
ejpam-3283	252	52	by	by	ADP
ejpam-3283	252	53	lemma	lemma	PROPN
ejpam-3283	252	54	5	5	NUM
ejpam-3283	252	55	,	,	PUNCT
ejpam-3283	252	56	we	we	PRON
ejpam-3283	252	57	gain	gain	VERB
ejpam-3283	252	58	that	that	SCONJ
ejpam-3283	252	59	i	i	PRON
ejpam-3283	252	60	∪	∪	VERB
ejpam-3283	252	61	f(sn−1	f(sn−1	PROPN
ejpam-3283	252	62	,	,	PUNCT
ejpam-3283	252	63	a	a	DET
ejpam-3283	252	64	)	)	PUNCT
ejpam-3283	252	65	is	be	AUX
ejpam-3283	252	66	an	an	DET
ejpam-3283	252	67	n	n	CCONJ
ejpam-3283	252	68	-	-	PUNCT
ejpam-3283	252	69	ideal	ideal	NOUN
ejpam-3283	252	70	of	of	ADP
ejpam-3283	252	71	s.	s.	PROPN
ejpam-3283	252	72	since	since	SCONJ
ejpam-3283	252	73	i	i	PRON
ejpam-3283	252	74	is	be	AUX
ejpam-3283	252	75	a	a	DET
ejpam-3283	252	76	maximal	maximal	ADJ
ejpam-3283	252	77	n	n	CCONJ
ejpam-3283	252	78	-	-	PUNCT
ejpam-3283	252	79	ideal	ideal	NOUN
ejpam-3283	252	80	of	of	ADP
ejpam-3283	252	81	s	s	PRON
ejpam-3283	252	82	and	and	CCONJ
ejpam-3283	252	83	i	i	PRON
ejpam-3283	252	84	(	(	PUNCT
ejpam-3283	252	85	i	i	PRON
ejpam-3283	252	86	∪	∪	VERB
ejpam-3283	252	87	f(sn−1	f(sn−1	PROPN
ejpam-3283	252	88	,	,	PUNCT
ejpam-3283	252	89	a	a	PRON
ejpam-3283	252	90	)	)	PUNCT
ejpam-3283	252	91	,	,	PUNCT
ejpam-3283	252	92	we	we	PRON
ejpam-3283	252	93	acquire	acquire	VERB
ejpam-3283	252	94	that	that	SCONJ
ejpam-3283	252	95	i	i	PRON
ejpam-3283	252	96	∪	∪	VERB
ejpam-3283	252	97	f(sn−1	f(sn−1	PROPN
ejpam-3283	252	98	,	,	PUNCT
ejpam-3283	252	99	a	a	DET
ejpam-3283	252	100	)	)	PUNCT
ejpam-3283	253	1	=	=	SYM
ejpam-3283	253	2	s.	s.	PROPN
ejpam-3283	253	3	hence	hence	ADV
ejpam-3283	253	4	a	a	DET
ejpam-3283	253	5	∈	∈	PROPN
ejpam-3283	253	6	f(sn−1	f(sn−1	NOUN
ejpam-3283	253	7	,	,	PUNCT
ejpam-3283	253	8	a	a	PRON
ejpam-3283	253	9	)	)	PUNCT
ejpam-3283	253	10	because	because	SCONJ
ejpam-3283	253	11	a	a	DET
ejpam-3283	253	12	∈	∈	PROPN
ejpam-3283	253	13	sr	sr	PROPN
ejpam-3283	253	14	i.	i.	PROPN
ejpam-3283	253	15	this	this	PRON
ejpam-3283	253	16	implies	imply	VERB
ejpam-3283	253	17	that	that	SCONJ
ejpam-3283	253	18	sr	sr	PROPN
ejpam-3283	253	19	i	i	PROPN
ejpam-3283	253	20	⊆	⊆	PROPN
ejpam-3283	253	21	f(sn−1	f(sn−1	ADP
ejpam-3283	253	22	,	,	PUNCT
ejpam-3283	253	23	a	a	PRON
ejpam-3283	253	24	)	)	PUNCT
ejpam-3283	253	25	for	for	ADP
ejpam-3283	253	26	all	all	DET
ejpam-3283	253	27	a	a	DET
ejpam-3283	253	28	∈	∈	PROPN
ejpam-3283	253	29	sr	sr	PROPN
ejpam-3283	253	30	i.	i.	PROPN
ejpam-3283	253	31	hence	hence	ADV
ejpam-3283	253	32	,	,	PUNCT
ejpam-3283	253	33	this	this	DET
ejpam-3283	253	34	case	case	NOUN
ejpam-3283	253	35	satisfies	satisfy	VERB
ejpam-3283	253	36	the	the	DET
ejpam-3283	253	37	statement	statement	NOUN
ejpam-3283	253	38	(	(	PUNCT
ejpam-3283	253	39	2	2	NUM
ejpam-3283	253	40	)	)	PUNCT
ejpam-3283	253	41	.	.	PUNCT
ejpam-3283	254	1	conversely	conversely	ADV
ejpam-3283	254	2	,	,	PUNCT
ejpam-3283	254	3	suppose	suppose	VERB
ejpam-3283	254	4	that	that	SCONJ
ejpam-3283	254	5	j	j	PROPN
ejpam-3283	254	6	is	be	AUX
ejpam-3283	254	7	an	an	DET
ejpam-3283	254	8	n	n	CCONJ
ejpam-3283	254	9	-	-	PUNCT
ejpam-3283	254	10	ideal	ideal	NOUN
ejpam-3283	254	11	of	of	ADP
ejpam-3283	254	12	s	s	PRON
ejpam-3283	254	13	such	such	ADJ
ejpam-3283	254	14	that	that	SCONJ
ejpam-3283	254	15	i	i	PRON
ejpam-3283	254	16	(	(	PUNCT
ejpam-3283	254	17	j	j	PROPN
ejpam-3283	254	18	.	.	PUNCT
ejpam-3283	255	1	then	then	ADV
ejpam-3283	255	2	j	j	PROPN
ejpam-3283	255	3	r	r	NOUN
ejpam-3283	255	4	i	i	PROPN
ejpam-3283	255	5	6=	6=	ADP
ejpam-3283	255	6	∅.	∅.	ADV
ejpam-3283	255	7	if	if	SCONJ
ejpam-3283	255	8	there	there	PRON
ejpam-3283	255	9	exists	exist	VERB
ejpam-3283	255	10	a	a	DET
ejpam-3283	255	11	∈	∈	NOUN
ejpam-3283	255	12	s	s	VERB
ejpam-3283	255	13	such	such	ADJ
ejpam-3283	255	14	that	that	SCONJ
ejpam-3283	255	15	sr	sr	PROPN
ejpam-3283	256	1	i	i	PRON
ejpam-3283	256	2	=	=	PUNCT
ejpam-3283	256	3	{	{	PUNCT
ejpam-3283	256	4	a	a	NOUN
ejpam-3283	256	5	}	}	PUNCT
ejpam-3283	256	6	and	and	CCONJ
ejpam-3283	256	7	f(a	f(a	PROPN
ejpam-3283	256	8	,	,	PUNCT
ejpam-3283	256	9	sn−2	sn−2	PROPN
ejpam-3283	256	10	,	,	PUNCT
ejpam-3283	256	11	a	a	PRON
ejpam-3283	256	12	)	)	PUNCT
ejpam-3283	256	13	⊆	⊆	NUM
ejpam-3283	256	14	i	i	PRON
ejpam-3283	256	15	,	,	PUNCT
ejpam-3283	256	16	then	then	ADV
ejpam-3283	256	17	j	j	PROPN
ejpam-3283	257	1	r	r	NOUN
ejpam-3283	257	2	i	i	PROPN
ejpam-3283	257	3	⊆	⊆	NUM
ejpam-3283	257	4	sr	sr	PROPN
ejpam-3283	257	5	i	i	PRON
ejpam-3283	257	6	=	=	PUNCT
ejpam-3283	257	7	{	{	PUNCT
ejpam-3283	257	8	a	a	NOUN
ejpam-3283	257	9	}	}	PUNCT
ejpam-3283	257	10	,	,	PUNCT
ejpam-3283	257	11	and	and	CCONJ
ejpam-3283	257	12	hence	hence	ADV
ejpam-3283	257	13	j	j	PROPN
ejpam-3283	258	1	r	r	NOUN
ejpam-3283	258	2	i	i	NOUN
ejpam-3283	258	3	=	=	PUNCT
ejpam-3283	258	4	{	{	PUNCT
ejpam-3283	258	5	a	a	NOUN
ejpam-3283	258	6	}	}	PUNCT
ejpam-3283	258	7	.	.	PUNCT
ejpam-3283	259	1	this	this	PRON
ejpam-3283	259	2	implies	imply	VERB
ejpam-3283	259	3	that	that	SCONJ
ejpam-3283	259	4	j	j	PROPN
ejpam-3283	260	1	=	=	PUNCT
ejpam-3283	260	2	i	i	PRON
ejpam-3283	260	3	∪	∪	VERB
ejpam-3283	260	4	{	{	PUNCT
ejpam-3283	260	5	a	a	PRON
ejpam-3283	260	6	}	}	PUNCT
ejpam-3283	260	7	=	=	SYM
ejpam-3283	260	8	s.	s.	PROPN
ejpam-3283	260	9	hence	hence	ADV
ejpam-3283	260	10	we	we	PRON
ejpam-3283	260	11	obtain	obtain	VERB
ejpam-3283	260	12	that	that	SCONJ
ejpam-3283	260	13	i	i	PRON
ejpam-3283	260	14	is	be	AUX
ejpam-3283	260	15	a	a	DET
ejpam-3283	260	16	maximal	maximal	ADJ
ejpam-3283	260	17	n	n	CCONJ
ejpam-3283	260	18	-	-	PUNCT
ejpam-3283	260	19	ideal	ideal	NOUN
ejpam-3283	260	20	of	of	ADP
ejpam-3283	260	21	s.	s.	PROPN
ejpam-3283	260	22	next	next	ADV
ejpam-3283	260	23	,	,	PUNCT
ejpam-3283	260	24	if	if	SCONJ
ejpam-3283	260	25	s	s	VERB
ejpam-3283	260	26	r	r	NOUN
ejpam-3283	260	27	i	i	PROPN
ejpam-3283	260	28	⊆	⊆	PROPN
ejpam-3283	260	29	f(sn−1	f(sn−1	ADP
ejpam-3283	260	30	,	,	PUNCT
ejpam-3283	260	31	a	a	PRON
ejpam-3283	260	32	)	)	PUNCT
ejpam-3283	260	33	for	for	ADP
ejpam-3283	260	34	all	all	DET
ejpam-3283	260	35	a	a	DET
ejpam-3283	260	36	∈	∈	NOUN
ejpam-3283	260	37	s	s	PART
ejpam-3283	260	38	r	r	NOUN
ejpam-3283	260	39	i	i	PRON
ejpam-3283	260	40	,	,	PUNCT
ejpam-3283	260	41	then	then	ADV
ejpam-3283	260	42	s	s	VERB
ejpam-3283	260	43	r	r	NOUN
ejpam-3283	260	44	i	i	PROPN
ejpam-3283	260	45	⊆	⊆	NUM
ejpam-3283	260	46	f(sn−1	f(sn−1	ADP
ejpam-3283	260	47	,	,	PUNCT
ejpam-3283	260	48	x	x	X
ejpam-3283	260	49	)	)	PUNCT
ejpam-3283	260	50	⊆	⊆	NUM
ejpam-3283	260	51	f(sn−1	f(sn−1	PROPN
ejpam-3283	260	52	,	,	PUNCT
ejpam-3283	260	53	j	j	PROPN
ejpam-3283	260	54	)	)	PUNCT
ejpam-3283	260	55	⊆	⊆	NUM
ejpam-3283	260	56	j	j	PROPN
ejpam-3283	260	57	for	for	ADP
ejpam-3283	260	58	all	all	DET
ejpam-3283	260	59	x	x	SYM
ejpam-3283	260	60	∈	∈	PROPN
ejpam-3283	260	61	j	j	PROPN
ejpam-3283	260	62	r	r	NOUN
ejpam-3283	260	63	i.	i.	NOUN
ejpam-3283	260	64	hence	hence	ADV
ejpam-3283	260	65	s	s	VERB
ejpam-3283	260	66	=	=	PUNCT
ejpam-3283	260	67	(	(	PUNCT
ejpam-3283	260	68	s	s	NOUN
ejpam-3283	260	69	r	r	NOUN
ejpam-3283	260	70	i	i	NOUN
ejpam-3283	260	71	)	)	PUNCT
ejpam-3283	260	72	∪	∪	ADP
ejpam-3283	260	73	i	i	PRON
ejpam-3283	260	74	⊆	⊆	NUM
ejpam-3283	260	75	j	j	PROPN
ejpam-3283	260	76	∪	∪	X
ejpam-3283	260	77	j	j	PROPN
ejpam-3283	260	78	=	=	SYM
ejpam-3283	260	79	j	j	PROPN
ejpam-3283	260	80	⊆	⊆	NUM
ejpam-3283	260	81	s	s	NOUN
ejpam-3283	260	82	,	,	PUNCT
ejpam-3283	260	83	and	and	CCONJ
ejpam-3283	260	84	so	so	ADV
ejpam-3283	260	85	j	j	PROPN
ejpam-3283	260	86	=	=	PROPN
ejpam-3283	260	87	s.	s.	PROPN
ejpam-3283	260	88	therefore	therefore	ADV
ejpam-3283	260	89	,	,	PUNCT
ejpam-3283	260	90	i	i	PRON
ejpam-3283	260	91	is	be	AUX
ejpam-3283	260	92	a	a	DET
ejpam-3283	260	93	maximal	maximal	ADJ
ejpam-3283	260	94	n	n	CCONJ
ejpam-3283	260	95	-	-	PUNCT
ejpam-3283	260	96	ideal	ideal	NOUN
ejpam-3283	260	97	of	of	ADP
ejpam-3283	260	98	s.	s.	PROPN
ejpam-3283	260	99	hence	hence	ADV
ejpam-3283	260	100	the	the	DET
ejpam-3283	260	101	proof	proof	NOUN
ejpam-3283	260	102	of	of	ADP
ejpam-3283	260	103	this	this	DET
ejpam-3283	260	104	theorem	theorem	NOUN
ejpam-3283	260	105	is	be	AUX
ejpam-3283	260	106	completed	complete	VERB
ejpam-3283	260	107	.	.	PUNCT
ejpam-3283	261	1	example	example	NOUN
ejpam-3283	261	2	5	5	NUM
ejpam-3283	261	3	.	.	PUNCT
ejpam-3283	262	1	(	(	PUNCT
ejpam-3283	262	2	1	1	X
ejpam-3283	262	3	)	)	PUNCT
ejpam-3283	262	4	let	let	VERB
ejpam-3283	262	5	s	s	NOUN
ejpam-3283	262	6	=	=	VERB
ejpam-3283	262	7	n.	n.	NOUN
ejpam-3283	262	8	define	define	VERB
ejpam-3283	262	9	f	f	PROPN
ejpam-3283	262	10	:	:	PUNCT
ejpam-3283	262	11	sn	sn	PROPN
ejpam-3283	262	12	→	→	SYM
ejpam-3283	262	13	s	s	X
ejpam-3283	262	14	by	by	ADP
ejpam-3283	262	15	f(xn1	f(xn1	NOUN
ejpam-3283	262	16	)	)	PUNCT
ejpam-3283	262	17	=	=	PUNCT
ejpam-3283	263	1	x1	x1	PROPN
ejpam-3283	264	1	+	+	NUM
ejpam-3283	264	2	x2	x2	PROPN
ejpam-3283	265	1	+	+	CCONJ
ejpam-3283	265	2	.	.	PUNCT
ejpam-3283	265	3	.	.	PUNCT
ejpam-3283	265	4	.	.	PUNCT
ejpam-3283	266	1	+	+	CCONJ
ejpam-3283	266	2	xn	xn	PROPN
ejpam-3283	266	3	p.	p.	NOUN
ejpam-3283	266	4	petchkaew	petchkaew	NOUN
ejpam-3283	266	5	,	,	PUNCT
ejpam-3283	266	6	r.	r.	PROPN
ejpam-3283	266	7	chinram	chinram	PROPN
ejpam-3283	266	8	/	/	SYM
ejpam-3283	266	9	eur	eur	PROPN
ejpam-3283	266	10	.	.	PUNCT
ejpam-3283	267	1	j.	j.	PROPN
ejpam-3283	267	2	pure	pure	PROPN
ejpam-3283	267	3	appl	appl	PROPN
ejpam-3283	267	4	.	.	PROPN
ejpam-3283	267	5	math	math	PROPN
ejpam-3283	267	6	,	,	PUNCT
ejpam-3283	267	7	11	11	NUM
ejpam-3283	267	8	(	(	PUNCT
ejpam-3283	267	9	3	3	NUM
ejpam-3283	267	10	)	)	PUNCT
ejpam-3283	267	11	(	(	PUNCT
ejpam-3283	267	12	2018	2018	NUM
ejpam-3283	267	13	)	)	PUNCT
ejpam-3283	267	14	,	,	PUNCT
ejpam-3283	267	15	762	762	NUM
ejpam-3283	267	16	-	-	SYM
ejpam-3283	267	17	773	773	NUM
ejpam-3283	267	18	770	770	NUM
ejpam-3283	267	19	for	for	ADP
ejpam-3283	267	20	all	all	DET
ejpam-3283	267	21	x1	x1	PROPN
ejpam-3283	267	22	,	,	PUNCT
ejpam-3283	267	23	x2	x2	PROPN
ejpam-3283	267	24	,	,	PUNCT
ejpam-3283	267	25	.	.	PUNCT
ejpam-3283	267	26	.	.	PUNCT
ejpam-3283	268	1	.	.	PUNCT
ejpam-3283	269	1	,	,	PUNCT
ejpam-3283	269	2	xn	xn	PUNCT
ejpam-3283	269	3	∈	∈	PROPN
ejpam-3283	269	4	s	s	VERB
ejpam-3283	269	5	where	where	SCONJ
ejpam-3283	269	6	+	+	PRON
ejpam-3283	269	7	is	be	AUX
ejpam-3283	269	8	the	the	DET
ejpam-3283	269	9	usual	usual	ADJ
ejpam-3283	269	10	addition	addition	NOUN
ejpam-3283	269	11	of	of	ADP
ejpam-3283	269	12	n.	n.	NOUN
ejpam-3283	269	13	let	let	VERB
ejpam-3283	269	14	i	i	PRON
ejpam-3283	269	15	=	=	SYM
ejpam-3283	269	16	n	n	CCONJ
ejpam-3283	269	17	r	r	NOUN
ejpam-3283	269	18	{	{	PUNCT
ejpam-3283	269	19	1	1	NUM
ejpam-3283	269	20	}	}	PUNCT
ejpam-3283	269	21	.	.	PUNCT
ejpam-3283	270	1	thus	thus	ADV
ejpam-3283	270	2	s	s	VERB
ejpam-3283	270	3	r	r	NOUN
ejpam-3283	270	4	i	i	NOUN
ejpam-3283	270	5	=	=	PUNCT
ejpam-3283	270	6	{	{	PUNCT
ejpam-3283	270	7	1	1	NUM
ejpam-3283	270	8	}	}	PUNCT
ejpam-3283	270	9	and	and	CCONJ
ejpam-3283	270	10	f(1	f(1	PROPN
ejpam-3283	270	11	,	,	PUNCT
ejpam-3283	270	12	sn−2	sn−2	PROPN
ejpam-3283	270	13	,	,	PUNCT
ejpam-3283	270	14	1	1	NUM
ejpam-3283	270	15	)	)	PUNCT
ejpam-3283	270	16	⊆	⊆	NUM
ejpam-3283	270	17	i.	i.	NOUN
ejpam-3283	270	18	by	by	ADP
ejpam-3283	270	19	theorem	theorem	ADJ
ejpam-3283	270	20	7(1	7(1	NUM
ejpam-3283	270	21	)	)	PUNCT
ejpam-3283	270	22	,	,	PUNCT
ejpam-3283	270	23	i	i	PRON
ejpam-3283	270	24	is	be	AUX
ejpam-3283	270	25	a	a	DET
ejpam-3283	270	26	maximal	maximal	ADJ
ejpam-3283	270	27	n	n	CCONJ
ejpam-3283	270	28	-	-	PUNCT
ejpam-3283	270	29	ideal	ideal	NOUN
ejpam-3283	270	30	of	of	ADP
ejpam-3283	270	31	s.	s.	PROPN
ejpam-3283	270	32	(	(	PUNCT
ejpam-3283	270	33	2	2	X
ejpam-3283	270	34	)	)	PUNCT
ejpam-3283	270	35	let	let	VERB
ejpam-3283	270	36	s	s	PRON
ejpam-3283	270	37	=	=	PUNCT
ejpam-3283	270	38	{	{	PUNCT
ejpam-3283	270	39	0,−1	0,−1	PROPN
ejpam-3283	270	40	,	,	PUNCT
ejpam-3283	270	41	1	1	NUM
ejpam-3283	270	42	}	}	PUNCT
ejpam-3283	270	43	.	.	PUNCT
ejpam-3283	271	1	define	define	VERB
ejpam-3283	271	2	f	f	PROPN
ejpam-3283	271	3	:	:	PUNCT
ejpam-3283	271	4	sn	sn	PROPN
ejpam-3283	271	5	→	→	SYM
ejpam-3283	271	6	s	s	X
ejpam-3283	271	7	by	by	ADP
ejpam-3283	271	8	f(xn1	f(xn1	NOUN
ejpam-3283	271	9	)	)	PUNCT
ejpam-3283	271	10	=	=	PUNCT
ejpam-3283	272	1	x1	x1	PROPN
ejpam-3283	272	2	·	·	PUNCT
ejpam-3283	272	3	x2	x2	X
ejpam-3283	272	4	·	·	PUNCT
ejpam-3283	272	5	.	.	PUNCT
ejpam-3283	272	6	.	.	PUNCT
ejpam-3283	272	7	.	.	PUNCT
ejpam-3283	273	1	·	·	PUNCT
ejpam-3283	273	2	xn	xn	PUNCT
ejpam-3283	274	1	for	for	ADP
ejpam-3283	274	2	all	all	DET
ejpam-3283	274	3	x1	x1	PROPN
ejpam-3283	274	4	,	,	PUNCT
ejpam-3283	274	5	x2	x2	PROPN
ejpam-3283	274	6	,	,	PUNCT
ejpam-3283	274	7	.	.	PUNCT
ejpam-3283	274	8	.	.	PUNCT
ejpam-3283	274	9	.	.	PUNCT
ejpam-3283	275	1	,	,	PUNCT
ejpam-3283	275	2	xn	xn	PUNCT
ejpam-3283	275	3	∈	∈	PROPN
ejpam-3283	275	4	s	s	VERB
ejpam-3283	275	5	where	where	SCONJ
ejpam-3283	275	6	·	·	PUNCT
ejpam-3283	275	7	is	be	AUX
ejpam-3283	275	8	the	the	DET
ejpam-3283	275	9	usual	usual	ADJ
ejpam-3283	275	10	multiplication	multiplication	NOUN
ejpam-3283	275	11	.	.	PUNCT
ejpam-3283	276	1	let	let	VERB
ejpam-3283	276	2	i	i	PRON
ejpam-3283	276	3	=	=	PUNCT
ejpam-3283	276	4	{	{	PUNCT
ejpam-3283	276	5	0	0	NUM
ejpam-3283	276	6	}	}	PUNCT
ejpam-3283	276	7	.	.	PUNCT
ejpam-3283	277	1	then	then	ADV
ejpam-3283	277	2	sri	sri	VERB
ejpam-3283	277	3	⊆	⊆	NUM
ejpam-3283	277	4	f(sn−1	f(sn−1	NOUN
ejpam-3283	277	5	,	,	PUNCT
ejpam-3283	277	6	1	1	NUM
ejpam-3283	277	7	)	)	PUNCT
ejpam-3283	277	8	and	and	CCONJ
ejpam-3283	277	9	sri	sri	VERB
ejpam-3283	277	10	⊆	⊆	NUM
ejpam-3283	277	11	f(sn−1,−1	f(sn−1,−1	NOUN
ejpam-3283	277	12	)	)	PUNCT
ejpam-3283	277	13	.	.	PUNCT
ejpam-3283	278	1	by	by	ADP
ejpam-3283	278	2	theorem	theorem	NOUN
ejpam-3283	278	3	7(2	7(2	NOUN
ejpam-3283	278	4	)	)	PUNCT
ejpam-3283	278	5	,	,	PUNCT
ejpam-3283	278	6	i	i	PRON
ejpam-3283	278	7	is	be	AUX
ejpam-3283	278	8	a	a	DET
ejpam-3283	278	9	maximal	maximal	ADJ
ejpam-3283	278	10	n	n	CCONJ
ejpam-3283	278	11	-	-	PUNCT
ejpam-3283	278	12	ideal	ideal	NOUN
ejpam-3283	278	13	of	of	ADP
ejpam-3283	278	14	s.	s.	PROPN
ejpam-3283	278	15	for	for	ADP
ejpam-3283	278	16	an	an	DET
ejpam-3283	278	17	n	n	CCONJ
ejpam-3283	278	18	-	-	PUNCT
ejpam-3283	278	19	ary	ary	NOUN
ejpam-3283	278	20	semigroup	semigroup	PROPN
ejpam-3283	278	21	s	s	PROPN
ejpam-3283	278	22	,	,	PUNCT
ejpam-3283	278	23	the	the	DET
ejpam-3283	278	24	notation	notation	NOUN
ejpam-3283	278	25	u	u	NOUN
ejpam-3283	278	26	is	be	AUX
ejpam-3283	278	27	assumed	assume	VERB
ejpam-3283	278	28	to	to	PART
ejpam-3283	278	29	be	be	AUX
ejpam-3283	278	30	the	the	DET
ejpam-3283	278	31	union	union	NOUN
ejpam-3283	278	32	of	of	ADP
ejpam-3283	278	33	all	all	DET
ejpam-3283	278	34	nonzero	nonzero	ADJ
ejpam-3283	279	1	proper	proper	ADJ
ejpam-3283	279	2	n	n	CCONJ
ejpam-3283	279	3	-	-	PUNCT
ejpam-3283	279	4	ideals	ideal	NOUN
ejpam-3283	279	5	of	of	ADP
ejpam-3283	279	6	s	s	PRON
ejpam-3283	279	7	if	if	SCONJ
ejpam-3283	279	8	s	s	PROPN
ejpam-3283	279	9	has	have	VERB
ejpam-3283	279	10	a	a	DET
ejpam-3283	279	11	zero	zero	NUM
ejpam-3283	279	12	element	element	NOUN
ejpam-3283	279	13	and	and	CCONJ
ejpam-3283	279	14	the	the	DET
ejpam-3283	279	15	notation	notation	NOUN
ejpam-3283	279	16	u	u	NOUN
ejpam-3283	279	17	is	be	AUX
ejpam-3283	279	18	assumed	assume	VERB
ejpam-3283	279	19	to	to	PART
ejpam-3283	279	20	be	be	AUX
ejpam-3283	279	21	the	the	DET
ejpam-3283	279	22	the	the	DET
ejpam-3283	279	23	union	union	NOUN
ejpam-3283	279	24	of	of	ADP
ejpam-3283	279	25	all	all	DET
ejpam-3283	279	26	proper	proper	ADJ
ejpam-3283	279	27	n	n	CCONJ
ejpam-3283	279	28	-	-	PUNCT
ejpam-3283	279	29	ideals	ideal	NOUN
ejpam-3283	279	30	of	of	ADP
ejpam-3283	279	31	s	s	PRON
ejpam-3283	279	32	if	if	SCONJ
ejpam-3283	279	33	s	s	NOUN
ejpam-3283	279	34	has	have	VERB
ejpam-3283	279	35	no	no	DET
ejpam-3283	279	36	a	a	DET
ejpam-3283	279	37	zero	zero	NUM
ejpam-3283	279	38	element	element	NOUN
ejpam-3283	279	39	,	,	PUNCT
ejpam-3283	279	40	from	from	ADP
ejpam-3283	279	41	now	now	ADV
ejpam-3283	279	42	on	on	ADV
ejpam-3283	279	43	.	.	PUNCT
ejpam-3283	280	1	lemma	lemma	PROPN
ejpam-3283	280	2	8	8	NUM
ejpam-3283	280	3	.	.	PUNCT
ejpam-3283	281	1	u	u	NOUN
ejpam-3283	282	1	=	=	SYM
ejpam-3283	282	2	s	s	X
ejpam-3283	282	3	if	if	SCONJ
ejpam-3283	282	4	and	and	CCONJ
ejpam-3283	282	5	only	only	ADV
ejpam-3283	282	6	if	if	SCONJ
ejpam-3283	282	7	in(a	in(a	NOUN
ejpam-3283	282	8	)	)	PUNCT
ejpam-3283	283	1	6=	6=	ADP
ejpam-3283	283	2	s	s	VERB
ejpam-3283	283	3	for	for	ADP
ejpam-3283	283	4	all	all	DET
ejpam-3283	283	5	a	a	DET
ejpam-3283	283	6	∈	∈	PROPN
ejpam-3283	283	7	s.	s.	PROPN
ejpam-3283	283	8	proof	proof	PROPN
ejpam-3283	283	9	.	.	PUNCT
ejpam-3283	284	1	assume	assume	VERB
ejpam-3283	284	2	that	that	SCONJ
ejpam-3283	284	3	u	u	PRON
ejpam-3283	284	4	=	=	PUNCT
ejpam-3283	284	5	s.	s.	PROPN
ejpam-3283	284	6	if	if	SCONJ
ejpam-3283	284	7	in(a	in(a	NOUN
ejpam-3283	284	8	)	)	PUNCT
ejpam-3283	284	9	=	=	SYM
ejpam-3283	284	10	s	s	PROPN
ejpam-3283	284	11	for	for	ADP
ejpam-3283	284	12	some	some	PRON
ejpam-3283	284	13	a	a	DET
ejpam-3283	284	14	∈	∈	NOUN
ejpam-3283	284	15	s.	s.	PROPN
ejpam-3283	284	16	then	then	ADV
ejpam-3283	284	17	a	a	DET
ejpam-3283	284	18	6∈	6∈	NOUN
ejpam-3283	284	19	iγ	iγ	NOUN
ejpam-3283	284	20	for	for	ADP
ejpam-3283	284	21	all	all	DET
ejpam-3283	284	22	proper	proper	ADJ
ejpam-3283	284	23	n	n	CCONJ
ejpam-3283	284	24	-	-	PUNCT
ejpam-3283	284	25	ideal	ideal	NOUN
ejpam-3283	284	26	iγ	iγ	NOUN
ejpam-3283	284	27	of	of	ADP
ejpam-3283	284	28	s.	s.	PROPN
ejpam-3283	284	29	hence	hence	ADV
ejpam-3283	284	30	a	a	DET
ejpam-3283	284	31	6∈	6∈	NOUN
ejpam-3283	284	32	u	u	NOUN
ejpam-3283	284	33	=	=	SYM
ejpam-3283	284	34	s	s	PROPN
ejpam-3283	284	35	,	,	PUNCT
ejpam-3283	284	36	which	which	PRON
ejpam-3283	284	37	is	be	AUX
ejpam-3283	284	38	a	a	DET
ejpam-3283	284	39	contradiction	contradiction	NOUN
ejpam-3283	284	40	.	.	PUNCT
ejpam-3283	285	1	therefore	therefore	ADV
ejpam-3283	285	2	,	,	PUNCT
ejpam-3283	285	3	in(a	in(a	NUM
ejpam-3283	285	4	)	)	PUNCT
ejpam-3283	285	5	6=	6=	ADP
ejpam-3283	285	6	s	s	VERB
ejpam-3283	285	7	for	for	ADP
ejpam-3283	285	8	all	all	DET
ejpam-3283	285	9	a	a	DET
ejpam-3283	285	10	∈	∈	NOUN
ejpam-3283	285	11	s.	s.	PROPN
ejpam-3283	285	12	conversely	conversely	ADV
ejpam-3283	285	13	,	,	PUNCT
ejpam-3283	285	14	suppose	suppose	VERB
ejpam-3283	285	15	that	that	SCONJ
ejpam-3283	285	16	in(a	in(a	NUM
ejpam-3283	285	17	)	)	PUNCT
ejpam-3283	285	18	6=	6=	ADP
ejpam-3283	285	19	s	s	VERB
ejpam-3283	285	20	for	for	ADP
ejpam-3283	285	21	all	all	DET
ejpam-3283	285	22	a	a	DET
ejpam-3283	285	23	∈	∈	NOUN
ejpam-3283	285	24	s.	s.	PROPN
ejpam-3283	285	25	this	this	PRON
ejpam-3283	285	26	implies	imply	VERB
ejpam-3283	285	27	that	that	SCONJ
ejpam-3283	285	28	in(a	in(a	NUM
ejpam-3283	285	29	)	)	PUNCT
ejpam-3283	285	30	is	be	AUX
ejpam-3283	285	31	a	a	DET
ejpam-3283	285	32	proper	proper	ADJ
ejpam-3283	285	33	ideal	ideal	NOUN
ejpam-3283	285	34	for	for	ADP
ejpam-3283	285	35	all	all	DET
ejpam-3283	285	36	a	a	DET
ejpam-3283	285	37	∈	∈	ADJ
ejpam-3283	285	38	s	s	NOUN
ejpam-3283	285	39	,	,	PUNCT
ejpam-3283	285	40	and	and	CCONJ
ejpam-3283	285	41	so	so	ADV
ejpam-3283	285	42	s	s	VERB
ejpam-3283	285	43	⊆	⊆	NUM
ejpam-3283	285	44	⋃	⋃	NOUN
ejpam-3283	285	45	a∈s	a∈s	NOUN
ejpam-3283	285	46	in(a	in(a	NUM
ejpam-3283	285	47	)	)	PUNCT
ejpam-3283	285	48	⊆	⊆	NUM
ejpam-3283	285	49	u	u	NOUN
ejpam-3283	285	50	⊆	⊆	NUM
ejpam-3283	285	51	s.	s.	PROPN
ejpam-3283	285	52	therefore	therefore	ADV
ejpam-3283	285	53	,	,	PUNCT
ejpam-3283	285	54	we	we	PRON
ejpam-3283	285	55	obtain	obtain	VERB
ejpam-3283	285	56	that	that	DET
ejpam-3283	285	57	u	u	NOUN
ejpam-3283	285	58	=	=	PROPN
ejpam-3283	285	59	s.	s.	PROPN
ejpam-3283	285	60	theorem	theorem	VERB
ejpam-3283	285	61	8	8	NUM
ejpam-3283	285	62	.	.	PUNCT
ejpam-3283	286	1	if	if	SCONJ
ejpam-3283	286	2	s	s	PROPN
ejpam-3283	286	3	has	have	VERB
ejpam-3283	286	4	no	no	DET
ejpam-3283	286	5	zero	zero	NUM
ejpam-3283	286	6	element	element	NOUN
ejpam-3283	286	7	,	,	PUNCT
ejpam-3283	286	8	then	then	ADV
ejpam-3283	286	9	the	the	DET
ejpam-3283	286	10	exactly	exactly	ADV
ejpam-3283	286	11	one	one	NUM
ejpam-3283	286	12	of	of	ADP
ejpam-3283	286	13	the	the	DET
ejpam-3283	286	14	following	following	ADJ
ejpam-3283	286	15	statements	statement	NOUN
ejpam-3283	286	16	is	be	AUX
ejpam-3283	286	17	satisfied	satisfied	ADJ
ejpam-3283	286	18	:	:	PUNCT
ejpam-3283	286	19	(	(	PUNCT
ejpam-3283	286	20	1	1	X
ejpam-3283	286	21	)	)	PUNCT
ejpam-3283	286	22	s	s	VERB
ejpam-3283	286	23	is	be	AUX
ejpam-3283	286	24	n	n	PRON
ejpam-3283	286	25	-	-	PUNCT
ejpam-3283	286	26	simple	simple	NOUN
ejpam-3283	286	27	.	.	PUNCT
ejpam-3283	287	1	(	(	PUNCT
ejpam-3283	287	2	2	2	NUM
ejpam-3283	287	3	)	)	PUNCT
ejpam-3283	287	4	in(a	in(a	PUNCT
ejpam-3283	287	5	)	)	PUNCT
ejpam-3283	288	1	6=	6=	ADP
ejpam-3283	288	2	s	s	VERB
ejpam-3283	288	3	for	for	ADP
ejpam-3283	288	4	all	all	DET
ejpam-3283	288	5	a	a	DET
ejpam-3283	288	6	∈	∈	NOUN
ejpam-3283	288	7	s.	s.	PROPN
ejpam-3283	288	8	(	(	PUNCT
ejpam-3283	288	9	3	3	X
ejpam-3283	288	10	)	)	PUNCT
ejpam-3283	288	11	there	there	PRON
ejpam-3283	288	12	exists	exist	VERB
ejpam-3283	288	13	a	a	DET
ejpam-3283	288	14	∈	∈	NOUN
ejpam-3283	288	15	s	s	VERB
ejpam-3283	288	16	such	such	ADJ
ejpam-3283	288	17	that	that	SCONJ
ejpam-3283	288	18	in(a	in(a	NUM
ejpam-3283	288	19	)	)	PUNCT
ejpam-3283	288	20	=	=	SYM
ejpam-3283	288	21	s	s	PROPN
ejpam-3283	288	22	,	,	PUNCT
ejpam-3283	288	23	a	a	PRON
ejpam-3283	288	24	/∈	/∈	NOUN
ejpam-3283	288	25	f(sn−1	f(sn−1	NOUN
ejpam-3283	288	26	,	,	PUNCT
ejpam-3283	288	27	a	a	PRON
ejpam-3283	288	28	)	)	PUNCT
ejpam-3283	288	29	,	,	PUNCT
ejpam-3283	288	30	f(a	f(a	PROPN
ejpam-3283	288	31	,	,	PUNCT
ejpam-3283	288	32	sn−2	sn−2	PROPN
ejpam-3283	288	33	,	,	PUNCT
ejpam-3283	288	34	a	a	PRON
ejpam-3283	288	35	)	)	PUNCT
ejpam-3283	288	36	⊆	⊆	NUM
ejpam-3283	288	37	u	u	NOUN
ejpam-3283	288	38	=	=	SYM
ejpam-3283	288	39	s	s	PART
ejpam-3283	288	40	r	r	NOUN
ejpam-3283	288	41	{	{	PUNCT
ejpam-3283	288	42	a	a	NOUN
ejpam-3283	288	43	}	}	PUNCT
ejpam-3283	288	44	and	and	CCONJ
ejpam-3283	288	45	u	u	NOUN
ejpam-3283	288	46	is	be	AUX
ejpam-3283	288	47	the	the	DET
ejpam-3283	288	48	unique	unique	ADJ
ejpam-3283	288	49	maximal	maximal	ADJ
ejpam-3283	288	50	n	n	CCONJ
ejpam-3283	288	51	-	-	PUNCT
ejpam-3283	288	52	ideal	ideal	NOUN
ejpam-3283	288	53	of	of	ADP
ejpam-3283	288	54	s.	s.	PROPN
ejpam-3283	288	55	(	(	PUNCT
ejpam-3283	288	56	4	4	NUM
ejpam-3283	288	57	)	)	PUNCT
ejpam-3283	288	58	s	s	PART
ejpam-3283	288	59	r	r	NOUN
ejpam-3283	288	60	u	u	NOUN
ejpam-3283	288	61	=	=	PUNCT
ejpam-3283	288	62	{	{	PUNCT
ejpam-3283	288	63	a	a	DET
ejpam-3283	288	64	∈	∈	NOUN
ejpam-3283	288	65	s	s	VERB
ejpam-3283	288	66	|	|	ADV
ejpam-3283	288	67	f(sn−1	f(sn−1	ADJ
ejpam-3283	288	68	,	,	PUNCT
ejpam-3283	288	69	a	a	PRON
ejpam-3283	288	70	)	)	PUNCT
ejpam-3283	288	71	=	=	SYM
ejpam-3283	288	72	s	s	X
ejpam-3283	288	73	}	}	PUNCT
ejpam-3283	288	74	and	and	CCONJ
ejpam-3283	288	75	u	u	NOUN
ejpam-3283	288	76	is	be	AUX
ejpam-3283	288	77	the	the	DET
ejpam-3283	288	78	unique	unique	ADJ
ejpam-3283	288	79	maximal	maximal	ADJ
ejpam-3283	288	80	n	n	CCONJ
ejpam-3283	288	81	-	-	PUNCT
ejpam-3283	288	82	ideal	ideal	NOUN
ejpam-3283	288	83	of	of	ADP
ejpam-3283	288	84	s.	s.	PROPN
ejpam-3283	288	85	proof	proof	PROPN
ejpam-3283	288	86	.	.	PUNCT
ejpam-3283	289	1	assume	assume	VERB
ejpam-3283	289	2	that	that	SCONJ
ejpam-3283	289	3	s	s	VERB
ejpam-3283	289	4	is	be	AUX
ejpam-3283	289	5	not	not	PART
ejpam-3283	289	6	n	n	CCONJ
ejpam-3283	289	7	-	-	PUNCT
ejpam-3283	289	8	simple	simple	ADJ
ejpam-3283	289	9	.	.	PUNCT
ejpam-3283	290	1	this	this	PRON
ejpam-3283	290	2	implies	imply	VERB
ejpam-3283	290	3	that	that	SCONJ
ejpam-3283	290	4	there	there	PRON
ejpam-3283	290	5	exists	exist	VERB
ejpam-3283	290	6	a	a	DET
ejpam-3283	290	7	proper	proper	ADJ
ejpam-3283	290	8	n	n	CCONJ
ejpam-3283	290	9	-	-	PUNCT
ejpam-3283	290	10	ideal	ideal	NOUN
ejpam-3283	290	11	i	i	PRON
ejpam-3283	290	12	of	of	ADP
ejpam-3283	290	13	s.	s.	PROPN
ejpam-3283	290	14	hence	hence	ADV
ejpam-3283	290	15	u	u	PROPN
ejpam-3283	290	16	is	be	AUX
ejpam-3283	290	17	an	an	DET
ejpam-3283	290	18	n	n	CCONJ
ejpam-3283	290	19	-	-	PUNCT
ejpam-3283	290	20	ideal	ideal	NOUN
ejpam-3283	290	21	of	of	ADP
ejpam-3283	290	22	s.	s.	PROPN
ejpam-3283	290	23	we	we	PRON
ejpam-3283	290	24	divide	divide	VERB
ejpam-3283	290	25	into	into	ADP
ejpam-3283	290	26	two	two	NUM
ejpam-3283	290	27	cases	case	NOUN
ejpam-3283	290	28	:	:	PUNCT
ejpam-3283	290	29	case	case	NOUN
ejpam-3283	290	30	1	1	NUM
ejpam-3283	290	31	:	:	PUNCT
ejpam-3283	290	32	if	if	SCONJ
ejpam-3283	290	33	u	u	PROPN
ejpam-3283	290	34	=	=	SYM
ejpam-3283	290	35	s	s	PROPN
ejpam-3283	290	36	,	,	PUNCT
ejpam-3283	290	37	then	then	ADV
ejpam-3283	290	38	in(a	in(a	NUM
ejpam-3283	290	39	)	)	PUNCT
ejpam-3283	290	40	6=	6=	ADP
ejpam-3283	290	41	s	s	VERB
ejpam-3283	290	42	for	for	ADP
ejpam-3283	290	43	all	all	DET
ejpam-3283	290	44	a	a	DET
ejpam-3283	290	45	∈	∈	NOUN
ejpam-3283	290	46	s	s	PUNCT
ejpam-3283	290	47	by	by	ADP
ejpam-3283	290	48	lemma	lemma	PROPN
ejpam-3283	290	49	8	8	NUM
ejpam-3283	290	50	.	.	PUNCT
ejpam-3283	291	1	in	in	ADP
ejpam-3283	291	2	this	this	DET
ejpam-3283	291	3	case	case	NOUN
ejpam-3283	291	4	,	,	PUNCT
ejpam-3283	291	5	the	the	DET
ejpam-3283	291	6	statement	statement	NOUN
ejpam-3283	291	7	(	(	PUNCT
ejpam-3283	291	8	2	2	X
ejpam-3283	291	9	)	)	PUNCT
ejpam-3283	291	10	is	be	AUX
ejpam-3283	291	11	satisfied	satisfied	ADJ
ejpam-3283	291	12	.	.	PUNCT
ejpam-3283	292	1	case	case	NOUN
ejpam-3283	292	2	2	2	NUM
ejpam-3283	292	3	:	:	PUNCT
ejpam-3283	292	4	if	if	SCONJ
ejpam-3283	292	5	u	u	PROPN
ejpam-3283	292	6	6=	6=	SYM
ejpam-3283	292	7	s	s	PROPN
ejpam-3283	292	8	,	,	PUNCT
ejpam-3283	292	9	then	then	ADV
ejpam-3283	292	10	u	u	NOUN
ejpam-3283	292	11	is	be	AUX
ejpam-3283	292	12	a	a	DET
ejpam-3283	292	13	maximal	maximal	ADJ
ejpam-3283	292	14	n	n	CCONJ
ejpam-3283	292	15	-	-	PUNCT
ejpam-3283	292	16	ideal	ideal	NOUN
ejpam-3283	292	17	of	of	ADP
ejpam-3283	292	18	s.	s.	PROPN
ejpam-3283	292	19	we	we	PRON
ejpam-3283	292	20	would	would	AUX
ejpam-3283	292	21	like	like	VERB
ejpam-3283	292	22	to	to	PART
ejpam-3283	292	23	show	show	VERB
ejpam-3283	292	24	that	that	SCONJ
ejpam-3283	292	25	u	u	PRON
ejpam-3283	292	26	is	be	AUX
ejpam-3283	292	27	the	the	DET
ejpam-3283	292	28	unique	unique	ADJ
ejpam-3283	292	29	maximal	maximal	ADJ
ejpam-3283	292	30	n	n	CCONJ
ejpam-3283	292	31	-	-	PUNCT
ejpam-3283	292	32	ideal	ideal	NOUN
ejpam-3283	292	33	of	of	ADP
ejpam-3283	292	34	s.	s.	PROPN
ejpam-3283	292	35	suppose	suppose	VERB
ejpam-3283	292	36	that	that	SCONJ
ejpam-3283	292	37	i	i	PRON
ejpam-3283	292	38	is	be	AUX
ejpam-3283	292	39	a	a	DET
ejpam-3283	292	40	maximal	maximal	ADJ
ejpam-3283	292	41	n	n	CCONJ
ejpam-3283	292	42	-	-	PUNCT
ejpam-3283	292	43	ideal	ideal	NOUN
ejpam-3283	292	44	of	of	ADP
ejpam-3283	292	45	s	s	PROPN
ejpam-3283	292	46	,	,	PUNCT
ejpam-3283	292	47	and	and	CCONJ
ejpam-3283	292	48	so	so	ADV
ejpam-3283	292	49	i	i	PRON
ejpam-3283	292	50	is	be	AUX
ejpam-3283	292	51	a	a	DET
ejpam-3283	292	52	proper	proper	ADJ
ejpam-3283	292	53	n	n	CCONJ
ejpam-3283	292	54	-	-	PUNCT
ejpam-3283	292	55	ideal	ideal	NOUN
ejpam-3283	292	56	of	of	ADP
ejpam-3283	292	57	s.	s.	PROPN
ejpam-3283	292	58	hence	hence	ADV
ejpam-3283	292	59	i	i	PROPN
ejpam-3283	292	60	⊆	⊆	NUM
ejpam-3283	292	61	u	u	NOUN
ejpam-3283	292	62	(	(	PUNCT
ejpam-3283	292	63	s.	s.	PROPN
ejpam-3283	292	64	since	since	SCONJ
ejpam-3283	292	65	i	i	PRON
ejpam-3283	292	66	is	be	AUX
ejpam-3283	292	67	a	a	DET
ejpam-3283	292	68	maximal	maximal	ADJ
ejpam-3283	292	69	n	n	CCONJ
ejpam-3283	292	70	-	-	PUNCT
ejpam-3283	292	71	ideal	ideal	NOUN
ejpam-3283	292	72	of	of	ADP
ejpam-3283	292	73	s	s	PROPN
ejpam-3283	292	74	,	,	PUNCT
ejpam-3283	292	75	we	we	PRON
ejpam-3283	292	76	obtain	obtain	VERB
ejpam-3283	292	77	i	i	PRON
ejpam-3283	292	78	=	=	NOUN
ejpam-3283	292	79	u	u	PROPN
ejpam-3283	292	80	.	.	PUNCT
ejpam-3283	293	1	therefore	therefore	ADV
ejpam-3283	293	2	,	,	PUNCT
ejpam-3283	293	3	u	u	NOUN
ejpam-3283	293	4	is	be	AUX
ejpam-3283	293	5	the	the	DET
ejpam-3283	293	6	unique	unique	ADJ
ejpam-3283	293	7	maximal	maximal	ADJ
ejpam-3283	293	8	n	n	CCONJ
ejpam-3283	293	9	-	-	PUNCT
ejpam-3283	293	10	ideal	ideal	NOUN
ejpam-3283	293	11	of	of	ADP
ejpam-3283	293	12	s	s	PRON
ejpam-3283	293	13	as	as	ADP
ejpam-3283	293	14	desire	desire	NOUN
ejpam-3283	293	15	.	.	PUNCT
ejpam-3283	294	1	furthermore	furthermore	ADV
ejpam-3283	294	2	,	,	PUNCT
ejpam-3283	294	3	by	by	ADP
ejpam-3283	294	4	theorem	theorem	NOUN
ejpam-3283	294	5	7	7	NUM
ejpam-3283	294	6	,	,	PUNCT
ejpam-3283	294	7	we	we	PRON
ejpam-3283	294	8	acquire	acquire	VERB
ejpam-3283	294	9	(	(	PUNCT
ejpam-3283	294	10	1	1	NUM
ejpam-3283	294	11	)	)	PUNCT
ejpam-3283	294	12	s	s	PART
ejpam-3283	294	13	r	r	NOUN
ejpam-3283	294	14	u	u	NOUN
ejpam-3283	294	15	=	=	PUNCT
ejpam-3283	294	16	{	{	PUNCT
ejpam-3283	294	17	a	a	NOUN
ejpam-3283	294	18	}	}	PUNCT
ejpam-3283	294	19	and	and	CCONJ
ejpam-3283	294	20	f(a	f(a	PROPN
ejpam-3283	294	21	,	,	PUNCT
ejpam-3283	294	22	sn−2	sn−2	PROPN
ejpam-3283	294	23	,	,	PUNCT
ejpam-3283	294	24	a	a	PRON
ejpam-3283	294	25	)	)	PUNCT
ejpam-3283	294	26	⊆	⊆	NUM
ejpam-3283	294	27	u	u	NOUN
ejpam-3283	294	28	for	for	ADP
ejpam-3283	294	29	some	some	DET
ejpam-3283	294	30	a	a	DET
ejpam-3283	294	31	∈	∈	NOUN
ejpam-3283	294	32	s	s	PART
ejpam-3283	294	33	or	or	CCONJ
ejpam-3283	294	34	(	(	PUNCT
ejpam-3283	294	35	2	2	NUM
ejpam-3283	294	36	)	)	PUNCT
ejpam-3283	294	37	s	s	PART
ejpam-3283	294	38	r	r	NOUN
ejpam-3283	294	39	u	u	NOUN
ejpam-3283	294	40	⊆	⊆	NUM
ejpam-3283	294	41	f(sn−1	f(sn−1	ADP
ejpam-3283	294	42	,	,	PUNCT
ejpam-3283	294	43	a	a	PRON
ejpam-3283	294	44	)	)	PUNCT
ejpam-3283	294	45	for	for	ADP
ejpam-3283	294	46	all	all	DET
ejpam-3283	294	47	a	a	DET
ejpam-3283	294	48	∈	∈	NOUN
ejpam-3283	294	49	s	s	PART
ejpam-3283	294	50	r	r	NOUN
ejpam-3283	294	51	u	u	NOUN
ejpam-3283	294	52	.	.	PUNCT
ejpam-3283	295	1	p.	p.	NOUN
ejpam-3283	295	2	petchkaew	petchkaew	PROPN
ejpam-3283	295	3	,	,	PUNCT
ejpam-3283	295	4	r.	r.	PROPN
ejpam-3283	295	5	chinram	chinram	PROPN
ejpam-3283	295	6	/	/	SYM
ejpam-3283	295	7	eur	eur	PROPN
ejpam-3283	295	8	.	.	PUNCT
ejpam-3283	296	1	j.	j.	PROPN
ejpam-3283	296	2	pure	pure	PROPN
ejpam-3283	296	3	appl	appl	PROPN
ejpam-3283	296	4	.	.	PROPN
ejpam-3283	296	5	math	math	PROPN
ejpam-3283	296	6	,	,	PUNCT
ejpam-3283	296	7	11	11	NUM
ejpam-3283	296	8	(	(	PUNCT
ejpam-3283	296	9	3	3	NUM
ejpam-3283	296	10	)	)	PUNCT
ejpam-3283	296	11	(	(	PUNCT
ejpam-3283	296	12	2018	2018	NUM
ejpam-3283	296	13	)	)	PUNCT
ejpam-3283	296	14	,	,	PUNCT
ejpam-3283	296	15	762	762	NUM
ejpam-3283	296	16	-	-	SYM
ejpam-3283	296	17	773	773	NUM
ejpam-3283	296	18	771	771	NUM
ejpam-3283	296	19	first	first	ADV
ejpam-3283	296	20	,	,	PUNCT
ejpam-3283	296	21	we	we	PRON
ejpam-3283	296	22	assume	assume	VERB
ejpam-3283	296	23	that	that	SCONJ
ejpam-3283	296	24	s	s	VERB
ejpam-3283	296	25	r	r	NOUN
ejpam-3283	296	26	u	u	NOUN
ejpam-3283	296	27	=	=	PUNCT
ejpam-3283	296	28	{	{	PUNCT
ejpam-3283	296	29	a	a	NOUN
ejpam-3283	296	30	}	}	PUNCT
ejpam-3283	296	31	and	and	CCONJ
ejpam-3283	296	32	f(a	f(a	PROPN
ejpam-3283	296	33	,	,	PUNCT
ejpam-3283	296	34	sn−2	sn−2	PROPN
ejpam-3283	296	35	,	,	PUNCT
ejpam-3283	296	36	a	a	PRON
ejpam-3283	296	37	)	)	PUNCT
ejpam-3283	296	38	⊆	⊆	NUM
ejpam-3283	296	39	u	u	NOUN
ejpam-3283	296	40	for	for	ADP
ejpam-3283	296	41	some	some	PRON
ejpam-3283	296	42	a	a	DET
ejpam-3283	296	43	∈	∈	NOUN
ejpam-3283	296	44	s.	s.	PROPN
ejpam-3283	296	45	since	since	SCONJ
ejpam-3283	296	46	s	s	PROPN
ejpam-3283	296	47	r	r	NOUN
ejpam-3283	296	48	u	u	NOUN
ejpam-3283	296	49	=	=	PUNCT
ejpam-3283	296	50	{	{	PUNCT
ejpam-3283	296	51	a	a	NOUN
ejpam-3283	296	52	}	}	PUNCT
ejpam-3283	296	53	,	,	PUNCT
ejpam-3283	296	54	we	we	PRON
ejpam-3283	296	55	have	have	VERB
ejpam-3283	296	56	f(a	f(a	NOUN
ejpam-3283	296	57	,	,	PUNCT
ejpam-3283	296	58	sn−2	sn−2	PROPN
ejpam-3283	296	59	,	,	PUNCT
ejpam-3283	296	60	a	a	PRON
ejpam-3283	296	61	)	)	PUNCT
ejpam-3283	296	62	⊆	⊆	NUM
ejpam-3283	296	63	u	u	NOUN
ejpam-3283	296	64	=	=	SYM
ejpam-3283	296	65	s	s	PART
ejpam-3283	296	66	r	r	NOUN
ejpam-3283	296	67	{	{	PUNCT
ejpam-3283	296	68	a	a	NOUN
ejpam-3283	296	69	}	}	PUNCT
ejpam-3283	296	70	.	.	PUNCT
ejpam-3283	297	1	since	since	SCONJ
ejpam-3283	297	2	a	a	DET
ejpam-3283	297	3	/∈	/∈	SYM
ejpam-3283	297	4	u	u	NOUN
ejpam-3283	297	5	,	,	PUNCT
ejpam-3283	297	6	we	we	PRON
ejpam-3283	297	7	have	have	VERB
ejpam-3283	297	8	in(a	in(a	PUNCT
ejpam-3283	297	9	)	)	PUNCT
ejpam-3283	298	1	=	=	VERB
ejpam-3283	298	2	s.	s.	PROPN
ejpam-3283	298	3	if	if	SCONJ
ejpam-3283	298	4	a	a	DET
ejpam-3283	298	5	∈	∈	PROPN
ejpam-3283	298	6	f(sn−1	f(sn−1	PROPN
ejpam-3283	298	7	,	,	PUNCT
ejpam-3283	298	8	a	a	PRON
ejpam-3283	298	9	)	)	PUNCT
ejpam-3283	298	10	,	,	PUNCT
ejpam-3283	298	11	then	then	ADV
ejpam-3283	298	12	{	{	PUNCT
ejpam-3283	298	13	a	a	NOUN
ejpam-3283	298	14	}	}	PUNCT
ejpam-3283	298	15	⊆	⊆	NUM
ejpam-3283	298	16	f(sn−1	f(sn−1	NOUN
ejpam-3283	298	17	,	,	PUNCT
ejpam-3283	298	18	a	a	PRON
ejpam-3283	298	19	)	)	PUNCT
ejpam-3283	298	20	,	,	PUNCT
ejpam-3283	298	21	and	and	CCONJ
ejpam-3283	298	22	hence	hence	ADV
ejpam-3283	298	23	s	s	X
ejpam-3283	298	24	=	=	NOUN
ejpam-3283	298	25	in(a	in(a	NUM
ejpam-3283	298	26	)	)	PUNCT
ejpam-3283	298	27	=	=	SYM
ejpam-3283	298	28	f(sn−1	f(sn−1	PROPN
ejpam-3283	298	29	,	,	PUNCT
ejpam-3283	298	30	a	a	PRON
ejpam-3283	298	31	)	)	PUNCT
ejpam-3283	298	32	∪	∪	NOUN
ejpam-3283	298	33	{	{	PUNCT
ejpam-3283	298	34	a	a	DET
ejpam-3283	298	35	}	}	PUNCT
ejpam-3283	298	36	=	=	SYM
ejpam-3283	298	37	f(sn−1	f(sn−1	NOUN
ejpam-3283	298	38	,	,	PUNCT
ejpam-3283	298	39	a	a	PRON
ejpam-3283	298	40	)	)	PUNCT
ejpam-3283	298	41	by	by	ADP
ejpam-3283	298	42	corollary	corollary	ADJ
ejpam-3283	298	43	1	1	NUM
ejpam-3283	298	44	.	.	PUNCT
ejpam-3283	299	1	this	this	PRON
ejpam-3283	299	2	implies	imply	VERB
ejpam-3283	299	3	that	that	SCONJ
ejpam-3283	299	4	a	a	DET
ejpam-3283	299	5	=	=	X
ejpam-3283	299	6	f(sn−1	f(sn−1	ADJ
ejpam-3283	299	7	1	1	NUM
ejpam-3283	299	8	,	,	PUNCT
ejpam-3283	299	9	a	a	NOUN
ejpam-3283	299	10	)	)	PUNCT
ejpam-3283	299	11	and	and	CCONJ
ejpam-3283	299	12	s1	s1	PROPN
ejpam-3283	299	13	=	=	SYM
ejpam-3283	299	14	f(s2n−2	f(s2n−2	PROPN
ejpam-3283	299	15	n	n	CCONJ
ejpam-3283	299	16	,	,	PUNCT
ejpam-3283	299	17	a	a	X
ejpam-3283	299	18	)	)	PUNCT
ejpam-3283	299	19	for	for	ADP
ejpam-3283	299	20	some	some	DET
ejpam-3283	299	21	s1	s1	NOUN
ejpam-3283	299	22	,	,	PUNCT
ejpam-3283	299	23	s2	s2	NOUN
ejpam-3283	299	24	,	,	PUNCT
ejpam-3283	299	25	.	.	PUNCT
ejpam-3283	299	26	.	.	PUNCT
ejpam-3283	300	1	.	.	PUNCT
ejpam-3283	301	1	,	,	PUNCT
ejpam-3283	301	2	s2n−2	s2n−2	PROPN
ejpam-3283	301	3	∈	∈	PROPN
ejpam-3283	301	4	s.	s.	PROPN
ejpam-3283	301	5	hence	hence	ADV
ejpam-3283	301	6	a	a	DET
ejpam-3283	301	7	=	=	X
ejpam-3283	301	8	f(sn−1	f(sn−1	ADJ
ejpam-3283	301	9	1	1	NUM
ejpam-3283	301	10	,	,	PUNCT
ejpam-3283	301	11	a	a	PRON
ejpam-3283	301	12	)	)	PUNCT
ejpam-3283	301	13	=	=	SYM
ejpam-3283	301	14	f(s1	f(s1	NOUN
ejpam-3283	301	15	,	,	PUNCT
ejpam-3283	301	16	s	s	PROPN
ejpam-3283	301	17	n−1	n−1	PROPN
ejpam-3283	301	18	2	2	NUM
ejpam-3283	301	19	,	,	PUNCT
ejpam-3283	301	20	a	a	PRON
ejpam-3283	301	21	)	)	PUNCT
ejpam-3283	301	22	=	=	SYM
ejpam-3283	302	1	f(f(s2n−2	f(f(s2n−2	PROPN
ejpam-3283	302	2	n	n	CCONJ
ejpam-3283	302	3	,	,	PUNCT
ejpam-3283	302	4	a	a	X
ejpam-3283	302	5	)	)	PUNCT
ejpam-3283	302	6	,	,	PUNCT
ejpam-3283	302	7	sn−1	sn−1	PROPN
ejpam-3283	302	8	2	2	NUM
ejpam-3283	302	9	,	,	PUNCT
ejpam-3283	302	10	a	a	PRON
ejpam-3283	302	11	)	)	PUNCT
ejpam-3283	302	12	=	=	SYM
ejpam-3283	302	13	f(s2n−2	f(s2n−2	PROPN
ejpam-3283	302	14	n	n	CCONJ
ejpam-3283	302	15	,	,	PUNCT
ejpam-3283	302	16	f(a	f(a	PROPN
ejpam-3283	302	17	,	,	PUNCT
ejpam-3283	302	18	sn−1	sn−1	PROPN
ejpam-3283	302	19	2	2	NUM
ejpam-3283	302	20	,	,	PUNCT
ejpam-3283	302	21	a	a	PRON
ejpam-3283	302	22	)	)	PUNCT
ejpam-3283	302	23	)	)	PUNCT
ejpam-3283	302	24	.	.	PUNCT
ejpam-3283	303	1	since	since	SCONJ
ejpam-3283	303	2	f(a	f(a	PROPN
ejpam-3283	303	3	,	,	PUNCT
ejpam-3283	303	4	sn−2	sn−2	PROPN
ejpam-3283	303	5	,	,	PUNCT
ejpam-3283	303	6	a	a	PRON
ejpam-3283	303	7	)	)	PUNCT
ejpam-3283	303	8	⊆	⊆	NUM
ejpam-3283	303	9	u	u	NOUN
ejpam-3283	303	10	and	and	CCONJ
ejpam-3283	303	11	u	u	NOUN
ejpam-3283	303	12	is	be	AUX
ejpam-3283	303	13	an	an	DET
ejpam-3283	303	14	n	n	CCONJ
ejpam-3283	303	15	-	-	PUNCT
ejpam-3283	303	16	ideal	ideal	NOUN
ejpam-3283	303	17	of	of	ADP
ejpam-3283	303	18	s	s	PROPN
ejpam-3283	303	19	,	,	PUNCT
ejpam-3283	303	20	we	we	PRON
ejpam-3283	303	21	have	have	VERB
ejpam-3283	303	22	that	that	PRON
ejpam-3283	303	23	a	a	DET
ejpam-3283	303	24	=	=	SYM
ejpam-3283	303	25	f(s2n−2	f(s2n−2	PROPN
ejpam-3283	303	26	n	n	CCONJ
ejpam-3283	303	27	,	,	PUNCT
ejpam-3283	303	28	f(a	f(a	PROPN
ejpam-3283	303	29	,	,	PUNCT
ejpam-3283	303	30	sn−1	sn−1	PROPN
ejpam-3283	303	31	2	2	NUM
ejpam-3283	303	32	,	,	PUNCT
ejpam-3283	303	33	a	a	PRON
ejpam-3283	303	34	)	)	PUNCT
ejpam-3283	303	35	)	)	PUNCT
ejpam-3283	304	1	∈	∈	PROPN
ejpam-3283	304	2	u	u	NOUN
ejpam-3283	304	3	,	,	PUNCT
ejpam-3283	304	4	which	which	PRON
ejpam-3283	304	5	is	be	AUX
ejpam-3283	304	6	a	a	DET
ejpam-3283	304	7	contradiction	contradiction	NOUN
ejpam-3283	304	8	.	.	PUNCT
ejpam-3283	305	1	hence	hence	ADV
ejpam-3283	305	2	a	a	DET
ejpam-3283	305	3	/∈	/∈	INTJ
ejpam-3283	305	4	f(sn−1	f(sn−1	NOUN
ejpam-3283	305	5	,	,	PUNCT
ejpam-3283	305	6	a	a	PRON
ejpam-3283	305	7	)	)	PUNCT
ejpam-3283	305	8	.	.	PUNCT
ejpam-3283	306	1	in	in	ADP
ejpam-3283	306	2	this	this	DET
ejpam-3283	306	3	case	case	NOUN
ejpam-3283	306	4	,	,	PUNCT
ejpam-3283	306	5	the	the	DET
ejpam-3283	306	6	statement	statement	NOUN
ejpam-3283	306	7	(	(	PUNCT
ejpam-3283	306	8	3	3	X
ejpam-3283	306	9	)	)	PUNCT
ejpam-3283	306	10	is	be	AUX
ejpam-3283	306	11	satisfied	satisfied	ADJ
ejpam-3283	306	12	.	.	PUNCT
ejpam-3283	307	1	finally	finally	ADV
ejpam-3283	307	2	,	,	PUNCT
ejpam-3283	307	3	suppose	suppose	VERB
ejpam-3283	307	4	that	that	SCONJ
ejpam-3283	307	5	s	s	VERB
ejpam-3283	307	6	r	r	NOUN
ejpam-3283	307	7	u	u	NOUN
ejpam-3283	307	8	⊆	⊆	NUM
ejpam-3283	307	9	f(sn−1	f(sn−1	ADP
ejpam-3283	307	10	,	,	PUNCT
ejpam-3283	307	11	a	a	PRON
ejpam-3283	307	12	)	)	PUNCT
ejpam-3283	307	13	for	for	ADP
ejpam-3283	307	14	all	all	DET
ejpam-3283	307	15	a	a	DET
ejpam-3283	307	16	∈	∈	NOUN
ejpam-3283	307	17	s	s	PART
ejpam-3283	307	18	r	r	NOUN
ejpam-3283	307	19	u	u	NOUN
ejpam-3283	307	20	.	.	PUNCT
ejpam-3283	308	1	we	we	PRON
ejpam-3283	308	2	would	would	AUX
ejpam-3283	308	3	like	like	VERB
ejpam-3283	308	4	to	to	PART
ejpam-3283	308	5	show	show	VERB
ejpam-3283	308	6	that	that	SCONJ
ejpam-3283	308	7	s	s	VERB
ejpam-3283	308	8	r	r	NOUN
ejpam-3283	308	9	u	u	NOUN
ejpam-3283	308	10	=	=	PUNCT
ejpam-3283	308	11	{	{	PUNCT
ejpam-3283	308	12	a	a	DET
ejpam-3283	308	13	∈	∈	NOUN
ejpam-3283	308	14	s	s	VERB
ejpam-3283	308	15	|	|	ADV
ejpam-3283	308	16	f(sn−1	f(sn−1	ADJ
ejpam-3283	308	17	,	,	PUNCT
ejpam-3283	308	18	a	a	PRON
ejpam-3283	308	19	)	)	PUNCT
ejpam-3283	308	20	=	=	SYM
ejpam-3283	308	21	s	s	X
ejpam-3283	308	22	}	}	PUNCT
ejpam-3283	308	23	.	.	PUNCT
ejpam-3283	309	1	let	let	VERB
ejpam-3283	309	2	a	a	DET
ejpam-3283	309	3	∈	∈	NOUN
ejpam-3283	309	4	s	s	PART
ejpam-3283	309	5	r	r	NOUN
ejpam-3283	309	6	u	u	NOUN
ejpam-3283	309	7	.	.	PUNCT
ejpam-3283	310	1	by	by	ADP
ejpam-3283	310	2	the	the	DET
ejpam-3283	310	3	hypothesis	hypothesis	NOUN
ejpam-3283	310	4	,	,	PUNCT
ejpam-3283	310	5	we	we	PRON
ejpam-3283	310	6	have	have	VERB
ejpam-3283	310	7	that	that	PRON
ejpam-3283	310	8	a	a	DET
ejpam-3283	310	9	∈	∈	PROPN
ejpam-3283	310	10	f(sn−1	f(sn−1	PROPN
ejpam-3283	310	11	,	,	PUNCT
ejpam-3283	310	12	a	a	PRON
ejpam-3283	310	13	)	)	PUNCT
ejpam-3283	310	14	,	,	PUNCT
ejpam-3283	310	15	and	and	CCONJ
ejpam-3283	310	16	so	so	ADV
ejpam-3283	310	17	{	{	PUNCT
ejpam-3283	310	18	a	a	NOUN
ejpam-3283	310	19	}	}	PUNCT
ejpam-3283	310	20	⊆	⊆	NUM
ejpam-3283	310	21	f(sn−1	f(sn−1	NOUN
ejpam-3283	310	22	,	,	PUNCT
ejpam-3283	310	23	a	a	PRON
ejpam-3283	310	24	)	)	PUNCT
ejpam-3283	310	25	.	.	PUNCT
ejpam-3283	311	1	then	then	ADV
ejpam-3283	311	2	in(a	in(a	PUNCT
ejpam-3283	311	3	)	)	PUNCT
ejpam-3283	311	4	=	=	SYM
ejpam-3283	311	5	f(sn−1	f(sn−1	PROPN
ejpam-3283	311	6	,	,	PUNCT
ejpam-3283	311	7	a	a	PRON
ejpam-3283	311	8	)	)	PUNCT
ejpam-3283	311	9	∪	∪	NOUN
ejpam-3283	311	10	{	{	PUNCT
ejpam-3283	311	11	a	a	DET
ejpam-3283	311	12	}	}	PUNCT
ejpam-3283	311	13	=	=	SYM
ejpam-3283	311	14	f(sn−1	f(sn−1	NOUN
ejpam-3283	311	15	,	,	PUNCT
ejpam-3283	311	16	a	a	PRON
ejpam-3283	311	17	)	)	PUNCT
ejpam-3283	311	18	by	by	ADP
ejpam-3283	311	19	corollary	corollary	ADJ
ejpam-3283	311	20	1	1	NUM
ejpam-3283	311	21	.	.	PUNCT
ejpam-3283	312	1	since	since	SCONJ
ejpam-3283	312	2	a	a	DET
ejpam-3283	312	3	/∈	/∈	SYM
ejpam-3283	312	4	u	u	NOUN
ejpam-3283	312	5	,	,	PUNCT
ejpam-3283	312	6	we	we	PRON
ejpam-3283	312	7	obtain	obtain	VERB
ejpam-3283	312	8	in(a	in(a	PUNCT
ejpam-3283	312	9	)	)	PUNCT
ejpam-3283	313	1	=	=	SYM
ejpam-3283	313	2	s.	s.	PROPN
ejpam-3283	313	3	hence	hence	ADV
ejpam-3283	313	4	s	s	PART
ejpam-3283	313	5	=	=	NOUN
ejpam-3283	313	6	in(a	in(a	X
ejpam-3283	313	7	)	)	PUNCT
ejpam-3283	313	8	=	=	SYM
ejpam-3283	313	9	f(sn−1	f(sn−1	PROPN
ejpam-3283	313	10	,	,	PUNCT
ejpam-3283	313	11	a	a	PRON
ejpam-3283	313	12	)	)	PUNCT
ejpam-3283	313	13	.	.	PUNCT
ejpam-3283	314	1	now	now	ADV
ejpam-3283	314	2	,	,	PUNCT
ejpam-3283	314	3	we	we	PRON
ejpam-3283	314	4	get	get	VERB
ejpam-3283	314	5	s	s	PROPN
ejpam-3283	314	6	ru	ru	NOUN
ejpam-3283	314	7	⊆	⊆	NUM
ejpam-3283	314	8	{	{	PUNCT
ejpam-3283	314	9	a	a	DET
ejpam-3283	314	10	∈	∈	NOUN
ejpam-3283	314	11	s	s	VERB
ejpam-3283	314	12	|	|	ADV
ejpam-3283	314	13	f(sn−1	f(sn−1	ADJ
ejpam-3283	314	14	,	,	PUNCT
ejpam-3283	314	15	a	a	PRON
ejpam-3283	314	16	)	)	PUNCT
ejpam-3283	314	17	=	=	SYM
ejpam-3283	314	18	s	s	X
ejpam-3283	314	19	}	}	PUNCT
ejpam-3283	314	20	.	.	PUNCT
ejpam-3283	315	1	conversely	conversely	ADV
ejpam-3283	315	2	,	,	PUNCT
ejpam-3283	315	3	let	let	VERB
ejpam-3283	315	4	a	a	DET
ejpam-3283	315	5	∈	∈	NOUN
ejpam-3283	315	6	s	s	AUX
ejpam-3283	315	7	be	be	AUX
ejpam-3283	315	8	such	such	ADJ
ejpam-3283	315	9	that	that	PRON
ejpam-3283	315	10	s	s	NOUN
ejpam-3283	315	11	=	=	NOUN
ejpam-3283	315	12	f(sn−1	f(sn−1	PROPN
ejpam-3283	315	13	,	,	PUNCT
ejpam-3283	315	14	a	a	PRON
ejpam-3283	315	15	)	)	PUNCT
ejpam-3283	315	16	.	.	PUNCT
ejpam-3283	316	1	if	if	SCONJ
ejpam-3283	316	2	a	a	DET
ejpam-3283	316	3	∈	∈	PROPN
ejpam-3283	316	4	u	u	NOUN
ejpam-3283	316	5	,	,	PUNCT
ejpam-3283	316	6	then	then	ADV
ejpam-3283	316	7	in(a	in(a	PUNCT
ejpam-3283	316	8	)	)	PUNCT
ejpam-3283	316	9	⊆	⊆	NUM
ejpam-3283	316	10	u	u	NOUN
ejpam-3283	316	11	(	(	PUNCT
ejpam-3283	316	12	s.	s.	PROPN
ejpam-3283	316	13	by	by	ADP
ejpam-3283	316	14	corollary	corollary	ADJ
ejpam-3283	316	15	1	1	NUM
ejpam-3283	316	16	,	,	PUNCT
ejpam-3283	316	17	we	we	PRON
ejpam-3283	316	18	have	have	VERB
ejpam-3283	316	19	in(a	in(a	PUNCT
ejpam-3283	316	20	)	)	PUNCT
ejpam-3283	316	21	=	=	SYM
ejpam-3283	316	22	f(sn−1	f(sn−1	PROPN
ejpam-3283	316	23	,	,	PUNCT
ejpam-3283	316	24	a	a	PRON
ejpam-3283	316	25	)	)	PUNCT
ejpam-3283	316	26	∪	∪	NOUN
ejpam-3283	316	27	{	{	PUNCT
ejpam-3283	316	28	a	a	NOUN
ejpam-3283	316	29	}	}	PUNCT
ejpam-3283	316	30	=	=	SYM
ejpam-3283	316	31	s	s	NOUN
ejpam-3283	316	32	∪	∪	X
ejpam-3283	316	33	{	{	PUNCT
ejpam-3283	316	34	a	a	NOUN
ejpam-3283	316	35	}	}	PUNCT
ejpam-3283	316	36	=	=	SYM
ejpam-3283	316	37	s	s	NOUN
ejpam-3283	316	38	,	,	PUNCT
ejpam-3283	316	39	which	which	PRON
ejpam-3283	316	40	is	be	AUX
ejpam-3283	316	41	a	a	DET
ejpam-3283	316	42	contradiction	contradiction	NOUN
ejpam-3283	316	43	.	.	PUNCT
ejpam-3283	317	1	this	this	PRON
ejpam-3283	317	2	implies	imply	VERB
ejpam-3283	317	3	that	that	SCONJ
ejpam-3283	317	4	a	a	DET
ejpam-3283	317	5	∈	∈	NOUN
ejpam-3283	317	6	s	s	PART
ejpam-3283	317	7	r	r	NOUN
ejpam-3283	317	8	u	u	NOUN
ejpam-3283	317	9	.	.	PUNCT
ejpam-3283	318	1	this	this	PRON
ejpam-3283	318	2	implies	imply	VERB
ejpam-3283	318	3	that	that	SCONJ
ejpam-3283	318	4	{	{	PUNCT
ejpam-3283	318	5	a	a	DET
ejpam-3283	318	6	∈	∈	NOUN
ejpam-3283	318	7	s	s	VERB
ejpam-3283	318	8	|	|	ADV
ejpam-3283	318	9	f(sn−1	f(sn−1	ADJ
ejpam-3283	318	10	,	,	PUNCT
ejpam-3283	318	11	a	a	PRON
ejpam-3283	318	12	)	)	PUNCT
ejpam-3283	318	13	=	=	SYM
ejpam-3283	318	14	s	s	X
ejpam-3283	318	15	}	}	PUNCT
ejpam-3283	318	16	⊆	⊆	NUM
ejpam-3283	318	17	s	s	NOUN
ejpam-3283	318	18	r	r	NOUN
ejpam-3283	318	19	u	u	NOUN
ejpam-3283	318	20	.	.	PUNCT
ejpam-3283	319	1	therefore	therefore	ADV
ejpam-3283	319	2	,	,	PUNCT
ejpam-3283	319	3	s	s	VERB
ejpam-3283	319	4	r	r	NOUN
ejpam-3283	319	5	u	u	NOUN
ejpam-3283	319	6	=	=	PUNCT
ejpam-3283	319	7	{	{	PUNCT
ejpam-3283	319	8	a	a	DET
ejpam-3283	319	9	∈	∈	NOUN
ejpam-3283	319	10	s	s	VERB
ejpam-3283	319	11	|	|	ADV
ejpam-3283	319	12	f(sn−1	f(sn−1	ADJ
ejpam-3283	319	13	,	,	PUNCT
ejpam-3283	319	14	a	a	PRON
ejpam-3283	319	15	)	)	PUNCT
ejpam-3283	319	16	=	=	SYM
ejpam-3283	320	1	s	s	X
ejpam-3283	320	2	}	}	PUNCT
ejpam-3283	320	3	,	,	PUNCT
ejpam-3283	320	4	as	as	SCONJ
ejpam-3283	320	5	desired	desire	VERB
ejpam-3283	320	6	.	.	PUNCT
ejpam-3283	321	1	in	in	ADP
ejpam-3283	321	2	this	this	DET
ejpam-3283	321	3	case	case	NOUN
ejpam-3283	321	4	,	,	PUNCT
ejpam-3283	321	5	the	the	DET
ejpam-3283	321	6	statement	statement	NOUN
ejpam-3283	321	7	(	(	PUNCT
ejpam-3283	321	8	4	4	X
ejpam-3283	321	9	)	)	PUNCT
ejpam-3283	321	10	is	be	AUX
ejpam-3283	321	11	satisfied	satisfied	ADJ
ejpam-3283	321	12	.	.	PUNCT
ejpam-3283	322	1	hence	hence	ADV
ejpam-3283	322	2	the	the	DET
ejpam-3283	322	3	proof	proof	NOUN
ejpam-3283	322	4	is	be	AUX
ejpam-3283	322	5	completed	complete	VERB
ejpam-3283	322	6	.	.	PUNCT
ejpam-3283	323	1	example	example	NOUN
ejpam-3283	324	1	6	6	NUM
ejpam-3283	324	2	.	.	PUNCT
ejpam-3283	325	1	(	(	PUNCT
ejpam-3283	325	2	1	1	X
ejpam-3283	325	3	)	)	PUNCT
ejpam-3283	325	4	let	let	VERB
ejpam-3283	325	5	s	s	PRON
ejpam-3283	325	6	=	=	NOUN
ejpam-3283	325	7	{	{	PUNCT
ejpam-3283	325	8	−1	−1	NOUN
ejpam-3283	325	9	,	,	PUNCT
ejpam-3283	325	10	1	1	NUM
ejpam-3283	325	11	}	}	PUNCT
ejpam-3283	325	12	.	.	PUNCT
ejpam-3283	326	1	define	define	VERB
ejpam-3283	326	2	f	f	PROPN
ejpam-3283	326	3	:	:	PUNCT
ejpam-3283	326	4	sn	sn	PROPN
ejpam-3283	326	5	→	→	SYM
ejpam-3283	326	6	s	s	X
ejpam-3283	326	7	by	by	ADP
ejpam-3283	326	8	f(xn1	f(xn1	NOUN
ejpam-3283	326	9	)	)	PUNCT
ejpam-3283	326	10	=	=	PUNCT
ejpam-3283	327	1	x1	x1	PROPN
ejpam-3283	327	2	·	·	PUNCT
ejpam-3283	327	3	x2	x2	X
ejpam-3283	327	4	·	·	PUNCT
ejpam-3283	327	5	.	.	PUNCT
ejpam-3283	327	6	.	.	PUNCT
ejpam-3283	327	7	.	.	PUNCT
ejpam-3283	328	1	·	·	PUNCT
ejpam-3283	328	2	xn	xn	PUNCT
ejpam-3283	329	1	for	for	ADP
ejpam-3283	329	2	all	all	DET
ejpam-3283	329	3	x1	x1	PROPN
ejpam-3283	329	4	,	,	PUNCT
ejpam-3283	329	5	x2	x2	PROPN
ejpam-3283	329	6	,	,	PUNCT
ejpam-3283	329	7	.	.	PUNCT
ejpam-3283	329	8	.	.	PUNCT
ejpam-3283	329	9	.	.	PUNCT
ejpam-3283	330	1	,	,	PUNCT
ejpam-3283	330	2	xn	xn	PUNCT
ejpam-3283	330	3	∈	∈	PROPN
ejpam-3283	330	4	s	s	VERB
ejpam-3283	330	5	where	where	SCONJ
ejpam-3283	330	6	·	·	PUNCT
ejpam-3283	330	7	is	be	AUX
ejpam-3283	330	8	the	the	DET
ejpam-3283	330	9	usual	usual	ADJ
ejpam-3283	330	10	multiplication	multiplication	NOUN
ejpam-3283	330	11	.	.	PUNCT
ejpam-3283	331	1	then	then	ADV
ejpam-3283	331	2	s	s	VERB
ejpam-3283	331	3	is	be	AUX
ejpam-3283	331	4	n	n	PRON
ejpam-3283	331	5	-	-	PUNCT
ejpam-3283	331	6	simple	simple	ADJ
ejpam-3283	331	7	,	,	PUNCT
ejpam-3283	331	8	this	this	PRON
ejpam-3283	331	9	implies	imply	VERB
ejpam-3283	331	10	that	that	SCONJ
ejpam-3283	331	11	u	u	NOUN
ejpam-3283	331	12	=	=	PUNCT
ejpam-3283	331	13	∅.	∅.	VERB
ejpam-3283	331	14	so	so	ADV
ejpam-3283	331	15	,	,	PUNCT
ejpam-3283	331	16	s	s	AUX
ejpam-3283	331	17	satisfies	satisfie	NOUN
ejpam-3283	331	18	the	the	DET
ejpam-3283	331	19	condition	condition	NOUN
ejpam-3283	331	20	(	(	PUNCT
ejpam-3283	331	21	1	1	NUM
ejpam-3283	331	22	)	)	PUNCT
ejpam-3283	331	23	of	of	ADP
ejpam-3283	331	24	theorem	theorem	ADJ
ejpam-3283	331	25	8	8	NUM
ejpam-3283	331	26	.	.	PUNCT
ejpam-3283	332	1	(	(	PUNCT
ejpam-3283	332	2	2	2	X
ejpam-3283	332	3	)	)	PUNCT
ejpam-3283	332	4	let	let	VERB
ejpam-3283	332	5	s	s	PRON
ejpam-3283	332	6	=	=	VERB
ejpam-3283	332	7	nr	nr	PROPN
ejpam-3283	332	8	{	{	PUNCT
ejpam-3283	332	9	1	1	NUM
ejpam-3283	332	10	}	}	PUNCT
ejpam-3283	332	11	.	.	PUNCT
ejpam-3283	333	1	define	define	VERB
ejpam-3283	333	2	f	f	PROPN
ejpam-3283	333	3	:	:	PUNCT
ejpam-3283	333	4	sn	sn	PROPN
ejpam-3283	333	5	→	→	SYM
ejpam-3283	333	6	s	s	X
ejpam-3283	333	7	by	by	ADP
ejpam-3283	333	8	f(xn1	f(xn1	NOUN
ejpam-3283	333	9	)	)	PUNCT
ejpam-3283	333	10	=	=	PUNCT
ejpam-3283	334	1	x1	x1	PROPN
ejpam-3283	334	2	·	·	PUNCT
ejpam-3283	334	3	x2	x2	X
ejpam-3283	334	4	·	·	PUNCT
ejpam-3283	334	5	.	.	PUNCT
ejpam-3283	334	6	.	.	PUNCT
ejpam-3283	334	7	.	.	PUNCT
ejpam-3283	335	1	·	·	PUNCT
ejpam-3283	335	2	xn	xn	PUNCT
ejpam-3283	336	1	for	for	ADP
ejpam-3283	336	2	all	all	DET
ejpam-3283	336	3	x1	x1	PROPN
ejpam-3283	336	4	,	,	PUNCT
ejpam-3283	336	5	x2	x2	PROPN
ejpam-3283	336	6	,	,	PUNCT
ejpam-3283	336	7	.	.	PUNCT
ejpam-3283	336	8	.	.	PUNCT
ejpam-3283	336	9	.	.	PUNCT
ejpam-3283	337	1	,	,	PUNCT
ejpam-3283	337	2	xn	xn	PUNCT
ejpam-3283	337	3	∈	∈	PROPN
ejpam-3283	337	4	s	s	VERB
ejpam-3283	337	5	where	where	SCONJ
ejpam-3283	337	6	·	·	PUNCT
ejpam-3283	337	7	is	be	AUX
ejpam-3283	337	8	the	the	DET
ejpam-3283	337	9	usual	usual	ADJ
ejpam-3283	337	10	multiplication	multiplication	NOUN
ejpam-3283	337	11	.	.	PUNCT
ejpam-3283	338	1	it	it	PRON
ejpam-3283	338	2	is	be	AUX
ejpam-3283	338	3	easy	easy	ADJ
ejpam-3283	338	4	to	to	PART
ejpam-3283	338	5	verify	verify	VERB
ejpam-3283	338	6	that	that	PRON
ejpam-3283	338	7	in(a	in(a	NUM
ejpam-3283	338	8	)	)	PUNCT
ejpam-3283	338	9	6=	6=	ADP
ejpam-3283	338	10	s	s	VERB
ejpam-3283	338	11	for	for	ADP
ejpam-3283	338	12	all	all	DET
ejpam-3283	338	13	a	a	DET
ejpam-3283	338	14	∈	∈	NOUN
ejpam-3283	338	15	s.	s.	PROPN
ejpam-3283	338	16	hence	hence	ADV
ejpam-3283	338	17	s	s	PART
ejpam-3283	338	18	satisfies	satisfie	NOUN
ejpam-3283	338	19	the	the	DET
ejpam-3283	338	20	condition	condition	NOUN
ejpam-3283	338	21	(	(	PUNCT
ejpam-3283	338	22	2	2	NUM
ejpam-3283	338	23	)	)	PUNCT
ejpam-3283	338	24	of	of	ADP
ejpam-3283	338	25	theorem	theorem	NOUN
ejpam-3283	338	26	8	8	NUM
ejpam-3283	338	27	.	.	PUNCT
ejpam-3283	339	1	(	(	PUNCT
ejpam-3283	339	2	3	3	X
ejpam-3283	339	3	)	)	PUNCT
ejpam-3283	339	4	consider	consider	VERB
ejpam-3283	339	5	z2n+1	z2n+1	NOUN
ejpam-3283	339	6	,	,	PUNCT
ejpam-3283	339	7	let	let	VERB
ejpam-3283	339	8	s	s	PRON
ejpam-3283	339	9	=	=	VERB
ejpam-3283	339	10	{	{	PUNCT
ejpam-3283	339	11	0	0	NUM
ejpam-3283	339	12	,	,	PUNCT
ejpam-3283	339	13	2	2	NUM
ejpam-3283	339	14	,	,	PUNCT
ejpam-3283	339	15	2n	2n	NUM
ejpam-3283	339	16	}	}	PUNCT
ejpam-3283	339	17	.	.	PUNCT
ejpam-3283	340	1	define	define	VERB
ejpam-3283	340	2	f	f	PROPN
ejpam-3283	340	3	:	:	PUNCT
ejpam-3283	340	4	sn	sn	PROPN
ejpam-3283	340	5	→	→	SYM
ejpam-3283	340	6	s	s	X
ejpam-3283	340	7	by	by	ADP
ejpam-3283	340	8	f(xn1	f(xn1	NOUN
ejpam-3283	340	9	)	)	PUNCT
ejpam-3283	340	10	=	=	PUNCT
ejpam-3283	341	1	x1	x1	PROPN
ejpam-3283	341	2	·	·	PUNCT
ejpam-3283	341	3	x2	x2	X
ejpam-3283	341	4	·	·	PUNCT
ejpam-3283	341	5	.	.	PUNCT
ejpam-3283	341	6	.	.	PUNCT
ejpam-3283	341	7	.	.	PUNCT
ejpam-3283	342	1	·	·	PUNCT
ejpam-3283	342	2	xn	xn	PUNCT
ejpam-3283	343	1	for	for	ADP
ejpam-3283	343	2	all	all	DET
ejpam-3283	343	3	x1	x1	PROPN
ejpam-3283	343	4	,	,	PUNCT
ejpam-3283	343	5	x2	x2	PROPN
ejpam-3283	343	6	,	,	PUNCT
ejpam-3283	343	7	.	.	PUNCT
ejpam-3283	343	8	.	.	PUNCT
ejpam-3283	343	9	.	.	PUNCT
ejpam-3283	344	1	,	,	PUNCT
ejpam-3283	344	2	xn	xn	PUNCT
ejpam-3283	344	3	∈	∈	PROPN
ejpam-3283	344	4	s	s	VERB
ejpam-3283	344	5	where	where	SCONJ
ejpam-3283	344	6	·	·	PUNCT
ejpam-3283	344	7	is	be	AUX
ejpam-3283	344	8	the	the	DET
ejpam-3283	344	9	usual	usual	ADJ
ejpam-3283	344	10	multiplication	multiplication	NOUN
ejpam-3283	344	11	.	.	PUNCT
ejpam-3283	345	1	thus	thus	ADV
ejpam-3283	345	2	u	u	X
ejpam-3283	345	3	=	=	PUNCT
ejpam-3283	345	4	{	{	PUNCT
ejpam-3283	345	5	0	0	NUM
ejpam-3283	345	6	,	,	PUNCT
ejpam-3283	345	7	2n	2n	NUM
ejpam-3283	345	8	}	}	PUNCT
ejpam-3283	345	9	.	.	PUNCT
ejpam-3283	346	1	it	it	PRON
ejpam-3283	346	2	is	be	AUX
ejpam-3283	346	3	easy	easy	ADJ
ejpam-3283	346	4	to	to	PART
ejpam-3283	346	5	verify	verify	VERB
ejpam-3283	346	6	that	that	SCONJ
ejpam-3283	346	7	s	s	ADP
ejpam-3283	346	8	satisfies	satisfie	NOUN
ejpam-3283	346	9	the	the	DET
ejpam-3283	346	10	condition	condition	NOUN
ejpam-3283	346	11	(	(	PUNCT
ejpam-3283	346	12	3	3	NUM
ejpam-3283	346	13	)	)	PUNCT
ejpam-3283	346	14	of	of	ADP
ejpam-3283	346	15	theorem	theorem	NOUN
ejpam-3283	346	16	8	8	NUM
ejpam-3283	346	17	by	by	ADP
ejpam-3283	346	18	use	use	NOUN
ejpam-3283	346	19	a	a	DET
ejpam-3283	346	20	=	=	SYM
ejpam-3283	346	21	2	2	NUM
ejpam-3283	346	22	.	.	PUNCT
ejpam-3283	347	1	(	(	PUNCT
ejpam-3283	347	2	4	4	X
ejpam-3283	347	3	)	)	PUNCT
ejpam-3283	347	4	let	let	VERB
ejpam-3283	347	5	s	s	NOUN
ejpam-3283	347	6	=	=	VERB
ejpam-3283	347	7	n.	n.	NOUN
ejpam-3283	347	8	define	define	VERB
ejpam-3283	347	9	f	f	PROPN
ejpam-3283	347	10	:	:	PUNCT
ejpam-3283	347	11	sn	sn	PROPN
ejpam-3283	347	12	→	→	SYM
ejpam-3283	347	13	s	s	X
ejpam-3283	347	14	by	by	ADP
ejpam-3283	347	15	f(xn1	f(xn1	NOUN
ejpam-3283	347	16	)	)	PUNCT
ejpam-3283	347	17	=	=	PUNCT
ejpam-3283	348	1	x1	x1	PROPN
ejpam-3283	348	2	·	·	PUNCT
ejpam-3283	348	3	x2	x2	X
ejpam-3283	348	4	·	·	PUNCT
ejpam-3283	348	5	.	.	PUNCT
ejpam-3283	348	6	.	.	PUNCT
ejpam-3283	348	7	.	.	PUNCT
ejpam-3283	349	1	·	·	PUNCT
ejpam-3283	349	2	xn	xn	PUNCT
ejpam-3283	350	1	for	for	ADP
ejpam-3283	350	2	all	all	DET
ejpam-3283	350	3	x1	x1	PROPN
ejpam-3283	350	4	,	,	PUNCT
ejpam-3283	350	5	x2	x2	PROPN
ejpam-3283	350	6	,	,	PUNCT
ejpam-3283	350	7	.	.	PUNCT
ejpam-3283	350	8	.	.	PUNCT
ejpam-3283	350	9	.	.	PUNCT
ejpam-3283	351	1	,	,	PUNCT
ejpam-3283	351	2	xn	xn	PUNCT
ejpam-3283	351	3	∈	∈	PROPN
ejpam-3283	351	4	s	s	VERB
ejpam-3283	351	5	where	where	SCONJ
ejpam-3283	351	6	·	·	PUNCT
ejpam-3283	351	7	is	be	AUX
ejpam-3283	351	8	the	the	DET
ejpam-3283	351	9	usual	usual	ADJ
ejpam-3283	351	10	multiplication	multiplication	NOUN
ejpam-3283	351	11	.	.	PUNCT
ejpam-3283	352	1	then	then	ADV
ejpam-3283	352	2	u	u	X
ejpam-3283	353	1	=	=	PROPN
ejpam-3283	353	2	s	s	PART
ejpam-3283	353	3	\	\	X
ejpam-3283	353	4	{	{	PUNCT
ejpam-3283	353	5	1	1	NUM
ejpam-3283	353	6	}	}	PUNCT
ejpam-3283	353	7	.	.	PUNCT
ejpam-3283	354	1	it	it	PRON
ejpam-3283	354	2	is	be	AUX
ejpam-3283	354	3	easy	easy	ADJ
ejpam-3283	354	4	to	to	PART
ejpam-3283	354	5	verify	verify	VERB
ejpam-3283	354	6	that	that	SCONJ
ejpam-3283	354	7	s	s	ADP
ejpam-3283	354	8	satisfies	satisfie	NOUN
ejpam-3283	354	9	the	the	DET
ejpam-3283	354	10	condition	condition	NOUN
ejpam-3283	354	11	(	(	PUNCT
ejpam-3283	354	12	4	4	NUM
ejpam-3283	354	13	)	)	PUNCT
ejpam-3283	354	14	of	of	ADP
ejpam-3283	354	15	theorem	theorem	ADJ
ejpam-3283	354	16	8	8	NUM
ejpam-3283	354	17	.	.	PUNCT
ejpam-3283	354	18	theorem	theorem	NOUN
ejpam-3283	354	19	9	9	NUM
ejpam-3283	354	20	.	.	PUNCT
ejpam-3283	355	1	if	if	SCONJ
ejpam-3283	355	2	s	s	PROPN
ejpam-3283	355	3	has	have	VERB
ejpam-3283	355	4	a	a	DET
ejpam-3283	355	5	zero	zero	NUM
ejpam-3283	355	6	element	element	NOUN
ejpam-3283	355	7	and	and	CCONJ
ejpam-3283	355	8	f(sn	f(sn	NOUN
ejpam-3283	355	9	)	)	PUNCT
ejpam-3283	355	10	6=	6=	PUNCT
ejpam-3283	355	11	{	{	PUNCT
ejpam-3283	355	12	0	0	NUM
ejpam-3283	355	13	}	}	PUNCT
ejpam-3283	355	14	,	,	PUNCT
ejpam-3283	355	15	then	then	ADV
ejpam-3283	355	16	the	the	DET
ejpam-3283	355	17	exactly	exactly	ADV
ejpam-3283	355	18	one	one	NUM
ejpam-3283	355	19	of	of	ADP
ejpam-3283	355	20	the	the	DET
ejpam-3283	355	21	following	following	ADJ
ejpam-3283	355	22	statements	statement	NOUN
ejpam-3283	355	23	is	be	AUX
ejpam-3283	355	24	satisfied	satisfied	ADJ
ejpam-3283	355	25	:	:	PUNCT
ejpam-3283	355	26	p.	p.	NOUN
ejpam-3283	355	27	petchkaew	petchkaew	NOUN
ejpam-3283	355	28	,	,	PUNCT
ejpam-3283	355	29	r.	r.	PROPN
ejpam-3283	355	30	chinram	chinram	PROPN
ejpam-3283	355	31	/	/	SYM
ejpam-3283	355	32	eur	eur	PROPN
ejpam-3283	355	33	.	.	PUNCT
ejpam-3283	356	1	j.	j.	PROPN
ejpam-3283	356	2	pure	pure	PROPN
ejpam-3283	356	3	appl	appl	PROPN
ejpam-3283	356	4	.	.	PROPN
ejpam-3283	356	5	math	math	PROPN
ejpam-3283	356	6	,	,	PUNCT
ejpam-3283	356	7	11	11	NUM
ejpam-3283	356	8	(	(	PUNCT
ejpam-3283	356	9	3	3	NUM
ejpam-3283	356	10	)	)	PUNCT
ejpam-3283	356	11	(	(	PUNCT
ejpam-3283	356	12	2018	2018	NUM
ejpam-3283	356	13	)	)	PUNCT
ejpam-3283	356	14	,	,	PUNCT
ejpam-3283	356	15	762	762	NUM
ejpam-3283	356	16	-	-	SYM
ejpam-3283	356	17	773	773	NUM
ejpam-3283	356	18	772	772	NUM
ejpam-3283	356	19	(	(	PUNCT
ejpam-3283	356	20	1	1	NUM
ejpam-3283	356	21	)	)	PUNCT
ejpam-3283	356	22	s	s	VERB
ejpam-3283	356	23	is	be	AUX
ejpam-3283	356	24	0	0	NUM
ejpam-3283	356	25	-	-	PUNCT
ejpam-3283	356	26	n	n	CCONJ
ejpam-3283	356	27	-	-	PUNCT
ejpam-3283	356	28	simple	simple	NOUN
ejpam-3283	356	29	.	.	PUNCT
ejpam-3283	357	1	(	(	PUNCT
ejpam-3283	357	2	2	2	NUM
ejpam-3283	357	3	)	)	PUNCT
ejpam-3283	357	4	in(a	in(a	PUNCT
ejpam-3283	357	5	)	)	PUNCT
ejpam-3283	358	1	6=	6=	ADP
ejpam-3283	358	2	s	s	VERB
ejpam-3283	358	3	for	for	ADP
ejpam-3283	358	4	all	all	DET
ejpam-3283	358	5	a	a	DET
ejpam-3283	358	6	∈	∈	NOUN
ejpam-3283	358	7	s.	s.	PROPN
ejpam-3283	358	8	(	(	PUNCT
ejpam-3283	358	9	3	3	X
ejpam-3283	358	10	)	)	PUNCT
ejpam-3283	358	11	there	there	PRON
ejpam-3283	358	12	exists	exist	VERB
ejpam-3283	358	13	a	a	DET
ejpam-3283	358	14	∈	∈	NOUN
ejpam-3283	358	15	s	s	VERB
ejpam-3283	358	16	such	such	ADJ
ejpam-3283	358	17	that	that	SCONJ
ejpam-3283	358	18	in(a	in(a	NUM
ejpam-3283	358	19	)	)	PUNCT
ejpam-3283	358	20	=	=	SYM
ejpam-3283	358	21	s	s	PROPN
ejpam-3283	358	22	,	,	PUNCT
ejpam-3283	358	23	a	a	PRON
ejpam-3283	358	24	/∈	/∈	NOUN
ejpam-3283	358	25	f(sn−1	f(sn−1	NOUN
ejpam-3283	358	26	,	,	PUNCT
ejpam-3283	358	27	a	a	PRON
ejpam-3283	358	28	)	)	PUNCT
ejpam-3283	358	29	,	,	PUNCT
ejpam-3283	358	30	f(a	f(a	PROPN
ejpam-3283	358	31	,	,	PUNCT
ejpam-3283	358	32	sn−2	sn−2	PROPN
ejpam-3283	358	33	,	,	PUNCT
ejpam-3283	358	34	a	a	PRON
ejpam-3283	358	35	)	)	PUNCT
ejpam-3283	358	36	⊆	⊆	NUM
ejpam-3283	358	37	u	u	NOUN
ejpam-3283	358	38	=	=	SYM
ejpam-3283	358	39	s	s	PART
ejpam-3283	358	40	r	r	NOUN
ejpam-3283	358	41	{	{	PUNCT
ejpam-3283	358	42	a	a	NOUN
ejpam-3283	358	43	}	}	PUNCT
ejpam-3283	358	44	and	and	CCONJ
ejpam-3283	358	45	u	u	NOUN
ejpam-3283	358	46	is	be	AUX
ejpam-3283	358	47	the	the	DET
ejpam-3283	358	48	unique	unique	ADJ
ejpam-3283	358	49	maximal	maximal	ADJ
ejpam-3283	358	50	n	n	CCONJ
ejpam-3283	358	51	-	-	PUNCT
ejpam-3283	358	52	ideal	ideal	NOUN
ejpam-3283	358	53	of	of	ADP
ejpam-3283	358	54	s.	s.	PROPN
ejpam-3283	358	55	(	(	PUNCT
ejpam-3283	358	56	4	4	NUM
ejpam-3283	358	57	)	)	PUNCT
ejpam-3283	358	58	s	s	PART
ejpam-3283	358	59	r	r	NOUN
ejpam-3283	358	60	u	u	NOUN
ejpam-3283	358	61	=	=	PUNCT
ejpam-3283	358	62	{	{	PUNCT
ejpam-3283	358	63	a	a	DET
ejpam-3283	358	64	∈	∈	NOUN
ejpam-3283	358	65	s	s	VERB
ejpam-3283	358	66	|	|	ADV
ejpam-3283	358	67	f(sn−1	f(sn−1	ADJ
ejpam-3283	358	68	,	,	PUNCT
ejpam-3283	358	69	a	a	PRON
ejpam-3283	358	70	)	)	PUNCT
ejpam-3283	358	71	=	=	SYM
ejpam-3283	358	72	s	s	X
ejpam-3283	358	73	}	}	PUNCT
ejpam-3283	358	74	and	and	CCONJ
ejpam-3283	358	75	u	u	NOUN
ejpam-3283	358	76	is	be	AUX
ejpam-3283	358	77	the	the	DET
ejpam-3283	358	78	unique	unique	ADJ
ejpam-3283	358	79	maximal	maximal	ADJ
ejpam-3283	358	80	n	n	CCONJ
ejpam-3283	358	81	-	-	PUNCT
ejpam-3283	358	82	ideal	ideal	NOUN
ejpam-3283	358	83	of	of	ADP
ejpam-3283	358	84	s.	s.	PROPN
ejpam-3283	358	85	proof	proof	PROPN
ejpam-3283	358	86	.	.	PUNCT
ejpam-3283	359	1	this	this	PRON
ejpam-3283	359	2	follows	follow	VERB
ejpam-3283	359	3	from	from	ADP
ejpam-3283	359	4	theorem	theorem	ADJ
ejpam-3283	359	5	8	8	NUM
ejpam-3283	359	6	.	.	NOUN
ejpam-3283	359	7	6	6	NUM
ejpam-3283	359	8	.	.	X
ejpam-3283	360	1	discussion	discussion	NOUN
ejpam-3283	360	2	in	in	ADP
ejpam-3283	360	3	this	this	DET
ejpam-3283	360	4	paper	paper	NOUN
ejpam-3283	360	5	,	,	PUNCT
ejpam-3283	360	6	we	we	PRON
ejpam-3283	360	7	introduce	introduce	VERB
ejpam-3283	360	8	many	many	ADJ
ejpam-3283	360	9	algebraic	algebraic	ADJ
ejpam-3283	360	10	structures	structure	NOUN
ejpam-3283	360	11	of	of	ADP
ejpam-3283	360	12	n	n	CCONJ
ejpam-3283	360	13	-	-	PUNCT
ejpam-3283	360	14	ary	ary	NOUN
ejpam-3283	360	15	semigroups	semigroup	NOUN
ejpam-3283	360	16	and	and	CCONJ
ejpam-3283	360	17	ones	one	NOUN
ejpam-3283	360	18	of	of	ADP
ejpam-3283	360	19	those	those	DET
ejpam-3283	360	20	important	important	ADJ
ejpam-3283	360	21	are	be	AUX
ejpam-3283	360	22	n	n	PRON
ejpam-3283	360	23	-	-	PUNCT
ejpam-3283	360	24	ideals	ideal	NOUN
ejpam-3283	360	25	,	,	PUNCT
ejpam-3283	360	26	n	n	CCONJ
ejpam-3283	360	27	-	-	PUNCT
ejpam-3283	360	28	simple	simple	ADJ
ejpam-3283	360	29	,	,	PUNCT
ejpam-3283	360	30	0	0	NUM
ejpam-3283	360	31	-	-	PUNCT
ejpam-3283	360	32	n	n	CCONJ
ejpam-3283	360	33	-	-	PUNCT
ejpam-3283	360	34	simple	simple	ADJ
ejpam-3283	360	35	,	,	PUNCT
ejpam-3283	360	36	minimal	minimal	ADJ
ejpam-3283	360	37	n	n	CCONJ
ejpam-3283	360	38	-	-	PUNCT
ejpam-3283	360	39	ideals	ideal	NOUN
ejpam-3283	360	40	,	,	PUNCT
ejpam-3283	360	41	0	0	NUM
ejpam-3283	360	42	-	-	PUNCT
ejpam-3283	360	43	minimal	minimal	ADJ
ejpam-3283	360	44	nideals	nideal	NOUN
ejpam-3283	360	45	,	,	PUNCT
ejpam-3283	360	46	and	and	CCONJ
ejpam-3283	360	47	maximal	maximal	ADJ
ejpam-3283	360	48	n	n	CCONJ
ejpam-3283	360	49	-	-	PUNCT
ejpam-3283	360	50	ideals	ideal	NOUN
ejpam-3283	360	51	.	.	PUNCT
ejpam-3283	361	1	the	the	DET
ejpam-3283	361	2	concept	concept	NOUN
ejpam-3283	361	3	of	of	ADP
ejpam-3283	361	4	n	n	CCONJ
ejpam-3283	361	5	-	-	PUNCT
ejpam-3283	361	6	ideals	ideal	NOUN
ejpam-3283	361	7	(	(	PUNCT
ejpam-3283	361	8	n	n	CCONJ
ejpam-3283	361	9	-	-	PUNCT
ejpam-3283	361	10	simple	simple	ADJ
ejpam-3283	361	11	,	,	PUNCT
ejpam-3283	361	12	0	0	NUM
ejpam-3283	361	13	-	-	PUNCT
ejpam-3283	361	14	n	n	CCONJ
ejpam-3283	361	15	-	-	PUNCT
ejpam-3283	361	16	simple	simple	ADJ
ejpam-3283	361	17	,	,	PUNCT
ejpam-3283	361	18	minimal	minimal	ADJ
ejpam-3283	361	19	n	n	CCONJ
ejpam-3283	361	20	-	-	PUNCT
ejpam-3283	361	21	ideals	ideal	NOUN
ejpam-3283	361	22	,	,	PUNCT
ejpam-3283	361	23	0	0	NUM
ejpam-3283	361	24	-	-	PUNCT
ejpam-3283	361	25	minimal	minimal	ADJ
ejpam-3283	361	26	n	n	CCONJ
ejpam-3283	361	27	-	-	PUNCT
ejpam-3283	361	28	ideals	ideal	NOUN
ejpam-3283	361	29	,	,	PUNCT
ejpam-3283	361	30	and	and	CCONJ
ejpam-3283	361	31	maximal	maximal	ADJ
ejpam-3283	361	32	n	n	CCONJ
ejpam-3283	361	33	-	-	PUNCT
ejpam-3283	361	34	ideals	ideal	NOUN
ejpam-3283	361	35	,	,	PUNCT
ejpam-3283	361	36	respectively	respectively	ADV
ejpam-3283	361	37	)	)	PUNCT
ejpam-3283	361	38	of	of	ADP
ejpam-3283	361	39	n	n	CCONJ
ejpam-3283	361	40	-	-	PUNCT
ejpam-3283	361	41	ary	ary	NOUN
ejpam-3283	361	42	semigroups	semigroup	NOUN
ejpam-3283	361	43	that	that	SCONJ
ejpam-3283	361	44	we	we	PRON
ejpam-3283	361	45	studied	study	VERB
ejpam-3283	361	46	supports	support	VERB
ejpam-3283	361	47	the	the	DET
ejpam-3283	361	48	concept	concept	NOUN
ejpam-3283	361	49	of	of	ADP
ejpam-3283	361	50	left	left	ADJ
ejpam-3283	361	51	ideals	ideal	NOUN
ejpam-3283	361	52	(	(	PUNCT
ejpam-3283	361	53	left	leave	VERB
ejpam-3283	361	54	-	-	PUNCT
ejpam-3283	361	55	simple	simple	ADJ
ejpam-3283	361	56	,	,	PUNCT
ejpam-3283	361	57	left	leave	VERB
ejpam-3283	361	58	0	0	NOUN
ejpam-3283	361	59	-	-	PUNCT
ejpam-3283	361	60	simple	simple	ADJ
ejpam-3283	361	61	,	,	PUNCT
ejpam-3283	361	62	minimal	minimal	ADJ
ejpam-3283	361	63	left	leave	VERB
ejpam-3283	361	64	ideals	ideal	NOUN
ejpam-3283	361	65	,	,	PUNCT
ejpam-3283	361	66	0	0	NUM
ejpam-3283	361	67	-	-	PUNCT
ejpam-3283	361	68	minimal	minimal	ADJ
ejpam-3283	361	69	left	leave	VERB
ejpam-3283	361	70	ideals	ideal	NOUN
ejpam-3283	361	71	,	,	PUNCT
ejpam-3283	361	72	maximal	maximal	ADJ
ejpam-3283	361	73	left	leave	VERB
ejpam-3283	361	74	ideals	ideal	NOUN
ejpam-3283	361	75	,	,	PUNCT
ejpam-3283	361	76	respectively	respectively	ADV
ejpam-3283	361	77	)	)	PUNCT
ejpam-3283	361	78	of	of	ADP
ejpam-3283	361	79	semigroups	semigroup	NOUN
ejpam-3283	361	80	in	in	ADP
ejpam-3283	361	81	case	case	NOUN
ejpam-3283	361	82	n	n	NOUN
ejpam-3283	361	83	=	=	SYM
ejpam-3283	361	84	2	2	NUM
ejpam-3283	361	85	and	and	CCONJ
ejpam-3283	361	86	of	of	ADP
ejpam-3283	361	87	ternary	ternary	ADJ
ejpam-3283	361	88	semigroups	semigroup	NOUN
ejpam-3283	361	89	in	in	ADP
ejpam-3283	361	90	case	case	NOUN
ejpam-3283	361	91	n	n	NOUN
ejpam-3283	361	92	=	=	SYM
ejpam-3283	361	93	3	3	NUM
ejpam-3283	361	94	that	that	PRON
ejpam-3283	361	95	are	be	AUX
ejpam-3283	361	96	investigated	investigate	VERB
ejpam-3283	361	97	by	by	ADP
ejpam-3283	361	98	several	several	ADJ
ejpam-3283	361	99	researchers	researcher	NOUN
ejpam-3283	361	100	before	before	ADV
ejpam-3283	361	101	.	.	PUNCT
ejpam-3283	362	1	of	of	ADP
ejpam-3283	362	2	course	course	NOUN
ejpam-3283	362	3	,	,	PUNCT
ejpam-3283	362	4	the	the	DET
ejpam-3283	362	5	study	study	NOUN
ejpam-3283	362	6	of	of	ADP
ejpam-3283	362	7	left	left	ADJ
ejpam-3283	362	8	ideals	ideal	NOUN
ejpam-3283	362	9	(	(	PUNCT
ejpam-3283	362	10	left	leave	VERB
ejpam-3283	362	11	-	-	PUNCT
ejpam-3283	362	12	simple	simple	ADJ
ejpam-3283	362	13	,	,	PUNCT
ejpam-3283	362	14	left	leave	VERB
ejpam-3283	362	15	0	0	NOUN
ejpam-3283	362	16	-	-	PUNCT
ejpam-3283	362	17	simple	simple	ADJ
ejpam-3283	362	18	,	,	PUNCT
ejpam-3283	362	19	minimal	minimal	ADJ
ejpam-3283	362	20	left	leave	VERB
ejpam-3283	362	21	ideals	ideal	NOUN
ejpam-3283	362	22	,	,	PUNCT
ejpam-3283	362	23	0	0	NUM
ejpam-3283	362	24	-	-	PUNCT
ejpam-3283	362	25	minimal	minimal	ADJ
ejpam-3283	362	26	left	leave	VERB
ejpam-3283	362	27	ideals	ideal	NOUN
ejpam-3283	362	28	,	,	PUNCT
ejpam-3283	362	29	maximal	maximal	ADJ
ejpam-3283	362	30	left	leave	VERB
ejpam-3283	362	31	ideals	ideal	NOUN
ejpam-3283	362	32	,	,	PUNCT
ejpam-3283	362	33	respectively	respectively	ADV
ejpam-3283	362	34	)	)	PUNCT
ejpam-3283	362	35	is	be	AUX
ejpam-3283	362	36	always	always	ADV
ejpam-3283	362	37	come	come	VERB
ejpam-3283	362	38	together	together	ADV
ejpam-3283	362	39	with	with	ADP
ejpam-3283	362	40	the	the	DET
ejpam-3283	362	41	study	study	NOUN
ejpam-3283	362	42	of	of	ADP
ejpam-3283	362	43	right	right	ADJ
ejpam-3283	362	44	ideals	ideal	NOUN
ejpam-3283	362	45	(	(	PUNCT
ejpam-3283	362	46	right	right	ADV
ejpam-3283	362	47	-	-	PUNCT
ejpam-3283	362	48	simple	simple	ADJ
ejpam-3283	362	49	,	,	PUNCT
ejpam-3283	362	50	right	right	ADJ
ejpam-3283	362	51	0	0	NUM
ejpam-3283	362	52	-	-	NOUN
ejpam-3283	362	53	simple	simple	ADJ
ejpam-3283	362	54	,	,	PUNCT
ejpam-3283	362	55	minimal	minimal	ADJ
ejpam-3283	362	56	right	right	ADJ
ejpam-3283	362	57	ideals	ideal	NOUN
ejpam-3283	362	58	,	,	PUNCT
ejpam-3283	362	59	0	0	NUM
ejpam-3283	362	60	-	-	PUNCT
ejpam-3283	362	61	minimal	minimal	ADJ
ejpam-3283	362	62	right	right	ADJ
ejpam-3283	362	63	ideals	ideal	NOUN
ejpam-3283	362	64	,	,	PUNCT
ejpam-3283	362	65	maximal	maximal	ADJ
ejpam-3283	362	66	right	right	ADJ
ejpam-3283	362	67	ideals	ideal	NOUN
ejpam-3283	362	68	,	,	PUNCT
ejpam-3283	362	69	respectively	respectively	ADV
ejpam-3283	362	70	)	)	PUNCT
ejpam-3283	362	71	because	because	SCONJ
ejpam-3283	362	72	they	they	PRON
ejpam-3283	362	73	have	have	VERB
ejpam-3283	362	74	the	the	DET
ejpam-3283	362	75	similar	similar	ADJ
ejpam-3283	362	76	structures	structure	NOUN
ejpam-3283	362	77	,	,	PUNCT
ejpam-3283	362	78	no	no	ADV
ejpam-3283	362	79	matter	matter	ADV
ejpam-3283	362	80	what	what	PRON
ejpam-3283	362	81	we	we	PRON
ejpam-3283	362	82	consider	consider	VERB
ejpam-3283	362	83	in	in	ADP
ejpam-3283	362	84	groups	group	NOUN
ejpam-3283	362	85	,	,	PUNCT
ejpam-3283	362	86	semigroups	semigroup	NOUN
ejpam-3283	362	87	,	,	PUNCT
ejpam-3283	362	88	or	or	CCONJ
ejpam-3283	362	89	ternary	ternary	ADJ
ejpam-3283	362	90	semigroups	semigroup	NOUN
ejpam-3283	362	91	,	,	PUNCT
ejpam-3283	362	92	and	and	CCONJ
ejpam-3283	362	93	so	so	SCONJ
ejpam-3283	362	94	that	that	SCONJ
ejpam-3283	362	95	many	many	ADJ
ejpam-3283	362	96	researchers	researcher	NOUN
ejpam-3283	362	97	usually	usually	ADV
ejpam-3283	362	98	show	show	VERB
ejpam-3283	362	99	in	in	ADP
ejpam-3283	362	100	the	the	DET
ejpam-3283	362	101	only	only	ADJ
ejpam-3283	362	102	one	one	NUM
ejpam-3283	362	103	case	case	NOUN
ejpam-3283	362	104	between	between	ADP
ejpam-3283	362	105	left	left	ADJ
ejpam-3283	362	106	and	and	CCONJ
ejpam-3283	362	107	right	right	ADJ
ejpam-3283	362	108	.	.	PUNCT
ejpam-3283	363	1	in	in	ADP
ejpam-3283	363	2	case	case	NOUN
ejpam-3283	363	3	of	of	ADP
ejpam-3283	363	4	right	right	NOUN
ejpam-3283	363	5	,	,	PUNCT
ejpam-3283	363	6	the	the	DET
ejpam-3283	363	7	right	right	ADJ
ejpam-3283	363	8	ideals	ideal	NOUN
ejpam-3283	363	9	(	(	PUNCT
ejpam-3283	363	10	right	right	ADV
ejpam-3283	363	11	-	-	PUNCT
ejpam-3283	363	12	simple	simple	ADJ
ejpam-3283	363	13	,	,	PUNCT
ejpam-3283	363	14	right	right	ADJ
ejpam-3283	363	15	0	0	NUM
ejpam-3283	363	16	-	-	NOUN
ejpam-3283	363	17	simple	simple	ADJ
ejpam-3283	363	18	,	,	PUNCT
ejpam-3283	363	19	minimal	minimal	ADJ
ejpam-3283	363	20	right	right	ADJ
ejpam-3283	363	21	ideals	ideal	NOUN
ejpam-3283	363	22	,	,	PUNCT
ejpam-3283	363	23	0	0	NUM
ejpam-3283	363	24	-	-	PUNCT
ejpam-3283	363	25	minimal	minimal	ADJ
ejpam-3283	363	26	right	right	ADJ
ejpam-3283	363	27	ideals	ideal	NOUN
ejpam-3283	363	28	,	,	PUNCT
ejpam-3283	363	29	maximal	maximal	ADJ
ejpam-3283	363	30	right	right	ADJ
ejpam-3283	363	31	ideals	ideal	NOUN
ejpam-3283	363	32	,	,	PUNCT
ejpam-3283	363	33	respectively	respectively	ADV
ejpam-3283	363	34	)	)	PUNCT
ejpam-3283	363	35	of	of	ADP
ejpam-3283	363	36	semigroups	semigroup	NOUN
ejpam-3283	363	37	/	/	SYM
ejpam-3283	363	38	ternary	ternary	ADJ
ejpam-3283	363	39	semigroups	semigroup	NOUN
ejpam-3283	363	40	are	be	AUX
ejpam-3283	363	41	just	just	ADV
ejpam-3283	363	42	the	the	DET
ejpam-3283	363	43	1	1	NUM
ejpam-3283	363	44	-	-	PUNCT
ejpam-3283	363	45	ideals	ideal	NOUN
ejpam-3283	363	46	(	(	PUNCT
ejpam-3283	363	47	1	1	NUM
ejpam-3283	363	48	-	-	NOUN
ejpam-3283	363	49	simple	simple	ADJ
ejpam-3283	363	50	,	,	PUNCT
ejpam-3283	363	51	0	0	NUM
ejpam-3283	363	52	-	-	SYM
ejpam-3283	363	53	1simple	1simple	NUM
ejpam-3283	363	54	,	,	PUNCT
ejpam-3283	363	55	minimal	minimal	ADJ
ejpam-3283	363	56	1	1	NUM
ejpam-3283	363	57	-	-	PUNCT
ejpam-3283	363	58	ideals	ideal	NOUN
ejpam-3283	363	59	,	,	PUNCT
ejpam-3283	363	60	0	0	NUM
ejpam-3283	363	61	-	-	PUNCT
ejpam-3283	363	62	minimal	minimal	ADJ
ejpam-3283	363	63	1	1	NUM
ejpam-3283	363	64	-	-	PUNCT
ejpam-3283	363	65	ideals	ideal	NOUN
ejpam-3283	363	66	,	,	PUNCT
ejpam-3283	363	67	and	and	CCONJ
ejpam-3283	363	68	maximal	maximal	ADJ
ejpam-3283	363	69	1	1	NUM
ejpam-3283	363	70	-	-	PUNCT
ejpam-3283	363	71	ideals	ideal	NOUN
ejpam-3283	363	72	,	,	PUNCT
ejpam-3283	363	73	respectively	respectively	ADV
ejpam-3283	363	74	)	)	PUNCT
ejpam-3283	363	75	of	of	ADP
ejpam-3283	363	76	2	2	NUM
ejpam-3283	363	77	-	-	PUNCT
ejpam-3283	363	78	ary	ary	NOUN
ejpam-3283	363	79	semigroups/3	semigroups/3	PROPN
ejpam-3283	363	80	-	-	PUNCT
ejpam-3283	363	81	ary	ary	PROPN
ejpam-3283	363	82	semigroups	semigroup	NOUN
ejpam-3283	363	83	.	.	PUNCT
ejpam-3283	364	1	for	for	ADP
ejpam-3283	364	2	any	any	DET
ejpam-3283	364	3	results	result	NOUN
ejpam-3283	364	4	of	of	ADP
ejpam-3283	364	5	this	this	DET
ejpam-3283	364	6	research	research	NOUN
ejpam-3283	364	7	,	,	PUNCT
ejpam-3283	364	8	we	we	PRON
ejpam-3283	364	9	can	can	AUX
ejpam-3283	364	10	place	place	VERB
ejpam-3283	364	11	1	1	NUM
ejpam-3283	364	12	-	-	PUNCT
ejpam-3283	364	13	ideal	ideal	NOUN
ejpam-3283	364	14	,	,	PUNCT
ejpam-3283	364	15	1simple	1simple	NUM
ejpam-3283	364	16	,	,	PUNCT
ejpam-3283	364	17	0	0	NUM
ejpam-3283	364	18	-	-	SYM
ejpam-3283	364	19	1	1	NUM
ejpam-3283	364	20	-	-	PUNCT
ejpam-3283	364	21	simple	simple	ADJ
ejpam-3283	364	22	,	,	PUNCT
ejpam-3283	364	23	minimal	minimal	ADJ
ejpam-3283	364	24	1	1	NUM
ejpam-3283	364	25	-	-	PUNCT
ejpam-3283	364	26	ideal	ideal	ADJ
ejpam-3283	364	27	,	,	PUNCT
ejpam-3283	364	28	0	0	NUM
ejpam-3283	364	29	-	-	PUNCT
ejpam-3283	364	30	minimal	minimal	ADJ
ejpam-3283	364	31	1	1	NUM
ejpam-3283	364	32	-	-	PUNCT
ejpam-3283	364	33	ideal	ideal	NOUN
ejpam-3283	364	34	,	,	PUNCT
ejpam-3283	364	35	and	and	CCONJ
ejpam-3283	364	36	maximal	maximal	ADJ
ejpam-3283	364	37	1	1	NUM
ejpam-3283	364	38	-	-	PUNCT
ejpam-3283	364	39	ideal	ideal	ADJ
ejpam-3283	364	40	,	,	PUNCT
ejpam-3283	364	41	respectively	respectively	ADV
ejpam-3283	364	42	,	,	PUNCT
ejpam-3283	364	43	instead	instead	ADV
ejpam-3283	364	44	of	of	ADP
ejpam-3283	364	45	n	n	CCONJ
ejpam-3283	364	46	-	-	PUNCT
ejpam-3283	364	47	ideal	ideal	ADJ
ejpam-3283	364	48	,	,	PUNCT
ejpam-3283	364	49	n	n	CCONJ
ejpam-3283	364	50	-	-	PUNCT
ejpam-3283	364	51	simple	simple	ADJ
ejpam-3283	364	52	,	,	PUNCT
ejpam-3283	364	53	0	0	NUM
ejpam-3283	364	54	-	-	PUNCT
ejpam-3283	364	55	n	n	CCONJ
ejpam-3283	364	56	-	-	PUNCT
ejpam-3283	364	57	simple	simple	ADJ
ejpam-3283	364	58	,	,	PUNCT
ejpam-3283	364	59	minimal	minimal	ADJ
ejpam-3283	364	60	n	n	CCONJ
ejpam-3283	364	61	-	-	PUNCT
ejpam-3283	364	62	ideal	ideal	ADJ
ejpam-3283	364	63	,	,	PUNCT
ejpam-3283	364	64	0	0	NUM
ejpam-3283	364	65	-	-	PUNCT
ejpam-3283	364	66	minimal	minimal	ADJ
ejpam-3283	364	67	n	n	CCONJ
ejpam-3283	364	68	-	-	PUNCT
ejpam-3283	364	69	ideal	ideal	NOUN
ejpam-3283	364	70	,	,	PUNCT
ejpam-3283	364	71	and	and	CCONJ
ejpam-3283	364	72	maximal	maximal	ADJ
ejpam-3283	364	73	n	n	CCONJ
ejpam-3283	364	74	-	-	PUNCT
ejpam-3283	364	75	ideal	ideal	ADJ
ejpam-3283	364	76	,	,	PUNCT
ejpam-3283	364	77	respectively	respectively	ADV
ejpam-3283	364	78	,	,	PUNCT
ejpam-3283	364	79	and	and	CCONJ
ejpam-3283	364	80	then	then	ADV
ejpam-3283	364	81	we	we	PRON
ejpam-3283	364	82	will	will	AUX
ejpam-3283	364	83	obtain	obtain	VERB
ejpam-3283	364	84	the	the	DET
ejpam-3283	364	85	similar	similar	ADJ
ejpam-3283	364	86	results	result	NOUN
ejpam-3283	364	87	.	.	PUNCT
ejpam-3283	365	1	finally	finally	ADV
ejpam-3283	365	2	,	,	PUNCT
ejpam-3283	365	3	we	we	PRON
ejpam-3283	365	4	present	present	VERB
ejpam-3283	365	5	some	some	DET
ejpam-3283	365	6	ideas	idea	NOUN
ejpam-3283	365	7	for	for	ADP
ejpam-3283	365	8	extending	extend	VERB
ejpam-3283	365	9	our	our	PRON
ejpam-3283	365	10	results	result	NOUN
ejpam-3283	365	11	.	.	PUNCT
ejpam-3283	366	1	we	we	PRON
ejpam-3283	366	2	know	know	VERB
ejpam-3283	366	3	that	that	SCONJ
ejpam-3283	366	4	an	an	DET
ejpam-3283	366	5	n	n	CCONJ
ejpam-3283	366	6	-	-	PUNCT
ejpam-3283	366	7	ary	ary	PROPN
ejpam-3283	366	8	semigroup	semigroup	PROPN
ejpam-3283	366	9	is	be	AUX
ejpam-3283	366	10	just	just	ADV
ejpam-3283	366	11	a	a	DET
ejpam-3283	366	12	ternary	ternary	ADJ
ejpam-3283	366	13	semigroup	semigroup	NOUN
ejpam-3283	366	14	if	if	SCONJ
ejpam-3283	366	15	n	n	NOUN
ejpam-3283	366	16	=	=	SYM
ejpam-3283	366	17	3	3	X
ejpam-3283	366	18	.	.	X
ejpam-3283	367	1	there	there	PRON
ejpam-3283	367	2	are	be	VERB
ejpam-3283	367	3	many	many	ADJ
ejpam-3283	367	4	researchs	research	NOUN
ejpam-3283	367	5	about	about	ADP
ejpam-3283	367	6	the	the	DET
ejpam-3283	367	7	lateral	lateral	ADJ
ejpam-3283	367	8	ideals	ideal	NOUN
ejpam-3283	367	9	of	of	ADP
ejpam-3283	367	10	ternary	ternary	ADJ
ejpam-3283	367	11	semigroups	semigroup	NOUN
ejpam-3283	367	12	which	which	PRON
ejpam-3283	367	13	are	be	AUX
ejpam-3283	367	14	just	just	ADV
ejpam-3283	367	15	2	2	NUM
ejpam-3283	367	16	-	-	PUNCT
ejpam-3283	367	17	ideals	ideal	NOUN
ejpam-3283	367	18	of	of	ADP
ejpam-3283	367	19	3	3	NUM
ejpam-3283	367	20	-	-	PUNCT
ejpam-3283	367	21	ary	ary	NOUN
ejpam-3283	367	22	semigroups	semigroup	NOUN
ejpam-3283	367	23	.	.	PUNCT
ejpam-3283	368	1	for	for	ADP
ejpam-3283	368	2	the	the	DET
ejpam-3283	368	3	more	more	ADV
ejpam-3283	368	4	generally	generally	ADJ
ejpam-3283	368	5	cases	case	NOUN
ejpam-3283	368	6	of	of	ADP
ejpam-3283	368	7	our	our	PRON
ejpam-3283	368	8	results	result	NOUN
ejpam-3283	368	9	that	that	SCONJ
ejpam-3283	368	10	we	we	PRON
ejpam-3283	368	11	let	let	VERB
ejpam-3283	368	12	them	they	PRON
ejpam-3283	368	13	be	be	AUX
ejpam-3283	368	14	the	the	DET
ejpam-3283	368	15	open	open	ADJ
ejpam-3283	368	16	problems	problem	NOUN
ejpam-3283	368	17	are	be	AUX
ejpam-3283	368	18	the	the	DET
ejpam-3283	368	19	minimality	minimality	NOUN
ejpam-3283	368	20	and	and	CCONJ
ejpam-3283	368	21	maximality	maximality	PROPN
ejpam-3283	368	22	of	of	ADP
ejpam-3283	368	23	i	i	PROPN
ejpam-3283	368	24	-	-	PUNCT
ejpam-3283	368	25	ideals	ideal	NOUN
ejpam-3283	368	26	where	where	SCONJ
ejpam-3283	368	27	1	1	NUM
ejpam-3283	368	28	<	<	X
ejpam-3283	368	29	i	i	X
ejpam-3283	368	30	<	<	X
ejpam-3283	368	31	n	n	X
ejpam-3283	368	32	in	in	ADP
ejpam-3283	368	33	n	n	CCONJ
ejpam-3283	368	34	-	-	PUNCT
ejpam-3283	368	35	ary	ary	NOUN
ejpam-3283	368	36	semigroups	semigroup	NOUN
ejpam-3283	368	37	.	.	PUNCT
ejpam-3283	369	1	acknowledgements	acknowledgement	NOUN
ejpam-3283	369	2	the	the	DET
ejpam-3283	369	3	second	second	ADJ
ejpam-3283	369	4	author	author	NOUN
ejpam-3283	369	5	was	be	AUX
ejpam-3283	369	6	supported	support	VERB
ejpam-3283	369	7	by	by	ADP
ejpam-3283	369	8	algebra	algebra	NOUN
ejpam-3283	369	9	and	and	CCONJ
ejpam-3283	369	10	applications	application	NOUN
ejpam-3283	369	11	research	research	NOUN
ejpam-3283	369	12	unit	unit	NOUN
ejpam-3283	369	13	,	,	PUNCT
ejpam-3283	369	14	prince	prince	NOUN
ejpam-3283	369	15	of	of	ADP
ejpam-3283	369	16	songkla	songkla	PROPN
ejpam-3283	369	17	university	university	PROPN
ejpam-3283	369	18	.	.	PUNCT
ejpam-3283	370	1	references	reference	NOUN
ejpam-3283	370	2	773	773	NUM
ejpam-3283	370	3	references	reference	NOUN
ejpam-3283	370	4	[	[	X
ejpam-3283	370	5	1	1	NUM
ejpam-3283	370	6	]	]	PUNCT
ejpam-3283	370	7	m	m	VERB
ejpam-3283	370	8	arslanov	arslanov	NOUN
ejpam-3283	370	9	and	and	CCONJ
ejpam-3283	370	10	n	n	PRON
ejpam-3283	370	11	kehayopulu	kehayopulu	ADJ
ejpam-3283	370	12	.	.	PUNCT
ejpam-3283	371	1	a	a	DET
ejpam-3283	371	2	note	note	NOUN
ejpam-3283	371	3	on	on	ADP
ejpam-3283	371	4	minimal	minimal	ADJ
ejpam-3283	371	5	and	and	CCONJ
ejpam-3283	371	6	maximal	maximal	ADJ
ejpam-3283	371	7	ideals	ideal	NOUN
ejpam-3283	371	8	of	of	ADP
ejpam-3283	371	9	ordered	order	VERB
ejpam-3283	371	10	semigroups	semigroup	NOUN
ejpam-3283	371	11	.	.	PUNCT
ejpam-3283	372	1	lobachevskii	lobachevskii	PROPN
ejpam-3283	372	2	journal	journal	PROPN
ejpam-3283	372	3	of	of	ADP
ejpam-3283	372	4	mathematics	mathematic	NOUN
ejpam-3283	372	5	,	,	PUNCT
ejpam-3283	372	6	11:3–6	11:3–6	NUM
ejpam-3283	372	7	,	,	PUNCT
ejpam-3283	372	8	2002	2002	NUM
ejpam-3283	372	9	.	.	PUNCT
ejpam-3283	373	1	[	[	X
ejpam-3283	373	2	2	2	NUM
ejpam-3283	373	3	]	]	X
ejpam-3283	373	4	y	y	PROPN
ejpam-3283	373	5	cao	cao	PROPN
ejpam-3283	373	6	and	and	CCONJ
ejpam-3283	373	7	x	x	PROPN
ejpam-3283	373	8	xu	xu	PROPN
ejpam-3283	373	9	.	.	PUNCT
ejpam-3283	374	1	on	on	ADP
ejpam-3283	374	2	minimal	minimal	ADJ
ejpam-3283	374	3	and	and	CCONJ
ejpam-3283	374	4	maximal	maximal	ADJ
ejpam-3283	374	5	left	leave	VERB
ejpam-3283	374	6	ideal	ideal	NOUN
ejpam-3283	374	7	in	in	ADP
ejpam-3283	374	8	ordered	order	VERB
ejpam-3283	374	9	semigroups	semigroup	NOUN
ejpam-3283	374	10	.	.	PUNCT
ejpam-3283	375	1	semigroup	semigroup	PROPN
ejpam-3283	375	2	forum	forum	PROPN
ejpam-3283	375	3	,	,	PUNCT
ejpam-3283	375	4	60:202–207	60:202–207	PROPN
ejpam-3283	375	5	,	,	PUNCT
ejpam-3283	375	6	2000	2000	NUM
ejpam-3283	375	7	.	.	PUNCT
ejpam-3283	376	1	[	[	X
ejpam-3283	376	2	3	3	X
ejpam-3283	376	3	]	]	X
ejpam-3283	376	4	w	w	ADP
ejpam-3283	376	5	a	a	DET
ejpam-3283	376	6	dudek	dudek	PROPN
ejpam-3283	376	7	and	and	CCONJ
ejpam-3283	376	8	i	i	PROPN
ejpam-3283	376	9	grozdinska	grozdinska	NOUN
ejpam-3283	376	10	.	.	PUNCT
ejpam-3283	377	1	on	on	ADP
ejpam-3283	377	2	ideals	ideal	NOUN
ejpam-3283	377	3	in	in	ADP
ejpam-3283	377	4	regular	regular	ADJ
ejpam-3283	377	5	n	n	CCONJ
ejpam-3283	377	6	-	-	PUNCT
ejpam-3283	377	7	ary	ary	PROPN
ejpam-3283	377	8	semigroups	semigroup	NOUN
ejpam-3283	377	9	.	.	PUNCT
ejpam-3283	378	1	matematichki	matematichki	PROPN
ejpam-3283	378	2	bilten	bilten	VERB
ejpam-3283	378	3	,	,	PUNCT
ejpam-3283	378	4	4:25–44	4:25–44	NUM
ejpam-3283	378	5	,	,	PUNCT
ejpam-3283	378	6	1980	1980	NUM
ejpam-3283	378	7	.	.	PUNCT
ejpam-3283	379	1	[	[	X
ejpam-3283	379	2	4	4	X
ejpam-3283	379	3	]	]	X
ejpam-3283	379	4	w	w	ADP
ejpam-3283	379	5	a	a	DET
ejpam-3283	379	6	dudek	dudek	PROPN
ejpam-3283	379	7	.	.	PUNCT
ejpam-3283	380	1	remarks	remark	NOUN
ejpam-3283	380	2	on	on	ADP
ejpam-3283	380	3	n	n	NOUN
ejpam-3283	380	4	-	-	PUNCT
ejpam-3283	380	5	groups	group	NOUN
ejpam-3283	380	6	.	.	PUNCT
ejpam-3283	381	1	demonstratio	demonstratio	PROPN
ejpam-3283	381	2	mathematica	mathematica	PROPN
ejpam-3283	381	3	,	,	PUNCT
ejpam-3283	381	4	13	13	NUM
ejpam-3283	381	5	:	:	PUNCT
ejpam-3283	381	6	165–181	165–181	NUM
ejpam-3283	381	7	,	,	PUNCT
ejpam-3283	381	8	1980	1980	NUM
ejpam-3283	381	9	.	.	PUNCT
ejpam-3283	382	1	[	[	X
ejpam-3283	382	2	5	5	NUM
ejpam-3283	382	3	]	]	PUNCT
ejpam-3283	382	4	w	w	ADP
ejpam-3283	382	5	a	a	DET
ejpam-3283	382	6	dudek	dudek	PROPN
ejpam-3283	382	7	.	.	PUNCT
ejpam-3283	383	1	autodistributive	autodistributive	ADJ
ejpam-3283	383	2	n	n	CCONJ
ejpam-3283	383	3	-	-	PUNCT
ejpam-3283	383	4	groups	group	NOUN
ejpam-3283	383	5	.	.	PUNCT
ejpam-3283	384	1	annales	annales	PROPN
ejpam-3283	384	2	societatis	societatis	PROPN
ejpam-3283	384	3	mathematicae	mathematicae	PROPN
ejpam-3283	384	4	polonae	polonae	PROPN
ejpam-3283	384	5	,	,	PUNCT
ejpam-3283	384	6	commentationes	commentatione	NOUN
ejpam-3283	384	7	mathematicae	mathematicae	PROPN
ejpam-3283	384	8	,	,	PUNCT
ejpam-3283	384	9	23:1–11	23:1–11	NUM
ejpam-3283	384	10	,	,	PUNCT
ejpam-3283	384	11	1983	1983	NUM
ejpam-3283	384	12	.	.	PUNCT
ejpam-3283	385	1	[	[	X
ejpam-3283	385	2	6	6	NUM
ejpam-3283	385	3	]	]	PUNCT
ejpam-3283	385	4	w	w	ADP
ejpam-3283	385	5	a	a	DET
ejpam-3283	385	6	dudek	dudek	PROPN
ejpam-3283	385	7	.	.	PUNCT
ejpam-3283	386	1	on	on	ADP
ejpam-3283	386	2	(	(	PUNCT
ejpam-3283	386	3	i	i	PROPN
ejpam-3283	386	4	,	,	PUNCT
ejpam-3283	386	5	j)-associative	j)-associative	ADJ
ejpam-3283	386	6	n	n	CCONJ
ejpam-3283	386	7	-	-	PUNCT
ejpam-3283	386	8	groupoids	groupoid	NOUN
ejpam-3283	386	9	with	with	ADP
ejpam-3283	386	10	the	the	DET
ejpam-3283	386	11	non	non	ADJ
ejpam-3283	386	12	-	-	ADJ
ejpam-3283	386	13	empty	empty	ADJ
ejpam-3283	386	14	center	center	NOUN
ejpam-3283	386	15	.	.	PUNCT
ejpam-3283	387	1	ricerche	ricerche	PROPN
ejpam-3283	387	2	di	di	PROPN
ejpam-3283	387	3	matematica	matematica	PROPN
ejpam-3283	387	4	,	,	PUNCT
ejpam-3283	387	5	35:105–111	35:105–111	PROPN
ejpam-3283	387	6	,	,	PUNCT
ejpam-3283	387	7	1986	1986	NUM
ejpam-3283	387	8	.	.	PUNCT
ejpam-3283	388	1	[	[	X
ejpam-3283	388	2	7	7	X
ejpam-3283	388	3	]	]	X
ejpam-3283	388	4	w	w	ADP
ejpam-3283	388	5	a	a	DET
ejpam-3283	388	6	dudek	dudek	PROPN
ejpam-3283	388	7	.	.	PUNCT
ejpam-3283	388	8	idempotents	idempotent	NOUN
ejpam-3283	388	9	in	in	ADP
ejpam-3283	388	10	n	n	CCONJ
ejpam-3283	388	11	-	-	PUNCT
ejpam-3283	388	12	ary	ary	NOUN
ejpam-3283	388	13	semigroups	semigroup	NOUN
ejpam-3283	388	14	.	.	PUNCT
ejpam-3283	389	1	southeast	southeast	ADJ
ejpam-3283	389	2	asian	asian	ADJ
ejpam-3283	389	3	bulletin	bulletin	NOUN
ejpam-3283	389	4	of	of	ADP
ejpam-3283	389	5	mathematics	mathematic	NOUN
ejpam-3283	389	6	,	,	PUNCT
ejpam-3283	389	7	25:97–104	25:97–104	NUM
ejpam-3283	389	8	,	,	PUNCT
ejpam-3283	389	9	2011	2011	NUM
ejpam-3283	389	10	.	.	PUNCT
ejpam-3283	390	1	[	[	X
ejpam-3283	390	2	8	8	NUM
ejpam-3283	390	3	]	]	X
ejpam-3283	390	4	j	j	PROPN
ejpam-3283	390	5	w	w	PROPN
ejpam-3283	390	6	grzymala	grzymala	NOUN
ejpam-3283	390	7	-	-	PUNCT
ejpam-3283	390	8	busse	busse	NOUN
ejpam-3283	390	9	.	.	PUNCT
ejpam-3283	391	1	automorphisms	automorphisms	PROPN
ejpam-3283	391	2	of	of	ADP
ejpam-3283	391	3	polyadic	polyadic	ADJ
ejpam-3283	391	4	automata	automata	PROPN
ejpam-3283	391	5	.	.	PUNCT
ejpam-3283	392	1	journal	journal	NOUN
ejpam-3283	392	2	of	of	ADP
ejpam-3283	392	3	the	the	DET
ejpam-3283	392	4	association	association	NOUN
ejpam-3283	392	5	for	for	ADP
ejpam-3283	392	6	computing	computing	NOUN
ejpam-3283	392	7	machinery	machinery	NOUN
ejpam-3283	392	8	,	,	PUNCT
ejpam-3283	392	9	16:208–219	16:208–219	PROPN
ejpam-3283	392	10	,	,	PUNCT
ejpam-3283	392	11	1969	1969	NUM
ejpam-3283	392	12	.	.	PUNCT
ejpam-3283	393	1	[	[	X
ejpam-3283	393	2	9	9	NUM
ejpam-3283	393	3	]	]	X
ejpam-3283	393	4	a	a	DET
ejpam-3283	393	5	iampan	iampan	NOUN
ejpam-3283	393	6	.	.	PUNCT
ejpam-3283	394	1	the	the	DET
ejpam-3283	394	2	minimality	minimality	NOUN
ejpam-3283	394	3	and	and	CCONJ
ejpam-3283	394	4	maximality	maximality	NOUN
ejpam-3283	394	5	of	of	ADP
ejpam-3283	394	6	left	left	NOUN
ejpam-3283	394	7	(	(	PUNCT
ejpam-3283	394	8	right	right	ADJ
ejpam-3283	394	9	)	)	PUNCT
ejpam-3283	394	10	ideals	ideal	NOUN
ejpam-3283	394	11	in	in	ADP
ejpam-3283	394	12	ternary	ternary	ADJ
ejpam-3283	394	13	semigroups	semigroup	NOUN
ejpam-3283	394	14	.	.	PUNCT
ejpam-3283	395	1	international	international	ADJ
ejpam-3283	395	2	journal	journal	PROPN
ejpam-3283	395	3	of	of	ADP
ejpam-3283	395	4	contemporary	contemporary	PROPN
ejpam-3283	395	5	mathematical	mathematical	PROPN
ejpam-3283	395	6	sciences	sciences	PROPN
ejpam-3283	395	7	,	,	PUNCT
ejpam-3283	395	8	5:2409–2417	5:2409–2417	NUM
ejpam-3283	395	9	,	,	PUNCT
ejpam-3283	395	10	2010	2010	NUM
ejpam-3283	395	11	.	.	PUNCT
ejpam-3283	396	1	[	[	X
ejpam-3283	396	2	10	10	NUM
ejpam-3283	396	3	]	]	X
ejpam-3283	396	4	r	r	NOUN
ejpam-3283	396	5	kasner	kasner	NOUN
ejpam-3283	396	6	.	.	PUNCT
ejpam-3283	397	1	an	an	DET
ejpam-3283	397	2	extension	extension	NOUN
ejpam-3283	397	3	of	of	ADP
ejpam-3283	397	4	the	the	DET
ejpam-3283	397	5	group	group	NOUN
ejpam-3283	397	6	concepts	concept	NOUN
ejpam-3283	397	7	.	.	PUNCT
ejpam-3283	398	1	bulletin	bulletin	NOUN
ejpam-3283	398	2	of	of	ADP
ejpam-3283	398	3	the	the	DET
ejpam-3283	398	4	american	american	PROPN
ejpam-3283	398	5	mathematical	mathematical	PROPN
ejpam-3283	398	6	society	society	NOUN
ejpam-3283	398	7	,	,	PUNCT
ejpam-3283	398	8	10:290–291	10:290–291	NUM
ejpam-3283	398	9	,	,	PUNCT
ejpam-3283	398	10	1904	1904	NUM
ejpam-3283	398	11	.	.	PUNCT
ejpam-3283	399	1	[	[	X
ejpam-3283	399	2	11	11	NUM
ejpam-3283	399	3	]	]	X
ejpam-3283	399	4	y	y	PROPN
ejpam-3283	399	5	nambu	nambu	PROPN
ejpam-3283	399	6	.	.	PUNCT
ejpam-3283	400	1	generalized	generalized	ADJ
ejpam-3283	400	2	hamiltonian	hamiltonian	ADJ
ejpam-3283	400	3	mechanics	mechanic	NOUN
ejpam-3283	400	4	.	.	PUNCT
ejpam-3283	401	1	physical	physical	ADJ
ejpam-3283	401	2	review	review	NOUN
ejpam-3283	401	3	journals	journal	NOUN
ejpam-3283	401	4	,	,	PUNCT
ejpam-3283	401	5	d7:2405	d7:2405	PROPN
ejpam-3283	401	6	–	–	PUNCT
ejpam-3283	401	7	2412	2412	NUM
ejpam-3283	401	8	,	,	PUNCT
ejpam-3283	401	9	1973	1973	NUM
ejpam-3283	401	10	.	.	PUNCT
ejpam-3283	402	1	[	[	X
ejpam-3283	402	2	12	12	NUM
ejpam-3283	402	3	]	]	X
ejpam-3283	402	4	f	f	PROPN
ejpam-3283	402	5	m	m	VERB
ejpam-3283	402	6	sioson	sioson	PROPN
ejpam-3283	402	7	.	.	PUNCT
ejpam-3283	403	1	on	on	ADP
ejpam-3283	403	2	regular	regular	ADJ
ejpam-3283	403	3	algebraic	algebraic	ADJ
ejpam-3283	403	4	systems	system	NOUN
ejpam-3283	403	5	.	.	PUNCT
ejpam-3283	404	1	proceedings	proceeding	NOUN
ejpam-3283	404	2	of	of	ADP
ejpam-3283	404	3	the	the	DET
ejpam-3283	404	4	japan	japan	PROPN
ejpam-3283	404	5	academy	academy	PROPN
ejpam-3283	404	6	,	,	PUNCT
ejpam-3283	404	7	39:283	39:283	NUM
ejpam-3283	404	8	–	–	PUNCT
ejpam-3283	404	9	286	286	NUM
ejpam-3283	404	10	,	,	PUNCT
ejpam-3283	404	11	1963	1963	NUM
ejpam-3283	404	12	.	.	PUNCT
ejpam-3283	405	1	[	[	X
ejpam-3283	405	2	13	13	NUM
ejpam-3283	405	3	]	]	SYM
ejpam-3283	405	4	j	j	PROPN
ejpam-3283	405	5	p	p	PROPN
ejpam-3283	405	6	solano	solano	PROPN
ejpam-3283	405	7	,	,	PUNCT
ejpam-3283	405	8	s	s	PART
ejpam-3283	405	9	suebsung	suebsung	NOUN
ejpam-3283	405	10	and	and	CCONJ
ejpam-3283	405	11	r	r	NOUN
ejpam-3283	405	12	chinram	chinram	NOUN
ejpam-3283	405	13	.	.	PUNCT
ejpam-3283	406	1	on	on	ADP
ejpam-3283	406	2	ideals	ideal	NOUN
ejpam-3283	406	3	of	of	ADP
ejpam-3283	406	4	fuzzy	fuzzy	ADJ
ejpam-3283	406	5	points	point	NOUN
ejpam-3283	406	6	n	n	CCONJ
ejpam-3283	406	7	-	-	PUNCT
ejpam-3283	406	8	ary	ary	NOUN
ejpam-3283	406	9	semigroups	semigroup	NOUN
ejpam-3283	406	10	.	.	PUNCT
ejpam-3283	407	1	international	international	ADJ
ejpam-3283	407	2	journal	journal	NOUN
ejpam-3283	407	3	of	of	ADP
ejpam-3283	407	4	mathematics	mathematic	NOUN
ejpam-3283	407	5	and	and	CCONJ
ejpam-3283	407	6	computer	computer	NOUN
ejpam-3283	407	7	science	science	NOUN
ejpam-3283	407	8	,	,	PUNCT
ejpam-3283	407	9	13:179	13:179	NUM
ejpam-3283	407	10	-	-	SYM
ejpam-3283	407	11	186	186	NUM
ejpam-3283	407	12	,	,	PUNCT
ejpam-3283	407	13	2018	2018	NUM
ejpam-3283	407	14	.	.	PUNCT
ejpam-3283	408	1	[	[	X
ejpam-3283	408	2	14	14	NUM
ejpam-3283	408	3	]	]	PUNCT
ejpam-3283	408	4	l	l	NOUN
ejpam-3283	408	5	vainerman	vainerman	NOUN
ejpam-3283	408	6	and	and	CCONJ
ejpam-3283	408	7	r	r	PROPN
ejpam-3283	408	8	kerner	kerner	PROPN
ejpam-3283	408	9	.	.	PUNCT
ejpam-3283	409	1	on	on	ADP
ejpam-3283	409	2	special	special	ADJ
ejpam-3283	409	3	classes	class	NOUN
ejpam-3283	409	4	of	of	ADP
ejpam-3283	409	5	n	n	CCONJ
ejpam-3283	409	6	-	-	PUNCT
ejpam-3283	409	7	algebras	algebras	PROPN
ejpam-3283	409	8	.	.	PUNCT
ejpam-3283	410	1	journal	journal	PROPN
ejpam-3283	410	2	of	of	ADP
ejpam-3283	410	3	mathematical	mathematical	ADJ
ejpam-3283	410	4	physics	physics	NOUN
ejpam-3283	410	5	,	,	PUNCT
ejpam-3283	410	6	37:2553–2565	37:2553–2565	NUM
ejpam-3283	410	7	,	,	PUNCT
ejpam-3283	410	8	1996	1996	NUM
ejpam-3283	410	9	.	.	PUNCT
ejpam-3283	411	1	[	[	X
ejpam-3283	411	2	15	15	NUM
ejpam-3283	411	3	]	]	X
ejpam-3283	411	4	q	q	X
ejpam-3283	411	5	wang	wang	PROPN
ejpam-3283	411	6	,	,	PUNCT
ejpam-3283	411	7	x	x	PROPN
ejpam-3283	411	8	zhou	zhou	PROPN
ejpam-3283	411	9	and	and	CCONJ
ejpam-3283	411	10	j	j	PROPN
ejpam-3283	411	11	zhan	zhan	PROPN
ejpam-3283	411	12	.	.	PUNCT
ejpam-3283	412	1	a	a	DET
ejpam-3283	412	2	novel	novel	ADJ
ejpam-3283	412	3	study	study	NOUN
ejpam-3283	412	4	of	of	ADP
ejpam-3283	412	5	soft	soft	ADJ
ejpam-3283	412	6	sets	set	NOUN
ejpam-3283	412	7	over	over	ADP
ejpam-3283	412	8	n	n	CCONJ
ejpam-3283	412	9	-	-	PUNCT
ejpam-3283	412	10	ary	ary	NOUN
ejpam-3283	412	11	semigroups	semigroup	NOUN
ejpam-3283	412	12	.	.	PUNCT
ejpam-3283	413	1	italian	italian	ADJ
ejpam-3283	413	2	journal	journal	NOUN
ejpam-3283	413	3	of	of	ADP
ejpam-3283	413	4	pure	pure	ADJ
ejpam-3283	413	5	and	and	CCONJ
ejpam-3283	413	6	applied	applied	ADJ
ejpam-3283	413	7	mathematics	mathematic	NOUN
ejpam-3283	413	8	,	,	PUNCT
ejpam-3283	413	9	37:583–594	37:583–594	PROPN
ejpam-3283	413	10	,	,	PUNCT
ejpam-3283	413	11	2017	2017	NUM
ejpam-3283	413	12	.	.	PUNCT