id	sid	tid	token	lemma	pos
iajs-1139	1	1	ibn	ibn	PROPN
iajs-1139	1	2	alhaitham	alhaitham	NOUN
iajs-1139	1	3	j.	j.	PROPN
iajs-1139	1	4	for	for	ADP
iajs-1139	1	5	pure	pure	ADJ
iajs-1139	1	6	&	&	CCONJ
iajs-1139	1	7	appl	appl	PROPN
iajs-1139	1	8	.	.	PUNCT
iajs-1139	2	1	sci	sci	PROPN
iajs-1139	2	2	.	.	PUNCT
iajs-1139	3	1	vol.22	vol.22	PROPN
iajs-1139	3	2	(	(	PUNCT
iajs-1139	3	3	4	4	NUM
iajs-1139	3	4	)	)	PUNCT
iajs-1139	3	5	2009	2009	NUM
iajs-1139	3	6	semiessential	semiessential	ADJ
iajs-1139	3	7	fuzzy	fuzzy	ADJ
iajs-1139	3	8	ideals	ideal	NOUN
iajs-1139	3	9	and	and	CCONJ
iajs-1139	3	10	semiuniform	semiuniform	VERB
iajs-1139	3	11	fuzzy	fuzzy	ADJ
iajs-1139	3	12	rings	ring	NOUN
iajs-1139	3	13	m.	m.	NOUN
iajs-1139	3	14	a.hamil	a.hamil	PROPN
iajs-1139	3	15	department	department	PROPN
iajs-1139	3	16	of	of	ADP
iajs-1139	3	17	mathematics	mathematics	PROPN
iajs-1139	3	18	,	,	PUNCT
iajs-1139	3	19	college	college	NOUN
iajs-1139	3	20	of	of	ADP
iajs-1139	3	21	education	education	PROPN
iajs-1139	3	22	ibn	ibn	PROPN
iajs-1139	3	23	-	-	PUNCT
iajs-1139	3	24	al	al	PROPN
iajs-1139	3	25	-	-	PUNCT
iajs-1139	3	26	haitham	haitham	PROPN
iajs-1139	3	27	,	,	PUNCT
iajs-1139	3	28	university	university	PROPN
iajs-1139	3	29	of	of	ADP
iajs-1139	3	30	baghdad	baghdad	PROPN
iajs-1139	3	31	abstract	abstract	ADV
iajs-1139	3	32	in	in	ADP
iajs-1139	3	33	this	this	DET
iajs-1139	3	34	paper	paper	NOUN
iajs-1139	3	35	,	,	PUNCT
iajs-1139	3	36	we	we	PRON
iajs-1139	3	37	introduce	introduce	VERB
iajs-1139	3	38	and	and	CCONJ
iajs-1139	3	39	study	study	VERB
iajs-1139	3	40	semiessential	semiessential	ADJ
iajs-1139	3	41	fuzzy	fuzzy	ADJ
iajs-1139	3	42	ideals	ideal	NOUN
iajs-1139	3	43	of	of	ADP
iajs-1139	3	44	fuzzy	fuzzy	ADJ
iajs-1139	3	45	rings	ring	NOUN
iajs-1139	3	46	,	,	PUNCT
iajs-1139	3	47	uniform	uniform	ADJ
iajs-1139	3	48	fuzzy	fuzzy	ADJ
iajs-1139	3	49	rings	ring	NOUN
iajs-1139	3	50	and	and	CCONJ
iajs-1139	3	51	semiuniform	semiuniform	VERB
iajs-1139	3	52	fuzzy	fuzzy	ADJ
iajs-1139	3	53	rings	ring	NOUN
iajs-1139	3	54	.	.	PUNCT
iajs-1139	4	1	introduction	introduction	NOUN
iajs-1139	4	2	zadah	zadah	PROPN
iajs-1139	4	3	in	in	ADP
iajs-1139	4	4	[	[	X
iajs-1139	4	5	1	1	NUM
iajs-1139	4	6	]	]	PUNCT
iajs-1139	4	7	introduced	introduce	VERB
iajs-1139	4	8	the	the	DET
iajs-1139	4	9	notion	notion	NOUN
iajs-1139	4	10	of	of	ADP
iajs-1139	4	11	a	a	DET
iajs-1139	4	12	fuzzy	fuzzy	NOUN
iajs-1139	4	13	subset	subset	VERB
iajs-1139	4	14	a	a	PRON
iajs-1139	4	15	of	of	ADP
iajs-1139	4	16	a	a	DET
iajs-1139	4	17	nonempty	nonempty	ADJ
iajs-1139	4	18	set	set	VERB
iajs-1139	4	19	s	s	PRON
iajs-1139	4	20	as	as	ADP
iajs-1139	4	21	a	a	DET
iajs-1139	4	22	mapping	mapping	NOUN
iajs-1139	4	23	from	from	ADP
iajs-1139	4	24	s	s	PRON
iajs-1139	4	25	into	into	ADP
iajs-1139	4	26	[	[	X
iajs-1139	4	27	0,1	0,1	NUM
iajs-1139	4	28	]	]	PUNCT
iajs-1139	4	29	,	,	PUNCT
iajs-1139	4	30	liu	liu	PROPN
iajs-1139	4	31	in	in	ADP
iajs-1139	4	32	[	[	X
iajs-1139	4	33	2	2	NUM
iajs-1139	4	34	]	]	PUNCT
iajs-1139	4	35	introduced	introduce	VERB
iajs-1139	4	36	the	the	DET
iajs-1139	4	37	concept	concept	NOUN
iajs-1139	4	38	of	of	ADP
iajs-1139	4	39	a	a	DET
iajs-1139	4	40	fuzzy	fuzzy	ADJ
iajs-1139	4	41	ring	ring	NOUN
iajs-1139	4	42	,	,	PUNCT
iajs-1139	4	43	martines	martine	VERB
iajs-1139	4	44	[	[	X
iajs-1139	4	45	3	3	NUM
iajs-1139	4	46	]	]	PUNCT
iajs-1139	4	47	introduced	introduce	VERB
iajs-1139	4	48	the	the	DET
iajs-1139	4	49	notion	notion	NOUN
iajs-1139	4	50	of	of	ADP
iajs-1139	4	51	a	a	DET
iajs-1139	4	52	fuzzy	fuzzy	ADJ
iajs-1139	4	53	ideal	ideal	NOUN
iajs-1139	4	54	of	of	ADP
iajs-1139	4	55	a	a	DET
iajs-1139	4	56	fuzzy	fuzzy	ADJ
iajs-1139	4	57	ring	ring	NOUN
iajs-1139	4	58	.	.	PUNCT
iajs-1139	5	1	a	a	DET
iajs-1139	5	2	non	non	ADJ
iajs-1139	5	3	zero	zero	NUM
iajs-1139	5	4	proper	proper	ADJ
iajs-1139	5	5	ideal	ideal	NOUN
iajs-1139	5	6	i	i	PRON
iajs-1139	5	7	of	of	ADP
iajs-1139	5	8	a	a	DET
iajs-1139	5	9	ring	ring	NOUN
iajs-1139	5	10	r	r	NOUN
iajs-1139	5	11	is	be	AUX
iajs-1139	5	12	called	call	VERB
iajs-1139	5	13	an	an	DET
iajs-1139	5	14	essential	essential	ADJ
iajs-1139	5	15	ideal	ideal	NOUN
iajs-1139	5	16	if	if	SCONJ
iajs-1139	5	17	i	i	PRON
iajs-1139	5	18			PUNCT
iajs-1139	5	19	j	j	PROPN
iajs-1139	5	20			PROPN
iajs-1139	5	21	(	(	PUNCT
iajs-1139	5	22	0	0	NUM
iajs-1139	5	23	)	)	PUNCT
iajs-1139	5	24	,	,	PUNCT
iajs-1139	5	25	for	for	ADP
iajs-1139	5	26	any	any	DET
iajs-1139	5	27	non	non	ADJ
iajs-1139	5	28	zero	zero	NUM
iajs-1139	5	29	ideal	ideal	ADJ
iajs-1139	5	30	j	j	PROPN
iajs-1139	5	31	of	of	ADP
iajs-1139	5	32	r	r	PROPN
iajs-1139	5	33	,	,	PUNCT
iajs-1139	5	34	[	[	X
iajs-1139	5	35	4	4	NUM
iajs-1139	5	36	]	]	PUNCT
iajs-1139	5	37	.	.	PUNCT
iajs-1139	6	1	inaam	inaam	NOUN
iajs-1139	6	2	in	in	ADP
iajs-1139	6	3	[	[	X
iajs-1139	6	4	5	5	NUM
iajs-1139	6	5	]	]	PUNCT
iajs-1139	6	6	fuzzified	fuzzifie	VERB
iajs-1139	6	7	this	this	DET
iajs-1139	6	8	concept	concept	NOUN
iajs-1139	6	9	to	to	ADP
iajs-1139	6	10	essential	essential	ADJ
iajs-1139	6	11	fuzzy	fuzzy	ADJ
iajs-1139	6	12	ideal	ideal	NOUN
iajs-1139	6	13	of	of	ADP
iajs-1139	6	14	fuzzy	fuzzy	ADJ
iajs-1139	6	15	ring	ring	NOUN
iajs-1139	6	16	and	and	CCONJ
iajs-1139	6	17	gave	give	VERB
iajs-1139	6	18	its	its	PRON
iajs-1139	6	19	basic	basic	ADJ
iajs-1139	6	20	properties	property	NOUN
iajs-1139	6	21	.	.	PUNCT
iajs-1139	7	1	nada	nada	PROPN
iajs-1139	7	2	in	in	ADP
iajs-1139	7	3	[	[	X
iajs-1139	7	4	6	6	NUM
iajs-1139	7	5	]	]	PUNCT
iajs-1139	7	6	introduced	introduce	VERB
iajs-1139	7	7	and	and	CCONJ
iajs-1139	7	8	studied	study	VERB
iajs-1139	7	9	notion	notion	NOUN
iajs-1139	7	10	of	of	ADP
iajs-1139	7	11	semiessential	semiessential	ADJ
iajs-1139	7	12	ideal	ideal	NOUN
iajs-1139	7	13	in	in	ADP
iajs-1139	7	14	a	a	DET
iajs-1139	7	15	ring	ring	NOUN
iajs-1139	7	16	r	r	NOUN
iajs-1139	7	17	,	,	PUNCT
iajs-1139	7	18	where	where	SCONJ
iajs-1139	7	19	a	a	DET
iajs-1139	7	20	non	non	ADJ
iajs-1139	7	21	zero	zero	NUM
iajs-1139	7	22	ideal	ideal	NOUN
iajs-1139	7	23	i	i	PRON
iajs-1139	7	24	of	of	ADP
iajs-1139	7	25	r	r	NOUN
iajs-1139	7	26	is	be	AUX
iajs-1139	7	27	called	call	VERB
iajs-1139	7	28	semiessential	semiessential	ADJ
iajs-1139	7	29	if	if	SCONJ
iajs-1139	7	30	i	i	PRON
iajs-1139	7	31			VERB
iajs-1139	7	32	p	p	PROPN
iajs-1139	7	33			PROPN
iajs-1139	7	34	(	(	PUNCT
iajs-1139	7	35	0	0	NUM
iajs-1139	7	36	)	)	PUNCT
iajs-1139	7	37	for	for	ADP
iajs-1139	7	38	all	all	DET
iajs-1139	7	39	non	non	ADJ
iajs-1139	7	40	zero	zero	NUM
iajs-1139	7	41	prime	prime	ADJ
iajs-1139	7	42	ideals	ideal	NOUN
iajs-1139	7	43	of	of	ADP
iajs-1139	7	44	r	r	NOUN
iajs-1139	7	45	,	,	PUNCT
iajs-1139	7	46	[	[	X
iajs-1139	7	47	4	4	NUM
iajs-1139	7	48	]	]	PUNCT
iajs-1139	7	49	.	.	PUNCT
iajs-1139	8	1	a	a	DET
iajs-1139	8	2	ring	ring	NOUN
iajs-1139	8	3	r	r	NOUN
iajs-1139	8	4	is	be	AUX
iajs-1139	8	5	called	call	VERB
iajs-1139	8	6	uniform	uniform	ADJ
iajs-1139	8	7	if	if	SCONJ
iajs-1139	8	8	every	every	DET
iajs-1139	8	9	ideal	ideal	NOUN
iajs-1139	8	10	of	of	ADP
iajs-1139	8	11	r	r	NOUN
iajs-1139	8	12	is	be	AUX
iajs-1139	8	13	essential	essential	ADJ
iajs-1139	8	14	.	.	PUNCT
iajs-1139	9	1	nada	nada	PROPN
iajs-1139	9	2	in	in	ADP
iajs-1139	9	3	[	[	X
iajs-1139	9	4	6	6	NUM
iajs-1139	9	5	]	]	PUNCT
iajs-1139	9	6	introduced	introduce	VERB
iajs-1139	9	7	and	and	CCONJ
iajs-1139	9	8	studied	study	VERB
iajs-1139	9	9	the	the	DET
iajs-1139	9	10	notion	notion	NOUN
iajs-1139	9	11	semiuniform	semiuniform	NOUN
iajs-1139	9	12	ring	ring	NOUN
iajs-1139	9	13	where	where	SCONJ
iajs-1139	9	14	a	a	DET
iajs-1139	9	15	ring	ring	NOUN
iajs-1139	9	16	r	r	NOUN
iajs-1139	9	17	is	be	AUX
iajs-1139	9	18	called	call	VERB
iajs-1139	9	19	semiuniform	semiuniform	NOUN
iajs-1139	9	20	ring	ring	NOUN
iajs-1139	9	21	if	if	SCONJ
iajs-1139	9	22	every	every	DET
iajs-1139	9	23	ideal	ideal	NOUN
iajs-1139	9	24	of	of	ADP
iajs-1139	9	25	r	r	NOUN
iajs-1139	9	26	is	be	AUX
iajs-1139	9	27	semiessential	semiessential	ADJ
iajs-1139	9	28	ideal	ideal	NOUN
iajs-1139	9	29	.	.	PUNCT
iajs-1139	10	1	in	in	ADP
iajs-1139	10	2	this	this	DET
iajs-1139	10	3	paper	paper	NOUN
iajs-1139	10	4	we	we	PRON
iajs-1139	10	5	fuzzify	fuzzify	VERB
iajs-1139	10	6	the	the	DET
iajs-1139	10	7	concepts	concept	NOUN
iajs-1139	10	8	semiessential	semiessential	ADJ
iajs-1139	10	9	ideal	ideal	NOUN
iajs-1139	10	10	of	of	ADP
iajs-1139	10	11	a	a	DET
iajs-1139	10	12	ring	ring	NOUN
iajs-1139	10	13	,	,	PUNCT
iajs-1139	10	14	uniform	uniform	ADJ
iajs-1139	10	15	ring	ring	NOUN
iajs-1139	10	16	and	and	CCONJ
iajs-1139	10	17	semiuniform	semiuniform	VERB
iajs-1139	10	18	ring	ring	NOUN
iajs-1139	10	19	into	into	ADP
iajs-1139	10	20	semiessential	semiessential	ADJ
iajs-1139	10	21	fuzzy	fuzzy	ADJ
iajs-1139	10	22	ideal	ideal	NOUN
iajs-1139	10	23	of	of	ADP
iajs-1139	10	24	fuzzy	fuzzy	ADJ
iajs-1139	10	25	ring	ring	NOUN
iajs-1139	10	26	,	,	PUNCT
iajs-1139	10	27	uniform	uniform	ADJ
iajs-1139	10	28	fuzzy	fuzzy	ADJ
iajs-1139	10	29	ring	ring	NOUN
iajs-1139	10	30	and	and	CCONJ
iajs-1139	10	31	semiuniform	semiuniform	VERB
iajs-1139	10	32	fuzzy	fuzzy	ADJ
iajs-1139	10	33	ring	ring	NOUN
iajs-1139	10	34	.	.	PUNCT
iajs-1139	11	1	where	where	SCONJ
iajs-1139	11	2	a	a	DET
iajs-1139	11	3	fuzzy	fuzzy	ADJ
iajs-1139	11	4	ideal	ideal	NOUN
iajs-1139	11	5	a	a	PRON
iajs-1139	11	6	of	of	ADP
iajs-1139	11	7	a	a	DET
iajs-1139	11	8	fuzzy	fuzzy	ADJ
iajs-1139	11	9	ring	ring	NOUN
iajs-1139	11	10	x	x	VERB
iajs-1139	11	11	is	be	AUX
iajs-1139	11	12	semiessential	semiessential	ADJ
iajs-1139	11	13	if	if	SCONJ
iajs-1139	11	14	i	i	PRON
iajs-1139	11	15			VERB
iajs-1139	11	16	p	p	PROPN
iajs-1139	11	17			PROPN
iajs-1139	11	18	(	(	PUNCT
iajs-1139	11	19	0	0	NUM
iajs-1139	11	20	)	)	PUNCT
iajs-1139	11	21	for	for	ADP
iajs-1139	11	22	any	any	DET
iajs-1139	11	23	prime	prime	ADJ
iajs-1139	11	24	fuzzy	fuzzy	ADJ
iajs-1139	11	25	ideal	ideal	NOUN
iajs-1139	11	26	p	p	NOUN
iajs-1139	11	27	of	of	ADP
iajs-1139	11	28	x.	x.	NOUN
iajs-1139	11	29	a	a	DET
iajs-1139	11	30	fuzzy	fuzzy	ADJ
iajs-1139	11	31	ring	ring	NOUN
iajs-1139	11	32	x	x	VERB
iajs-1139	11	33	is	be	AUX
iajs-1139	11	34	called	call	VERB
iajs-1139	11	35	uniform	uniform	ADJ
iajs-1139	11	36	(	(	PUNCT
iajs-1139	11	37	semiuniform	semiuniform	NOUN
iajs-1139	11	38	)	)	PUNCT
iajs-1139	11	39	if	if	SCONJ
iajs-1139	11	40	every	every	DET
iajs-1139	11	41	fuzzy	fuzzy	ADJ
iajs-1139	11	42	ideal	ideal	NOUN
iajs-1139	11	43	of	of	ADP
iajs-1139	11	44	x	x	PUNCT
iajs-1139	11	45	is	be	AUX
iajs-1139	11	46	essential	essential	ADJ
iajs-1139	11	47	(	(	PUNCT
iajs-1139	11	48	semiessential	semiessential	NOUN
iajs-1139	11	49	)	)	PUNCT
iajs-1139	11	50	respectively	respectively	ADV
iajs-1139	11	51	.	.	PUNCT
iajs-1139	12	1	in	in	ADP
iajs-1139	12	2	s.1	s.1	NUM
iajs-1139	12	3	,	,	PUNCT
iajs-1139	12	4	some	some	DET
iajs-1139	12	5	basic	basic	ADJ
iajs-1139	12	6	definitions	definition	NOUN
iajs-1139	12	7	and	and	CCONJ
iajs-1139	12	8	results	result	NOUN
iajs-1139	12	9	are	be	AUX
iajs-1139	12	10	collected	collect	VERB
iajs-1139	12	11	.	.	PUNCT
iajs-1139	13	1	in	in	ADP
iajs-1139	13	2	s.2	s.2	PROPN
iajs-1139	13	3	,	,	PUNCT
iajs-1139	13	4	we	we	PRON
iajs-1139	13	5	study	study	VERB
iajs-1139	13	6	semiesential	semiesential	ADJ
iajs-1139	13	7	fuzzy	fuzzy	ADJ
iajs-1139	13	8	ideals	ideal	NOUN
iajs-1139	13	9	of	of	ADP
iajs-1139	13	10	fuzzy	fuzzy	ADJ
iajs-1139	13	11	ring	ring	NOUN
iajs-1139	13	12	,	,	PUNCT
iajs-1139	13	13	we	we	PRON
iajs-1139	13	14	give	give	VERB
iajs-1139	13	15	some	some	DET
iajs-1139	13	16	basic	basic	ADJ
iajs-1139	13	17	properties	property	NOUN
iajs-1139	13	18	about	about	ADP
iajs-1139	13	19	this	this	DET
iajs-1139	13	20	concept	concept	NOUN
iajs-1139	13	21	.	.	PUNCT
iajs-1139	14	1	in	in	ADP
iajs-1139	14	2	s.3	s.3	NOUN
iajs-1139	14	3	,	,	PUNCT
iajs-1139	14	4	we	we	PRON
iajs-1139	14	5	study	study	VERB
iajs-1139	14	6	the	the	DET
iajs-1139	14	7	notion	notion	NOUN
iajs-1139	14	8	of	of	ADP
iajs-1139	14	9	uniform	uniform	ADJ
iajs-1139	14	10	fuzzy	fuzzy	ADJ
iajs-1139	14	11	rings	ring	NOUN
iajs-1139	14	12	and	and	CCONJ
iajs-1139	14	13	semiuniform	semiuniform	VERB
iajs-1139	14	14	fuzzy	fuzzy	ADJ
iajs-1139	14	15	rings	ring	NOUN
iajs-1139	14	16	.	.	PUNCT
iajs-1139	15	1	several	several	ADJ
iajs-1139	15	2	properties	property	NOUN
iajs-1139	15	3	about	about	ADP
iajs-1139	15	4	them	they	PRON
iajs-1139	15	5	are	be	AUX
iajs-1139	15	6	given	give	VERB
iajs-1139	15	7	.	.	PUNCT
iajs-1139	16	1	throughout	throughout	ADP
iajs-1139	16	2	this	this	DET
iajs-1139	16	3	paper	paper	NOUN
iajs-1139	16	4	,	,	PUNCT
iajs-1139	16	5	r	r	NOUN
iajs-1139	16	6	is	be	AUX
iajs-1139	16	7	commutative	commutative	ADJ
iajs-1139	16	8	ring	ring	NOUN
iajs-1139	16	9	with	with	ADP
iajs-1139	16	10	unity	unity	NOUN
iajs-1139	16	11	,	,	PUNCT
iajs-1139	16	12	and	and	CCONJ
iajs-1139	16	13	x(0	x(0	PROPN
iajs-1139	16	14	)	)	PUNCT
iajs-1139	17	1	=	=	SYM
iajs-1139	17	2	1	1	NUM
iajs-1139	17	3	,	,	PUNCT
iajs-1139	17	4	for	for	ADP
iajs-1139	17	5	any	any	DET
iajs-1139	17	6	fuzzy	fuzzy	ADJ
iajs-1139	17	7	ring	ring	NOUN
iajs-1139	17	8	.	.	PUNCT
iajs-1139	18	1	s.1	s.1	DET
iajs-1139	18	2	preliminaries	preliminary	NOUN
iajs-1139	18	3	let	let	VERB
iajs-1139	18	4	r	r	PRON
iajs-1139	18	5	be	be	AUX
iajs-1139	18	6	a	a	DET
iajs-1139	18	7	commutative	commutative	ADJ
iajs-1139	18	8	ring	ring	NOUN
iajs-1139	18	9	with	with	ADP
iajs-1139	18	10	identity	identity	NOUN
iajs-1139	18	11	.	.	PUNCT
iajs-1139	19	1	a	a	DET
iajs-1139	19	2	fuzzy	fuzzy	ADJ
iajs-1139	19	3	subset	subset	NOUN
iajs-1139	19	4	of	of	ADP
iajs-1139	19	5	r	r	NOUN
iajs-1139	19	6	is	be	AUX
iajs-1139	19	7	a	a	DET
iajs-1139	19	8	function	function	NOUN
iajs-1139	19	9	from	from	ADP
iajs-1139	19	10	r	r	NOUN
iajs-1139	19	11	into	into	ADP
iajs-1139	19	12	[	[	X
iajs-1139	19	13	0,1	0,1	NUM
iajs-1139	19	14	]	]	PUNCT
iajs-1139	19	15	.	.	PUNCT
iajs-1139	20	1	let	let	VERB
iajs-1139	20	2	a	a	PRON
iajs-1139	20	3	and	and	CCONJ
iajs-1139	20	4	b	b	NOUN
iajs-1139	20	5	be	be	AUX
iajs-1139	20	6	a	a	DET
iajs-1139	20	7	fuzzy	fuzzy	ADJ
iajs-1139	20	8	subsets	subset	NOUN
iajs-1139	20	9	of	of	ADP
iajs-1139	20	10	r	r	NOUN
iajs-1139	20	11	we	we	PRON
iajs-1139	20	12	write	write	VERB
iajs-1139	20	13	a	a	DET
iajs-1139	20	14			PROPN
iajs-1139	20	15	b	b	PROPN
iajs-1139	20	16	if	if	SCONJ
iajs-1139	20	17	a(x	a(x	NOUN
iajs-1139	20	18	)	)	PUNCT
iajs-1139	20	19			NOUN
iajs-1139	20	20	b(x	b(x	NOUN
iajs-1139	20	21	)	)	PUNCT
iajs-1139	20	22	,	,	PUNCT
iajs-1139	20	23	for	for	ADP
iajs-1139	20	24	all	all	DET
iajs-1139	20	25	x	x	ADJ
iajs-1139	20	26			NOUN
iajs-1139	20	27	r	r	NOUN
iajs-1139	20	28	,	,	PUNCT
iajs-1139	20	29	(	(	PUNCT
iajs-1139	20	30	1	1	NUM
iajs-1139	20	31	)	)	PUNCT
iajs-1139	20	32	and	and	CCONJ
iajs-1139	20	33	(	(	PUNCT
iajs-1139	20	34	a	a	DET
iajs-1139	20	35			PUNCT
iajs-1139	20	36	b)(x	b)(x	NOUN
iajs-1139	20	37	)	)	PUNCT
iajs-1139	20	38	=	=	SYM
iajs-1139	20	39	min	min	NOUN
iajs-1139	20	40	{	{	PUNCT
iajs-1139	20	41	a(x	a(x	NOUN
iajs-1139	20	42	)	)	PUNCT
iajs-1139	20	43	,	,	PUNCT
iajs-1139	20	44	b(x	b(x	NOUN
iajs-1139	20	45	)	)	PUNCT
iajs-1139	20	46	}	}	PUNCT
iajs-1139	20	47	,	,	PUNCT
iajs-1139	20	48			NOUN
iajs-1139	20	49	x	x	SYM
iajs-1139	20	50			PROPN
iajs-1139	20	51	r.	r.	PROPN
iajs-1139	20	52	for	for	ADP
iajs-1139	20	53	each	each	DET
iajs-1139	20	54	t	t	NOUN
iajs-1139	20	55			NOUN
iajs-1139	21	1	[	[	X
iajs-1139	21	2	0,1	0,1	NUM
iajs-1139	21	3	]	]	PUNCT
iajs-1139	21	4	,	,	PUNCT
iajs-1139	21	5	the	the	DET
iajs-1139	21	6	set	set	NOUN
iajs-1139	21	7	{	{	PUNCT
iajs-1139	21	8	x	x	SYM
iajs-1139	21	9			NOUN
iajs-1139	21	10	r	r	NOUN
iajs-1139	21	11	;	;	PUNCT
iajs-1139	21	12	a(x	a(x	PROPN
iajs-1139	21	13	)	)	PUNCT
iajs-1139	21	14			NUM
iajs-1139	21	15	t	t	PROPN
iajs-1139	21	16	}	}	PUNCT
iajs-1139	21	17	is	be	AUX
iajs-1139	21	18	called	call	VERB
iajs-1139	21	19	the	the	DET
iajs-1139	21	20	level	level	NOUN
iajs-1139	21	21	subset	subset	NOUN
iajs-1139	21	22	of	of	ADP
iajs-1139	21	23	r	r	NOUN
iajs-1139	21	24	,	,	PUNCT
iajs-1139	21	25	[	[	X
iajs-1139	21	26	7	7	NUM
iajs-1139	21	27	]	]	PUNCT
iajs-1139	21	28	.	.	PUNCT
iajs-1139	22	1	if	if	SCONJ
iajs-1139	22	2	a	a	PRON
iajs-1139	22	3	and	and	CCONJ
iajs-1139	22	4	b	b	NOUN
iajs-1139	22	5	are	be	AUX
iajs-1139	22	6	fuzzy	fuzzy	ADJ
iajs-1139	22	7	subsets	subset	NOUN
iajs-1139	22	8	of	of	ADP
iajs-1139	22	9	r	r	NOUN
iajs-1139	22	10	,	,	PUNCT
iajs-1139	22	11	then	then	ADV
iajs-1139	22	12			PROPN
iajs-1139	22	13	t	t	PROPN
iajs-1139	22	14			NOUN
iajs-1139	22	15	[	[	X
iajs-1139	22	16	0,1	0,1	NUM
iajs-1139	22	17	]	]	SYM
iajs-1139	22	18	1	1	NUM
iajs-1139	22	19	.	.	PUNCT
iajs-1139	23	1	(	(	PUNCT
iajs-1139	23	2	a	a	DET
iajs-1139	23	3			X
iajs-1139	23	4	b)t	b)t	NOUN
iajs-1139	23	5	=	=	SYM
iajs-1139	23	6	at	at	ADP
iajs-1139	23	7			NOUN
iajs-1139	23	8	bt	bt	NOUN
iajs-1139	23	9	,	,	PUNCT
iajs-1139	23	10	[	[	X
iajs-1139	23	11	1	1	NUM
iajs-1139	23	12	]	]	PUNCT
iajs-1139	23	13	,	,	PUNCT
iajs-1139	23	14	2	2	X
iajs-1139	23	15	.	.	X
iajs-1139	23	16	a	a	DET
iajs-1139	23	17	=	=	SYM
iajs-1139	23	18	b	b	X
iajs-1139	23	19	iff	iff	PROPN
iajs-1139	23	20	at	at	ADP
iajs-1139	23	21	=	=	SYM
iajs-1139	23	22	bt	bt	PROPN
iajs-1139	23	23	,	,	PUNCT
iajs-1139	23	24	[	[	X
iajs-1139	23	25	1	1	NUM
iajs-1139	23	26	]	]	PUNCT
iajs-1139	23	27	.	.	PUNCT
iajs-1139	24	1	let	let	VERB
iajs-1139	24	2	f	f	PRON
iajs-1139	24	3	be	be	AUX
iajs-1139	24	4	a	a	DET
iajs-1139	24	5	mapping	mapping	NOUN
iajs-1139	24	6	from	from	ADP
iajs-1139	24	7	a	a	DET
iajs-1139	24	8	set	set	NOUN
iajs-1139	24	9	m	m	NOUN
iajs-1139	24	10	into	into	ADP
iajs-1139	24	11	a	a	DET
iajs-1139	24	12	set	set	NOUN
iajs-1139	24	13	n	n	CCONJ
iajs-1139	24	14	,	,	PUNCT
iajs-1139	24	15	let	let	VERB
iajs-1139	24	16	b	b	X
iajs-1139	24	17	be	be	AUX
iajs-1139	24	18	a	a	DET
iajs-1139	24	19	fuzzy	fuzzy	ADJ
iajs-1139	24	20	subset	subset	NOUN
iajs-1139	24	21	of	of	ADP
iajs-1139	24	22	n.	n.	NOUN
iajs-1139	24	23	the	the	DET
iajs-1139	24	24	inverse	inverse	NOUN
iajs-1139	24	25	image	image	NOUN
iajs-1139	24	26	of	of	ADP
iajs-1139	24	27	b	b	PROPN
iajs-1139	24	28	is	be	AUX
iajs-1139	24	29	a	a	DET
iajs-1139	24	30	fuzzy	fuzzy	ADJ
iajs-1139	24	31	subset	subset	NOUN
iajs-1139	24	32	of	of	ADP
iajs-1139	24	33	m	m	AUX
iajs-1139	24	34	defined	define	VERB
iajs-1139	24	35	by	by	ADP
iajs-1139	24	36	f	f	PROPN
iajs-1139	24	37	–	–	PUNCT
iajs-1139	24	38	1	1	NUM
iajs-1139	24	39	(	(	PUNCT
iajs-1139	24	40	b)(x	b)(x	PROPN
iajs-1139	24	41	)	)	PUNCT
iajs-1139	24	42	=	=	SYM
iajs-1139	25	1	b(f	b(f	PROPN
iajs-1139	25	2	(	(	PUNCT
iajs-1139	25	3	x	x	NOUN
iajs-1139	25	4	)	)	PUNCT
iajs-1139	25	5	)	)	PUNCT
iajs-1139	25	6	,	,	PUNCT
iajs-1139	25	7			NOUN
iajs-1139	25	8	x	x	PROPN
iajs-1139	25	9	m	m	PROPN
iajs-1139	25	10	,	,	PUNCT
iajs-1139	25	11	[	[	X
iajs-1139	25	12	1	1	NUM
iajs-1139	25	13	]	]	PUNCT
iajs-1139	25	14	.	.	PUNCT
iajs-1139	26	1	let	let	VERB
iajs-1139	26	2	a	a	DET
iajs-1139	26	3	be	be	AUX
iajs-1139	26	4	a	a	DET
iajs-1139	26	5	fuzzy	fuzzy	ADJ
iajs-1139	26	6	subset	subset	NOUN
iajs-1139	26	7	of	of	ADP
iajs-1139	26	8	a	a	DET
iajs-1139	26	9	set	set	ADJ
iajs-1139	26	10	m.	m.	NOUN
iajs-1139	26	11	a	a	PRON
iajs-1139	26	12	is	be	AUX
iajs-1139	26	13	called	call	VERB
iajs-1139	26	14	an	an	DET
iajs-1139	26	15	f	f	NOUN
iajs-1139	26	16	-	-	PUNCT
iajs-1139	26	17	invariant	invariant	ADJ
iajs-1139	26	18	if	if	SCONJ
iajs-1139	26	19	a(x	a(x	NOUN
iajs-1139	26	20	)	)	PUNCT
iajs-1139	26	21	=	=	SYM
iajs-1139	26	22	a(y	a(y	PROPN
iajs-1139	26	23	)	)	PUNCT
iajs-1139	26	24	,	,	PUNCT
iajs-1139	27	1	whenever	whenever	SCONJ
iajs-1139	27	2	f	f	PROPN
iajs-1139	27	3	(	(	PUNCT
iajs-1139	27	4	x	x	X
iajs-1139	27	5	)	)	PUNCT
iajs-1139	27	6	=	=	SYM
iajs-1139	27	7	f	f	PROPN
iajs-1139	27	8	(	(	PUNCT
iajs-1139	27	9	y	y	PROPN
iajs-1139	27	10	)	)	PUNCT
iajs-1139	27	11	,	,	PUNCT
iajs-1139	27	12	where	where	SCONJ
iajs-1139	27	13	x	x	X
iajs-1139	27	14	,	,	PUNCT
iajs-1139	27	15	y	y	PROPN
iajs-1139	27	16	m	m	PRON
iajs-1139	27	17	,	,	PUNCT
iajs-1139	27	18	[	[	X
iajs-1139	27	19	8	8	NUM
iajs-1139	27	20	]	]	PUNCT
iajs-1139	27	21	.	.	PUNCT
iajs-1139	28	1	ibn	ibn	PROPN
iajs-1139	28	2	alhaitham	alhaitham	PROPN
iajs-1139	28	3	j.	j.	PROPN
iajs-1139	28	4	for	for	ADP
iajs-1139	28	5	pure	pure	ADJ
iajs-1139	28	6	&	&	CCONJ
iajs-1139	28	7	appl	appl	PROPN
iajs-1139	28	8	.	.	PUNCT
iajs-1139	29	1	sci	sci	PROPN
iajs-1139	29	2	.	.	PUNCT
iajs-1139	30	1	vol.22	vol.22	PROPN
iajs-1139	30	2	(	(	PUNCT
iajs-1139	30	3	4	4	NUM
iajs-1139	30	4	)	)	PUNCT
iajs-1139	30	5	2009	2009	NUM
iajs-1139	30	6	if	if	SCONJ
iajs-1139	30	7	f	f	PROPN
iajs-1139	30	8	is	be	AUX
iajs-1139	30	9	a	a	DET
iajs-1139	30	10	function	function	NOUN
iajs-1139	30	11	from	from	ADP
iajs-1139	30	12	a	a	DET
iajs-1139	30	13	set	set	NOUN
iajs-1139	30	14	m	m	NOUN
iajs-1139	30	15	into	into	ADP
iajs-1139	30	16	a	a	DET
iajs-1139	30	17	set	set	NOUN
iajs-1139	30	18	n	n	CCONJ
iajs-1139	30	19	,	,	PUNCT
iajs-1139	30	20	let	let	VERB
iajs-1139	30	21	a1	a1	NOUN
iajs-1139	30	22	and	and	CCONJ
iajs-1139	30	23	a2	a2	PROPN
iajs-1139	30	24	be	be	VERB
iajs-1139	30	25	fuzzy	fuzzy	ADJ
iajs-1139	30	26	subsets	subset	NOUN
iajs-1139	30	27	of	of	ADP
iajs-1139	30	28	m	m	NOUN
iajs-1139	30	29	and	and	CCONJ
iajs-1139	30	30	b1	b1	PROPN
iajs-1139	30	31	,	,	PUNCT
iajs-1139	30	32	b2	b2	NOUN
iajs-1139	30	33	be	be	VERB
iajs-1139	30	34	fuzzy	fuzzy	ADJ
iajs-1139	30	35	subsets	subset	NOUN
iajs-1139	30	36	of	of	ADP
iajs-1139	30	37	n	n	CCONJ
iajs-1139	30	38	,	,	PUNCT
iajs-1139	30	39	then	then	ADV
iajs-1139	30	40	1	1	X
iajs-1139	30	41	.	.	PUNCT
iajs-1139	31	1	f	f	PROPN
iajs-1139	31	2	(	(	PUNCT
iajs-1139	31	3	a1	a1	NOUN
iajs-1139	31	4			NOUN
iajs-1139	31	5	a2	a2	NOUN
iajs-1139	31	6	)	)	PUNCT
iajs-1139	31	7	=	=	SYM
iajs-1139	32	1	f	f	PROPN
iajs-1139	32	2	(	(	PUNCT
iajs-1139	32	3	a1	a1	PROPN
iajs-1139	32	4	)	)	PUNCT
iajs-1139	32	5			ADJ
iajs-1139	32	6	f	f	X
iajs-1139	32	7	(	(	PUNCT
iajs-1139	32	8	a2	a2	PROPN
iajs-1139	32	9	)	)	PUNCT
iajs-1139	32	10	,	,	PUNCT
iajs-1139	32	11	a1	a1	PROPN
iajs-1139	32	12	,	,	PUNCT
iajs-1139	32	13	a2	a2	PROPN
iajs-1139	32	14	are	be	AUX
iajs-1139	32	15	f	f	X
iajs-1139	32	16	-	-	PUNCT
iajs-1139	32	17	invariant	invariant	ADJ
iajs-1139	32	18	,	,	PUNCT
iajs-1139	32	19	[	[	X
iajs-1139	32	20	8	8	NUM
iajs-1139	32	21	]	]	SYM
iajs-1139	32	22	2	2	NUM
iajs-1139	32	23	.	.	X
iajs-1139	32	24	f	f	PROPN
iajs-1139	33	1	–	–	PUNCT
iajs-1139	33	2	1	1	NUM
iajs-1139	33	3	(	(	PUNCT
iajs-1139	33	4	b1	b1	NOUN
iajs-1139	33	5			VERB
iajs-1139	33	6	b2	b2	NOUN
iajs-1139	33	7	)	)	PUNCT
iajs-1139	34	1	=	=	SYM
iajs-1139	34	2	f	f	X
iajs-1139	34	3	–	–	PUNCT
iajs-1139	34	4	1	1	NUM
iajs-1139	34	5	(	(	PUNCT
iajs-1139	34	6	b1	b1	NOUN
iajs-1139	34	7	)	)	PUNCT
iajs-1139	34	8			PUNCT
iajs-1139	34	9	f	f	X
iajs-1139	34	10	–	–	PUNCT
iajs-1139	34	11	1	1	NUM
iajs-1139	34	12	(	(	PUNCT
iajs-1139	34	13	b2	b2	NOUN
iajs-1139	34	14	)	)	PUNCT
iajs-1139	34	15	,	,	PUNCT
iajs-1139	35	1	[	[	X
iajs-1139	35	2	8	8	NUM
iajs-1139	35	3	]	]	PUNCT
iajs-1139	35	4	.	.	PUNCT
iajs-1139	36	1	3	3	X
iajs-1139	36	2	.	.	X
iajs-1139	36	3	f	f	PROPN
iajs-1139	36	4	–	–	PUNCT
iajs-1139	36	5	1	1	NUM
iajs-1139	36	6	(	(	PUNCT
iajs-1139	36	7	f	f	PROPN
iajs-1139	36	8	(	(	PUNCT
iajs-1139	36	9	a1	a1	PROPN
iajs-1139	36	10	)	)	PUNCT
iajs-1139	36	11	)	)	PUNCT
iajs-1139	37	1	=	=	SYM
iajs-1139	37	2	a1	a1	NOUN
iajs-1139	37	3	,	,	PUNCT
iajs-1139	37	4	whenever	whenever	SCONJ
iajs-1139	37	5	a1	a1	NOUN
iajs-1139	37	6	is	be	AUX
iajs-1139	37	7	f	f	NOUN
iajs-1139	37	8	-	-	PUNCT
iajs-1139	37	9	invariant	invariant	ADJ
iajs-1139	37	10	,	,	PUNCT
iajs-1139	37	11	[	[	X
iajs-1139	37	12	8	8	NUM
iajs-1139	37	13	]	]	SYM
iajs-1139	37	14	4	4	NUM
iajs-1139	37	15	.	.	X
iajs-1139	38	1	f	f	PROPN
iajs-1139	38	2	(	(	PUNCT
iajs-1139	38	3	f	f	X
iajs-1139	38	4	–	–	PUNCT
iajs-1139	38	5	1	1	NUM
iajs-1139	38	6	(	(	PUNCT
iajs-1139	38	7	b1	b1	NOUN
iajs-1139	38	8	)	)	PUNCT
iajs-1139	38	9	)	)	PUNCT
iajs-1139	39	1	=	=	SYM
iajs-1139	39	2	b1	b1	NOUN
iajs-1139	39	3	,	,	PUNCT
iajs-1139	39	4	[	[	X
iajs-1139	39	5	8	8	NUM
iajs-1139	39	6	]	]	PUNCT
iajs-1139	39	7	.	.	PUNCT
iajs-1139	40	1	moreover	moreover	ADV
iajs-1139	40	2	the	the	DET
iajs-1139	40	3	following	follow	VERB
iajs-1139	40	4	definitions	definition	NOUN
iajs-1139	40	5	and	and	CCONJ
iajs-1139	40	6	properties	property	NOUN
iajs-1139	40	7	are	be	AUX
iajs-1139	40	8	needed	need	VERB
iajs-1139	40	9	later	later	ADV
iajs-1139	40	10	1.1	1.1	NUM
iajs-1139	40	11	definition	definition	NOUN
iajs-1139	40	12	[	[	X
iajs-1139	40	13	2	2	X
iajs-1139	40	14	]	]	PUNCT
iajs-1139	40	15	let	let	VERB
iajs-1139	40	16	x	x	PRON
iajs-1139	40	17	be	be	AUX
iajs-1139	40	18	a	a	DET
iajs-1139	40	19	fuzzy	fuzzy	ADJ
iajs-1139	40	20	subset	subset	NOUN
iajs-1139	40	21	of	of	ADP
iajs-1139	40	22	a	a	DET
iajs-1139	40	23	ring	ring	NOUN
iajs-1139	40	24	r.	r.	PROPN
iajs-1139	40	25	x	x	PUNCT
iajs-1139	40	26	is	be	AUX
iajs-1139	40	27	called	call	VERB
iajs-1139	40	28	a	a	DET
iajs-1139	40	29	fuzzy	fuzzy	ADJ
iajs-1139	40	30	ring	ring	NOUN
iajs-1139	40	31	of	of	ADP
iajs-1139	40	32	r	r	NOUN
iajs-1139	40	33	if	if	SCONJ
iajs-1139	40	34	x	x	PROPN
iajs-1139	40	35			NOUN
iajs-1139	40	36	o1	o1	NOUN
iajs-1139	40	37	and	and	CCONJ
iajs-1139	40	38	for	for	ADP
iajs-1139	40	39	each	each	DET
iajs-1139	40	40	x	x	NOUN
iajs-1139	40	41	,	,	PUNCT
iajs-1139	40	42	y	y	PROPN
iajs-1139	40	43			NOUN
iajs-1139	40	44	r	r	NOUN
iajs-1139	40	45	1	1	NUM
iajs-1139	40	46	.	.	PUNCT
iajs-1139	40	47	x(x	x(x	PROPN
iajs-1139	40	48	–	–	PUNCT
iajs-1139	40	49	y	y	PROPN
iajs-1139	40	50	)	)	PUNCT
iajs-1139	40	51			NUM
iajs-1139	40	52	min	min	PROPN
iajs-1139	40	53	{	{	PUNCT
iajs-1139	40	54	x(x),x(y	x(x),x(y	PROPN
iajs-1139	40	55	)	)	PUNCT
iajs-1139	40	56	}	}	PUNCT
iajs-1139	40	57	,	,	PUNCT
iajs-1139	40	58	2	2	X
iajs-1139	40	59	.	.	PUNCT
iajs-1139	40	60	x(x	x(x	PROPN
iajs-1139	41	1	y	y	PROPN
iajs-1139	41	2	)	)	PUNCT
iajs-1139	41	3			PROPN
iajs-1139	41	4	max	max	PROPN
iajs-1139	41	5	{	{	PUNCT
iajs-1139	41	6	x(x),x(y	x(x),x(y	PROPN
iajs-1139	41	7	)	)	PUNCT
iajs-1139	41	8	}	}	PUNCT
iajs-1139	41	9	.	.	PUNCT
iajs-1139	42	1	1.2	1.2	NUM
iajs-1139	42	2	definition	definition	NOUN
iajs-1139	42	3	[	[	X
iajs-1139	42	4	3	3	X
iajs-1139	42	5	]	]	PUNCT
iajs-1139	42	6	let	let	VERB
iajs-1139	42	7	x	x	PRON
iajs-1139	42	8	be	be	AUX
iajs-1139	42	9	a	a	DET
iajs-1139	42	10	fuzzy	fuzzy	ADJ
iajs-1139	42	11	ring	ring	NOUN
iajs-1139	42	12	of	of	ADP
iajs-1139	42	13	r	r	NOUN
iajs-1139	42	14	,	,	PUNCT
iajs-1139	42	15	a	a	DET
iajs-1139	42	16	fuzzy	fuzzy	NOUN
iajs-1139	42	17	subset	subset	VERB
iajs-1139	42	18	a	a	PRON
iajs-1139	42	19	of	of	ADP
iajs-1139	42	20	r	r	NOUN
iajs-1139	42	21	is	be	AUX
iajs-1139	42	22	called	call	VERB
iajs-1139	42	23	a	a	DET
iajs-1139	42	24	fuzzy	fuzzy	ADJ
iajs-1139	42	25	ideal	ideal	NOUN
iajs-1139	42	26	of	of	ADP
iajs-1139	42	27	x	x	PRON
iajs-1139	42	28	if	if	SCONJ
iajs-1139	42	29	1	1	NUM
iajs-1139	42	30	.	.	PUNCT
iajs-1139	43	1	a	a	DET
iajs-1139	43	2			PROPN
iajs-1139	43	3	x	x	SYM
iajs-1139	43	4	,	,	PUNCT
iajs-1139	43	5	2	2	NUM
iajs-1139	43	6	.	.	X
iajs-1139	43	7	a(x	a(x	PROPN
iajs-1139	43	8	–	–	PUNCT
iajs-1139	43	9	y	y	NOUN
iajs-1139	43	10	)	)	PUNCT
iajs-1139	43	11			NUM
iajs-1139	43	12	min	min	NOUN
iajs-1139	43	13	{	{	PUNCT
iajs-1139	43	14	a(x	a(x	PROPN
iajs-1139	43	15	)	)	PUNCT
iajs-1139	43	16	,	,	PUNCT
iajs-1139	43	17	a(y	a(y	PROPN
iajs-1139	43	18	)	)	PUNCT
iajs-1139	43	19	}	}	PUNCT
iajs-1139	43	20	,	,	PUNCT
iajs-1139	43	21			NOUN
iajs-1139	43	22	x	x	PROPN
iajs-1139	43	23	,	,	PUNCT
iajs-1139	43	24	y	y	PROPN
iajs-1139	43	25			PROPN
iajs-1139	43	26	r	r	NOUN
iajs-1139	43	27	,	,	PUNCT
iajs-1139	43	28	3	3	NUM
iajs-1139	43	29	.	.	PUNCT
iajs-1139	43	30	a(x	a(x	PROPN
iajs-1139	43	31	y	y	PROPN
iajs-1139	43	32	)	)	PUNCT
iajs-1139	43	33			PROPN
iajs-1139	43	34	max	max	PROPN
iajs-1139	43	35	{	{	PUNCT
iajs-1139	43	36	a(x	a(x	PROPN
iajs-1139	43	37	)	)	PUNCT
iajs-1139	43	38	,	,	PUNCT
iajs-1139	43	39	x(y	x(y	PROPN
iajs-1139	43	40	)	)	PUNCT
iajs-1139	43	41	}	}	PUNCT
iajs-1139	43	42	,	,	PUNCT
iajs-1139	43	43			NOUN
iajs-1139	43	44	x	x	PROPN
iajs-1139	43	45	,	,	PUNCT
iajs-1139	43	46	y	y	PROPN
iajs-1139	43	47			PROPN
iajs-1139	43	48	r.	r.	PROPN
iajs-1139	43	49	note	note	VERB
iajs-1139	43	50	that	that	SCONJ
iajs-1139	43	51	if	if	SCONJ
iajs-1139	43	52	x	x	PRON
iajs-1139	43	53	is	be	AUX
iajs-1139	43	54	a	a	DET
iajs-1139	43	55	fuzzy	fuzzy	ADJ
iajs-1139	43	56	ring	ring	NOUN
iajs-1139	43	57	of	of	ADP
iajs-1139	43	58	r	r	NOUN
iajs-1139	43	59	,	,	PUNCT
iajs-1139	43	60	then	then	ADV
iajs-1139	43	61	x(a	x(a	NOUN
iajs-1139	43	62	)	)	PUNCT
iajs-1139	43	63			NOUN
iajs-1139	43	64	x(0	x(0	NOUN
iajs-1139	43	65	)	)	PUNCT
iajs-1139	43	66	,	,	PUNCT
iajs-1139	43	67			VERB
iajs-1139	43	68	a	a	DET
iajs-1139	43	69			NOUN
iajs-1139	43	70	r	r	NOUN
iajs-1139	43	71	,	,	PUNCT
iajs-1139	43	72	[	[	X
iajs-1139	43	73	3	3	NUM
iajs-1139	43	74	,	,	PUNCT
iajs-1139	43	75	proposition	proposition	NOUN
iajs-1139	43	76	2.7	2.7	NUM
iajs-1139	43	77	]	]	PUNCT
iajs-1139	43	78	if	if	SCONJ
iajs-1139	43	79	a	a	PRON
iajs-1139	43	80	is	be	AUX
iajs-1139	43	81	a	a	DET
iajs-1139	43	82	fuzzy	fuzzy	ADJ
iajs-1139	43	83	ideal	ideal	NOUN
iajs-1139	43	84	of	of	ADP
iajs-1139	43	85	x	x	PRON
iajs-1139	43	86	,	,	PUNCT
iajs-1139	43	87	then	then	ADV
iajs-1139	43	88	a(a	a(a	PROPN
iajs-1139	43	89	)	)	PUNCT
iajs-1139	43	90			NOUN
iajs-1139	43	91	a(0	a(0	PROPN
iajs-1139	43	92	)	)	PUNCT
iajs-1139	43	93	,	,	PUNCT
iajs-1139	43	94			VERB
iajs-1139	43	95	a	a	DET
iajs-1139	43	96			NOUN
iajs-1139	43	97	r	r	NOUN
iajs-1139	43	98	,	,	PUNCT
iajs-1139	43	99	[	[	X
iajs-1139	43	100	3	3	NUM
iajs-1139	43	101	,	,	PUNCT
iajs-1139	43	102	proposition	proposition	NOUN
iajs-1139	43	103	2.9	2.9	NUM
iajs-1139	43	104	]	]	PUNCT
iajs-1139	43	105	.	.	PUNCT
iajs-1139	44	1	1.3	1.3	NUM
iajs-1139	44	2	proposition	proposition	NOUN
iajs-1139	44	3	a	a	DET
iajs-1139	44	4	fuzzy	fuzzy	ADJ
iajs-1139	44	5	subset	subset	NOUN
iajs-1139	44	6	x	x	X
iajs-1139	44	7	:	:	PUNCT
iajs-1139	44	8	r	r	NOUN
iajs-1139	44	9			X
iajs-1139	45	1	[	[	X
iajs-1139	45	2	0,1	0,1	NUM
iajs-1139	45	3	]	]	PUNCT
iajs-1139	45	4	,	,	PUNCT
iajs-1139	45	5	is	be	AUX
iajs-1139	45	6	a	a	DET
iajs-1139	45	7	fuzzy	fuzzy	ADJ
iajs-1139	45	8	ring	ring	NOUN
iajs-1139	45	9	if	if	SCONJ
iajs-1139	45	10	xt	xt	PROPN
iajs-1139	45	11	is	be	AUX
iajs-1139	45	12	a	a	DET
iajs-1139	45	13	subring	subring	NOUN
iajs-1139	45	14	of	of	ADP
iajs-1139	45	15	r	r	NOUN
iajs-1139	45	16	,	,	PUNCT
iajs-1139	45	17			PROPN
iajs-1139	45	18	t	t	PROPN
iajs-1139	45	19			PROPN
iajs-1139	45	20	[	[	X
iajs-1139	45	21	0,x(0	0,x(0	NUM
iajs-1139	45	22	)	)	PUNCT
iajs-1139	45	23	]	]	PUNCT
iajs-1139	45	24	,	,	PUNCT
iajs-1139	45	25	(	(	PUNCT
iajs-1139	45	26	3	3	NUM
iajs-1139	45	27	,	,	PUNCT
iajs-1139	45	28	proposition	proposition	NOUN
iajs-1139	45	29	2.10	2.10	NUM
iajs-1139	45	30	(	(	PUNCT
iajs-1139	45	31	i	i	NOUN
iajs-1139	45	32	)	)	PUNCT
iajs-1139	45	33	)	)	PUNCT
iajs-1139	45	34	.	.	PUNCT
iajs-1139	46	1	given	give	VERB
iajs-1139	46	2	a	a	DET
iajs-1139	46	3	fuzzy	fuzzy	ADJ
iajs-1139	46	4	ring	ring	NOUN
iajs-1139	46	5	x	x	PUNCT
iajs-1139	46	6	and	and	CCONJ
iajs-1139	46	7	a	a	DET
iajs-1139	46	8	fuzzy	fuzzy	ADJ
iajs-1139	46	9	set	set	VERB
iajs-1139	46	10	a	a	PRON
iajs-1139	46	11	:	:	PUNCT
iajs-1139	46	12	r	r	NOUN
iajs-1139	46	13			X
iajs-1139	47	1	[	[	X
iajs-1139	47	2	0,1	0,1	NUM
iajs-1139	47	3	]	]	PUNCT
iajs-1139	47	4	with	with	ADP
iajs-1139	47	5	a	a	DET
iajs-1139	47	6			NOUN
iajs-1139	47	7	o1	o1	NOUN
iajs-1139	47	8	,	,	PUNCT
iajs-1139	47	9	then	then	ADV
iajs-1139	47	10	a	a	PRON
iajs-1139	47	11	is	be	AUX
iajs-1139	47	12	a	a	DET
iajs-1139	47	13	fuzzy	fuzzy	ADJ
iajs-1139	47	14	ideal	ideal	NOUN
iajs-1139	47	15	of	of	ADP
iajs-1139	47	16	x	x	SYM
iajs-1139	47	17	iff	iff	PROPN
iajs-1139	47	18	at	at	ADP
iajs-1139	47	19	is	be	AUX
iajs-1139	47	20	an	an	DET
iajs-1139	47	21	ideal	ideal	NOUN
iajs-1139	47	22	of	of	ADP
iajs-1139	47	23	xt	xt	PROPN
iajs-1139	47	24			PROPN
iajs-1139	47	25	t	t	PROPN
iajs-1139	47	26			PROPN
iajs-1139	47	27	[	[	X
iajs-1139	47	28	0,a(0	0,a(0	NUM
iajs-1139	47	29	)	)	PUNCT
iajs-1139	47	30	]	]	PUNCT
iajs-1139	47	31	,	,	PUNCT
iajs-1139	47	32	[	[	X
iajs-1139	47	33	3	3	NUM
iajs-1139	47	34	,	,	PUNCT
iajs-1139	47	35	p	p	PRON
iajs-1139	47	36	roposition	roposition	NOUN
iajs-1139	47	37	2.10	2.10	NUM
iajs-1139	47	38	(	(	PUNCT
iajs-1139	47	39	iii	iii	NOUN
iajs-1139	47	40	)	)	PUNCT
iajs-1139	47	41	]	]	PUNCT
iajs-1139	47	42	.	.	PUNCT
iajs-1139	48	1	1.4	1.4	NUM
iajs-1139	48	2	definition	definition	NOUN
iajs-1139	48	3	[	[	X
iajs-1139	48	4	9	9	NUM
iajs-1139	48	5	]	]	X
iajs-1139	48	6	a	a	DET
iajs-1139	48	7	fuzzy	fuzzy	ADJ
iajs-1139	48	8	ideal	ideal	NOUN
iajs-1139	48	9	p	p	NOUN
iajs-1139	48	10	of	of	ADP
iajs-1139	48	11	a	a	DET
iajs-1139	48	12	fuzzy	fuzzy	ADJ
iajs-1139	48	13	ring	ring	NOUN
iajs-1139	48	14	x	x	AUX
iajs-1139	48	15	is	be	AUX
iajs-1139	48	16	called	call	VERB
iajs-1139	48	17	a	a	DET
iajs-1139	48	18	prime	prime	ADJ
iajs-1139	48	19	fuzzy	fuzzy	ADJ
iajs-1139	48	20	ideal	ideal	NOUN
iajs-1139	48	21	if	if	SCONJ
iajs-1139	48	22	p	p	PROPN
iajs-1139	48	23			NOUN
iajs-1139	48	24	r	r	PROPN
iajs-1139	48	25	(	(	PUNCT
iajs-1139	48	26	where	where	SCONJ
iajs-1139	48	27	r	r	PROPN
iajs-1139	48	28	denotes	denote	VERB
iajs-1139	48	29	the	the	DET
iajs-1139	48	30	characteristic	characteristic	ADJ
iajs-1139	48	31	function	function	NOUN
iajs-1139	48	32	of	of	ADP
iajs-1139	48	33	r	r	NOUN
iajs-1139	48	34	such	such	ADJ
iajs-1139	48	35	that	that	SCONJ
iajs-1139	48	36	r(x	r(x	PROPN
iajs-1139	48	37	)	)	PUNCT
iajs-1139	48	38	=	=	SYM
iajs-1139	48	39	1	1	NUM
iajs-1139	48	40	,	,	PUNCT
iajs-1139	48	41			NOUN
iajs-1139	48	42	x	x	SYM
iajs-1139	48	43			PROPN
iajs-1139	48	44	r	r	NOUN
iajs-1139	48	45	)	)	PUNCT
iajs-1139	48	46	and	and	CCONJ
iajs-1139	48	47	it	it	PRON
iajs-1139	48	48	satisfies	satisfy	VERB
iajs-1139	48	49	:	:	PUNCT
iajs-1139	48	50	min	min	X
iajs-1139	48	51	{	{	PUNCT
iajs-1139	48	52	p(x	p(x	PROPN
iajs-1139	48	53	y),x(x),x(y	y),x(x),x(y	NOUN
iajs-1139	48	54	)	)	PUNCT
iajs-1139	48	55	}	}	PUNCT
iajs-1139	48	56			NUM
iajs-1139	48	57	max	max	NOUN
iajs-1139	48	58	{	{	PUNCT
iajs-1139	48	59	p(x),p(y	p(x),p(y	PROPN
iajs-1139	48	60	)	)	PUNCT
iajs-1139	48	61	}	}	PUNCT
iajs-1139	48	62	for	for	ADP
iajs-1139	48	63	all	all	DET
iajs-1139	48	64	x	x	NOUN
iajs-1139	48	65	,	,	PUNCT
iajs-1139	48	66	y	y	PROPN
iajs-1139	48	67	r.	r.	PROPN
iajs-1139	48	68	ifx	ifx	PROPN
iajs-1139	49	1	=	=	PUNCT
iajs-1139	49	2	r	r	PROPN
iajs-1139	49	3	,	,	PUNCT
iajs-1139	49	4	then	then	ADV
iajs-1139	49	5	p(x	p(x	VERB
iajs-1139	49	6	y	y	PROPN
iajs-1139	49	7	)	)	PUNCT
iajs-1139	49	8			PROPN
iajs-1139	49	9	max	max	NOUN
iajs-1139	49	10	{	{	PUNCT
iajs-1139	49	11	p(x),p(y	p(x),p(y	PROPN
iajs-1139	49	12	)	)	PUNCT
iajs-1139	49	13	}	}	PUNCT
iajs-1139	49	14	for	for	ADP
iajs-1139	49	15	all	all	DET
iajs-1139	49	16	x	x	NOUN
iajs-1139	49	17	,	,	PUNCT
iajs-1139	49	18	y	y	PROPN
iajs-1139	49	19			PROPN
iajs-1139	49	20	r.	r.	PROPN
iajs-1139	49	21	1.5	1.5	NUM
iajs-1139	49	22	definition	definition	NOUN
iajs-1139	49	23	[	[	X
iajs-1139	49	24	10	10	NUM
iajs-1139	49	25	]	]	PUNCT
iajs-1139	49	26	let	let	VERB
iajs-1139	49	27	x	x	PRON
iajs-1139	49	28	and	and	CCONJ
iajs-1139	49	29	y	y	PROPN
iajs-1139	49	30	be	be	AUX
iajs-1139	49	31	fuzzy	fuzzy	ADJ
iajs-1139	49	32	rings	ring	NOUN
iajs-1139	49	33	of	of	ADP
iajs-1139	49	34	r1	r1	PROPN
iajs-1139	49	35	and	and	CCONJ
iajs-1139	49	36	r2	r2	NOUN
iajs-1139	49	37	respectively	respectively	ADV
iajs-1139	49	38	.	.	PUNCT
iajs-1139	50	1	then	then	ADV
iajs-1139	50	2	the	the	DET
iajs-1139	50	3	direct	direct	ADJ
iajs-1139	50	4	sum	sum	NOUN
iajs-1139	50	5	of	of	ADP
iajs-1139	50	6	x	x	PROPN
iajs-1139	50	7	and	and	CCONJ
iajs-1139	50	8	y	y	PROPN
iajs-1139	50	9	(	(	PUNCT
iajs-1139	50	10	denoted	denote	VERB
iajs-1139	50	11	by	by	ADP
iajs-1139	50	12	xy	xy	PROPN
iajs-1139	50	13	)	)	PUNCT
iajs-1139	50	14	is	be	AUX
iajs-1139	50	15	defined	define	VERB
iajs-1139	50	16	by	by	ADP
iajs-1139	50	17	:	:	PUNCT
iajs-1139	50	18	xy	xy	PROPN
iajs-1139	50	19	:	:	PUNCT
iajs-1139	50	20	r1r2	r1r2	PROPN
iajs-1139	50	21			X
iajs-1139	51	1	[	[	X
iajs-1139	51	2	0,1	0,1	NUM
iajs-1139	51	3	]	]	PUNCT
iajs-1139	51	4	such	such	ADJ
iajs-1139	51	5	that	that	SCONJ
iajs-1139	51	6	(	(	PUNCT
iajs-1139	51	7	xy)(a	xy)(a	PROPN
iajs-1139	51	8	,	,	PUNCT
iajs-1139	51	9	b	b	NOUN
iajs-1139	51	10	)	)	PUNCT
iajs-1139	51	11	=	=	SYM
iajs-1139	51	12	min{x(a),y(b	min{x(a),y(b	NOUN
iajs-1139	51	13	)	)	PUNCT
iajs-1139	51	14	}	}	PUNCT
iajs-1139	51	15	,	,	PUNCT
iajs-1139	51	16			NOUN
iajs-1139	51	17	(	(	PUNCT
iajs-1139	51	18	a	a	DET
iajs-1139	51	19	,	,	PUNCT
iajs-1139	51	20	b	b	NOUN
iajs-1139	51	21	)	)	PUNCT
iajs-1139	51	22			NOUN
iajs-1139	51	23	r1r2	r1r2	PROPN
iajs-1139	51	24	.	.	PUNCT
iajs-1139	52	1	if	if	SCONJ
iajs-1139	52	2	a	a	DET
iajs-1139	52	3	,	,	PUNCT
iajs-1139	52	4	b	b	NOUN
iajs-1139	52	5	are	be	AUX
iajs-1139	52	6	fuzzy	fuzzy	ADJ
iajs-1139	52	7	ideals	ideal	NOUN
iajs-1139	52	8	of	of	ADP
iajs-1139	52	9	x	x	X
iajs-1139	52	10	and	and	CCONJ
iajs-1139	52	11	y	y	PROPN
iajs-1139	52	12	respectively	respectively	ADV
iajs-1139	52	13	,	,	PUNCT
iajs-1139	52	14	then	then	ADV
iajs-1139	52	15	a	a	DET
iajs-1139	52	16			ADJ
iajs-1139	52	17	b	b	NOUN
iajs-1139	52	18	:	:	PUNCT
iajs-1139	52	19	r1r2	r1r2	PROPN
iajs-1139	52	20			X
iajs-1139	53	1	[	[	X
iajs-1139	53	2	0,1	0,1	NUM
iajs-1139	53	3	]	]	PUNCT
iajs-1139	53	4	defined	define	VERB
iajs-1139	53	5	by	by	ADP
iajs-1139	53	6	:	:	PUNCT
iajs-1139	53	7	(	(	PUNCT
iajs-1139	53	8	a	a	DET
iajs-1139	53	9			ADJ
iajs-1139	53	10	b)(a	b)(a	NOUN
iajs-1139	53	11	,	,	PUNCT
iajs-1139	53	12	b	b	NOUN
iajs-1139	53	13	)	)	PUNCT
iajs-1139	53	14	=	=	SYM
iajs-1139	53	15	min{a(a),b(b	min{a(a),b(b	PROPN
iajs-1139	53	16	)	)	PUNCT
iajs-1139	53	17	}	}	PUNCT
iajs-1139	53	18	,	,	PUNCT
iajs-1139	53	19			NOUN
iajs-1139	53	20	(	(	PUNCT
iajs-1139	53	21	a	a	DET
iajs-1139	53	22	,	,	PUNCT
iajs-1139	53	23	b	b	NOUN
iajs-1139	53	24	)	)	PUNCT
iajs-1139	53	25			NOUN
iajs-1139	53	26	r1r2	r1r2	PROPN
iajs-1139	53	27	.	.	PUNCT
iajs-1139	54	1	1.6	1.6	NUM
iajs-1139	54	2	proposition	proposition	NOUN
iajs-1139	54	3	[	[	X
iajs-1139	54	4	9	9	NUM
iajs-1139	54	5	]	]	PUNCT
iajs-1139	54	6	let	let	VERB
iajs-1139	54	7	x	x	PRON
iajs-1139	54	8	,	,	PUNCT
iajs-1139	54	9	y	y	PROPN
iajs-1139	54	10	be	be	VERB
iajs-1139	54	11	two	two	NUM
iajs-1139	54	12	fuzzy	fuzzy	ADJ
iajs-1139	54	13	rings	ring	NOUN
iajs-1139	54	14	of	of	ADP
iajs-1139	54	15	r	r	NOUN
iajs-1139	54	16	and	and	CCONJ
iajs-1139	54	17	r	r	NOUN
iajs-1139	54	18	respectively	respectively	ADV
iajs-1139	54	19	and	and	CCONJ
iajs-1139	54	20	f	f	NOUN
iajs-1139	54	21	:	:	PUNCT
iajs-1139	54	22	rr	rr	NOUN
iajs-1139	54	23	is	be	AUX
iajs-1139	54	24	a	a	DET
iajs-1139	54	25	homomorphism	homomorphism	NOUN
iajs-1139	54	26	,	,	PUNCT
iajs-1139	54	27	then	then	ADV
iajs-1139	54	28	if	if	SCONJ
iajs-1139	54	29	a	a	PRON
iajs-1139	54	30	is	be	AUX
iajs-1139	54	31	a	a	DET
iajs-1139	54	32	prime	prime	ADJ
iajs-1139	54	33	fuzzy	fuzzy	ADJ
iajs-1139	54	34	ideal	ideal	NOUN
iajs-1139	54	35	of	of	ADP
iajs-1139	54	36	x	x	X
iajs-1139	54	37	and	and	CCONJ
iajs-1139	54	38	a	a	PRON
iajs-1139	54	39	is	be	AUX
iajs-1139	54	40	f	f	NOUN
iajs-1139	54	41	-	-	PUNCT
iajs-1139	54	42	invariant	invariant	ADJ
iajs-1139	54	43	,	,	PUNCT
iajs-1139	54	44	then	then	ADV
iajs-1139	54	45	f	f	X
iajs-1139	54	46	(	(	PUNCT
iajs-1139	54	47	a	a	PRON
iajs-1139	54	48	)	)	PUNCT
iajs-1139	54	49	is	be	AUX
iajs-1139	54	50	a	a	DET
iajs-1139	54	51	prime	prime	ADJ
iajs-1139	54	52	fuzzy	fuzzy	ADJ
iajs-1139	54	53	ideal	ideal	NOUN
iajs-1139	54	54	of	of	ADP
iajs-1139	54	55	y.	y.	PROPN
iajs-1139	54	56	1.7	1.7	NUM
iajs-1139	54	57	definition	definition	NOUN
iajs-1139	54	58	[	[	X
iajs-1139	54	59	10	10	NUM
iajs-1139	54	60	]	]	PUNCT
iajs-1139	54	61	let	let	VERB
iajs-1139	54	62	a	a	PRON
iajs-1139	54	63	be	be	AUX
iajs-1139	54	64	a	a	DET
iajs-1139	54	65	fuzzy	fuzzy	ADJ
iajs-1139	54	66	ideal	ideal	NOUN
iajs-1139	54	67	of	of	ADP
iajs-1139	54	68	a	a	DET
iajs-1139	54	69	fuzzy	fuzzy	ADJ
iajs-1139	54	70	ring	ring	NOUN
iajs-1139	54	71	x	x	INTJ
iajs-1139	54	72	of	of	ADP
iajs-1139	54	73	a	a	DET
iajs-1139	54	74	ring	ring	NOUN
iajs-1139	54	75	r	r	NOUN
iajs-1139	54	76	,	,	PUNCT
iajs-1139	54	77	a	a	PRON
iajs-1139	54	78	is	be	AUX
iajs-1139	54	79	called	call	VERB
iajs-1139	54	80	an	an	DET
iajs-1139	54	81	essential	essential	ADJ
iajs-1139	54	82	fuzzy	fuzzy	ADJ
iajs-1139	54	83	ideal	ideal	NOUN
iajs-1139	54	84	if	if	SCONJ
iajs-1139	54	85	a	a	DET
iajs-1139	54	86			PUNCT
iajs-1139	54	87	k	k	PROPN
iajs-1139	54	88			NOUN
iajs-1139	54	89	o1	o1	NOUN
iajs-1139	54	90	for	for	ADP
iajs-1139	54	91	each	each	DET
iajs-1139	54	92	fuzzy	fuzzy	ADJ
iajs-1139	54	93	ideal	ideal	NOUN
iajs-1139	54	94	k	k	PROPN
iajs-1139	54	95	of	of	ADP
iajs-1139	54	96	x	x	PROPN
iajs-1139	54	97	,	,	PUNCT
iajs-1139	54	98	k	k	PROPN
iajs-1139	54	99			PROPN
iajs-1139	54	100	o1	o1	PROPN
iajs-1139	54	101	.	.	PUNCT
iajs-1139	55	1	ibn	ibn	PROPN
iajs-1139	55	2	alhaitham	alhaitham	PROPN
iajs-1139	55	3	j.	j.	PROPN
iajs-1139	55	4	for	for	ADP
iajs-1139	55	5	pure	pure	ADJ
iajs-1139	55	6	&	&	CCONJ
iajs-1139	55	7	appl	appl	PROPN
iajs-1139	55	8	.	.	PUNCT
iajs-1139	56	1	sci	sci	PROPN
iajs-1139	56	2	.	.	PUNCT
iajs-1139	57	1	vol.22	vol.22	PROPN
iajs-1139	57	2	(	(	PUNCT
iajs-1139	57	3	4	4	NUM
iajs-1139	57	4	)	)	PUNCT
iajs-1139	57	5	2009	2009	NUM
iajs-1139	57	6	s.2	s.2	X
iajs-1139	57	7	semiessential	semiessential	ADJ
iajs-1139	57	8	fuzzy	fuzzy	ADJ
iajs-1139	57	9	ideals	ideal	NOUN
iajs-1139	57	10	in	in	ADP
iajs-1139	57	11	this	this	DET
iajs-1139	57	12	section	section	NOUN
iajs-1139	57	13	,	,	PUNCT
iajs-1139	57	14	we	we	PRON
iajs-1139	57	15	introduce	introduce	VERB
iajs-1139	57	16	the	the	DET
iajs-1139	57	17	notion	notion	NOUN
iajs-1139	57	18	of	of	ADP
iajs-1139	57	19	semiessential	semiessential	ADJ
iajs-1139	57	20	fuzzy	fuzzy	ADJ
iajs-1139	57	21	ideals	ideal	NOUN
iajs-1139	57	22	of	of	ADP
iajs-1139	57	23	fuzzy	fuzzy	ADJ
iajs-1139	57	24	ring	ring	NOUN
iajs-1139	57	25	as	as	ADP
iajs-1139	57	26	a	a	DET
iajs-1139	57	27	generalization	generalization	NOUN
iajs-1139	57	28	of	of	ADP
iajs-1139	57	29	(	(	PUNCT
iajs-1139	57	30	ordinary	ordinary	ADJ
iajs-1139	57	31	)	)	PUNCT
iajs-1139	57	32	notion	notion	NOUN
iajs-1139	57	33	semiessential	semiessential	ADJ
iajs-1139	57	34	ideals	ideal	NOUN
iajs-1139	57	35	of	of	ADP
iajs-1139	57	36	a	a	DET
iajs-1139	57	37	ring	ring	NOUN
iajs-1139	57	38	.	.	PUNCT
iajs-1139	58	1	we	we	PRON
iajs-1139	58	2	shall	shall	AUX
iajs-1139	58	3	give	give	VERB
iajs-1139	58	4	many	many	ADJ
iajs-1139	58	5	properties	property	NOUN
iajs-1139	58	6	of	of	ADP
iajs-1139	58	7	this	this	DET
iajs-1139	58	8	concept	concept	NOUN
iajs-1139	58	9	.	.	PUNCT
iajs-1139	59	1	2.1	2.1	NUM
iajs-1139	59	2	definition	definition	NOUN
iajs-1139	59	3	let	let	VERB
iajs-1139	59	4	x	x	PRON
iajs-1139	59	5	be	be	AUX
iajs-1139	59	6	a	a	DET
iajs-1139	59	7	fuzzy	fuzzy	ADJ
iajs-1139	59	8	ring	ring	NOUN
iajs-1139	59	9	of	of	ADP
iajs-1139	59	10	a	a	DET
iajs-1139	59	11	ring	ring	NOUN
iajs-1139	59	12	r.	r.	PROPN
iajs-1139	59	13	let	let	VERB
iajs-1139	59	14	a	a	PRON
iajs-1139	59	15	be	be	AUX
iajs-1139	59	16	a	a	DET
iajs-1139	59	17	fuzzy	fuzzy	ADJ
iajs-1139	59	18	ideal	ideal	NOUN
iajs-1139	59	19	of	of	ADP
iajs-1139	59	20	x	x	SYM
iajs-1139	59	21	such	such	ADJ
iajs-1139	59	22	that	that	SCONJ
iajs-1139	59	23	a	a	DET
iajs-1139	59	24	ox(0)=o1	ox(0)=o1	NOUN
iajs-1139	59	25	.	.	PUNCT
iajs-1139	60	1	a	a	PRON
iajs-1139	60	2	is	be	AUX
iajs-1139	60	3	called	call	VERB
iajs-1139	60	4	a	a	DET
iajs-1139	60	5	semiessential	semiessential	ADJ
iajs-1139	60	6	fuzzy	fuzzy	ADJ
iajs-1139	60	7	ideal	ideal	NOUN
iajs-1139	60	8	of	of	ADP
iajs-1139	60	9	x	x	PRON
iajs-1139	60	10	if	if	SCONJ
iajs-1139	60	11	abo1	abo1	NOUN
iajs-1139	60	12	,	,	PUNCT
iajs-1139	60	13	for	for	ADP
iajs-1139	60	14	any	any	DET
iajs-1139	60	15	prime	prime	ADJ
iajs-1139	60	16	fuzzy	fuzzy	ADJ
iajs-1139	60	17	ideal	ideal	NOUN
iajs-1139	60	18	b	b	PROPN
iajs-1139	60	19	of	of	ADP
iajs-1139	60	20	x.	x.	NOUN
iajs-1139	60	21	2.2	2.2	NUM
iajs-1139	60	22	remark	remark	NOUN
iajs-1139	60	23	let	let	VERB
iajs-1139	60	24	x	x	PRON
iajs-1139	60	25	be	be	AUX
iajs-1139	60	26	a	a	DET
iajs-1139	60	27	fuzzy	fuzzy	ADJ
iajs-1139	60	28	ring	ring	NOUN
iajs-1139	60	29	of	of	ADP
iajs-1139	60	30	a	a	DET
iajs-1139	60	31	ring	ring	NOUN
iajs-1139	60	32	r.	r.	NOUN
iajs-1139	60	33	it	it	PRON
iajs-1139	60	34	is	be	AUX
iajs-1139	60	35	clear	clear	ADJ
iajs-1139	60	36	that	that	SCONJ
iajs-1139	60	37	if	if	SCONJ
iajs-1139	60	38	a	a	PRON
iajs-1139	60	39	is	be	AUX
iajs-1139	60	40	an	an	DET
iajs-1139	60	41	essential	essential	ADJ
iajs-1139	60	42	fuzzy	fuzzy	ADJ
iajs-1139	60	43	ideal	ideal	NOUN
iajs-1139	60	44	of	of	ADP
iajs-1139	60	45	x	x	PRON
iajs-1139	60	46	,	,	PUNCT
iajs-1139	60	47	then	then	ADV
iajs-1139	60	48	a	a	PRON
iajs-1139	60	49	is	be	AUX
iajs-1139	60	50	a	a	DET
iajs-1139	60	51	semiessential	semiessential	ADJ
iajs-1139	60	52	fuzzy	fuzzy	ADJ
iajs-1139	60	53	ideal	ideal	NOUN
iajs-1139	60	54	of	of	ADP
iajs-1139	60	55	x.	x.	NOUN
iajs-1139	60	56	proof	proof	NOUN
iajs-1139	60	57	:	:	PUNCT
iajs-1139	60	58	it	it	PRON
iajs-1139	60	59	is	be	AUX
iajs-1139	60	60	easy	easy	ADJ
iajs-1139	60	61	,	,	PUNCT
iajs-1139	60	62	so	so	CCONJ
iajs-1139	60	63	it	it	PRON
iajs-1139	60	64	is	be	AUX
iajs-1139	60	65	omitted	omit	VERB
iajs-1139	60	66	.	.	PUNCT
iajs-1139	61	1	the	the	DET
iajs-1139	61	2	converse	converse	NOUN
iajs-1139	61	3	of	of	ADP
iajs-1139	61	4	remark	remark	NOUN
iajs-1139	61	5	2.2	2.2	NUM
iajs-1139	61	6	is	be	AUX
iajs-1139	61	7	not	not	PART
iajs-1139	61	8	true	true	ADJ
iajs-1139	61	9	in	in	ADP
iajs-1139	61	10	general	general	ADJ
iajs-1139	61	11	.	.	PUNCT
iajs-1139	62	1	however	however	ADV
iajs-1139	62	2	an	an	DET
iajs-1139	62	3	example	example	NOUN
iajs-1139	62	4	which	which	PRON
iajs-1139	62	5	will	will	AUX
iajs-1139	62	6	explain	explain	VERB
iajs-1139	62	7	this	this	DET
iajs-1139	62	8	dependens	dependen	NOUN
iajs-1139	62	9	on	on	ADP
iajs-1139	62	10	theorem	theorem	ADJ
iajs-1139	62	11	2.3	2.3	NUM
iajs-1139	62	12	,	,	PUNCT
iajs-1139	62	13	so	so	SCONJ
iajs-1139	62	14	we	we	PRON
iajs-1139	62	15	shall	shall	AUX
iajs-1139	62	16	give	give	VERB
iajs-1139	62	17	it	it	PRON
iajs-1139	62	18	later	later	ADV
iajs-1139	62	19	(	(	PUNCT
iajs-1139	62	20	see	see	VERB
iajs-1139	62	21	remark	remark	NOUN
iajs-1139	62	22	2.5	2.5	NUM
iajs-1139	62	23	)	)	PUNCT
iajs-1139	62	24	.	.	PUNCT
iajs-1139	63	1	2.3	2.3	NUM
iajs-1139	63	2	theorem	theorem	NOUN
iajs-1139	63	3	let	let	VERB
iajs-1139	63	4	x	x	PRON
iajs-1139	63	5	be	be	AUX
iajs-1139	63	6	a	a	DET
iajs-1139	63	7	fuzzy	fuzzy	ADJ
iajs-1139	63	8	ring	ring	NOUN
iajs-1139	63	9	of	of	ADP
iajs-1139	63	10	r	r	NOUN
iajs-1139	63	11	,	,	PUNCT
iajs-1139	63	12	let	let	VERB
iajs-1139	63	13	a	a	PRON
iajs-1139	63	14	be	be	AUX
iajs-1139	63	15	a	a	DET
iajs-1139	63	16	fuzzy	fuzzy	ADJ
iajs-1139	63	17	ideal	ideal	NOUN
iajs-1139	63	18	of	of	ADP
iajs-1139	63	19	x	x	PRON
iajs-1139	63	20	,	,	PUNCT
iajs-1139	63	21	if	if	SCONJ
iajs-1139	63	22	at	at	ADP
iajs-1139	63	23	is	be	AUX
iajs-1139	63	24	a	a	DET
iajs-1139	63	25	semiessential	semiessential	ADJ
iajs-1139	63	26	ideal	ideal	NOUN
iajs-1139	63	27	of	of	ADP
iajs-1139	63	28	xt	xt	PROPN
iajs-1139	63	29	,	,	PUNCT
iajs-1139	63	30			PROPN
iajs-1139	63	31	t	t	PROPN
iajs-1139	63	32			PROPN
iajs-1139	63	33	(	(	PUNCT
iajs-1139	63	34	0,1	0,1	NOUN
iajs-1139	63	35	]	]	PUNCT
iajs-1139	63	36	.	.	PUNCT
iajs-1139	64	1	then	then	ADV
iajs-1139	64	2	a	a	PRON
iajs-1139	64	3	is	be	AUX
iajs-1139	64	4	a	a	DET
iajs-1139	64	5	semiessential	semiessential	ADJ
iajs-1139	64	6	fuzzy	fuzzy	ADJ
iajs-1139	64	7	ideal	ideal	NOUN
iajs-1139	64	8	of	of	ADP
iajs-1139	64	9	x.	x.	NOUN
iajs-1139	64	10	proof	proof	NOUN
iajs-1139	64	11	:	:	PUNCT
iajs-1139	64	12	let	let	VERB
iajs-1139	64	13	b	b	X
iajs-1139	64	14	be	be	AUX
iajs-1139	64	15	a	a	DET
iajs-1139	64	16	prime	prime	ADJ
iajs-1139	64	17	fuzzy	fuzzy	ADJ
iajs-1139	64	18	ideal	ideal	NOUN
iajs-1139	64	19	of	of	ADP
iajs-1139	64	20	x	x	SYM
iajs-1139	64	21	such	such	ADJ
iajs-1139	64	22	that	that	DET
iajs-1139	64	23	bo1	bo1	NOUN
iajs-1139	64	24	.	.	PUNCT
iajs-1139	65	1	to	to	PART
iajs-1139	65	2	prove	prove	VERB
iajs-1139	65	3	a	a	DET
iajs-1139	65	4			NOUN
iajs-1139	65	5	b	b	PROPN
iajs-1139	65	6	o1	o1	PROPN
iajs-1139	65	7	,	,	PUNCT
iajs-1139	65	8	since	since	SCONJ
iajs-1139	65	9	b	b	NOUN
iajs-1139	65	10	is	be	AUX
iajs-1139	65	11	a	a	DET
iajs-1139	65	12	prime	prime	ADJ
iajs-1139	65	13	fuzzy	fuzzy	ADJ
iajs-1139	65	14	ideal	ideal	NOUN
iajs-1139	65	15	.	.	PUNCT
iajs-1139	66	1	hence	hence	ADV
iajs-1139	66	2	bt	bt	PROPN
iajs-1139	66	3	is	be	AUX
iajs-1139	66	4	a	a	DET
iajs-1139	66	5	prime	prime	ADJ
iajs-1139	66	6	ideal	ideal	NOUN
iajs-1139	66	7	of	of	ADP
iajs-1139	66	8	xt	xt	PROPN
iajs-1139	66	9	,	,	PUNCT
iajs-1139	66	10			PROPN
iajs-1139	66	11	t	t	PROPN
iajs-1139	66	12			PROPN
iajs-1139	66	13	(	(	PUNCT
iajs-1139	66	14	0,x(0	0,x(0	NUM
iajs-1139	66	15	)	)	PUNCT
iajs-1139	66	16	]	]	PUNCT
iajs-1139	66	17	by	by	ADP
iajs-1139	66	18	[	[	X
iajs-1139	66	19	11,proposition	11,proposition	NUM
iajs-1139	66	20	1.2.9	1.2.9	NUM
iajs-1139	66	21	]	]	PUNCT
iajs-1139	66	22	.	.	PUNCT
iajs-1139	66	23	which	which	PRON
iajs-1139	66	24	implies	imply	VERB
iajs-1139	66	25	at	at	ADP
iajs-1139	66	26			NOUN
iajs-1139	66	27	bt	bt	PROPN
iajs-1139	66	28			PROPN
iajs-1139	66	29	(	(	PUNCT
iajs-1139	66	30	0	0	NUM
iajs-1139	66	31	)	)	PUNCT
iajs-1139	66	32	and	and	CCONJ
iajs-1139	66	33	at	at	ADP
iajs-1139	66	34			NOUN
iajs-1139	66	35	bt	bt	X
iajs-1139	67	1	=	=	SYM
iajs-1139	67	2	(	(	PUNCT
iajs-1139	67	3	a	a	DET
iajs-1139	67	4			NOUN
iajs-1139	67	5	b)t	b)t	NOUN
iajs-1139	67	6			NOUN
iajs-1139	67	7	(	(	PUNCT
iajs-1139	67	8	0	0	NUM
iajs-1139	67	9	)	)	PUNCT
iajs-1139	67	10	.	.	PUNCT
iajs-1139	68	1	hence	hence	ADV
iajs-1139	68	2	a	a	DET
iajs-1139	68	3			PROPN
iajs-1139	68	4	b	b	PROPN
iajs-1139	68	5	o1	o1	PROPN
iajs-1139	68	6	.	.	PUNCT
iajs-1139	69	1	thus	thus	ADV
iajs-1139	69	2	a	a	PRON
iajs-1139	69	3	is	be	AUX
iajs-1139	69	4	a	a	DET
iajs-1139	69	5	semiessential	semiessential	ADJ
iajs-1139	69	6	fuzzy	fuzzy	ADJ
iajs-1139	69	7	ideal	ideal	NOUN
iajs-1139	69	8	of	of	ADP
iajs-1139	69	9	x.	x.	NOUN
iajs-1139	69	10	the	the	DET
iajs-1139	69	11	following	follow	VERB
iajs-1139	69	12	remark	remark	NOUN
iajs-1139	69	13	shows	show	VERB
iajs-1139	69	14	that	that	SCONJ
iajs-1139	69	15	the	the	DET
iajs-1139	69	16	converse	converse	NOUN
iajs-1139	69	17	of	of	ADP
iajs-1139	69	18	this	this	DET
iajs-1139	69	19	theorem	theorem	NOUN
iajs-1139	69	20	is	be	AUX
iajs-1139	69	21	not	not	PART
iajs-1139	69	22	true	true	ADJ
iajs-1139	69	23	in	in	ADP
iajs-1139	69	24	general	general	ADJ
iajs-1139	69	25	.	.	PUNCT
iajs-1139	70	1	2.4	2.4	NUM
iajs-1139	70	2	remark	remark	NOUN
iajs-1139	70	3	if	if	SCONJ
iajs-1139	70	4	x	x	PRON
iajs-1139	70	5	is	be	AUX
iajs-1139	70	6	a	a	DET
iajs-1139	70	7	fuzzy	fuzzy	ADJ
iajs-1139	70	8	ring	ring	NOUN
iajs-1139	70	9	of	of	ADP
iajs-1139	70	10	a	a	DET
iajs-1139	70	11	ring	ring	NOUN
iajs-1139	70	12	r	r	NOUN
iajs-1139	70	13	,	,	PUNCT
iajs-1139	70	14	a	a	PRON
iajs-1139	70	15	is	be	AUX
iajs-1139	70	16	a	a	DET
iajs-1139	70	17	semiessential	semiessential	ADJ
iajs-1139	70	18	fuzzy	fuzzy	ADJ
iajs-1139	70	19	ideal	ideal	NOUN
iajs-1139	70	20	of	of	ADP
iajs-1139	70	21	x	x	PRON
iajs-1139	70	22	,	,	PUNCT
iajs-1139	70	23	then	then	ADV
iajs-1139	70	24	it	it	PRON
iajs-1139	70	25	is	be	AUX
iajs-1139	70	26	not	not	PART
iajs-1139	70	27	necessarily	necessarily	ADV
iajs-1139	70	28	that	that	PRON
iajs-1139	70	29	a	a	DET
iajs-1139	70	30	t	t	NOUN
iajs-1139	70	31	is	be	AUX
iajs-1139	70	32	a	a	DET
iajs-1139	70	33	semiessential	semiessential	ADJ
iajs-1139	70	34	ideal	ideal	NOUN
iajs-1139	70	35	of	of	ADP
iajs-1139	70	36	xt	xt	PROPN
iajs-1139	70	37	,	,	PUNCT
iajs-1139	70	38			PROPN
iajs-1139	70	39	t	t	PROPN
iajs-1139	70	40			NOUN
iajs-1139	70	41	[	[	X
iajs-1139	70	42	0,1	0,1	NUM
iajs-1139	70	43	]	]	PUNCT
iajs-1139	70	44	.	.	PUNCT
iajs-1139	71	1	as	as	SCONJ
iajs-1139	71	2	the	the	DET
iajs-1139	71	3	following	follow	VERB
iajs-1139	71	4	example	example	NOUN
iajs-1139	71	5	shows	show	VERB
iajs-1139	71	6	:	:	PUNCT
iajs-1139	71	7	example	example	NOUN
iajs-1139	71	8	:	:	PUNCT
iajs-1139	71	9	let	let	VERB
iajs-1139	71	10	r	r	NOUN
iajs-1139	71	11	=	=	SYM
iajs-1139	71	12	z6	z6	PROPN
iajs-1139	71	13	,	,	PUNCT
iajs-1139	71	14	define	define	VERB
iajs-1139	71	15	x	x	NOUN
iajs-1139	71	16	:	:	PUNCT
iajs-1139	71	17	z6	z6	PROPN
iajs-1139	71	18			PROPN
iajs-1139	72	1	[	[	X
iajs-1139	72	2	0,1	0,1	NUM
iajs-1139	72	3	]	]	PUNCT
iajs-1139	72	4	,	,	PUNCT
iajs-1139	72	5	a	a	DET
iajs-1139	72	6	:	:	PUNCT
iajs-1139	72	7	z6	z6	PROPN
iajs-1139	72	8			PROPN
iajs-1139	73	1	[	[	X
iajs-1139	73	2	0,1	0,1	NUM
iajs-1139	73	3	]	]	PUNCT
iajs-1139	73	4	by	by	ADP
iajs-1139	73	5	:	:	PUNCT
iajs-1139	73	6	1	1	NUM
iajs-1139	73	7	if	if	SCONJ
iajs-1139	73	8	0	0	NUM
iajs-1139	73	9	1	1	NUM
iajs-1139	73	10	x	x	X
iajs-1139	73	11	(	(	PUNCT
iajs-1139	73	12	)	)	PUNCT
iajs-1139	73	13	if	if	SCONJ
iajs-1139	73	14	2,4	2,4	NUM
iajs-1139	73	15	2	2	NUM
iajs-1139	73	16	0	0	NUM
iajs-1139	73	17	otherwise	otherwise	ADV
iajs-1139	73	18	.	.	PUNCT
iajs-1139	74	1			VERB
iajs-1139	74	2			NUM
iajs-1139	75	1			NUM
iajs-1139	75	2			PROPN
iajs-1139	75	3			NUM
iajs-1139	75	4			PROPN
iajs-1139	75	5	a	a	PROPN
iajs-1139	75	6	a	a	DET
iajs-1139	75	7	a	a	DET
iajs-1139	75	8	1	1	NUM
iajs-1139	75	9	if	if	SCONJ
iajs-1139	75	10	0	0	NUM
iajs-1139	75	11	1	1	NUM
iajs-1139	75	12	(	(	PUNCT
iajs-1139	75	13	)	)	PUNCT
iajs-1139	75	14	if	if	SCONJ
iajs-1139	75	15	2,4	2,4	NUM
iajs-1139	75	16	3	3	NUM
iajs-1139	75	17	0	0	NUM
iajs-1139	75	18	otherwise	otherwise	ADV
iajs-1139	75	19	.	.	PUNCT
iajs-1139	76	1			VERB
iajs-1139	76	2			NOUN
iajs-1139	77	1			PROPN
iajs-1139	77	2			PROPN
iajs-1139	77	3			PROPN
iajs-1139	77	4			NUM
iajs-1139	77	5			PROPN
iajs-1139	77	6	a	a	PROPN
iajs-1139	77	7	a	a	PROPN
iajs-1139	77	8	a	a	PRON
iajs-1139	77	9	it	it	PRON
iajs-1139	77	10	is	be	AUX
iajs-1139	77	11	clear	clear	ADJ
iajs-1139	77	12	x	x	PRON
iajs-1139	77	13	is	be	AUX
iajs-1139	77	14	a	a	DET
iajs-1139	77	15	fuzzy	fuzzy	ADJ
iajs-1139	77	16	ring	ring	NOUN
iajs-1139	77	17	of	of	ADP
iajs-1139	77	18	z6	z6	PROPN
iajs-1139	77	19	,	,	PUNCT
iajs-1139	77	20	a	a	PRON
iajs-1139	77	21	is	be	AUX
iajs-1139	77	22	a	a	DET
iajs-1139	77	23	fuzzy	fuzzy	ADJ
iajs-1139	77	24	ideal	ideal	NOUN
iajs-1139	77	25	of	of	ADP
iajs-1139	77	26	x	x	X
iajs-1139	77	27	and	and	CCONJ
iajs-1139	77	28	a	a	DET
iajs-1139	77	29			NOUN
iajs-1139	77	30	o1	o1	NOUN
iajs-1139	77	31	.	.	PUNCT
iajs-1139	78	1	a	a	PRON
iajs-1139	78	2	is	be	AUX
iajs-1139	78	3	an	an	DET
iajs-1139	78	4	essential	essential	ADJ
iajs-1139	78	5	fuzzy	fuzzy	ADJ
iajs-1139	78	6	ideal	ideal	NOUN
iajs-1139	78	7	of	of	ADP
iajs-1139	78	8	x	x	PUNCT
iajs-1139	78	9	see	see	VERB
iajs-1139	78	10	(	(	PUNCT
iajs-1139	78	11	5	5	NUM
iajs-1139	78	12	,	,	PUNCT
iajs-1139	78	13	remark	remark	NOUN
iajs-1139	78	14	2.3	2.3	NUM
iajs-1139	78	15	)	)	PUNCT
iajs-1139	78	16	.	.	PUNCT
iajs-1139	79	1	hence	hence	ADV
iajs-1139	79	2	a	a	PRON
iajs-1139	79	3	is	be	AUX
iajs-1139	79	4	semiessential	semiessential	ADJ
iajs-1139	79	5	in	in	ADP
iajs-1139	79	6	x.	x.	NOUN
iajs-1139	79	7	on	on	ADP
iajs-1139	79	8	the	the	DET
iajs-1139	79	9	other	other	ADJ
iajs-1139	79	10	hand	hand	NOUN
iajs-1139	79	11	,	,	PUNCT
iajs-1139	79	12	1	1	NUM
iajs-1139	79	13	2	2	NUM
iajs-1139	79	14			NOUN
iajs-1139	79	15	=	=	PUNCT
iajs-1139	79	16	{	{	PUNCT
iajs-1139	79	17	0	0	NUM
iajs-1139	79	18	}	}	PUNCT
iajs-1139	79	19	,	,	PUNCT
iajs-1139	79	20	1	1	NUM
iajs-1139	79	21	2	2	NUM
iajs-1139	79	22	x	x	X
iajs-1139	79	23	=	=	PRON
iajs-1139	79	24	{	{	PUNCT
iajs-1139	79	25	0,2,4	0,2,4	NOUN
iajs-1139	79	26	}	}	PUNCT
iajs-1139	79	27	.	.	PUNCT
iajs-1139	80	1	hence	hence	ADV
iajs-1139	80	2	a	a	DET
iajs-1139	80	3	1	1	NUM
iajs-1139	80	4	2	2	NUM
iajs-1139	80	5	is	be	AUX
iajs-1139	80	6	not	not	PART
iajs-1139	80	7	semiessential	semiessential	ADJ
iajs-1139	80	8	ideal	ideal	NOUN
iajs-1139	80	9	in	in	ADP
iajs-1139	80	10	x	x	SYM
iajs-1139	80	11	1	1	NUM
iajs-1139	80	12	2	2	NUM
iajs-1139	80	13	.	.	PUNCT
iajs-1139	81	1	2.5	2.5	NUM
iajs-1139	81	2	remark	remark	NOUN
iajs-1139	81	3	if	if	SCONJ
iajs-1139	81	4	x	x	PRON
iajs-1139	81	5	is	be	AUX
iajs-1139	81	6	a	a	DET
iajs-1139	81	7	fuzzy	fuzzy	ADJ
iajs-1139	81	8	ring	ring	NOUN
iajs-1139	81	9	of	of	ADP
iajs-1139	81	10	a	a	DET
iajs-1139	81	11	ring	ring	NOUN
iajs-1139	81	12	r	r	NOUN
iajs-1139	81	13	,	,	PUNCT
iajs-1139	81	14	a	a	PRON
iajs-1139	81	15	is	be	AUX
iajs-1139	81	16	a	a	DET
iajs-1139	81	17	semiessential	semiessential	ADJ
iajs-1139	81	18	fuzzy	fuzzy	ADJ
iajs-1139	81	19	ideal	ideal	NOUN
iajs-1139	81	20	of	of	ADP
iajs-1139	81	21	x	x	PRON
iajs-1139	81	22	,	,	PUNCT
iajs-1139	81	23	then	then	ADV
iajs-1139	81	24	it	it	PRON
iajs-1139	81	25	is	be	AUX
iajs-1139	81	26	not	not	PART
iajs-1139	81	27	necessarily	necessarily	ADV
iajs-1139	81	28	that	that	SCONJ
iajs-1139	81	29	a	a	PRON
iajs-1139	81	30	is	be	AUX
iajs-1139	81	31	an	an	DET
iajs-1139	81	32	essential	essential	ADJ
iajs-1139	81	33	fuzzy	fuzzy	ADJ
iajs-1139	81	34	ideal	ideal	NOUN
iajs-1139	81	35	of	of	ADP
iajs-1139	81	36	x	x	PRON
iajs-1139	81	37	,	,	PUNCT
iajs-1139	81	38	as	as	SCONJ
iajs-1139	81	39	the	the	DET
iajs-1139	81	40	following	follow	VERB
iajs-1139	81	41	example	example	NOUN
iajs-1139	81	42	shows	show	VERB
iajs-1139	81	43	:	:	PUNCT
iajs-1139	81	44	example	example	NOUN
iajs-1139	81	45	:	:	PUNCT
iajs-1139	81	46	let	let	VERB
iajs-1139	81	47	r	r	NOUN
iajs-1139	81	48	=	=	SYM
iajs-1139	81	49	z12	z12	PROPN
iajs-1139	81	50	,	,	PUNCT
iajs-1139	81	51	define	define	VERB
iajs-1139	81	52	x	x	NOUN
iajs-1139	81	53	:	:	PUNCT
iajs-1139	81	54	z12	z12	NUM
iajs-1139	81	55			X
iajs-1139	82	1	[	[	X
iajs-1139	82	2	0,1	0,1	NUM
iajs-1139	82	3	]	]	PUNCT
iajs-1139	82	4	by	by	ADP
iajs-1139	82	5	x(a	x(a	NOUN
iajs-1139	82	6	)	)	PUNCT
iajs-1139	82	7	=	=	SYM
iajs-1139	82	8	1	1	NUM
iajs-1139	82	9	,	,	PUNCT
iajs-1139	82	10			VERB
iajs-1139	82	11	a	a	DET
iajs-1139	82	12			PROPN
iajs-1139	82	13	z12	z12	NUM
iajs-1139	82	14	,	,	PUNCT
iajs-1139	82	15	let	let	VERB
iajs-1139	82	16	a	a	PRON
iajs-1139	82	17	:	:	PUNCT
iajs-1139	82	18	z12	z12	NUM
iajs-1139	82	19			PROPN
iajs-1139	83	1	[	[	X
iajs-1139	83	2	0,1	0,1	NUM
iajs-1139	83	3	]	]	PUNCT
iajs-1139	83	4	define	define	NOUN
iajs-1139	83	5	by	by	ADP
iajs-1139	83	6	:	:	PUNCT
iajs-1139	83	7	ibn	ibn	PROPN
iajs-1139	83	8	alhaitham	alhaitham	NOUN
iajs-1139	83	9	j.	j.	PROPN
iajs-1139	83	10	for	for	ADP
iajs-1139	83	11	pure	pure	ADJ
iajs-1139	83	12	&	&	CCONJ
iajs-1139	83	13	appl	appl	PROPN
iajs-1139	83	14	.	.	PUNCT
iajs-1139	84	1	sci	sci	PROPN
iajs-1139	84	2	.	.	PUNCT
iajs-1139	85	1	vol.22	vol.22	PROPN
iajs-1139	85	2	(	(	PUNCT
iajs-1139	85	3	4	4	NUM
iajs-1139	85	4	)	)	PUNCT
iajs-1139	85	5	2009	2009	NUM
iajs-1139	85	6	1	1	NUM
iajs-1139	85	7	if	if	SCONJ
iajs-1139	85	8	(	(	PUNCT
iajs-1139	85	9	6	6	NUM
iajs-1139	85	10	)	)	PUNCT
iajs-1139	85	11	(	(	PUNCT
iajs-1139	85	12	)	)	PUNCT
iajs-1139	85	13	0	0	PUNCT
iajs-1139	85	14	otherwise	otherwise	ADV
iajs-1139	85	15	.	.	PUNCT
iajs-1139	86	1			NOUN
iajs-1139	87	1			NOUN
iajs-1139	88	1			PROPN
iajs-1139	88	2			NUM
iajs-1139	89	1			NUM
iajs-1139	89	2			NOUN
iajs-1139	89	3	x	x	SYM
iajs-1139	90	1	x	x	X
iajs-1139	90	2	it	it	PRON
iajs-1139	90	3	is	be	AUX
iajs-1139	90	4	clear	clear	ADJ
iajs-1139	90	5	a	a	PRON
iajs-1139	90	6	is	be	AUX
iajs-1139	90	7	a	a	DET
iajs-1139	90	8	fuzzy	fuzzy	ADJ
iajs-1139	90	9	ideal	ideal	NOUN
iajs-1139	90	10	of	of	ADP
iajs-1139	90	11	a	a	DET
iajs-1139	90	12	fuzzy	fuzzy	ADJ
iajs-1139	90	13	ring	ring	NOUN
iajs-1139	90	14	x	x	NOUN
iajs-1139	90	15	,	,	PUNCT
iajs-1139	90	16	x	x	PROPN
iajs-1139	90	17	t	t	NOUN
iajs-1139	90	18	=	=	PUNCT
iajs-1139	90	19	z12	z12	PROPN
iajs-1139	90	20	and	and	CCONJ
iajs-1139	90	21	at	at	ADP
iajs-1139	90	22	=	=	PUNCT
iajs-1139	90	23	(	(	PUNCT
iajs-1139	90	24	6	6	NUM
iajs-1139	90	25	)	)	PUNCT
iajs-1139	90	26	,	,	PUNCT
iajs-1139	90	27			PROPN
iajs-1139	90	28	t	t	PROPN
iajs-1139	90	29	>	>	X
iajs-1139	90	30	0	0	PUNCT
iajs-1139	91	1	is	be	AUX
iajs-1139	91	2	a	a	DET
iajs-1139	91	3	semiessential	semiessential	ADJ
iajs-1139	91	4	ideal	ideal	NOUN
iajs-1139	91	5	in	in	ADP
iajs-1139	91	6	z12	z12	PROPN
iajs-1139	91	7	since	since	SCONJ
iajs-1139	91	8	(	(	PUNCT
iajs-1139	91	9	6	6	NUM
iajs-1139	91	10	)	)	PUNCT
iajs-1139	91	11			NOUN
iajs-1139	91	12	(	(	PUNCT
iajs-1139	91	13	3	3	NUM
iajs-1139	91	14	)	)	PUNCT
iajs-1139	91	15	=	=	SYM
iajs-1139	92	1	(	(	PUNCT
iajs-1139	92	2	6	6	NUM
iajs-1139	92	3	)	)	PUNCT
iajs-1139	92	4	and	and	CCONJ
iajs-1139	92	5	(	(	PUNCT
iajs-1139	92	6	6	6	NUM
iajs-1139	92	7	)	)	PUNCT
iajs-1139	92	8			NOUN
iajs-1139	92	9	(	(	PUNCT
iajs-1139	92	10	2	2	NUM
iajs-1139	92	11	)	)	PUNCT
iajs-1139	92	12	=	=	SYM
iajs-1139	92	13	(	(	PUNCT
iajs-1139	92	14	6	6	NUM
iajs-1139	92	15	)	)	PUNCT
iajs-1139	92	16	,	,	PUNCT
iajs-1139	92	17	(	(	PUNCT
iajs-1139	92	18	3	3	X
iajs-1139	92	19	)	)	PUNCT
iajs-1139	92	20	and	and	CCONJ
iajs-1139	92	21	(	(	PUNCT
iajs-1139	92	22	2	2	X
iajs-1139	92	23	)	)	PUNCT
iajs-1139	92	24	are	be	AUX
iajs-1139	92	25	prime	prime	ADJ
iajs-1139	92	26	ideals	ideal	NOUN
iajs-1139	92	27	of	of	ADP
iajs-1139	92	28	z12	z12	PROPN
iajs-1139	92	29	.	.	PUNCT
iajs-1139	93	1	thus	thus	ADV
iajs-1139	93	2	a	a	PRON
iajs-1139	93	3	is	be	AUX
iajs-1139	93	4	a	a	DET
iajs-1139	93	5	semiessential	semiessential	ADJ
iajs-1139	93	6	fuzzy	fuzzy	ADJ
iajs-1139	93	7	ideal	ideal	NOUN
iajs-1139	93	8	by	by	ADP
iajs-1139	93	9	theorem	theorem	NOUN
iajs-1139	93	10	2.3	2.3	NUM
iajs-1139	93	11	.	.	PUNCT
iajs-1139	94	1	but	but	CCONJ
iajs-1139	94	2	a	a	PRON
iajs-1139	94	3	is	be	AUX
iajs-1139	94	4	not	not	PART
iajs-1139	94	5	an	an	DET
iajs-1139	94	6	essential	essential	ADJ
iajs-1139	94	7	fuzzy	fuzzy	ADJ
iajs-1139	94	8	ideal	ideal	NOUN
iajs-1139	94	9	since	since	SCONJ
iajs-1139	94	10	there	there	PRON
iajs-1139	94	11	exists	exist	VERB
iajs-1139	94	12	fuzzy	fuzzy	ADJ
iajs-1139	94	13	ideal	ideal	PROPN
iajs-1139	94	14	b	b	PROPN
iajs-1139	94	15	of	of	ADP
iajs-1139	94	16	x	x	SYM
iajs-1139	94	17	defined	define	VERB
iajs-1139	94	18	by	by	ADP
iajs-1139	94	19	:	:	PUNCT
iajs-1139	94	20	1	1	NUM
iajs-1139	95	1	if	if	SCONJ
iajs-1139	95	2	(	(	PUNCT
iajs-1139	95	3	4	4	NUM
iajs-1139	95	4	)	)	PUNCT
iajs-1139	95	5	(	(	PUNCT
iajs-1139	95	6	)	)	PUNCT
iajs-1139	95	7	0	0	PUNCT
iajs-1139	95	8	otherwise	otherwise	ADV
iajs-1139	95	9	.	.	PUNCT
iajs-1139	96	1			ADP
iajs-1139	96	2			PROPN
iajs-1139	96	3			NOUN
iajs-1139	97	1			PROPN
iajs-1139	98	1			NUM
iajs-1139	98	2			NOUN
iajs-1139	98	3	x	x	X
iajs-1139	98	4	x	x	SYM
iajs-1139	98	5	b	b	PROPN
iajs-1139	98	6			NOUN
iajs-1139	98	7	o1.but	o1.but	CCONJ
iajs-1139	98	8	a	a	DET
iajs-1139	98	9			NOUN
iajs-1139	98	10	b	b	NOUN
iajs-1139	98	11	=	=	SYM
iajs-1139	98	12	o1	o1	NOUN
iajs-1139	98	13	.	.	PUNCT
iajs-1139	99	1	2.6	2.6	NUM
iajs-1139	99	2	remark	remark	NOUN
iajs-1139	99	3	let	let	VERB
iajs-1139	99	4	x	x	PRON
iajs-1139	99	5	be	be	AUX
iajs-1139	99	6	a	a	DET
iajs-1139	99	7	fuzzy	fuzzy	ADJ
iajs-1139	99	8	ring	ring	NOUN
iajs-1139	99	9	of	of	ADP
iajs-1139	99	10	r	r	NOUN
iajs-1139	99	11	,	,	PUNCT
iajs-1139	99	12	let	let	VERB
iajs-1139	99	13	a	a	PRON
iajs-1139	99	14	and	and	CCONJ
iajs-1139	99	15	b	b	NOUN
iajs-1139	99	16	be	be	AUX
iajs-1139	99	17	fuzzy	fuzzy	ADJ
iajs-1139	99	18	ideals	ideal	NOUN
iajs-1139	99	19	of	of	ADP
iajs-1139	99	20	x	x	PUNCT
iajs-1139	99	21	such	such	ADJ
iajs-1139	99	22	that	that	SCONJ
iajs-1139	99	23	a	a	DET
iajs-1139	99	24			PROPN
iajs-1139	99	25	b.	b.	PROPN
iajs-1139	99	26	if	if	SCONJ
iajs-1139	99	27	a	a	PRON
iajs-1139	99	28	is	be	AUX
iajs-1139	99	29	a	a	DET
iajs-1139	99	30	semiessential	semiessential	NOUN
iajs-1139	99	31	.	.	PUNCT
iajs-1139	100	1	then	then	ADV
iajs-1139	100	2	b	b	X
iajs-1139	100	3	is	be	AUX
iajs-1139	100	4	a	a	DET
iajs-1139	100	5	semiessential	semiessential	ADJ
iajs-1139	100	6	fuzzy	fuzzy	ADJ
iajs-1139	100	7	ideal	ideal	NOUN
iajs-1139	100	8	of	of	ADP
iajs-1139	100	9	x.	x.	NOUN
iajs-1139	100	10	proof	proof	NOUN
iajs-1139	100	11	:	:	PUNCT
iajs-1139	100	12	it	it	PRON
iajs-1139	100	13	is	be	AUX
iajs-1139	100	14	clear	clear	ADJ
iajs-1139	100	15	.	.	PUNCT
iajs-1139	101	1	2.7	2.7	NUM
iajs-1139	101	2	corollary	corollary	ADJ
iajs-1139	101	3	if	if	SCONJ
iajs-1139	101	4	a	a	PRON
iajs-1139	101	5	and	and	CCONJ
iajs-1139	101	6	b	b	NOUN
iajs-1139	101	7	are	be	AUX
iajs-1139	101	8	fuzzy	fuzzy	ADJ
iajs-1139	101	9	ideals	ideal	NOUN
iajs-1139	101	10	of	of	ADP
iajs-1139	101	11	a	a	DET
iajs-1139	101	12	fuzzy	fuzzy	ADJ
iajs-1139	101	13	ring	ring	NOUN
iajs-1139	101	14	x	x	INTJ
iajs-1139	101	15	of	of	ADP
iajs-1139	101	16	a	a	DET
iajs-1139	101	17	ring	ring	NOUN
iajs-1139	101	18	r	r	NOUN
iajs-1139	101	19	such	such	ADJ
iajs-1139	101	20	that	that	SCONJ
iajs-1139	101	21	a	a	DET
iajs-1139	101	22			NOUN
iajs-1139	101	23	b	b	NOUN
iajs-1139	101	24	is	be	AUX
iajs-1139	101	25	a	a	DET
iajs-1139	101	26	semiessential	semiessential	ADJ
iajs-1139	101	27	fuzzy	fuzzy	ADJ
iajs-1139	101	28	ideal	ideal	NOUN
iajs-1139	101	29	of	of	ADP
iajs-1139	101	30	x	x	PRON
iajs-1139	101	31	,	,	PUNCT
iajs-1139	101	32	then	then	ADV
iajs-1139	101	33	a	a	PRON
iajs-1139	101	34	and	and	CCONJ
iajs-1139	101	35	b	b	NOUN
iajs-1139	101	36	are	be	AUX
iajs-1139	101	37	semiessential	semiessential	ADJ
iajs-1139	101	38	fuzzy	fuzzy	ADJ
iajs-1139	101	39	ideals	ideal	NOUN
iajs-1139	101	40	of	of	ADP
iajs-1139	101	41	x.	x.	NOUN
iajs-1139	101	42	2.8	2.8	NUM
iajs-1139	101	43	remark	remark	NOUN
iajs-1139	101	44	let	let	VERB
iajs-1139	101	45	a	a	PRON
iajs-1139	101	46	and	and	CCONJ
iajs-1139	101	47	b	b	NOUN
iajs-1139	101	48	be	be	AUX
iajs-1139	101	49	fuzzy	fuzzy	ADJ
iajs-1139	101	50	ideals	ideal	NOUN
iajs-1139	101	51	of	of	ADP
iajs-1139	101	52	fuzzy	fuzzy	ADJ
iajs-1139	101	53	ring	ring	NOUN
iajs-1139	101	54	x	x	INTJ
iajs-1139	101	55	of	of	ADP
iajs-1139	101	56	a	a	DET
iajs-1139	101	57	ring	ring	NOUN
iajs-1139	101	58	r	r	NOUN
iajs-1139	101	59	such	such	ADJ
iajs-1139	101	60	that	that	SCONJ
iajs-1139	101	61	a	a	DET
iajs-1139	101	62			PROPN
iajs-1139	101	63	b	b	PROPN
iajs-1139	101	64	and	and	CCONJ
iajs-1139	101	65	b	b	PROPN
iajs-1139	101	66	is	be	AUX
iajs-1139	101	67	a	a	DET
iajs-1139	101	68	semiessential	semiessential	NOUN
iajs-1139	101	69	,	,	PUNCT
iajs-1139	101	70	then	then	ADV
iajs-1139	101	71	it	it	PRON
iajs-1139	101	72	not	not	PART
iajs-1139	101	73	necessarily	necessarily	ADV
iajs-1139	101	74	that	that	SCONJ
iajs-1139	101	75	a	a	PRON
iajs-1139	101	76	is	be	AUX
iajs-1139	101	77	semiessential	semiessential	ADJ
iajs-1139	101	78	fuzzy	fuzzy	ADJ
iajs-1139	101	79	ideal	ideal	NOUN
iajs-1139	101	80	of	of	ADP
iajs-1139	101	81	x	x	PRON
iajs-1139	101	82	,	,	PUNCT
iajs-1139	101	83	as	as	SCONJ
iajs-1139	101	84	the	the	DET
iajs-1139	101	85	following	follow	VERB
iajs-1139	101	86	example	example	NOUN
iajs-1139	101	87	shows	show	VERB
iajs-1139	101	88	:	:	PUNCT
iajs-1139	101	89	example	example	NOUN
iajs-1139	101	90	:	:	PUNCT
iajs-1139	101	91	let	let	VERB
iajs-1139	101	92	x	x	PRON
iajs-1139	101	93	:	:	PUNCT
iajs-1139	101	94	z12	z12	NUM
iajs-1139	101	95			X
iajs-1139	102	1	[	[	X
iajs-1139	102	2	0,1	0,1	NUM
iajs-1139	102	3	]	]	PUNCT
iajs-1139	102	4	,	,	PUNCT
iajs-1139	102	5	a	a	DET
iajs-1139	102	6	:	:	PUNCT
iajs-1139	102	7	z12	z12	NUM
iajs-1139	102	8			PROPN
iajs-1139	103	1	[	[	X
iajs-1139	103	2	0,1	0,1	NUM
iajs-1139	103	3	]	]	PUNCT
iajs-1139	103	4	,	,	PUNCT
iajs-1139	103	5	b	b	X
iajs-1139	103	6	:	:	PUNCT
iajs-1139	103	7	z12	z12	NUM
iajs-1139	103	8			X
iajs-1139	104	1	[	[	X
iajs-1139	104	2	0,1	0,1	NUM
iajs-1139	104	3	]	]	PUNCT
iajs-1139	104	4	defined	define	VERB
iajs-1139	104	5	by	by	ADP
iajs-1139	104	6	:	:	PUNCT
iajs-1139	104	7	x(a)=	x(a)=	PROPN
iajs-1139	104	8	1	1	NUM
iajs-1139	104	9	,	,	PUNCT
iajs-1139	104	10			VERB
iajs-1139	104	11	a	a	DET
iajs-1139	104	12			PROPN
iajs-1139	104	13	z12	z12	NUM
iajs-1139	104	14	,	,	PUNCT
iajs-1139	104	15	1	1	NUM
iajs-1139	104	16	if	if	SCONJ
iajs-1139	104	17	(	(	PUNCT
iajs-1139	104	18	4	4	NUM
iajs-1139	104	19	)	)	PUNCT
iajs-1139	104	20	(	(	PUNCT
iajs-1139	104	21	)	)	PUNCT
iajs-1139	104	22	0	0	PUNCT
iajs-1139	104	23	otherwise	otherwise	ADV
iajs-1139	104	24	.	.	PUNCT
iajs-1139	105	1	x	x	PUNCT
iajs-1139	105	2	x	x	PUNCT
iajs-1139	105	3			NOUN
iajs-1139	105	4			NOUN
iajs-1139	106	1			PROPN
iajs-1139	106	2			NUM
iajs-1139	107	1			NOUN
iajs-1139	107	2			INTJ
iajs-1139	107	3	and	and	CCONJ
iajs-1139	107	4	1	1	NUM
iajs-1139	107	5	if	if	SCONJ
iajs-1139	107	6	(	(	PUNCT
iajs-1139	107	7	2	2	NUM
iajs-1139	107	8	)	)	PUNCT
iajs-1139	107	9	(	(	PUNCT
iajs-1139	107	10	)	)	PUNCT
iajs-1139	107	11	0	0	PUNCT
iajs-1139	108	1	otherwise	otherwise	ADV
iajs-1139	108	2	.	.	PUNCT
iajs-1139	109	1	x	x	PUNCT
iajs-1139	109	2	x	x	PUNCT
iajs-1139	109	3			ADP
iajs-1139	109	4			PROPN
iajs-1139	109	5			NOUN
iajs-1139	109	6			PROPN
iajs-1139	110	1			NUM
iajs-1139	110	2			INTJ
iajs-1139	110	3	.	.	PUNCT
iajs-1139	111	1	it	it	PRON
iajs-1139	111	2	is	be	AUX
iajs-1139	111	3	clear	clear	ADJ
iajs-1139	111	4	that	that	SCONJ
iajs-1139	111	5	xt	xt	PUNCT
iajs-1139	111	6	=	=	SYM
iajs-1139	111	7	z12	z12	PROPN
iajs-1139	111	8	(	(	PUNCT
iajs-1139	111	9			NOUN
iajs-1139	111	10	t	t	PROPN
iajs-1139	111	11	>	>	X
iajs-1139	111	12	0	0	NUM
iajs-1139	111	13	)	)	PUNCT
iajs-1139	111	14	and	and	CCONJ
iajs-1139	111	15	a	a	DET
iajs-1139	111	16	,	,	PUNCT
iajs-1139	111	17	b	b	NOUN
iajs-1139	111	18	are	be	AUX
iajs-1139	111	19	fuzzy	fuzzy	ADJ
iajs-1139	111	20	ideals	ideal	NOUN
iajs-1139	111	21	of	of	ADP
iajs-1139	111	22	fuzzy	fuzzy	ADJ
iajs-1139	111	23	ring	ring	NOUN
iajs-1139	111	24	x	x	NOUN
iajs-1139	111	25	,	,	PUNCT
iajs-1139	111	26	bt	bt	X
iajs-1139	111	27	=	=	SYM
iajs-1139	111	28	(	(	PUNCT
iajs-1139	111	29	2	2	NUM
iajs-1139	111	30	)	)	PUNCT
iajs-1139	111	31	is	be	AUX
iajs-1139	111	32	a	a	DET
iajs-1139	111	33	semiessential	semiessential	ADJ
iajs-1139	111	34	ideal	ideal	NOUN
iajs-1139	111	35	of	of	ADP
iajs-1139	111	36	xt	xt	PROPN
iajs-1139	111	37	,	,	PUNCT
iajs-1139	111	38			PROPN
iajs-1139	111	39	t	t	PROPN
iajs-1139	111	40	>	>	X
iajs-1139	111	41	0	0	NUM
iajs-1139	111	42	,	,	PUNCT
iajs-1139	111	43	since	since	SCONJ
iajs-1139	111	44	(	(	PUNCT
iajs-1139	111	45	2	2	X
iajs-1139	111	46	)	)	PUNCT
iajs-1139	111	47			NOUN
iajs-1139	111	48	(	(	PUNCT
iajs-1139	111	49	2	2	NUM
iajs-1139	111	50	)	)	PUNCT
iajs-1139	111	51	=	=	SYM
iajs-1139	111	52	(	(	PUNCT
iajs-1139	111	53	2	2	NUM
iajs-1139	111	54	)	)	PUNCT
iajs-1139	111	55	and	and	CCONJ
iajs-1139	111	56	(	(	PUNCT
iajs-1139	111	57	2	2	X
iajs-1139	111	58	)	)	PUNCT
iajs-1139	111	59			NOUN
iajs-1139	111	60	(	(	PUNCT
iajs-1139	111	61	3	3	NUM
iajs-1139	111	62	)	)	PUNCT
iajs-1139	111	63	=	=	SYM
iajs-1139	111	64	(	(	PUNCT
iajs-1139	111	65	6	6	NUM
iajs-1139	111	66	)	)	PUNCT
iajs-1139	111	67	where	where	SCONJ
iajs-1139	111	68	(	(	PUNCT
iajs-1139	111	69	2	2	NUM
iajs-1139	111	70	)	)	PUNCT
iajs-1139	111	71	and	and	CCONJ
iajs-1139	111	72	(	(	PUNCT
iajs-1139	111	73	3	3	X
iajs-1139	111	74	)	)	PUNCT
iajs-1139	111	75	are	be	AUX
iajs-1139	111	76	the	the	DET
iajs-1139	111	77	only	only	ADJ
iajs-1139	111	78	prime	prime	ADJ
iajs-1139	111	79	ideals	ideal	NOUN
iajs-1139	111	80	of	of	ADP
iajs-1139	111	81	z12	z12	PROPN
iajs-1139	111	82	=	=	SYM
iajs-1139	111	83	xt	xt	PROPN
iajs-1139	111	84	,	,	PUNCT
iajs-1139	111	85			NOUN
iajs-1139	111	86	t	t	PROPN
iajs-1139	111	87	>	>	X
iajs-1139	111	88	0	0	X
iajs-1139	111	89	.	.	PUNCT
iajs-1139	112	1	thus	thus	ADV
iajs-1139	112	2	b	b	X
iajs-1139	112	3	is	be	AUX
iajs-1139	112	4	a	a	DET
iajs-1139	112	5	semiessential	semiessential	ADJ
iajs-1139	112	6	fuzzy	fuzzy	ADJ
iajs-1139	112	7	ideal	ideal	NOUN
iajs-1139	112	8	of	of	ADP
iajs-1139	112	9	x.	x.	NOUN
iajs-1139	112	10	let	let	VERB
iajs-1139	112	11	1	1	NUM
iajs-1139	112	12	if	if	SCONJ
iajs-1139	112	13	(	(	PUNCT
iajs-1139	112	14	3	3	X
iajs-1139	112	15	)	)	PUNCT
iajs-1139	112	16	c	c	NOUN
iajs-1139	112	17	(	(	PUNCT
iajs-1139	112	18	)	)	PUNCT
iajs-1139	112	19	0	0	PUNCT
iajs-1139	113	1	otherwise	otherwise	ADV
iajs-1139	113	2	.	.	PUNCT
iajs-1139	114	1			ADP
iajs-1139	115	1			NOUN
iajs-1139	116	1			NUM
iajs-1139	117	1			NUM
iajs-1139	117	2			NOUN
iajs-1139	117	3	x	x	SYM
iajs-1139	117	4	x	x	X
iajs-1139	117	5	,	,	PUNCT
iajs-1139	117	6	c	c	NOUN
iajs-1139	117	7	is	be	AUX
iajs-1139	117	8	a	a	DET
iajs-1139	117	9	prime	prime	ADJ
iajs-1139	117	10	fuzzy	fuzzy	ADJ
iajs-1139	117	11	ideal	ideal	NOUN
iajs-1139	117	12	of	of	ADP
iajs-1139	117	13	x	x	PRON
iajs-1139	117	14	,	,	PUNCT
iajs-1139	117	15	since	since	SCONJ
iajs-1139	117	16	ct	ct	NUM
iajs-1139	117	17	=	=	SYM
iajs-1139	117	18	(	(	PUNCT
iajs-1139	117	19	3	3	NUM
iajs-1139	117	20	)	)	PUNCT
iajs-1139	117	21	is	be	AUX
iajs-1139	117	22	a	a	DET
iajs-1139	117	23	prime	prime	ADJ
iajs-1139	117	24	ideal	ideal	NOUN
iajs-1139	117	25	of	of	ADP
iajs-1139	117	26	xt	xt	PROPN
iajs-1139	117	27	,	,	PUNCT
iajs-1139	117	28			PROPN
iajs-1139	117	29	t	t	PROPN
iajs-1139	117	30	>	>	X
iajs-1139	117	31	0	0	NUM
iajs-1139	117	32	.	.	PUNCT
iajs-1139	118	1	but	but	CCONJ
iajs-1139	118	2	a	a	DET
iajs-1139	118	3			PUNCT
iajs-1139	118	4	c=	c=	NOUN
iajs-1139	118	5	o1	o1	NOUN
iajs-1139	118	6	.	.	PUNCT
iajs-1139	119	1	thus	thus	ADV
iajs-1139	119	2	a	a	PRON
iajs-1139	119	3	is	be	AUX
iajs-1139	119	4	not	not	PART
iajs-1139	119	5	a	a	DET
iajs-1139	119	6	semiessential	semiessential	ADJ
iajs-1139	119	7	fuzzy	fuzzy	ADJ
iajs-1139	119	8	ideal	ideal	NOUN
iajs-1139	119	9	of	of	ADP
iajs-1139	119	10	x.	x.	PROPN
iajs-1139	119	11	2.9	2.9	NUM
iajs-1139	119	12	remark	remark	NOUN
iajs-1139	119	13	if	if	SCONJ
iajs-1139	119	14	a	a	PRON
iajs-1139	119	15	and	and	CCONJ
iajs-1139	119	16	b	b	NOUN
iajs-1139	119	17	are	be	AUX
iajs-1139	119	18	semiessential	semiessential	ADJ
iajs-1139	119	19	fuzzy	fuzzy	ADJ
iajs-1139	119	20	ideals	ideal	NOUN
iajs-1139	119	21	of	of	ADP
iajs-1139	119	22	fuzzy	fuzzy	ADJ
iajs-1139	119	23	ring	ring	NOUN
iajs-1139	119	24	x	x	INTJ
iajs-1139	119	25	of	of	ADP
iajs-1139	119	26	a	a	DET
iajs-1139	119	27	ring	ring	NOUN
iajs-1139	119	28	r.	r.	PROPN
iajs-1139	119	29	then	then	ADV
iajs-1139	119	30	it	it	PRON
iajs-1139	119	31	not	not	PART
iajs-1139	119	32	necessarily	necessarily	ADV
iajs-1139	119	33	that	that	SCONJ
iajs-1139	119	34	a	a	DET
iajs-1139	119	35			NOUN
iajs-1139	119	36	b	b	NOUN
iajs-1139	119	37	is	be	AUX
iajs-1139	119	38	semiessential	semiessential	ADJ
iajs-1139	119	39	fuzzy	fuzzy	ADJ
iajs-1139	119	40	ideal	ideal	NOUN
iajs-1139	119	41	of	of	ADP
iajs-1139	119	42	x.	x.	NOUN
iajs-1139	119	43	we	we	PRON
iajs-1139	119	44	can	can	AUX
iajs-1139	119	45	give	give	VERB
iajs-1139	119	46	the	the	DET
iajs-1139	119	47	following	follow	VERB
iajs-1139	119	48	example	example	NOUN
iajs-1139	119	49	:	:	PUNCT
iajs-1139	119	50	example	example	NOUN
iajs-1139	119	51	:	:	PUNCT
iajs-1139	119	52	let	let	VERB
iajs-1139	119	53	x	x	PRON
iajs-1139	119	54	:	:	PUNCT
iajs-1139	119	55	z36	z36	PROPN
iajs-1139	119	56			PROPN
iajs-1139	120	1	[	[	X
iajs-1139	120	2	0,1	0,1	NUM
iajs-1139	120	3	]	]	PUNCT
iajs-1139	120	4	define	define	NOUN
iajs-1139	120	5	by	by	ADP
iajs-1139	120	6	x(a	x(a	NOUN
iajs-1139	120	7	)	)	PUNCT
iajs-1139	120	8	=	=	SYM
iajs-1139	120	9	1	1	NUM
iajs-1139	120	10	,	,	PUNCT
iajs-1139	120	11			VERB
iajs-1139	120	12	a	a	DET
iajs-1139	120	13			NOUN
iajs-1139	120	14	z36	z36	NOUN
iajs-1139	120	15	and	and	CCONJ
iajs-1139	120	16	let	let	VERB
iajs-1139	120	17	a	a	DET
iajs-1139	120	18	:	:	PUNCT
iajs-1139	120	19	z36	z36	PROPN
iajs-1139	120	20			NOUN
iajs-1139	121	1	[	[	X
iajs-1139	121	2	0,1	0,1	NUM
iajs-1139	121	3	]	]	PUNCT
iajs-1139	121	4	,	,	PUNCT
iajs-1139	121	5	b	b	X
iajs-1139	121	6	:	:	PUNCT
iajs-1139	121	7	z36	z36	PROPN
iajs-1139	121	8			PROPN
iajs-1139	122	1	[	[	X
iajs-1139	122	2	0,1	0,1	NUM
iajs-1139	122	3	]	]	PUNCT
iajs-1139	122	4	defined	define	VERB
iajs-1139	122	5	by	by	ADP
iajs-1139	122	6	:	:	PUNCT
iajs-1139	122	7	1	1	NUM
iajs-1139	122	8	if	if	SCONJ
iajs-1139	122	9	(	(	PUNCT
iajs-1139	122	10	12	12	NUM
iajs-1139	122	11	)	)	PUNCT
iajs-1139	122	12	(	(	PUNCT
iajs-1139	122	13	)	)	PUNCT
iajs-1139	122	14	0	0	PUNCT
iajs-1139	122	15	otherwise	otherwise	ADV
iajs-1139	122	16	.	.	PUNCT
iajs-1139	123	1			NOUN
iajs-1139	124	1			NOUN
iajs-1139	125	1			PROPN
iajs-1139	125	2			NUM
iajs-1139	126	1			NUM
iajs-1139	126	2			NOUN
iajs-1139	126	3	x	x	SYM
iajs-1139	126	4	x	x	X
iajs-1139	126	5	1	1	NUM
iajs-1139	126	6	if	if	SCONJ
iajs-1139	126	7	(	(	PUNCT
iajs-1139	126	8	18	18	NUM
iajs-1139	126	9	)	)	PUNCT
iajs-1139	126	10	(	(	PUNCT
iajs-1139	126	11	)	)	PUNCT
iajs-1139	126	12	0	0	PUNCT
iajs-1139	127	1	otherwise	otherwise	ADV
iajs-1139	127	2	.	.	PUNCT
iajs-1139	128	1			ADP
iajs-1139	128	2			PROPN
iajs-1139	128	3			NOUN
iajs-1139	129	1			PROPN
iajs-1139	130	1			NUM
iajs-1139	130	2			NOUN
iajs-1139	130	3	x	x	SYM
iajs-1139	130	4	x	x	X
iajs-1139	130	5	ibn	ibn	PROPN
iajs-1139	130	6	alhaitham	alhaitham	NOUN
iajs-1139	130	7	j.	j.	PROPN
iajs-1139	130	8	for	for	ADP
iajs-1139	130	9	pure	pure	ADJ
iajs-1139	130	10	&	&	CCONJ
iajs-1139	130	11	appl	appl	PROPN
iajs-1139	130	12	.	.	PUNCT
iajs-1139	131	1	sci	sci	PROPN
iajs-1139	131	2	.	.	PUNCT
iajs-1139	132	1	vol.22	vol.22	PROPN
iajs-1139	132	2	(	(	PUNCT
iajs-1139	132	3	4	4	NUM
iajs-1139	132	4	)	)	PUNCT
iajs-1139	132	5	2009	2009	NUM
iajs-1139	132	6	a	a	PRON
iajs-1139	132	7	and	and	CCONJ
iajs-1139	132	8	b	b	NOUN
iajs-1139	132	9	are	be	AUX
iajs-1139	132	10	fuzzy	fuzzy	ADJ
iajs-1139	132	11	ideals	ideal	NOUN
iajs-1139	132	12	of	of	ADP
iajs-1139	132	13	x.	x.	NOUN
iajs-1139	132	14	but	but	CCONJ
iajs-1139	132	15	for	for	ADP
iajs-1139	132	16	each	each	DET
iajs-1139	132	17	t	t	NOUN
iajs-1139	132	18			NOUN
iajs-1139	132	19	(	(	PUNCT
iajs-1139	132	20	0,1	0,1	NOUN
iajs-1139	132	21	]	]	PUNCT
iajs-1139	132	22	,	,	PUNCT
iajs-1139	132	23	at	at	ADP
iajs-1139	132	24	=	=	SYM
iajs-1139	132	25	(	(	PUNCT
iajs-1139	132	26	12	12	NUM
iajs-1139	132	27	)	)	PUNCT
iajs-1139	132	28	,	,	PUNCT
iajs-1139	132	29	bt	bt	NOUN
iajs-1139	132	30	=	=	SYM
iajs-1139	132	31	(	(	PUNCT
iajs-1139	132	32	18	18	NUM
iajs-1139	132	33	)	)	PUNCT
iajs-1139	132	34	.	.	PUNCT
iajs-1139	133	1	it	it	PRON
iajs-1139	133	2	is	be	AUX
iajs-1139	133	3	easy	easy	ADJ
iajs-1139	133	4	to	to	PART
iajs-1139	133	5	show	show	VERB
iajs-1139	133	6	that	that	SCONJ
iajs-1139	133	7	at	at	ADP
iajs-1139	133	8	and	and	CCONJ
iajs-1139	133	9	bt	bt	NOUN
iajs-1139	133	10	are	be	AUX
iajs-1139	133	11	semiessential	semiessential	ADJ
iajs-1139	133	12	ideals	ideal	NOUN
iajs-1139	133	13	of	of	ADP
iajs-1139	133	14	xt	xt	PROPN
iajs-1139	133	15	=	=	SYM
iajs-1139	133	16	z36	z36	PROPN
iajs-1139	133	17	.	.	PUNCT
iajs-1139	134	1	thus	thus	ADV
iajs-1139	134	2	a	a	PRON
iajs-1139	134	3	and	and	CCONJ
iajs-1139	134	4	b	b	NOUN
iajs-1139	134	5	are	be	AUX
iajs-1139	134	6	semiessential	semiessential	ADJ
iajs-1139	134	7	fuzzy	fuzzy	ADJ
iajs-1139	134	8	ideals	ideal	NOUN
iajs-1139	134	9	of	of	ADP
iajs-1139	134	10	x	x	PUNCT
iajs-1139	134	11	by	by	ADP
iajs-1139	134	12	theorem	theorem	NOUN
iajs-1139	134	13	2.3	2.3	NUM
iajs-1139	134	14	.	.	PUNCT
iajs-1139	135	1	but	but	CCONJ
iajs-1139	135	2	a	a	DET
iajs-1139	135	3			PROPN
iajs-1139	135	4	b	b	NOUN
iajs-1139	135	5	=	=	SYM
iajs-1139	135	6	o1	o1	NOUN
iajs-1139	135	7	.	.	PUNCT
iajs-1139	136	1	thus	thus	ADV
iajs-1139	136	2	a	a	DET
iajs-1139	136	3			NOUN
iajs-1139	136	4	b	b	NOUN
iajs-1139	136	5	is	be	AUX
iajs-1139	136	6	not	not	PART
iajs-1139	136	7	a	a	DET
iajs-1139	136	8	semiessential	semiessential	ADJ
iajs-1139	136	9	fuzzy	fuzzy	ADJ
iajs-1139	136	10	ideal	ideal	NOUN
iajs-1139	136	11	of	of	ADP
iajs-1139	136	12	x.	x.	NOUN
iajs-1139	136	13	2.10	2.10	NUM
iajs-1139	136	14	proposition	proposition	NOUN
iajs-1139	136	15	let	let	VERB
iajs-1139	136	16	a	a	PRON
iajs-1139	136	17	and	and	CCONJ
iajs-1139	136	18	b	b	NOUN
iajs-1139	136	19	be	be	AUX
iajs-1139	136	20	fuzzy	fuzzy	ADJ
iajs-1139	136	21	ideals	ideal	NOUN
iajs-1139	136	22	of	of	ADP
iajs-1139	136	23	fuzzy	fuzzy	ADJ
iajs-1139	136	24	ring	ring	NOUN
iajs-1139	136	25	x	x	INTJ
iajs-1139	136	26	of	of	ADP
iajs-1139	136	27	a	a	DET
iajs-1139	136	28	ring	ring	NOUN
iajs-1139	136	29	r	r	NOUN
iajs-1139	136	30	such	such	ADJ
iajs-1139	136	31	that	that	SCONJ
iajs-1139	136	32	a	a	PRON
iajs-1139	136	33	is	be	AUX
iajs-1139	136	34	an	an	DET
iajs-1139	136	35	essential	essential	ADJ
iajs-1139	136	36	fuzzy	fuzzy	ADJ
iajs-1139	136	37	ideal	ideal	NOUN
iajs-1139	136	38	and	and	CCONJ
iajs-1139	136	39	b	b	NOUN
iajs-1139	136	40	is	be	AUX
iajs-1139	136	41	a	a	DET
iajs-1139	136	42	semiessential	semiessential	ADJ
iajs-1139	136	43	fuzzy	fuzzy	ADJ
iajs-1139	136	44	ideal	ideal	NOUN
iajs-1139	136	45	.	.	PUNCT
iajs-1139	137	1	then	then	ADV
iajs-1139	137	2	a	a	DET
iajs-1139	137	3			NOUN
iajs-1139	137	4	b	b	NOUN
iajs-1139	137	5	is	be	AUX
iajs-1139	137	6	a	a	DET
iajs-1139	137	7	semiessential	semiessential	ADJ
iajs-1139	137	8	fuzzy	fuzzy	ADJ
iajs-1139	137	9	ideal	ideal	NOUN
iajs-1139	137	10	of	of	ADP
iajs-1139	137	11	x.	x.	NOUN
iajs-1139	137	12	proof	proof	NOUN
iajs-1139	137	13	:	:	PUNCT
iajs-1139	137	14	let	let	VERB
iajs-1139	137	15	p	p	PRON
iajs-1139	137	16	be	be	AUX
iajs-1139	137	17	a	a	DET
iajs-1139	137	18	non	non	ADJ
iajs-1139	137	19	-	-	ADJ
iajs-1139	137	20	zero	zero	ADJ
iajs-1139	137	21	prime	prime	ADJ
iajs-1139	137	22	fuzzy	fuzzy	ADJ
iajs-1139	137	23	ideal	ideal	NOUN
iajs-1139	137	24	of	of	ADP
iajs-1139	137	25	x	x	PRON
iajs-1139	137	26	,	,	PUNCT
iajs-1139	137	27	since	since	SCONJ
iajs-1139	137	28	b	b	NOUN
iajs-1139	137	29	is	be	AUX
iajs-1139	137	30	a	a	DET
iajs-1139	137	31	semiessential	semiessential	ADJ
iajs-1139	137	32	fuzzy	fuzzy	ADJ
iajs-1139	137	33	ideal	ideal	NOUN
iajs-1139	137	34	of	of	ADP
iajs-1139	137	35	x	x	PRON
iajs-1139	137	36	,	,	PUNCT
iajs-1139	137	37	then	then	ADV
iajs-1139	137	38	b	b	X
iajs-1139	137	39			PUNCT
iajs-1139	137	40	p	p	NOUN
iajs-1139	137	41	≠	≠	PROPN
iajs-1139	137	42	o1.also	o1.also	SCONJ
iajs-1139	137	43	a	a	PRON
iajs-1139	137	44	is	be	AUX
iajs-1139	137	45	an	an	DET
iajs-1139	137	46	essential	essential	ADJ
iajs-1139	137	47	fuzzy	fuzzy	ADJ
iajs-1139	137	48	ideal	ideal	NOUN
iajs-1139	137	49	of	of	ADP
iajs-1139	137	50	x	x	PRON
iajs-1139	137	51	,	,	PUNCT
iajs-1139	137	52	we	we	PRON
iajs-1139	137	53	get	get	VERB
iajs-1139	137	54	(	(	PUNCT
iajs-1139	137	55	a	a	DET
iajs-1139	137	56			ADJ
iajs-1139	137	57	b	b	NOUN
iajs-1139	137	58	)	)	PUNCT
iajs-1139	137	59			PUNCT
iajs-1139	137	60	p	p	PROPN
iajs-1139	137	61			NOUN
iajs-1139	137	62	o1	o1	PROPN
iajs-1139	137	63	.	.	PUNCT
iajs-1139	138	1	which	which	PRON
iajs-1139	138	2	implies	imply	VERB
iajs-1139	138	3	a	a	DET
iajs-1139	138	4			NOUN
iajs-1139	138	5	b	b	NOUN
iajs-1139	138	6	is	be	AUX
iajs-1139	138	7	a	a	DET
iajs-1139	138	8	semiessential	semiessential	ADJ
iajs-1139	138	9	fuzzy	fuzzy	ADJ
iajs-1139	138	10	ideal	ideal	NOUN
iajs-1139	138	11	of	of	ADP
iajs-1139	138	12	x.	x.	NOUN
iajs-1139	138	13	2.11	2.11	NUM
iajs-1139	138	14	proposition	proposition	NOUN
iajs-1139	138	15	let	let	VERB
iajs-1139	138	16	x.	x.	NOUN
iajs-1139	138	17	be	be	AUX
iajs-1139	138	18	a	a	DET
iajs-1139	138	19	fuzzy	fuzzy	ADJ
iajs-1139	138	20	ring	ring	NOUN
iajs-1139	138	21	of	of	ADP
iajs-1139	138	22	r	r	NOUN
iajs-1139	138	23	such	such	ADJ
iajs-1139	138	24	that	that	SCONJ
iajs-1139	138	25	x(a	x(a	NOUN
iajs-1139	138	26	)	)	PUNCT
iajs-1139	138	27	=	=	SYM
iajs-1139	138	28	1	1	NUM
iajs-1139	138	29	,	,	PUNCT
iajs-1139	138	30			VERB
iajs-1139	138	31	a	a	DET
iajs-1139	138	32			PROPN
iajs-1139	138	33	r.	r.	NOUN
iajs-1139	138	34	let	let	VERB
iajs-1139	138	35	i	i	PRON
iajs-1139	138	36	be	be	AUX
iajs-1139	138	37	a	a	DET
iajs-1139	138	38	semiessential	semiessential	ADJ
iajs-1139	138	39	ideal	ideal	NOUN
iajs-1139	138	40	of	of	ADP
iajs-1139	138	41	r.	r.	PROPN
iajs-1139	138	42	if	if	SCONJ
iajs-1139	138	43	a	a	PRON
iajs-1139	138	44	:	:	PUNCT
iajs-1139	139	1	r	r	NOUN
iajs-1139	139	2			X
iajs-1139	140	1	[	[	X
iajs-1139	140	2	0,1	0,1	NUM
iajs-1139	140	3	]	]	PUNCT
iajs-1139	140	4	defined	define	VERB
iajs-1139	140	5	by	by	ADP
iajs-1139	140	6	:	:	PUNCT
iajs-1139	140	7	1	1	NUM
iajs-1139	140	8	if	if	SCONJ
iajs-1139	140	9	i	i	PRON
iajs-1139	140	10	(	(	PUNCT
iajs-1139	140	11	)	)	PUNCT
iajs-1139	140	12	r	r	NOUN
iajs-1139	140	13	if	if	SCONJ
iajs-1139	140	14	a	a	DET
iajs-1139	140	15	i	i	NOUN
iajs-1139	140	16			ADP
iajs-1139	141	1			PROPN
iajs-1139	141	2			NUM
iajs-1139	142	1			NUM
iajs-1139	142	2			NOUN
iajs-1139	143	1	a	a	DET
iajs-1139	143	2	a	a	DET
iajs-1139	143	3	where	where	SCONJ
iajs-1139	143	4	r	r	NOUN
iajs-1139	143	5			NOUN
iajs-1139	143	6	(	(	PUNCT
iajs-1139	143	7	0,1	0,1	NUM
iajs-1139	143	8	)	)	PUNCT
iajs-1139	143	9	.	.	PUNCT
iajs-1139	144	1	then	then	ADV
iajs-1139	144	2	a	a	PRON
iajs-1139	144	3	is	be	AUX
iajs-1139	144	4	a	a	DET
iajs-1139	144	5	semiessential	semiessential	ADJ
iajs-1139	144	6	fuzzy	fuzzy	ADJ
iajs-1139	144	7	ideal	ideal	NOUN
iajs-1139	144	8	of	of	ADP
iajs-1139	144	9	x.	x.	NOUN
iajs-1139	144	10	proof	proof	NOUN
iajs-1139	144	11	:	:	PUNCT
iajs-1139	144	12	it	it	PRON
iajs-1139	144	13	is	be	AUX
iajs-1139	144	14	easy	easy	ADJ
iajs-1139	144	15	,	,	PUNCT
iajs-1139	144	16	so	so	ADV
iajs-1139	144	17	it	it	PRON
iajs-1139	144	18	omitted	omit	VERB
iajs-1139	144	19	.	.	PUNCT
iajs-1139	145	1	2.12	2.12	NUM
iajs-1139	145	2	proposition	proposition	NOUN
iajs-1139	145	3	let	let	VERB
iajs-1139	145	4	x.	x.	NOUN
iajs-1139	145	5	be	be	AUX
iajs-1139	145	6	a	a	DET
iajs-1139	145	7	fuzzy	fuzzy	ADJ
iajs-1139	145	8	ring	ring	NOUN
iajs-1139	145	9	of	of	ADP
iajs-1139	145	10	r	r	NOUN
iajs-1139	145	11	such	such	ADJ
iajs-1139	145	12	that	that	SCONJ
iajs-1139	145	13	x(a	x(a	NOUN
iajs-1139	145	14	)	)	PUNCT
iajs-1139	145	15	=	=	SYM
iajs-1139	145	16	1	1	NUM
iajs-1139	145	17	,	,	PUNCT
iajs-1139	145	18			VERB
iajs-1139	145	19	a	a	DET
iajs-1139	145	20			PROPN
iajs-1139	145	21	r.	r.	NOUN
iajs-1139	145	22	let	let	VERB
iajs-1139	145	23	i	i	PRON
iajs-1139	145	24	be	be	AUX
iajs-1139	145	25	a	a	DET
iajs-1139	145	26	ideal	ideal	NOUN
iajs-1139	145	27	of	of	ADP
iajs-1139	145	28	r.	r.	PROPN
iajs-1139	146	1	then	then	ADV
iajs-1139	146	2	i	i	PRON
iajs-1139	146	3	is	be	AUX
iajs-1139	146	4	a	a	DET
iajs-1139	146	5	semiessential	semiessential	ADJ
iajs-1139	146	6	ideal	ideal	NOUN
iajs-1139	146	7	of	of	ADP
iajs-1139	146	8	r	r	NOUN
iajs-1139	146	9	if	if	SCONJ
iajs-1139	146	10	i	i	PROPN
iajs-1139	146	11	is	be	AUX
iajs-1139	146	12	a	a	DET
iajs-1139	146	13	semiessential	semiessential	ADJ
iajs-1139	146	14	fuzzy	fuzzy	ADJ
iajs-1139	146	15	ideal	ideal	NOUN
iajs-1139	146	16	of	of	ADP
iajs-1139	146	17	x	x	INTJ
iajs-1139	147	1	where	where	SCONJ
iajs-1139	147	2	i	i	PRON
iajs-1139	147	3	1	1	NUM
iajs-1139	147	4	if	if	SCONJ
iajs-1139	147	5	i	i	PRON
iajs-1139	147	6	(	(	PUNCT
iajs-1139	147	7	)	)	PUNCT
iajs-1139	147	8	0	0	NUM
iajs-1139	147	9	otherwise	otherwise	ADV
iajs-1139	147	10			X
iajs-1139	147	11			X
iajs-1139	147	12			NUM
iajs-1139	147	13			NUM
iajs-1139	147	14			NOUN
iajs-1139	147	15	x	x	NOUN
iajs-1139	147	16	x	x	SYM
iajs-1139	147	17	proof	proof	NOUN
iajs-1139	147	18	:	:	PUNCT
iajs-1139	147	19	it	it	PRON
iajs-1139	147	20	is	be	AUX
iajs-1139	147	21	easy	easy	ADJ
iajs-1139	147	22	,	,	PUNCT
iajs-1139	147	23	so	so	CCONJ
iajs-1139	147	24	it	it	PRON
iajs-1139	147	25	is	be	AUX
iajs-1139	147	26	omitted	omit	VERB
iajs-1139	147	27	.	.	PUNCT
iajs-1139	148	1	before	before	ADP
iajs-1139	148	2	studying	study	VERB
iajs-1139	148	3	the	the	DET
iajs-1139	148	4	direct	direct	ADJ
iajs-1139	148	5	sum	sum	NOUN
iajs-1139	148	6	of	of	ADP
iajs-1139	148	7	semiessential	semiessential	ADJ
iajs-1139	148	8	fuzzy	fuzzy	ADJ
iajs-1139	148	9	ideals	ideal	NOUN
iajs-1139	148	10	,	,	PUNCT
iajs-1139	148	11	we	we	PRON
iajs-1139	148	12	need	need	VERB
iajs-1139	148	13	the	the	DET
iajs-1139	148	14	following	follow	VERB
iajs-1139	148	15	lemma	lemma	PROPN
iajs-1139	148	16	.	.	PROPN
iajs-1139	149	1	2.13	2.13	NUM
iajs-1139	149	2	lemma	lemma	PROPN
iajs-1139	149	3	let	let	VERB
iajs-1139	149	4	x	x	PRON
iajs-1139	149	5	and	and	CCONJ
iajs-1139	149	6	y	y	PROPN
iajs-1139	149	7	be	be	AUX
iajs-1139	149	8	fuzzy	fuzzy	ADJ
iajs-1139	149	9	rings	ring	NOUN
iajs-1139	149	10	of	of	ADP
iajs-1139	149	11	rings	ring	NOUN
iajs-1139	149	12	r1	r1	PROPN
iajs-1139	149	13	,	,	PUNCT
iajs-1139	149	14	r2	r2	PROPN
iajs-1139	149	15	respectively	respectively	ADV
iajs-1139	149	16	.	.	PUNCT
iajs-1139	150	1	let	let	VERB
iajs-1139	150	2	w	w	NOUN
iajs-1139	150	3	be	be	AUX
iajs-1139	150	4	a	a	DET
iajs-1139	150	5	fuzzy	fuzzy	ADJ
iajs-1139	150	6	ideal	ideal	NOUN
iajs-1139	150	7	of	of	ADP
iajs-1139	150	8	t	t	PROPN
iajs-1139	151	1	=	=	PUNCT
iajs-1139	151	2	x	x	PROPN
iajs-1139	151	3			PROPN
iajs-1139	151	4	y	y	PROPN
iajs-1139	151	5	,	,	PUNCT
iajs-1139	151	6	then	then	ADV
iajs-1139	151	7	w	w	PROPN
iajs-1139	151	8	is	be	AUX
iajs-1139	151	9	a	a	DET
iajs-1139	151	10	prime	prime	ADJ
iajs-1139	151	11	fuzzy	fuzzy	ADJ
iajs-1139	151	12	ideal	ideal	NOUN
iajs-1139	151	13	of	of	ADP
iajs-1139	151	14	t	t	PROPN
iajs-1139	151	15	if	if	SCONJ
iajs-1139	151	16	there	there	PRON
iajs-1139	151	17	exists	exist	VERB
iajs-1139	151	18	a	a	DET
iajs-1139	151	19	and	and	CCONJ
iajs-1139	151	20	b	b	NOUN
iajs-1139	151	21	prime	prime	ADJ
iajs-1139	151	22	fuzzy	fuzzy	ADJ
iajs-1139	151	23	ideals	ideal	NOUN
iajs-1139	151	24	of	of	ADP
iajs-1139	151	25	x	x	PRON
iajs-1139	151	26	,	,	PUNCT
iajs-1139	151	27	y	y	PROPN
iajs-1139	151	28	respectively	respectively	ADV
iajs-1139	152	1	such	such	ADJ
iajs-1139	152	2	that	that	SCONJ
iajs-1139	152	3	w	w	NOUN
iajs-1139	152	4	=	=	PUNCT
iajs-1139	152	5	a	a	DET
iajs-1139	152	6			ADJ
iajs-1139	152	7	y	y	NOUN
iajs-1139	152	8	or	or	CCONJ
iajs-1139	152	9	w	w	NOUN
iajs-1139	152	10	=	=	NOUN
iajs-1139	152	11	x	x	PROPN
iajs-1139	152	12			PROPN
iajs-1139	152	13	b.	b.	PROPN
iajs-1139	152	14	proof	proof	NOUN
iajs-1139	152	15	:	:	PUNCT
iajs-1139	152	16	if	if	SCONJ
iajs-1139	152	17	w	w	NOUN
iajs-1139	152	18	is	be	AUX
iajs-1139	152	19	a	a	DET
iajs-1139	152	20	prime	prime	ADJ
iajs-1139	152	21	fuzzy	fuzzy	ADJ
iajs-1139	152	22	ideal	ideal	NOUN
iajs-1139	152	23	in	in	ADP
iajs-1139	152	24	t	t	PROPN
iajs-1139	152	25	=	=	PUNCT
iajs-1139	152	26	x	x	PROPN
iajs-1139	152	27			PROPN
iajs-1139	152	28	y.	y.	PROPN
iajs-1139	152	29	since	since	SCONJ
iajs-1139	152	30	w	w	PROPN
iajs-1139	152	31	is	be	AUX
iajs-1139	152	32	a	a	DET
iajs-1139	152	33	fuzzy	fuzzy	ADJ
iajs-1139	152	34	ideal	ideal	NOUN
iajs-1139	152	35	in	in	ADP
iajs-1139	152	36	xy	xy	PROPN
iajs-1139	152	37	,	,	PUNCT
iajs-1139	152	38	there	there	PRON
iajs-1139	152	39	exists	exist	VERB
iajs-1139	152	40	fuzzy	fuzzy	ADJ
iajs-1139	152	41	ideal	ideal	NOUN
iajs-1139	152	42	a	a	PRON
iajs-1139	152	43	and	and	CCONJ
iajs-1139	152	44	b	b	NOUN
iajs-1139	152	45	of	of	ADP
iajs-1139	152	46	x	x	PRON
iajs-1139	152	47	,	,	PUNCT
iajs-1139	152	48	y	y	PROPN
iajs-1139	152	49	respectively	respectively	ADV
iajs-1139	152	50	such	such	ADJ
iajs-1139	152	51	that	that	SCONJ
iajs-1139	152	52	w	w	NOUN
iajs-1139	152	53	=	=	PUNCT
iajs-1139	152	54	a	a	DET
iajs-1139	152	55			PROPN
iajs-1139	152	56	b	b	PROPN
iajs-1139	152	57	by	by	ADP
iajs-1139	152	58	(	(	PUNCT
iajs-1139	152	59	10,theorem	10,theorem	NUM
iajs-1139	152	60	2.4.1.9	2.4.1.9	NUM
iajs-1139	152	61	)	)	PUNCT
iajs-1139	152	62	.	.	PUNCT
iajs-1139	153	1	thus	thus	ADV
iajs-1139	153	2	wt	wt	X
iajs-1139	153	3	=	=	PUNCT
iajs-1139	153	4	at	at	ADP
iajs-1139	153	5			PROPN
iajs-1139	153	6	bt	bt	PROPN
iajs-1139	153	7	,	,	PUNCT
iajs-1139	153	8			PROPN
iajs-1139	153	9	t	t	PROPN
iajs-1139	153	10			PROPN
iajs-1139	153	11	(	(	PUNCT
iajs-1139	153	12	0,1	0,1	NOUN
iajs-1139	153	13	]	]	PUNCT
iajs-1139	153	14	.	.	PUNCT
iajs-1139	154	1	but	but	CCONJ
iajs-1139	154	2	w	w	NOUN
iajs-1139	154	3	is	be	AUX
iajs-1139	154	4	a	a	DET
iajs-1139	154	5	prime	prime	NOUN
iajs-1139	154	6	so	so	ADV
iajs-1139	154	7	wt	wt	PROPN
iajs-1139	154	8	is	be	AUX
iajs-1139	154	9	prime	prime	ADJ
iajs-1139	154	10	in	in	ADP
iajs-1139	154	11	t	t	PROPN
iajs-1139	154	12	t	t	NOUN
iajs-1139	154	13	=	=	SYM
iajs-1139	154	14	xt	xt	PROPN
iajs-1139	154	15			PROPN
iajs-1139	154	16	yt	yt	PROPN
iajs-1139	154	17	,	,	PUNCT
iajs-1139	154	18			PROPN
iajs-1139	154	19	t	t	PROPN
iajs-1139	154	20			PROPN
iajs-1139	154	21	(	(	PUNCT
iajs-1139	154	22	0,1	0,1	NOUN
iajs-1139	154	23	]	]	PUNCT
iajs-1139	154	24	.	.	PUNCT
iajs-1139	155	1	hence	hence	ADV
iajs-1139	155	2	either	either	CCONJ
iajs-1139	155	3	wt	wt	PROPN
iajs-1139	155	4	=	=	NOUN
iajs-1139	156	1	i	i	PRON
iajs-1139	156	2			VERB
iajs-1139	156	3	yt	yt	VERB
iajs-1139	156	4	or	or	CCONJ
iajs-1139	156	5	wt	wt	PROPN
iajs-1139	156	6	=	=	PUNCT
iajs-1139	156	7	xt	xt	PROPN
iajs-1139	156	8			PROPN
iajs-1139	156	9	j	j	PROPN
iajs-1139	156	10	where	where	SCONJ
iajs-1139	156	11	i	i	PRON
iajs-1139	156	12	,	,	PUNCT
iajs-1139	156	13	j	j	PROPN
iajs-1139	156	14	are	be	AUX
iajs-1139	156	15	prime	prime	ADJ
iajs-1139	156	16	ideals	ideal	NOUN
iajs-1139	156	17	in	in	ADP
iajs-1139	156	18	xt	xt	PROPN
iajs-1139	156	19	,	,	PUNCT
iajs-1139	156	20	yt	yt	X
iajs-1139	156	21	respectively	respectively	ADV
iajs-1139	156	22	,	,	PUNCT
iajs-1139	156	23	by	by	ADP
iajs-1139	156	24	(	(	PUNCT
iajs-1139	156	25	12	12	NUM
iajs-1139	156	26	,	,	PUNCT
iajs-1139	156	27	page	page	NOUN
iajs-1139	156	28	53	53	NUM
iajs-1139	156	29	)	)	PUNCT
iajs-1139	156	30	.	.	PUNCT
iajs-1139	157	1	therefore	therefore	ADV
iajs-1139	157	2	i=	i=	PROPN
iajs-1139	157	3	a	a	DET
iajs-1139	157	4	t	t	NOUN
iajs-1139	157	5	or	or	CCONJ
iajs-1139	157	6	j	j	PROPN
iajs-1139	157	7	=	=	SYM
iajs-1139	157	8	bt	bt	PROPN
iajs-1139	157	9	and	and	CCONJ
iajs-1139	157	10	hence	hence	ADV
iajs-1139	157	11	wt	wt	PROPN
iajs-1139	157	12	=	=	PUNCT
iajs-1139	157	13	at	at	ADP
iajs-1139	157	14			PROPN
iajs-1139	157	15	yt	yt	PROPN
iajs-1139	157	16	or	or	CCONJ
iajs-1139	157	17	wt	wt	PROPN
iajs-1139	157	18	=	=	PUNCT
iajs-1139	157	19	xt	xt	PROPN
iajs-1139	157	20			PROPN
iajs-1139	157	21	bt	bt	PROPN
iajs-1139	157	22	.	.	PUNCT
iajs-1139	158	1	it	it	PRON
iajs-1139	158	2	follows	follow	VERB
iajs-1139	158	3	that	that	PRON
iajs-1139	158	4	wt	wt	PROPN
iajs-1139	158	5	=	=	SYM
iajs-1139	158	6	(	(	PUNCT
iajs-1139	158	7	a	a	DET
iajs-1139	158	8			ADJ
iajs-1139	158	9	y)t	y)t	NOUN
iajs-1139	158	10	or	or	CCONJ
iajs-1139	158	11	wt	wt	PROPN
iajs-1139	158	12	=(	=(	PROPN
iajs-1139	158	13	x	x	PROPN
iajs-1139	158	14	b)t	b)t	PROPN
iajs-1139	158	15	.	.	PUNCT
iajs-1139	159	1	thus	thus	ADV
iajs-1139	159	2	w	w	X
iajs-1139	159	3	=	=	PUNCT
iajs-1139	159	4	a	a	DET
iajs-1139	159	5			ADJ
iajs-1139	159	6	y	y	NOUN
iajs-1139	159	7	or	or	CCONJ
iajs-1139	159	8	w	w	NOUN
iajs-1139	159	9	=	=	NOUN
iajs-1139	159	10	x	x	PROPN
iajs-1139	159	11			PROPN
iajs-1139	159	12	b.	b.	PROPN
iajs-1139	160	1	conversely	conversely	ADV
iajs-1139	160	2	;	;	PUNCT
iajs-1139	160	3	if	if	SCONJ
iajs-1139	160	4	w	w	PROPN
iajs-1139	160	5	=	=	PUNCT
iajs-1139	160	6	a	a	DET
iajs-1139	160	7			ADJ
iajs-1139	160	8	y	y	NOUN
iajs-1139	160	9	or	or	CCONJ
iajs-1139	160	10	w	w	NOUN
iajs-1139	160	11	=	=	NOUN
iajs-1139	160	12	x	x	PROPN
iajs-1139	160	13			PROPN
iajs-1139	160	14	b	b	PROPN
iajs-1139	160	15	,	,	PUNCT
iajs-1139	160	16	where	where	SCONJ
iajs-1139	160	17	a	a	PRON
iajs-1139	160	18	and	and	CCONJ
iajs-1139	160	19	b	b	NOUN
iajs-1139	160	20	are	be	AUX
iajs-1139	160	21	prime	prime	ADJ
iajs-1139	160	22	fuzzy	fuzzy	ADJ
iajs-1139	160	23	ideals	ideal	NOUN
iajs-1139	160	24	of	of	ADP
iajs-1139	160	25	x	x	PROPN
iajs-1139	160	26	,	,	PUNCT
iajs-1139	160	27	y	y	PROPN
iajs-1139	160	28	respectively	respectively	ADV
iajs-1139	160	29	.	.	PUNCT
iajs-1139	161	1	if	if	SCONJ
iajs-1139	161	2	w	w	NOUN
iajs-1139	161	3	=	=	PUNCT
iajs-1139	161	4	a	a	DET
iajs-1139	161	5			PROPN
iajs-1139	161	6	y	y	NOUN
iajs-1139	161	7	,	,	PUNCT
iajs-1139	161	8	then	then	ADV
iajs-1139	161	9	wt	wt	ADP
iajs-1139	161	10	=	=	PUNCT
iajs-1139	161	11	(	(	PUNCT
iajs-1139	161	12	a	a	DET
iajs-1139	161	13			ADJ
iajs-1139	161	14	y)t	y)t	PUNCT
iajs-1139	162	1	=	=	PUNCT
iajs-1139	162	2	at	at	ADP
iajs-1139	162	3			PROPN
iajs-1139	162	4	yt	yt	PROPN
iajs-1139	162	5	but	but	CCONJ
iajs-1139	162	6	at	at	ADP
iajs-1139	162	7	is	be	AUX
iajs-1139	162	8	a	a	DET
iajs-1139	162	9	prime	prime	ADJ
iajs-1139	162	10	ideal	ideal	NOUN
iajs-1139	162	11	in	in	ADP
iajs-1139	162	12	xt	xt	PROPN
iajs-1139	162	13	,	,	PUNCT
iajs-1139	162	14			PROPN
iajs-1139	162	15	t	t	PROPN
iajs-1139	162	16			PROPN
iajs-1139	162	17	(	(	PUNCT
iajs-1139	162	18	0,1	0,1	NUM
iajs-1139	162	19	]	]	PUNCT
iajs-1139	162	20	by	by	ADP
iajs-1139	162	21	(	(	PUNCT
iajs-1139	162	22	11,proposition	11,proposition	NUM
iajs-1139	162	23	1.2.9	1.2.9	NUM
iajs-1139	162	24	)	)	PUNCT
iajs-1139	162	25	.	.	PUNCT
iajs-1139	163	1	hence	hence	ADV
iajs-1139	163	2	at	at	ADP
iajs-1139	163	3			PROPN
iajs-1139	163	4	yt	yt	PROPN
iajs-1139	163	5	is	be	AUX
iajs-1139	163	6	prime	prime	ADJ
iajs-1139	163	7	in	in	ADP
iajs-1139	163	8	t	t	PROPN
iajs-1139	163	9	t	t	NOUN
iajs-1139	163	10	by	by	ADP
iajs-1139	163	11	(	(	PUNCT
iajs-1139	163	12	12,page	12,page	NUM
iajs-1139	163	13	53	53	NUM
iajs-1139	163	14	)	)	PUNCT
iajs-1139	163	15	.	.	PUNCT
iajs-1139	164	1	that	that	PRON
iajs-1139	164	2	is	be	AUX
iajs-1139	164	3	wt	wt	NOUN
iajs-1139	164	4	is	be	AUX
iajs-1139	164	5	a	a	DET
iajs-1139	164	6	prime	prime	ADJ
iajs-1139	164	7	ideal	ideal	NOUN
iajs-1139	164	8	in	in	ADP
iajs-1139	164	9	(	(	PUNCT
iajs-1139	164	10	x	x	SYM
iajs-1139	164	11			ADJ
iajs-1139	164	12	y)t	y)t	PUNCT
iajs-1139	165	1	=	=	PUNCT
iajs-1139	165	2	t	t	PROPN
iajs-1139	165	3	t	t	NOUN
iajs-1139	165	4	.	.	PUNCT
iajs-1139	166	1	thus	thus	ADV
iajs-1139	166	2	w	w	NOUN
iajs-1139	166	3	is	be	AUX
iajs-1139	166	4	a	a	DET
iajs-1139	166	5	prime	prime	ADJ
iajs-1139	166	6	fuzzy	fuzzy	ADJ
iajs-1139	166	7	ideal	ideal	NOUN
iajs-1139	166	8	of	of	ADP
iajs-1139	166	9	x	x	PROPN
iajs-1139	166	10			PROPN
iajs-1139	166	11	y	y	PROPN
iajs-1139	166	12	=	=	SYM
iajs-1139	166	13	t	t	PROPN
iajs-1139	166	14	by	by	ADP
iajs-1139	166	15	(	(	PUNCT
iajs-1139	166	16	11	11	NUM
iajs-1139	166	17	,	,	PUNCT
iajs-1139	166	18	proposition	proposition	NOUN
iajs-1139	166	19	1.2.9	1.2.9	NUM
iajs-1139	166	20	)	)	PUNCT
iajs-1139	166	21	.	.	PUNCT
iajs-1139	167	1	now	now	ADV
iajs-1139	167	2	we	we	PRON
iajs-1139	167	3	can	can	AUX
iajs-1139	167	4	give	give	VERB
iajs-1139	167	5	the	the	DET
iajs-1139	167	6	following	follow	VERB
iajs-1139	167	7	main	main	ADJ
iajs-1139	167	8	result	result	NOUN
iajs-1139	167	9	.	.	PUNCT
iajs-1139	168	1	2.14	2.14	NUM
iajs-1139	168	2	theorem	theorem	VERB
iajs-1139	168	3	let	let	VERB
iajs-1139	168	4	x	x	PRON
iajs-1139	168	5	and	and	CCONJ
iajs-1139	168	6	y	y	PROPN
iajs-1139	168	7	be	be	AUX
iajs-1139	168	8	fuzzy	fuzzy	ADJ
iajs-1139	168	9	rings	ring	NOUN
iajs-1139	168	10	of	of	ADP
iajs-1139	168	11	r1	r1	PROPN
iajs-1139	168	12	,	,	PUNCT
iajs-1139	168	13	r2	r2	PROPN
iajs-1139	168	14	respectively	respectively	ADV
iajs-1139	168	15	.	.	PUNCT
iajs-1139	169	1	if	if	SCONJ
iajs-1139	169	2	a	a	PRON
iajs-1139	169	3	and	and	CCONJ
iajs-1139	169	4	b	b	NOUN
iajs-1139	169	5	are	be	AUX
iajs-1139	169	6	semiessential	semiessential	ADJ
iajs-1139	169	7	fuzzy	fuzzy	ADJ
iajs-1139	169	8	ideals	ideal	NOUN
iajs-1139	169	9	of	of	ADP
iajs-1139	169	10	x	x	PROPN
iajs-1139	169	11	,	,	PUNCT
iajs-1139	169	12	y	y	PROPN
iajs-1139	169	13	respectively	respectively	ADV
iajs-1139	169	14	.	.	PUNCT
iajs-1139	170	1	then	then	ADV
iajs-1139	170	2	ab	ab	PROPN
iajs-1139	170	3	is	be	AUX
iajs-1139	170	4	a	a	DET
iajs-1139	170	5	semiessential	semiessential	ADJ
iajs-1139	170	6	fuzzy	fuzzy	ADJ
iajs-1139	170	7	ideal	ideal	NOUN
iajs-1139	170	8	of	of	ADP
iajs-1139	170	9	x	x	PROPN
iajs-1139	170	10			PROPN
iajs-1139	170	11	y.	y.	PROPN
iajs-1139	170	12	ibn	ibn	PROPN
iajs-1139	170	13	alhaitham	alhaitham	PROPN
iajs-1139	170	14	j.	j.	PROPN
iajs-1139	170	15	for	for	ADP
iajs-1139	170	16	pure	pure	ADJ
iajs-1139	170	17	&	&	CCONJ
iajs-1139	170	18	appl	appl	PROPN
iajs-1139	170	19	.	.	PUNCT
iajs-1139	171	1	sci	sci	PROPN
iajs-1139	171	2	.	.	PUNCT
iajs-1139	172	1	vol.22	vol.22	PROPN
iajs-1139	172	2	(	(	PUNCT
iajs-1139	172	3	4	4	NUM
iajs-1139	172	4	)	)	PUNCT
iajs-1139	172	5	2009	2009	NUM
iajs-1139	172	6	proof	proof	NOUN
iajs-1139	172	7	:	:	PUNCT
iajs-1139	172	8	to	to	PART
iajs-1139	172	9	prove	prove	VERB
iajs-1139	172	10	a	a	DET
iajs-1139	172	11			ADJ
iajs-1139	172	12	b	b	NOUN
iajs-1139	172	13	is	be	AUX
iajs-1139	172	14	semiessential	semiessential	ADJ
iajs-1139	172	15	fuzzy	fuzzy	ADJ
iajs-1139	172	16	ideal	ideal	NOUN
iajs-1139	172	17	of	of	ADP
iajs-1139	172	18	x	x	PROPN
iajs-1139	172	19			PROPN
iajs-1139	172	20	y.	y.	PROPN
iajs-1139	172	21	since	since	SCONJ
iajs-1139	172	22	a	a	DET
iajs-1139	172	23			PROPN
iajs-1139	172	24	b	b	NOUN
iajs-1139	172	25	is	be	AUX
iajs-1139	172	26	a	a	DET
iajs-1139	172	27	non	non	ADJ
iajs-1139	172	28	-	-	ADJ
iajs-1139	172	29	zero	zero	ADJ
iajs-1139	172	30	fuzzy	fuzzy	ADJ
iajs-1139	172	31	ideal	ideal	NOUN
iajs-1139	172	32	of	of	ADP
iajs-1139	172	33	xy	xy	PROPN
iajs-1139	172	34	by	by	ADP
iajs-1139	172	35	(	(	PUNCT
iajs-1139	172	36	10	10	NUM
iajs-1139	172	37	,	,	PUNCT
iajs-1139	172	38	theorem	theorem	ADJ
iajs-1139	172	39	2.4.1.9	2.4.1.9	NUM
iajs-1139	172	40	)	)	PUNCT
iajs-1139	172	41	,	,	PUNCT
iajs-1139	172	42	there	there	PRON
iajs-1139	172	43	exists	exist	VERB
iajs-1139	172	44	(	(	PUNCT
iajs-1139	172	45	a	a	DET
iajs-1139	172	46	,	,	PUNCT
iajs-1139	172	47	b	b	NOUN
iajs-1139	172	48	)	)	PUNCT
iajs-1139	172	49			NOUN
iajs-1139	172	50	r1	r1	PROPN
iajs-1139	172	51			PROPN
iajs-1139	172	52	r2	r2	PROPN
iajs-1139	172	53	such	such	ADJ
iajs-1139	172	54	that	that	SCONJ
iajs-1139	172	55	(	(	PUNCT
iajs-1139	172	56	a	a	DET
iajs-1139	172	57			ADJ
iajs-1139	172	58	b)(a	b)(a	NOUN
iajs-1139	172	59	,	,	PUNCT
iajs-1139	172	60	b	b	NOUN
iajs-1139	172	61	)	)	PUNCT
iajs-1139	172	62	=	=	SYM
iajs-1139	172	63	min{a(a),b(b	min{a(a),b(b	PROPN
iajs-1139	172	64	)	)	PUNCT
iajs-1139	172	65	}	}	PUNCT
iajs-1139	172	66			NOUN
iajs-1139	172	67	0	0	NUM
iajs-1139	172	68	.	.	PUNCT
iajs-1139	173	1	thus	thus	ADV
iajs-1139	173	2	a(a	a(a	PROPN
iajs-1139	173	3	)	)	PUNCT
iajs-1139	173	4			NOUN
iajs-1139	173	5	0	0	NUM
iajs-1139	173	6	and	and	CCONJ
iajs-1139	173	7	b(b	b(b	PROPN
iajs-1139	173	8	)	)	PUNCT
iajs-1139	173	9	}	}	PUNCT
iajs-1139	173	10			NOUN
iajs-1139	173	11	0	0	NUM
iajs-1139	173	12	.	.	PUNCT
iajs-1139	174	1	now	now	ADV
iajs-1139	174	2	,	,	PUNCT
iajs-1139	174	3	let	let	VERB
iajs-1139	174	4	w	w	NOUN
iajs-1139	174	5	be	be	AUX
iajs-1139	174	6	a	a	DET
iajs-1139	174	7	non	non	ADJ
iajs-1139	174	8	-	-	ADJ
iajs-1139	174	9	zero	zero	ADJ
iajs-1139	174	10	prime	prime	ADJ
iajs-1139	174	11	fuzzy	fuzzy	ADJ
iajs-1139	174	12	ideal	ideal	NOUN
iajs-1139	174	13	of	of	ADP
iajs-1139	174	14	x	x	PROPN
iajs-1139	174	15			PROPN
iajs-1139	174	16	y	y	PROPN
iajs-1139	174	17	,	,	PUNCT
iajs-1139	174	18	hence	hence	ADV
iajs-1139	174	19	either	either	CCONJ
iajs-1139	175	1	w	w	X
iajs-1139	175	2	=	=	PUNCT
iajs-1139	175	3	c	c	PROPN
iajs-1139	175	4			VERB
iajs-1139	175	5	y	y	PROPN
iajs-1139	175	6	or	or	CCONJ
iajs-1139	175	7	w	w	NOUN
iajs-1139	175	8	=	=	NOUN
iajs-1139	175	9	x	x	SYM
iajs-1139	175	10			PROPN
iajs-1139	175	11	d	d	PROPN
iajs-1139	175	12	,	,	PUNCT
iajs-1139	175	13	for	for	ADP
iajs-1139	175	14	some	some	DET
iajs-1139	175	15	prime	prime	ADJ
iajs-1139	175	16	fuzzy	fuzzy	ADJ
iajs-1139	175	17	ideals	ideal	NOUN
iajs-1139	175	18	c	c	NOUN
iajs-1139	175	19	,	,	PUNCT
iajs-1139	175	20	d	d	NOUN
iajs-1139	175	21	of	of	ADP
iajs-1139	175	22	x	x	PROPN
iajs-1139	175	23	,	,	PUNCT
iajs-1139	175	24	y	y	PROPN
iajs-1139	175	25	respectively	respectively	ADV
iajs-1139	175	26	.	.	PUNCT
iajs-1139	176	1	assume	assume	VERB
iajs-1139	176	2	w	w	NOUN
iajs-1139	176	3	=	=	PUNCT
iajs-1139	176	4	c	c	PROPN
iajs-1139	176	5			PROPN
iajs-1139	176	6	y.	y.	NOUN
iajs-1139	176	7	if	if	SCONJ
iajs-1139	176	8	c	c	NOUN
iajs-1139	176	9	=	=	SYM
iajs-1139	176	10	o1	o1	PROPN
iajs-1139	176	11	,	,	PUNCT
iajs-1139	176	12	then	then	ADV
iajs-1139	176	13	(	(	PUNCT
iajs-1139	176	14	a	a	DET
iajs-1139	176	15			ADJ
iajs-1139	176	16	b	b	NOUN
iajs-1139	176	17	)	)	PUNCT
iajs-1139	176	18			PUNCT
iajs-1139	177	1	w	w	NOUN
iajs-1139	177	2	=	=	SYM
iajs-1139	177	3	(	(	PUNCT
iajs-1139	177	4	a	a	DET
iajs-1139	177	5			ADJ
iajs-1139	177	6	b	b	NOUN
iajs-1139	177	7	)	)	PUNCT
iajs-1139	177	8			NOUN
iajs-1139	177	9	(	(	PUNCT
iajs-1139	177	10	c	c	PROPN
iajs-1139	177	11			PROPN
iajs-1139	177	12	y	y	PROPN
iajs-1139	177	13	)	)	PUNCT
iajs-1139	177	14	=	=	PUNCT
iajs-1139	178	1	(	(	PUNCT
iajs-1139	178	2	a	a	DET
iajs-1139	178	3			ADJ
iajs-1139	178	4	c	c	NOUN
iajs-1139	178	5	)	)	PUNCT
iajs-1139	178	6			NOUN
iajs-1139	178	7	(	(	PUNCT
iajs-1139	178	8	b	b	PROPN
iajs-1139	178	9			X
iajs-1139	178	10	y	y	NOUN
iajs-1139	178	11	)	)	PUNCT
iajs-1139	178	12	=	=	SYM
iajs-1139	178	13	o1	o1	PROPN
iajs-1139	178	14			PROPN
iajs-1139	178	15	b	b	PROPN
iajs-1139	178	16	but	but	CCONJ
iajs-1139	178	17	(	(	PUNCT
iajs-1139	178	18	o1	o1	NOUN
iajs-1139	178	19			VERB
iajs-1139	178	20	b)(a	b)(a	PROPN
iajs-1139	178	21	,	,	PUNCT
iajs-1139	178	22	b	b	NOUN
iajs-1139	178	23	)	)	PUNCT
iajs-1139	178	24	=	=	SYM
iajs-1139	178	25	min{o1(0),b(b	min{o1(0),b(b	NOUN
iajs-1139	178	26	)	)	PUNCT
iajs-1139	178	27	}	}	PUNCT
iajs-1139	178	28	=	=	SYM
iajs-1139	178	29	min	min	NOUN
iajs-1139	178	30	{	{	PUNCT
iajs-1139	178	31	1	1	NUM
iajs-1139	178	32	,	,	PUNCT
iajs-1139	178	33	b(b	b(b	PROPN
iajs-1139	178	34	)	)	PUNCT
iajs-1139	178	35	}	}	PUNCT
iajs-1139	178	36	=	=	SYM
iajs-1139	178	37	b(b	b(b	PROPN
iajs-1139	178	38	)	)	PUNCT
iajs-1139	178	39			NOUN
iajs-1139	178	40	0	0	NUM
iajs-1139	179	1	thus	thus	ADV
iajs-1139	179	2	(	(	PUNCT
iajs-1139	179	3	a	a	DET
iajs-1139	179	4			ADJ
iajs-1139	179	5	b	b	NOUN
iajs-1139	179	6	)	)	PUNCT
iajs-1139	179	7			PUNCT
iajs-1139	179	8	w	w	PROPN
iajs-1139	179	9			NOUN
iajs-1139	179	10	o1	o1	NOUN
iajs-1139	179	11	.	.	PUNCT
iajs-1139	180	1	if	if	SCONJ
iajs-1139	180	2	c	c	PROPN
iajs-1139	180	3			NOUN
iajs-1139	180	4	o1	o1	PROPN
iajs-1139	180	5	,	,	PUNCT
iajs-1139	180	6	then	then	ADV
iajs-1139	180	7	a	a	DET
iajs-1139	180	8			PROPN
iajs-1139	180	9	c	c	PROPN
iajs-1139	180	10			NOUN
iajs-1139	180	11	o1	o1	PROPN
iajs-1139	180	12	,	,	PUNCT
iajs-1139	180	13	since	since	SCONJ
iajs-1139	180	14	a	a	PRON
iajs-1139	180	15	is	be	AUX
iajs-1139	180	16	semiessential	semiessential	ADJ
iajs-1139	180	17	in	in	ADP
iajs-1139	180	18	x.	x.	NOUN
iajs-1139	180	19	hence	hence	ADV
iajs-1139	180	20	there	there	PRON
iajs-1139	180	21	exists	exist	VERB
iajs-1139	180	22	a1	a1	PROPN
iajs-1139	180	23			NOUN
iajs-1139	180	24	r1	r1	NOUN
iajs-1139	180	25	such	such	ADJ
iajs-1139	180	26	that	that	SCONJ
iajs-1139	180	27	(	(	PUNCT
iajs-1139	180	28	a	a	DET
iajs-1139	180	29			NOUN
iajs-1139	180	30	c)(a1	c)(a1	NOUN
iajs-1139	180	31	)	)	PUNCT
iajs-1139	180	32	o1	o1	PROPN
iajs-1139	180	33	,	,	PUNCT
iajs-1139	180	34	so	so	ADV
iajs-1139	180	35	min{a(a1),c(a1)}0	min{a(a1),c(a1)}0	PROPN
iajs-1139	180	36	,	,	PUNCT
iajs-1139	180	37	since	since	SCONJ
iajs-1139	180	38	(	(	PUNCT
iajs-1139	180	39	ab)(cy)=	ab)(cy)=	PROPN
iajs-1139	180	40	(	(	PUNCT
iajs-1139	180	41	a	a	DET
iajs-1139	180	42			ADJ
iajs-1139	180	43	c	c	NOUN
iajs-1139	180	44	)	)	PUNCT
iajs-1139	180	45			NOUN
iajs-1139	180	46	(	(	PUNCT
iajs-1139	180	47	b	b	PROPN
iajs-1139	180	48			X
iajs-1139	180	49	y	y	NOUN
iajs-1139	180	50	)	)	PUNCT
iajs-1139	180	51	=	=	NOUN
iajs-1139	181	1	(	(	PUNCT
iajs-1139	181	2	a	a	DET
iajs-1139	181	3			ADJ
iajs-1139	181	4	c	c	NOUN
iajs-1139	181	5	)	)	PUNCT
iajs-1139	181	6			PROPN
iajs-1139	181	7	b.	b.	PROPN
iajs-1139	182	1	it	it	PRON
iajs-1139	182	2	follows	follow	VERB
iajs-1139	182	3	that	that	SCONJ
iajs-1139	183	1	[	[	X
iajs-1139	183	2	(	(	PUNCT
iajs-1139	183	3	a	a	DET
iajs-1139	183	4			ADJ
iajs-1139	183	5	c	c	NOUN
iajs-1139	183	6	)	)	PUNCT
iajs-1139	183	7			ADJ
iajs-1139	183	8	b](a	b](a	NOUN
iajs-1139	183	9	,	,	PUNCT
iajs-1139	183	10	b	b	NOUN
iajs-1139	183	11	)	)	PUNCT
iajs-1139	183	12	=	=	SYM
iajs-1139	183	13	min{a(a1),c(a1),b(b	min{a(a1),c(a1),b(b	NOUN
iajs-1139	183	14	)	)	PUNCT
iajs-1139	183	15	}	}	PUNCT
iajs-1139	183	16			NOUN
iajs-1139	183	17	0	0	NUM
iajs-1139	183	18	.	.	PUNCT
iajs-1139	184	1	that	that	SCONJ
iajs-1139	184	2	[	[	X
iajs-1139	184	3	(	(	PUNCT
iajs-1139	184	4	a	a	DET
iajs-1139	184	5			ADJ
iajs-1139	184	6	b	b	NOUN
iajs-1139	184	7	)	)	PUNCT
iajs-1139	184	8			NOUN
iajs-1139	184	9	(	(	PUNCT
iajs-1139	184	10	c	c	PROPN
iajs-1139	184	11			PROPN
iajs-1139	184	12	y	y	PROPN
iajs-1139	184	13	)	)	PUNCT
iajs-1139	184	14			PROPN
iajs-1139	184	15	o(x	o(x	ADJ
iajs-1139	184	16			PROPN
iajs-1139	184	17	y	y	PROPN
iajs-1139	184	18	)	)	PUNCT
iajs-1139	184	19			NOUN
iajs-1139	184	20	o1(0,0	o1(0,0	NOUN
iajs-1139	184	21	)	)	PUNCT
iajs-1139	184	22	.	.	PUNCT
iajs-1139	185	1	similarly	similarly	ADV
iajs-1139	185	2	,	,	PUNCT
iajs-1139	185	3	if	if	SCONJ
iajs-1139	185	4	w	w	ADP
iajs-1139	185	5	=	=	PUNCT
iajs-1139	185	6	x	x	SYM
iajs-1139	185	7			PROPN
iajs-1139	185	8	d	d	NOUN
iajs-1139	185	9	,	,	PUNCT
iajs-1139	185	10	then	then	ADV
iajs-1139	185	11	(	(	PUNCT
iajs-1139	185	12	a	a	PROPN
iajs-1139	185	13	b	b	X
iajs-1139	185	14	)	)	PUNCT
iajs-1139	185	15	wo1	wo1	PROPN
iajs-1139	185	16	.	.	PUNCT
iajs-1139	186	1	therefore	therefore	ADV
iajs-1139	186	2	,	,	PUNCT
iajs-1139	186	3	a	a	DET
iajs-1139	186	4			ADJ
iajs-1139	186	5	b	b	NOUN
iajs-1139	186	6	is	be	AUX
iajs-1139	186	7	semiessential	semiessential	ADJ
iajs-1139	186	8	.	.	PUNCT
iajs-1139	187	1	the	the	DET
iajs-1139	187	2	converse	converse	NOUN
iajs-1139	187	3	of	of	ADP
iajs-1139	187	4	theorem	theorem	NOUN
iajs-1139	187	5	2.14	2.14	NUM
iajs-1139	187	6	is	be	AUX
iajs-1139	187	7	not	not	PART
iajs-1139	187	8	true	true	ADJ
iajs-1139	187	9	in	in	ADP
iajs-1139	187	10	general	general	ADJ
iajs-1139	187	11	as	as	ADP
iajs-1139	187	12	the	the	DET
iajs-1139	187	13	following	follow	VERB
iajs-1139	187	14	example	example	NOUN
iajs-1139	187	15	shows	show	VERB
iajs-1139	187	16	;	;	PUNCT
iajs-1139	187	17	2.15	2.15	NUM
iajs-1139	187	18	example	example	NOUN
iajs-1139	187	19	let	let	VERB
iajs-1139	187	20	x	x	PRON
iajs-1139	187	21	:	:	PUNCT
iajs-1139	187	22	z6	z6	PROPN
iajs-1139	187	23			PROPN
iajs-1139	188	1	[	[	X
iajs-1139	188	2	0,1	0,1	NUM
iajs-1139	188	3	]	]	PUNCT
iajs-1139	188	4	,	,	PUNCT
iajs-1139	188	5	y	y	PROPN
iajs-1139	188	6	:	:	PUNCT
iajs-1139	188	7	z12	z12	NUM
iajs-1139	188	8			X
iajs-1139	189	1	[	[	X
iajs-1139	189	2	0,1	0,1	NUM
iajs-1139	189	3	]	]	PUNCT
iajs-1139	189	4	define	define	NOUN
iajs-1139	189	5	by	by	ADP
iajs-1139	189	6	x(a	x(a	NOUN
iajs-1139	189	7	)	)	PUNCT
iajs-1139	189	8	=	=	SYM
iajs-1139	189	9	1	1	NUM
iajs-1139	189	10	,	,	PUNCT
iajs-1139	189	11			VERB
iajs-1139	189	12	a	a	DET
iajs-1139	189	13			PROPN
iajs-1139	189	14	z6	z6	PROPN
iajs-1139	189	15	,	,	PUNCT
iajs-1139	189	16	y(b	y(b	PROPN
iajs-1139	189	17	)	)	PUNCT
iajs-1139	190	1	=	=	SYM
iajs-1139	190	2	1	1	NUM
iajs-1139	190	3	,	,	PUNCT
iajs-1139	190	4			NOUN
iajs-1139	190	5	b	b	PROPN
iajs-1139	190	6			PROPN
iajs-1139	190	7	z12	z12	NUM
iajs-1139	190	8	,	,	PUNCT
iajs-1139	190	9	let	let	VERB
iajs-1139	190	10	a	a	DET
iajs-1139	190	11	:	:	PUNCT
iajs-1139	190	12	z6	z6	PROPN
iajs-1139	190	13			PROPN
iajs-1139	191	1	[	[	X
iajs-1139	191	2	0,1	0,1	NUM
iajs-1139	191	3	]	]	PUNCT
iajs-1139	191	4	,	,	PUNCT
iajs-1139	191	5	b	b	X
iajs-1139	191	6	:	:	PUNCT
iajs-1139	191	7	z12	z12	NUM
iajs-1139	191	8			X
iajs-1139	192	1	[	[	X
iajs-1139	192	2	0,1	0,1	NUM
iajs-1139	192	3	]	]	PUNCT
iajs-1139	192	4	defined	define	VERB
iajs-1139	192	5	by	by	ADP
iajs-1139	192	6	:	:	PUNCT
iajs-1139	192	7	1	1	NUM
iajs-1139	192	8	if	if	SCONJ
iajs-1139	192	9	{	{	PUNCT
iajs-1139	192	10	0,3	0,3	NOUN
iajs-1139	192	11	}	}	PUNCT
iajs-1139	192	12	(	(	PUNCT
iajs-1139	192	13	)	)	PUNCT
iajs-1139	192	14	0	0	PUNCT
iajs-1139	192	15	otherwise	otherwise	ADV
iajs-1139	192	16	.	.	PUNCT
iajs-1139	193	1	x	x	PUNCT
iajs-1139	193	2	x	x	PUNCT
iajs-1139	193	3			NOUN
iajs-1139	193	4			NOUN
iajs-1139	194	1			PROPN
iajs-1139	194	2			NUM
iajs-1139	195	1			NUM
iajs-1139	195	2			INTJ
iajs-1139	195	3	,	,	PUNCT
iajs-1139	195	4			NOUN
iajs-1139	195	5	x	x	SYM
iajs-1139	195	6			PROPN
iajs-1139	195	7	z6	z6	PROPN
iajs-1139	195	8	1	1	NUM
iajs-1139	195	9	if	if	SCONJ
iajs-1139	195	10	{	{	PUNCT
iajs-1139	195	11	0,3,6,9	0,3,6,9	NUM
iajs-1139	195	12	}	}	PUNCT
iajs-1139	195	13	(	(	PUNCT
iajs-1139	195	14	)	)	PUNCT
iajs-1139	195	15	0	0	PUNCT
iajs-1139	195	16	otherwise	otherwise	ADV
iajs-1139	195	17	.	.	PUNCT
iajs-1139	196	1	x	x	PUNCT
iajs-1139	196	2	x	x	PUNCT
iajs-1139	196	3			ADP
iajs-1139	196	4			PROPN
iajs-1139	196	5			NOUN
iajs-1139	196	6			PROPN
iajs-1139	197	1			NUM
iajs-1139	197	2			INTJ
iajs-1139	197	3	,	,	PUNCT
iajs-1139	197	4			NOUN
iajs-1139	197	5	x	x	SYM
iajs-1139	197	6			PROPN
iajs-1139	197	7	z12	z12	NUM
iajs-1139	197	8	it	it	PRON
iajs-1139	197	9	is	be	AUX
iajs-1139	197	10	easy	easy	ADJ
iajs-1139	197	11	to	to	PART
iajs-1139	197	12	check	check	VERB
iajs-1139	197	13	that	that	PRON
iajs-1139	197	14	a	a	PRON
iajs-1139	197	15	and	and	CCONJ
iajs-1139	197	16	b	b	NOUN
iajs-1139	197	17	are	be	AUX
iajs-1139	197	18	fuzzy	fuzzy	ADJ
iajs-1139	197	19	ideals	ideal	NOUN
iajs-1139	197	20	of	of	ADP
iajs-1139	197	21	the	the	DET
iajs-1139	197	22	fuzzy	fuzzy	ADJ
iajs-1139	197	23	rings	ring	NOUN
iajs-1139	197	24	x	x	PUNCT
iajs-1139	197	25	and	and	CCONJ
iajs-1139	197	26	y	y	PROPN
iajs-1139	197	27	respectively	respectively	ADV
iajs-1139	197	28	.	.	PUNCT
iajs-1139	198	1	1	1	NUM
iajs-1139	199	1	if	if	SCONJ
iajs-1139	199	2	{	{	PUNCT
iajs-1139	199	3	0,3	0,3	NOUN
iajs-1139	199	4	}	}	PUNCT
iajs-1139	199	5	,	,	PUNCT
iajs-1139	199	6	{	{	PUNCT
iajs-1139	199	7	0,3,6,9	0,3,6,9	NUM
iajs-1139	199	8	}	}	PUNCT
iajs-1139	199	9	(	(	PUNCT
iajs-1139	199	10	)	)	PUNCT
iajs-1139	199	11	(	(	PUNCT
iajs-1139	199	12	,	,	PUNCT
iajs-1139	199	13	)	)	PUNCT
iajs-1139	199	14	0	0	PUNCT
iajs-1139	199	15	otherwise	otherwise	ADV
iajs-1139	199	16	.	.	PUNCT
iajs-1139	200	1			ADP
iajs-1139	200	2			PROPN
iajs-1139	200	3			PROPN
iajs-1139	200	4			VERB
iajs-1139	200	5			PROPN
iajs-1139	201	1			NUM
iajs-1139	201	2			NOUN
iajs-1139	201	3	x	x	X
iajs-1139	201	4	y	y	NOUN
iajs-1139	201	5	x	x	SYM
iajs-1139	201	6	y	y	PROPN
iajs-1139	201	7	(	(	PUNCT
iajs-1139	201	8	a	a	DET
iajs-1139	201	9			ADJ
iajs-1139	201	10	b)t	b)t	NOUN
iajs-1139	201	11	=	=	PUNCT
iajs-1139	201	12	a	a	DET
iajs-1139	201	13	t	t	NOUN
iajs-1139	201	14			VERB
iajs-1139	201	15	bt	bt	NOUN
iajs-1139	202	1	=	=	PUNCT
iajs-1139	202	2	<3	<3	X
iajs-1139	202	3	>	>	X
iajs-1139	202	4			PROPN
iajs-1139	203	1	<3	<3	X
iajs-1139	203	2	>	>	X
iajs-1139	203	3	is	be	AUX
iajs-1139	203	4	a	a	DET
iajs-1139	203	5	semiessential	semiessential	ADJ
iajs-1139	203	6	ideal	ideal	NOUN
iajs-1139	203	7	in	in	ADP
iajs-1139	203	8	z6	z6	PROPN
iajs-1139	203	9			PROPN
iajs-1139	203	10	z12	z12	PROPN
iajs-1139	203	11	.	.	PUNCT
iajs-1139	204	1	since	since	SCONJ
iajs-1139	204	2	the	the	DET
iajs-1139	204	3	prime	prime	ADJ
iajs-1139	204	4	ideals	ideal	NOUN
iajs-1139	204	5	in	in	ADP
iajs-1139	204	6	z6	z6	PROPN
iajs-1139	204	7			PROPN
iajs-1139	204	8	z12	z12	PROPN
iajs-1139	204	9	are	be	AUX
iajs-1139	204	10	:	:	PUNCT
iajs-1139	204	11	<	<	X
iajs-1139	204	12	2	2	X
iajs-1139	204	13	>	>	X
iajs-1139	204	14			PROPN
iajs-1139	204	15	z12	z12	PROPN
iajs-1139	204	16	,	,	PUNCT
iajs-1139	204	17	<3	<3	PROPN
iajs-1139	204	18	>	>	X
iajs-1139	204	19			PROPN
iajs-1139	204	20	z12	z12	PROPN
iajs-1139	204	21	,	,	PUNCT
iajs-1139	204	22	z6	z6	PROPN
iajs-1139	204	23			PROPN
iajs-1139	204	24	<	<	X
iajs-1139	204	25	2	2	NUM
iajs-1139	204	26	>	>	PUNCT
iajs-1139	204	27	,	,	PUNCT
iajs-1139	204	28	z6	z6	PROPN
iajs-1139	204	29			VERB
iajs-1139	204	30	<3	<3	X
iajs-1139	204	31	>	>	PUNCT
iajs-1139	204	32	.	.	PUNCT
iajs-1139	205	1	hence	hence	ADV
iajs-1139	205	2	(	(	PUNCT
iajs-1139	205	3	(	(	PUNCT
iajs-1139	205	4	3	3	X
iajs-1139	205	5	)	)	PUNCT
iajs-1139	205	6			NOUN
iajs-1139	205	7	(	(	PUNCT
iajs-1139	205	8	3	3	NUM
iajs-1139	205	9	)	)	PUNCT
iajs-1139	205	10	)	)	PUNCT
iajs-1139	206	1			X
iajs-1139	206	2	(	(	PUNCT
iajs-1139	206	3	(	(	PUNCT
iajs-1139	206	4	2	2	X
iajs-1139	206	5	)	)	PUNCT
iajs-1139	206	6			PROPN
iajs-1139	206	7	z12)0	z12)0	PROPN
iajs-1139	206	8	,	,	PUNCT
iajs-1139	206	9	(	(	PUNCT
iajs-1139	206	10	(	(	PUNCT
iajs-1139	206	11	3	3	X
iajs-1139	206	12	)	)	PUNCT
iajs-1139	206	13			NOUN
iajs-1139	206	14	(	(	PUNCT
iajs-1139	206	15	3	3	NUM
iajs-1139	206	16	)	)	PUNCT
iajs-1139	206	17	)	)	PUNCT
iajs-1139	207	1			NOUN
iajs-1139	207	2	(	(	PUNCT
iajs-1139	207	3	(	(	PUNCT
iajs-1139	207	4	3	3	X
iajs-1139	207	5	)	)	PUNCT
iajs-1139	207	6			PROPN
iajs-1139	207	7	z12	z12	PROPN
iajs-1139	207	8	)	)	PUNCT
iajs-1139	207	9			PROPN
iajs-1139	207	10	0	0	NUM
iajs-1139	207	11	,	,	PUNCT
iajs-1139	207	12	(	(	PUNCT
iajs-1139	207	13	(	(	PUNCT
iajs-1139	207	14	3	3	X
iajs-1139	207	15	)	)	PUNCT
iajs-1139	207	16			NOUN
iajs-1139	207	17	(	(	PUNCT
iajs-1139	207	18	3	3	NUM
iajs-1139	207	19	)	)	PUNCT
iajs-1139	207	20	)	)	PUNCT
iajs-1139	208	1			NOUN
iajs-1139	208	2	(	(	PUNCT
iajs-1139	208	3	z6	z6	PROPN
iajs-1139	208	4			PROPN
iajs-1139	208	5	(	(	PUNCT
iajs-1139	208	6	2	2	NUM
iajs-1139	208	7	)	)	PUNCT
iajs-1139	208	8	)	)	PUNCT
iajs-1139	208	9			NOUN
iajs-1139	208	10	0	0	NUM
iajs-1139	209	1	and	and	CCONJ
iajs-1139	209	2	(	(	PUNCT
iajs-1139	209	3	(	(	PUNCT
iajs-1139	209	4	3	3	X
iajs-1139	209	5	)	)	PUNCT
iajs-1139	209	6			NOUN
iajs-1139	209	7	(	(	PUNCT
iajs-1139	209	8	3	3	NUM
iajs-1139	209	9	)	)	PUNCT
iajs-1139	209	10	)	)	PUNCT
iajs-1139	210	1			NOUN
iajs-1139	210	2	(	(	PUNCT
iajs-1139	210	3	z6	z6	PROPN
iajs-1139	210	4	(3	(3	X
iajs-1139	210	5	)	)	PUNCT
iajs-1139	210	6	)	)	PUNCT
iajs-1139	210	7			NOUN
iajs-1139	210	8	0	0	NUM
iajs-1139	210	9	.	.	NOUN
iajs-1139	211	1	which	which	PRON
iajs-1139	211	2	implies	imply	VERB
iajs-1139	211	3	(	(	PUNCT
iajs-1139	211	4	a	a	DET
iajs-1139	211	5			ADJ
iajs-1139	211	6	b)t	b)t	NOUN
iajs-1139	211	7	is	be	AUX
iajs-1139	211	8	semiessential	semiessential	ADJ
iajs-1139	211	9	in	in	ADP
iajs-1139	211	10	(	(	PUNCT
iajs-1139	211	11	x	x	X
iajs-1139	211	12			PROPN
iajs-1139	211	13	y)t	y)t	NOUN
iajs-1139	211	14	,	,	PUNCT
iajs-1139	211	15			NOUN
iajs-1139	211	16	t.	t.	NOUN
iajs-1139	211	17	thus	thus	ADV
iajs-1139	211	18	a	a	DET
iajs-1139	211	19			ADJ
iajs-1139	211	20	b	b	NOUN
iajs-1139	211	21	is	be	AUX
iajs-1139	211	22	semiessential	semiessential	ADJ
iajs-1139	211	23	fuzzy	fuzzy	ADJ
iajs-1139	211	24	ideal	ideal	NOUN
iajs-1139	211	25	of	of	ADP
iajs-1139	211	26	x	x	PROPN
iajs-1139	211	27			PROPN
iajs-1139	211	28	y	y	PROPN
iajs-1139	211	29	by	by	ADP
iajs-1139	211	30	theorem	theorem	NOUN
iajs-1139	211	31	2.3	2.3	NUM
iajs-1139	211	32	.	.	PUNCT
iajs-1139	212	1	but	but	CCONJ
iajs-1139	212	2	a	a	PRON
iajs-1139	212	3	is	be	AUX
iajs-1139	212	4	not	not	PART
iajs-1139	212	5	semiessential	semiessential	ADJ
iajs-1139	212	6	fuzzy	fuzzy	ADJ
iajs-1139	212	7	ideal	ideal	NOUN
iajs-1139	212	8	of	of	ADP
iajs-1139	212	9	x	x	PRON
iajs-1139	212	10	,	,	PUNCT
iajs-1139	212	11	since	since	SCONJ
iajs-1139	212	12	there	there	PRON
iajs-1139	212	13	exists	exist	VERB
iajs-1139	212	14	prime	prime	ADJ
iajs-1139	212	15	fuzzy	fuzzy	ADJ
iajs-1139	212	16	ideal	ideal	NOUN
iajs-1139	212	17	c	c	PROPN
iajs-1139	212	18	of	of	ADP
iajs-1139	212	19	x	x	INTJ
iajs-1139	212	20	such	such	ADJ
iajs-1139	212	21	that	that	SCONJ
iajs-1139	212	22	a	a	DET
iajs-1139	212	23			NOUN
iajs-1139	212	24	c	c	NOUN
iajs-1139	212	25	=	=	SYM
iajs-1139	212	26	o1	o1	PROPN
iajs-1139	212	27	,	,	PUNCT
iajs-1139	212	28	where	where	SCONJ
iajs-1139	212	29	1	1	NUM
iajs-1139	212	30	if	if	SCONJ
iajs-1139	212	31	{	{	PUNCT
iajs-1139	212	32	0,2,4	0,2,4	NOUN
iajs-1139	212	33	}	}	SYM
iajs-1139	212	34	c	c	NOUN
iajs-1139	212	35	(	(	PUNCT
iajs-1139	212	36	)	)	PUNCT
iajs-1139	212	37	0	0	PUNCT
iajs-1139	212	38	otherwise	otherwise	ADV
iajs-1139	212	39	.	.	PUNCT
iajs-1139	213	1			ADP
iajs-1139	214	1			NOUN
iajs-1139	215	1			NUM
iajs-1139	216	1			NUM
iajs-1139	216	2			NOUN
iajs-1139	216	3	x	x	SYM
iajs-1139	216	4	x	x	PUNCT
iajs-1139	216	5			NOUN
iajs-1139	216	6	x	x	SYM
iajs-1139	216	7			PROPN
iajs-1139	216	8	z6	z6	PROPN
iajs-1139	216	9	.	.	PUNCT
iajs-1139	217	1	similarly	similarly	ADV
iajs-1139	217	2	,	,	PUNCT
iajs-1139	217	3	we	we	PRON
iajs-1139	217	4	can	can	AUX
iajs-1139	217	5	show	show	VERB
iajs-1139	217	6	that	that	SCONJ
iajs-1139	217	7	b	b	NOUN
iajs-1139	217	8	is	be	AUX
iajs-1139	217	9	not	not	PART
iajs-1139	217	10	semiessential	semiessential	ADJ
iajs-1139	217	11	fuzzy	fuzzy	ADJ
iajs-1139	217	12	ideal	ideal	NOUN
iajs-1139	217	13	of	of	ADP
iajs-1139	217	14	y.	y.	PROPN
iajs-1139	217	15	next	next	ADV
iajs-1139	217	16	,	,	PUNCT
iajs-1139	217	17	we	we	PRON
iajs-1139	217	18	have	have	VERB
iajs-1139	217	19	the	the	DET
iajs-1139	217	20	following	follow	VERB
iajs-1139	217	21	proposition	proposition	NOUN
iajs-1139	217	22	about	about	ADP
iajs-1139	217	23	the	the	DET
iajs-1139	217	24	inverse	inverse	NOUN
iajs-1139	217	25	image	image	NOUN
iajs-1139	217	26	of	of	ADP
iajs-1139	217	27	semiessential	semiessential	ADJ
iajs-1139	217	28	fuzzy	fuzzy	ADJ
iajs-1139	217	29	ideals	ideal	NOUN
iajs-1139	217	30	.	.	PUNCT
iajs-1139	218	1	2.16	2.16	NUM
iajs-1139	218	2	proposition	proposition	NOUN
iajs-1139	218	3	let	let	VERB
iajs-1139	218	4	x	x	PRON
iajs-1139	218	5	and	and	CCONJ
iajs-1139	218	6	y	y	PROPN
iajs-1139	218	7	be	be	AUX
iajs-1139	218	8	fuzzy	fuzzy	ADJ
iajs-1139	218	9	rings	ring	NOUN
iajs-1139	218	10	of	of	ADP
iajs-1139	218	11	rings	ring	NOUN
iajs-1139	218	12	r1	r1	PROPN
iajs-1139	218	13	,	,	PUNCT
iajs-1139	218	14	r2	r2	PROPN
iajs-1139	218	15	respectively	respectively	ADV
iajs-1139	218	16	.	.	PUNCT
iajs-1139	219	1	let	let	VERB
iajs-1139	219	2	f	f	NOUN
iajs-1139	219	3	:	:	PUNCT
iajs-1139	219	4	r1	r1	PROPN
iajs-1139	219	5			PROPN
iajs-1139	219	6	r2	r2	PROPN
iajs-1139	219	7	be	be	VERB
iajs-1139	219	8	a	a	DET
iajs-1139	219	9	homomorphism	homomorphism	NOUN
iajs-1139	219	10	.	.	PUNCT
iajs-1139	220	1	if	if	SCONJ
iajs-1139	220	2	a	a	PRON
iajs-1139	220	3	is	be	AUX
iajs-1139	220	4	a	a	DET
iajs-1139	220	5	semiessential	semiessential	ADJ
iajs-1139	220	6	fuzzy	fuzzy	ADJ
iajs-1139	220	7	ideal	ideal	NOUN
iajs-1139	220	8	of	of	ADP
iajs-1139	220	9	y	y	PROPN
iajs-1139	220	10	,	,	PUNCT
iajs-1139	220	11	then	then	ADV
iajs-1139	220	12	f	f	PROPN
iajs-1139	220	13	1(a	1(a	NUM
iajs-1139	220	14	)	)	PUNCT
iajs-1139	220	15	is	be	AUX
iajs-1139	220	16	a	a	DET
iajs-1139	220	17	semiessential	semiessential	ADJ
iajs-1139	220	18	fuzzy	fuzzy	ADJ
iajs-1139	220	19	ideal	ideal	NOUN
iajs-1139	220	20	of	of	ADP
iajs-1139	220	21	x.	x.	PROPN
iajs-1139	220	22	ibn	ibn	PROPN
iajs-1139	220	23	alhaitham	alhaitham	PROPN
iajs-1139	220	24	j.	j.	PROPN
iajs-1139	220	25	for	for	ADP
iajs-1139	220	26	pure	pure	ADJ
iajs-1139	220	27	&	&	CCONJ
iajs-1139	220	28	appl	appl	PROPN
iajs-1139	220	29	.	.	PUNCT
iajs-1139	221	1	sci	sci	PROPN
iajs-1139	221	2	.	.	PUNCT
iajs-1139	222	1	vol.22	vol.22	PROPN
iajs-1139	222	2	(	(	PUNCT
iajs-1139	222	3	4	4	NUM
iajs-1139	222	4	)	)	PUNCT
iajs-1139	222	5	2009	2009	NUM
iajs-1139	222	6	proof	proof	NOUN
iajs-1139	222	7	:	:	PUNCT
iajs-1139	222	8	by	by	ADP
iajs-1139	222	9	(	(	PUNCT
iajs-1139	222	10	3,proposition	3,proposition	NUM
iajs-1139	222	11	3.3	3.3	NUM
iajs-1139	222	12	)	)	PUNCT
iajs-1139	222	13	,	,	PUNCT
iajs-1139	222	14	f	f	PROPN
iajs-1139	222	15	1	1	NUM
iajs-1139	222	16	(	(	PUNCT
iajs-1139	222	17	a	a	NOUN
iajs-1139	222	18	)	)	PUNCT
iajs-1139	222	19	is	be	AUX
iajs-1139	222	20	a	a	DET
iajs-1139	222	21	fuzzy	fuzzy	ADJ
iajs-1139	222	22	ideal	ideal	NOUN
iajs-1139	222	23	of	of	ADP
iajs-1139	222	24	x.	x.	NOUN
iajs-1139	222	25	let	let	VERB
iajs-1139	222	26	b	b	X
iajs-1139	222	27	be	be	AUX
iajs-1139	222	28	a	a	DET
iajs-1139	222	29	prime	prime	ADJ
iajs-1139	222	30	fuzzy	fuzzy	ADJ
iajs-1139	222	31	ideal	ideal	NOUN
iajs-1139	222	32	of	of	ADP
iajs-1139	222	33	x	x	X
iajs-1139	222	34	and	and	CCONJ
iajs-1139	222	35	b	b	PROPN
iajs-1139	222	36	is	be	AUX
iajs-1139	222	37	f	f	NOUN
iajs-1139	222	38	-	-	PUNCT
iajs-1139	222	39	invariant	invariant	ADJ
iajs-1139	222	40	.	.	PUNCT
iajs-1139	223	1	to	to	PART
iajs-1139	223	2	prove	prove	VERB
iajs-1139	223	3	that	that	SCONJ
iajs-1139	223	4	f	f	PROPN
iajs-1139	223	5	1	1	NUM
iajs-1139	223	6	(	(	PUNCT
iajs-1139	223	7	a	a	NOUN
iajs-1139	223	8	)	)	PUNCT
iajs-1139	223	9			PROPN
iajs-1139	223	10	b	b	PROPN
iajs-1139	223	11			PROPN
iajs-1139	223	12	ox(0	ox(0	PROPN
iajs-1139	223	13	)	)	PUNCT
iajs-1139	223	14	=	=	SYM
iajs-1139	223	15	o1	o1	NOUN
iajs-1139	223	16	.	.	PUNCT
iajs-1139	224	1	f	f	PROPN
iajs-1139	224	2	(	(	PUNCT
iajs-1139	224	3	f	f	PROPN
iajs-1139	224	4	1(a	1(a	NUM
iajs-1139	224	5	)	)	PUNCT
iajs-1139	224	6			NOUN
iajs-1139	224	7	b	b	NOUN
iajs-1139	224	8	)	)	PUNCT
iajs-1139	224	9	=	=	SYM
iajs-1139	224	10	f	f	PROPN
iajs-1139	224	11	(	(	PUNCT
iajs-1139	224	12	f	f	PROPN
iajs-1139	224	13	1(a	1(a	NUM
iajs-1139	224	14	)	)	PUNCT
iajs-1139	224	15			PUNCT
iajs-1139	225	1	f	f	X
iajs-1139	225	2	(	(	PUNCT
iajs-1139	225	3	b	b	NOUN
iajs-1139	225	4	)	)	PUNCT
iajs-1139	225	5	)	)	PUNCT
iajs-1139	226	1	=	=	PUNCT
iajs-1139	226	2	a	a	DET
iajs-1139	226	3			ADJ
iajs-1139	226	4	f	f	X
iajs-1139	226	5	(	(	PUNCT
iajs-1139	226	6	b	b	NOUN
iajs-1139	226	7	)	)	PUNCT
iajs-1139	226	8	since	since	SCONJ
iajs-1139	226	9	a	a	DET
iajs-1139	226	10	,	,	PUNCT
iajs-1139	226	11	b	b	NOUN
iajs-1139	226	12	are	be	AUX
iajs-1139	226	13	f	f	NOUN
iajs-1139	226	14	-	-	PUNCT
iajs-1139	226	15	invariant	invariant	ADJ
iajs-1139	226	16	.	.	PUNCT
iajs-1139	227	1	however	however	ADV
iajs-1139	227	2	f	f	X
iajs-1139	227	3	(	(	PUNCT
iajs-1139	227	4	b	b	NOUN
iajs-1139	227	5	)	)	PUNCT
iajs-1139	227	6	is	be	AUX
iajs-1139	227	7	prime	prime	ADJ
iajs-1139	227	8	fuzzy	fuzzy	ADJ
iajs-1139	227	9	ideal	ideal	NOUN
iajs-1139	227	10	of	of	ADP
iajs-1139	227	11	y	y	PROPN
iajs-1139	227	12	by	by	ADP
iajs-1139	227	13	proposition	proposition	NOUN
iajs-1139	227	14	1.6	1.6	NUM
iajs-1139	227	15	.	.	PUNCT
iajs-1139	228	1	therefore	therefore	ADV
iajs-1139	228	2	,	,	PUNCT
iajs-1139	228	3	a	a	DET
iajs-1139	228	4			ADJ
iajs-1139	228	5	f	f	NOUN
iajs-1139	228	6	(	(	PUNCT
iajs-1139	228	7	b	b	NOUN
iajs-1139	228	8	)	)	PUNCT
iajs-1139	228	9			PROPN
iajs-1139	228	10	oy(0	oy(0	PROPN
iajs-1139	228	11	)	)	PUNCT
iajs-1139	228	12	,	,	PUNCT
iajs-1139	228	13	since	since	SCONJ
iajs-1139	228	14	a	a	PRON
iajs-1139	228	15	is	be	AUX
iajs-1139	228	16	semiessential	semiessential	ADJ
iajs-1139	228	17	.	.	PUNCT
iajs-1139	229	1	on	on	ADP
iajs-1139	229	2	the	the	DET
iajs-1139	229	3	other	other	ADJ
iajs-1139	229	4	hand	hand	NOUN
iajs-1139	229	5	,	,	PUNCT
iajs-1139	229	6	f	f	PROPN
iajs-1139	229	7	1(a	1(a	NUM
iajs-1139	229	8			PROPN
iajs-1139	229	9	f	f	X
iajs-1139	229	10	(	(	PUNCT
iajs-1139	229	11	b	b	NOUN
iajs-1139	229	12	)	)	PUNCT
iajs-1139	229	13	)	)	PUNCT
iajs-1139	230	1	=	=	PUNCT
iajs-1139	230	2	f	f	PROPN
iajs-1139	230	3	1(a	1(a	NUM
iajs-1139	230	4	)	)	PUNCT
iajs-1139	230	5			PUNCT
iajs-1139	230	6	f	f	X
iajs-1139	230	7	–	–	PUNCT
iajs-1139	230	8	1(f	1(f	NUM
iajs-1139	230	9	(	(	PUNCT
iajs-1139	230	10	b	b	NOUN
iajs-1139	230	11	)	)	PUNCT
iajs-1139	230	12	)	)	PUNCT
iajs-1139	231	1	=	=	PUNCT
iajs-1139	231	2	f	f	X
iajs-1139	231	3	1	1	NUM
iajs-1139	231	4	(	(	PUNCT
iajs-1139	231	5	a	a	NOUN
iajs-1139	231	6	)	)	PUNCT
iajs-1139	231	7			PROPN
iajs-1139	231	8	b	b	NOUN
iajs-1139	231	9	,	,	PUNCT
iajs-1139	231	10	since	since	SCONJ
iajs-1139	231	11	b	b	PROPN
iajs-1139	231	12	f	f	X
iajs-1139	231	13	-	-	PUNCT
iajs-1139	231	14	invariant	invariant	ADJ
iajs-1139	231	15			NOUN
iajs-1139	231	16	ox(0	ox(0	PROPN
iajs-1139	231	17	)	)	PUNCT
iajs-1139	231	18			NOUN
iajs-1139	231	19	o1	o1	NOUN
iajs-1139	231	20	.	.	PUNCT
iajs-1139	232	1	thus	thus	ADV
iajs-1139	232	2	f	f	PROPN
iajs-1139	232	3	1(a	1(a	NUM
iajs-1139	232	4	)	)	PUNCT
iajs-1139	232	5	is	be	AUX
iajs-1139	232	6	a	a	DET
iajs-1139	232	7	semiessential	semiessential	ADJ
iajs-1139	232	8	fuzzy	fuzzy	ADJ
iajs-1139	232	9	ideal	ideal	NOUN
iajs-1139	232	10	of	of	ADP
iajs-1139	232	11	x.	x.	NOUN
iajs-1139	232	12	s.3	s.3	NOUN
iajs-1139	232	13	uniform	uniform	NOUN
iajs-1139	232	14	and	and	CCONJ
iajs-1139	232	15	semiuniform	semiuniform	VERB
iajs-1139	232	16	fuzzy	fuzzy	ADJ
iajs-1139	232	17	rings	ring	NOUN
iajs-1139	232	18	recall	recall	VERB
iajs-1139	232	19	that	that	SCONJ
iajs-1139	232	20	a	a	DET
iajs-1139	232	21	ring	ring	NOUN
iajs-1139	232	22	r	r	NOUN
iajs-1139	232	23	is	be	AUX
iajs-1139	232	24	called	call	VERB
iajs-1139	232	25	a	a	DET
iajs-1139	232	26	uniform	uniform	ADJ
iajs-1139	232	27	ring	ring	NOUN
iajs-1139	232	28	if	if	SCONJ
iajs-1139	232	29	every	every	DET
iajs-1139	232	30	non	non	ADJ
iajs-1139	232	31	zero	zero	NUM
iajs-1139	232	32	ideal	ideal	NOUN
iajs-1139	232	33	i	i	PRON
iajs-1139	232	34	of	of	ADP
iajs-1139	232	35	r	r	NOUN
iajs-1139	232	36	is	be	AUX
iajs-1139	232	37	an	an	DET
iajs-1139	232	38	essential	essential	ADJ
iajs-1139	232	39	ideal	ideal	NOUN
iajs-1139	232	40	,	,	PUNCT
iajs-1139	232	41	[	[	X
iajs-1139	232	42	4	4	X
iajs-1139	232	43	]	]	PUNCT
iajs-1139	232	44	and	and	CCONJ
iajs-1139	232	45	a	a	DET
iajs-1139	232	46	ring	ring	NOUN
iajs-1139	232	47	r	r	NOUN
iajs-1139	232	48	is	be	AUX
iajs-1139	232	49	called	call	VERB
iajs-1139	232	50	semiuniform	semiuniform	NOUN
iajs-1139	232	51	if	if	SCONJ
iajs-1139	232	52	every	every	DET
iajs-1139	232	53	non	non	ADJ
iajs-1139	232	54	zero	zero	NUM
iajs-1139	232	55	ideal	ideal	NOUN
iajs-1139	232	56	i	i	PRON
iajs-1139	232	57	of	of	ADP
iajs-1139	232	58	r	r	NOUN
iajs-1139	232	59	is	be	AUX
iajs-1139	232	60	a	a	DET
iajs-1139	232	61	semiessential	semiessential	ADJ
iajs-1139	232	62	ideal	ideal	NOUN
iajs-1139	232	63	of	of	ADP
iajs-1139	232	64	r	r	NOUN
iajs-1139	232	65	,	,	PUNCT
iajs-1139	232	66	[	[	X
iajs-1139	232	67	6	6	NUM
iajs-1139	232	68	]	]	PUNCT
iajs-1139	232	69	.	.	PUNCT
iajs-1139	233	1	in	in	ADP
iajs-1139	233	2	this	this	DET
iajs-1139	233	3	section	section	NOUN
iajs-1139	233	4	,	,	PUNCT
iajs-1139	233	5	we	we	PRON
iajs-1139	233	6	introduce	introduce	VERB
iajs-1139	233	7	and	and	CCONJ
iajs-1139	233	8	study	study	VERB
iajs-1139	233	9	the	the	DET
iajs-1139	233	10	notion	notion	NOUN
iajs-1139	233	11	of	of	ADP
iajs-1139	233	12	uniform	uniform	NOUN
iajs-1139	233	13	and	and	CCONJ
iajs-1139	233	14	semiuniform	semiuniform	VERB
iajs-1139	233	15	fuzzy	fuzzy	ADJ
iajs-1139	233	16	rings	ring	NOUN
iajs-1139	233	17	and	and	CCONJ
iajs-1139	233	18	give	give	VERB
iajs-1139	233	19	many	many	ADJ
iajs-1139	233	20	properties	property	NOUN
iajs-1139	233	21	about	about	ADP
iajs-1139	233	22	them	they	PRON
iajs-1139	233	23	.	.	PUNCT
iajs-1139	234	1	3.1	3.1	NUM
iajs-1139	234	2	definition	definition	NOUN
iajs-1139	234	3	let	let	VERB
iajs-1139	234	4	x	x	PRON
iajs-1139	234	5	be	be	AUX
iajs-1139	234	6	a	a	DET
iajs-1139	234	7	fuzzy	fuzzy	ADJ
iajs-1139	234	8	ring	ring	NOUN
iajs-1139	234	9	of	of	ADP
iajs-1139	234	10	a	a	DET
iajs-1139	234	11	ring	ring	NOUN
iajs-1139	234	12	r.	r.	PROPN
iajs-1139	234	13	x	x	PUNCT
iajs-1139	234	14	is	be	AUX
iajs-1139	234	15	called	call	VERB
iajs-1139	234	16	uniform	uniform	ADJ
iajs-1139	234	17	(	(	PUNCT
iajs-1139	234	18	semiuniform	semiuniform	NOUN
iajs-1139	234	19	)	)	PUNCT
iajs-1139	234	20	if	if	SCONJ
iajs-1139	234	21	every	every	DET
iajs-1139	234	22	non	non	ADJ
iajs-1139	234	23	zero	zero	NUM
iajs-1139	234	24	fuzzy	fuzzy	ADJ
iajs-1139	234	25	ideal	ideal	NOUN
iajs-1139	234	26	a	a	PRON
iajs-1139	234	27	of	of	ADP
iajs-1139	234	28	x	x	NOUN
iajs-1139	234	29	is	be	AUX
iajs-1139	234	30	an	an	DET
iajs-1139	234	31	essential	essential	ADJ
iajs-1139	234	32	(	(	PUNCT
iajs-1139	234	33	semiessential	semiessential	ADJ
iajs-1139	234	34	)	)	PUNCT
iajs-1139	234	35	fuzzy	fuzzy	ADJ
iajs-1139	234	36	ideal	ideal	NOUN
iajs-1139	234	37	of	of	ADP
iajs-1139	234	38	x.	x.	NOUN
iajs-1139	234	39	3.2	3.2	NUM
iajs-1139	234	40	proposition	proposition	NOUN
iajs-1139	234	41	let	let	VERB
iajs-1139	234	42	x	x	PRON
iajs-1139	234	43	be	be	AUX
iajs-1139	234	44	a	a	DET
iajs-1139	234	45	uniform	uniform	ADJ
iajs-1139	234	46	fuzzy	fuzzy	ADJ
iajs-1139	234	47	ring	ring	NOUN
iajs-1139	234	48	,	,	PUNCT
iajs-1139	234	49	then	then	ADV
iajs-1139	234	50	x	x	PUNCT
iajs-1139	234	51	is	be	AUX
iajs-1139	234	52	semiuniform	semiuniform	VERB
iajs-1139	234	53	fuzzy	fuzzy	ADJ
iajs-1139	234	54	ring	ring	NOUN
iajs-1139	234	55	.	.	PUNCT
iajs-1139	235	1	proof	proof	NOUN
iajs-1139	235	2	:	:	PUNCT
iajs-1139	235	3	it	it	PRON
iajs-1139	235	4	is	be	AUX
iajs-1139	235	5	follows	follow	VERB
iajs-1139	235	6	by	by	ADP
iajs-1139	235	7	remark	remark	NOUN
iajs-1139	235	8	2.2	2.2	NUM
iajs-1139	235	9	.	.	PUNCT
iajs-1139	236	1	but	but	CCONJ
iajs-1139	236	2	the	the	DET
iajs-1139	236	3	converse	converse	NOUN
iajs-1139	236	4	is	be	AUX
iajs-1139	236	5	not	not	PART
iajs-1139	236	6	true	true	ADJ
iajs-1139	236	7	in	in	ADP
iajs-1139	236	8	general	general	ADJ
iajs-1139	236	9	.	.	PUNCT
iajs-1139	237	1	3.3	3.3	NUM
iajs-1139	237	2	remark	remark	NOUN
iajs-1139	237	3	if	if	SCONJ
iajs-1139	237	4	x	x	PRON
iajs-1139	237	5	is	be	AUX
iajs-1139	237	6	a	a	DET
iajs-1139	237	7	semiuniform	semiuniform	NOUN
iajs-1139	237	8	fuzzy	fuzzy	ADJ
iajs-1139	237	9	ring	ring	NOUN
iajs-1139	237	10	.	.	PUNCT
iajs-1139	238	1	then	then	ADV
iajs-1139	238	2	it	it	PRON
iajs-1139	238	3	is	be	AUX
iajs-1139	238	4	not	not	PART
iajs-1139	238	5	necessarily	necessarily	ADV
iajs-1139	238	6	that	that	PRON
iajs-1139	238	7	x	x	PRON
iajs-1139	238	8	is	be	AUX
iajs-1139	238	9	uniform	uniform	ADJ
iajs-1139	238	10	fuzzy	fuzzy	ADJ
iajs-1139	238	11	ring	ring	NOUN
iajs-1139	238	12	as	as	ADP
iajs-1139	238	13	the	the	DET
iajs-1139	238	14	following	follow	VERB
iajs-1139	238	15	example	example	NOUN
iajs-1139	238	16	shows	show	VERB
iajs-1139	238	17	:	:	PUNCT
iajs-1139	238	18	example	example	NOUN
iajs-1139	238	19	:	:	PUNCT
iajs-1139	238	20	let	let	VERB
iajs-1139	238	21	x	x	PRON
iajs-1139	238	22	:	:	PUNCT
iajs-1139	238	23	z36	z36	PROPN
iajs-1139	238	24			PROPN
iajs-1139	239	1	[	[	X
iajs-1139	239	2	0,1	0,1	NUM
iajs-1139	239	3	]	]	PUNCT
iajs-1139	239	4	define	define	NOUN
iajs-1139	239	5	by	by	ADP
iajs-1139	239	6	x(a	x(a	NOUN
iajs-1139	239	7	)	)	PUNCT
iajs-1139	239	8	=	=	SYM
iajs-1139	239	9	1	1	NUM
iajs-1139	239	10	,	,	PUNCT
iajs-1139	239	11			VERB
iajs-1139	239	12	a	a	DET
iajs-1139	239	13			PROPN
iajs-1139	239	14	z36	z36	NOUN
iajs-1139	239	15	.	.	PUNCT
iajs-1139	240	1	it	it	PRON
iajs-1139	240	2	is	be	AUX
iajs-1139	240	3	easy	easy	ADJ
iajs-1139	240	4	to	to	PART
iajs-1139	240	5	check	check	VERB
iajs-1139	240	6	that	that	PRON
iajs-1139	240	7	x	x	PRON
iajs-1139	240	8	is	be	AUX
iajs-1139	240	9	a	a	DET
iajs-1139	240	10	fuzzy	fuzzy	ADJ
iajs-1139	240	11	ring	ring	NOUN
iajs-1139	240	12	of	of	ADP
iajs-1139	240	13	z36	z36	PROPN
iajs-1139	240	14	.	.	PUNCT
iajs-1139	241	1	for	for	ADP
iajs-1139	241	2	each	each	DET
iajs-1139	241	3	fuzzy	fuzzy	ADJ
iajs-1139	241	4	ideal	ideal	NOUN
iajs-1139	241	5	a	a	PRON
iajs-1139	241	6	of	of	ADP
iajs-1139	241	7	x	x	PRON
iajs-1139	241	8	,	,	PUNCT
iajs-1139	241	9	a	a	DET
iajs-1139	241	10	t	t	NOUN
iajs-1139	241	11	is	be	AUX
iajs-1139	241	12	an	an	DET
iajs-1139	241	13	ideal	ideal	NOUN
iajs-1139	241	14	of	of	ADP
iajs-1139	241	15	xt	xt	PROPN
iajs-1139	241	16	,	,	PUNCT
iajs-1139	241	17			PROPN
iajs-1139	241	18	t	t	PROPN
iajs-1139	241	19	[0,1	[0,1	NOUN
iajs-1139	241	20	]	]	PUNCT
iajs-1139	241	21	and	and	CCONJ
iajs-1139	241	22	xt	xt	X
iajs-1139	241	23	=	=	NOUN
iajs-1139	241	24	z36	z36	PROPN
iajs-1139	241	25	.	.	PUNCT
iajs-1139	242	1	the	the	DET
iajs-1139	242	2	ideals	ideal	NOUN
iajs-1139	242	3	of	of	ADP
iajs-1139	242	4	z36	z36	PROPN
iajs-1139	242	5	are	be	AUX
iajs-1139	242	6	:	:	PUNCT
iajs-1139	242	7	(	(	PUNCT
iajs-1139	242	8	0	0	NUM
iajs-1139	242	9	)	)	PUNCT
iajs-1139	242	10	,	,	PUNCT
iajs-1139	242	11	(	(	PUNCT
iajs-1139	242	12	2	2	NUM
iajs-1139	242	13	)	)	PUNCT
iajs-1139	242	14	,	,	PUNCT
iajs-1139	242	15	(	(	PUNCT
iajs-1139	242	16	3	3	NUM
iajs-1139	242	17	)	)	PUNCT
iajs-1139	242	18	,	,	PUNCT
iajs-1139	242	19	(	(	PUNCT
iajs-1139	242	20	4),(6),(12	4),(6),(12	NUM
iajs-1139	242	21	)	)	PUNCT
iajs-1139	242	22	and	and	CCONJ
iajs-1139	242	23	(	(	PUNCT
iajs-1139	242	24	18)which	18)which	PROPN
iajs-1139	242	25	are	be	AUX
iajs-1139	242	26	semiessential	semiessential	ADJ
iajs-1139	242	27	ideals	ideal	NOUN
iajs-1139	242	28	in	in	ADP
iajs-1139	242	29	z36	z36	PROPN
iajs-1139	242	30	.	.	PUNCT
iajs-1139	243	1	so	so	ADV
iajs-1139	243	2	a	a	PRON
iajs-1139	243	3	is	be	AUX
iajs-1139	243	4	a	a	DET
iajs-1139	243	5	semiessential	semiessential	ADJ
iajs-1139	243	6	fuzzy	fuzzy	ADJ
iajs-1139	243	7	ideal	ideal	NOUN
iajs-1139	243	8	of	of	ADP
iajs-1139	243	9	x	x	PUNCT
iajs-1139	243	10	by	by	ADP
iajs-1139	243	11	theorem	theorem	NOUN
iajs-1139	243	12	2.3	2.3	NUM
iajs-1139	243	13	.	.	PUNCT
iajs-1139	244	1	thus	thus	ADV
iajs-1139	244	2	x	x	PRON
iajs-1139	244	3	is	be	AUX
iajs-1139	244	4	a	a	DET
iajs-1139	244	5	semiuniform	semiuniform	NOUN
iajs-1139	244	6	fuzzy	fuzzy	ADJ
iajs-1139	244	7	ring	ring	NOUN
iajs-1139	244	8	.	.	PUNCT
iajs-1139	245	1	but	but	CCONJ
iajs-1139	245	2	x	x	X
iajs-1139	245	3	is	be	AUX
iajs-1139	245	4	not	not	PART
iajs-1139	245	5	uniform	uniform	ADJ
iajs-1139	245	6	fuzzy	fuzzy	ADJ
iajs-1139	245	7	ring	ring	NOUN
iajs-1139	245	8	since	since	SCONJ
iajs-1139	245	9	there	there	PRON
iajs-1139	245	10	exist	exist	VERB
iajs-1139	245	11	fuzzy	fuzzy	ADJ
iajs-1139	245	12	ideals	ideal	NOUN
iajs-1139	245	13	a	a	PRON
iajs-1139	245	14	and	and	CCONJ
iajs-1139	245	15	b	b	NOUN
iajs-1139	245	16	of	of	ADP
iajs-1139	245	17	x	x	SYM
iajs-1139	245	18	such	such	ADJ
iajs-1139	245	19	that	that	SCONJ
iajs-1139	245	20	1	1	NUM
iajs-1139	245	21	if	if	SCONJ
iajs-1139	245	22	(	(	PUNCT
iajs-1139	245	23	12	12	NUM
iajs-1139	245	24	)	)	PUNCT
iajs-1139	245	25	(	(	PUNCT
iajs-1139	245	26	)	)	PUNCT
iajs-1139	245	27	0	0	PUNCT
iajs-1139	245	28	otherwise	otherwise	ADV
iajs-1139	245	29	.	.	PUNCT
iajs-1139	246	1			NOUN
iajs-1139	247	1			NOUN
iajs-1139	248	1			PROPN
iajs-1139	248	2			NUM
iajs-1139	249	1			NUM
iajs-1139	249	2			NOUN
iajs-1139	249	3	x	x	SYM
iajs-1139	249	4	x	x	X
iajs-1139	249	5	1	1	NUM
iajs-1139	249	6	if	if	SCONJ
iajs-1139	249	7	(	(	PUNCT
iajs-1139	249	8	18	18	NUM
iajs-1139	249	9	)	)	PUNCT
iajs-1139	249	10	(	(	PUNCT
iajs-1139	249	11	)	)	PUNCT
iajs-1139	249	12	0	0	PUNCT
iajs-1139	250	1	otherwise	otherwise	ADV
iajs-1139	250	2	.	.	PUNCT
iajs-1139	251	1			ADP
iajs-1139	251	2			PROPN
iajs-1139	251	3			NOUN
iajs-1139	252	1			PROPN
iajs-1139	253	1			NUM
iajs-1139	253	2			NOUN
iajs-1139	253	3	x	x	SYM
iajs-1139	253	4	x	x	X
iajs-1139	253	5	,	,	PUNCT
iajs-1139	253	6			NOUN
iajs-1139	253	7	x	x	SYM
iajs-1139	253	8			PROPN
iajs-1139	253	9	z36	z36	PROPN
iajs-1139	253	10	.	.	PUNCT
iajs-1139	254	1	a	a	PRON
iajs-1139	254	2	is	be	AUX
iajs-1139	254	3	not	not	PART
iajs-1139	254	4	essential	essential	ADJ
iajs-1139	254	5	fuzzy	fuzzy	ADJ
iajs-1139	254	6	ideal	ideal	NOUN
iajs-1139	254	7	since	since	SCONJ
iajs-1139	254	8	a	a	DET
iajs-1139	254	9			NOUN
iajs-1139	254	10	b	b	NOUN
iajs-1139	254	11	=	=	SYM
iajs-1139	254	12	o1	o1	NOUN
iajs-1139	254	13	.	.	PUNCT
iajs-1139	255	1	thus	thus	ADV
iajs-1139	255	2	x	x	PRON
iajs-1139	255	3	is	be	AUX
iajs-1139	255	4	not	not	PART
iajs-1139	255	5	uniform	uniform	ADJ
iajs-1139	255	6	fuzzy	fuzzy	ADJ
iajs-1139	255	7	ring	ring	NOUN
iajs-1139	255	8	.	.	PUNCT
iajs-1139	256	1	3.4	3.4	NUM
iajs-1139	256	2	remark	remark	NOUN
iajs-1139	256	3	let	let	VERB
iajs-1139	256	4	x	x	PRON
iajs-1139	256	5	be	be	AUX
iajs-1139	256	6	a	a	DET
iajs-1139	256	7	fuzzy	fuzzy	ADJ
iajs-1139	256	8	ring	ring	NOUN
iajs-1139	256	9	of	of	ADP
iajs-1139	256	10	a	a	DET
iajs-1139	256	11	ring	ring	NOUN
iajs-1139	256	12	r	r	NOUN
iajs-1139	256	13	such	such	ADJ
iajs-1139	256	14	that	that	DET
iajs-1139	256	15	xt	xt	PROPN
iajs-1139	256	16	is	be	AUX
iajs-1139	256	17	a	a	DET
iajs-1139	256	18	uniform	uniform	ADJ
iajs-1139	256	19	ring	ring	NOUN
iajs-1139	256	20	,	,	PUNCT
iajs-1139	256	21			PROPN
iajs-1139	256	22	t	t	PROPN
iajs-1139	256	23			PROPN
iajs-1139	256	24	(	(	PUNCT
iajs-1139	256	25	0,1	0,1	NOUN
iajs-1139	256	26	]	]	PUNCT
iajs-1139	256	27	.	.	PUNCT
iajs-1139	257	1	then	then	ADV
iajs-1139	257	2	x	x	X
iajs-1139	257	3	is	be	AUX
iajs-1139	257	4	a	a	DET
iajs-1139	257	5	uniform	uniform	ADJ
iajs-1139	257	6	fuzzy	fuzzy	ADJ
iajs-1139	257	7	ring	ring	NOUN
iajs-1139	257	8	.	.	PUNCT
iajs-1139	258	1	proof	proof	NOUN
iajs-1139	258	2	:	:	PUNCT
iajs-1139	258	3	it	it	PRON
iajs-1139	258	4	is	be	AUX
iajs-1139	258	5	follow	follow	VERB
iajs-1139	258	6	by	by	ADP
iajs-1139	258	7	[	[	X
iajs-1139	258	8	5	5	NUM
iajs-1139	258	9	,	,	PUNCT
iajs-1139	258	10	proposition	proposition	NOUN
iajs-1139	258	11	2.4	2.4	NUM
iajs-1139	258	12	]	]	PUNCT
iajs-1139	258	13	.	.	PUNCT
iajs-1139	259	1	3.5	3.5	NUM
iajs-1139	259	2	remark	remark	NOUN
iajs-1139	259	3	let	let	VERB
iajs-1139	259	4	x	x	PRON
iajs-1139	259	5	be	be	AUX
iajs-1139	259	6	a	a	DET
iajs-1139	259	7	fuzzy	fuzzy	ADJ
iajs-1139	259	8	ring	ring	NOUN
iajs-1139	259	9	of	of	ADP
iajs-1139	259	10	a	a	DET
iajs-1139	259	11	ring	ring	NOUN
iajs-1139	259	12	r	r	NOUN
iajs-1139	259	13	such	such	ADJ
iajs-1139	259	14	that	that	DET
iajs-1139	259	15	xt	xt	PROPN
iajs-1139	259	16	is	be	AUX
iajs-1139	259	17	a	a	DET
iajs-1139	259	18	semiuniform	semiuniform	NOUN
iajs-1139	259	19	ring	ring	NOUN
iajs-1139	259	20	,	,	PUNCT
iajs-1139	259	21			NOUN
iajs-1139	259	22	t	t	PROPN
iajs-1139	259	23	,	,	PUNCT
iajs-1139	259	24	then	then	ADV
iajs-1139	259	25	x	x	PUNCT
iajs-1139	259	26	is	be	AUX
iajs-1139	259	27	a	a	DET
iajs-1139	259	28	semiuniform	semiuniform	NOUN
iajs-1139	259	29	fuzzy	fuzzy	ADJ
iajs-1139	259	30	ring	ring	NOUN
iajs-1139	259	31	.	.	PUNCT
iajs-1139	260	1	proof	proof	NOUN
iajs-1139	260	2	:	:	PUNCT
iajs-1139	260	3	let	let	VERB
iajs-1139	260	4	a	a	PRON
iajs-1139	260	5	be	be	AUX
iajs-1139	260	6	a	a	DET
iajs-1139	260	7	non	non	ADJ
iajs-1139	260	8	-	-	ADJ
iajs-1139	260	9	zero	zero	ADJ
iajs-1139	260	10	fuzzy	fuzzy	ADJ
iajs-1139	260	11	ideal	ideal	NOUN
iajs-1139	260	12	of	of	ADP
iajs-1139	260	13	x.	x.	NOUN
iajs-1139	260	14	to	to	PART
iajs-1139	260	15	prove	prove	VERB
iajs-1139	260	16	a	a	PRON
iajs-1139	260	17	is	be	AUX
iajs-1139	260	18	a	a	DET
iajs-1139	260	19	semiessential	semiessential	NOUN
iajs-1139	260	20	in	in	ADP
iajs-1139	260	21	x.	x.	PROPN
iajs-1139	260	22	ibn	ibn	PROPN
iajs-1139	260	23	alhaitham	alhaitham	PROPN
iajs-1139	260	24	j.	j.	PROPN
iajs-1139	260	25	for	for	ADP
iajs-1139	260	26	pure	pure	ADJ
iajs-1139	260	27	&	&	CCONJ
iajs-1139	260	28	appl	appl	PROPN
iajs-1139	260	29	.	.	PUNCT
iajs-1139	261	1	sci	sci	PROPN
iajs-1139	261	2	.	.	PUNCT
iajs-1139	262	1	vol.22	vol.22	PROPN
iajs-1139	262	2	(	(	PUNCT
iajs-1139	262	3	4	4	NUM
iajs-1139	262	4	)	)	PUNCT
iajs-1139	262	5	2009	2009	NUM
iajs-1139	262	6	suppose	suppose	VERB
iajs-1139	262	7	there	there	PRON
iajs-1139	262	8	exists	exist	VERB
iajs-1139	262	9	a	a	DET
iajs-1139	262	10	non	non	ADJ
iajs-1139	262	11	-	-	ADJ
iajs-1139	262	12	zero	zero	ADJ
iajs-1139	262	13	prime	prime	ADJ
iajs-1139	262	14	fuzzy	fuzzy	ADJ
iajs-1139	262	15	ideal	ideal	NOUN
iajs-1139	262	16	p	p	NOUN
iajs-1139	262	17	of	of	ADP
iajs-1139	262	18	x	x	INTJ
iajs-1139	262	19	such	such	ADJ
iajs-1139	262	20	that	that	SCONJ
iajs-1139	262	21	a	a	DET
iajs-1139	262	22			NOUN
iajs-1139	262	23	p	p	NOUN
iajs-1139	262	24	=	=	NOUN
iajs-1139	262	25	o1	o1	NOUN
iajs-1139	262	26	since	since	SCONJ
iajs-1139	262	27	p	p	PROPN
iajs-1139	262	28			NOUN
iajs-1139	262	29	o1	o1	NOUN
iajs-1139	262	30	,	,	PUNCT
iajs-1139	262	31	there	there	PRON
iajs-1139	262	32	exist	exist	VERB
iajs-1139	262	33	t1	t1	NOUN
iajs-1139	262	34			NOUN
iajs-1139	262	35	(	(	PUNCT
iajs-1139	262	36	0,1	0,1	NOUN
iajs-1139	262	37	]	]	PUNCT
iajs-1139	262	38	such	such	ADJ
iajs-1139	262	39	that	that	SCONJ
iajs-1139	262	40	1	1	NUM
iajs-1139	262	41	pt	pt	NOUN
iajs-1139	262	42			NOUN
iajs-1139	262	43	{	{	PUNCT
iajs-1139	262	44	0	0	NUM
iajs-1139	262	45	}	}	PUNCT
iajs-1139	262	46	.	.	PUNCT
iajs-1139	263	1	on	on	ADP
iajs-1139	263	2	the	the	DET
iajs-1139	263	3	other	other	ADJ
iajs-1139	263	4	hand	hand	NOUN
iajs-1139	263	5	,	,	PUNCT
iajs-1139	263	6	(	(	PUNCT
iajs-1139	263	7	a	a	SYM
iajs-1139	263	8	p	p	X
iajs-1139	263	9	)	)	PUNCT
iajs-1139	263	10	=	=	SYM
iajs-1139	263	11	o1	o1	NOUN
iajs-1139	263	12	.	.	PUNCT
iajs-1139	263	13	implies	imply	VERB
iajs-1139	263	14	1	1	NUM
iajs-1139	263	15	1	1	NUM
iajs-1139	263	16	11p	11p	NOUN
iajs-1139	263	17	(	(	PUNCT
iajs-1139	263	18	o	o	NOUN
iajs-1139	263	19	)	)	PUNCT
iajs-1139	263	20	{	{	PUNCT
iajs-1139	263	21	0}	0}	X
iajs-1139	263	22			VERB
iajs-1139	263	23			PROPN
iajs-1139	263	24	t	t	NOUN
iajs-1139	263	25	t	t	PROPN
iajs-1139	263	26	t	t	PROPN
iajs-1139	263	27	.	.	PUNCT
iajs-1139	264	1	but	but	CCONJ
iajs-1139	264	2	this	this	PRON
iajs-1139	264	3	is	be	AUX
iajs-1139	264	4	a	a	DET
iajs-1139	264	5	contradiction	contradiction	NOUN
iajs-1139	264	6	since	since	SCONJ
iajs-1139	264	7	1	1	NUM
iajs-1139	264	8	pt	pt	PROPN
iajs-1139	264	9			NOUN
iajs-1139	264	10	{	{	PUNCT
iajs-1139	264	11	0	0	NUM
iajs-1139	264	12	}	}	PUNCT
iajs-1139	264	13	,	,	PUNCT
iajs-1139	264	14	1	1	NUM
iajs-1139	264	15	pt	pt	NOUN
iajs-1139	264	16	is	be	AUX
iajs-1139	264	17	a	a	DET
iajs-1139	264	18	prime	prime	NOUN
iajs-1139	264	19	in	in	ADP
iajs-1139	264	20	xt	xt	PROPN
iajs-1139	264	21	,	,	PUNCT
iajs-1139	264	22	(	(	PUNCT
iajs-1139	264	23	11	11	NUM
iajs-1139	264	24	,	,	PUNCT
iajs-1139	264	25	proposition	proposition	NOUN
iajs-1139	264	26	1.2.9	1.2.9	NUM
iajs-1139	264	27	)	)	PUNCT
iajs-1139	264	28	and	and	CCONJ
iajs-1139	264	29	1	1	NUM
iajs-1139	264	30	t	t	NOUN
iajs-1139	264	31	is	be	AUX
iajs-1139	264	32	semiessential	semiessential	ADJ
iajs-1139	264	33	in	in	ADP
iajs-1139	264	34	xt	xt	PROPN
iajs-1139	264	35	.	.	PUNCT
iajs-1139	265	1	hence	hence	ADV
iajs-1139	265	2	ap	ap	PROPN
iajs-1139	265	3			NOUN
iajs-1139	265	4	o1	o1	PROPN
iajs-1139	265	5	and	and	CCONJ
iajs-1139	265	6	a	a	PRON
iajs-1139	265	7	is	be	AUX
iajs-1139	265	8	a	a	DET
iajs-1139	265	9	semiessential	semiessential	ADJ
iajs-1139	265	10	fuzzy	fuzzy	ADJ
iajs-1139	265	11	ideal	ideal	NOUN
iajs-1139	265	12	in	in	ADP
iajs-1139	265	13	x.	x.	NOUN
iajs-1139	265	14	thus	thus	ADV
iajs-1139	265	15	x	x	PRON
iajs-1139	265	16	is	be	AUX
iajs-1139	265	17	a	a	DET
iajs-1139	265	18	semiuniform	semiuniform	NOUN
iajs-1139	265	19	fuzy	fuzy	ADJ
iajs-1139	265	20	ring	ring	NOUN
iajs-1139	265	21	.	.	PUNCT
iajs-1139	266	1	the	the	DET
iajs-1139	266	2	notion	notion	NOUN
iajs-1139	266	3	of	of	ADP
iajs-1139	266	4	semiessential	semiessential	ADJ
iajs-1139	266	5	fuzzy	fuzzy	ADJ
iajs-1139	266	6	ideal	ideal	NOUN
iajs-1139	266	7	of	of	ADP
iajs-1139	266	8	fuzzy	fuzzy	ADJ
iajs-1139	266	9	ring	ring	NOUN
iajs-1139	266	10	can	can	AUX
iajs-1139	266	11	be	be	AUX
iajs-1139	266	12	generalized	generalize	VERB
iajs-1139	266	13	to	to	ADP
iajs-1139	266	14	semiessential	semiessential	ADJ
iajs-1139	266	15	fuzzy	fuzzy	ADJ
iajs-1139	266	16	submodules	submodule	NOUN
iajs-1139	266	17	similarly	similarly	ADV
iajs-1139	266	18	we	we	PRON
iajs-1139	266	19	can	can	AUX
iajs-1139	266	20	obtain	obtain	VERB
iajs-1139	266	21	similar	similar	ADJ
iajs-1139	266	22	results	result	NOUN
iajs-1139	266	23	except	except	SCONJ
iajs-1139	266	24	the	the	DET
iajs-1139	266	25	direct	direct	ADJ
iajs-1139	266	26	sum	sum	NOUN
iajs-1139	266	27	of	of	ADP
iajs-1139	266	28	essential	essential	ADJ
iajs-1139	266	29	fuzzy	fuzzy	ADJ
iajs-1139	266	30	ideals	ideal	NOUN
iajs-1139	266	31	.	.	PUNCT
iajs-1139	267	1	references	reference	NOUN
iajs-1139	267	2	1	1	NUM
iajs-1139	267	3	.	.	X
iajs-1139	268	1	zadah	zadah	PROPN
iajs-1139	268	2	,	,	PUNCT
iajs-1139	268	3	l.a	l.a	PROPN
iajs-1139	268	4	.	.	PROPN
iajs-1139	268	5	,	,	PUNCT
iajs-1139	268	6	(	(	PUNCT
iajs-1139	268	7	1965	1965	NUM
iajs-1139	268	8	)	)	PUNCT
iajs-1139	268	9	,	,	PUNCT
iajs-1139	268	10	"	"	PUNCT
iajs-1139	268	11	fuzzy	fuzzy	ADJ
iajs-1139	268	12	sets	set	NOUN
iajs-1139	268	13	"	"	PUNCT
iajs-1139	268	14	,	,	PUNCT
iajs-1139	268	15	inform	inform	NOUN
iajs-1139	268	16	and	and	CCONJ
iajs-1139	268	17	control	control	NOUN
iajs-1139	268	18	,	,	PUNCT
iajs-1139	268	19	8	8	NUM
iajs-1139	268	20	,	,	PUNCT
iajs-1139	268	21	338	338	NUM
iajs-1139	268	22	-	-	SYM
iajs-1139	268	23	353	353	NUM
iajs-1139	268	24	.	.	NOUN
iajs-1139	269	1	2	2	NUM
iajs-1139	269	2	.	.	X
iajs-1139	269	3	liu	liu	PROPN
iajs-1139	269	4	,	,	PUNCT
iajs-1139	269	5	w.j	w.j	PROPN
iajs-1139	269	6	.	.	PROPN
iajs-1139	269	7	,	,	PUNCT
iajs-1139	269	8	(	(	PUNCT
iajs-1139	269	9	1982	1982	NUM
iajs-1139	269	10	)	)	PUNCT
iajs-1139	269	11	,	,	PUNCT
iajs-1139	269	12	"	"	PUNCT
iajs-1139	269	13	fuzzy	fuzzy	ADJ
iajs-1139	269	14	invariant	invariant	ADJ
iajs-1139	269	15	subgroup	subgroup	NOUN
iajs-1139	269	16	and	and	CCONJ
iajs-1139	269	17	fuzzy	fuzzy	ADJ
iajs-1139	269	18	ideals	ideal	NOUN
iajs-1139	269	19	"	"	PUNCT
iajs-1139	269	20	,	,	PUNCT
iajs-1139	269	21	fuzzy	fuzzy	ADJ
iajs-1139	269	22	sets	set	NOUN
iajs-1139	269	23	and	and	CCONJ
iajs-1139	269	24	systems	system	NOUN
iajs-1139	269	25	,	,	PUNCT
iajs-1139	269	26	8	8	NUM
iajs-1139	269	27	,	,	PUNCT
iajs-1139	269	28	133	133	NUM
iajs-1139	269	29	-	-	SYM
iajs-1139	269	30	139	139	NUM
iajs-1139	269	31	.	.	PUNCT
iajs-1139	270	1	3	3	X
iajs-1139	270	2	.	.	NUM
iajs-1139	270	3	martines	martine	NOUN
iajs-1139	270	4	,	,	PUNCT
iajs-1139	270	5	l.	l.	PROPN
iajs-1139	270	6	,	,	PUNCT
iajs-1139	270	7	(	(	PUNCT
iajs-1139	270	8	1995	1995	NUM
iajs-1139	270	9	)	)	PUNCT
iajs-1139	270	10	,	,	PUNCT
iajs-1139	270	11	"	"	PUNCT
iajs-1139	270	12	fuzzy	fuzzy	ADJ
iajs-1139	270	13	subgroups	subgroup	NOUN
iajs-1139	270	14	of	of	ADP
iajs-1139	270	15	fuzzy	fuzzy	ADJ
iajs-1139	270	16	groups	group	NOUN
iajs-1139	270	17	and	and	CCONJ
iajs-1139	270	18	ideals	ideal	NOUN
iajs-1139	270	19	of	of	ADP
iajs-1139	270	20	fuzzy	fuzzy	ADJ
iajs-1139	270	21	rings	ring	NOUN
iajs-1139	270	22	"	"	PUNCT
iajs-1139	270	23	,	,	PUNCT
iajs-1139	270	24	the	the	DET
iajs-1139	270	25	journal	journal	NOUN
iajs-1139	270	26	of	of	ADP
iajs-1139	270	27	fuzzy	fuzzy	ADJ
iajs-1139	270	28	math	math	NOUN
iajs-1139	270	29	.	.	PUNCT
iajs-1139	270	30	,	,	PUNCT
iajs-1139	270	31	3	3	X
iajs-1139	270	32	,	,	PUNCT
iajs-1139	270	33	no	no	INTJ
iajs-1139	270	34	.	.	NOUN
iajs-1139	270	35	4	4	NUM
iajs-1139	270	36	,	,	PUNCT
iajs-1139	270	37	833	833	NUM
iajs-1139	270	38	-	-	SYM
iajs-1139	270	39	849	849	NUM
iajs-1139	270	40	.	.	NOUN
iajs-1139	270	41	4	4	NUM
iajs-1139	270	42	.	.	X
iajs-1139	271	1	kasch	kasch	PROPN
iajs-1139	271	2	,	,	PUNCT
iajs-1139	271	3	f.	f.	PROPN
iajs-1139	271	4	,	,	PUNCT
iajs-1139	271	5	(	(	PUNCT
iajs-1139	271	6	1982	1982	NUM
iajs-1139	271	7	)	)	PUNCT
iajs-1139	271	8	,	,	PUNCT
iajs-1139	271	9	"	"	PUNCT
iajs-1139	271	10	m	m	VERB
iajs-1139	271	11	odules	odule	NOUN
iajs-1139	271	12	and	and	CCONJ
iajs-1139	271	13	rings	ring	NOUN
iajs-1139	271	14	"	"	PUNCT
iajs-1139	271	15	,	,	PUNCT
iajs-1139	271	16	academic	academic	ADJ
iajs-1139	271	17	press	press	NOUN
iajs-1139	271	18	,	,	PUNCT
iajs-1139	271	19	london	london	PROPN
iajs-1139	271	20	,	,	PUNCT
iajs-1139	271	21	new	new	PROPN
iajs-1139	271	22	york	york	PROPN
iajs-1139	271	23	.	.	PUNCT
iajs-1139	272	1	5	5	X
iajs-1139	272	2	.	.	X
iajs-1139	272	3	hadi	hadi	PROPN
iajs-1139	272	4	,	,	PUNCT
iajs-1139	272	5	m.a	m.a	PROPN
iajs-1139	272	6	.	.	PROPN
iajs-1139	272	7	inaam	inaam	PROPN
iajs-1139	272	8	,	,	PUNCT
iajs-1139	272	9	(	(	PUNCT
iajs-1139	272	10	2001	2001	NUM
iajs-1139	272	11	)	)	PUNCT
iajs-1139	272	12	,	,	PUNCT
iajs-1139	272	13	"	"	PUNCT
iajs-1139	272	14	some	some	DET
iajs-1139	272	15	special	special	ADJ
iajs-1139	272	16	fuzzy	fuzzy	ADJ
iajs-1139	272	17	ideals	ideal	NOUN
iajs-1139	272	18	of	of	ADP
iajs-1139	272	19	fuzzy	fuzzy	ADJ
iajs-1139	272	20	rings	ring	NOUN
iajs-1139	272	21	'	'	PART
iajs-1139	272	22	,	,	PUNCT
iajs-1139	272	23	j.	j.	PROPN
iajs-1139	272	24	math	math	PROPN
iajs-1139	272	25	.	.	PUNCT
iajs-1139	273	1	and	and	CCONJ
iajs-1139	273	2	physic	physic	NOUN
iajs-1139	273	3	,	,	PUNCT
iajs-1139	273	4	6	6	NUM
iajs-1139	273	5	,	,	PUNCT
iajs-1139	273	6	no.2	no.2	PROPN
iajs-1139	273	7	.	.	PROPN
iajs-1139	274	1	6	6	NUM
iajs-1139	274	2	.	.	X
iajs-1139	274	3	al	al	PROPN
iajs-1139	274	4	-	-	PUNCT
iajs-1139	274	5	daban	daban	PROPN
iajs-1139	274	6	,	,	PUNCT
iajs-1139	274	7	k.a.nada	k.a.nada	PROPN
iajs-1139	274	8	,	,	PUNCT
iajs-1139	274	9	(	(	PUNCT
iajs-1139	274	10	2005	2005	NUM
iajs-1139	274	11	)	)	PUNCT
iajs-1139	274	12	,	,	PUNCT
iajs-1139	274	13	"	"	PUNCT
iajs-1139	274	14	semiessential	semiessential	ADJ
iajs-1139	274	15	submodules	submodule	NOUN
iajs-1139	274	16	and	and	CCONJ
iajs-1139	274	17	semiuniform	semiuniform	NOUN
iajs-1139	274	18	modules	module	NOUN
iajs-1139	274	19	"	"	PUNCT
iajs-1139	274	20	,	,	PUNCT
iajs-1139	274	21	m.sc	m.sc	PROPN
iajs-1139	274	22	.	.	PUNCT
iajs-1139	275	1	thesis	thesis	NOUN
iajs-1139	275	2	,	,	PUNCT
iajs-1139	275	3	tukirt	tukirt	PROPN
iajs-1139	275	4	university	university	NOUN
iajs-1139	275	5	.	.	PUNCT
iajs-1139	276	1	7	7	X
iajs-1139	276	2	.	.	X
iajs-1139	276	3	al	al	PROPN
iajs-1139	276	4	-	-	PUNCT
iajs-1139	276	5	khamees	khamees	PROPN
iajs-1139	276	6	,	,	PUNCT
iajs-1139	276	7	y.	y.	NOUN
iajs-1139	276	8	and	and	CCONJ
iajs-1139	276	9	mordeson	mordeson	NOUN
iajs-1139	276	10	,	,	PUNCT
iajs-1139	276	11	(	(	PUNCT
iajs-1139	276	12	1998	1998	NUM
iajs-1139	276	13	)	)	PUNCT
iajs-1139	276	14	,	,	PUNCT
iajs-1139	276	15	"	"	PUNCT
iajs-1139	276	16	fuzzy	fuzzy	ADJ
iajs-1139	276	17	principal	principal	ADJ
iajs-1139	276	18	ideals	ideal	NOUN
iajs-1139	276	19	and	and	CCONJ
iajs-1139	276	20	fuzzy	fuzzy	ADJ
iajs-1139	276	21	simple	simple	ADJ
iajs-1139	276	22	field	field	NOUN
iajs-1139	276	23	extensions	extension	NOUN
iajs-1139	276	24	"	"	PUNCT
iajs-1139	276	25	,	,	PUNCT
iajs-1139	276	26	fuzzy	fuzzy	ADJ
iajs-1139	276	27	sets	set	NOUN
iajs-1139	276	28	and	and	CCONJ
iajs-1139	276	29	systems	system	NOUN
iajs-1139	276	30	,	,	PUNCT
iajs-1139	276	31	96	96	NUM
iajs-1139	276	32	,	,	PUNCT
iajs-1139	276	33	147	147	NUM
iajs-1139	276	34	-	-	SYM
iajs-1139	276	35	253	253	NUM
iajs-1139	276	36	.	.	PUNCT
iajs-1139	276	37	8	8	NUM
iajs-1139	276	38	.	.	X
iajs-1139	277	1	kumar	kumar	PROPN
iajs-1139	277	2	,	,	PUNCT
iajs-1139	277	3	r.	r.	PROPN
iajs-1139	277	4	,	,	PUNCT
iajs-1139	277	5	(	(	PUNCT
iajs-1139	277	6	1991	1991	NUM
iajs-1139	277	7	)	)	PUNCT
iajs-1139	277	8	,	,	PUNCT
iajs-1139	277	9	"	"	PUNCT
iajs-1139	277	10	fuzzy	fuzzy	ADJ
iajs-1139	277	11	semiprimary	semiprimary	ADJ
iajs-1139	277	12	ideals	ideal	NOUN
iajs-1139	277	13	of	of	ADP
iajs-1139	277	14	ring	ring	NOUN
iajs-1139	277	15	"	"	PUNCT
iajs-1139	277	16	,	,	PUNCT
iajs-1139	277	17	fuzzy	fuzzy	ADJ
iajs-1139	277	18	sets	set	NOUN
iajs-1139	277	19	and	and	CCONJ
iajs-1139	277	20	systems	system	NOUN
iajs-1139	277	21	,	,	PUNCT
iajs-1139	277	22	42	42	NUM
iajs-1139	277	23	,	,	PUNCT
iajs-1139	277	24	263	263	NUM
iajs-1139	277	25	-	-	SYM
iajs-1139	277	26	272	272	NUM
iajs-1139	277	27	.	.	NOUN
iajs-1139	277	28	9	9	NUM
iajs-1139	277	29	.	.	PUNCT
iajs-1139	277	30	martines	martine	NOUN
iajs-1139	277	31	,	,	PUNCT
iajs-1139	277	32	l.	l.	PROPN
iajs-1139	277	33	,	,	PUNCT
iajs-1139	277	34	(	(	PUNCT
iajs-1139	277	35	1999	1999	NUM
iajs-1139	277	36	)	)	PUNCT
iajs-1139	277	37	,	,	PUNCT
iajs-1139	277	38	"	"	PUNCT
iajs-1139	277	39	prime	prime	ADJ
iajs-1139	277	40	and	and	CCONJ
iajs-1139	277	41	primary	primary	ADJ
iajs-1139	277	42	l	l	ADJ
iajs-1139	277	43	-	-	ADJ
iajs-1139	277	44	fuzzy	fuzzy	ADJ
iajs-1139	277	45	ideals	ideal	NOUN
iajs-1139	277	46	of	of	ADP
iajs-1139	277	47	l	l	ADJ
iajs-1139	277	48	-	-	ADJ
iajs-1139	277	49	fuzzy	fuzzy	ADJ
iajs-1139	277	50	rings	ring	NOUN
iajs-1139	277	51	"	"	PUNCT
iajs-1139	277	52	,	,	PUNCT
iajs-1139	277	53	fuzzy	fuzzy	ADJ
iajs-1139	277	54	sets	set	NOUN
iajs-1139	277	55	and	and	CCONJ
iajs-1139	277	56	systems	system	NOUN
iajs-1139	277	57	,	,	PUNCT
iajs-1139	277	58	101	101	NUM
iajs-1139	277	59	,	,	PUNCT
iajs-1139	277	60	489	489	NUM
iajs-1139	277	61	-	-	SYM
iajs-1139	277	62	494	494	NUM
iajs-1139	277	63	.	.	NOUN
iajs-1139	277	64	10	10	NUM
iajs-1139	277	65	.	.	PUNCT
iajs-1139	277	66	abo	abo	NOUN
iajs-1139	277	67	-	-	PUNCT
iajs-1139	277	68	drab	drab	NOUN
iajs-1139	277	69	,	,	PUNCT
iajs-1139	277	70	a.t	a.t	PROPN
iajs-1139	277	71	.	.	PROPN
iajs-1139	277	72	,	,	PUNCT
iajs-1139	277	73	(	(	PUNCT
iajs-1139	277	74	2000	2000	NUM
iajs-1139	277	75	)	)	PUNCT
iajs-1139	277	76	,	,	PUNCT
iajs-1139	277	77	"	"	PUNCT
iajs-1139	277	78	almost	almost	ADV
iajs-1139	277	79	quasi	quasi	ADJ
iajs-1139	277	80	-	-	ADJ
iajs-1139	277	81	forbenius	forbenius	ADJ
iajs-1139	277	82	fuzzy	fuzzy	ADJ
iajs-1139	277	83	rings	ring	NOUN
iajs-1139	277	84	"	"	PUNCT
iajs-1139	277	85	,	,	PUNCT
iajs-1139	277	86	m.sc	m.sc	PROPN
iajs-1139	277	87	.	.	PUNCT
iajs-1139	278	1	thesis	thesis	NOUN
iajs-1139	278	2	,	,	PUNCT
iajs-1139	278	3	university	university	NOUN
iajs-1139	278	4	of	of	ADP
iajs-1139	278	5	baghdad	baghdad	PROPN
iajs-1139	278	6	,	,	PUNCT
iajs-1139	278	7	college	college	NOUN
iajs-1139	278	8	of	of	ADP
iajs-1139	278	9	education	education	NOUN
iajs-1139	278	10	,	,	PUNCT
iajs-1139	278	11	ibn	ibn	NOUN
iajs-1139	278	12	-	-	PUNCT
iajs-1139	278	13	alhaitham	alhaitham	NOUN
iajs-1139	278	14	.	.	PUNCT
iajs-1139	279	1	11	11	NUM
iajs-1139	279	2	.	.	X
iajs-1139	279	3	megeed	megeed	PROPN
iajs-1139	279	4	,	,	PUNCT
iajs-1139	279	5	n.r	n.r	PROPN
iajs-1139	279	6	.	.	PROPN
iajs-1139	279	7	,	,	PUNCT
iajs-1139	279	8	(	(	PUNCT
iajs-1139	279	9	2000	2000	NUM
iajs-1139	279	10	)	)	PUNCT
iajs-1139	279	11	,	,	PUNCT
iajs-1139	279	12	"	"	PUNCT
iajs-1139	279	13	some	some	DET
iajs-1139	279	14	results	result	NOUN
iajs-1139	279	15	on	on	ADP
iajs-1139	279	16	ctegories	ctegorie	NOUN
iajs-1139	279	17	of	of	ADP
iajs-1139	279	18	rings	ring	NOUN
iajs-1139	279	19	"	"	PUNCT
iajs-1139	279	20	,	,	PUNCT
iajs-1139	279	21	fuzzy	fuzzy	ADJ
iajs-1139	279	22	ring	ring	NOUN
iajs-1139	279	23	and	and	CCONJ
iajs-1139	279	24	its	its	PRON
iajs-1139	279	25	spectrum	spectrum	NOUN
iajs-1139	279	26	,	,	PUNCT
iajs-1139	279	27	m.sc	m.sc	PROPN
iajs-1139	279	28	.	.	PUNCT
iajs-1139	280	1	thesis	thesis	NOUN
iajs-1139	280	2	,	,	PUNCT
iajs-1139	280	3	university	university	NOUN
iajs-1139	280	4	of	of	ADP
iajs-1139	280	5	baghdad	baghdad	PROPN
iajs-1139	280	6	,	,	PUNCT
iajs-1139	280	7	college	college	NOUN
iajs-1139	280	8	of	of	ADP
iajs-1139	280	9	education	education	NOUN
iajs-1139	280	10	,	,	PUNCT
iajs-1139	280	11	ibn	ibn	NOUN
iajs-1139	280	12	-	-	PUNCT
iajs-1139	280	13	alhaitham	alhaitham	NOUN
iajs-1139	280	14	.	.	PUNCT
iajs-1139	281	1	12	12	NUM
iajs-1139	281	2	.	.	PUNCT
iajs-1139	282	1	larsen	larsen	PROPN
iajs-1139	282	2	,	,	PUNCT
iajs-1139	282	3	m.d	m.d	PROPN
iajs-1139	282	4	.	.	PROPN
iajs-1139	282	5	and	and	CCONJ
iajs-1139	282	6	mc.carthy	mc.carthy	PROPN
iajs-1139	282	7	p.j	p.j	PROPN
iajs-1139	282	8	.	.	PROPN
iajs-1139	282	9	,	,	PUNCT
iajs-1139	282	10	(	(	PUNCT
iajs-1139	282	11	1971	1971	NUM
iajs-1139	282	12	)	)	PUNCT
iajs-1139	282	13	,	,	PUNCT
iajs-1139	282	14	"	"	PUNCT
iajs-1139	282	15	multiplicative	multiplicative	ADJ
iajs-1139	282	16	theory	theory	NOUN
iajs-1139	282	17	of	of	ADP
iajs-1139	282	18	ideals	ideal	NOUN
iajs-1139	282	19	"	"	PUNCT
iajs-1139	282	20	,	,	PUNCT
iajs-1139	282	21	academic	academic	ADJ
iajs-1139	282	22	press	press	NOUN
iajs-1139	282	23	,	,	PUNCT
iajs-1139	282	24	new	new	PROPN
iajs-1139	282	25	york	york	PROPN
iajs-1139	282	26	.	.	PUNCT
iajs-1139	282	27	2009	2009	NUM
iajs-1139	282	28	)	)	PUNCT
iajs-1139	282	29	4	4	NUM
iajs-1139	282	30	(	(	PUNCT
iajs-1139	282	31	22مجلة	22مجلة	NUM
iajs-1139	282	32	ابن	ابن	PROPN
iajs-1139	282	33	الھیثم	الھیثم	PROPN
iajs-1139	282	34	للعلوم	للعلوم	PROPN
iajs-1139	282	35	الصرفة	الصرفة	PROPN
iajs-1139	282	36	والتطبیقیة	والتطبیقیة	PROPN
iajs-1139	282	37	المجلد	المجلد	PROPN
iajs-1139	282	38	المثالیات	المثالیات	VERB
iajs-1139	282	39	شبه	شبه	PROPN
iajs-1139	282	40	الجوهریة	الجوهریة	VERB
iajs-1139	282	41	الضبابیة	الضبابیة	PROPN
iajs-1139	282	42	والحلقات	والحلقات	PROPN
iajs-1139	282	43	شبه	شبه	PROPN
iajs-1139	282	44	المنتظمة	المنتظمة	PROPN
iajs-1139	282	45	الضبابیة	الضبابیة	NOUN
iajs-1139	282	46	میسون	میسون	NOUN
iajs-1139	282	47	عبد	عبد	VERB
iajs-1139	282	48	هامل	هامل	NOUN
iajs-1139	282	49	ابن	ابن	VERB
iajs-1139	282	50	الهیثم	الهیثم	PROPN
iajs-1139	282	51	،	،	PROPN
iajs-1139	282	52	جامعة	جامعة	PROPN
iajs-1139	282	53	بغداد	بغداد	PROPN
iajs-1139	282	54	-	-	PUNCT
iajs-1139	282	55	كلیة	كلیة	PROPN
iajs-1139	282	56	التربیة	التربیة	NOUN
iajs-1139	282	57	،	،	PROPN
iajs-1139	283	1	قسم	قسم	PROPN
iajs-1139	283	2	الریاضیات	الریاضیات	PROPN
iajs-1139	283	3	الخالصة	الخالصة	PROPN
iajs-1139	283	4	فـي	فـي	VERB
iajs-1139	284	1	هــذا	هــذا	PROPN
iajs-1139	284	2	البحــث	البحــث	PROPN
iajs-1139	284	3	قــدمنا	قــدمنا	PROPN
iajs-1139	284	4	ودرســنا	ودرســنا	VERB
iajs-1139	284	5	المثالیـات	المثالیـات	PROPN
iajs-1139	284	6	شــبه	شــبه	PROPN
iajs-1139	284	7	الجوهریــة	الجوهریــة	PROPN
iajs-1139	284	8	الــضبابیة	الــضبابیة	PROPN
iajs-1139	284	9	فــي	فــي	ADJ
iajs-1139	284	10	حلقـة	حلقـة	PROPN
iajs-1139	284	11	ضــبابیة	ضــبابیة	NOUN
iajs-1139	284	12	،	،	PROPN
iajs-1139	284	13	الحلقــات	الحلقــات	PROPN
iajs-1139	284	14	المنتظمــة	المنتظمــة	PROPN
iajs-1139	284	15	الــضبابیة	الــضبابیة	PROPN
iajs-1139	284	16	.والحلقات	.والحلقات	PROPN
iajs-1139	284	17	شبه	شبه	VERB
iajs-1139	284	18	المنتظمة	المنتظمة	PROPN
iajs-1139	284	19	الضبابیة	الضبابیة	NOUN