id	sid	tid	token	lemma	pos
iajs-1241	1	1	n−primary	n−primary	PROPN
iajs-1241	1	2	submodules	submodules	PROPN
iajs-1241	1	3	ibn	ibn	PROPN
iajs-1241	1	4	alhaitham	alhaitham	PROPN
iajs-1241	1	5	j.	j.	PROPN
iajs-1241	1	6	for	for	ADP
iajs-1241	1	7	pure	pure	ADJ
iajs-1241	1	8	&	&	CCONJ
iajs-1241	1	9	appl	appl	PROPN
iajs-1241	1	10	.	.	PUNCT
iajs-1241	2	1	sci	sci	PROPN
iajs-1241	2	2	vol.22	vol.22	PROPN
iajs-1241	2	3	(	(	PUNCT
iajs-1241	2	4	2	2	NUM
iajs-1241	2	5	)	)	PUNCT
iajs-1241	2	6	2009	2009	NUM
iajs-1241	2	7	fuzzy	fuzzy	ADJ
iajs-1241	2	8	semimaximal	semimaximal	ADJ
iajs-1241	2	9	ideals	ideal	NOUN
iajs-1241	2	10	i	i	PRON
iajs-1241	2	11	,	,	PUNCT
iajs-1241	2	12	m.a.hadi	m.a.hadi	NOUN
iajs-1241	2	13	,	,	PUNCT
iajs-1241	2	14	m.	m.	NOUN
iajs-1241	2	15	a.hamil	a.hamil	PROPN
iajs-1241	2	16	department	department	PROPN
iajs-1241	2	17	of	of	ADP
iajs-1241	2	18	mathematics	mathematics	PROPN
iajs-1241	2	19	,	,	PUNCT
iajs-1241	2	20	college	college	NOUN
iajs-1241	2	21	of	of	ADP
iajs-1241	2	22	education	education	PROPN
iajs-1241	2	23	ibn	ibn	PROPN
iajs-1241	2	24	-	-	PUNCT
iajs-1241	2	25	al	al	PROPN
iajs-1241	2	26	-	-	PUNCT
iajs-1241	2	27	haitham	haitham	PROPN
iajs-1241	2	28	university	university	PROPN
iajs-1241	2	29	of	of	ADP
iajs-1241	2	30	baghdad	baghdad	PROPN
iajs-1241	2	31	abstract	abstract	ADV
iajs-1241	2	32	let	let	VERB
iajs-1241	2	33	r	r	PRON
iajs-1241	2	34	be	be	AUX
iajs-1241	2	35	a	a	DET
iajs-1241	2	36	commutative	commutative	ADJ
iajs-1241	2	37	ring	ring	NOUN
iajs-1241	2	38	with	with	ADP
iajs-1241	2	39	identity	identity	NOUN
iajs-1241	2	40	.	.	PUNCT
iajs-1241	3	1	a	a	DET
iajs-1241	3	2	proper	proper	ADJ
iajs-1241	3	3	ideal	ideal	NOUN
iajs-1241	3	4	i	i	PRON
iajs-1241	3	5	of	of	ADP
iajs-1241	3	6	r	r	NOUN
iajs-1241	3	7	is	be	AUX
iajs-1241	3	8	called	call	VERB
iajs-1241	3	9	semimaximal	semimaximal	ADJ
iajs-1241	3	10	if	if	SCONJ
iajs-1241	3	11	i	i	PRON
iajs-1241	3	12	is	be	AUX
iajs-1241	3	13	a	a	DET
iajs-1241	3	14	finite	finite	ADJ
iajs-1241	3	15	intersection	intersection	NOUN
iajs-1241	3	16	of	of	ADP
iajs-1241	3	17	maximal	maximal	ADJ
iajs-1241	3	18	ideals	ideal	NOUN
iajs-1241	3	19	of	of	ADP
iajs-1241	3	20	r.	r.	PROPN
iajs-1241	3	21	in	in	ADP
iajs-1241	3	22	this	this	DET
iajs-1241	3	23	paper	paper	NOUN
iajs-1241	3	24	we	we	PRON
iajs-1241	3	25	fuzzify	fuzzify	VERB
iajs-1241	3	26	this	this	DET
iajs-1241	3	27	concept	concept	NOUN
iajs-1241	3	28	to	to	ADP
iajs-1241	3	29	fuzzy	fuzzy	ADJ
iajs-1241	3	30	ideals	ideal	NOUN
iajs-1241	3	31	of	of	ADP
iajs-1241	3	32	r	r	NOUN
iajs-1241	3	33	,	,	PUNCT
iajs-1241	3	34	where	where	SCONJ
iajs-1241	3	35	a	a	DET
iajs-1241	3	36	fuzzy	fuzzy	ADJ
iajs-1241	3	37	ideal	ideal	NOUN
iajs-1241	3	38	a	a	PRON
iajs-1241	3	39	of	of	ADP
iajs-1241	3	40	r	r	NOUN
iajs-1241	3	41	is	be	AUX
iajs-1241	3	42	called	call	VERB
iajs-1241	3	43	semimaximal	semimaximal	PROPN
iajs-1241	3	44	if	if	SCONJ
iajs-1241	3	45	a	a	PRON
iajs-1241	3	46	is	be	AUX
iajs-1241	3	47	a	a	DET
iajs-1241	3	48	finite	finite	ADJ
iajs-1241	3	49	intersection	intersection	NOUN
iajs-1241	3	50	of	of	ADP
iajs-1241	3	51	fuzzy	fuzzy	ADJ
iajs-1241	3	52	maximal	maximal	ADJ
iajs-1241	3	53	ideals	ideal	NOUN
iajs-1241	3	54	.	.	PUNCT
iajs-1241	4	1	various	various	ADJ
iajs-1241	4	2	basic	basic	ADJ
iajs-1241	4	3	properties	property	NOUN
iajs-1241	4	4	are	be	AUX
iajs-1241	4	5	given	give	VERB
iajs-1241	4	6	.	.	PUNCT
iajs-1241	5	1	moreover	moreover	ADV
iajs-1241	5	2	some	some	DET
iajs-1241	5	3	examples	example	NOUN
iajs-1241	5	4	are	be	AUX
iajs-1241	5	5	given	give	VERB
iajs-1241	5	6	to	to	PART
iajs-1241	5	7	illustrate	illustrate	VERB
iajs-1241	5	8	this	this	DET
iajs-1241	5	9	concept	concept	NOUN
iajs-1241	5	10	.	.	PUNCT
iajs-1241	6	1	introduction	introduction	NOUN
iajs-1241	6	2	let	let	VERB
iajs-1241	6	3	r	r	PRON
iajs-1241	6	4	be	be	AUX
iajs-1241	6	5	a	a	DET
iajs-1241	6	6	commutative	commutative	ADJ
iajs-1241	6	7	ring	ring	NOUN
iajs-1241	6	8	with	with	ADP
iajs-1241	6	9	unity	unity	NOUN
iajs-1241	6	10	.	.	PUNCT
iajs-1241	7	1	it	it	PRON
iajs-1241	7	2	is	be	AUX
iajs-1241	7	3	well	well	ADV
iajs-1241	7	4	-	-	PUNCT
iajs-1241	7	5	known	know	VERB
iajs-1241	7	6	that	that	SCONJ
iajs-1241	7	7	a	a	DET
iajs-1241	7	8	proper	proper	ADJ
iajs-1241	7	9	ideal	ideal	NOUN
iajs-1241	7	10	m	m	NOUN
iajs-1241	7	11	of	of	ADP
iajs-1241	7	12	a	a	DET
iajs-1241	7	13	ring	ring	NOUN
iajs-1241	7	14	r	r	NOUN
iajs-1241	7	15	is	be	AUX
iajs-1241	7	16	called	call	VERB
iajs-1241	7	17	maximal	maximal	ADJ
iajs-1241	7	18	if	if	SCONJ
iajs-1241	7	19	for	for	ADP
iajs-1241	7	20	every	every	DET
iajs-1241	7	21	ideal	ideal	ADJ
iajs-1241	7	22	b	b	PROPN
iajs-1241	7	23	of	of	ADP
iajs-1241	7	24	r	r	NOUN
iajs-1241	7	25	,	,	PUNCT
iajs-1241	7	26	m	m	VERB
iajs-1241	7	27			PROPN
iajs-1241	7	28	b	b	PROPN
iajs-1241	7	29			PROPN
iajs-1241	7	30	r	r	NOUN
iajs-1241	7	31	implies	imply	VERB
iajs-1241	7	32	b	b	X
iajs-1241	7	33	=	=	ADJ
iajs-1241	7	34	r.	r.	ADJ
iajs-1241	7	35	d.s.malik	d.s.malik	NOUN
iajs-1241	7	36	and	and	CCONJ
iajs-1241	7	37	j.n.mordeson	j.n.mordeson	ADJ
iajs-1241	7	38	in	in	ADP
iajs-1241	7	39	(	(	PUNCT
iajs-1241	7	40	1	1	X
iajs-1241	7	41	)	)	PUNCT
iajs-1241	7	42	introduced	introduce	VERB
iajs-1241	7	43	and	and	CCONJ
iajs-1241	7	44	studied	study	VERB
iajs-1241	7	45	the	the	DET
iajs-1241	7	46	concept	concept	NOUN
iajs-1241	7	47	of	of	ADP
iajs-1241	7	48	fuzzy	fuzzy	ADJ
iajs-1241	7	49	maximal	maximal	ADJ
iajs-1241	7	50	ideal	ideal	NOUN
iajs-1241	7	51	of	of	ADP
iajs-1241	7	52	r	r	NOUN
iajs-1241	7	53	,	,	PUNCT
iajs-1241	7	54	where	where	SCONJ
iajs-1241	7	55	a	a	DET
iajs-1241	7	56	fuzzy	fuzzy	ADJ
iajs-1241	7	57	ideal	ideal	NOUN
iajs-1241	7	58	a	a	PRON
iajs-1241	7	59	of	of	ADP
iajs-1241	7	60	r	r	NOUN
iajs-1241	7	61	is	be	AUX
iajs-1241	7	62	maximal	maximal	ADJ
iajs-1241	7	63	if	if	SCONJ
iajs-1241	7	64	1	1	NUM
iajs-1241	7	65	.	.	X
iajs-1241	8	1	a	a	PRON
iajs-1241	8	2	is	be	AUX
iajs-1241	8	3	not	not	PART
iajs-1241	8	4	constant	constant	ADJ
iajs-1241	8	5	,	,	PUNCT
iajs-1241	8	6	2	2	X
iajs-1241	8	7	.	.	X
iajs-1241	9	1	for	for	ADP
iajs-1241	9	2	any	any	DET
iajs-1241	9	3	fuzzy	fuzzy	ADJ
iajs-1241	9	4	ideal	ideal	NOUN
iajs-1241	9	5	b	b	PROPN
iajs-1241	9	6	of	of	ADP
iajs-1241	9	7	r	r	NOUN
iajs-1241	9	8	if	if	SCONJ
iajs-1241	9	9	a	a	DET
iajs-1241	9	10			PROPN
iajs-1241	9	11	b	b	PROPN
iajs-1241	9	12	,	,	PUNCT
iajs-1241	9	13	then	then	ADV
iajs-1241	9	14	either	either	CCONJ
iajs-1241	9	15	a	a	ADJ
iajs-1241	9	16	=	=	PUNCT
iajs-1241	9	17	b	b	PROPN
iajs-1241	9	18	or	or	CCONJ
iajs-1241	9	19	b	b	NOUN
iajs-1241	9	20	=	=	SYM
iajs-1241	9	21	r	r	PROPN
iajs-1241	9	22	in	in	ADP
iajs-1241	9	23	fact	fact	NOUN
iajs-1241	9	24	d.s.malik	d.s.malik	NOUN
iajs-1241	9	25	and	and	CCONJ
iajs-1241	9	26	j.n.mordeson	j.n.mordeson	ADJ
iajs-1241	9	27	in	in	ADP
iajs-1241	9	28	(	(	PUNCT
iajs-1241	9	29	1	1	X
iajs-1241	9	30	)	)	PUNCT
iajs-1241	9	31	explained	explain	VERB
iajs-1241	9	32	that	that	SCONJ
iajs-1241	9	33	a	a	DET
iajs-1241	9	34	fuzzy	fuzzy	ADJ
iajs-1241	9	35	maximal	maximal	ADJ
iajs-1241	9	36	ideal	ideal	NOUN
iajs-1241	9	37	a	a	PRON
iajs-1241	9	38	on	on	ADP
iajs-1241	9	39	r	r	NOUN
iajs-1241	9	40	can	can	AUX
iajs-1241	9	41	not	not	PART
iajs-1241	9	42	be	be	AUX
iajs-1241	9	43	defined	define	VERB
iajs-1241	9	44	as	as	ADP
iajs-1241	9	45	a	a	DET
iajs-1241	9	46	fuzzy	fuzzy	ADJ
iajs-1241	9	47	ideal	ideal	NOUN
iajs-1241	9	48	a	a	DET
iajs-1241	9	49			NOUN
iajs-1241	9	50	r	r	PUNCT
iajs-1241	9	51	such	such	ADJ
iajs-1241	9	52	that	that	PRON
iajs-1241	9	53	for	for	ADP
iajs-1241	9	54	each	each	DET
iajs-1241	9	55	fuzzy	fuzzy	ADJ
iajs-1241	9	56	ideal	ideal	PROPN
iajs-1241	9	57	b	b	PROPN
iajs-1241	9	58	of	of	ADP
iajs-1241	9	59	r	r	NOUN
iajs-1241	9	60	,	,	PUNCT
iajs-1241	9	61	if	if	SCONJ
iajs-1241	9	62	a	a	DET
iajs-1241	9	63			PROPN
iajs-1241	9	64	b	b	PROPN
iajs-1241	9	65			PROPN
iajs-1241	9	66	r	r	PROPN
iajs-1241	9	67	implies	imply	VERB
iajs-1241	9	68	b	b	PROPN
iajs-1241	9	69	=	=	SYM
iajs-1241	9	70	r	r	PROPN
iajs-1241	9	71	.	.	PROPN
iajs-1241	9	72	goodreal	goodreal	NOUN
iajs-1241	9	73	in	in	ADP
iajs-1241	9	74	(	(	PUNCT
iajs-1241	9	75	2	2	X
iajs-1241	9	76	)	)	PUNCT
iajs-1241	9	77	introduced	introduce	VERB
iajs-1241	9	78	the	the	DET
iajs-1241	9	79	concept	concept	NOUN
iajs-1241	9	80	of	of	ADP
iajs-1241	9	81	semimaximal	semimaximal	ADJ
iajs-1241	9	82	ideals	ideal	NOUN
iajs-1241	9	83	,	,	PUNCT
iajs-1241	9	84	where	where	SCONJ
iajs-1241	9	85	an	an	DET
iajs-1241	9	86	ideal	ideal	NOUN
iajs-1241	9	87	i	i	PRON
iajs-1241	9	88	of	of	ADP
iajs-1241	9	89	r	r	NOUN
iajs-1241	9	90	is	be	AUX
iajs-1241	9	91	called	call	VERB
iajs-1241	9	92	semimaximal	semimaximal	ADJ
iajs-1241	9	93	if	if	SCONJ
iajs-1241	9	94	it	it	PRON
iajs-1241	9	95	is	be	AUX
iajs-1241	9	96	a	a	DET
iajs-1241	9	97	finite	finite	ADJ
iajs-1241	9	98	intersection	intersection	NOUN
iajs-1241	9	99	of	of	ADP
iajs-1241	9	100	maximal	maximal	ADJ
iajs-1241	9	101	ideals	ideal	NOUN
iajs-1241	9	102	.	.	PUNCT
iajs-1241	10	1	also	also	ADV
iajs-1241	10	2	,	,	PUNCT
iajs-1241	10	3	this	this	DET
iajs-1241	10	4	concept	concept	NOUN
iajs-1241	10	5	was	be	AUX
iajs-1241	10	6	studied	study	VERB
iajs-1241	10	7	by	by	ADP
iajs-1241	10	8	hatem	hatem	PROPN
iajs-1241	10	9	in	in	ADP
iajs-1241	10	10	(	(	PUNCT
iajs-1241	10	11	3	3	NUM
iajs-1241	10	12	)	)	PUNCT
iajs-1241	10	13	.	.	PUNCT
iajs-1241	11	1	in	in	ADP
iajs-1241	11	2	this	this	DET
iajs-1241	11	3	paper	paper	NOUN
iajs-1241	11	4	,	,	PUNCT
iajs-1241	11	5	we	we	PRON
iajs-1241	11	6	fuzzify	fuzzify	VERB
iajs-1241	11	7	this	this	DET
iajs-1241	11	8	concept	concept	NOUN
iajs-1241	11	9	to	to	ADP
iajs-1241	11	10	fuzzy	fuzzy	ADJ
iajs-1241	11	11	ideals	ideal	NOUN
iajs-1241	11	12	of	of	ADP
iajs-1241	11	13	r	r	NOUN
iajs-1241	11	14	,	,	PUNCT
iajs-1241	11	15	where	where	SCONJ
iajs-1241	11	16	a	a	DET
iajs-1241	11	17	fuzzy	fuzzy	ADJ
iajs-1241	11	18	ideal	ideal	NOUN
iajs-1241	11	19	a	a	PRON
iajs-1241	11	20	of	of	ADP
iajs-1241	11	21	r	r	NOUN
iajs-1241	11	22	is	be	AUX
iajs-1241	11	23	called	call	VERB
iajs-1241	11	24	semimaximal	semimaximal	PROPN
iajs-1241	11	25	if	if	SCONJ
iajs-1241	11	26	a	a	PRON
iajs-1241	11	27	is	be	AUX
iajs-1241	11	28	a	a	DET
iajs-1241	11	29	finite	finite	ADJ
iajs-1241	11	30	intersection	intersection	NOUN
iajs-1241	11	31	of	of	ADP
iajs-1241	11	32	fuzzy	fuzzy	ADJ
iajs-1241	11	33	maximal	maximal	ADJ
iajs-1241	11	34	ideals	ideal	NOUN
iajs-1241	11	35	of	of	ADP
iajs-1241	11	36	r.	r.	PROPN
iajs-1241	11	37	moreover	moreover	ADV
iajs-1241	11	38	,	,	PUNCT
iajs-1241	11	39	we	we	PRON
iajs-1241	11	40	generalize	generalize	VERB
iajs-1241	11	41	many	many	ADJ
iajs-1241	11	42	properties	property	NOUN
iajs-1241	11	43	of	of	ADP
iajs-1241	11	44	maximal	maximal	ADJ
iajs-1241	11	45	and	and	CCONJ
iajs-1241	11	46	semimaximal	semimaximal	ADJ
iajs-1241	11	47	ideals	ideal	NOUN
iajs-1241	11	48	in	in	ADP
iajs-1241	11	49	to	to	ADP
iajs-1241	11	50	fuzzy	fuzzy	ADJ
iajs-1241	11	51	semimaximal	semimaximal	ADJ
iajs-1241	11	52	ideals	ideal	NOUN
iajs-1241	11	53	of	of	ADP
iajs-1241	11	54	a	a	DET
iajs-1241	11	55	ring	ring	NOUN
iajs-1241	11	56	.	.	PUNCT
iajs-1241	12	1	this	this	DET
iajs-1241	12	2	paper	paper	NOUN
iajs-1241	12	3	consists	consist	VERB
iajs-1241	12	4	of	of	ADP
iajs-1241	12	5	four	four	NUM
iajs-1241	12	6	sections	section	NOUN
iajs-1241	12	7	.	.	PUNCT
iajs-1241	13	1	in	in	ADP
iajs-1241	13	2	s.1	s.1	NUM
iajs-1241	13	3	,	,	PUNCT
iajs-1241	13	4	we	we	PRON
iajs-1241	13	5	recall	recall	VERB
iajs-1241	13	6	many	many	ADJ
iajs-1241	13	7	definitions	definition	NOUN
iajs-1241	13	8	and	and	CCONJ
iajs-1241	13	9	properties	property	NOUN
iajs-1241	13	10	which	which	PRON
iajs-1241	13	11	are	be	AUX
iajs-1241	13	12	needed	need	VERB
iajs-1241	13	13	in	in	ADP
iajs-1241	13	14	our	our	PRON
iajs-1241	13	15	work	work	NOUN
iajs-1241	13	16	.	.	PUNCT
iajs-1241	14	1	in	in	ADP
iajs-1241	14	2	s.2	s.2	ADP
iajs-1241	14	3	,	,	PUNCT
iajs-1241	14	4	various	various	ADJ
iajs-1241	14	5	basic	basic	ADJ
iajs-1241	14	6	properties	property	NOUN
iajs-1241	14	7	about	about	ADP
iajs-1241	14	8	fuzzy	fuzzy	ADJ
iajs-1241	14	9	semimaximal	semimaximal	ADJ
iajs-1241	14	10	ideals	ideal	NOUN
iajs-1241	14	11	are	be	AUX
iajs-1241	14	12	discussed	discuss	VERB
iajs-1241	14	13	.	.	PUNCT
iajs-1241	15	1	in	in	ADP
iajs-1241	15	2	s.3	s.3	NOUN
iajs-1241	15	3	,	,	PUNCT
iajs-1241	15	4	the	the	DET
iajs-1241	15	5	image	image	NOUN
iajs-1241	15	6	and	and	CCONJ
iajs-1241	15	7	inverse	inverse	NOUN
iajs-1241	15	8	image	image	NOUN
iajs-1241	15	9	of	of	ADP
iajs-1241	15	10	fuzzy	fuzzy	ADJ
iajs-1241	15	11	semimaximal	semimaximal	ADJ
iajs-1241	15	12	ideals	ideal	NOUN
iajs-1241	15	13	are	be	AUX
iajs-1241	15	14	studied	study	VERB
iajs-1241	15	15	.	.	PUNCT
iajs-1241	16	1	in	in	ADP
iajs-1241	16	2	s.4	s.4	PROPN
iajs-1241	16	3	,	,	PUNCT
iajs-1241	16	4	we	we	PRON
iajs-1241	16	5	study	study	VERB
iajs-1241	16	6	the	the	DET
iajs-1241	16	7	behavior	behavior	NOUN
iajs-1241	16	8	of	of	ADP
iajs-1241	16	9	fuzzy	fuzzy	ADJ
iajs-1241	16	10	semimaximal	semimaximal	ADJ
iajs-1241	16	11	ideals	ideal	NOUN
iajs-1241	16	12	in	in	ADP
iajs-1241	16	13	a	a	DET
iajs-1241	16	14	ring	ring	NOUN
iajs-1241	16	15	r	r	NOUN
iajs-1241	16	16	,	,	PUNCT
iajs-1241	16	17	where	where	SCONJ
iajs-1241	16	18	r	r	NOUN
iajs-1241	16	19	=	=	SYM
iajs-1241	16	20	r1r2	r1r2	PROPN
iajs-1241	16	21	(	(	PUNCT
iajs-1241	16	22	direct	direct	ADJ
iajs-1241	16	23	sum	sum	NOUN
iajs-1241	16	24	of	of	ADP
iajs-1241	16	25	two	two	NUM
iajs-1241	16	26	rings	ring	NOUN
iajs-1241	16	27	r1	r1	NOUN
iajs-1241	16	28	,	,	PUNCT
iajs-1241	16	29	r2	r2	PROPN
iajs-1241	16	30	)	)	PUNCT
iajs-1241	16	31	.	.	PUNCT
iajs-1241	17	1	s.1	s.1	NOUN
iajs-1241	17	2	preliminaries	preliminary	NOUN
iajs-1241	17	3	this	this	DET
iajs-1241	17	4	section	section	NOUN
iajs-1241	17	5	contains	contain	VERB
iajs-1241	17	6	some	some	DET
iajs-1241	17	7	definitions	definition	NOUN
iajs-1241	17	8	and	and	CCONJ
iajs-1241	17	9	properties	property	NOUN
iajs-1241	17	10	of	of	ADP
iajs-1241	17	11	fuzzy	fuzzy	ADJ
iajs-1241	17	12	subset	subset	NOUN
iajs-1241	17	13	,	,	PUNCT
iajs-1241	17	14	fuzzy	fuzzy	ADJ
iajs-1241	17	15	ideals	ideal	NOUN
iajs-1241	17	16	and	and	CCONJ
iajs-1241	17	17	fuzzy	fuzzy	ADJ
iajs-1241	17	18	rings	ring	NOUN
iajs-1241	17	19	,	,	PUNCT
iajs-1241	17	20	which	which	PRON
iajs-1241	17	21	we	we	PRON
iajs-1241	17	22	used	use	VERB
iajs-1241	17	23	in	in	ADP
iajs-1241	17	24	the	the	DET
iajs-1241	17	25	next	next	ADJ
iajs-1241	17	26	section	section	NOUN
iajs-1241	17	27	.	.	PUNCT
iajs-1241	18	1	first	first	ADV
iajs-1241	18	2	we	we	PRON
iajs-1241	18	3	give	give	VERB
iajs-1241	18	4	some	some	DET
iajs-1241	18	5	basic	basic	ADJ
iajs-1241	18	6	definitions	definition	NOUN
iajs-1241	18	7	and	and	CCONJ
iajs-1241	18	8	properties	property	NOUN
iajs-1241	18	9	of	of	ADP
iajs-1241	18	10	fuzzy	fuzzy	ADJ
iajs-1241	18	11	subsets	subset	NOUN
iajs-1241	18	12	.	.	PUNCT
iajs-1241	19	1	let	let	VERB
iajs-1241	19	2	r	r	PRON
iajs-1241	19	3	be	be	AUX
iajs-1241	19	4	a	a	DET
iajs-1241	19	5	commutative	commutative	ADJ
iajs-1241	19	6	ring	ring	NOUN
iajs-1241	19	7	with	with	ADP
iajs-1241	19	8	unity	unity	NOUN
iajs-1241	19	9	,	,	PUNCT
iajs-1241	19	10	a	a	DET
iajs-1241	19	11	fuzzy	fuzzy	ADJ
iajs-1241	19	12	subset	subset	NOUN
iajs-1241	19	13	of	of	ADP
iajs-1241	19	14	r	r	NOUN
iajs-1241	19	15	is	be	AUX
iajs-1241	19	16	a	a	DET
iajs-1241	19	17	function	function	NOUN
iajs-1241	19	18	from	from	ADP
iajs-1241	19	19	r	r	NOUN
iajs-1241	19	20	into	into	ADP
iajs-1241	19	21	[	[	X
iajs-1241	19	22	0,1	0,1	NUM
iajs-1241	19	23	]	]	PUNCT
iajs-1241	19	24	,	,	PUNCT
iajs-1241	19	25	(	(	PUNCT
iajs-1241	19	26	4	4	NUM
iajs-1241	19	27	)	)	PUNCT
iajs-1241	19	28	.	.	PUNCT
iajs-1241	20	1	a	a	DET
iajs-1241	20	2	fuzzy	fuzzy	NOUN
iajs-1241	20	3	subset	subset	VERB
iajs-1241	20	4	a	a	PRON
iajs-1241	20	5	is	be	AUX
iajs-1241	20	6	called	call	VERB
iajs-1241	20	7	a	a	DET
iajs-1241	20	8	fuzzy	fuzzy	ADJ
iajs-1241	20	9	constant	constant	ADJ
iajs-1241	20	10	if	if	SCONJ
iajs-1241	20	11	a(x	a(x	NOUN
iajs-1241	20	12	)	)	PUNCT
iajs-1241	20	13	=	=	SYM
iajs-1241	20	14	t	t	PROPN
iajs-1241	20	15	,	,	PUNCT
iajs-1241	20	16			NOUN
iajs-1241	20	17	x	x	PROPN
iajs-1241	20	18	r	r	PROPN
iajs-1241	20	19	,	,	PUNCT
iajs-1241	20	20	t	t	X
iajs-1241	20	21			NOUN
iajs-1241	21	1	[	[	X
iajs-1241	21	2	0,1	0,1	NUM
iajs-1241	21	3	]	]	PUNCT
iajs-1241	21	4	,	,	PUNCT
iajs-1241	21	5	(	(	PUNCT
iajs-1241	21	6	4	4	NUM
iajs-1241	21	7	)	)	PUNCT
iajs-1241	21	8	.	.	PUNCT
iajs-1241	22	1	for	for	ADP
iajs-1241	22	2	each	each	DET
iajs-1241	22	3	t	t	NOUN
iajs-1241	22	4			NOUN
iajs-1241	22	5	[	[	X
iajs-1241	22	6	0,1	0,1	NUM
iajs-1241	22	7	]	]	PUNCT
iajs-1241	22	8	,	,	PUNCT
iajs-1241	22	9	the	the	DET
iajs-1241	22	10	set	set	NOUN
iajs-1241	22	11	at	at	ADP
iajs-1241	22	12	=	=	PUNCT
iajs-1241	22	13	{	{	PUNCT
iajs-1241	22	14	x	x	PROPN
iajs-1241	22	15	r	r	PROPN
iajs-1241	22	16	,	,	PUNCT
iajs-1241	22	17	a(x	a(x	PROPN
iajs-1241	22	18	)	)	PUNCT
iajs-1241	22	19			NUM
iajs-1241	22	20	t	t	PROPN
iajs-1241	22	21	}	}	PUNCT
iajs-1241	22	22	is	be	AUX
iajs-1241	22	23	called	call	VERB
iajs-1241	22	24	a	a	DET
iajs-1241	22	25	level	level	NOUN
iajs-1241	22	26	subset	subset	NOUN
iajs-1241	22	27	of	of	ADP
iajs-1241	22	28	a	a	PRON
iajs-1241	22	29	,	,	PUNCT
iajs-1241	22	30	(	(	PUNCT
iajs-1241	22	31	4	4	NUM
iajs-1241	22	32	)	)	PUNCT
iajs-1241	22	33	.	.	PUNCT
iajs-1241	23	1	a	a	PROPN
iajs-1241	23	2	denoted	denote	VERB
iajs-1241	23	3	the	the	DET
iajs-1241	23	4	set	set	NOUN
iajs-1241	23	5	{	{	PUNCT
iajs-1241	23	6	x	x	PROPN
iajs-1241	23	7			PROPN
iajs-1241	23	8	r	r	NOUN
iajs-1241	23	9	,	,	PUNCT
iajs-1241	23	10	a(x	a(x	PROPN
iajs-1241	23	11	)	)	PUNCT
iajs-1241	23	12	=	=	PUNCT
iajs-1241	23	13	a(0	a(0	PROPN
iajs-1241	23	14	)	)	PUNCT
iajs-1241	23	15	}	}	PUNCT
iajs-1241	23	16	,	,	PUNCT
iajs-1241	23	17	(	(	PUNCT
iajs-1241	23	18	1	1	NUM
iajs-1241	23	19	)	)	PUNCT
iajs-1241	23	20	.	.	PUNCT
iajs-1241	24	1	if	if	SCONJ
iajs-1241	24	2	x	x	PROPN
iajs-1241	24	3	r	r	PROPN
iajs-1241	24	4	and	and	CCONJ
iajs-1241	24	5	t	t	PROPN
iajs-1241	24	6			NOUN
iajs-1241	24	7	[	[	X
iajs-1241	24	8	0,1	0,1	NUM
iajs-1241	24	9	]	]	PUNCT
iajs-1241	24	10	,	,	PUNCT
iajs-1241	24	11	we	we	PRON
iajs-1241	24	12	let	let	VERB
iajs-1241	24	13	xt	xt	PART
iajs-1241	24	14	denote	denote	VERB
iajs-1241	24	15	the	the	DET
iajs-1241	24	16	fuzzy	fuzzy	ADJ
iajs-1241	24	17	subset	subset	NOUN
iajs-1241	24	18	of	of	ADP
iajs-1241	24	19	a	a	DET
iajs-1241	24	20	define	define	NOUN
iajs-1241	24	21	by	by	ADP
iajs-1241	24	22	xt(y	xt(y	NOUN
iajs-1241	24	23	)	)	PUNCT
iajs-1241	25	1	=	=	SYM
iajs-1241	25	2	0	0	PUNCT
iajs-1241	26	1	if	if	SCONJ
iajs-1241	26	2	x	x	PROPN
iajs-1241	26	3			VERB
iajs-1241	26	4	y	y	PROPN
iajs-1241	26	5	and	and	CCONJ
iajs-1241	26	6	xt(y	xt(y	NUM
iajs-1241	26	7	)	)	PUNCT
iajs-1241	26	8	=	=	SYM
iajs-1241	26	9	t	t	NOUN
iajs-1241	26	10	if	if	SCONJ
iajs-1241	26	11	x	x	PROPN
iajs-1241	26	12	=	=	SYM
iajs-1241	26	13	y	y	PROPN
iajs-1241	26	14	,	,	PUNCT
iajs-1241	26	15	xt	xt	PROPN
iajs-1241	26	16	is	be	AUX
iajs-1241	26	17	called	call	VERB
iajs-1241	26	18	a	a	DET
iajs-1241	26	19	fuzzy	fuzzy	ADJ
iajs-1241	26	20	singleton	singleton	NOUN
iajs-1241	26	21	,	,	PUNCT
iajs-1241	26	22	(	(	PUNCT
iajs-1241	26	23	5	5	NUM
iajs-1241	26	24	)	)	PUNCT
iajs-1241	26	25	.	.	PUNCT
iajs-1241	27	1	if	if	SCONJ
iajs-1241	27	2	a	a	PRON
iajs-1241	27	3	and	and	CCONJ
iajs-1241	27	4	b	b	NOUN
iajs-1241	27	5	are	be	AUX
iajs-1241	27	6	fuzzy	fuzzy	ADJ
iajs-1241	27	7	subsets	subset	NOUN
iajs-1241	27	8	of	of	ADP
iajs-1241	27	9	r	r	NOUN
iajs-1241	27	10	,	,	PUNCT
iajs-1241	27	11	then	then	ADV
iajs-1241	27	12	:	:	PUNCT
iajs-1241	27	13	ibn	ibn	PROPN
iajs-1241	27	14	alhaitham	alhaitham	NOUN
iajs-1241	27	15	j.	j.	PROPN
iajs-1241	27	16	for	for	ADP
iajs-1241	27	17	pure	pure	ADJ
iajs-1241	27	18	&	&	CCONJ
iajs-1241	27	19	appl	appl	PROPN
iajs-1241	27	20	.	.	PUNCT
iajs-1241	28	1	sci	sci	PROPN
iajs-1241	28	2	vol.22	vol.22	PROPN
iajs-1241	28	3	(	(	PUNCT
iajs-1241	28	4	2	2	NUM
iajs-1241	28	5	)	)	PUNCT
iajs-1241	28	6	2009	2009	NUM
iajs-1241	28	7	1	1	NUM
iajs-1241	28	8	.	.	PUNCT
iajs-1241	29	1	a	a	DET
iajs-1241	29	2			PROPN
iajs-1241	29	3	b	b	PROPN
iajs-1241	29	4	if	if	SCONJ
iajs-1241	29	5	a(x	a(x	NOUN
iajs-1241	29	6	)	)	PUNCT
iajs-1241	29	7			NOUN
iajs-1241	29	8	b(x	b(x	NOUN
iajs-1241	29	9	)	)	PUNCT
iajs-1241	29	10	,	,	PUNCT
iajs-1241	29	11	for	for	ADP
iajs-1241	29	12	all	all	DET
iajs-1241	29	13	x	x	PROPN
iajs-1241	29	14	r	r	NOUN
iajs-1241	29	15	,	,	PUNCT
iajs-1241	29	16	(	(	PUNCT
iajs-1241	29	17	6	6	NUM
iajs-1241	29	18	)	)	PUNCT
iajs-1241	29	19	,	,	PUNCT
iajs-1241	29	20	2	2	X
iajs-1241	29	21	.	.	X
iajs-1241	30	1	a	a	DET
iajs-1241	30	2	=	=	SYM
iajs-1241	30	3	b	b	NOUN
iajs-1241	30	4	if	if	SCONJ
iajs-1241	30	5	a(x	a(x	NOUN
iajs-1241	30	6	)	)	PUNCT
iajs-1241	30	7	=	=	SYM
iajs-1241	30	8	b(x	b(x	NOUN
iajs-1241	30	9	)	)	PUNCT
iajs-1241	30	10	,	,	PUNCT
iajs-1241	30	11	for	for	ADP
iajs-1241	30	12	all	all	DET
iajs-1241	30	13	x	x	PROPN
iajs-1241	30	14	r	r	NOUN
iajs-1241	30	15	,	,	PUNCT
iajs-1241	30	16	(	(	PUNCT
iajs-1241	30	17	6	6	NUM
iajs-1241	30	18	)	)	PUNCT
iajs-1241	30	19	.	.	PUNCT
iajs-1241	31	1	we	we	PRON
iajs-1241	31	2	define	define	VERB
iajs-1241	31	3	a	a	DET
iajs-1241	31	4			NOUN
iajs-1241	31	5	b	b	NOUN
iajs-1241	31	6	by	by	ADP
iajs-1241	31	7	,	,	PUNCT
iajs-1241	31	8	(	(	PUNCT
iajs-1241	31	9	a	a	DET
iajs-1241	31	10			PUNCT
iajs-1241	31	11	b)(x	b)(x	NOUN
iajs-1241	31	12	)	)	PUNCT
iajs-1241	31	13	=	=	SYM
iajs-1241	31	14	min{a(x	min{a(x	NOUN
iajs-1241	31	15	)	)	PUNCT
iajs-1241	31	16	,	,	PUNCT
iajs-1241	31	17	b(x	b(x	NOUN
iajs-1241	31	18	)	)	PUNCT
iajs-1241	31	19	}	}	PUNCT
iajs-1241	31	20	,	,	PUNCT
iajs-1241	31	21			NOUN
iajs-1241	31	22	x	x	PROPN
iajs-1241	31	23	r	r	NOUN
iajs-1241	31	24	,	,	PUNCT
iajs-1241	31	25	and	and	CCONJ
iajs-1241	31	26	let	let	VERB
iajs-1241	31	27	{	{	PUNCT
iajs-1241	31	28	a:	a:	VERB
iajs-1241	31	29	}	}	PUNCT
iajs-1241	31	30	be	be	AUX
iajs-1241	31	31	a	a	DET
iajs-1241	31	32	collection	collection	NOUN
iajs-1241	31	33	of	of	ADP
iajs-1241	31	34	fuzzy	fuzzy	ADJ
iajs-1241	31	35	subsets	subset	NOUN
iajs-1241	31	36	of	of	ADP
iajs-1241	31	37	r.	r.	PROPN
iajs-1241	31	38	define	define	VERB
iajs-1241	31	39	the	the	DET
iajs-1241	31	40	fuzzy	fuzzy	ADJ
iajs-1241	31	41	subset	subset	NOUN
iajs-1241	31	42	of	of	ADP
iajs-1241	31	43	r	r	NOUN
iajs-1241	31	44	(	(	PUNCT
iajs-1241	31	45	intersection	intersection	NOUN
iajs-1241	31	46	)	)	PUNCT
iajs-1241	31	47	by	by	ADP
iajs-1241	31	48	(	(	PUNCT
iajs-1241	31	49	)	)	PUNCT
iajs-1241	31	50			X
iajs-1241	31	51			VERB
iajs-1241	31	52			PROPN
iajs-1241	31	53	x	x	PUNCT
iajs-1241	31	54	=	=	PUNCT
iajs-1241	31	55	inf	inf	NOUN
iajs-1241	31	56	{	{	PUNCT
iajs-1241	31	57	a:	a:	PROPN
iajs-1241	31	58	}	}	PUNCT
iajs-1241	31	59	,	,	PUNCT
iajs-1241	31	60	for	for	ADP
iajs-1241	31	61	all	all	DET
iajs-1241	31	62	x	x	PROPN
iajs-1241	31	63	r	r	NOUN
iajs-1241	31	64	,	,	PUNCT
iajs-1241	31	65	(	(	PUNCT
iajs-1241	31	66	6	6	NUM
iajs-1241	31	67	)	)	PUNCT
iajs-1241	31	68	.	.	PUNCT
iajs-1241	32	1	let	let	VERB
iajs-1241	32	2	a	a	PRON
iajs-1241	32	3	and	and	CCONJ
iajs-1241	32	4	b	b	NOUN
iajs-1241	32	5	be	be	AUX
iajs-1241	32	6	fuzzy	fuzzy	ADJ
iajs-1241	32	7	subsets	subset	NOUN
iajs-1241	32	8	of	of	ADP
iajs-1241	32	9	r	r	NOUN
iajs-1241	32	10	,	,	PUNCT
iajs-1241	32	11	then	then	ADV
iajs-1241	32	12	for	for	ADP
iajs-1241	32	13	all	all	DET
iajs-1241	32	14	t	t	NOUN
iajs-1241	32	15	[0,1	[0,1	NOUN
iajs-1241	32	16	]	]	PUNCT
iajs-1241	32	17	,	,	PUNCT
iajs-1241	32	18	(	(	PUNCT
iajs-1241	32	19	ab)t	ab)t	PROPN
iajs-1241	32	20	=	=	X
iajs-1241	32	21	at	at	ADP
iajs-1241	32	22	bt,(3	bt,(3	NOUN
iajs-1241	32	23	)	)	PUNCT
iajs-1241	32	24	.	.	PUNCT
iajs-1241	33	1	let	let	VERB
iajs-1241	33	2	f	f	PRON
iajs-1241	33	3	be	be	AUX
iajs-1241	33	4	a	a	DET
iajs-1241	33	5	mapping	mapping	NOUN
iajs-1241	33	6	from	from	ADP
iajs-1241	33	7	a	a	DET
iajs-1241	33	8	set	set	NOUN
iajs-1241	33	9	m	m	NOUN
iajs-1241	33	10	into	into	ADP
iajs-1241	33	11	a	a	DET
iajs-1241	33	12	set	set	NOUN
iajs-1241	33	13	n.	n.	NOUN
iajs-1241	33	14	let	let	VERB
iajs-1241	33	15	a	a	PRON
iajs-1241	33	16	be	be	AUX
iajs-1241	33	17	a	a	DET
iajs-1241	33	18	fuzzy	fuzzy	ADJ
iajs-1241	33	19	subset	subset	NOUN
iajs-1241	33	20	of	of	ADP
iajs-1241	33	21	m	m	PROPN
iajs-1241	33	22	and	and	CCONJ
iajs-1241	33	23	b	b	X
iajs-1241	33	24	be	be	AUX
iajs-1241	33	25	a	a	DET
iajs-1241	33	26	fuzzy	fuzzy	ADJ
iajs-1241	33	27	subset	subset	NOUN
iajs-1241	33	28	of	of	ADP
iajs-1241	33	29	n.	n.	NOUN
iajs-1241	33	30	the	the	DET
iajs-1241	33	31	image	image	NOUN
iajs-1241	33	32	of	of	ADP
iajs-1241	33	33	a	a	DET
iajs-1241	33	34	denoted	denote	VERB
iajs-1241	33	35	by	by	ADP
iajs-1241	33	36	f	f	PROPN
iajs-1241	33	37	(	(	PUNCT
iajs-1241	33	38	a	a	NOUN
iajs-1241	33	39	)	)	PUNCT
iajs-1241	33	40	is	be	AUX
iajs-1241	33	41	the	the	DET
iajs-1241	33	42	fuzzy	fuzzy	ADJ
iajs-1241	33	43	subset	subset	NOUN
iajs-1241	33	44	of	of	ADP
iajs-1241	33	45	n	n	CCONJ
iajs-1241	33	46	defined	define	VERB
iajs-1241	33	47	by	by	ADP
iajs-1241	33	48	:	:	PUNCT
iajs-1241	33	49	1	1	NUM
iajs-1241	33	50	1sup	1sup	NUM
iajs-1241	33	51	{	{	PUNCT
iajs-1241	33	52	(	(	PUNCT
iajs-1241	33	53	)	)	PUNCT
iajs-1241	33	54	(	(	PUNCT
iajs-1241	33	55	)	)	PUNCT
iajs-1241	33	56	,	,	PUNCT
iajs-1241	33	57	if	if	SCONJ
iajs-1241	33	58	(	(	PUNCT
iajs-1241	33	59	y	y	NOUN
iajs-1241	33	60	)	)	PUNCT
iajs-1241	33	61	,	,	PUNCT
iajs-1241	33	62	forall	forall	NOUN
iajs-1241	33	63	}	}	PUNCT
iajs-1241	33	64	,	,	PUNCT
iajs-1241	33	65	(	(	PUNCT
iajs-1241	33	66	)	)	PUNCT
iajs-1241	33	67	0	0	NUM
iajs-1241	34	1	otherwise	otherwise	ADV
iajs-1241	34	2			NOUN
iajs-1241	34	3			NOUN
iajs-1241	34	4			NOUN
iajs-1241	34	5			PROPN
iajs-1241	34	6			PROPN
iajs-1241	34	7			NOUN
iajs-1241	34	8			VERB
iajs-1241	34	9			NOUN
iajs-1241	34	10			PROPN
iajs-1241	35	1			NUM
iajs-1241	36	1			NUM
iajs-1241	37	1			INTJ
iajs-1241	37	2	z	z	NOUN
iajs-1241	37	3	z	z	PUNCT
iajs-1241	38	1	f	f	X
iajs-1241	38	2	y	y	PROPN
iajs-1241	38	3	f	f	PROPN
iajs-1241	38	4	y	y	PROPN
iajs-1241	38	5	f	f	PROPN
iajs-1241	39	1	where	where	SCONJ
iajs-1241	39	2	f	f	PROPN
iajs-1241	39	3	-1	-1	INTJ
iajs-1241	39	4	(	(	PUNCT
iajs-1241	39	5	y	y	X
iajs-1241	39	6	)	)	PUNCT
iajs-1241	39	7	=	=	NOUN
iajs-1241	39	8	{	{	PUNCT
iajs-1241	39	9	x	x	PROPN
iajs-1241	39	10			PROPN
iajs-1241	39	11	m	m	PROPN
iajs-1241	39	12	,	,	PUNCT
iajs-1241	39	13	f	f	PROPN
iajs-1241	39	14	(	(	PUNCT
iajs-1241	39	15	x	x	X
iajs-1241	39	16	)	)	PUNCT
iajs-1241	39	17	=	=	SYM
iajs-1241	39	18	y	y	NOUN
iajs-1241	39	19	}	}	PUNCT
iajs-1241	39	20	.	.	PUNCT
iajs-1241	40	1	and	and	CCONJ
iajs-1241	40	2	the	the	DET
iajs-1241	40	3	inverse	inverse	ADJ
iajs-1241	40	4	image	image	NOUN
iajs-1241	40	5	of	of	ADP
iajs-1241	40	6	b	b	NOUN
iajs-1241	40	7	,	,	PUNCT
iajs-1241	40	8	denoted	denote	VERB
iajs-1241	40	9	by	by	ADP
iajs-1241	40	10	f	f	PROPN
iajs-1241	40	11	-1	-1	PUNCT
iajs-1241	40	12	(	(	PUNCT
iajs-1241	40	13	b	b	X
iajs-1241	40	14	)	)	PUNCT
iajs-1241	40	15	is	be	AUX
iajs-1241	40	16	the	the	DET
iajs-1241	40	17	fuzzy	fuzzy	ADJ
iajs-1241	40	18	subset	subset	NOUN
iajs-1241	40	19	of	of	ADP
iajs-1241	40	20	m	m	PROPN
iajs-1241	40	21	,	,	PUNCT
iajs-1241	40	22	denoted	denote	VERB
iajs-1241	40	23	by	by	ADP
iajs-1241	40	24	f	f	PROPN
iajs-1241	40	25	-1	-1	PROPN
iajs-1241	40	26	(	(	PUNCT
iajs-1241	40	27	b)(x	b)(x	PROPN
iajs-1241	40	28	)	)	PUNCT
iajs-1241	41	1	=	=	SYM
iajs-1241	41	2	b(f	b(f	PROPN
iajs-1241	41	3	(	(	PUNCT
iajs-1241	41	4	x	x	NOUN
iajs-1241	41	5	)	)	PUNCT
iajs-1241	41	6	)	)	PUNCT
iajs-1241	41	7	,	,	PUNCT
iajs-1241	41	8	for	for	ADP
iajs-1241	41	9	all	all	DET
iajs-1241	41	10	x	x	ADJ
iajs-1241	41	11			PROPN
iajs-1241	41	12	m	m	PROPN
iajs-1241	41	13	,	,	PUNCT
iajs-1241	41	14	(	(	PUNCT
iajs-1241	41	15	4	4	NUM
iajs-1241	41	16	)	)	PUNCT
iajs-1241	41	17	.	.	PUNCT
iajs-1241	42	1	let	let	VERB
iajs-1241	42	2	f	f	PRON
iajs-1241	42	3	be	be	AUX
iajs-1241	42	4	a	a	DET
iajs-1241	42	5	function	function	NOUN
iajs-1241	42	6	from	from	ADP
iajs-1241	42	7	a	a	DET
iajs-1241	42	8	set	set	NOUN
iajs-1241	42	9	m	m	NOUN
iajs-1241	42	10	into	into	ADP
iajs-1241	42	11	a	a	DET
iajs-1241	42	12	set	set	NOUN
iajs-1241	42	13	n.	n.	NOUN
iajs-1241	42	14	a	a	DET
iajs-1241	42	15	fuzzy	fuzzy	NOUN
iajs-1241	42	16	subset	subset	VERB
iajs-1241	42	17	a	a	PRON
iajs-1241	42	18	of	of	ADP
iajs-1241	42	19	m	m	PROPN
iajs-1241	42	20	is	be	AUX
iajs-1241	42	21	called	call	VERB
iajs-1241	42	22	f	f	PROPN
iajs-1241	42	23	–	–	PUNCT
iajs-1241	42	24	invariant	invariant	ADJ
iajs-1241	42	25	if	if	SCONJ
iajs-1241	42	26	a(x	a(x	NOUN
iajs-1241	42	27	)	)	PUNCT
iajs-1241	42	28	=	=	SYM
iajs-1241	42	29	a(y	a(y	PROPN
iajs-1241	42	30	)	)	PUNCT
iajs-1241	42	31	,	,	PUNCT
iajs-1241	42	32	whenever	whenever	SCONJ
iajs-1241	42	33	f	f	PROPN
iajs-1241	42	34	(	(	PUNCT
iajs-1241	42	35	x	x	X
iajs-1241	42	36	)	)	PUNCT
iajs-1241	42	37	=	=	SYM
iajs-1241	42	38	f	f	PROPN
iajs-1241	42	39	(	(	PUNCT
iajs-1241	42	40	y	y	NOUN
iajs-1241	42	41	)	)	PUNCT
iajs-1241	42	42	where	where	SCONJ
iajs-1241	42	43	x	x	X
iajs-1241	42	44	,	,	PUNCT
iajs-1241	42	45	y	y	PROPN
iajs-1241	42	46			PROPN
iajs-1241	42	47	m	m	PROPN
iajs-1241	42	48	,	,	PUNCT
iajs-1241	42	49	(	(	PUNCT
iajs-1241	42	50	7	7	NUM
iajs-1241	42	51	)	)	PUNCT
iajs-1241	42	52	.	.	PUNCT
iajs-1241	43	1	if	if	SCONJ
iajs-1241	43	2	f	f	PROPN
iajs-1241	43	3	a	a	DET
iajs-1241	43	4	function	function	NOUN
iajs-1241	43	5	from	from	ADP
iajs-1241	43	6	a	a	DET
iajs-1241	43	7	set	set	NOUN
iajs-1241	43	8	m	m	NOUN
iajs-1241	43	9	into	into	ADP
iajs-1241	43	10	a	a	DET
iajs-1241	43	11	set	set	NOUN
iajs-1241	43	12	n	n	CCONJ
iajs-1241	43	13	,	,	PUNCT
iajs-1241	43	14	a1	a1	NOUN
iajs-1241	43	15	and	and	CCONJ
iajs-1241	43	16	a2	a2	PROPN
iajs-1241	43	17	are	be	AUX
iajs-1241	43	18	fuzzy	fuzzy	ADJ
iajs-1241	43	19	subsets	subset	NOUN
iajs-1241	43	20	of	of	ADP
iajs-1241	43	21	m	m	NOUN
iajs-1241	43	22	and	and	CCONJ
iajs-1241	43	23	b1	b1	NOUN
iajs-1241	43	24	,	,	PUNCT
iajs-1241	43	25	b2	b2	NOUN
iajs-1241	43	26	are	be	AUX
iajs-1241	43	27	fuzzy	fuzzy	ADJ
iajs-1241	43	28	subsets	subset	NOUN
iajs-1241	43	29	of	of	ADP
iajs-1241	43	30	n	n	CCONJ
iajs-1241	43	31	,	,	PUNCT
iajs-1241	43	32	then	then	ADV
iajs-1241	43	33	1	1	X
iajs-1241	43	34	.	.	X
iajs-1241	43	35	f	f	PROPN
iajs-1241	43	36	(	(	PUNCT
iajs-1241	43	37	a1a2	a1a2	PROPN
iajs-1241	43	38	)	)	PUNCT
iajs-1241	43	39	=	=	SYM
iajs-1241	43	40	f	f	PROPN
iajs-1241	43	41	(	(	PUNCT
iajs-1241	43	42	a1	a1	PROPN
iajs-1241	43	43	)	)	PUNCT
iajs-1241	43	44			ADJ
iajs-1241	43	45	f	f	X
iajs-1241	43	46	(	(	PUNCT
iajs-1241	43	47	a2	a2	PROPN
iajs-1241	43	48	)	)	PUNCT
iajs-1241	43	49	,	,	PUNCT
iajs-1241	43	50	whenever	whenever	SCONJ
iajs-1241	43	51	a1	a1	PROPN
iajs-1241	43	52	,	,	PUNCT
iajs-1241	43	53	a2	a2	PROPN
iajs-1241	43	54	f	f	PROPN
iajs-1241	43	55	–	–	PUNCT
iajs-1241	43	56	invariant	invariant	ADJ
iajs-1241	43	57	,	,	PUNCT
iajs-1241	43	58	(	(	PUNCT
iajs-1241	43	59	8)	8)	NOUN
iajs-1241	43	60	2	2	NUM
iajs-1241	43	61	.	.	PUNCT
iajs-1241	44	1	f	f	PROPN
iajs-1241	44	2	-1	-1	PUNCT
iajs-1241	44	3	(	(	PUNCT
iajs-1241	44	4	b1b2	b1b2	PROPN
iajs-1241	44	5	)	)	PUNCT
iajs-1241	44	6	=	=	SYM
iajs-1241	45	1	f	f	X
iajs-1241	45	2	-1	-1	INTJ
iajs-1241	45	3	(	(	PUNCT
iajs-1241	45	4	b1	b1	NOUN
iajs-1241	45	5	)	)	PUNCT
iajs-1241	45	6			PROPN
iajs-1241	45	7	f	f	X
iajs-1241	45	8	-1	-1	PRON
iajs-1241	45	9	(	(	PUNCT
iajs-1241	45	10	b2	b2	NOUN
iajs-1241	45	11	)	)	PUNCT
iajs-1241	45	12	,	,	PUNCT
iajs-1241	45	13	(	(	PUNCT
iajs-1241	45	14	8)	8)	NUM
iajs-1241	45	15	.	.	PUNCT
iajs-1241	46	1	moreover	moreover	ADV
iajs-1241	46	2	the	the	DET
iajs-1241	46	3	following	follow	VERB
iajs-1241	46	4	definitions	definition	NOUN
iajs-1241	46	5	and	and	CCONJ
iajs-1241	46	6	properties	property	NOUN
iajs-1241	46	7	are	be	AUX
iajs-1241	46	8	needed	need	VERB
iajs-1241	46	9	later	later	ADV
iajs-1241	46	10	definition	definition	NOUN
iajs-1241	46	11	1.1	1.1	NUM
iajs-1241	46	12	:	:	PUNCT
iajs-1241	46	13	(	(	PUNCT
iajs-1241	46	14	9	9	X
iajs-1241	46	15	)	)	PUNCT
iajs-1241	46	16	a	a	DET
iajs-1241	46	17	fuzzy	fuzzy	ADJ
iajs-1241	46	18	subset	subset	NOUN
iajs-1241	46	19	k	k	PROPN
iajs-1241	46	20	of	of	ADP
iajs-1241	46	21	r	r	NOUN
iajs-1241	46	22	is	be	AUX
iajs-1241	46	23	called	call	VERB
iajs-1241	46	24	a	a	DET
iajs-1241	46	25	fuzzy	fuzzy	ADJ
iajs-1241	46	26	ideal	ideal	NOUN
iajs-1241	46	27	of	of	ADP
iajs-1241	46	28	r	r	NOUN
iajs-1241	46	29	if	if	SCONJ
iajs-1241	46	30	for	for	ADP
iajs-1241	46	31	each	each	DET
iajs-1241	46	32	x	x	NOUN
iajs-1241	46	33	,	,	PUNCT
iajs-1241	46	34	y	y	PROPN
iajs-1241	46	35			PROPN
iajs-1241	46	36	r	r	NOUN
iajs-1241	46	37	,	,	PUNCT
iajs-1241	46	38	then	then	ADV
iajs-1241	46	39	:	:	PUNCT
iajs-1241	46	40	1	1	X
iajs-1241	46	41	.	.	X
iajs-1241	47	1	k(x	k(x	PROPN
iajs-1241	47	2	–	–	PUNCT
iajs-1241	47	3	y	y	NOUN
iajs-1241	47	4	)	)	PUNCT
iajs-1241	47	5			NUM
iajs-1241	47	6	min	min	PROPN
iajs-1241	47	7	{	{	PUNCT
iajs-1241	47	8	k(x	k(x	PROPN
iajs-1241	47	9	)	)	PUNCT
iajs-1241	47	10	,	,	PUNCT
iajs-1241	47	11	k(y	k(y	PROPN
iajs-1241	47	12	)	)	PUNCT
iajs-1241	47	13	}	}	PUNCT
iajs-1241	47	14	,	,	PUNCT
iajs-1241	47	15	2	2	X
iajs-1241	47	16	.	.	PUNCT
iajs-1241	47	17	k(x	k(x	PROPN
iajs-1241	47	18	y	y	PROPN
iajs-1241	47	19	)	)	PUNCT
iajs-1241	47	20			PROPN
iajs-1241	47	21	max	max	PROPN
iajs-1241	47	22	{	{	PUNCT
iajs-1241	47	23	k(x	k(x	PROPN
iajs-1241	47	24	)	)	PUNCT
iajs-1241	47	25	,	,	PUNCT
iajs-1241	47	26	k(y	k(y	PROPN
iajs-1241	47	27	)	)	PUNCT
iajs-1241	47	28	}	}	PUNCT
iajs-1241	47	29	.	.	PUNCT
iajs-1241	48	1	definition	definition	NOUN
iajs-1241	48	2	1.2	1.2	NUM
iajs-1241	48	3	:	:	PUNCT
iajs-1241	48	4	(	(	PUNCT
iajs-1241	48	5	10	10	X
iajs-1241	48	6	)	)	PUNCT
iajs-1241	48	7	let	let	VERB
iajs-1241	48	8	x	x	PRON
iajs-1241	48	9	be	be	AUX
iajs-1241	48	10	a	a	DET
iajs-1241	48	11	fuzzy	fuzzy	ADJ
iajs-1241	48	12	subset	subset	NOUN
iajs-1241	48	13	of	of	ADP
iajs-1241	48	14	a	a	DET
iajs-1241	48	15	ring	ring	NOUN
iajs-1241	48	16	r	r	NOUN
iajs-1241	48	17	,	,	PUNCT
iajs-1241	48	18	then	then	ADV
iajs-1241	48	19	x	x	PUNCT
iajs-1241	48	20	is	be	AUX
iajs-1241	48	21	called	call	VERB
iajs-1241	48	22	fuzzy	fuzzy	ADJ
iajs-1241	48	23	ring	ring	NOUN
iajs-1241	48	24	of	of	ADP
iajs-1241	48	25	r	r	NOUN
iajs-1241	48	26	if	if	SCONJ
iajs-1241	48	27	for	for	ADP
iajs-1241	48	28	each	each	DET
iajs-1241	48	29	x	x	NOUN
iajs-1241	48	30	,	,	PUNCT
iajs-1241	48	31	y	y	PROPN
iajs-1241	48	32			PROPN
iajs-1241	48	33	r	r	NOUN
iajs-1241	48	34	,	,	PUNCT
iajs-1241	48	35	then	then	ADV
iajs-1241	48	36	1	1	NUM
iajs-1241	48	37	.	.	PUNCT
iajs-1241	48	38	x	x	SYM
iajs-1241	48	39			PROPN
iajs-1241	48	40	0	0	NUM
iajs-1241	48	41	,	,	PUNCT
iajs-1241	48	42	2	2	NUM
iajs-1241	48	43	.	.	PUNCT
iajs-1241	48	44	x(x	x(x	PROPN
iajs-1241	48	45	–	–	PUNCT
iajs-1241	48	46	y	y	PROPN
iajs-1241	48	47	)	)	PUNCT
iajs-1241	48	48			NUM
iajs-1241	48	49	min	min	PROPN
iajs-1241	48	50	{	{	PUNCT
iajs-1241	48	51	x(x	x(x	PROPN
iajs-1241	48	52	)	)	PUNCT
iajs-1241	48	53	,	,	PUNCT
iajs-1241	48	54	x(y	x(y	PROPN
iajs-1241	48	55	)	)	PUNCT
iajs-1241	48	56	}	}	PUNCT
iajs-1241	48	57	,	,	PUNCT
iajs-1241	48	58	3	3	X
iajs-1241	48	59	.	.	PUNCT
iajs-1241	48	60	x(x	x(x	PROPN
iajs-1241	49	1	y	y	PROPN
iajs-1241	49	2	)	)	PUNCT
iajs-1241	49	3			PROPN
iajs-1241	49	4	max	max	PROPN
iajs-1241	49	5	{	{	PUNCT
iajs-1241	49	6	x(x	x(x	PROPN
iajs-1241	49	7	)	)	PUNCT
iajs-1241	49	8	,	,	PUNCT
iajs-1241	49	9	x(y	x(y	PROPN
iajs-1241	49	10	)	)	PUNCT
iajs-1241	49	11	}	}	PUNCT
iajs-1241	49	12	.	.	PUNCT
iajs-1241	50	1	proposition	proposition	NOUN
iajs-1241	50	2	1.3	1.3	NUM
iajs-1241	50	3	:	:	PUNCT
iajs-1241	50	4	(	(	PUNCT
iajs-1241	50	5	1	1	X
iajs-1241	50	6	)	)	PUNCT
iajs-1241	50	7	let	let	AUX
iajs-1241	50	8	{	{	PUNCT
iajs-1241	50	9	a,	a,	PART
iajs-1241	50	10	}	}	PUNCT
iajs-1241	50	11	be	be	AUX
iajs-1241	50	12	a	a	DET
iajs-1241	50	13	family	family	NOUN
iajs-1241	50	14	of	of	ADP
iajs-1241	50	15	fuzzy	fuzzy	ADJ
iajs-1241	50	16	ideals	ideal	NOUN
iajs-1241	50	17	of	of	ADP
iajs-1241	50	18	r	r	NOUN
iajs-1241	50	19	,	,	PUNCT
iajs-1241	50	20	then	then	ADV
iajs-1241	50	21			PROPN
iajs-1241	50	22			PROPN
iajs-1241	50	23			PUNCT
iajs-1241	50	24	is	be	AUX
iajs-1241	50	25	a	a	DET
iajs-1241	50	26	fuzzy	fuzzy	ADJ
iajs-1241	50	27	ideal	ideal	NOUN
iajs-1241	50	28	of	of	ADP
iajs-1241	50	29	r.	r.	PROPN
iajs-1241	50	30	definition	definition	NOUN
iajs-1241	50	31	1.4	1.4	NUM
iajs-1241	50	32	:	:	PUNCT
iajs-1241	50	33	(	(	PUNCT
iajs-1241	50	34	11	11	X
iajs-1241	50	35	)	)	PUNCT
iajs-1241	50	36	let	let	VERB
iajs-1241	50	37	x	x	PRON
iajs-1241	50	38	be	be	AUX
iajs-1241	50	39	a	a	DET
iajs-1241	50	40	fuzzy	fuzzy	ADJ
iajs-1241	50	41	ring	ring	NOUN
iajs-1241	50	42	of	of	ADP
iajs-1241	50	43	a	a	DET
iajs-1241	50	44	ring	ring	NOUN
iajs-1241	50	45	r	r	NOUN
iajs-1241	50	46	,	,	PUNCT
iajs-1241	50	47	let	let	VERB
iajs-1241	50	48	a	a	PRON
iajs-1241	50	49	be	be	AUX
iajs-1241	50	50	a	a	DET
iajs-1241	50	51	fuzzy	fuzzy	ADJ
iajs-1241	50	52	subset	subset	NOUN
iajs-1241	50	53	of	of	ADP
iajs-1241	50	54	x	x	SYM
iajs-1241	50	55	such	such	ADJ
iajs-1241	50	56	that	that	SCONJ
iajs-1241	50	57	a	a	DET
iajs-1241	50	58			PROPN
iajs-1241	50	59	x.	x.	NOUN
iajs-1241	50	60	then	then	ADV
iajs-1241	50	61	a	a	PRON
iajs-1241	50	62	is	be	AUX
iajs-1241	50	63	called	call	VERB
iajs-1241	50	64	a	a	DET
iajs-1241	50	65	fuzzy	fuzzy	ADJ
iajs-1241	50	66	ideal	ideal	NOUN
iajs-1241	50	67	of	of	ADP
iajs-1241	50	68	a	a	DET
iajs-1241	50	69	fuzzy	fuzzy	ADJ
iajs-1241	50	70	ring	ring	NOUN
iajs-1241	50	71	x	x	INTJ
iajs-1241	50	72	if	if	SCONJ
iajs-1241	50	73	for	for	ADP
iajs-1241	50	74	each	each	DET
iajs-1241	50	75	x	x	NOUN
iajs-1241	50	76	,	,	PUNCT
iajs-1241	50	77	y	y	PROPN
iajs-1241	50	78			NOUN
iajs-1241	50	79	r	r	NOUN
iajs-1241	50	80	1	1	NUM
iajs-1241	50	81	.	.	PUNCT
iajs-1241	51	1	a(x	a(x	PROPN
iajs-1241	51	2	–	–	PUNCT
iajs-1241	51	3	y	y	NOUN
iajs-1241	51	4	)	)	PUNCT
iajs-1241	51	5			NUM
iajs-1241	51	6	min	min	NOUN
iajs-1241	51	7	{	{	PUNCT
iajs-1241	51	8	a(x	a(x	PROPN
iajs-1241	51	9	)	)	PUNCT
iajs-1241	51	10	,	,	PUNCT
iajs-1241	51	11	a(y	a(y	PROPN
iajs-1241	51	12	)	)	PUNCT
iajs-1241	51	13	}	}	PUNCT
iajs-1241	51	14	,	,	PUNCT
iajs-1241	51	15	2	2	X
iajs-1241	51	16	.	.	X
iajs-1241	51	17	a(x	a(x	PROPN
iajs-1241	51	18	y	y	PROPN
iajs-1241	51	19	)	)	PUNCT
iajs-1241	51	20			NUM
iajs-1241	51	21	min{max	min{max	NOUN
iajs-1241	51	22	{	{	PUNCT
iajs-1241	51	23	a(x	a(x	NOUN
iajs-1241	51	24	)	)	PUNCT
iajs-1241	51	25	,	,	PUNCT
iajs-1241	51	26	a(y	a(y	PROPN
iajs-1241	51	27	)	)	PUNCT
iajs-1241	51	28	,	,	PUNCT
iajs-1241	51	29	x(x	x(x	PROPN
iajs-1241	51	30	y	y	PROPN
iajs-1241	51	31	)	)	PUNCT
iajs-1241	51	32	}	}	PUNCT
iajs-1241	51	33	.	.	PUNCT
iajs-1241	52	1	note	note	VERB
iajs-1241	52	2	1.5	1.5	NUM
iajs-1241	52	3	:	:	PUNCT
iajs-1241	52	4	it	it	PRON
iajs-1241	52	5	is	be	AUX
iajs-1241	52	6	clear	clear	ADJ
iajs-1241	52	7	that	that	SCONJ
iajs-1241	52	8	any	any	DET
iajs-1241	52	9	fuzzy	fuzzy	ADJ
iajs-1241	52	10	ideal	ideal	NOUN
iajs-1241	52	11	of	of	ADP
iajs-1241	52	12	a	a	DET
iajs-1241	52	13	ring	ring	NOUN
iajs-1241	52	14	r	r	NOUN
iajs-1241	52	15	is	be	AUX
iajs-1241	52	16	a	a	DET
iajs-1241	52	17	fuzzy	fuzzy	ADJ
iajs-1241	52	18	ideal	ideal	NOUN
iajs-1241	52	19	of	of	ADP
iajs-1241	52	20	a	a	DET
iajs-1241	52	21	fuzzy	fuzzy	ADJ
iajs-1241	52	22	ring	ring	NOUN
iajs-1241	52	23	x	x	INTJ
iajs-1241	52	24	of	of	ADP
iajs-1241	52	25	r	r	NOUN
iajs-1241	52	26	such	such	ADJ
iajs-1241	52	27	that	that	SCONJ
iajs-1241	52	28	x(a	x(a	NOUN
iajs-1241	52	29	)	)	PUNCT
iajs-1241	52	30	=	=	SYM
iajs-1241	52	31	1	1	NUM
iajs-1241	52	32	,	,	PUNCT
iajs-1241	52	33			VERB
iajs-1241	52	34	a	a	DET
iajs-1241	52	35			PROPN
iajs-1241	52	36	r.	r.	NOUN
iajs-1241	52	37	proposition	proposition	NOUN
iajs-1241	52	38	1.6	1.6	NUM
iajs-1241	52	39	:	:	PUNCT
iajs-1241	52	40	let	let	VERB
iajs-1241	52	41	f	f	PRON
iajs-1241	52	42	be	be	AUX
iajs-1241	52	43	a	a	DET
iajs-1241	52	44	homomorphism	homomorphism	NOUN
iajs-1241	52	45	from	from	ADP
iajs-1241	52	46	a	a	DET
iajs-1241	52	47	ring	ring	NOUN
iajs-1241	52	48	r1	r1	NOUN
iajs-1241	52	49	into	into	ADP
iajs-1241	52	50	a	a	DET
iajs-1241	52	51	ring	ring	NOUN
iajs-1241	52	52	r2	r2	NOUN
iajs-1241	52	53	,	,	PUNCT
iajs-1241	52	54	then	then	ADV
iajs-1241	52	55	the	the	DET
iajs-1241	52	56	following	follow	VERB
iajs-1241	52	57	are	be	AUX
iajs-1241	52	58	true	true	ADJ
iajs-1241	52	59	:	:	PUNCT
iajs-1241	52	60	1	1	NUM
iajs-1241	52	61	.	.	X
iajs-1241	52	62	f	f	X
iajs-1241	52	63	(	(	PUNCT
iajs-1241	52	64	a	a	X
iajs-1241	52	65	)	)	PUNCT
iajs-1241	52	66	is	be	AUX
iajs-1241	52	67	a	a	DET
iajs-1241	52	68	fuzzy	fuzzy	ADJ
iajs-1241	52	69	ideal	ideal	NOUN
iajs-1241	52	70	of	of	ADP
iajs-1241	52	71	r2	r2	PROPN
iajs-1241	52	72	,	,	PUNCT
iajs-1241	52	73	for	for	ADP
iajs-1241	52	74	each	each	DET
iajs-1241	52	75	fuzzy	fuzzy	ADJ
iajs-1241	52	76	ideal	ideal	NOUN
iajs-1241	52	77	a	a	PRON
iajs-1241	52	78	of	of	ADP
iajs-1241	52	79	r1	r1	NOUN
iajs-1241	52	80	,	,	PUNCT
iajs-1241	52	81	(	(	PUNCT
iajs-1241	52	82	6	6	NUM
iajs-1241	52	83	)	)	PUNCT
iajs-1241	52	84	.	.	PUNCT
iajs-1241	53	1	2	2	X
iajs-1241	53	2	.	.	X
iajs-1241	53	3	f	f	PROPN
iajs-1241	53	4	-1	-1	PUNCT
iajs-1241	54	1	(	(	PUNCT
iajs-1241	54	2	b	b	X
iajs-1241	54	3	)	)	PUNCT
iajs-1241	54	4	is	be	AUX
iajs-1241	54	5	a	a	DET
iajs-1241	54	6	fuzzy	fuzzy	ADJ
iajs-1241	54	7	ideal	ideal	NOUN
iajs-1241	54	8	of	of	ADP
iajs-1241	54	9	r1	r1	PROPN
iajs-1241	54	10	,	,	PUNCT
iajs-1241	54	11	for	for	ADP
iajs-1241	54	12	each	each	DET
iajs-1241	54	13	fuzzy	fuzzy	ADJ
iajs-1241	54	14	ideal	ideal	PROPN
iajs-1241	54	15	b	b	PROPN
iajs-1241	54	16	of	of	ADP
iajs-1241	54	17	r2	r2	PROPN
iajs-1241	54	18	,	,	PUNCT
iajs-1241	54	19	(	(	PUNCT
iajs-1241	54	20	7	7	X
iajs-1241	54	21	)	)	PUNCT
iajs-1241	54	22	proposition	proposition	NOUN
iajs-1241	54	23	1.7	1.7	NUM
iajs-1241	54	24	:	:	PUNCT
iajs-1241	54	25	(	(	PUNCT
iajs-1241	54	26	1	1	X
iajs-1241	54	27	)	)	PUNCT
iajs-1241	54	28	ibn	ibn	NOUN
iajs-1241	54	29	alhaitham	alhaitham	NOUN
iajs-1241	54	30	j.	j.	PROPN
iajs-1241	54	31	for	for	ADP
iajs-1241	54	32	pure	pure	ADJ
iajs-1241	54	33	&	&	CCONJ
iajs-1241	54	34	appl	appl	PROPN
iajs-1241	54	35	.	.	PUNCT
iajs-1241	55	1	sci	sci	PROPN
iajs-1241	55	2	vol.22	vol.22	PROPN
iajs-1241	55	3	(	(	PUNCT
iajs-1241	55	4	2	2	NUM
iajs-1241	55	5	)	)	PUNCT
iajs-1241	55	6	2009	2009	NUM
iajs-1241	55	7	let	let	VERB
iajs-1241	55	8	a	a	DET
iajs-1241	55	9	,	,	PUNCT
iajs-1241	55	10	b	b	NOUN
iajs-1241	55	11	be	be	AUX
iajs-1241	55	12	fuzzy	fuzzy	ADJ
iajs-1241	55	13	ideals	ideal	NOUN
iajs-1241	55	14	of	of	ADP
iajs-1241	55	15	a	a	DET
iajs-1241	55	16	ring	ring	NOUN
iajs-1241	55	17	r	r	NOUN
iajs-1241	55	18	such	such	ADJ
iajs-1241	55	19	that	that	DET
iajs-1241	55	20	a(0	a(0	PROPN
iajs-1241	55	21	)	)	PUNCT
iajs-1241	55	22	=	=	SYM
iajs-1241	55	23	1	1	NUM
iajs-1241	55	24	=	=	SYM
iajs-1241	55	25	b(0	b(0	PROPN
iajs-1241	55	26	)	)	PUNCT
iajs-1241	55	27	.	.	PUNCT
iajs-1241	56	1	then	then	ADV
iajs-1241	56	2	(	(	PUNCT
iajs-1241	56	3	ab)	ab)	PROPN
iajs-1241	56	4	=	=	PUNCT
iajs-1241	56	5	a	a	PROPN
iajs-1241	56	6			PUNCT
iajs-1241	56	7	b.	b.	NOUN
iajs-1241	56	8	proposition	proposition	NOUN
iajs-1241	56	9	1.8	1.8	NUM
iajs-1241	56	10	:	:	PUNCT
iajs-1241	56	11	(	(	PUNCT
iajs-1241	56	12	1	1	X
iajs-1241	56	13	)	)	PUNCT
iajs-1241	56	14	let	let	AUX
iajs-1241	56	15	{	{	PUNCT
iajs-1241	56	16	a:	a:	VERB
iajs-1241	56	17	}	}	PUNCT
iajs-1241	56	18	be	be	AUX
iajs-1241	56	19	a	a	DET
iajs-1241	56	20	family	family	NOUN
iajs-1241	56	21	of	of	ADP
iajs-1241	56	22	fuzzy	fuzzy	ADJ
iajs-1241	56	23	ideals	ideal	NOUN
iajs-1241	56	24	of	of	ADP
iajs-1241	56	25	r	r	NOUN
iajs-1241	56	26	such	such	ADJ
iajs-1241	56	27	that	that	DET
iajs-1241	56	28	a(0	a(0	NOUN
iajs-1241	56	29	)	)	PUNCT
iajs-1241	57	1	=	=	SYM
iajs-1241	57	2	1	1	NUM
iajs-1241	57	3	,	,	PUNCT
iajs-1241	57	4	for	for	ADP
iajs-1241	57	5	all	all	DET
iajs-1241	57	6	.	.	NOUN
iajs-1241	57	7	then	then	ADV
iajs-1241	57	8	(	(	PUNCT
iajs-1241	57	9	)	)	PUNCT
iajs-1241	57	10			PROPN
iajs-1241	57	11			PROPN
iajs-1241	57	12			PROPN
iajs-1241	57	13			NOUN
iajs-1241	57	14	=	=	SYM
iajs-1241	57	15			NOUN
iajs-1241	57	16			PROPN
iajs-1241	57	17			PROPN
iajs-1241	57	18			PROPN
iajs-1241	57	19			PROPN
iajs-1241	57	20			PROPN
iajs-1241	57	21	.	.	PUNCT
iajs-1241	58	1	s.2	s.2	PROPN
iajs-1241	58	2	basic	basic	ADJ
iajs-1241	58	3	properties	property	NOUN
iajs-1241	58	4	of	of	ADP
iajs-1241	58	5	fuzzy	fuzzy	ADJ
iajs-1241	58	6	semimaximal	semimaximal	ADJ
iajs-1241	58	7	ideals	ideal	NOUN
iajs-1241	58	8	first	first	ADV
iajs-1241	58	9	,	,	PUNCT
iajs-1241	58	10	we	we	PRON
iajs-1241	58	11	give	give	VERB
iajs-1241	58	12	the	the	DET
iajs-1241	58	13	following	follow	VERB
iajs-1241	58	14	lemma	lemma	PROPN
iajs-1241	58	15	which	which	PRON
iajs-1241	58	16	summarized	summarize	VERB
iajs-1241	58	17	the	the	DET
iajs-1241	58	18	basic	basic	ADJ
iajs-1241	58	19	properties	property	NOUN
iajs-1241	58	20	of	of	ADP
iajs-1241	58	21	fuzzy	fuzzy	ADJ
iajs-1241	58	22	maximal	maximal	ADJ
iajs-1241	58	23	ideals	ideal	NOUN
iajs-1241	58	24	.	.	PUNCT
iajs-1241	59	1	lemma	lemma	PROPN
iajs-1241	59	2	2.1	2.1	NUM
iajs-1241	59	3	:	:	PUNCT
iajs-1241	59	4	(	(	PUNCT
iajs-1241	59	5	1	1	X
iajs-1241	59	6	)	)	PUNCT
iajs-1241	59	7	1	1	NUM
iajs-1241	59	8	.	.	PUNCT
iajs-1241	60	1	let	let	VERB
iajs-1241	60	2	a	a	DET
iajs-1241	60	3	be	be	AUX
iajs-1241	60	4	a	a	DET
iajs-1241	60	5	fuzzy	fuzzy	ADJ
iajs-1241	60	6	maximal	maximal	ADJ
iajs-1241	60	7	ideal	ideal	NOUN
iajs-1241	60	8	of	of	ADP
iajs-1241	60	9	r	r	NOUN
iajs-1241	60	10	,	,	PUNCT
iajs-1241	60	11	then	then	ADV
iajs-1241	60	12	a(0	a(0	PROPN
iajs-1241	60	13	)	)	PUNCT
iajs-1241	60	14	=	=	SYM
iajs-1241	60	15	1	1	NUM
iajs-1241	60	16	(	(	PUNCT
iajs-1241	60	17	see	see	VERB
iajs-1241	60	18	th.3.3	th.3.3	NOUN
iajs-1241	60	19	)	)	PUNCT
iajs-1241	60	20	.	.	PUNCT
iajs-1241	61	1	2	2	X
iajs-1241	61	2	.	.	X
iajs-1241	61	3	let	let	VERB
iajs-1241	61	4	a	a	DET
iajs-1241	61	5	be	be	AUX
iajs-1241	61	6	a	a	DET
iajs-1241	61	7	fuzzy	fuzzy	ADJ
iajs-1241	61	8	maximal	maximal	ADJ
iajs-1241	61	9	ideal	ideal	NOUN
iajs-1241	61	10	of	of	ADP
iajs-1241	61	11	r	r	NOUN
iajs-1241	61	12	,	,	PUNCT
iajs-1241	61	13	then	then	ADV
iajs-1241	61	14	im(a)	im(a)	NOUN
iajs-1241	61	15	=	=	SYM
iajs-1241	61	16	2	2	NUM
iajs-1241	61	17	;	;	PUNCT
iajs-1241	61	18	that	that	PRON
iajs-1241	61	19	is	be	AUX
iajs-1241	61	20	a	a	PRON
iajs-1241	61	21	is	be	AUX
iajs-1241	61	22	a	a	DET
iajs-1241	61	23	two	two	NUM
iajs-1241	61	24	valued	value	VERB
iajs-1241	61	25	,	,	PUNCT
iajs-1241	61	26	where	where	SCONJ
iajs-1241	61	27	im(a	im(a	NOUN
iajs-1241	61	28	)	)	PUNCT
iajs-1241	61	29	denotes	denote	VERB
iajs-1241	61	30	image	image	NOUN
iajs-1241	61	31	of	of	ADP
iajs-1241	61	32	a	a	PRON
iajs-1241	61	33	and	and	CCONJ
iajs-1241	61	34	im(a)	im(a)	PROPN
iajs-1241	61	35	denotes	denote	VERB
iajs-1241	61	36	the	the	DET
iajs-1241	61	37	cardinality	cardinality	NOUN
iajs-1241	61	38	of	of	ADP
iajs-1241	61	39	im(a	im(a	PROPN
iajs-1241	61	40	)	)	PUNCT
iajs-1241	61	41	(	(	PUNCT
iajs-1241	61	42	see	see	VERB
iajs-1241	61	43	th.3.4	th.3.4	NOUN
iajs-1241	61	44	)	)	PUNCT
iajs-1241	61	45	.	.	PUNCT
iajs-1241	62	1	3	3	X
iajs-1241	62	2	.	.	X
iajs-1241	62	3	if	if	SCONJ
iajs-1241	62	4	a	a	PRON
iajs-1241	62	5	is	be	AUX
iajs-1241	62	6	a	a	DET
iajs-1241	62	7	fuzzy	fuzzy	ADJ
iajs-1241	62	8	maximal	maximal	ADJ
iajs-1241	62	9	ideal	ideal	NOUN
iajs-1241	62	10	of	of	ADP
iajs-1241	62	11	r	r	NOUN
iajs-1241	62	12	,	,	PUNCT
iajs-1241	62	13	then	then	ADV
iajs-1241	62	14	a	a	PROPN
iajs-1241	62	15	is	be	AUX
iajs-1241	62	16	a	a	DET
iajs-1241	62	17	maximal	maximal	ADJ
iajs-1241	62	18	ideal	ideal	NOUN
iajs-1241	62	19	of	of	ADP
iajs-1241	62	20	r	r	NOUN
iajs-1241	62	21	(	(	PUNCT
iajs-1241	62	22	see	see	VERB
iajs-1241	62	23	th.3.5	th.3.5	PROPN
iajs-1241	62	24	)	)	PUNCT
iajs-1241	62	25	.	.	PUNCT
iajs-1241	63	1	4	4	X
iajs-1241	63	2	.	.	X
iajs-1241	63	3	if	if	SCONJ
iajs-1241	63	4	a	a	PRON
iajs-1241	63	5	is	be	AUX
iajs-1241	63	6	a	a	DET
iajs-1241	63	7	fuzzy	fuzzy	ADJ
iajs-1241	63	8	ideal	ideal	NOUN
iajs-1241	63	9	of	of	ADP
iajs-1241	63	10	r	r	NOUN
iajs-1241	63	11	and	and	CCONJ
iajs-1241	63	12	a	a	PROPN
iajs-1241	63	13	is	be	AUX
iajs-1241	63	14	a	a	DET
iajs-1241	63	15	maximal	maximal	ADJ
iajs-1241	63	16	ideal	ideal	NOUN
iajs-1241	63	17	of	of	ADP
iajs-1241	63	18	r	r	NOUN
iajs-1241	63	19	,	,	PUNCT
iajs-1241	63	20	then	then	ADV
iajs-1241	63	21	a	a	PRON
iajs-1241	63	22	is	be	AUX
iajs-1241	63	23	two	two	NUM
iajs-1241	63	24	valued	value	VERB
iajs-1241	63	25	(	(	PUNCT
iajs-1241	63	26	see	see	VERB
iajs-1241	63	27	th	th	X
iajs-1241	63	28	.	.	PUNCT
iajs-1241	63	29	3.6	3.6	NUM
iajs-1241	63	30	)	)	PUNCT
iajs-1241	63	31	.	.	PUNCT
iajs-1241	64	1	5	5	X
iajs-1241	64	2	.	.	X
iajs-1241	64	3	if	if	SCONJ
iajs-1241	64	4	a	a	PRON
iajs-1241	64	5	is	be	AUX
iajs-1241	64	6	a	a	DET
iajs-1241	64	7	fuzzy	fuzzy	ADJ
iajs-1241	64	8	ideal	ideal	NOUN
iajs-1241	64	9	of	of	ADP
iajs-1241	64	10	r	r	NOUN
iajs-1241	64	11	and	and	CCONJ
iajs-1241	64	12	a	a	PROPN
iajs-1241	64	13	is	be	AUX
iajs-1241	64	14	a	a	DET
iajs-1241	64	15	maximal	maximal	ADJ
iajs-1241	64	16	ideal	ideal	NOUN
iajs-1241	64	17	of	of	ADP
iajs-1241	64	18	r	r	NOUN
iajs-1241	64	19	such	such	ADJ
iajs-1241	64	20	that	that	DET
iajs-1241	64	21	a(0	a(0	PROPN
iajs-1241	64	22	)	)	PUNCT
iajs-1241	64	23	=	=	SYM
iajs-1241	65	1	1	1	X
iajs-1241	65	2	.	.	PUNCT
iajs-1241	65	3	then	then	ADV
iajs-1241	65	4	a	a	PROPN
iajs-1241	65	5	is	be	AUX
iajs-1241	65	6	a	a	DET
iajs-1241	65	7	fuzzy	fuzzy	ADJ
iajs-1241	65	8	maximal	maximal	ADJ
iajs-1241	65	9	ideal	ideal	NOUN
iajs-1241	65	10	of	of	ADP
iajs-1241	65	11	r	r	NOUN
iajs-1241	65	12	(	(	PUNCT
iajs-1241	65	13	see	see	VERB
iajs-1241	65	14	th	th	X
iajs-1241	65	15	.	.	PROPN
iajs-1241	65	16	3.7	3.7	NUM
iajs-1241	65	17	)	)	PUNCT
iajs-1241	65	18	.	.	PUNCT
iajs-1241	66	1	6	6	X
iajs-1241	66	2	.	.	X
iajs-1241	67	1	if	if	SCONJ
iajs-1241	67	2	i	i	PRON
iajs-1241	67	3			VERB
iajs-1241	67	4	r	r	NOUN
iajs-1241	67	5	be	be	VERB
iajs-1241	67	6	an	an	DET
iajs-1241	67	7	ideal	ideal	NOUN
iajs-1241	67	8	of	of	ADP
iajs-1241	67	9	r.	r.	PROPN
iajs-1241	67	10	then	then	ADV
iajs-1241	67	11	i	i	PRON
iajs-1241	67	12	is	be	AUX
iajs-1241	67	13	a	a	DET
iajs-1241	67	14	maximal	maximal	ADJ
iajs-1241	67	15	ideal	ideal	NOUN
iajs-1241	67	16	of	of	ADP
iajs-1241	67	17	r	r	NOUN
iajs-1241	67	18	if	if	SCONJ
iajs-1241	68	1	and	and	CCONJ
iajs-1241	68	2	only	only	ADV
iajs-1241	68	3	if	if	SCONJ
iajs-1241	68	4	i	i	PROPN
iajs-1241	68	5	is	be	AUX
iajs-1241	68	6	a	a	DET
iajs-1241	68	7	fuzzy	fuzzy	ADJ
iajs-1241	68	8	maximal	maximal	ADJ
iajs-1241	68	9	ideal	ideal	NOUN
iajs-1241	68	10	of	of	ADP
iajs-1241	68	11	r	r	NOUN
iajs-1241	68	12	(	(	PUNCT
iajs-1241	68	13	see	see	VERB
iajs-1241	68	14	cor	cor	PROPN
iajs-1241	68	15	.	.	PROPN
iajs-1241	68	16	3.8	3.8	NUM
iajs-1241	68	17	)	)	PUNCT
iajs-1241	68	18	.	.	PUNCT
iajs-1241	69	1	thus	thus	ADV
iajs-1241	69	2	we	we	PRON
iajs-1241	69	3	introduce	introduce	VERB
iajs-1241	69	4	the	the	DET
iajs-1241	69	5	following	following	NOUN
iajs-1241	69	6	:	:	PUNCT
iajs-1241	69	7	definition	definition	NOUN
iajs-1241	69	8	2.2	2.2	NUM
iajs-1241	69	9	:	:	PUNCT
iajs-1241	69	10	let	let	VERB
iajs-1241	69	11	a	a	PRON
iajs-1241	69	12	be	be	AUX
iajs-1241	69	13	a	a	DET
iajs-1241	69	14	fuzzy	fuzzy	ADJ
iajs-1241	69	15	ideal	ideal	NOUN
iajs-1241	69	16	of	of	ADP
iajs-1241	69	17	r	r	NOUN
iajs-1241	69	18	,	,	PUNCT
iajs-1241	69	19	a	a	PRON
iajs-1241	69	20	is	be	AUX
iajs-1241	69	21	called	call	VERB
iajs-1241	69	22	a	a	DET
iajs-1241	69	23	fuzzy	fuzzy	ADJ
iajs-1241	69	24	semimaximal	semimaximal	NOUN
iajs-1241	69	25	ideal	ideal	NOUN
iajs-1241	69	26	if	if	SCONJ
iajs-1241	69	27	a	a	PRON
iajs-1241	69	28	is	be	AUX
iajs-1241	69	29	a	a	DET
iajs-1241	69	30	finite	finite	ADJ
iajs-1241	69	31	intersection	intersection	NOUN
iajs-1241	69	32	of	of	ADP
iajs-1241	69	33	fuzzy	fuzzy	ADJ
iajs-1241	69	34	maximal	maximal	ADJ
iajs-1241	69	35	ideals	ideal	NOUN
iajs-1241	69	36	of	of	ADP
iajs-1241	69	37	r.	r.	PROPN
iajs-1241	69	38	remarks	remark	VERB
iajs-1241	69	39	2.3	2.3	NUM
iajs-1241	69	40	:	:	PUNCT
iajs-1241	70	1	1	1	X
iajs-1241	70	2	.	.	X
iajs-1241	70	3	it	it	PRON
iajs-1241	70	4	is	be	AUX
iajs-1241	70	5	clear	clear	ADJ
iajs-1241	70	6	that	that	SCONJ
iajs-1241	70	7	every	every	DET
iajs-1241	70	8	fuzzy	fuzzy	ADJ
iajs-1241	70	9	maximal	maximal	ADJ
iajs-1241	70	10	ideal	ideal	NOUN
iajs-1241	70	11	is	be	AUX
iajs-1241	70	12	fuzzy	fuzzy	ADJ
iajs-1241	70	13	semimaximal	semimaximal	ADJ
iajs-1241	70	14	ideal	ideal	NOUN
iajs-1241	70	15	.	.	PUNCT
iajs-1241	71	1	however	however	ADV
iajs-1241	71	2	the	the	DET
iajs-1241	71	3	converse	converse	NOUN
iajs-1241	71	4	is	be	AUX
iajs-1241	71	5	not	not	PART
iajs-1241	71	6	true	true	ADJ
iajs-1241	71	7	as	as	SCONJ
iajs-1241	71	8	the	the	DET
iajs-1241	71	9	following	follow	VERB
iajs-1241	71	10	example	example	NOUN
iajs-1241	71	11	shows	show	VERB
iajs-1241	71	12	:	:	PUNCT
iajs-1241	71	13	let	let	VERB
iajs-1241	71	14	a	a	PRON
iajs-1241	71	15	:	:	PUNCT
iajs-1241	71	16	z	z	NOUN
iajs-1241	71	17			X
iajs-1241	72	1	[	[	X
iajs-1241	72	2	0,1	0,1	NUM
iajs-1241	72	3	]	]	PUNCT
iajs-1241	72	4	defined	define	VERB
iajs-1241	72	5	by	by	ADP
iajs-1241	72	6	:	:	PUNCT
iajs-1241	72	7	1	1	NUM
iajs-1241	72	8	6	6	NUM
iajs-1241	72	9	,	,	PUNCT
iajs-1241	72	10	(	(	PUNCT
iajs-1241	72	11	)	)	PUNCT
iajs-1241	72	12	1	1	NUM
iajs-1241	72	13	otherwise	otherwise	ADV
iajs-1241	72	14	2	2	NUM
iajs-1241	72	15			NOUN
iajs-1241	72	16			NOUN
iajs-1241	72	17			NUM
iajs-1241	72	18			PROPN
iajs-1241	72	19			PROPN
iajs-1241	72	20			NUM
iajs-1241	72	21			NOUN
iajs-1241	72	22	x	x	X
iajs-1241	73	1	x	x	X
iajs-1241	73	2	it	it	PRON
iajs-1241	73	3	is	be	AUX
iajs-1241	73	4	clear	clear	ADJ
iajs-1241	73	5	that	that	SCONJ
iajs-1241	73	6	a	a	PRON
iajs-1241	73	7	is	be	AUX
iajs-1241	73	8	fuzzy	fuzzy	ADJ
iajs-1241	73	9	ideal	ideal	NOUN
iajs-1241	73	10	of	of	ADP
iajs-1241	73	11	z	z	PROPN
iajs-1241	73	12	and	and	CCONJ
iajs-1241	73	13	a	a	PRON
iajs-1241	73	14	is	be	AUX
iajs-1241	73	15	not	not	PART
iajs-1241	73	16	fuzzy	fuzzy	ADJ
iajs-1241	73	17	maximal	maximal	ADJ
iajs-1241	73	18	ideal	ideal	NOUN
iajs-1241	73	19	since	since	SCONJ
iajs-1241	73	20	a	a	NOUN
iajs-1241	74	1	=	=	PROPN
iajs-1241	74	2	6z	6z	NOUN
iajs-1241	74	3	is	be	AUX
iajs-1241	74	4	not	not	PART
iajs-1241	74	5	maximal	maximal	ADJ
iajs-1241	74	6	ideal	ideal	NOUN
iajs-1241	74	7	(	(	PUNCT
iajs-1241	74	8	see	see	VERB
iajs-1241	74	9	lemma	lemma	PROPN
iajs-1241	74	10	2.1(3	2.1(3	NUM
iajs-1241	74	11	)	)	PUNCT
iajs-1241	74	12	)	)	PUNCT
iajs-1241	74	13	.	.	PUNCT
iajs-1241	75	1	however	however	ADV
iajs-1241	75	2	a	a	DET
iajs-1241	75	3	=	=	PUNCT
iajs-1241	75	4	a1	a1	NOUN
iajs-1241	75	5			NOUN
iajs-1241	75	6	a2	a2	NOUN
iajs-1241	75	7	,	,	PUNCT
iajs-1241	75	8	where	where	SCONJ
iajs-1241	75	9	a1	a1	NOUN
iajs-1241	75	10	and	and	CCONJ
iajs-1241	75	11	a2	a2	PROPN
iajs-1241	75	12	are	be	AUX
iajs-1241	75	13	fuzzy	fuzzy	ADJ
iajs-1241	75	14	ideals	ideal	NOUN
iajs-1241	75	15	defined	define	VERB
iajs-1241	75	16	by	by	ADP
iajs-1241	75	17	:	:	PUNCT
iajs-1241	75	18	a1	a1	NOUN
iajs-1241	75	19	:	:	PUNCT
iajs-1241	75	20	z	z	NOUN
iajs-1241	75	21			X
iajs-1241	76	1	[	[	X
iajs-1241	76	2	0,1	0,1	NUM
iajs-1241	76	3	]	]	X
iajs-1241	76	4	a2	a2	PROPN
iajs-1241	76	5	:	:	PUNCT
iajs-1241	76	6	z	z	PROPN
iajs-1241	76	7			X
iajs-1241	77	1	[	[	X
iajs-1241	77	2	0,1	0,1	NUM
iajs-1241	77	3	]	]	SYM
iajs-1241	77	4	1	1	NUM
iajs-1241	77	5	1	1	NUM
iajs-1241	77	6	2	2	NUM
iajs-1241	77	7	,	,	PUNCT
iajs-1241	77	8	(	(	PUNCT
iajs-1241	77	9	)	)	PUNCT
iajs-1241	77	10	0	0	PUNCT
iajs-1241	78	1	otherwise	otherwise	ADV
iajs-1241	78	2			NOUN
iajs-1241	78	3			PUNCT
iajs-1241	79	1			PROPN
iajs-1241	79	2			NUM
iajs-1241	80	1			NUM
iajs-1241	80	2			NOUN
iajs-1241	80	3	x	x	SYM
iajs-1241	80	4	x	x	SYM
iajs-1241	80	5	2	2	NUM
iajs-1241	80	6	1	1	NUM
iajs-1241	80	7	3	3	NUM
iajs-1241	80	8	,	,	PUNCT
iajs-1241	80	9	(	(	PUNCT
iajs-1241	80	10	)	)	PUNCT
iajs-1241	80	11	0	0	PUNCT
iajs-1241	81	1	otherwise	otherwise	ADV
iajs-1241	81	2			NOUN
iajs-1241	81	3			PUNCT
iajs-1241	82	1			PROPN
iajs-1241	82	2			NUM
iajs-1241	83	1			NUM
iajs-1241	83	2			NOUN
iajs-1241	83	3	x	x	SYM
iajs-1241	83	4	x	x	SYM
iajs-1241	83	5	a1	a1	NOUN
iajs-1241	83	6	and	and	CCONJ
iajs-1241	83	7	a2	a2	PROPN
iajs-1241	83	8	are	be	AUX
iajs-1241	83	9	fuzzy	fuzzy	ADJ
iajs-1241	83	10	maximal	maximal	ADJ
iajs-1241	83	11	ideals	ideal	NOUN
iajs-1241	83	12	of	of	ADP
iajs-1241	83	13	z	z	NOUN
iajs-1241	83	14	since	since	SCONJ
iajs-1241	83	15	(	(	PUNCT
iajs-1241	83	16	a1)	a1)	PUNCT
iajs-1241	83	17	=	=	SYM
iajs-1241	83	18	2z	2z	NOUN
iajs-1241	83	19	and	and	CCONJ
iajs-1241	83	20	(	(	PUNCT
iajs-1241	83	21	a1)	a1)	ADP
iajs-1241	83	22	=	=	SYM
iajs-1241	83	23	3z	3z	NUM
iajs-1241	83	24	are	be	AUX
iajs-1241	83	25	maximal	maximal	ADJ
iajs-1241	83	26	ideals	ideal	NOUN
iajs-1241	83	27	(	(	PUNCT
iajs-1241	83	28	see	see	VERB
iajs-1241	83	29	lemma	lemma	PROPN
iajs-1241	83	30	2.1(5	2.1(5	NUM
iajs-1241	83	31	)	)	PUNCT
iajs-1241	83	32	)	)	PUNCT
iajs-1241	83	33	.	.	PUNCT
iajs-1241	84	1	2	2	X
iajs-1241	84	2	.	.	X
iajs-1241	84	3	if	if	SCONJ
iajs-1241	84	4	a	a	PRON
iajs-1241	84	5	is	be	AUX
iajs-1241	84	6	a	a	DET
iajs-1241	84	7	fuzzy	fuzzy	ADJ
iajs-1241	84	8	semimaximal	semimaximal	ADJ
iajs-1241	84	9	ideal	ideal	NOUN
iajs-1241	84	10	of	of	ADP
iajs-1241	84	11	r	r	NOUN
iajs-1241	84	12	,	,	PUNCT
iajs-1241	84	13	then	then	ADV
iajs-1241	84	14	a(0	a(0	PROPN
iajs-1241	84	15	)	)	PUNCT
iajs-1241	84	16	=	=	SYM
iajs-1241	85	1	1	1	X
iajs-1241	85	2	.	.	PUNCT
iajs-1241	85	3	proof	proof	NOUN
iajs-1241	85	4	.	.	PUNCT
iajs-1241	86	1	since	since	SCONJ
iajs-1241	86	2	a	a	PRON
iajs-1241	86	3	is	be	AUX
iajs-1241	86	4	a	a	DET
iajs-1241	86	5	fuzzy	fuzzy	ADJ
iajs-1241	86	6	semimaximal	semimaximal	ADJ
iajs-1241	86	7	ideal	ideal	NOUN
iajs-1241	86	8	of	of	ADP
iajs-1241	86	9	r	r	NOUN
iajs-1241	86	10	,	,	PUNCT
iajs-1241	86	11	a	a	DET
iajs-1241	86	12	=	=	PUNCT
iajs-1241	86	13	a1	a1	NOUN
iajs-1241	86	14			PUNCT
iajs-1241	86	15	a2	a2	PROPN
iajs-1241	86	16			PUNCT
iajs-1241	86	17			X
iajs-1241	86	18	an	an	PRON
iajs-1241	86	19	,	,	PUNCT
iajs-1241	86	20	where	where	SCONJ
iajs-1241	86	21	ai	ai	NOUN
iajs-1241	86	22	is	be	AUX
iajs-1241	86	23	a	a	DET
iajs-1241	86	24	fuzzy	fuzzy	ADJ
iajs-1241	86	25	maximal	maximal	ADJ
iajs-1241	86	26	ideal	ideal	NOUN
iajs-1241	86	27	of	of	ADP
iajs-1241	86	28	r	r	NOUN
iajs-1241	86	29	,	,	PUNCT
iajs-1241	86	30	for	for	ADP
iajs-1241	86	31	all	all	DET
iajs-1241	86	32	i	i	PRON
iajs-1241	86	33	=	=	NOUN
iajs-1241	86	34	1	1	NUM
iajs-1241	86	35	,	,	PUNCT
iajs-1241	86	36	2	2	NUM
iajs-1241	86	37	,	,	PUNCT
iajs-1241	86	38			PROPN
iajs-1241	86	39	,	,	PUNCT
iajs-1241	86	40	n.	n.	NOUN
iajs-1241	86	41	since	since	SCONJ
iajs-1241	86	42	ai	ai	PROPN
iajs-1241	86	43	(	(	PUNCT
iajs-1241	86	44	0	0	NUM
iajs-1241	86	45	)	)	PUNCT
iajs-1241	86	46	=	=	SYM
iajs-1241	86	47	1	1	NUM
iajs-1241	86	48	by	by	ADP
iajs-1241	86	49	lemma	lemma	PROPN
iajs-1241	86	50	2.1(1	2.1(1	NUM
iajs-1241	86	51	)	)	PUNCT
iajs-1241	86	52	,	,	PUNCT
iajs-1241	86	53	then	then	ADV
iajs-1241	86	54	a(0	a(0	PROPN
iajs-1241	86	55	)	)	PUNCT
iajs-1241	86	56	=	=	SYM
iajs-1241	86	57	1	1	NUM
iajs-1241	86	58	(	(	PUNCT
iajs-1241	86	59	0	0	NUM
iajs-1241	86	60	)	)	PUNCT
iajs-1241	87	1			NOUN
iajs-1241	88	1			PROPN
iajs-1241	88	2	n	n	CCONJ
iajs-1241	89	1	i	i	PRON
iajs-1241	89	2	i	i	NOUN
iajs-1241	89	3	=	=	SYM
iajs-1241	89	4	min	min	PROPN
iajs-1241	89	5	{	{	PUNCT
iajs-1241	89	6	ai	ai	NOUN
iajs-1241	89	7	(	(	PUNCT
iajs-1241	89	8	0	0	NUM
iajs-1241	89	9	)	)	PUNCT
iajs-1241	89	10	,	,	PUNCT
iajs-1241	89	11	i	i	PRON
iajs-1241	89	12	=	=	NOUN
iajs-1241	89	13	1	1	NUM
iajs-1241	89	14	,	,	PUNCT
iajs-1241	89	15	2	2	NUM
iajs-1241	89	16	,	,	PUNCT
iajs-1241	89	17			PROPN
iajs-1241	89	18	,	,	PUNCT
iajs-1241	89	19	n	n	CCONJ
iajs-1241	89	20	}	}	PUNCT
iajs-1241	89	21	=	=	SYM
iajs-1241	89	22	1	1	NUM
iajs-1241	89	23	3	3	NUM
iajs-1241	89	24	.	.	PUNCT
iajs-1241	90	1	if	if	SCONJ
iajs-1241	90	2	a	a	PRON
iajs-1241	90	3	and	and	CCONJ
iajs-1241	90	4	b	b	NOUN
iajs-1241	90	5	are	be	AUX
iajs-1241	90	6	fuzzy	fuzzy	ADJ
iajs-1241	90	7	semimaximal	semimaximal	ADJ
iajs-1241	90	8	ideals	ideal	NOUN
iajs-1241	90	9	of	of	ADP
iajs-1241	90	10	r	r	NOUN
iajs-1241	90	11	,	,	PUNCT
iajs-1241	90	12	then	then	ADV
iajs-1241	90	13	a	a	DET
iajs-1241	90	14			NOUN
iajs-1241	90	15	b	b	NOUN
iajs-1241	90	16	is	be	AUX
iajs-1241	90	17	a	a	DET
iajs-1241	90	18	fuzzy	fuzzy	ADJ
iajs-1241	90	19	semimaximal	semimaximal	ADJ
iajs-1241	90	20	ideal	ideal	NOUN
iajs-1241	90	21	of	of	ADP
iajs-1241	90	22	r.	r.	PROPN
iajs-1241	90	23	ibn	ibn	PROPN
iajs-1241	90	24	alhaitham	alhaitham	PROPN
iajs-1241	90	25	j.	j.	PROPN
iajs-1241	90	26	for	for	ADP
iajs-1241	90	27	pure	pure	ADJ
iajs-1241	90	28	&	&	CCONJ
iajs-1241	90	29	appl	appl	PROPN
iajs-1241	90	30	.	.	PUNCT
iajs-1241	91	1	sci	sci	PROPN
iajs-1241	91	2	vol.22	vol.22	PROPN
iajs-1241	91	3	(	(	PUNCT
iajs-1241	91	4	2	2	NUM
iajs-1241	91	5	)	)	PUNCT
iajs-1241	91	6	2009	2009	NUM
iajs-1241	91	7	proof	proof	NOUN
iajs-1241	91	8	.	.	PUNCT
iajs-1241	92	1	since	since	SCONJ
iajs-1241	92	2	a	a	PRON
iajs-1241	92	3	and	and	CCONJ
iajs-1241	92	4	b	b	NOUN
iajs-1241	92	5	are	be	AUX
iajs-1241	92	6	fuzzy	fuzzy	ADJ
iajs-1241	92	7	semimaximal	semimaximal	ADJ
iajs-1241	92	8	ideal	ideal	NOUN
iajs-1241	92	9	of	of	ADP
iajs-1241	92	10	r	r	NOUN
iajs-1241	92	11	,	,	PUNCT
iajs-1241	92	12	then	then	ADV
iajs-1241	92	13	a	a	PRON
iajs-1241	92	14	=	=	SYM
iajs-1241	92	15	1	1	NUM
iajs-1241	92	16			PROPN
iajs-1241	92	17	n	n	NUM
iajs-1241	92	18	i	i	PRON
iajs-1241	92	19	i	i	INTJ
iajs-1241	92	20	,	,	PUNCT
iajs-1241	93	1	b	b	X
iajs-1241	93	2	=	=	SYM
iajs-1241	93	3	1	1	NUM
iajs-1241	93	4			NOUN
iajs-1241	93	5	m	m	VERB
iajs-1241	94	1	i	i	PRON
iajs-1241	94	2	i	i	PRON
iajs-1241	94	3	,	,	PUNCT
iajs-1241	94	4	where	where	SCONJ
iajs-1241	94	5	ai	ai	NOUN
iajs-1241	94	6	is	be	AUX
iajs-1241	94	7	a	a	DET
iajs-1241	94	8	fuzzy	fuzzy	ADJ
iajs-1241	94	9	maximal	maximal	ADJ
iajs-1241	94	10	ideal	ideal	NOUN
iajs-1241	94	11	,	,	PUNCT
iajs-1241	94	12	for	for	ADP
iajs-1241	94	13	all	all	DET
iajs-1241	94	14	i	i	PRON
iajs-1241	94	15	=	=	NOUN
iajs-1241	94	16	1	1	NUM
iajs-1241	94	17	,	,	PUNCT
iajs-1241	94	18	2	2	NUM
iajs-1241	94	19	,	,	PUNCT
iajs-1241	94	20			PROPN
iajs-1241	94	21	,	,	PUNCT
iajs-1241	94	22	n	n	PROPN
iajs-1241	94	23	and	and	CCONJ
iajs-1241	94	24	bi	bi	NOUN
iajs-1241	94	25	is	be	AUX
iajs-1241	94	26	a	a	DET
iajs-1241	94	27	fuzzy	fuzzy	ADJ
iajs-1241	94	28	maximal	maximal	ADJ
iajs-1241	94	29	ideal	ideal	NOUN
iajs-1241	94	30	,	,	PUNCT
iajs-1241	94	31	for	for	ADP
iajs-1241	94	32	all	all	DET
iajs-1241	94	33	i	i	PRON
iajs-1241	94	34	=	=	NOUN
iajs-1241	94	35	1	1	NUM
iajs-1241	94	36	,	,	PUNCT
iajs-1241	94	37	2	2	NUM
iajs-1241	94	38	,	,	PUNCT
iajs-1241	94	39			PROPN
iajs-1241	94	40	,	,	PUNCT
iajs-1241	94	41	m.	m.	NOUN
iajs-1241	94	42	thus	thus	ADV
iajs-1241	94	43	a	a	DET
iajs-1241	94	44			NOUN
iajs-1241	94	45	b	b	NOUN
iajs-1241	94	46	=	=	SYM
iajs-1241	94	47	a1	a1	NOUN
iajs-1241	94	48			PUNCT
iajs-1241	94	49	a2	a2	PROPN
iajs-1241	94	50			PUNCT
iajs-1241	94	51			ADP
iajs-1241	94	52	an	an	DET
iajs-1241	94	53			NOUN
iajs-1241	94	54	b1	b1	NOUN
iajs-1241	94	55			VERB
iajs-1241	94	56	b2	b2	NOUN
iajs-1241	94	57			PUNCT
iajs-1241	94	58			NUM
iajs-1241	94	59	bm	bm	PROPN
iajs-1241	94	60	;	;	PUNCT
iajs-1241	94	61	that	that	PRON
iajs-1241	94	62	is	be	AUX
iajs-1241	94	63	a	a	DET
iajs-1241	94	64			NOUN
iajs-1241	94	65	b	b	NOUN
iajs-1241	94	66	is	be	AUX
iajs-1241	94	67	a	a	DET
iajs-1241	94	68	finite	finite	ADJ
iajs-1241	94	69	intersection	intersection	NOUN
iajs-1241	94	70	of	of	ADP
iajs-1241	94	71	fuzzy	fuzzy	ADJ
iajs-1241	94	72	maximal	maximal	ADJ
iajs-1241	94	73	ideals	ideal	NOUN
iajs-1241	94	74	of	of	ADP
iajs-1241	94	75	r.	r.	PROPN
iajs-1241	94	76	4	4	NUM
iajs-1241	94	77	.	.	PUNCT
iajs-1241	95	1	if	if	SCONJ
iajs-1241	95	2	{	{	PUNCT
iajs-1241	95	3	ai	ai	VERB
iajs-1241	95	4	,	,	PUNCT
iajs-1241	95	5	i	i	PRON
iajs-1241	95	6	=	=	NOUN
iajs-1241	95	7	1	1	NUM
iajs-1241	95	8	,	,	PUNCT
iajs-1241	95	9	2	2	NUM
iajs-1241	95	10	,	,	PUNCT
iajs-1241	95	11			PROPN
iajs-1241	95	12	,	,	PUNCT
iajs-1241	95	13	n	n	CCONJ
iajs-1241	95	14	}	}	PUNCT
iajs-1241	95	15	be	be	AUX
iajs-1241	95	16	a	a	DET
iajs-1241	95	17	family	family	NOUN
iajs-1241	95	18	of	of	ADP
iajs-1241	95	19	fuzzy	fuzzy	ADJ
iajs-1241	95	20	semimaximal	semimaximal	ADJ
iajs-1241	95	21	ideals	ideal	NOUN
iajs-1241	95	22	of	of	ADP
iajs-1241	95	23	r	r	NOUN
iajs-1241	95	24	,	,	PUNCT
iajs-1241	95	25	then	then	ADV
iajs-1241	95	26	1	1	NUM
iajs-1241	95	27			PROPN
iajs-1241	96	1	n	n	PRON
iajs-1241	96	2	i	i	PRON
iajs-1241	97	1	i	i	PRON
iajs-1241	97	2	is	be	AUX
iajs-1241	97	3	a	a	DET
iajs-1241	97	4	fuzzy	fuzzy	ADJ
iajs-1241	97	5	semimaximal	semimaximal	ADJ
iajs-1241	97	6	ideal	ideal	NOUN
iajs-1241	97	7	of	of	ADP
iajs-1241	97	8	r.	r.	PROPN
iajs-1241	97	9	proof	proof	NOUN
iajs-1241	97	10	.	.	PUNCT
iajs-1241	98	1	it	it	PRON
iajs-1241	98	2	is	be	AUX
iajs-1241	98	3	easy	easy	ADJ
iajs-1241	98	4	,	,	PUNCT
iajs-1241	98	5	so	so	CCONJ
iajs-1241	98	6	it	it	PRON
iajs-1241	98	7	is	be	AUX
iajs-1241	98	8	omitted	omit	VERB
iajs-1241	98	9	.	.	PUNCT
iajs-1241	99	1	compare	compare	VERB
iajs-1241	99	2	the	the	DET
iajs-1241	99	3	following	following	ADJ
iajs-1241	99	4	result	result	NOUN
iajs-1241	99	5	with	with	ADP
iajs-1241	99	6	lemma	lemma	PROPN
iajs-1241	99	7	2.1(3	2.1(3	NUM
iajs-1241	99	8	)	)	PUNCT
iajs-1241	99	9	proposition	proposition	NOUN
iajs-1241	99	10	2.4	2.4	NUM
iajs-1241	99	11	:	:	PUNCT
iajs-1241	99	12	if	if	SCONJ
iajs-1241	99	13	a	a	PRON
iajs-1241	99	14	is	be	AUX
iajs-1241	99	15	a	a	DET
iajs-1241	99	16	fuzzy	fuzzy	ADJ
iajs-1241	99	17	semimaximal	semimaximal	ADJ
iajs-1241	99	18	ideal	ideal	NOUN
iajs-1241	99	19	of	of	ADP
iajs-1241	99	20	r	r	NOUN
iajs-1241	99	21	,	,	PUNCT
iajs-1241	99	22	then	then	ADV
iajs-1241	99	23	a	a	PROPN
iajs-1241	99	24	is	be	AUX
iajs-1241	99	25	semimaximal	semimaximal	ADJ
iajs-1241	99	26	ideal	ideal	NOUN
iajs-1241	99	27	of	of	ADP
iajs-1241	99	28	r.	r.	PROPN
iajs-1241	99	29	proof	proof	NOUN
iajs-1241	99	30	.	.	PUNCT
iajs-1241	100	1	since	since	SCONJ
iajs-1241	100	2	a	a	PRON
iajs-1241	100	3	is	be	AUX
iajs-1241	100	4	a	a	DET
iajs-1241	100	5	fuzzy	fuzzy	ADJ
iajs-1241	100	6	semimaximal	semimaximal	NOUN
iajs-1241	100	7	ideal	ideal	NOUN
iajs-1241	100	8	,	,	PUNCT
iajs-1241	100	9	so	so	ADV
iajs-1241	101	1	a	a	PRON
iajs-1241	101	2	=	=	SYM
iajs-1241	101	3	1	1	NUM
iajs-1241	101	4			PROPN
iajs-1241	101	5	n	n	NUM
iajs-1241	102	1	i	i	PRON
iajs-1241	102	2	i	i	PRON
iajs-1241	102	3	,	,	PUNCT
iajs-1241	102	4	where	where	SCONJ
iajs-1241	102	5	ai	ai	NOUN
iajs-1241	102	6	is	be	AUX
iajs-1241	102	7	a	a	DET
iajs-1241	102	8	fuzzy	fuzzy	ADJ
iajs-1241	102	9	maximal	maximal	ADJ
iajs-1241	102	10	ideal	ideal	NOUN
iajs-1241	102	11	for	for	ADP
iajs-1241	102	12	all	all	PRON
iajs-1241	102	13	i	i	PRON
iajs-1241	102	14	=	=	NOUN
iajs-1241	102	15	1	1	NUM
iajs-1241	102	16	,	,	PUNCT
iajs-1241	102	17	2	2	NUM
iajs-1241	102	18	,	,	PUNCT
iajs-1241	102	19			PROPN
iajs-1241	102	20	,	,	PUNCT
iajs-1241	102	21	n.	n.	NOUN
iajs-1241	102	22	since	since	SCONJ
iajs-1241	102	23	ai	ai	PROPN
iajs-1241	102	24	(	(	PUNCT
iajs-1241	102	25	0	0	NUM
iajs-1241	102	26	)	)	PUNCT
iajs-1241	102	27	=	=	SYM
iajs-1241	102	28	1	1	NUM
iajs-1241	102	29	(	(	PUNCT
iajs-1241	102	30	by	by	ADP
iajs-1241	102	31	lemma2.1(1	lemma2.1(1	PROPN
iajs-1241	102	32	)	)	PUNCT
iajs-1241	102	33	)	)	PUNCT
iajs-1241	102	34	,	,	PUNCT
iajs-1241	102	35	so	so	SCONJ
iajs-1241	102	36	that	that	SCONJ
iajs-1241	102	37	a	a	NOUN
iajs-1241	102	38	=	=	SYM
iajs-1241	102	39	1	1	PROPN
iajs-1241	102	40			PROPN
iajs-1241	102	41			NOUN
iajs-1241	102	42			PROPN
iajs-1241	102	43			PROPN
iajs-1241	102	44			PROPN
iajs-1241	103	1			ADJ
iajs-1241	103	2			NOUN
iajs-1241	103	3	n	n	VERB
iajs-1241	104	1	i	i	PRON
iajs-1241	104	2	i	i	PRON
iajs-1241	104	3	=	=	VERB
iajs-1241	104	4			NOUN
iajs-1241	104	5			PROPN
iajs-1241	104	6	1	1	NUM
iajs-1241	104	7			NUM
iajs-1241	104	8			NOUN
iajs-1241	105	1			PROPN
iajs-1241	106	1	n	n	PRON
iajs-1241	107	1	i	i	PRON
iajs-1241	108	1	i	i	INTJ
iajs-1241	108	2	(	(	PUNCT
iajs-1241	108	3	by	by	ADP
iajs-1241	108	4	prop.1.8	prop.1.8	NOUN
iajs-1241	108	5	)	)	PUNCT
iajs-1241	108	6	but	but	CCONJ
iajs-1241	108	7	(	(	PUNCT
iajs-1241	108	8	ai)	ai)	PROPN
iajs-1241	108	9	is	be	AUX
iajs-1241	108	10	maximal	maximal	ADJ
iajs-1241	108	11	ideal	ideal	ADJ
iajs-1241	108	12	,	,	PUNCT
iajs-1241	108	13			NOUN
iajs-1241	108	14	i	i	NOUN
iajs-1241	108	15	=	=	NOUN
iajs-1241	108	16	1	1	NUM
iajs-1241	108	17	,	,	PUNCT
iajs-1241	108	18	2	2	NUM
iajs-1241	108	19	,	,	PUNCT
iajs-1241	108	20			PROPN
iajs-1241	108	21	,	,	PUNCT
iajs-1241	108	22	n	n	CCONJ
iajs-1241	108	23	by	by	ADP
iajs-1241	108	24	lemma	lemma	PROPN
iajs-1241	108	25	2.1(3	2.1(3	NUM
iajs-1241	108	26	)	)	PUNCT
iajs-1241	108	27	.	.	PUNCT
iajs-1241	109	1	hence	hence	ADV
iajs-1241	109	2	a	a	VERB
iajs-1241	109	3	=	=	PUNCT
iajs-1241	110	1			NOUN
iajs-1241	110	2			PROPN
iajs-1241	110	3	1	1	NUM
iajs-1241	110	4			NUM
iajs-1241	110	5			NOUN
iajs-1241	110	6			PROPN
iajs-1241	110	7	n	n	NUM
iajs-1241	110	8	i	i	PRON
iajs-1241	110	9	i	i	PRON
iajs-1241	110	10	.	.	PUNCT
iajs-1241	111	1	thus	thus	ADV
iajs-1241	111	2	a	a	PROPN
iajs-1241	111	3	is	be	AUX
iajs-1241	111	4	a	a	DET
iajs-1241	111	5	maximal	maximal	ADJ
iajs-1241	111	6	ideal	ideal	NOUN
iajs-1241	111	7	.	.	PUNCT
iajs-1241	112	1	the	the	DET
iajs-1241	112	2	converse	converse	NOUN
iajs-1241	112	3	of	of	ADP
iajs-1241	112	4	this	this	DET
iajs-1241	112	5	proposition	proposition	NOUN
iajs-1241	112	6	is	be	AUX
iajs-1241	112	7	not	not	PART
iajs-1241	112	8	true	true	ADJ
iajs-1241	112	9	in	in	ADP
iajs-1241	112	10	general	general	ADJ
iajs-1241	112	11	.	.	PUNCT
iajs-1241	113	1	however	however	ADV
iajs-1241	113	2	an	an	DET
iajs-1241	113	3	example	example	NOUN
iajs-1241	113	4	which	which	PRON
iajs-1241	113	5	will	will	AUX
iajs-1241	113	6	explain	explain	VERB
iajs-1241	113	7	this	this	PRON
iajs-1241	113	8	depend	depend	VERB
iajs-1241	113	9	on	on	ADP
iajs-1241	113	10	theorem	theorem	ADJ
iajs-1241	113	11	2.10	2.10	NUM
iajs-1241	113	12	.	.	PUNCT
iajs-1241	114	1	so	so	ADV
iajs-1241	114	2	we	we	PRON
iajs-1241	114	3	shall	shall	AUX
iajs-1241	114	4	give	give	VERB
iajs-1241	114	5	it	it	PRON
iajs-1241	114	6	later	later	ADV
iajs-1241	114	7	(	(	PUNCT
iajs-1241	114	8	see	see	VERB
iajs-1241	114	9	remark	remark	NOUN
iajs-1241	114	10	2.11	2.11	NUM
iajs-1241	114	11	)	)	PUNCT
iajs-1241	114	12	.	.	PUNCT
iajs-1241	115	1	before	before	ADP
iajs-1241	115	2	giving	give	VERB
iajs-1241	115	3	our	our	PRON
iajs-1241	115	4	next	next	ADJ
iajs-1241	115	5	result	result	NOUN
iajs-1241	115	6	,	,	PUNCT
iajs-1241	115	7	we	we	PRON
iajs-1241	115	8	need	need	VERB
iajs-1241	115	9	to	to	PART
iajs-1241	115	10	recall	recall	VERB
iajs-1241	115	11	the	the	DET
iajs-1241	115	12	following	following	NOUN
iajs-1241	115	13	:	:	PUNCT
iajs-1241	115	14	definition	definition	NOUN
iajs-1241	115	15	2.5	2.5	NUM
iajs-1241	115	16	:	:	PUNCT
iajs-1241	115	17	(	(	PUNCT
iajs-1241	115	18	1	1	X
iajs-1241	115	19	)	)	PUNCT
iajs-1241	115	20	let	let	VERB
iajs-1241	115	21	a	a	PRON
iajs-1241	115	22	be	be	AUX
iajs-1241	115	23	a	a	DET
iajs-1241	115	24	fuzzy	fuzzy	ADJ
iajs-1241	115	25	ideal	ideal	NOUN
iajs-1241	115	26	of	of	ADP
iajs-1241	115	27	r	r	NOUN
iajs-1241	115	28	,	,	PUNCT
iajs-1241	115	29	then	then	ADV
iajs-1241	115	30	a	a	PRON
iajs-1241	115	31	is	be	AUX
iajs-1241	115	32	called	call	VERB
iajs-1241	115	33	fuzzy	fuzzy	ADJ
iajs-1241	115	34	prime	prime	NOUN
iajs-1241	115	35	if	if	SCONJ
iajs-1241	115	36	either	either	CCONJ
iajs-1241	115	37	a	a	DET
iajs-1241	115	38	=	=	X
iajs-1241	115	39	r	r	PROPN
iajs-1241	115	40	or	or	CCONJ
iajs-1241	115	41	1	1	NUM
iajs-1241	115	42	.	.	PUNCT
iajs-1241	116	1	a	a	PRON
iajs-1241	116	2	is	be	AUX
iajs-1241	116	3	not	not	PART
iajs-1241	116	4	constant	constant	ADJ
iajs-1241	116	5	and	and	CCONJ
iajs-1241	116	6	2	2	NUM
iajs-1241	116	7	.	.	X
iajs-1241	117	1	for	for	ADP
iajs-1241	117	2	any	any	DET
iajs-1241	117	3	fuzzy	fuzzy	ADJ
iajs-1241	117	4	ideals	ideal	NOUN
iajs-1241	117	5	b	b	NOUN
iajs-1241	117	6	and	and	CCONJ
iajs-1241	117	7	c	c	PROPN
iajs-1241	117	8	of	of	ADP
iajs-1241	117	9	r	r	NOUN
iajs-1241	117	10	,	,	PUNCT
iajs-1241	117	11	if	if	SCONJ
iajs-1241	117	12	bc	bc	ADJ
iajs-1241	117	13			PROPN
iajs-1241	117	14	a	a	X
iajs-1241	117	15	,	,	PUNCT
iajs-1241	117	16	then	then	ADV
iajs-1241	117	17	either	either	CCONJ
iajs-1241	117	18	b	b	VERB
iajs-1241	117	19	a	a	PRON
iajs-1241	117	20	or	or	CCONJ
iajs-1241	117	21	c	c	PROPN
iajs-1241	117	22			PROPN
iajs-1241	117	23	a.	a.	NOUN
iajs-1241	117	24	definition	definition	NOUN
iajs-1241	117	25	2.6	2.6	NUM
iajs-1241	117	26	:	:	PUNCT
iajs-1241	117	27	(	(	PUNCT
iajs-1241	117	28	1	1	X
iajs-1241	117	29	)	)	PUNCT
iajs-1241	117	30	let	let	VERB
iajs-1241	117	31	a	a	PRON
iajs-1241	117	32	be	be	AUX
iajs-1241	117	33	a	a	DET
iajs-1241	117	34	fuzzy	fuzzy	ADJ
iajs-1241	117	35	ideal	ideal	NOUN
iajs-1241	117	36	of	of	ADP
iajs-1241	117	37	r.	r.	PROPN
iajs-1241	117	38	the	the	DET
iajs-1241	117	39	fuzzy	fuzzy	ADJ
iajs-1241	117	40	radical	radical	NOUN
iajs-1241	117	41	of	of	ADP
iajs-1241	117	42	a	a	DET
iajs-1241	117	43	denoted	denote	VERB
iajs-1241	117	44	by	by	ADP
iajs-1241	117	45			PROPN
iajs-1241	117	46	defined	define	VERB
iajs-1241	117	47	by	by	ADP
iajs-1241	117	48			PROPN
iajs-1241	117	49	=	=	PUNCT
iajs-1241	117	50			NOUN
iajs-1241	117	51	{	{	PUNCT
iajs-1241	117	52	p	p	X
iajs-1241	117	53	:p	:p	NOUN
iajs-1241	117	54			NOUN
iajs-1241	117	55	£	£	SYM
iajs-1241	117	56	(	(	PUNCT
iajs-1241	117	57	a	a	NOUN
iajs-1241	117	58	)	)	PUNCT
iajs-1241	117	59	}	}	PUNCT
iajs-1241	117	60	,	,	PUNCT
iajs-1241	117	61	where	where	SCONJ
iajs-1241	117	62	£	£	SYM
iajs-1241	117	63	(	(	PUNCT
iajs-1241	117	64	a	a	PRON
iajs-1241	117	65	)	)	PUNCT
iajs-1241	117	66	denotes	denote	VERB
iajs-1241	117	67	the	the	DET
iajs-1241	117	68	set	set	NOUN
iajs-1241	117	69	of	of	ADP
iajs-1241	117	70	all	all	DET
iajs-1241	117	71	fuzzy	fuzzy	ADJ
iajs-1241	117	72	prime	prime	ADJ
iajs-1241	117	73	ideals	ideal	NOUN
iajs-1241	117	74	of	of	ADP
iajs-1241	117	75	r	r	NOUN
iajs-1241	117	76	which	which	PRON
iajs-1241	117	77	contains	contain	VERB
iajs-1241	117	78	a.	a.	NOUN
iajs-1241	117	79	proposition	proposition	NOUN
iajs-1241	117	80	2.7	2.7	NUM
iajs-1241	117	81	:	:	PUNCT
iajs-1241	117	82	if	if	SCONJ
iajs-1241	117	83	a	a	PRON
iajs-1241	117	84	is	be	AUX
iajs-1241	117	85	a	a	DET
iajs-1241	117	86	fuzzy	fuzzy	ADJ
iajs-1241	117	87	semimaximal	semimaximal	ADJ
iajs-1241	117	88	ideal	ideal	NOUN
iajs-1241	117	89	of	of	ADP
iajs-1241	117	90	r	r	NOUN
iajs-1241	117	91	,	,	PUNCT
iajs-1241	117	92	then	then	ADV
iajs-1241	117	93			PROPN
iajs-1241	117	94	=	=	PUNCT
iajs-1241	117	95	a	a	DET
iajs-1241	117	96	proof	proof	NOUN
iajs-1241	117	97	.	.	PUNCT
iajs-1241	118	1	since	since	SCONJ
iajs-1241	118	2	a	a	PRON
iajs-1241	118	3	is	be	AUX
iajs-1241	118	4	a	a	DET
iajs-1241	118	5	fuzzy	fuzzy	ADJ
iajs-1241	118	6	semimaximal	semimaximal	NOUN
iajs-1241	118	7	ideal	ideal	NOUN
iajs-1241	118	8	,	,	PUNCT
iajs-1241	118	9	then	then	ADV
iajs-1241	118	10	a	a	DET
iajs-1241	118	11	=	=	PUNCT
iajs-1241	118	12	a1	a1	NOUN
iajs-1241	118	13			PUNCT
iajs-1241	118	14	a2	a2	PROPN
iajs-1241	118	15			PUNCT
iajs-1241	118	16			X
iajs-1241	118	17	an	an	PRON
iajs-1241	118	18	,	,	PUNCT
iajs-1241	118	19	where	where	SCONJ
iajs-1241	118	20	a1	a1	NOUN
iajs-1241	118	21	,	,	PUNCT
iajs-1241	118	22	a2	a2	PROPN
iajs-1241	118	23	,	,	PUNCT
iajs-1241	118	24			PROPN
iajs-1241	118	25	,	,	PUNCT
iajs-1241	118	26	an	an	DET
iajs-1241	118	27	are	be	AUX
iajs-1241	118	28	fuzzy	fuzzy	ADJ
iajs-1241	118	29	maximal	maximal	ADJ
iajs-1241	118	30	ideals	ideal	NOUN
iajs-1241	118	31	of	of	ADP
iajs-1241	118	32	r.	r.	PROPN
iajs-1241	118	33	but	but	CCONJ
iajs-1241	118	34	for	for	ADP
iajs-1241	118	35	each	each	DET
iajs-1241	118	36	i	i	NOUN
iajs-1241	118	37	=	=	NOUN
iajs-1241	118	38	1	1	NUM
iajs-1241	118	39	,	,	PUNCT
iajs-1241	118	40	2	2	NUM
iajs-1241	118	41	,	,	PUNCT
iajs-1241	118	42			PROPN
iajs-1241	118	43	,	,	PUNCT
iajs-1241	118	44	n	n	CCONJ
iajs-1241	118	45	ai	ai	VERB
iajs-1241	118	46	is	be	AUX
iajs-1241	118	47	a	a	DET
iajs-1241	118	48	fuzzy	fuzzy	ADJ
iajs-1241	118	49	prime	prime	ADJ
iajs-1241	118	50	ideal	ideal	NOUN
iajs-1241	118	51	,	,	PUNCT
iajs-1241	118	52	hence	hence	ADV
iajs-1241	118	53			PROPN
iajs-1241	118	54	i	i	PRON
iajs-1241	118	55	=	=	VERB
iajs-1241	118	56	ai	ai	VERB
iajs-1241	118	57	by	by	ADP
iajs-1241	118	58	(	(	PUNCT
iajs-1241	118	59	theorem	theorem	ADJ
iajs-1241	118	60	5.13,(1	5.13,(1	NOUN
iajs-1241	118	61	)	)	PUNCT
iajs-1241	118	62	)	)	PUNCT
iajs-1241	118	63	.	.	PUNCT
iajs-1241	119	1	thus	thus	ADV
iajs-1241	119	2			PROPN
iajs-1241	119	3	=	=	PUNCT
iajs-1241	120	1	1	1	NUM
iajs-1241	121	1			PROPN
iajs-1241	121	2	n	n	NUM
iajs-1241	122	1	i	i	PRON
iajs-1241	122	2	i	i	PRON
iajs-1241	123	1	and	and	CCONJ
iajs-1241	124	1	so	so	ADV
iajs-1241	124	2			PROPN
iajs-1241	124	3	=	=	PUNCT
iajs-1241	124	4	a	a	DET
iajs-1241	124	5	remark	remark	NOUN
iajs-1241	124	6	2.8	2.8	NUM
iajs-1241	124	7	:	:	PUNCT
iajs-1241	124	8	if	if	SCONJ
iajs-1241	124	9	a	a	PRON
iajs-1241	124	10	is	be	AUX
iajs-1241	124	11	a	a	DET
iajs-1241	124	12	fuzzy	fuzzy	ADJ
iajs-1241	124	13	semimaximal	semimaximal	NOUN
iajs-1241	124	14	ideal	ideal	NOUN
iajs-1241	124	15	,	,	PUNCT
iajs-1241	124	16	then	then	ADV
iajs-1241	124	17	it	it	PRON
iajs-1241	124	18	is	be	AUX
iajs-1241	124	19	not	not	PART
iajs-1241	124	20	necessary	necessary	ADJ
iajs-1241	124	21	that	that	SCONJ
iajs-1241	124	22	a	a	PRON
iajs-1241	124	23	is	be	AUX
iajs-1241	124	24	a	a	DET
iajs-1241	124	25	fuzzy	fuzzy	ADJ
iajs-1241	124	26	prime	prime	ADJ
iajs-1241	124	27	ideal	ideal	NOUN
iajs-1241	124	28	.	.	PUNCT
iajs-1241	125	1	we	we	PRON
iajs-1241	125	2	can	can	AUX
iajs-1241	125	3	give	give	VERB
iajs-1241	125	4	the	the	DET
iajs-1241	125	5	following	follow	VERB
iajs-1241	125	6	example	example	NOUN
iajs-1241	125	7	:	:	PUNCT
iajs-1241	125	8	example	example	NOUN
iajs-1241	125	9	:	:	PUNCT
iajs-1241	125	10	let	let	VERB
iajs-1241	125	11	a	a	PRON
iajs-1241	125	12	:	:	PUNCT
iajs-1241	125	13	z	z	NOUN
iajs-1241	125	14			X
iajs-1241	126	1	[	[	X
iajs-1241	126	2	0,1	0,1	NUM
iajs-1241	126	3	]	]	PUNCT
iajs-1241	126	4	defined	define	VERB
iajs-1241	126	5	by	by	ADP
iajs-1241	126	6	ibn	ibn	PROPN
iajs-1241	126	7	alhaitham	alhaitham	NOUN
iajs-1241	126	8	j.	j.	PROPN
iajs-1241	126	9	for	for	ADP
iajs-1241	126	10	pure	pure	ADJ
iajs-1241	126	11	&	&	CCONJ
iajs-1241	126	12	appl	appl	PROPN
iajs-1241	126	13	.	.	PUNCT
iajs-1241	127	1	sci	sci	PROPN
iajs-1241	127	2	vol.22	vol.22	PROPN
iajs-1241	127	3	(	(	PUNCT
iajs-1241	127	4	2	2	NUM
iajs-1241	127	5	)	)	PUNCT
iajs-1241	127	6	2009	2009	NUM
iajs-1241	127	7	1	1	NUM
iajs-1241	127	8	0	0	NUM
iajs-1241	127	9	,	,	PUNCT
iajs-1241	127	10	(	(	PUNCT
iajs-1241	127	11	)	)	PUNCT
iajs-1241	127	12	0	0	NUM
iajs-1241	127	13	otherwise	otherwise	ADV
iajs-1241	127	14			VERB
iajs-1241	127	15			PROPN
iajs-1241	127	16			NUM
iajs-1241	128	1			NUM
iajs-1241	128	2			NOUN
iajs-1241	128	3	x	x	SYM
iajs-1241	128	4	x	x	X
iajs-1241	128	5	by	by	ADP
iajs-1241	128	6	(	(	PUNCT
iajs-1241	128	7	theorem	theorem	ADJ
iajs-1241	128	8	2.4(12	2.4(12	NUM
iajs-1241	128	9	)	)	PUNCT
iajs-1241	128	10	)	)	PUNCT
iajs-1241	128	11	.	.	PUNCT
iajs-1241	129	1	a	a	PRON
iajs-1241	129	2	is	be	AUX
iajs-1241	129	3	a	a	DET
iajs-1241	129	4	fuzzy	fuzzy	ADJ
iajs-1241	129	5	prime	prime	ADJ
iajs-1241	129	6	ideal	ideal	NOUN
iajs-1241	129	7	of	of	ADP
iajs-1241	129	8	z	z	PROPN
iajs-1241	129	9	,	,	PUNCT
iajs-1241	129	10	but	but	CCONJ
iajs-1241	129	11	a	a	PROPN
iajs-1241	129	12	=	=	SYM
iajs-1241	129	13	(	(	PUNCT
iajs-1241	129	14	0	0	NUM
iajs-1241	129	15	)	)	PUNCT
iajs-1241	129	16	is	be	AUX
iajs-1241	129	17	not	not	PART
iajs-1241	129	18	a	a	DET
iajs-1241	129	19	semimaximal	semimaximal	ADJ
iajs-1241	129	20	ideal	ideal	NOUN
iajs-1241	129	21	in	in	ADP
iajs-1241	129	22	z.	z.	PROPN
iajs-1241	129	23	thus	thus	ADV
iajs-1241	129	24	a	a	PRON
iajs-1241	129	25	is	be	AUX
iajs-1241	129	26	not	not	PART
iajs-1241	129	27	fuzzy	fuzzy	ADJ
iajs-1241	129	28	semimaximal	semimaximal	ADJ
iajs-1241	129	29	ideal	ideal	NOUN
iajs-1241	129	30	(	(	PUNCT
iajs-1241	129	31	by	by	ADP
iajs-1241	129	32	prop.2.4	prop.2.4	NOUN
iajs-1241	129	33	)	)	PUNCT
iajs-1241	129	34	.	.	PUNCT
iajs-1241	130	1	compare	compare	VERB
iajs-1241	130	2	the	the	DET
iajs-1241	130	3	following	following	NOUN
iajs-1241	130	4	with	with	ADP
iajs-1241	130	5	(	(	PUNCT
iajs-1241	130	6	lemma	lemma	PROPN
iajs-1241	130	7	2.1(6	2.1(6	NUM
iajs-1241	130	8	)	)	PUNCT
iajs-1241	130	9	)	)	PUNCT
iajs-1241	130	10	.	.	PUNCT
iajs-1241	131	1	proposition	proposition	NOUN
iajs-1241	131	2	2.9	2.9	NUM
iajs-1241	131	3	:	:	PUNCT
iajs-1241	131	4	let	let	VERB
iajs-1241	131	5	i	i	PRON
iajs-1241	131	6	be	be	AUX
iajs-1241	131	7	an	an	DET
iajs-1241	131	8	ideal	ideal	NOUN
iajs-1241	131	9	of	of	ADP
iajs-1241	131	10	r	r	NOUN
iajs-1241	131	11	,	,	PUNCT
iajs-1241	131	12	then	then	ADV
iajs-1241	131	13	i	i	PRON
iajs-1241	131	14	is	be	AUX
iajs-1241	131	15	a	a	DET
iajs-1241	131	16	semimaximal	semimaximal	ADJ
iajs-1241	131	17	ideal	ideal	NOUN
iajs-1241	131	18	of	of	ADP
iajs-1241	131	19	r	r	NOUN
iajs-1241	131	20	if	if	SCONJ
iajs-1241	132	1	and	and	CCONJ
iajs-1241	132	2	only	only	ADV
iajs-1241	132	3	if	if	SCONJ
iajs-1241	132	4	i	i	PROPN
iajs-1241	132	5	is	be	AUX
iajs-1241	132	6	a	a	DET
iajs-1241	132	7	fuzzy	fuzzy	ADJ
iajs-1241	132	8	semimaximal	semimaximal	ADJ
iajs-1241	132	9	ideal	ideal	NOUN
iajs-1241	132	10	of	of	ADP
iajs-1241	132	11	r	r	NOUN
iajs-1241	132	12	,	,	PUNCT
iajs-1241	132	13	where	where	SCONJ
iajs-1241	132	14	1	1	NUM
iajs-1241	132	15	,	,	PUNCT
iajs-1241	132	16	(	(	PUNCT
iajs-1241	132	17	)	)	PUNCT
iajs-1241	132	18	0	0	PUNCT
iajs-1241	133	1	otherwise	otherwise	ADV
iajs-1241	133	2			X
iajs-1241	133	3			PROPN
iajs-1241	133	4			NUM
iajs-1241	134	1			NUM
iajs-1241	134	2			PROPN
iajs-1241	134	3	x	x	X
iajs-1241	135	1	x	x	PROPN
iajs-1241	135	2	proof	proof	NOUN
iajs-1241	135	3	.	.	PUNCT
iajs-1241	136	1	since	since	SCONJ
iajs-1241	136	2	i	i	PRON
iajs-1241	136	3	is	be	AUX
iajs-1241	136	4	a	a	DET
iajs-1241	136	5	semimaximal	semimaximal	ADJ
iajs-1241	136	6	ideal	ideal	NOUN
iajs-1241	136	7	,	,	PUNCT
iajs-1241	136	8	then	then	ADV
iajs-1241	136	9	1	1	NUM
iajs-1241	136	10			ADP
iajs-1241	136	11			NUM
iajs-1241	136	12			PROPN
iajs-1241	136	13	n	n	CCONJ
iajs-1241	137	1	i	i	PRON
iajs-1241	137	2	i	i	PRON
iajs-1241	137	3	,	,	PUNCT
iajs-1241	137	4	ii	ii	PROPN
iajs-1241	137	5	is	be	AUX
iajs-1241	137	6	a	a	DET
iajs-1241	137	7	maximal	maximal	ADJ
iajs-1241	137	8	ideal	ideal	NOUN
iajs-1241	137	9	,	,	PUNCT
iajs-1241	137	10			NOUN
iajs-1241	137	11	i	i	NOUN
iajs-1241	137	12	=	=	NOUN
iajs-1241	137	13	1	1	NUM
iajs-1241	137	14	,	,	PUNCT
iajs-1241	137	15	2	2	NUM
iajs-1241	137	16	,	,	PUNCT
iajs-1241	137	17			PROPN
iajs-1241	137	18	,	,	PUNCT
iajs-1241	137	19	n.	n.	NOUN
iajs-1241	137	20	it	it	PRON
iajs-1241	137	21	is	be	AUX
iajs-1241	137	22	clear	clear	ADJ
iajs-1241	137	23	that	that	SCONJ
iajs-1241	137	24	1	1	NUM
iajs-1241	137	25	2	2	NUM
iajs-1241	137	26			NOUN
iajs-1241	137	27			PROPN
iajs-1241	137	28			PART
iajs-1241	137	29			PROPN
iajs-1241	137	30			PUNCT
iajs-1241	137	31			X
iajs-1241	137	32	n	n	NUM
iajs-1241	137	33			X
iajs-1241	137	34			ADJ
iajs-1241	137	35			ADJ
iajs-1241	137	36			NOUN
iajs-1241	137	37	.	.	PUNCT
iajs-1241	138	1	but	but	CCONJ
iajs-1241	138	2	for	for	ADP
iajs-1241	138	3	each	each	DET
iajs-1241	138	4	i	i	NOUN
iajs-1241	138	5	=	=	NOUN
iajs-1241	138	6	1	1	NUM
iajs-1241	138	7	,	,	PUNCT
iajs-1241	138	8	2	2	NUM
iajs-1241	138	9	,	,	PUNCT
iajs-1241	138	10			PROPN
iajs-1241	138	11	,	,	PUNCT
iajs-1241	138	12	n	n	CCONJ
iajs-1241	138	13	,	,	PUNCT
iajs-1241	138	14			PROPN
iajs-1241	138	15			X
iajs-1241	138	16	i	i	ADJ
iajs-1241	138	17			X
iajs-1241	138	18	=	=	SYM
iajs-1241	138	19	ii	ii	NOUN
iajs-1241	138	20	so	so	ADV
iajs-1241	138	21	1	1	NUM
iajs-1241	138	22	2	2	NUM
iajs-1241	138	23	,	,	PUNCT
iajs-1241	138	24	,	,	PUNCT
iajs-1241	138	25			PROPN
iajs-1241	138	26			PROPN
iajs-1241	138	27	n	n	VERB
iajs-1241	138	28			ADJ
iajs-1241	138	29			ADJ
iajs-1241	138	30			X
iajs-1241	138	31	are	be	AUX
iajs-1241	138	32	fuzzy	fuzzy	ADJ
iajs-1241	138	33	maximal	maximal	ADJ
iajs-1241	138	34	ideals	ideal	NOUN
iajs-1241	138	35	by	by	ADP
iajs-1241	138	36	(	(	PUNCT
iajs-1241	138	37	lemma	lemma	PROPN
iajs-1241	138	38	2.1(6)).thus	2.1(6)).thus	NUM
iajs-1241	138	39	1	1	NUM
iajs-1241	138	40			NOUN
iajs-1241	138	41			ADP
iajs-1241	139	1			NUM
iajs-1241	139	2			PROPN
iajs-1241	140	1	n	n	INTJ
iajs-1241	140	2	i	i	PRON
iajs-1241	140	3	i	i	PROPN
iajs-1241	140	4			VERB
iajs-1241	140	5			ADJ
iajs-1241	140	6	is	be	AUX
iajs-1241	140	7	a	a	DET
iajs-1241	140	8	fuzzy	fuzzy	ADJ
iajs-1241	140	9	semimaximal	semimaximal	NOUN
iajs-1241	140	10	ideal	ideal	NOUN
iajs-1241	140	11	.	.	PUNCT
iajs-1241	141	1	conversely	conversely	ADV
iajs-1241	141	2	;	;	PUNCT
iajs-1241	141	3	if	if	SCONJ
iajs-1241	141	4	i	i	PROPN
iajs-1241	141	5	is	be	AUX
iajs-1241	141	6	a	a	DET
iajs-1241	141	7	fuzzy	fuzzy	ADJ
iajs-1241	141	8	semimaximal	semimaximal	ADJ
iajs-1241	141	9	ideal	ideal	NOUN
iajs-1241	141	10	of	of	ADP
iajs-1241	141	11	r	r	NOUN
iajs-1241	141	12	,	,	PUNCT
iajs-1241	141	13	then	then	ADV
iajs-1241	141	14	by	by	ADP
iajs-1241	141	15	(	(	PUNCT
iajs-1241	141	16	lemma	lemma	PROPN
iajs-1241	141	17	2.1(3	2.1(3	NUM
iajs-1241	141	18	)	)	PUNCT
iajs-1241	141	19	)	)	PUNCT
iajs-1241	141	20	,	,	PUNCT
iajs-1241	141	21	(	(	PUNCT
iajs-1241	141	22	i)	i)	NOUN
iajs-1241	141	23	is	be	AUX
iajs-1241	141	24	semimaximal	semimaximal	ADJ
iajs-1241	141	25	,	,	PUNCT
iajs-1241	141	26	and	and	CCONJ
iajs-1241	141	27	since	since	SCONJ
iajs-1241	141	28	(	(	PUNCT
iajs-1241	141	29	i)	i)	NOUN
iajs-1241	141	30	=	=	SYM
iajs-1241	141	31	i.	i.	NOUN
iajs-1241	142	1	so	so	SCONJ
iajs-1241	142	2	the	the	DET
iajs-1241	142	3	result	result	NOUN
iajs-1241	142	4	is	be	AUX
iajs-1241	142	5	obtained	obtain	VERB
iajs-1241	142	6	.	.	PUNCT
iajs-1241	143	1	compare	compare	VERB
iajs-1241	143	2	the	the	DET
iajs-1241	143	3	following	following	NOUN
iajs-1241	143	4	with	with	ADP
iajs-1241	143	5	(	(	PUNCT
iajs-1241	143	6	lemma.2.1(2	lemma.2.1(2	NOUN
iajs-1241	143	7	)	)	PUNCT
iajs-1241	143	8	)	)	PUNCT
iajs-1241	143	9	.	.	PUNCT
iajs-1241	144	1	theorem	theorem	VERB
iajs-1241	144	2	2.10	2.10	NUM
iajs-1241	144	3	:	:	PUNCT
iajs-1241	144	4	let	let	VERB
iajs-1241	144	5	a	a	PRON
iajs-1241	144	6	be	be	AUX
iajs-1241	144	7	a	a	DET
iajs-1241	144	8	fuzzy	fuzzy	ADJ
iajs-1241	144	9	semimaximal	semimaximal	NOUN
iajs-1241	144	10	ideal	ideal	NOUN
iajs-1241	144	11	,	,	PUNCT
iajs-1241	144	12	then	then	ADV
iajs-1241	144	13	im	im	VERB
iajs-1241	144	14	a	a	NOUN
iajs-1241	144	15	=	=	SYM
iajs-1241	144	16	2	2	X
iajs-1241	144	17	.	.	PUNCT
iajs-1241	144	18	proof	proof	NOUN
iajs-1241	144	19	.	.	PUNCT
iajs-1241	145	1	1	1	NUM
iajs-1241	145	2			NOUN
iajs-1241	145	3	i	i	PRON
iajs-1241	145	4	m	m	VERB
iajs-1241	145	5	a	a	PRON
iajs-1241	145	6	since	since	SCONJ
iajs-1241	145	7	a(0	a(0	PROPN
iajs-1241	145	8	)	)	PUNCT
iajs-1241	145	9	=	=	SYM
iajs-1241	145	10	1	1	NUM
iajs-1241	145	11	(	(	PUNCT
iajs-1241	145	12	by	by	ADP
iajs-1241	145	13	rem.2.3(2	rem.2.3(2	NOUN
iajs-1241	145	14	)	)	PUNCT
iajs-1241	145	15	)	)	PUNCT
iajs-1241	145	16	.	.	PUNCT
iajs-1241	146	1	we	we	PRON
iajs-1241	146	2	claim	claim	VERB
iajs-1241	146	3	that	that	SCONJ
iajs-1241	146	4	for	for	ADP
iajs-1241	146	5	any	any	DET
iajs-1241	146	6	0	0	NUM
iajs-1241	146	7			NOUN
iajs-1241	146	8	t	t	NOUN
iajs-1241	146	9	<	<	X
iajs-1241	146	10	1	1	NUM
iajs-1241	146	11	,	,	PUNCT
iajs-1241	146	12	at	at	SCONJ
iajs-1241	146	13	=	=	SYM
iajs-1241	146	14	r.	r.	NOUN
iajs-1241	146	15	since	since	SCONJ
iajs-1241	146	16	a	a	PRON
iajs-1241	146	17	is	be	AUX
iajs-1241	146	18	a	a	DET
iajs-1241	146	19	fuzzy	fuzzy	ADJ
iajs-1241	146	20	semimaximal	semimaximal	NOUN
iajs-1241	146	21	ideal	ideal	NOUN
iajs-1241	146	22	,	,	PUNCT
iajs-1241	146	23	then	then	ADV
iajs-1241	146	24	a	a	DET
iajs-1241	146	25	=	=	PUNCT
iajs-1241	146	26	a1	a1	NOUN
iajs-1241	146	27			PUNCT
iajs-1241	146	28	a2	a2	PROPN
iajs-1241	146	29			PUNCT
iajs-1241	146	30			X
iajs-1241	146	31	an	an	PRON
iajs-1241	146	32	,	,	PUNCT
iajs-1241	146	33	where	where	SCONJ
iajs-1241	146	34	a1	a1	NOUN
iajs-1241	146	35	,	,	PUNCT
iajs-1241	146	36	a2	a2	PROPN
iajs-1241	146	37	,	,	PUNCT
iajs-1241	146	38			PROPN
iajs-1241	146	39	,	,	PUNCT
iajs-1241	147	1	an	an	PRON
iajs-1241	147	2	are	be	AUX
iajs-1241	147	3	fuzzy	fuzzy	ADJ
iajs-1241	147	4	maximal	maximal	ADJ
iajs-1241	147	5	ideals	ideal	NOUN
iajs-1241	147	6	of	of	ADP
iajs-1241	147	7	r.	r.	PROPN
iajs-1241	147	8	since	since	SCONJ
iajs-1241	147	9	0	0	NUM
iajs-1241	147	10			NUM
iajs-1241	147	11	t	t	NOUN
iajs-1241	147	12	<	<	X
iajs-1241	147	13	1	1	NUM
iajs-1241	147	14	,	,	PUNCT
iajs-1241	147	15	then	then	ADV
iajs-1241	147	16	by	by	ADP
iajs-1241	147	17	the	the	DET
iajs-1241	147	18	same	same	ADJ
iajs-1241	147	19	proof	proof	NOUN
iajs-1241	147	20	of	of	ADP
iajs-1241	147	21	theorem	theorem	ADJ
iajs-1241	147	22	3.4	3.4	NUM
iajs-1241	147	23	(	(	PUNCT
iajs-1241	147	24	1	1	NUM
iajs-1241	147	25	)	)	PUNCT
iajs-1241	147	26	,	,	PUNCT
iajs-1241	147	27	we	we	PRON
iajs-1241	147	28	have	have	VERB
iajs-1241	147	29	(	(	PUNCT
iajs-1241	147	30	a1)t	a1)t	NOUN
iajs-1241	147	31	=	=	SYM
iajs-1241	147	32	(	(	PUNCT
iajs-1241	148	1	a2)t	a2)t	PROPN
iajs-1241	148	2	=	=	SYM
iajs-1241	148	3			PROPN
iajs-1241	149	1	=	=	PUNCT
iajs-1241	150	1	(	(	PUNCT
iajs-1241	150	2	an)t	an)t	PROPN
iajs-1241	150	3	=	=	SYM
iajs-1241	150	4	r	r	NOUN
iajs-1241	150	5	but	but	CCONJ
iajs-1241	150	6	at	at	ADP
iajs-1241	150	7	=	=	PUNCT
iajs-1241	150	8	(	(	PUNCT
iajs-1241	150	9	a1)t	a1)t	NOUN
iajs-1241	150	10			PUNCT
iajs-1241	150	11	(	(	PUNCT
iajs-1241	150	12	a2)t	a2)t	ADJ
iajs-1241	150	13			PUNCT
iajs-1241	150	14			NOUN
iajs-1241	150	15			X
iajs-1241	150	16	(	(	PUNCT
iajs-1241	150	17	an)t	an)t	PROPN
iajs-1241	150	18	,	,	PUNCT
iajs-1241	150	19	so	so	CCONJ
iajs-1241	150	20	at	at	SCONJ
iajs-1241	150	21	=	=	SYM
iajs-1241	150	22	r	r	NOUN
iajs-1241	150	23	,	,	PUNCT
iajs-1241	150	24	for	for	ADP
iajs-1241	150	25	all	all	DET
iajs-1241	150	26	t	t	PROPN
iajs-1241	150	27	,	,	PUNCT
iajs-1241	150	28	0	0	NUM
iajs-1241	150	29			NUM
iajs-1241	150	30	t	t	X
iajs-1241	150	31	<	<	X
iajs-1241	150	32	1	1	NUM
iajs-1241	150	33	.	.	PUNCT
iajs-1241	150	34	suppose	suppose	VERB
iajs-1241	150	35	there	there	PRON
iajs-1241	150	36	exist	exist	VERB
iajs-1241	150	37	t1	t1	NOUN
iajs-1241	150	38	,	,	PUNCT
iajs-1241	150	39	t2	t2	NOUN
iajs-1241	150	40			NOUN
iajs-1241	151	1	[	[	X
iajs-1241	151	2	0,1	0,1	NUM
iajs-1241	151	3	]	]	PUNCT
iajs-1241	151	4	,	,	PUNCT
iajs-1241	151	5	t1	t1	PROPN
iajs-1241	151	6	,	,	PUNCT
iajs-1241	151	7	t2	t2	PROPN
iajs-1241	151	8			NOUN
iajs-1241	151	9	i	i	PRON
iajs-1241	151	10	m	m	VERB
iajs-1241	151	11	a.	a.	VERB
iajs-1241	151	12	then	then	ADV
iajs-1241	151	13	1	1	NUM
iajs-1241	151	14	2	2	NUM
iajs-1241	152	1			NOUN
iajs-1241	152	2			NUM
iajs-1241	153	1			PROPN
iajs-1241	153	2	t	t	NOUN
iajs-1241	153	3	t	t	NOUN
iajs-1241	153	4	=	=	PUNCT
iajs-1241	153	5	r	r	NOUN
iajs-1241	153	6	which	which	PRON
iajs-1241	153	7	implies	imply	VERB
iajs-1241	153	8	t1	t1	NOUN
iajs-1241	153	9	=	=	SYM
iajs-1241	153	10	t2	t2	NOUN
iajs-1241	153	11	.	.	PUNCT
iajs-1241	154	1	thus	thus	ADV
iajs-1241	154	2	im	im	NUM
iajs-1241	154	3	a	a	PRON
iajs-1241	154	4	has	have	AUX
iajs-1241	154	5	two	two	NUM
iajs-1241	154	6	valued	value	VERB
iajs-1241	154	7	namely	namely	ADV
iajs-1241	154	8	1	1	NUM
iajs-1241	154	9	,	,	PUNCT
iajs-1241	154	10	t.	t.	NOUN
iajs-1241	154	11	remark	remark	NOUN
iajs-1241	154	12	2.11	2.11	NUM
iajs-1241	154	13	:	:	PUNCT
iajs-1241	154	14	by	by	ADP
iajs-1241	154	15	using	use	VERB
iajs-1241	154	16	theorem	theorem	ADJ
iajs-1241	154	17	2.10	2.10	NUM
iajs-1241	154	18	,	,	PUNCT
iajs-1241	154	19	we	we	PRON
iajs-1241	154	20	can	can	AUX
iajs-1241	154	21	give	give	VERB
iajs-1241	154	22	an	an	DET
iajs-1241	154	23	example	example	NOUN
iajs-1241	154	24	which	which	PRON
iajs-1241	154	25	explains	explain	VERB
iajs-1241	154	26	that	that	SCONJ
iajs-1241	154	27	the	the	DET
iajs-1241	154	28	converse	converse	NOUN
iajs-1241	154	29	of	of	ADP
iajs-1241	154	30	proposition	proposition	NOUN
iajs-1241	154	31	2.4	2.4	NUM
iajs-1241	154	32	is	be	AUX
iajs-1241	154	33	not	not	PART
iajs-1241	154	34	true	true	ADJ
iajs-1241	154	35	in	in	ADP
iajs-1241	154	36	general	general	ADJ
iajs-1241	154	37	.	.	PUNCT
iajs-1241	155	1	example	example	NOUN
iajs-1241	155	2	:	:	PUNCT
iajs-1241	155	3	let	let	VERB
iajs-1241	155	4	a	a	PRON
iajs-1241	155	5	:	:	PUNCT
iajs-1241	155	6	z	z	NOUN
iajs-1241	155	7			X
iajs-1241	156	1	[	[	X
iajs-1241	156	2	0,1	0,1	NUM
iajs-1241	156	3	]	]	PUNCT
iajs-1241	156	4	defined	define	VERB
iajs-1241	156	5	by	by	ADP
iajs-1241	156	6	1	1	NUM
iajs-1241	156	7	6	6	NUM
iajs-1241	156	8	,	,	PUNCT
iajs-1241	156	9	1	1	NUM
iajs-1241	156	10	(	(	PUNCT
iajs-1241	156	11	)	)	PUNCT
iajs-1241	156	12	2	2	NUM
iajs-1241	156	13	6	6	NUM
iajs-1241	156	14	,	,	PUNCT
iajs-1241	156	15	2	2	NUM
iajs-1241	156	16	0	0	NUM
iajs-1241	156	17	otherwise	otherwise	ADV
iajs-1241	156	18			NOUN
iajs-1241	156	19			ADJ
iajs-1241	156	20			NUM
iajs-1241	156	21			NUM
iajs-1241	156	22			PROPN
iajs-1241	156	23			PROPN
iajs-1241	156	24			NOUN
iajs-1241	156	25			PROPN
iajs-1241	156	26			PROPN
iajs-1241	156	27			PROPN
iajs-1241	156	28			PRON
iajs-1241	156	29			NOUN
iajs-1241	156	30	x	x	PUNCT
iajs-1241	156	31	x	x	PUNCT
iajs-1241	156	32	x	x	X
iajs-1241	156	33	a	a	PRON
iajs-1241	156	34	is	be	AUX
iajs-1241	156	35	not	not	PART
iajs-1241	156	36	a	a	DET
iajs-1241	156	37	fuzzy	fuzzy	ADJ
iajs-1241	156	38	semimaximal	semimaximal	NOUN
iajs-1241	156	39	ideal	ideal	NOUN
iajs-1241	156	40	,	,	PUNCT
iajs-1241	156	41	since	since	SCONJ
iajs-1241	156	42	im	im	NUM
iajs-1241	156	43	a	a	NOUN
iajs-1241	156	44	=	=	SYM
iajs-1241	156	45	3	3	X
iajs-1241	156	46	.	.	PUNCT
iajs-1241	157	1	however	however	ADV
iajs-1241	157	2	a(0	a(0	PROPN
iajs-1241	157	3	)	)	PUNCT
iajs-1241	157	4	=	=	SYM
iajs-1241	157	5	1	1	NUM
iajs-1241	157	6	,	,	PUNCT
iajs-1241	157	7	a	a	NOUN
iajs-1241	157	8	=	=	NOUN
iajs-1241	158	1	6z	6z	NOUN
iajs-1241	158	2	is	be	AUX
iajs-1241	158	3	a	a	DET
iajs-1241	158	4	semimaximal	semimaximal	ADJ
iajs-1241	158	5	ideal	ideal	NOUN
iajs-1241	158	6	of	of	ADP
iajs-1241	158	7	z.	z.	PROPN
iajs-1241	158	8	ibn	ibn	PROPN
iajs-1241	158	9	alhaitham	alhaitham	PROPN
iajs-1241	158	10	j.	j.	PROPN
iajs-1241	158	11	for	for	ADP
iajs-1241	158	12	pure	pure	ADJ
iajs-1241	158	13	&	&	CCONJ
iajs-1241	158	14	appl	appl	PROPN
iajs-1241	158	15	.	.	PUNCT
iajs-1241	159	1	sci	sci	PROPN
iajs-1241	159	2	vol.22	vol.22	PROPN
iajs-1241	159	3	(	(	PUNCT
iajs-1241	159	4	2	2	NUM
iajs-1241	159	5	)	)	PUNCT
iajs-1241	159	6	2009	2009	NUM
iajs-1241	159	7	remark	remark	NOUN
iajs-1241	159	8	2.12	2.12	NUM
iajs-1241	159	9	:	:	PUNCT
iajs-1241	159	10	if	if	SCONJ
iajs-1241	159	11	a	a	DET
iajs-1241	159	12	a	a	DET
iajs-1241	159	13	fuzzy	fuzzy	ADJ
iajs-1241	159	14	semimaximal	semimaximal	ADJ
iajs-1241	159	15	ideal	ideal	NOUN
iajs-1241	159	16	of	of	ADP
iajs-1241	159	17	r	r	NOUN
iajs-1241	159	18	and	and	CCONJ
iajs-1241	159	19	b	b	NOUN
iajs-1241	159	20	is	be	AUX
iajs-1241	159	21	a	a	DET
iajs-1241	159	22	fuzzy	fuzzy	ADJ
iajs-1241	159	23	ideal	ideal	NOUN
iajs-1241	159	24	of	of	ADP
iajs-1241	159	25	r	r	NOUN
iajs-1241	160	1	such	such	ADJ
iajs-1241	160	2	that	that	DET
iajs-1241	160	3	b	b	PROPN
iajs-1241	160	4			NOUN
iajs-1241	160	5	r	r	PROPN
iajs-1241	160	6	and	and	CCONJ
iajs-1241	160	7	a	a	DET
iajs-1241	160	8			PROPN
iajs-1241	160	9	b.	b.	PROPN
iajs-1241	161	1	then	then	ADV
iajs-1241	161	2	it	it	PRON
iajs-1241	161	3	is	be	AUX
iajs-1241	161	4	not	not	PART
iajs-1241	161	5	necessary	necessary	ADJ
iajs-1241	161	6	that	that	SCONJ
iajs-1241	161	7	b	b	NOUN
iajs-1241	161	8	is	be	AUX
iajs-1241	161	9	a	a	DET
iajs-1241	161	10	fuzzy	fuzzy	ADJ
iajs-1241	161	11	semimaximal	semimaximal	NOUN
iajs-1241	161	12	ideal	ideal	NOUN
iajs-1241	161	13	.	.	PUNCT
iajs-1241	162	1	example	example	NOUN
iajs-1241	162	2	:	:	PUNCT
iajs-1241	162	3	let	let	VERB
iajs-1241	162	4	a	a	PRON
iajs-1241	162	5	:	:	PUNCT
iajs-1241	162	6	z	z	NOUN
iajs-1241	162	7			X
iajs-1241	163	1	[	[	X
iajs-1241	163	2	0,1	0,1	NUM
iajs-1241	163	3	]	]	PUNCT
iajs-1241	163	4	defined	define	VERB
iajs-1241	163	5	by	by	ADP
iajs-1241	163	6	1	1	NUM
iajs-1241	163	7	6	6	NUM
iajs-1241	163	8	,	,	PUNCT
iajs-1241	163	9	(	(	PUNCT
iajs-1241	163	10	)	)	PUNCT
iajs-1241	163	11	1	1	NUM
iajs-1241	163	12	otherwise	otherwise	ADV
iajs-1241	163	13	2	2	NUM
iajs-1241	163	14			NOUN
iajs-1241	163	15			NOUN
iajs-1241	163	16			NUM
iajs-1241	163	17			PROPN
iajs-1241	163	18			PROPN
iajs-1241	163	19			NUM
iajs-1241	163	20			NOUN
iajs-1241	163	21	x	x	X
iajs-1241	164	1	x	x	X
iajs-1241	164	2	a	a	PRON
iajs-1241	164	3	is	be	AUX
iajs-1241	164	4	a	a	DET
iajs-1241	164	5	fuzzy	fuzzy	ADJ
iajs-1241	164	6	semimaximal	semimaximal	ADJ
iajs-1241	164	7	ideal	ideal	NOUN
iajs-1241	164	8	(	(	PUNCT
iajs-1241	164	9	see	see	VERB
iajs-1241	164	10	remark	remark	NOUN
iajs-1241	164	11	2.3(1	2.3(1	ADV
iajs-1241	164	12	)	)	PUNCT
iajs-1241	164	13	)	)	PUNCT
iajs-1241	164	14	.	.	PUNCT
iajs-1241	165	1	let	let	VERB
iajs-1241	165	2	b	b	X
iajs-1241	165	3	:	:	PUNCT
iajs-1241	165	4	z	z	NOUN
iajs-1241	165	5			X
iajs-1241	166	1	[	[	X
iajs-1241	166	2	0,1	0,1	NUM
iajs-1241	166	3	]	]	PUNCT
iajs-1241	166	4	defined	define	VERB
iajs-1241	166	5	by	by	ADP
iajs-1241	166	6	1	1	NUM
iajs-1241	166	7	6	6	NUM
iajs-1241	166	8	,	,	PUNCT
iajs-1241	166	9	3	3	NUM
iajs-1241	166	10	(	(	PUNCT
iajs-1241	166	11	)	)	PUNCT
iajs-1241	166	12	2	2	NUM
iajs-1241	166	13	6	6	NUM
iajs-1241	166	14	,	,	PUNCT
iajs-1241	166	15	4	4	NUM
iajs-1241	166	16	1	1	NUM
iajs-1241	166	17	otherwise	otherwise	ADV
iajs-1241	166	18	2	2	NUM
iajs-1241	166	19			NOUN
iajs-1241	166	20			NOUN
iajs-1241	167	1			NOUN
iajs-1241	167	2			NUM
iajs-1241	167	3			NUM
iajs-1241	167	4			NOUN
iajs-1241	167	5			PROPN
iajs-1241	167	6			PROPN
iajs-1241	167	7			PROPN
iajs-1241	167	8			PROPN
iajs-1241	167	9			NOUN
iajs-1241	167	10			NUM
iajs-1241	167	11			DET
iajs-1241	167	12			NOUN
iajs-1241	167	13	x	x	PUNCT
iajs-1241	167	14	x	x	PUNCT
iajs-1241	167	15	x	x	X
iajs-1241	167	16	it	it	PRON
iajs-1241	167	17	is	be	AUX
iajs-1241	167	18	clear	clear	ADJ
iajs-1241	167	19	that	that	SCONJ
iajs-1241	167	20	a	a	DET
iajs-1241	167	21			PROPN
iajs-1241	167	22	b.	b.	PROPN
iajs-1241	167	23	however	however	ADV
iajs-1241	167	24	im	im	VERB
iajs-1241	167	25	a	a	ADJ
iajs-1241	167	26	=	=	SYM
iajs-1241	167	27	3	3	NUM
iajs-1241	167	28	,	,	PUNCT
iajs-1241	167	29	which	which	PRON
iajs-1241	167	30	implies	imply	VERB
iajs-1241	167	31	that	that	SCONJ
iajs-1241	167	32	b	b	NOUN
iajs-1241	167	33	is	be	AUX
iajs-1241	167	34	not	not	PART
iajs-1241	167	35	a	a	DET
iajs-1241	167	36	fuzzy	fuzzy	ADJ
iajs-1241	167	37	semimaximal	semimaximal	NOUN
iajs-1241	167	38	ideal	ideal	NOUN
iajs-1241	167	39	,	,	PUNCT
iajs-1241	167	40	by	by	ADP
iajs-1241	167	41	theorem	theorem	NOUN
iajs-1241	167	42	2.10	2.10	NUM
iajs-1241	167	43	.	.	PUNCT
iajs-1241	167	44	remark	remark	NOUN
iajs-1241	167	45	2.13	2.13	NUM
iajs-1241	167	46	:	:	PUNCT
iajs-1241	167	47	if	if	SCONJ
iajs-1241	167	48	a	a	DET
iajs-1241	167	49	a	a	DET
iajs-1241	167	50	fuzzy	fuzzy	ADJ
iajs-1241	167	51	semimaximal	semimaximal	NOUN
iajs-1241	167	52	ideal	ideal	NOUN
iajs-1241	167	53	and	and	CCONJ
iajs-1241	167	54	t	t	X
iajs-1241	167	55			NOUN
iajs-1241	167	56	[	[	X
iajs-1241	167	57	0,1	0,1	NUM
iajs-1241	167	58	)	)	PUNCT
iajs-1241	167	59	,	,	PUNCT
iajs-1241	167	60	then	then	ADV
iajs-1241	167	61	at	at	ADP
iajs-1241	167	62	does’nt	does’nt	NOUN
iajs-1241	167	63	need	need	VERB
iajs-1241	167	64	to	to	PART
iajs-1241	167	65	be	be	AUX
iajs-1241	167	66	a	a	DET
iajs-1241	167	67	semimaximal	semimaximal	ADJ
iajs-1241	167	68	ideal	ideal	NOUN
iajs-1241	167	69	of	of	ADP
iajs-1241	167	70	r.	r.	PROPN
iajs-1241	167	71	as	as	SCONJ
iajs-1241	167	72	can	can	AUX
iajs-1241	167	73	be	be	AUX
iajs-1241	167	74	seen	see	VERB
iajs-1241	167	75	by	by	ADP
iajs-1241	167	76	the	the	DET
iajs-1241	167	77	following	follow	VERB
iajs-1241	167	78	example	example	NOUN
iajs-1241	167	79	:	:	PUNCT
iajs-1241	167	80	example	example	NOUN
iajs-1241	167	81	:	:	PUNCT
iajs-1241	167	82	let	let	VERB
iajs-1241	167	83	a	a	PRON
iajs-1241	167	84	:	:	PUNCT
iajs-1241	167	85	z	z	NOUN
iajs-1241	167	86			X
iajs-1241	168	1	[	[	X
iajs-1241	168	2	0,1	0,1	NUM
iajs-1241	168	3	]	]	PUNCT
iajs-1241	168	4	defined	define	VERB
iajs-1241	168	5	by	by	ADP
iajs-1241	168	6	1	1	NUM
iajs-1241	168	7	2	2	NUM
iajs-1241	168	8	,	,	PUNCT
iajs-1241	168	9	(	(	PUNCT
iajs-1241	168	10	)	)	PUNCT
iajs-1241	168	11	1	1	NUM
iajs-1241	168	12	otherwise	otherwise	ADV
iajs-1241	168	13	2	2	NUM
iajs-1241	168	14			NOUN
iajs-1241	168	15			NOUN
iajs-1241	168	16			NUM
iajs-1241	168	17			PROPN
iajs-1241	168	18			PROPN
iajs-1241	168	19			NUM
iajs-1241	168	20			NOUN
iajs-1241	168	21	x	x	X
iajs-1241	169	1	x	x	X
iajs-1241	169	2	a	a	PRON
iajs-1241	169	3	is	be	AUX
iajs-1241	169	4	a	a	DET
iajs-1241	169	5	fuzzy	fuzzy	ADJ
iajs-1241	169	6	semimaximal	semimaximal	ADJ
iajs-1241	169	7	ideal	ideal	NOUN
iajs-1241	169	8	of	of	ADP
iajs-1241	169	9	z	z	NOUN
iajs-1241	169	10	and	and	CCONJ
iajs-1241	169	11	a	a	PROPN
iajs-1241	170	1	=	=	PUNCT
iajs-1241	170	2	2z	2z	NOUN
iajs-1241	170	3	which	which	PRON
iajs-1241	170	4	is	be	AUX
iajs-1241	170	5	amaximal	amaximal	ADJ
iajs-1241	170	6	ideal	ideal	NOUN
iajs-1241	170	7	and	and	CCONJ
iajs-1241	170	8	a(0	a(0	PROPN
iajs-1241	170	9	)	)	PUNCT
iajs-1241	170	10	=	=	SYM
iajs-1241	170	11	1	1	NUM
iajs-1241	170	12	,	,	PUNCT
iajs-1241	170	13	this	this	PRON
iajs-1241	170	14	implies	imply	VERB
iajs-1241	170	15	that	that	SCONJ
iajs-1241	170	16	a	a	PRON
iajs-1241	170	17	is	be	AUX
iajs-1241	170	18	a	a	DET
iajs-1241	170	19	fuzzy	fuzzy	ADJ
iajs-1241	170	20	maximal	maximal	ADJ
iajs-1241	170	21	ideal	ideal	NOUN
iajs-1241	170	22	,	,	PUNCT
iajs-1241	170	23	so	so	CCONJ
iajs-1241	170	24	it	it	PRON
iajs-1241	170	25	is	be	AUX
iajs-1241	170	26	semimaximal	semimaximal	ADJ
iajs-1241	170	27	.	.	PUNCT
iajs-1241	171	1	but	but	CCONJ
iajs-1241	171	2	a1/2	a1/2	NOUN
iajs-1241	171	3	=	=	PUNCT
iajs-1241	171	4	{	{	PUNCT
iajs-1241	171	5	x	x	NOUN
iajs-1241	171	6	:	:	PUNCT
iajs-1241	171	7	a(x	a(x	NOUN
iajs-1241	171	8	)	)	PUNCT
iajs-1241	171	9			NUM
iajs-1241	171	10	1	1	NUM
iajs-1241	171	11	2	2	NUM
iajs-1241	171	12	}	}	PUNCT
iajs-1241	171	13	=	=	PUNCT
iajs-1241	171	14	z	z	NOUN
iajs-1241	171	15	which	which	PRON
iajs-1241	171	16	is	be	AUX
iajs-1241	171	17	not	not	PART
iajs-1241	171	18	a	a	DET
iajs-1241	171	19	semimaximal	semimaximal	ADJ
iajs-1241	171	20	ideal	ideal	NOUN
iajs-1241	171	21	.	.	PUNCT
iajs-1241	172	1	recall	recall	VERB
iajs-1241	172	2	that	that	PRON
iajs-1241	172	3	,	,	PUNCT
iajs-1241	172	4	the	the	DET
iajs-1241	172	5	fuzzy	fuzzy	ADJ
iajs-1241	172	6	jacobson	jacobson	PROPN
iajs-1241	172	7	radical	radical	PROPN
iajs-1241	172	8	of	of	ADP
iajs-1241	172	9	a	a	DET
iajs-1241	172	10	ring	ring	NOUN
iajs-1241	172	11	r	r	NOUN
iajs-1241	172	12	denoted	denote	VERB
iajs-1241	172	13	by	by	ADP
iajs-1241	172	14	f	f	PROPN
iajs-1241	172	15	-	-	PUNCT
iajs-1241	172	16	j(r	j(r	PROPN
iajs-1241	172	17	)	)	PUNCT
iajs-1241	172	18	is	be	AUX
iajs-1241	172	19	the	the	DET
iajs-1241	172	20	intersection	intersection	NOUN
iajs-1241	172	21	of	of	ADP
iajs-1241	172	22	all	all	DET
iajs-1241	172	23	fuzzy	fuzzy	ADJ
iajs-1241	172	24	maximal	maximal	ADJ
iajs-1241	172	25	ideals	ideal	NOUN
iajs-1241	172	26	of	of	ADP
iajs-1241	172	27	r	r	NOUN
iajs-1241	172	28	(	(	PUNCT
iajs-1241	172	29	1	1	NUM
iajs-1241	172	30	)	)	PUNCT
iajs-1241	172	31	.	.	PUNCT
iajs-1241	173	1	f	f	X
iajs-1241	173	2	-	-	PUNCT
iajs-1241	173	3	j(r	j(r	PROPN
iajs-1241	173	4	)	)	PUNCT
iajs-1241	173	5	does’nt	does’nt	NOUN
iajs-1241	173	6	need	need	VERB
iajs-1241	173	7	to	to	PART
iajs-1241	173	8	be	be	AUX
iajs-1241	173	9	a	a	DET
iajs-1241	173	10	fuzzy	fuzzy	ADJ
iajs-1241	173	11	semimaximal	semimaximal	ADJ
iajs-1241	173	12	ideal	ideal	NOUN
iajs-1241	173	13	of	of	ADP
iajs-1241	173	14	r.	r.	PROPN
iajs-1241	173	15	example	example	PROPN
iajs-1241	173	16	:	:	PUNCT
iajs-1241	173	17	let	let	AUX
iajs-1241	173	18	{	{	PUNCT
iajs-1241	173	19	ai	ai	VERB
iajs-1241	173	20	,	,	PUNCT
iajs-1241	173	21	i	i	PRON
iajs-1241	173	22	=	=	NOUN
iajs-1241	173	23	1	1	NUM
iajs-1241	173	24	,	,	PUNCT
iajs-1241	173	25	2	2	NUM
iajs-1241	173	26	,	,	PUNCT
iajs-1241	173	27			PROPN
iajs-1241	173	28	,	,	PUNCT
iajs-1241	173	29	n	n	CCONJ
iajs-1241	173	30	}	}	PUNCT
iajs-1241	173	31	be	be	AUX
iajs-1241	173	32	the	the	DET
iajs-1241	173	33	collection	collection	NOUN
iajs-1241	173	34	of	of	ADP
iajs-1241	173	35	all	all	DET
iajs-1241	173	36	fuzzy	fuzzy	ADJ
iajs-1241	173	37	maximal	maximal	ADJ
iajs-1241	173	38	ideals	ideal	NOUN
iajs-1241	173	39	of	of	ADP
iajs-1241	173	40	z	z	NOUN
iajs-1241	173	41	,	,	PUNCT
iajs-1241	173	42	where	where	SCONJ
iajs-1241	173	43	1	1	NUM
iajs-1241	173	44	,	,	PUNCT
iajs-1241	173	45	(	(	PUNCT
iajs-1241	173	46	)	)	PUNCT
iajs-1241	173	47	1	1	X
iajs-1241	173	48	.	.	PUNCT
iajs-1241	174	1	i	i	PRON
iajs-1241	174	2			NOUN
iajs-1241	174	3			PUNCT
iajs-1241	174	4			NUM
iajs-1241	174	5			PROPN
iajs-1241	174	6			NUM
iajs-1241	174	7			NUM
iajs-1241	174	8			NOUN
iajs-1241	174	9			NOUN
iajs-1241	174	10	i	i	NOUN
iajs-1241	174	11	x	x	X
iajs-1241	174	12	p	p	X
iajs-1241	174	13	x	x	X
iajs-1241	174	14	x	x	X
iajs-1241	174	15	p	p	NOUN
iajs-1241	174	16	,	,	PUNCT
iajs-1241	174	17	p	p	NOUN
iajs-1241	174	18	is	be	AUX
iajs-1241	174	19	a	a	DET
iajs-1241	174	20	prime	prime	ADJ
iajs-1241	174	21	number	number	NOUN
iajs-1241	174	22	.	.	PUNCT
iajs-1241	175	1	ibn	ibn	PROPN
iajs-1241	175	2	alhaitham	alhaitham	PROPN
iajs-1241	175	3	j.	j.	PROPN
iajs-1241	175	4	for	for	ADP
iajs-1241	175	5	pure	pure	ADJ
iajs-1241	175	6	&	&	CCONJ
iajs-1241	175	7	appl	appl	PROPN
iajs-1241	175	8	.	.	PUNCT
iajs-1241	176	1	sci	sci	PROPN
iajs-1241	176	2	vol.22	vol.22	PROPN
iajs-1241	176	3	(	(	PUNCT
iajs-1241	176	4	2	2	NUM
iajs-1241	176	5	)	)	PUNCT
iajs-1241	176	6	2009	2009	NUM
iajs-1241	176	7	s	s	VERB
iajs-1241	176	8	a	a	DET
iajs-1241	176	9	prime	prime	ADJ
iajs-1241	176	10	no	no	NOUN
iajs-1241	176	11	.	.	NOUN
iajs-1241	176	12	1	1	NUM
iajs-1241	176	13	s	s	VERB
iajs-1241	176	14	a	a	DET
iajs-1241	176	15	prime	prime	ADJ
iajs-1241	176	16	no	no	NOUN
iajs-1241	176	17	.	.	PROPN
iajs-1241	176	18	1	1	NUM
iajs-1241	176	19	,	,	PUNCT
iajs-1241	176	20	f	f	PROPN
iajs-1241	176	21	j(r	j(r	PROPN
iajs-1241	176	22	)	)	PUNCT
iajs-1241	176	23	1	1	NUM
iajs-1241	176	24	inf	inf	NOUN
iajs-1241	176	25	{	{	PUNCT
iajs-1241	176	26	,	,	PUNCT
iajs-1241	176	27	}	}	PUNCT
iajs-1241	176	28	.	.	PUNCT
iajs-1241	177	1	i	i	PRON
iajs-1241	177	2			VERB
iajs-1241	177	3			PRON
iajs-1241	177	4			ADJ
iajs-1241	177	5			NOUN
iajs-1241	177	6			PUNCT
iajs-1241	177	7			NUM
iajs-1241	178	1			NOUN
iajs-1241	178	2			PROPN
iajs-1241	178	3			NUM
iajs-1241	178	4			NUM
iajs-1241	178	5			PROPN
iajs-1241	178	6			VERB
iajs-1241	178	7			NOUN
iajs-1241	179	1			NOUN
iajs-1241	179	2	p	p	NOUN
iajs-1241	180	1	i	i	PRON
iajs-1241	180	2	i	i	PRON
iajs-1241	181	1	i	i	PRON
iajs-1241	181	2	p	p	VERB
iajs-1241	182	1	i	i	PRON
iajs-1241	182	2	x	x	PROPN
iajs-1241	183	1	p	p	X
iajs-1241	183	2	i	i	NOUN
iajs-1241	183	3	x	x	PROPN
iajs-1241	184	1	p	p	X
iajs-1241	184	2	s	s	VERB
iajs-1241	184	3	a	a	DET
iajs-1241	184	4	prime	prime	ADJ
iajs-1241	184	5	no	no	NOUN
iajs-1241	184	6	.	.	NOUN
iajs-1241	184	7	1	1	NUM
iajs-1241	184	8	s	s	VERB
iajs-1241	184	9	a	a	DET
iajs-1241	184	10	prime	prime	ADJ
iajs-1241	184	11	no	no	NOUN
iajs-1241	184	12	.	.	PROPN
iajs-1241	184	13	1	1	NUM
iajs-1241	184	14	,	,	PUNCT
iajs-1241	184	15	f	f	PROPN
iajs-1241	184	16	j(r	j(r	PROPN
iajs-1241	184	17	)	)	PUNCT
iajs-1241	184	18	0	0	PUNCT
iajs-1241	184	19	.	.	PUNCT
iajs-1241	185	1			VERB
iajs-1241	185	2			PROPN
iajs-1241	185	3			NOUN
iajs-1241	185	4			PUNCT
iajs-1241	185	5			NUM
iajs-1241	186	1			NOUN
iajs-1241	186	2			PROPN
iajs-1241	186	3			NUM
iajs-1241	187	1			NOUN
iajs-1241	187	2			NOUN
iajs-1241	188	1			NOUN
iajs-1241	189	1			NOUN
iajs-1241	189	2	p	p	NOUN
iajs-1241	190	1	i	i	PRON
iajs-1241	190	2	i	i	PRON
iajs-1241	191	1	i	i	PRON
iajs-1241	191	2	p	p	VERB
iajs-1241	192	1	i	i	PRON
iajs-1241	192	2	x	x	PROPN
iajs-1241	193	1	p	p	X
iajs-1241	193	2	x	x	X
iajs-1241	193	3	p	p	X
iajs-1241	193	4	=	=	SYM
iajs-1241	193	5	01	01	NUM
iajs-1241	193	6	which	which	PRON
iajs-1241	193	7	is	be	AUX
iajs-1241	193	8	not	not	PART
iajs-1241	193	9	a	a	DET
iajs-1241	193	10	fuzzy	fuzzy	ADJ
iajs-1241	193	11	semimaximal	semimaximal	ADJ
iajs-1241	193	12	ideal	ideal	NOUN
iajs-1241	193	13	of	of	ADP
iajs-1241	193	14	z.	z.	PROPN
iajs-1241	193	15	now	now	ADV
iajs-1241	193	16	,	,	PUNCT
iajs-1241	193	17	let	let	VERB
iajs-1241	193	18	f	f	X
iajs-1241	193	19	-	-	PUNCT
iajs-1241	193	20	j(r	j(r	ADJ
iajs-1241	193	21	)	)	PUNCT
iajs-1241	193	22	denotes	denote	VERB
iajs-1241	193	23	the	the	DET
iajs-1241	193	24	intersection	intersection	NOUN
iajs-1241	193	25	of	of	ADP
iajs-1241	193	26	all	all	DET
iajs-1241	193	27	fuzzy	fuzzy	ADJ
iajs-1241	193	28	semimaximal	semimaximal	ADJ
iajs-1241	193	29	ideals	ideal	NOUN
iajs-1241	193	30	of	of	ADP
iajs-1241	193	31	r.	r.	PROPN
iajs-1241	193	32	then	then	ADV
iajs-1241	193	33	fj(r	fj(r	ADJ
iajs-1241	193	34	)	)	PUNCT
iajs-1241	193	35	is	be	AUX
iajs-1241	193	36	called	call	VERB
iajs-1241	193	37	a	a	DET
iajs-1241	193	38	fuzzy	fuzzy	ADJ
iajs-1241	193	39	semijacobson	semijacobson	NOUN
iajs-1241	193	40	radical	radical	ADJ
iajs-1241	193	41	of	of	ADP
iajs-1241	193	42	r.	r.	PROPN
iajs-1241	193	43	remark	remark	PROPN
iajs-1241	193	44	2.14	2.14	NUM
iajs-1241	193	45	:	:	PUNCT
iajs-1241	193	46	f	f	X
iajs-1241	193	47	-	-	PUNCT
iajs-1241	193	48	j(r	j(r	PROPN
iajs-1241	193	49	)	)	PUNCT
iajs-1241	194	1	=	=	SYM
iajs-1241	194	2	f	f	X
iajs-1241	194	3	-	-	PUNCT
iajs-1241	194	4	j(r	j(r	ADJ
iajs-1241	194	5	)	)	PUNCT
iajs-1241	194	6	.	.	PUNCT
iajs-1241	195	1	proof	proof	NOUN
iajs-1241	195	2	.	.	PUNCT
iajs-1241	196	1	it	it	PRON
iajs-1241	196	2	is	be	AUX
iajs-1241	196	3	clear	clear	ADJ
iajs-1241	196	4	that	that	SCONJ
iajs-1241	196	5	f	f	X
iajs-1241	196	6	-	-	PUNCT
iajs-1241	196	7	j(r	j(r	ADJ
iajs-1241	196	8	)	)	PUNCT
iajs-1241	196	9			PROPN
iajs-1241	197	1	f	f	PROPN
iajs-1241	197	2	-	-	PUNCT
iajs-1241	197	3	j(r	j(r	PROPN
iajs-1241	197	4	)	)	PUNCT
iajs-1241	197	5	.	.	PUNCT
iajs-1241	198	1	let	let	VERB
iajs-1241	198	2	xt	xt	PUNCT
iajs-1241	198	3			PROPN
iajs-1241	198	4	f	f	PROPN
iajs-1241	198	5	-	-	PUNCT
iajs-1241	198	6	j(r	j(r	PROPN
iajs-1241	198	7	)	)	PUNCT
iajs-1241	198	8	.	.	PUNCT
iajs-1241	199	1	then	then	ADV
iajs-1241	199	2	xt	xt	PROPN
iajs-1241	199	3	belongs	belong	VERB
iajs-1241	199	4	to	to	ADP
iajs-1241	199	5	any	any	DET
iajs-1241	199	6	fuzzy	fuzzy	ADJ
iajs-1241	199	7	maximal	maximal	ADJ
iajs-1241	199	8	ideal	ideal	NOUN
iajs-1241	199	9	.	.	PUNCT
iajs-1241	200	1	since	since	SCONJ
iajs-1241	200	2	any	any	DET
iajs-1241	200	3	fuzzy	fuzzy	ADJ
iajs-1241	200	4	semimaximal	semimaximal	NOUN
iajs-1241	200	5	ideal	ideal	NOUN
iajs-1241	200	6	a	a	PRON
iajs-1241	200	7	of	of	ADP
iajs-1241	200	8	r	r	NOUN
iajs-1241	200	9	is	be	AUX
iajs-1241	200	10	a	a	DET
iajs-1241	200	11	finite	finite	ADJ
iajs-1241	200	12	intersection	intersection	NOUN
iajs-1241	200	13	of	of	ADP
iajs-1241	200	14	fuzzy	fuzzy	ADJ
iajs-1241	200	15	maximal	maximal	ADJ
iajs-1241	200	16	ideals	ideal	NOUN
iajs-1241	200	17	,	,	PUNCT
iajs-1241	200	18	so	so	ADV
iajs-1241	200	19	xt	xt	PROPN
iajs-1241	200	20			PROPN
iajs-1241	200	21	a.	a.	NOUN
iajs-1241	201	1	it	it	PRON
iajs-1241	201	2	follows	follow	VERB
iajs-1241	201	3	that	that	PRON
iajs-1241	201	4	xt	xt	PUNCT
iajs-1241	201	5			NOUN
iajs-1241	201	6	f	f	NOUN
iajs-1241	201	7	-	-	PUNCT
iajs-1241	201	8	j(r	j(r	ADJ
iajs-1241	201	9	)	)	PUNCT
iajs-1241	201	10	.	.	PUNCT
iajs-1241	202	1	thus	thus	ADV
iajs-1241	202	2	f	f	X
iajs-1241	202	3	-	-	PUNCT
iajs-1241	202	4	j(r	j(r	PROPN
iajs-1241	202	5	)	)	PUNCT
iajs-1241	203	1	=	=	SYM
iajs-1241	203	2	f	f	X
iajs-1241	203	3	-	-	PUNCT
iajs-1241	203	4	j(r	j(r	ADJ
iajs-1241	203	5	)	)	PUNCT
iajs-1241	203	6	.	.	PUNCT
iajs-1241	204	1	s.3	s.3	NOUN
iajs-1241	204	2	image	image	NOUN
iajs-1241	204	3	and	and	CCONJ
iajs-1241	204	4	inverse	inverse	NOUN
iajs-1241	204	5	image	image	NOUN
iajs-1241	204	6	of	of	ADP
iajs-1241	204	7	fuzzy	fuzzy	ADJ
iajs-1241	204	8	semimaximal	semimaximal	ADJ
iajs-1241	204	9	ideals	ideal	NOUN
iajs-1241	204	10	in	in	ADP
iajs-1241	204	11	this	this	DET
iajs-1241	204	12	section	section	NOUN
iajs-1241	204	13	,	,	PUNCT
iajs-1241	204	14	we	we	PRON
iajs-1241	204	15	consider	consider	VERB
iajs-1241	204	16	the	the	DET
iajs-1241	204	17	homomorphic	homomorphic	ADJ
iajs-1241	204	18	image	image	NOUN
iajs-1241	204	19	and	and	CCONJ
iajs-1241	204	20	inverse	inverse	NOUN
iajs-1241	204	21	image	image	NOUN
iajs-1241	204	22	of	of	ADP
iajs-1241	204	23	fuzzy	fuzzy	ADJ
iajs-1241	204	24	semimaximal	semimaximal	ADJ
iajs-1241	204	25	ideals	ideal	NOUN
iajs-1241	204	26	.	.	PUNCT
iajs-1241	205	1	theorem	theorem	VERB
iajs-1241	205	2	3.1	3.1	NUM
iajs-1241	205	3	:	:	PUNCT
iajs-1241	205	4	let	let	VERB
iajs-1241	205	5	r1	r1	PROPN
iajs-1241	205	6	,	,	PUNCT
iajs-1241	205	7	r2	r2	PROPN
iajs-1241	205	8	be	be	VERB
iajs-1241	205	9	two	two	NUM
iajs-1241	205	10	rings	ring	NOUN
iajs-1241	205	11	,	,	PUNCT
iajs-1241	205	12	let	let	VERB
iajs-1241	205	13	f	f	PRON
iajs-1241	205	14	:	:	PUNCT
iajs-1241	205	15	r1	r1	PROPN
iajs-1241	205	16			PROPN
iajs-1241	205	17	r2	r2	PROPN
iajs-1241	205	18	be	be	VERB
iajs-1241	205	19	an	an	DET
iajs-1241	205	20	epimorphisim	epimorphisim	ADJ
iajs-1241	205	21	and	and	CCONJ
iajs-1241	205	22	every	every	DET
iajs-1241	205	23	fuzzy	fuzzy	ADJ
iajs-1241	205	24	ideal	ideal	NOUN
iajs-1241	205	25	of	of	ADP
iajs-1241	205	26	r1	r1	PROPN
iajs-1241	205	27	is	be	AUX
iajs-1241	205	28	f	f	NOUN
iajs-1241	205	29	-	-	PUNCT
iajs-1241	205	30	invariant	invariant	ADJ
iajs-1241	205	31	.	.	PUNCT
iajs-1241	206	1	then	then	ADV
iajs-1241	206	2	if	if	SCONJ
iajs-1241	206	3	a	a	PRON
iajs-1241	206	4	is	be	AUX
iajs-1241	206	5	a	a	DET
iajs-1241	206	6	fuzzy	fuzzy	ADJ
iajs-1241	206	7	semimaximal	semimaximal	ADJ
iajs-1241	206	8	ideal	ideal	NOUN
iajs-1241	206	9	of	of	ADP
iajs-1241	206	10	r1	r1	PROPN
iajs-1241	206	11	,	,	PUNCT
iajs-1241	206	12	then	then	ADV
iajs-1241	206	13	f	f	X
iajs-1241	206	14	(	(	PUNCT
iajs-1241	206	15	a	a	PRON
iajs-1241	206	16	)	)	PUNCT
iajs-1241	206	17	is	be	AUX
iajs-1241	206	18	a	a	DET
iajs-1241	206	19	fuzzy	fuzzy	ADJ
iajs-1241	206	20	semimaximal	semimaximal	NOUN
iajs-1241	206	21	of	of	ADP
iajs-1241	206	22	r2	r2	PROPN
iajs-1241	206	23	.	.	PUNCT
iajs-1241	207	1	proof	proof	NOUN
iajs-1241	207	2	.	.	PUNCT
iajs-1241	208	1	a	a	PRON
iajs-1241	208	2	is	be	AUX
iajs-1241	208	3	a	a	DET
iajs-1241	208	4	fuzzy	fuzzy	ADJ
iajs-1241	208	5	semimaximal	semimaximal	ADJ
iajs-1241	208	6	ideal	ideal	NOUN
iajs-1241	208	7	of	of	ADP
iajs-1241	208	8	r1	r1	PROPN
iajs-1241	208	9	,	,	PUNCT
iajs-1241	208	10	then	then	ADV
iajs-1241	208	11	a	a	DET
iajs-1241	208	12	=	=	PUNCT
iajs-1241	208	13	a1	a1	NOUN
iajs-1241	208	14			PUNCT
iajs-1241	208	15	a2	a2	PROPN
iajs-1241	208	16			PUNCT
iajs-1241	208	17			X
iajs-1241	208	18	an	an	PRON
iajs-1241	208	19	,	,	PUNCT
iajs-1241	208	20	where	where	SCONJ
iajs-1241	208	21	a1	a1	NOUN
iajs-1241	208	22	,	,	PUNCT
iajs-1241	208	23	a2	a2	PROPN
iajs-1241	208	24	,	,	PUNCT
iajs-1241	208	25			PROPN
iajs-1241	208	26	,	,	PUNCT
iajs-1241	209	1	an	an	PRON
iajs-1241	209	2	are	be	AUX
iajs-1241	209	3	fuzzy	fuzzy	ADJ
iajs-1241	209	4	maximal	maximal	ADJ
iajs-1241	209	5	ideals	ideal	NOUN
iajs-1241	209	6	of	of	ADP
iajs-1241	209	7	r1	r1	PROPN
iajs-1241	209	8	.	.	PUNCT
iajs-1241	210	1	also	also	ADV
iajs-1241	210	2	,	,	PUNCT
iajs-1241	210	3	since	since	SCONJ
iajs-1241	210	4	every	every	DET
iajs-1241	210	5	fuzzy	fuzzy	ADJ
iajs-1241	210	6	ideal	ideal	NOUN
iajs-1241	210	7	of	of	ADP
iajs-1241	210	8	r1	r1	PROPN
iajs-1241	210	9	is	be	AUX
iajs-1241	210	10	f	f	NOUN
iajs-1241	210	11	-	-	PUNCT
iajs-1241	210	12	invariant	invariant	ADJ
iajs-1241	210	13	.	.	PUNCT
iajs-1241	211	1	so	so	ADV
iajs-1241	211	2	f	f	PROPN
iajs-1241	211	3	(	(	PUNCT
iajs-1241	211	4	a	a	X
iajs-1241	211	5	)	)	PUNCT
iajs-1241	211	6	=	=	SYM
iajs-1241	211	7	f	f	X
iajs-1241	211	8	(	(	PUNCT
iajs-1241	211	9	a1	a1	PROPN
iajs-1241	211	10			PUNCT
iajs-1241	211	11	a2	a2	PROPN
iajs-1241	211	12			NOUN
iajs-1241	211	13	an	an	X
iajs-1241	211	14	)	)	PUNCT
iajs-1241	211	15	=	=	SYM
iajs-1241	211	16	f	f	PROPN
iajs-1241	211	17	(	(	PUNCT
iajs-1241	211	18	a1	a1	PROPN
iajs-1241	211	19	)	)	PUNCT
iajs-1241	211	20			ADJ
iajs-1241	211	21	f	f	X
iajs-1241	211	22	(	(	PUNCT
iajs-1241	211	23	a2	a2	PROPN
iajs-1241	211	24	)	)	PUNCT
iajs-1241	211	25			ADP
iajs-1241	211	26			X
iajs-1241	211	27	f	f	X
iajs-1241	211	28	(	(	PUNCT
iajs-1241	211	29	an	an	NOUN
iajs-1241	211	30	)	)	PUNCT
iajs-1241	211	31	.	.	PUNCT
iajs-1241	212	1	on	on	ADP
iajs-1241	212	2	the	the	DET
iajs-1241	212	3	other	other	ADJ
iajs-1241	212	4	hand	hand	NOUN
iajs-1241	212	5	,	,	PUNCT
iajs-1241	212	6	f	f	PROPN
iajs-1241	212	7	(	(	PUNCT
iajs-1241	212	8	ai	ai	PROPN
iajs-1241	212	9	)	)	PUNCT
iajs-1241	212	10	is	be	AUX
iajs-1241	212	11	a	a	DET
iajs-1241	212	12	fuzzy	fuzzy	ADJ
iajs-1241	212	13	maximal	maximal	ADJ
iajs-1241	212	14	ideal	ideal	NOUN
iajs-1241	212	15	of	of	ADP
iajs-1241	212	16	r2	r2	NOUN
iajs-1241	212	17	,	,	PUNCT
iajs-1241	212	18			NOUN
iajs-1241	212	19	i	i	NOUN
iajs-1241	212	20	=	=	NOUN
iajs-1241	212	21	1	1	NUM
iajs-1241	212	22	,	,	PUNCT
iajs-1241	212	23	2	2	NUM
iajs-1241	212	24	,	,	PUNCT
iajs-1241	212	25			PROPN
iajs-1241	212	26	,	,	PUNCT
iajs-1241	212	27	n	n	CCONJ
iajs-1241	212	28	by	by	ADP
iajs-1241	212	29	(	(	PUNCT
iajs-1241	212	30	th	th	NOUN
iajs-1241	212	31	.	.	PROPN
iajs-1241	212	32	3.2	3.2	NUM
iajs-1241	212	33	(	(	PUNCT
iajs-1241	212	34	1	1	NUM
iajs-1241	212	35	)	)	PUNCT
iajs-1241	212	36	)	)	PUNCT
iajs-1241	212	37	in(13	in(13	NOUN
iajs-1241	212	38	)	)	PUNCT
iajs-1241	212	39	and	and	CCONJ
iajs-1241	212	40	note	note	VERB
iajs-1241	212	41	1.5	1.5	NUM
iajs-1241	212	42	.	.	PUNCT
iajs-1241	213	1	hence	hence	ADV
iajs-1241	213	2	f	f	X
iajs-1241	213	3	(	(	PUNCT
iajs-1241	213	4	a	a	NOUN
iajs-1241	213	5	)	)	PUNCT
iajs-1241	213	6	is	be	AUX
iajs-1241	213	7	a	a	DET
iajs-1241	213	8	finite	finite	ADJ
iajs-1241	213	9	intersection	intersection	NOUN
iajs-1241	213	10	of	of	ADP
iajs-1241	213	11	fuzzy	fuzzy	ADJ
iajs-1241	213	12	maximal	maximal	ADJ
iajs-1241	213	13	ideals	ideal	NOUN
iajs-1241	213	14	.	.	PUNCT
iajs-1241	214	1	thus	thus	ADV
iajs-1241	214	2	f	f	X
iajs-1241	214	3	(	(	PUNCT
iajs-1241	214	4	a	a	NOUN
iajs-1241	214	5	)	)	PUNCT
iajs-1241	214	6	is	be	AUX
iajs-1241	214	7	a	a	DET
iajs-1241	214	8	fuzzy	fuzzy	ADJ
iajs-1241	214	9	semimaximal	semimaximal	ADJ
iajs-1241	214	10	ideal	ideal	NOUN
iajs-1241	214	11	of	of	ADP
iajs-1241	214	12	r2	r2	PROPN
iajs-1241	214	13	.	.	PUNCT
iajs-1241	215	1	theorem	theorem	VERB
iajs-1241	215	2	3.2	3.2	NUM
iajs-1241	215	3	:	:	PUNCT
iajs-1241	215	4	let	let	VERB
iajs-1241	215	5	r1	r1	PROPN
iajs-1241	215	6	,	,	PUNCT
iajs-1241	215	7	r2	r2	PROPN
iajs-1241	215	8	be	be	VERB
iajs-1241	215	9	two	two	NUM
iajs-1241	215	10	rings	ring	NOUN
iajs-1241	215	11	,	,	PUNCT
iajs-1241	215	12	let	let	VERB
iajs-1241	215	13	f	f	PRON
iajs-1241	215	14	:	:	PUNCT
iajs-1241	215	15	r1	r1	PROPN
iajs-1241	215	16			PROPN
iajs-1241	215	17	r2	r2	PROPN
iajs-1241	215	18	be	be	VERB
iajs-1241	215	19	an	an	DET
iajs-1241	215	20	epimorphisim	epimorphisim	NOUN
iajs-1241	215	21	.	.	PUNCT
iajs-1241	216	1	if	if	SCONJ
iajs-1241	216	2	b	b	PROPN
iajs-1241	216	3	is	be	AUX
iajs-1241	216	4	a	a	DET
iajs-1241	216	5	fuzzy	fuzzy	ADJ
iajs-1241	216	6	semimaximal	semimaximal	ADJ
iajs-1241	216	7	ideal	ideal	NOUN
iajs-1241	216	8	of	of	ADP
iajs-1241	216	9	r2	r2	PROPN
iajs-1241	216	10	,	,	PUNCT
iajs-1241	216	11	then	then	ADV
iajs-1241	216	12	f	f	PROPN
iajs-1241	216	13	-1	-1	INTJ
iajs-1241	216	14	(	(	PUNCT
iajs-1241	216	15	b	b	X
iajs-1241	216	16	)	)	PUNCT
iajs-1241	216	17	is	be	AUX
iajs-1241	216	18	a	a	DET
iajs-1241	216	19	fuzzy	fuzzy	ADJ
iajs-1241	216	20	semimaximal	semimaximal	ADJ
iajs-1241	216	21	ideal	ideal	NOUN
iajs-1241	216	22	of	of	ADP
iajs-1241	216	23	r1	r1	PROPN
iajs-1241	216	24	.	.	PUNCT
iajs-1241	217	1	proof	proof	NOUN
iajs-1241	217	2	.	.	PUNCT
iajs-1241	218	1	since	since	SCONJ
iajs-1241	218	2	b	b	PROPN
iajs-1241	218	3	is	be	AUX
iajs-1241	218	4	a	a	DET
iajs-1241	218	5	fuzzy	fuzzy	ADJ
iajs-1241	218	6	semimaximal	semimaximal	ADJ
iajs-1241	218	7	ideal	ideal	NOUN
iajs-1241	218	8	of	of	ADP
iajs-1241	218	9	r2	r2	PROPN
iajs-1241	218	10	,	,	PUNCT
iajs-1241	218	11	b	b	X
iajs-1241	218	12	=	=	SYM
iajs-1241	218	13	b1	b1	NOUN
iajs-1241	218	14			NOUN
iajs-1241	218	15	b2	b2	NOUN
iajs-1241	218	16			PUNCT
iajs-1241	218	17			X
iajs-1241	218	18	bn	bn	CCONJ
iajs-1241	218	19	,	,	PUNCT
iajs-1241	218	20	where	where	SCONJ
iajs-1241	218	21	bi	bi	NOUN
iajs-1241	218	22	is	be	AUX
iajs-1241	218	23	a	a	DET
iajs-1241	218	24	fuzzy	fuzzy	ADJ
iajs-1241	218	25	maximal	maximal	ADJ
iajs-1241	218	26	ideals	ideal	NOUN
iajs-1241	218	27	of	of	ADP
iajs-1241	218	28	r2	r2	NOUN
iajs-1241	218	29	,	,	PUNCT
iajs-1241	218	30	for	for	ADP
iajs-1241	218	31	all	all	DET
iajs-1241	218	32	i	i	PRON
iajs-1241	218	33	=	=	NOUN
iajs-1241	218	34	1	1	NUM
iajs-1241	218	35	,	,	PUNCT
iajs-1241	218	36	2	2	NUM
iajs-1241	218	37	,	,	PUNCT
iajs-1241	218	38			PROPN
iajs-1241	218	39	,	,	PUNCT
iajs-1241	218	40	n.	n.	NOUN
iajs-1241	218	41	but	but	CCONJ
iajs-1241	218	42	f	f	PROPN
iajs-1241	218	43	-1	-1	INTJ
iajs-1241	218	44	(	(	PUNCT
iajs-1241	218	45	b	b	X
iajs-1241	218	46	)	)	PUNCT
iajs-1241	218	47	=	=	SYM
iajs-1241	218	48	f	f	X
iajs-1241	219	1	-1	-1	PUNCT
iajs-1241	219	2	(	(	PUNCT
iajs-1241	219	3	b1	b1	NOUN
iajs-1241	219	4			ADJ
iajs-1241	219	5	b2	b2	NOUN
iajs-1241	219	6			NOUN
iajs-1241	219	7	bn	bn	NOUN
iajs-1241	219	8	)	)	PUNCT
iajs-1241	219	9	=	=	SYM
iajs-1241	220	1	f	f	PROPN
iajs-1241	220	2	-1	-1	INTJ
iajs-1241	220	3	(	(	PUNCT
iajs-1241	220	4	b1	b1	NOUN
iajs-1241	220	5	)	)	PUNCT
iajs-1241	220	6			PROPN
iajs-1241	220	7	f	f	X
iajs-1241	220	8	-1	-1	PRON
iajs-1241	220	9	(	(	PUNCT
iajs-1241	220	10	b2	b2	NOUN
iajs-1241	220	11	)	)	PUNCT
iajs-1241	220	12			ADP
iajs-1241	220	13			PROPN
iajs-1241	220	14	f	f	X
iajs-1241	220	15	-1	-1	PUNCT
iajs-1241	220	16	(	(	PUNCT
iajs-1241	220	17	bn	bn	NOUN
iajs-1241	220	18	)	)	PUNCT
iajs-1241	220	19	but	but	CCONJ
iajs-1241	220	20	for	for	ADP
iajs-1241	220	21	each	each	DET
iajs-1241	220	22	i	i	NOUN
iajs-1241	220	23	=	=	NOUN
iajs-1241	220	24	1	1	NUM
iajs-1241	220	25	,	,	PUNCT
iajs-1241	220	26	2	2	NUM
iajs-1241	220	27	,	,	PUNCT
iajs-1241	220	28			PROPN
iajs-1241	220	29	,	,	PUNCT
iajs-1241	220	30	n	n	CCONJ
iajs-1241	220	31	,	,	PUNCT
iajs-1241	220	32	f	f	PROPN
iajs-1241	220	33	-1	-1	PROPN
iajs-1241	220	34	(	(	PUNCT
iajs-1241	220	35	bi	bi	NOUN
iajs-1241	220	36	)	)	PUNCT
iajs-1241	220	37	is	be	AUX
iajs-1241	220	38	a	a	DET
iajs-1241	220	39	fuzzy	fuzzy	ADJ
iajs-1241	220	40	maximal	maximal	ADJ
iajs-1241	220	41	ideal	ideal	NOUN
iajs-1241	220	42	of	of	ADP
iajs-1241	220	43	r1	r1	PROPN
iajs-1241	220	44	by	by	ADP
iajs-1241	220	45	(	(	PUNCT
iajs-1241	220	46	th	th	X
iajs-1241	220	47	.	.	PUNCT
iajs-1241	220	48	3.2)(2	3.2)(2	NUM
iajs-1241	220	49	)	)	PUNCT
iajs-1241	220	50	in	in	ADP
iajs-1241	220	51	(	(	PUNCT
iajs-1241	220	52	13	13	NUM
iajs-1241	220	53	)	)	PUNCT
iajs-1241	220	54	and	and	CCONJ
iajs-1241	220	55	note	note	VERB
iajs-1241	220	56	1.5	1.5	NUM
iajs-1241	220	57	.	.	PUNCT
iajs-1241	221	1	hence	hence	ADV
iajs-1241	221	2	f	f	PROPN
iajs-1241	221	3	-1	-1	INTJ
iajs-1241	221	4	(	(	PUNCT
iajs-1241	221	5	b	b	X
iajs-1241	221	6	)	)	PUNCT
iajs-1241	221	7	is	be	AUX
iajs-1241	221	8	a	a	DET
iajs-1241	221	9	finite	finite	ADJ
iajs-1241	221	10	intersection	intersection	NOUN
iajs-1241	221	11	of	of	ADP
iajs-1241	221	12	fuzzy	fuzzy	ADJ
iajs-1241	221	13	maximal	maximal	ADJ
iajs-1241	221	14	ideals	ideal	NOUN
iajs-1241	221	15	.	.	PUNCT
iajs-1241	222	1	thus	thus	ADV
iajs-1241	222	2	f	f	X
iajs-1241	222	3	-1	-1	NOUN
iajs-1241	222	4	(	(	PUNCT
iajs-1241	222	5	b	b	X
iajs-1241	222	6	)	)	PUNCT
iajs-1241	222	7	is	be	AUX
iajs-1241	222	8	a	a	DET
iajs-1241	222	9	fuzzy	fuzzy	ADJ
iajs-1241	222	10	semimaximal	semimaximal	ADJ
iajs-1241	222	11	ideal	ideal	NOUN
iajs-1241	222	12	of	of	ADP
iajs-1241	222	13	r1	r1	PROPN
iajs-1241	222	14	.	.	PUNCT
iajs-1241	223	1	s.4	s.4	PROPN
iajs-1241	223	2	direct	direct	ADJ
iajs-1241	223	3	sum	sum	NOUN
iajs-1241	223	4	of	of	ADP
iajs-1241	223	5	fuzzy	fuzzy	ADJ
iajs-1241	223	6	semimaximal	semimaximal	ADJ
iajs-1241	223	7	ideals	ideal	NOUN
iajs-1241	223	8	ibn	ibn	PROPN
iajs-1241	223	9	alhaitham	alhaitham	PROPN
iajs-1241	223	10	j.	j.	PROPN
iajs-1241	223	11	for	for	ADP
iajs-1241	223	12	pure	pure	ADJ
iajs-1241	223	13	&	&	CCONJ
iajs-1241	223	14	appl	appl	PROPN
iajs-1241	223	15	.	.	PUNCT
iajs-1241	224	1	sci	sci	PROPN
iajs-1241	224	2	vol.22	vol.22	PROPN
iajs-1241	224	3	(	(	PUNCT
iajs-1241	224	4	2	2	NUM
iajs-1241	224	5	)	)	PUNCT
iajs-1241	224	6	2009	2009	NUM
iajs-1241	224	7	in	in	ADP
iajs-1241	224	8	this	this	DET
iajs-1241	224	9	section	section	NOUN
iajs-1241	224	10	,	,	PUNCT
iajs-1241	224	11	we	we	PRON
iajs-1241	224	12	turn	turn	VERB
iajs-1241	224	13	out	out	ADP
iajs-1241	224	14	attention	attention	NOUN
iajs-1241	224	15	to	to	PART
iajs-1241	224	16	study	study	VERB
iajs-1241	224	17	fuzzy	fuzzy	ADJ
iajs-1241	224	18	semimaximal	semimaximal	ADJ
iajs-1241	224	19	ideals	ideal	NOUN
iajs-1241	224	20	and	and	CCONJ
iajs-1241	224	21	direct	direct	ADJ
iajs-1241	224	22	sum	sum	NOUN
iajs-1241	224	23	.	.	PUNCT
iajs-1241	225	1	first	first	ADV
iajs-1241	225	2	we	we	PRON
iajs-1241	225	3	give	give	VERB
iajs-1241	225	4	the	the	DET
iajs-1241	225	5	following	follow	VERB
iajs-1241	225	6	lemmas	lemma	NOUN
iajs-1241	225	7	which	which	PRON
iajs-1241	225	8	are	be	AUX
iajs-1241	225	9	useful	useful	ADJ
iajs-1241	225	10	in	in	ADP
iajs-1241	225	11	our	our	PRON
iajs-1241	225	12	work	work	NOUN
iajs-1241	225	13	.	.	PUNCT
iajs-1241	226	1	lemma	lemma	PROPN
iajs-1241	226	2	4.1	4.1	NUM
iajs-1241	226	3	:	:	PUNCT
iajs-1241	226	4	let	let	VERB
iajs-1241	226	5	r1	r1	PROPN
iajs-1241	226	6	,	,	PUNCT
iajs-1241	226	7	r2	r2	PROPN
iajs-1241	226	8	be	be	VERB
iajs-1241	226	9	two	two	NUM
iajs-1241	226	10	rings	ring	NOUN
iajs-1241	226	11	,	,	PUNCT
iajs-1241	226	12	let	let	VERB
iajs-1241	226	13	a	a	PRON
iajs-1241	226	14	,	,	PUNCT
iajs-1241	226	15	b	b	NOUN
iajs-1241	226	16	be	be	AUX
iajs-1241	226	17	fuzzy	fuzzy	ADJ
iajs-1241	226	18	ideals	ideal	NOUN
iajs-1241	226	19	of	of	ADP
iajs-1241	226	20	r1	r1	NOUN
iajs-1241	226	21	,	,	PUNCT
iajs-1241	226	22	r2	r2	PROPN
iajs-1241	226	23	respectively	respectively	ADV
iajs-1241	226	24	.	.	PUNCT
iajs-1241	227	1	then	then	ADV
iajs-1241	227	2	ab	ab	PROPN
iajs-1241	227	3	is	be	AUX
iajs-1241	227	4	a	a	DET
iajs-1241	227	5	fuzzy	fuzzy	ADJ
iajs-1241	227	6	ideal	ideal	NOUN
iajs-1241	227	7	of	of	ADP
iajs-1241	227	8	r1r2	r1r2	PROPN
iajs-1241	227	9	,	,	PUNCT
iajs-1241	227	10	where	where	SCONJ
iajs-1241	227	11	(	(	PUNCT
iajs-1241	227	12	a	a	DET
iajs-1241	227	13			ADJ
iajs-1241	227	14	b)(a	b)(a	NOUN
iajs-1241	227	15	,	,	PUNCT
iajs-1241	227	16	b	b	NOUN
iajs-1241	227	17	)	)	PUNCT
iajs-1241	227	18	=	=	SYM
iajs-1241	227	19	min{a(a),b(b	min{a(a),b(b	PROPN
iajs-1241	227	20	)	)	PUNCT
iajs-1241	227	21	}	}	PUNCT
iajs-1241	227	22	,	,	PUNCT
iajs-1241	227	23	for	for	SCONJ
iajs-1241	227	24	all	all	DET
iajs-1241	227	25	(	(	PUNCT
iajs-1241	227	26	a	a	DET
iajs-1241	227	27	,	,	PUNCT
iajs-1241	227	28	b	b	NOUN
iajs-1241	227	29	)	)	PUNCT
iajs-1241	227	30			NOUN
iajs-1241	227	31	r1	r1	PROPN
iajs-1241	227	32			PROPN
iajs-1241	227	33	r2	r2	PROPN
iajs-1241	227	34	.	.	PUNCT
iajs-1241	228	1	proof	proof	NOUN
iajs-1241	228	2	.	.	PUNCT
iajs-1241	229	1	by	by	ADP
iajs-1241	229	2	using	use	VERB
iajs-1241	229	3	note	note	NOUN
iajs-1241	229	4	1.5	1.5	NUM
iajs-1241	229	5	and	and	CCONJ
iajs-1241	229	6	(	(	PUNCT
iajs-1241	229	7	th.2.4.1.8)(14	th.2.4.1.8)(14	NOUN
iajs-1241	229	8	)	)	PUNCT
iajs-1241	229	9	the	the	DET
iajs-1241	229	10	result	result	NOUN
iajs-1241	229	11	follows	follow	VERB
iajs-1241	229	12	directly	directly	ADV
iajs-1241	229	13	.	.	PUNCT
iajs-1241	230	1	lemma	lemma	PROPN
iajs-1241	230	2	4.2	4.2	NUM
iajs-1241	230	3	:	:	PUNCT
iajs-1241	230	4	let	let	VERB
iajs-1241	230	5	r1	r1	PROPN
iajs-1241	230	6	,	,	PUNCT
iajs-1241	230	7	r2	r2	PROPN
iajs-1241	230	8	be	be	VERB
iajs-1241	230	9	two	two	NUM
iajs-1241	230	10	rings	ring	NOUN
iajs-1241	230	11	,	,	PUNCT
iajs-1241	230	12	let	let	VERB
iajs-1241	230	13	a	a	PRON
iajs-1241	230	14	be	be	AUX
iajs-1241	230	15	a	a	DET
iajs-1241	230	16	fuzzy	fuzzy	ADJ
iajs-1241	230	17	ideals	ideal	NOUN
iajs-1241	230	18	of	of	ADP
iajs-1241	230	19	r1	r1	PROPN
iajs-1241	230	20			ADJ
iajs-1241	230	21	r2	r2	PROPN
iajs-1241	230	22	,	,	PUNCT
iajs-1241	230	23	then	then	ADV
iajs-1241	230	24	there	there	PRON
iajs-1241	230	25	exist	exist	VERB
iajs-1241	230	26	fuzzy	fuzzy	ADJ
iajs-1241	230	27	ideals	ideal	NOUN
iajs-1241	230	28	b1	b1	NOUN
iajs-1241	230	29	and	and	CCONJ
iajs-1241	230	30	b2	b2	NOUN
iajs-1241	230	31	of	of	ADP
iajs-1241	230	32	r1	r1	PROPN
iajs-1241	230	33	,	,	PUNCT
iajs-1241	230	34	r2	r2	PROPN
iajs-1241	230	35	respectively	respectively	ADV
iajs-1241	230	36	such	such	ADJ
iajs-1241	230	37	that	that	SCONJ
iajs-1241	230	38	a	a	DET
iajs-1241	230	39	=	=	SYM
iajs-1241	230	40	b1	b1	PROPN
iajs-1241	230	41			PROPN
iajs-1241	230	42	b2	b2	NOUN
iajs-1241	230	43	.	.	PUNCT
iajs-1241	231	1	proof	proof	NOUN
iajs-1241	231	2	.	.	PUNCT
iajs-1241	232	1	by	by	ADP
iajs-1241	232	2	using	use	VERB
iajs-1241	232	3	note	note	NOUN
iajs-1241	232	4	1.5	1.5	NUM
iajs-1241	232	5	and	and	CCONJ
iajs-1241	232	6	(	(	PUNCT
iajs-1241	232	7	th.2.4.1.9)(14	th.2.4.1.9)(14	NOUN
iajs-1241	232	8	)	)	PUNCT
iajs-1241	232	9	the	the	DET
iajs-1241	232	10	result	result	NOUN
iajs-1241	232	11	is	be	AUX
iajs-1241	232	12	obtained	obtain	VERB
iajs-1241	232	13	.	.	PUNCT
iajs-1241	233	1	lemma	lemma	PROPN
iajs-1241	233	2	4.3	4.3	NUM
iajs-1241	233	3	:	:	PUNCT
iajs-1241	233	4	if	if	SCONJ
iajs-1241	233	5	a	a	PRON
iajs-1241	233	6	and	and	CCONJ
iajs-1241	233	7	b	b	NOUN
iajs-1241	233	8	are	be	AUX
iajs-1241	233	9	fuzzy	fuzzy	ADJ
iajs-1241	233	10	ideals	ideal	NOUN
iajs-1241	233	11	of	of	ADP
iajs-1241	233	12	rings	ring	NOUN
iajs-1241	233	13	r1	r1	PROPN
iajs-1241	233	14	,	,	PUNCT
iajs-1241	233	15	r2	r2	PROPN
iajs-1241	233	16	respectively	respectively	ADV
iajs-1241	233	17	then	then	ADV
iajs-1241	233	18	(	(	PUNCT
iajs-1241	233	19	a	a	DET
iajs-1241	233	20			ADJ
iajs-1241	233	21	b)	b)	PROPN
iajs-1241	233	22	=	=	PROPN
iajs-1241	233	23	a	a	PROPN
iajs-1241	233	24			ADJ
iajs-1241	233	25	b.	b.	NOUN
iajs-1241	233	26	proof	proof	NOUN
iajs-1241	233	27	.	.	PUNCT
iajs-1241	234	1	let	let	VERB
iajs-1241	234	2	(	(	PUNCT
iajs-1241	234	3	x	x	NOUN
iajs-1241	234	4	,	,	PUNCT
iajs-1241	234	5	y	y	PROPN
iajs-1241	234	6	)	)	PUNCT
iajs-1241	234	7			NOUN
iajs-1241	234	8	(	(	PUNCT
iajs-1241	234	9	a	a	DET
iajs-1241	234	10			PROPN
iajs-1241	234	11	b)	b)	NOUN
iajs-1241	234	12	,	,	PUNCT
iajs-1241	234	13	then	then	ADV
iajs-1241	234	14	(	(	PUNCT
iajs-1241	234	15	a	a	DET
iajs-1241	234	16			ADJ
iajs-1241	234	17	b)(x	b)(x	PROPN
iajs-1241	234	18	,	,	PUNCT
iajs-1241	234	19	y	y	PROPN
iajs-1241	234	20	)	)	PUNCT
iajs-1241	234	21	=	=	SYM
iajs-1241	234	22	1	1	NUM
iajs-1241	234	23	and	and	CCONJ
iajs-1241	234	24	so	so	ADV
iajs-1241	234	25	min{a(x),b(y	min{a(x),b(y	ADJ
iajs-1241	234	26	)	)	PUNCT
iajs-1241	234	27	}	}	PUNCT
iajs-1241	234	28	=	=	SYM
iajs-1241	235	1	1	1	X
iajs-1241	235	2	.	.	PUNCT
iajs-1241	235	3	this	this	PRON
iajs-1241	235	4	implies	imply	VERB
iajs-1241	235	5	that	that	SCONJ
iajs-1241	235	6	a(x	a(x	NOUN
iajs-1241	235	7	)	)	PUNCT
iajs-1241	235	8	=	=	SYM
iajs-1241	235	9	1	1	NUM
iajs-1241	235	10	,	,	PUNCT
iajs-1241	235	11	b(y	b(y	PROPN
iajs-1241	235	12	)	)	PUNCT
iajs-1241	235	13	=	=	SYM
iajs-1241	236	1	1	1	X
iajs-1241	236	2	.	.	PUNCT
iajs-1241	236	3	hence	hence	ADV
iajs-1241	236	4	x	x	SYM
iajs-1241	236	5			PROPN
iajs-1241	236	6	a	a	PROPN
iajs-1241	236	7	and	and	CCONJ
iajs-1241	236	8	y	y	PROPN
iajs-1241	236	9			PROPN
iajs-1241	236	10	b.	b.	VERB
iajs-1241	236	11	thus	thus	ADV
iajs-1241	236	12	(	(	PUNCT
iajs-1241	236	13	x	x	X
iajs-1241	236	14	,	,	PUNCT
iajs-1241	236	15	y	y	PROPN
iajs-1241	236	16	)	)	PUNCT
iajs-1241	236	17			NOUN
iajs-1241	236	18	a	a	PROPN
iajs-1241	236	19			PROPN
iajs-1241	236	20	b	b	NOUN
iajs-1241	236	21	,	,	PUNCT
iajs-1241	236	22	so	so	CCONJ
iajs-1241	236	23	(	(	PUNCT
iajs-1241	236	24	a	a	DET
iajs-1241	236	25			ADJ
iajs-1241	236	26	b)	b)	PROPN
iajs-1241	236	27			PROPN
iajs-1241	236	28	a	a	PROPN
iajs-1241	236	29			ADJ
iajs-1241	236	30	b.	b.	NOUN
iajs-1241	236	31	conversely	conversely	ADV
iajs-1241	236	32	;	;	PUNCT
iajs-1241	236	33	let	let	VERB
iajs-1241	236	34	(	(	PUNCT
iajs-1241	236	35	x	x	NOUN
iajs-1241	236	36	,	,	PUNCT
iajs-1241	236	37	y	y	NOUN
iajs-1241	236	38	)	)	PUNCT
iajs-1241	236	39	a	a	PROPN
iajs-1241	236	40			ADJ
iajs-1241	236	41	b.	b.	NOUN
iajs-1241	237	1	then	then	ADV
iajs-1241	237	2	x	x	SYM
iajs-1241	237	3			PROPN
iajs-1241	237	4	a	a	PROPN
iajs-1241	237	5	and	and	CCONJ
iajs-1241	237	6	y	y	PROPN
iajs-1241	237	7			PROPN
iajs-1241	237	8	b.	b.	VERB
iajs-1241	237	9	hence	hence	ADV
iajs-1241	237	10	a(x	a(x	NOUN
iajs-1241	237	11	)	)	PUNCT
iajs-1241	237	12	=	=	SYM
iajs-1241	237	13	1	1	NUM
iajs-1241	237	14	,	,	PUNCT
iajs-1241	237	15	b(y)=1	b(y)=1	NUM
iajs-1241	237	16	.	.	PUNCT
iajs-1241	238	1	thus	thus	ADV
iajs-1241	238	2	min{a(x),b(y	min{a(x),b(y	NUM
iajs-1241	238	3	)	)	PUNCT
iajs-1241	238	4	}	}	PUNCT
iajs-1241	239	1	=	=	SYM
iajs-1241	239	2	1	1	NUM
iajs-1241	239	3	and	and	CCONJ
iajs-1241	239	4	so	so	ADV
iajs-1241	239	5	(	(	PUNCT
iajs-1241	239	6	a	a	PROPN
iajs-1241	239	7	b)(x	b)(x	PROPN
iajs-1241	239	8	,	,	PUNCT
iajs-1241	239	9	y	y	PROPN
iajs-1241	239	10	)	)	PUNCT
iajs-1241	239	11	=	=	SYM
iajs-1241	240	1	1	1	NUM
iajs-1241	240	2	;	;	PUNCT
iajs-1241	240	3	that	that	PRON
iajs-1241	240	4	is	is	ADV
iajs-1241	240	5	(	(	PUNCT
iajs-1241	240	6	x	x	X
iajs-1241	240	7	,	,	PUNCT
iajs-1241	240	8	y)(ab).	y)(ab).	PROPN
iajs-1241	240	9	thus	thus	ADV
iajs-1241	240	10	(	(	PUNCT
iajs-1241	240	11	a	a	DET
iajs-1241	240	12			ADJ
iajs-1241	240	13	b)	b)	PROPN
iajs-1241	240	14			PROPN
iajs-1241	240	15	a	a	PROPN
iajs-1241	240	16			ADJ
iajs-1241	240	17	b	b	NOUN
iajs-1241	240	18	and	and	CCONJ
iajs-1241	240	19	hence	hence	ADV
iajs-1241	240	20	(	(	PUNCT
iajs-1241	240	21	a	a	DET
iajs-1241	240	22			ADJ
iajs-1241	240	23	b)=a	b)=a	PROPN
iajs-1241	240	24			ADJ
iajs-1241	240	25	b.	b.	NOUN
iajs-1241	240	26	it	it	PRON
iajs-1241	240	27	is	be	AUX
iajs-1241	240	28	known	know	VERB
iajs-1241	240	29	that	that	SCONJ
iajs-1241	240	30	(	(	PUNCT
iajs-1241	240	31	see	see	VERB
iajs-1241	240	32	(	(	PUNCT
iajs-1241	240	33	15)p.53	15)p.53	NUM
iajs-1241	240	34	):	):	PUNCT
iajs-1241	240	35	if	if	SCONJ
iajs-1241	240	36	r1	r1	PROPN
iajs-1241	240	37	,	,	PUNCT
iajs-1241	240	38	r2	r2	PROPN
iajs-1241	240	39	be	be	VERB
iajs-1241	240	40	rings	ring	NOUN
iajs-1241	240	41	,	,	PUNCT
iajs-1241	240	42	r	r	NOUN
iajs-1241	240	43	=	=	SYM
iajs-1241	240	44	r1	r1	PROPN
iajs-1241	240	45			ADJ
iajs-1241	240	46	r2	r2	PROPN
iajs-1241	240	47	and	and	CCONJ
iajs-1241	240	48	a	a	PRON
iajs-1241	240	49	is	be	AUX
iajs-1241	240	50	an	an	DET
iajs-1241	240	51	ideal	ideal	NOUN
iajs-1241	240	52	of	of	ADP
iajs-1241	240	53	r	r	NOUN
iajs-1241	240	54	then	then	ADV
iajs-1241	240	55	a	a	PRON
iajs-1241	240	56	is	be	AUX
iajs-1241	240	57	a	a	DET
iajs-1241	240	58	maximal	maximal	ADJ
iajs-1241	240	59	ideal	ideal	NOUN
iajs-1241	240	60	of	of	ADP
iajs-1241	240	61	r	r	NOUN
iajs-1241	240	62	iff	iff	PROPN
iajs-1241	240	63	a	a	DET
iajs-1241	240	64	=	=	PUNCT
iajs-1241	240	65	a1	a1	NOUN
iajs-1241	240	66			ADJ
iajs-1241	240	67	r2	r2	PROPN
iajs-1241	240	68	or	or	CCONJ
iajs-1241	240	69	a	a	DET
iajs-1241	240	70	=	=	PROPN
iajs-1241	240	71	r1	r1	PROPN
iajs-1241	240	72			PROPN
iajs-1241	240	73	a2	a2	PROPN
iajs-1241	240	74	,	,	PUNCT
iajs-1241	240	75	where	where	SCONJ
iajs-1241	240	76	a1	a1	NOUN
iajs-1241	240	77	is	be	AUX
iajs-1241	240	78	a	a	DET
iajs-1241	240	79	maximal	maximal	ADJ
iajs-1241	240	80	ideal	ideal	NOUN
iajs-1241	240	81	of	of	ADP
iajs-1241	240	82	r1	r1	PROPN
iajs-1241	240	83	,	,	PUNCT
iajs-1241	240	84	a2	a2	PROPN
iajs-1241	240	85	is	be	AUX
iajs-1241	240	86	a	a	DET
iajs-1241	240	87	maximal	maximal	ADJ
iajs-1241	240	88	ideal	ideal	NOUN
iajs-1241	240	89	of	of	ADP
iajs-1241	240	90	r2	r2	PROPN
iajs-1241	240	91	.	.	PUNCT
iajs-1241	241	1	we	we	PRON
iajs-1241	241	2	generalize	generalize	VERB
iajs-1241	241	3	this	this	DET
iajs-1241	241	4	result	result	NOUN
iajs-1241	241	5	,	,	PUNCT
iajs-1241	241	6	to	to	ADP
iajs-1241	241	7	the	the	DET
iajs-1241	241	8	following	following	NOUN
iajs-1241	241	9	:	:	PUNCT
iajs-1241	241	10	lemma	lemma	PROPN
iajs-1241	241	11	4.4	4.4	NUM
iajs-1241	241	12	:	:	PUNCT
iajs-1241	241	13	let	let	VERB
iajs-1241	241	14	r1	r1	PROPN
iajs-1241	241	15	,	,	PUNCT
iajs-1241	241	16	r2	r2	PROPN
iajs-1241	241	17	be	be	VERB
iajs-1241	241	18	two	two	NUM
iajs-1241	241	19	rings	ring	NOUN
iajs-1241	241	20	,	,	PUNCT
iajs-1241	241	21	r	r	NOUN
iajs-1241	241	22	=	=	SYM
iajs-1241	241	23	r1	r1	PROPN
iajs-1241	241	24			ADJ
iajs-1241	241	25	r2	r2	PROPN
iajs-1241	241	26	and	and	CCONJ
iajs-1241	241	27	a	a	PRON
iajs-1241	241	28	is	be	AUX
iajs-1241	241	29	a	a	DET
iajs-1241	241	30	fuzzy	fuzzy	ADJ
iajs-1241	241	31	ideal	ideal	NOUN
iajs-1241	241	32	of	of	ADP
iajs-1241	241	33	r	r	NOUN
iajs-1241	241	34	then	then	ADV
iajs-1241	241	35	a	a	PRON
iajs-1241	241	36	is	be	AUX
iajs-1241	241	37	a	a	DET
iajs-1241	241	38	fuzzy	fuzzy	ADJ
iajs-1241	241	39	maximal	maximal	ADJ
iajs-1241	241	40	ideal	ideal	NOUN
iajs-1241	241	41	of	of	ADP
iajs-1241	241	42	r	r	NOUN
iajs-1241	241	43	if	if	SCONJ
iajs-1241	242	1	and	and	CCONJ
iajs-1241	242	2	only	only	ADV
iajs-1241	242	3	if	if	SCONJ
iajs-1241	242	4	either	either	CCONJ
iajs-1241	242	5	a	a	DET
iajs-1241	242	6	=	=	SYM
iajs-1241	242	7	b	b	PROPN
iajs-1241	242	8			ADJ
iajs-1241	242	9	r2	r2	PROPN
iajs-1241	242	10			NOUN
iajs-1241	242	11	,	,	PUNCT
iajs-1241	242	12	where	where	SCONJ
iajs-1241	242	13	b	b	NOUN
iajs-1241	242	14	is	be	AUX
iajs-1241	242	15	a	a	DET
iajs-1241	242	16	fuzzy	fuzzy	ADJ
iajs-1241	242	17	maximal	maximal	ADJ
iajs-1241	242	18	ideal	ideal	NOUN
iajs-1241	242	19	of	of	ADP
iajs-1241	242	20	r1	r1	NOUN
iajs-1241	242	21	or	or	CCONJ
iajs-1241	242	22	a	a	DET
iajs-1241	242	23	=	=	PROPN
iajs-1241	242	24	r1	r1	PROPN
iajs-1241	242	25			NOUN
iajs-1241	242	26			PROPN
iajs-1241	242	27	c	c	NOUN
iajs-1241	242	28	,	,	PUNCT
iajs-1241	242	29	where	where	SCONJ
iajs-1241	242	30	c	c	PROPN
iajs-1241	242	31	is	be	AUX
iajs-1241	242	32	a	a	DET
iajs-1241	242	33	fuzzy	fuzzy	ADJ
iajs-1241	242	34	maximal	maximal	ADJ
iajs-1241	242	35	ideal	ideal	NOUN
iajs-1241	242	36	of	of	ADP
iajs-1241	242	37	r2	r2	NOUN
iajs-1241	242	38	.	.	PUNCT
iajs-1241	243	1	proof	proof	NOUN
iajs-1241	243	2	.	.	PUNCT
iajs-1241	244	1	if	if	SCONJ
iajs-1241	244	2	a	a	PRON
iajs-1241	244	3	is	be	AUX
iajs-1241	244	4	a	a	DET
iajs-1241	244	5	fuzzy	fuzzy	ADJ
iajs-1241	244	6	maximal	maximal	ADJ
iajs-1241	244	7	ideal	ideal	NOUN
iajs-1241	244	8	of	of	ADP
iajs-1241	244	9	r.	r.	PROPN
iajs-1241	244	10	since	since	SCONJ
iajs-1241	244	11	a	a	PRON
iajs-1241	244	12	is	be	AUX
iajs-1241	244	13	a	a	DET
iajs-1241	244	14	fuzzy	fuzzy	ADJ
iajs-1241	244	15	ideal	ideal	NOUN
iajs-1241	244	16	of	of	ADP
iajs-1241	244	17	r	r	NOUN
iajs-1241	244	18	,	,	PUNCT
iajs-1241	244	19	so	so	ADV
iajs-1241	244	20	by	by	ADP
iajs-1241	244	21	lemma	lemma	PROPN
iajs-1241	244	22	4.2	4.2	NUM
iajs-1241	244	23	,	,	PUNCT
iajs-1241	244	24	a	a	DET
iajs-1241	244	25	=	=	SYM
iajs-1241	244	26	b	b	PROPN
iajs-1241	244	27			PROPN
iajs-1241	244	28	c	c	NOUN
iajs-1241	244	29	for	for	ADP
iajs-1241	244	30	some	some	DET
iajs-1241	244	31	fuzzy	fuzzy	ADJ
iajs-1241	244	32	ideals	ideal	NOUN
iajs-1241	244	33	b	b	NOUN
iajs-1241	244	34	and	and	CCONJ
iajs-1241	244	35	c	c	PROPN
iajs-1241	244	36	of	of	ADP
iajs-1241	244	37	r1	r1	PROPN
iajs-1241	244	38	,	,	PUNCT
iajs-1241	244	39	r2	r2	PROPN
iajs-1241	244	40	respectively	respectively	ADV
iajs-1241	244	41	.	.	PUNCT
iajs-1241	245	1	hence	hence	ADV
iajs-1241	245	2	a	a	PROPN
iajs-1241	245	3	=	=	PUNCT
iajs-1241	246	1	(	(	PUNCT
iajs-1241	246	2	b	b	PROPN
iajs-1241	246	3			PROPN
iajs-1241	246	4	c)	c)	PROPN
iajs-1241	246	5	=	=	SYM
iajs-1241	246	6	b	b	PROPN
iajs-1241	246	7			ADJ
iajs-1241	246	8	c	c	NOUN
iajs-1241	246	9	by	by	ADP
iajs-1241	246	10	lemma	lemma	PROPN
iajs-1241	246	11	4.3	4.3	NUM
iajs-1241	246	12	.	.	PUNCT
iajs-1241	247	1	then	then	ADV
iajs-1241	247	2	by	by	ADP
iajs-1241	247	3	(	(	PUNCT
iajs-1241	247	4	lemma	lemma	PROPN
iajs-1241	247	5	2.1.(3	2.1.(3	NUM
iajs-1241	247	6	)	)	PUNCT
iajs-1241	247	7	)	)	PUNCT
iajs-1241	247	8	,	,	PUNCT
iajs-1241	247	9	b	b	PROPN
iajs-1241	247	10			ADJ
iajs-1241	247	11	c	c	NOUN
iajs-1241	247	12	is	be	AUX
iajs-1241	247	13	a	a	DET
iajs-1241	247	14	maximal	maximal	ADJ
iajs-1241	247	15	ideal	ideal	NOUN
iajs-1241	247	16	.	.	PUNCT
iajs-1241	248	1	so	so	ADV
iajs-1241	248	2	either	either	CCONJ
iajs-1241	248	3	b	b	PROPN
iajs-1241	248	4			ADJ
iajs-1241	248	5	c	c	NOUN
iajs-1241	248	6	=	=	SYM
iajs-1241	248	7	r1	r1	PROPN
iajs-1241	248	8			ADJ
iajs-1241	248	9	c	c	NOUN
iajs-1241	248	10	or	or	CCONJ
iajs-1241	248	11	b	b	NOUN
iajs-1241	248	12			ADJ
iajs-1241	248	13	c	c	NOUN
iajs-1241	248	14	=	=	SYM
iajs-1241	248	15	b	b	PROPN
iajs-1241	248	16			ADJ
iajs-1241	248	17	r2	r2	PROPN
iajs-1241	248	18	.	.	PUNCT
iajs-1241	249	1	hat	hat	NOUN
iajs-1241	249	2	is	be	AUX
iajs-1241	249	3	either	either	CCONJ
iajs-1241	249	4	b	b	PROPN
iajs-1241	249	5	=	=	SYM
iajs-1241	249	6	r1	r1	PROPN
iajs-1241	249	7	or	or	CCONJ
iajs-1241	249	8	c	c	NOUN
iajs-1241	249	9	=	=	SYM
iajs-1241	249	10	r2	r2	PROPN
iajs-1241	249	11	.	.	PUNCT
iajs-1241	250	1	if	if	SCONJ
iajs-1241	250	2	b	b	PROPN
iajs-1241	250	3	=	=	SYM
iajs-1241	250	4	r1	r1	PROPN
iajs-1241	250	5	,	,	PUNCT
iajs-1241	250	6	then	then	ADV
iajs-1241	250	7	b	b	X
iajs-1241	250	8	=	=	PROPN
iajs-1241	250	9	r1	r1	PROPN
iajs-1241	250	10			NOUN
iajs-1241	250	11	.	.	PUNCT
iajs-1241	251	1	if	if	SCONJ
iajs-1241	251	2	c	c	NOUN
iajs-1241	251	3	=	=	SYM
iajs-1241	251	4	r2	r2	PROPN
iajs-1241	251	5	,	,	PUNCT
iajs-1241	251	6	then	then	ADV
iajs-1241	251	7	c	c	NOUN
iajs-1241	251	8	=	=	SYM
iajs-1241	251	9	r2	r2	PROPN
iajs-1241	251	10			X
iajs-1241	251	11	.	.	PUNCT
iajs-1241	252	1	hence	hence	ADV
iajs-1241	252	2	either	either	CCONJ
iajs-1241	252	3	b	b	PROPN
iajs-1241	252	4			PROPN
iajs-1241	252	5	c	c	NOUN
iajs-1241	252	6	=	=	SYM
iajs-1241	252	7	r1	r1	PROPN
iajs-1241	252	8			ADJ
iajs-1241	252	9	c	c	PROPN
iajs-1241	252	10	or	or	CCONJ
iajs-1241	252	11	b	b	NOUN
iajs-1241	252	12			ADJ
iajs-1241	252	13	c	c	NOUN
iajs-1241	252	14	=	=	SYM
iajs-1241	252	15	b	b	PROPN
iajs-1241	252	16			ADJ
iajs-1241	252	17	r2	r2	PROPN
iajs-1241	252	18			X
iajs-1241	252	19	.	.	PUNCT
iajs-1241	253	1	conversely	conversely	ADV
iajs-1241	253	2	;	;	PUNCT
iajs-1241	253	3	if	if	SCONJ
iajs-1241	253	4	a	a	DET
iajs-1241	253	5	=	=	SYM
iajs-1241	253	6	b	b	PROPN
iajs-1241	253	7			ADJ
iajs-1241	253	8	r2	r2	PROPN
iajs-1241	253	9			ADJ
iajs-1241	253	10	and	and	CCONJ
iajs-1241	253	11	b	b	NOUN
iajs-1241	253	12	is	be	AUX
iajs-1241	253	13	a	a	DET
iajs-1241	253	14	fuzzy	fuzzy	ADJ
iajs-1241	253	15	maximal	maximal	ADJ
iajs-1241	253	16	ideal	ideal	NOUN
iajs-1241	253	17	of	of	ADP
iajs-1241	253	18	r1	r1	PROPN
iajs-1241	253	19	.	.	PUNCT
iajs-1241	254	1	to	to	PART
iajs-1241	254	2	prove	prove	VERB
iajs-1241	254	3	a	a	PRON
iajs-1241	254	4	is	be	AUX
iajs-1241	254	5	a	a	DET
iajs-1241	254	6	fuzzy	fuzzy	ADJ
iajs-1241	254	7	maximal	maximal	ADJ
iajs-1241	254	8	ideal	ideal	NOUN
iajs-1241	254	9	of	of	ADP
iajs-1241	254	10	r.	r.	PROPN
iajs-1241	254	11	by	by	PROPN
iajs-1241	254	12	(	(	PUNCT
iajs-1241	254	13	lemma	lemma	PROPN
iajs-1241	254	14	4.3	4.3	NUM
iajs-1241	254	15	)	)	PUNCT
iajs-1241	254	16	,	,	PUNCT
iajs-1241	254	17	a	a	PROPN
iajs-1241	254	18	=	=	PUNCT
iajs-1241	254	19	b	b	PROPN
iajs-1241	254	20			PROPN
iajs-1241	254	21	(	(	PUNCT
iajs-1241	254	22	r2	r2	PROPN
iajs-1241	254	23			X
iajs-1241	254	24	)	)	PUNCT
iajs-1241	254	25			PROPN
iajs-1241	255	1	and	and	CCONJ
iajs-1241	255	2	so	so	ADV
iajs-1241	255	3	a	a	PROPN
iajs-1241	255	4	=	=	PUNCT
iajs-1241	255	5	b	b	PROPN
iajs-1241	255	6			ADJ
iajs-1241	255	7	r2	r2	PROPN
iajs-1241	255	8	.	.	PUNCT
iajs-1241	256	1	since	since	SCONJ
iajs-1241	256	2	b	b	PROPN
iajs-1241	256	3	is	be	AUX
iajs-1241	256	4	a	a	DET
iajs-1241	256	5	fuzzy	fuzzy	ADJ
iajs-1241	256	6	maximal	maximal	ADJ
iajs-1241	256	7	ideal	ideal	NOUN
iajs-1241	256	8	of	of	ADP
iajs-1241	256	9	r1	r1	PROPN
iajs-1241	256	10	,	,	PUNCT
iajs-1241	256	11	then	then	ADV
iajs-1241	256	12	by	by	ADP
iajs-1241	256	13	(	(	PUNCT
iajs-1241	256	14	lemma	lemma	PROPN
iajs-1241	256	15	2.1(3	2.1(3	NUM
iajs-1241	256	16	)	)	PUNCT
iajs-1241	256	17	)	)	PUNCT
iajs-1241	256	18	,	,	PUNCT
iajs-1241	256	19	b	b	PROPN
iajs-1241	256	20	is	be	AUX
iajs-1241	256	21	a	a	DET
iajs-1241	256	22	maximal	maximal	ADJ
iajs-1241	256	23	ideal	ideal	NOUN
iajs-1241	256	24	of	of	ADP
iajs-1241	256	25	r1	r1	PROPN
iajs-1241	256	26	.	.	PUNCT
iajs-1241	257	1	hence	hence	ADV
iajs-1241	257	2	b	b	VERB
iajs-1241	257	3			ADJ
iajs-1241	257	4	r2	r2	PROPN
iajs-1241	257	5	=	=	PROPN
iajs-1241	257	6	a	a	PROPN
iajs-1241	257	7	is	be	AUX
iajs-1241	257	8	a	a	DET
iajs-1241	257	9	maximal	maximal	ADJ
iajs-1241	257	10	ideal	ideal	NOUN
iajs-1241	257	11	of	of	ADP
iajs-1241	257	12	r.	r.	PROPN
iajs-1241	257	13	on	on	ADP
iajs-1241	257	14	the	the	DET
iajs-1241	257	15	other	other	ADJ
iajs-1241	257	16	hand	hand	NOUN
iajs-1241	257	17	,	,	PUNCT
iajs-1241	257	18	a(0,0	a(0,0	NOUN
iajs-1241	257	19	)	)	PUNCT
iajs-1241	257	20	=	=	SYM
iajs-1241	257	21	min{b(0	min{b(0	NOUN
iajs-1241	257	22	)	)	PUNCT
iajs-1241	257	23	,	,	PUNCT
iajs-1241	257	24	r2	r2	PROPN
iajs-1241	257	25			X
iajs-1241	257	26	(	(	PUNCT
iajs-1241	257	27	0	0	NUM
iajs-1241	257	28	)	)	PUNCT
iajs-1241	257	29	}	}	PUNCT
iajs-1241	257	30	.	.	PUNCT
iajs-1241	258	1	but	but	CCONJ
iajs-1241	258	2	b(0	b(0	NOUN
iajs-1241	258	3	)	)	PUNCT
iajs-1241	258	4	=	=	SYM
iajs-1241	258	5	1	1	NUM
iajs-1241	258	6	by	by	ADP
iajs-1241	258	7	(	(	PUNCT
iajs-1241	258	8	lemma	lemma	PROPN
iajs-1241	258	9	2.1(1	2.1(1	NUM
iajs-1241	258	10	)	)	PUNCT
iajs-1241	258	11	)	)	PUNCT
iajs-1241	258	12	,	,	PUNCT
iajs-1241	258	13	so	so	SCONJ
iajs-1241	258	14	a(0,0	a(0,0	NOUN
iajs-1241	258	15	)	)	PUNCT
iajs-1241	258	16	=	=	SYM
iajs-1241	258	17	min{1,1	min{1,1	NOUN
iajs-1241	258	18	}	}	PUNCT
iajs-1241	258	19	=	=	SYM
iajs-1241	259	1	1	1	X
iajs-1241	259	2	.	.	PUNCT
iajs-1241	259	3	then	then	ADV
iajs-1241	259	4	by	by	ADP
iajs-1241	259	5	(	(	PUNCT
iajs-1241	259	6	lemma.2.1(5	lemma.2.1(5	NOUN
iajs-1241	259	7	)	)	PUNCT
iajs-1241	259	8	)	)	PUNCT
iajs-1241	259	9	,	,	PUNCT
iajs-1241	259	10	a	a	PRON
iajs-1241	259	11	is	be	AUX
iajs-1241	259	12	a	a	DET
iajs-1241	259	13	fuzzy	fuzzy	ADJ
iajs-1241	259	14	maximal	maximal	ADJ
iajs-1241	259	15	ideal	ideal	NOUN
iajs-1241	259	16	of	of	ADP
iajs-1241	259	17	r.	r.	PROPN
iajs-1241	259	18	similarly	similarly	ADV
iajs-1241	259	19	,	,	PUNCT
iajs-1241	259	20	if	if	SCONJ
iajs-1241	259	21	a	a	DET
iajs-1241	259	22	=	=	PROPN
iajs-1241	259	23	r1	r1	PROPN
iajs-1241	259	24			NOUN
iajs-1241	259	25			PROPN
iajs-1241	259	26	c	c	NOUN
iajs-1241	259	27	,	,	PUNCT
iajs-1241	259	28	c	c	PROPN
iajs-1241	259	29	is	be	AUX
iajs-1241	259	30	a	a	DET
iajs-1241	259	31	fuzzy	fuzzy	ADJ
iajs-1241	259	32	maximal	maximal	ADJ
iajs-1241	259	33	ideal	ideal	NOUN
iajs-1241	259	34	of	of	ADP
iajs-1241	259	35	r2	r2	PROPN
iajs-1241	259	36	,	,	PUNCT
iajs-1241	259	37	then	then	ADV
iajs-1241	259	38	a	a	PRON
iajs-1241	259	39	is	be	AUX
iajs-1241	259	40	a	a	DET
iajs-1241	259	41	fuzzy	fuzzy	ADJ
iajs-1241	259	42	maximal	maximal	ADJ
iajs-1241	259	43	ideal	ideal	NOUN
iajs-1241	259	44	of	of	ADP
iajs-1241	259	45	r.	r.	PROPN
iajs-1241	259	46	now	now	ADV
iajs-1241	259	47	,	,	PUNCT
iajs-1241	259	48	we	we	PRON
iajs-1241	259	49	can	can	AUX
iajs-1241	259	50	give	give	VERB
iajs-1241	259	51	the	the	DET
iajs-1241	259	52	main	main	ADJ
iajs-1241	259	53	results	result	NOUN
iajs-1241	259	54	,	,	PUNCT
iajs-1241	259	55	first	first	ADV
iajs-1241	259	56	we	we	PRON
iajs-1241	259	57	have	have	VERB
iajs-1241	259	58	the	the	DET
iajs-1241	259	59	following	follow	VERB
iajs-1241	259	60	:	:	PUNCT
iajs-1241	259	61	ibn	ibn	NOUN
iajs-1241	259	62	alhaitham	alhaitham	NOUN
iajs-1241	259	63	j.	j.	PROPN
iajs-1241	259	64	for	for	ADP
iajs-1241	259	65	pure	pure	ADJ
iajs-1241	259	66	&	&	CCONJ
iajs-1241	259	67	appl	appl	PROPN
iajs-1241	259	68	.	.	PUNCT
iajs-1241	260	1	sci	sci	PROPN
iajs-1241	260	2	vol.22	vol.22	PROPN
iajs-1241	260	3	(	(	PUNCT
iajs-1241	260	4	2	2	NUM
iajs-1241	260	5	)	)	PUNCT
iajs-1241	260	6	2009	2009	NUM
iajs-1241	260	7	theorem	theorem	VERB
iajs-1241	260	8	4.5	4.5	NUM
iajs-1241	260	9	:	:	PUNCT
iajs-1241	260	10	let	let	VERB
iajs-1241	260	11	r1	r1	PROPN
iajs-1241	260	12	,	,	PUNCT
iajs-1241	260	13	r2	r2	PROPN
iajs-1241	260	14	be	be	VERB
iajs-1241	260	15	two	two	NUM
iajs-1241	260	16	rings	ring	NOUN
iajs-1241	260	17	,	,	PUNCT
iajs-1241	260	18	let	let	VERB
iajs-1241	260	19	r	r	NOUN
iajs-1241	260	20	=	=	SYM
iajs-1241	260	21	r1	r1	PROPN
iajs-1241	260	22			ADJ
iajs-1241	260	23	r2	r2	PROPN
iajs-1241	260	24	and	and	CCONJ
iajs-1241	260	25	a	a	PRON
iajs-1241	260	26	,	,	PUNCT
iajs-1241	260	27	b	b	NOUN
iajs-1241	260	28	be	be	AUX
iajs-1241	260	29	fuzzy	fuzzy	ADJ
iajs-1241	260	30	ideals	ideal	NOUN
iajs-1241	260	31	of	of	ADP
iajs-1241	260	32	r1	r1	NOUN
iajs-1241	260	33	,	,	PUNCT
iajs-1241	260	34	r2	r2	PROPN
iajs-1241	260	35	respectively	respectively	ADV
iajs-1241	260	36	.	.	PUNCT
iajs-1241	261	1	then	then	ADV
iajs-1241	261	2	(	(	PUNCT
iajs-1241	261	3	1	1	X
iajs-1241	261	4	)	)	PUNCT
iajs-1241	261	5	a	a	PRON
iajs-1241	261	6	is	be	AUX
iajs-1241	261	7	a	a	DET
iajs-1241	261	8	fuzzy	fuzzy	ADJ
iajs-1241	261	9	semimaximal	semimaximal	ADJ
iajs-1241	261	10	ideal	ideal	NOUN
iajs-1241	261	11	of	of	ADP
iajs-1241	261	12	r	r	NOUN
iajs-1241	261	13	if	if	SCONJ
iajs-1241	262	1	and	and	CCONJ
iajs-1241	262	2	only	only	ADV
iajs-1241	262	3	if	if	SCONJ
iajs-1241	262	4	a	a	DET
iajs-1241	262	5			ADJ
iajs-1241	262	6	r2	r2	PROPN
iajs-1241	262	7			X
iajs-1241	262	8	is	be	AUX
iajs-1241	262	9	a	a	DET
iajs-1241	262	10	fuzzy	fuzzy	ADJ
iajs-1241	262	11	semimaximal	semimaximal	ADJ
iajs-1241	262	12	ideal	ideal	NOUN
iajs-1241	262	13	of	of	ADP
iajs-1241	262	14	r.	r.	PROPN
iajs-1241	262	15	(	(	PUNCT
iajs-1241	262	16	2	2	NUM
iajs-1241	262	17	)	)	PUNCT
iajs-1241	262	18	b	b	NOUN
iajs-1241	262	19	is	be	AUX
iajs-1241	262	20	a	a	DET
iajs-1241	262	21	fuzzy	fuzzy	ADJ
iajs-1241	262	22	semimaximal	semimaximal	ADJ
iajs-1241	262	23	ideal	ideal	NOUN
iajs-1241	262	24	of	of	ADP
iajs-1241	262	25	r2	r2	PROPN
iajs-1241	262	26	if	if	SCONJ
iajs-1241	262	27	and	and	CCONJ
iajs-1241	262	28	only	only	ADV
iajs-1241	262	29	if	if	SCONJ
iajs-1241	262	30	r1	r1	PROPN
iajs-1241	262	31			VERB
iajs-1241	262	32			PROPN
iajs-1241	262	33	b	b	PROPN
iajs-1241	262	34	is	be	AUX
iajs-1241	262	35	a	a	DET
iajs-1241	262	36	fuzzy	fuzzy	ADJ
iajs-1241	262	37	semimaximal	semimaximal	ADJ
iajs-1241	262	38	ideal	ideal	NOUN
iajs-1241	262	39	of	of	ADP
iajs-1241	262	40	r.	r.	PROPN
iajs-1241	262	41	proof	proof	NOUN
iajs-1241	262	42	(	(	PUNCT
iajs-1241	262	43	1	1	NUM
iajs-1241	262	44	)	)	PUNCT
iajs-1241	262	45	.	.	PUNCT
iajs-1241	263	1	since	since	SCONJ
iajs-1241	263	2	a	a	PRON
iajs-1241	263	3	is	be	AUX
iajs-1241	263	4	a	a	DET
iajs-1241	263	5	fuzzy	fuzzy	ADJ
iajs-1241	263	6	semimaximal	semimaximal	ADJ
iajs-1241	263	7	ideal	ideal	NOUN
iajs-1241	263	8	of	of	ADP
iajs-1241	263	9	r1	r1	PROPN
iajs-1241	263	10	,	,	PUNCT
iajs-1241	264	1	a	a	PRON
iajs-1241	264	2	=	=	X
iajs-1241	264	3	1	1	NUM
iajs-1241	264	4			PROPN
iajs-1241	265	1	n	n	NUM
iajs-1241	266	1	i	i	PRON
iajs-1241	267	1	i	i	PRON
iajs-1241	267	2	,	,	PUNCT
iajs-1241	267	3	where	where	SCONJ
iajs-1241	267	4	ai	ai	NOUN
iajs-1241	267	5	is	be	AUX
iajs-1241	267	6	a	a	DET
iajs-1241	267	7	fuzzy	fuzzy	ADJ
iajs-1241	267	8	maximal	maximal	ADJ
iajs-1241	267	9	ideal	ideal	NOUN
iajs-1241	267	10	of	of	ADP
iajs-1241	267	11	r1	r1	PROPN
iajs-1241	267	12	,	,	PUNCT
iajs-1241	267	13	for	for	ADP
iajs-1241	267	14	all	all	DET
iajs-1241	267	15	i	i	PRON
iajs-1241	267	16	=	=	NOUN
iajs-1241	267	17	1	1	NUM
iajs-1241	267	18	,	,	PUNCT
iajs-1241	267	19	2	2	NUM
iajs-1241	267	20	,	,	PUNCT
iajs-1241	267	21			PROPN
iajs-1241	267	22	,	,	PUNCT
iajs-1241	267	23	n.	n.	PROPN
iajs-1241	267	24	hence	hence	ADV
iajs-1241	267	25	,	,	PUNCT
iajs-1241	267	26	a	a	DET
iajs-1241	267	27			ADJ
iajs-1241	267	28	r2	r2	NOUN
iajs-1241	267	29	=	=	SYM
iajs-1241	267	30	1	1	NUM
iajs-1241	268	1			PROPN
iajs-1241	268	2	n	n	PRON
iajs-1241	269	1	i	i	PRON
iajs-1241	269	2	i	i	PRON
iajs-1241	269	3			ADJ
iajs-1241	269	4	r2	r2	PROPN
iajs-1241	269	5			X
iajs-1241	269	6	=	=	PUNCT
iajs-1241	269	7	1	1	NUM
iajs-1241	270	1			PROPN
iajs-1241	270	2	n	n	PRON
iajs-1241	271	1	i	i	PRON
iajs-1241	271	2	i	i	PRON
iajs-1241	271	3			ADJ
iajs-1241	271	4	r2	r2	PROPN
iajs-1241	271	5	(	(	PUNCT
iajs-1241	271	6	)	)	PUNCT
iajs-1241	271	7	n	n	PROPN
iajs-1241	271	8	times	time	NOUN
iajs-1241	271	9			ADJ
iajs-1241	271	10	,	,	PUNCT
iajs-1241	271	11	since	since	SCONJ
iajs-1241	271	12	r2	r2	PROPN
iajs-1241	271	13			NOUN
iajs-1241	272	1	=	=	SYM
iajs-1241	272	2	r	r	NOUN
iajs-1241	272	3	r2	r2	NOUN
iajs-1241	272	4	2	2	NUM
iajs-1241	272	5			NOUN
iajs-1241	272	6			PUNCT
iajs-1241	272	7			PUNCT
iajs-1241	272	8	n	n	NUM
iajs-1241	272	9	times	time	NOUN
iajs-1241	272	10			X
iajs-1241	272	11			ADJ
iajs-1241	272	12	=	=	SYM
iajs-1241	272	13	r2	r2	PROPN
iajs-1241	272	14	1	1	NUM
iajs-1241	272	15	(	(	PUNCT
iajs-1241	272	16	)	)	PUNCT
iajs-1241	273	1			PROPN
iajs-1241	273	2			PROPN
iajs-1241	273	3			VERB
iajs-1241	273	4	n	n	CCONJ
iajs-1241	274	1	i	i	PRON
iajs-1241	275	1	i	i	PROPN
iajs-1241	275	2			ADJ
iajs-1241	275	3	,	,	PUNCT
iajs-1241	275	4	by	by	ADP
iajs-1241	275	5	lemma	lemma	PROPN
iajs-1241	275	6	2.4	2.4	NUM
iajs-1241	275	7	(	(	PUNCT
iajs-1241	275	8	16	16	NUM
iajs-1241	275	9	)	)	PUNCT
iajs-1241	275	10	but	but	CCONJ
iajs-1241	275	11	by	by	ADP
iajs-1241	275	12	lemma	lemma	PROPN
iajs-1241	275	13	4.4	4.4	NUM
iajs-1241	275	14	,	,	PUNCT
iajs-1241	275	15	ai	ai	VERB
iajs-1241	275	16			PROPN
iajs-1241	275	17	r2	r2	PROPN
iajs-1241	275	18			X
iajs-1241	275	19	is	be	AUX
iajs-1241	275	20	a	a	DET
iajs-1241	275	21	fuzzy	fuzzy	ADJ
iajs-1241	275	22	maximal	maximal	ADJ
iajs-1241	275	23	ideal	ideal	NOUN
iajs-1241	275	24	of	of	ADP
iajs-1241	275	25	r	r	NOUN
iajs-1241	275	26	,	,	PUNCT
iajs-1241	275	27	for	for	ADP
iajs-1241	275	28	all	all	DET
iajs-1241	275	29	i	i	PRON
iajs-1241	275	30	=	=	NOUN
iajs-1241	275	31	1	1	NUM
iajs-1241	275	32	,	,	PUNCT
iajs-1241	275	33	2	2	NUM
iajs-1241	275	34	,	,	PUNCT
iajs-1241	275	35			PROPN
iajs-1241	275	36	,	,	PUNCT
iajs-1241	275	37	n.	n.	NOUN
iajs-1241	275	38	thus	thus	ADV
iajs-1241	275	39	a	a	DET
iajs-1241	275	40			ADJ
iajs-1241	275	41	r2	r2	PROPN
iajs-1241	275	42			X
iajs-1241	275	43	is	be	AUX
iajs-1241	275	44	a	a	DET
iajs-1241	275	45	fuzzy	fuzzy	ADJ
iajs-1241	275	46	semimaximal	semimaximal	NOUN
iajs-1241	275	47	ideal	ideal	NOUN
iajs-1241	275	48	.	.	PUNCT
iajs-1241	276	1	conversely	conversely	ADV
iajs-1241	276	2	;	;	PUNCT
iajs-1241	276	3	if	if	SCONJ
iajs-1241	276	4	a	a	DET
iajs-1241	276	5			ADJ
iajs-1241	276	6	r2	r2	PROPN
iajs-1241	276	7			X
iajs-1241	276	8	is	be	AUX
iajs-1241	276	9	a	a	DET
iajs-1241	276	10	fuzzy	fuzzy	ADJ
iajs-1241	276	11	semimaximal	semimaximal	ADJ
iajs-1241	276	12	ideal	ideal	NOUN
iajs-1241	276	13	of	of	ADP
iajs-1241	276	14	r	r	NOUN
iajs-1241	276	15	,	,	PUNCT
iajs-1241	276	16	then	then	ADV
iajs-1241	276	17	a	a	DET
iajs-1241	276	18			ADJ
iajs-1241	276	19	r2	r2	NOUN
iajs-1241	276	20			X
iajs-1241	276	21	=	=	NOUN
iajs-1241	276	22	1	1	NUM
iajs-1241	276	23	d	d	NOUN
iajs-1241	276	24			VERB
iajs-1241	276	25	n	n	INTJ
iajs-1241	276	26	i	i	PRON
iajs-1241	277	1	i	i	PRON
iajs-1241	277	2	,	,	PUNCT
iajs-1241	277	3	where	where	SCONJ
iajs-1241	277	4	di	di	NOUN
iajs-1241	277	5	is	be	AUX
iajs-1241	277	6	a	a	DET
iajs-1241	277	7	fuzzy	fuzzy	ADJ
iajs-1241	277	8	maximal	maximal	ADJ
iajs-1241	277	9	ideal	ideal	NOUN
iajs-1241	277	10	of	of	ADP
iajs-1241	277	11	r	r	NOUN
iajs-1241	277	12	for	for	ADP
iajs-1241	277	13	all	all	DET
iajs-1241	277	14	i	i	PRON
iajs-1241	277	15	=	=	NOUN
iajs-1241	277	16	1	1	NUM
iajs-1241	277	17	,	,	PUNCT
iajs-1241	277	18	2	2	NUM
iajs-1241	277	19	,	,	PUNCT
iajs-1241	277	20			PROPN
iajs-1241	277	21	,	,	PUNCT
iajs-1241	277	22	n.	n.	NOUN
iajs-1241	277	23	by	by	ADP
iajs-1241	277	24	lemma	lemma	PROPN
iajs-1241	277	25	4.2	4.2	NUM
iajs-1241	277	26	,	,	PUNCT
iajs-1241	277	27	for	for	ADP
iajs-1241	277	28	each	each	DET
iajs-1241	277	29	i	i	NOUN
iajs-1241	277	30	=	=	NOUN
iajs-1241	277	31	1	1	NUM
iajs-1241	277	32	,	,	PUNCT
iajs-1241	277	33	2	2	NUM
iajs-1241	277	34	,	,	PUNCT
iajs-1241	277	35			PROPN
iajs-1241	277	36	,	,	PUNCT
iajs-1241	277	37	n	n	CCONJ
iajs-1241	277	38	,	,	PUNCT
iajs-1241	277	39	di	di	X
iajs-1241	277	40	=	=	SYM
iajs-1241	277	41	bi	bi	PROPN
iajs-1241	277	42			PROPN
iajs-1241	277	43	ci	ci	PROPN
iajs-1241	277	44	,	,	PUNCT
iajs-1241	277	45	where	where	SCONJ
iajs-1241	277	46	bi	bi	NOUN
iajs-1241	277	47	is	be	AUX
iajs-1241	277	48	a	a	DET
iajs-1241	277	49	fuzzy	fuzzy	ADJ
iajs-1241	277	50	ideal	ideal	NOUN
iajs-1241	277	51	of	of	ADP
iajs-1241	277	52	r1	r1	PROPN
iajs-1241	277	53	,	,	PUNCT
iajs-1241	277	54	ci	ci	PROPN
iajs-1241	277	55	is	be	AUX
iajs-1241	277	56	a	a	DET
iajs-1241	277	57	fuzzy	fuzzy	ADJ
iajs-1241	277	58	ideal	ideal	NOUN
iajs-1241	277	59	of	of	ADP
iajs-1241	277	60	r2	r2	PROPN
iajs-1241	277	61	.	.	PUNCT
iajs-1241	278	1	hence	hence	ADV
iajs-1241	278	2	a	a	DET
iajs-1241	278	3			ADJ
iajs-1241	278	4	r2	r2	NOUN
iajs-1241	278	5			X
iajs-1241	278	6	=	=	X
iajs-1241	278	7	1	1	NUM
iajs-1241	278	8	(	(	PUNCT
iajs-1241	278	9	c	c	NOUN
iajs-1241	278	10	)	)	PUNCT
iajs-1241	279	1			PROPN
iajs-1241	279	2			NOUN
iajs-1241	279	3			ADJ
iajs-1241	280	1	n	n	INTJ
iajs-1241	281	1	i	i	PRON
iajs-1241	282	1	i	i	PRON
iajs-1241	282	2	i	i	VERB
iajs-1241	282	3	=	=	PUNCT
iajs-1241	283	1	1	1	NUM
iajs-1241	283	2			NOUN
iajs-1241	283	3	n	n	CCONJ
iajs-1241	284	1	i	i	PRON
iajs-1241	284	2	i	i	PRON
iajs-1241	284	3			VERB
iajs-1241	284	4	1	1	NUM
iajs-1241	284	5	c	c	NOUN
iajs-1241	284	6			VERB
iajs-1241	285	1	n	n	CCONJ
iajs-1241	285	2	i	i	PRON
iajs-1241	286	1	i	i	PRON
iajs-1241	286	2	,	,	PUNCT
iajs-1241	286	3	by	by	ADP
iajs-1241	286	4	lemma2.4	lemma2.4	NOUN
iajs-1241	286	5	(	(	PUNCT
iajs-1241	286	6	16	16	NUM
iajs-1241	286	7	)	)	PUNCT
iajs-1241	286	8	it	it	PRON
iajs-1241	286	9	follows	follow	VERB
iajs-1241	286	10	that	that	SCONJ
iajs-1241	286	11	a	a	DET
iajs-1241	286	12	=	=	SYM
iajs-1241	286	13	1	1	NUM
iajs-1241	286	14			NOUN
iajs-1241	286	15	n	n	CCONJ
iajs-1241	286	16	i	i	PRON
iajs-1241	286	17	i	i	INTJ
iajs-1241	286	18	,	,	PUNCT
iajs-1241	286	19	1	1	NUM
iajs-1241	286	20	c	c	X
iajs-1241	286	21			VERB
iajs-1241	287	1	n	n	CCONJ
iajs-1241	288	1	i	i	PRON
iajs-1241	289	1	i	i	PRON
iajs-1241	289	2	=	=	SYM
iajs-1241	289	3	r2	r2	PROPN
iajs-1241	289	4			X
iajs-1241	289	5	.	.	PUNCT
iajs-1241	290	1	but	but	CCONJ
iajs-1241	290	2	1	1	NUM
iajs-1241	290	3	c	c	X
iajs-1241	290	4			VERB
iajs-1241	291	1	n	n	CCONJ
iajs-1241	292	1	i	i	PRON
iajs-1241	293	1	i	i	PRON
iajs-1241	293	2	=	=	SYM
iajs-1241	293	3	r2	r2	PROPN
iajs-1241	293	4			X
iajs-1241	293	5	implies	imply	VERB
iajs-1241	293	6	that	that	SCONJ
iajs-1241	293	7	ci	ci	NOUN
iajs-1241	293	8	=	=	PROPN
iajs-1241	293	9	r2	r2	PROPN
iajs-1241	293	10			NOUN
iajs-1241	293	11	,	,	PUNCT
iajs-1241	293	12			NOUN
iajs-1241	293	13	i	i	NOUN
iajs-1241	293	14	=	=	NOUN
iajs-1241	293	15	1	1	NUM
iajs-1241	293	16	,	,	PUNCT
iajs-1241	293	17	2	2	NUM
iajs-1241	293	18	,	,	PUNCT
iajs-1241	293	19			PROPN
iajs-1241	293	20	,	,	PUNCT
iajs-1241	293	21	n.	n.	PROPN
iajs-1241	293	22	hence	hence	ADV
iajs-1241	293	23	bi	bi	PROPN
iajs-1241	293	24			PROPN
iajs-1241	293	25	ci	ci	PROPN
iajs-1241	293	26	=	=	PUNCT
iajs-1241	293	27	bi	bi	PROPN
iajs-1241	293	28			PROPN
iajs-1241	293	29	r2	r2	PROPN
iajs-1241	293	30			ADJ
iajs-1241	293	31	;	;	PUNCT
iajs-1241	293	32	that	that	PRON
iajs-1241	293	33	is	be	AUX
iajs-1241	293	34	di	di	NOUN
iajs-1241	293	35	=	=	ADJ
iajs-1241	293	36	bi	bi	ADJ
iajs-1241	293	37			PROPN
iajs-1241	293	38	r2	r2	PROPN
iajs-1241	293	39			PROPN
iajs-1241	293	40	.	.	PUNCT
iajs-1241	294	1	then	then	ADV
iajs-1241	294	2	by	by	ADP
iajs-1241	294	3	lemma	lemma	PROPN
iajs-1241	294	4	4.4	4.4	NUM
iajs-1241	294	5	,	,	PUNCT
iajs-1241	294	6	bi	bi	NOUN
iajs-1241	294	7	is	be	AUX
iajs-1241	294	8	a	a	DET
iajs-1241	294	9	fuzzy	fuzzy	ADJ
iajs-1241	294	10	maximal	maximal	ADJ
iajs-1241	294	11	ideal	ideal	NOUN
iajs-1241	294	12	of	of	ADP
iajs-1241	294	13	r1	r1	PROPN
iajs-1241	294	14	and	and	CCONJ
iajs-1241	294	15	so	so	ADV
iajs-1241	294	16	a	a	DET
iajs-1241	294	17	=	=	SYM
iajs-1241	294	18	1	1	NUM
iajs-1241	294	19			NOUN
iajs-1241	294	20	n	n	CCONJ
iajs-1241	295	1	i	i	PRON
iajs-1241	295	2	i	i	PRON
iajs-1241	295	3	is	be	AUX
iajs-1241	295	4	a	a	DET
iajs-1241	295	5	fuzzy	fuzzy	ADJ
iajs-1241	295	6	semimaximal	semimaximal	ADJ
iajs-1241	295	7	ideal	ideal	NOUN
iajs-1241	295	8	of	of	ADP
iajs-1241	295	9	r1	r1	PROPN
iajs-1241	295	10	.	.	PUNCT
iajs-1241	296	1	(	(	PUNCT
iajs-1241	296	2	2	2	NUM
iajs-1241	296	3	)	)	PUNCT
iajs-1241	296	4	.	.	PUNCT
iajs-1241	297	1	the	the	DET
iajs-1241	297	2	proof	proof	NOUN
iajs-1241	297	3	is	be	AUX
iajs-1241	297	4	similarly	similarly	ADV
iajs-1241	297	5	.	.	PUNCT
iajs-1241	298	1	next	next	ADV
iajs-1241	298	2	,	,	PUNCT
iajs-1241	298	3	we	we	PRON
iajs-1241	298	4	can	can	AUX
iajs-1241	298	5	give	give	VERB
iajs-1241	298	6	the	the	DET
iajs-1241	298	7	following	following	NOUN
iajs-1241	298	8	:	:	PUNCT
iajs-1241	298	9	ibn	ibn	NOUN
iajs-1241	298	10	alhaitham	alhaitham	NOUN
iajs-1241	298	11	j.	j.	PROPN
iajs-1241	298	12	for	for	ADP
iajs-1241	298	13	pure	pure	ADJ
iajs-1241	298	14	&	&	CCONJ
iajs-1241	298	15	appl	appl	PROPN
iajs-1241	298	16	.	.	PUNCT
iajs-1241	299	1	sci	sci	PROPN
iajs-1241	299	2	vol.22	vol.22	PROPN
iajs-1241	299	3	(	(	PUNCT
iajs-1241	299	4	2	2	NUM
iajs-1241	299	5	)	)	PUNCT
iajs-1241	299	6	2009	2009	NUM
iajs-1241	299	7	theorem	theorem	VERB
iajs-1241	299	8	4.6	4.6	NUM
iajs-1241	299	9	:	:	PUNCT
iajs-1241	299	10	let	let	VERB
iajs-1241	299	11	r1	r1	PROPN
iajs-1241	299	12	,	,	PUNCT
iajs-1241	299	13	r2	r2	PROPN
iajs-1241	299	14	be	be	VERB
iajs-1241	299	15	two	two	NUM
iajs-1241	299	16	rings	ring	NOUN
iajs-1241	299	17	,	,	PUNCT
iajs-1241	299	18	let	let	VERB
iajs-1241	299	19	r	r	NOUN
iajs-1241	299	20	=	=	SYM
iajs-1241	299	21	r1	r1	PROPN
iajs-1241	299	22			ADJ
iajs-1241	299	23	r2	r2	PROPN
iajs-1241	299	24	and	and	CCONJ
iajs-1241	299	25	let	let	VERB
iajs-1241	299	26	a	a	PRON
iajs-1241	299	27	be	be	AUX
iajs-1241	299	28	a	a	DET
iajs-1241	299	29	fuzzy	fuzzy	ADJ
iajs-1241	299	30	ideal	ideal	NOUN
iajs-1241	299	31	of	of	ADP
iajs-1241	299	32	r.	r.	PROPN
iajs-1241	299	33	if	if	SCONJ
iajs-1241	299	34	a	a	PRON
iajs-1241	299	35	is	be	AUX
iajs-1241	299	36	a	a	DET
iajs-1241	299	37	fuzzy	fuzzy	ADJ
iajs-1241	299	38	semimaximal	semimaximal	NOUN
iajs-1241	299	39	of	of	ADP
iajs-1241	299	40	r	r	NOUN
iajs-1241	299	41	,	,	PUNCT
iajs-1241	299	42	then	then	ADV
iajs-1241	299	43	either	either	ADV
iajs-1241	299	44	:	:	PUNCT
iajs-1241	299	45	(	(	PUNCT
iajs-1241	299	46	1	1	X
iajs-1241	299	47	)	)	PUNCT
iajs-1241	299	48	there	there	PRON
iajs-1241	299	49	exists	exist	VERB
iajs-1241	299	50	fuzzy	fuzzy	ADJ
iajs-1241	299	51	semimaximal	semimaximal	ADJ
iajs-1241	299	52	ideals	ideal	NOUN
iajs-1241	299	53	b	b	NUM
iajs-1241	299	54	,	,	PUNCT
iajs-1241	299	55	c	c	PROPN
iajs-1241	299	56	of	of	ADP
iajs-1241	299	57	r1	r1	PROPN
iajs-1241	299	58	,	,	PUNCT
iajs-1241	299	59	r2	r2	PROPN
iajs-1241	299	60	respectively	respectively	ADV
iajs-1241	299	61	such	such	ADJ
iajs-1241	299	62	that	that	SCONJ
iajs-1241	299	63	a	a	DET
iajs-1241	299	64	=	=	SYM
iajs-1241	299	65	b	b	PROPN
iajs-1241	299	66			ADJ
iajs-1241	299	67	c	c	NOUN
iajs-1241	299	68	,	,	PUNCT
iajs-1241	299	69	or	or	CCONJ
iajs-1241	299	70	(	(	PUNCT
iajs-1241	299	71	2	2	X
iajs-1241	299	72	)	)	PUNCT
iajs-1241	299	73	there	there	PRON
iajs-1241	299	74	exists	exist	VERB
iajs-1241	299	75	a	a	DET
iajs-1241	299	76	fuzzy	fuzzy	ADJ
iajs-1241	299	77	semimaximal	semimaximal	NOUN
iajs-1241	299	78	ideal	ideal	PROPN
iajs-1241	299	79	b	b	PROPN
iajs-1241	299	80	of	of	ADP
iajs-1241	299	81	r1	r1	NOUN
iajs-1241	299	82	such	such	ADJ
iajs-1241	299	83	that	that	SCONJ
iajs-1241	299	84	a	a	DET
iajs-1241	299	85	=	=	SYM
iajs-1241	299	86	b	b	PROPN
iajs-1241	299	87			ADJ
iajs-1241	299	88	r2	r2	PROPN
iajs-1241	299	89			NOUN
iajs-1241	299	90	,	,	PUNCT
iajs-1241	299	91	or	or	CCONJ
iajs-1241	299	92	(	(	PUNCT
iajs-1241	299	93	3	3	X
iajs-1241	299	94	)	)	PUNCT
iajs-1241	299	95	there	there	PRON
iajs-1241	299	96	exists	exist	VERB
iajs-1241	299	97	a	a	DET
iajs-1241	299	98	fuzzy	fuzzy	ADJ
iajs-1241	299	99	semimaximal	semimaximal	ADJ
iajs-1241	299	100	ideal	ideal	NOUN
iajs-1241	299	101	c	c	PROPN
iajs-1241	299	102	of	of	ADP
iajs-1241	299	103	r2	r2	PROPN
iajs-1241	299	104	such	such	ADJ
iajs-1241	299	105	that	that	SCONJ
iajs-1241	299	106	a	a	DET
iajs-1241	299	107	=	=	PROPN
iajs-1241	299	108	r1	r1	PROPN
iajs-1241	299	109			ADJ
iajs-1241	299	110			PROPN
iajs-1241	299	111	c.	c.	PROPN
iajs-1241	299	112	proof	proof	NOUN
iajs-1241	299	113	.	.	PUNCT
iajs-1241	300	1	if	if	SCONJ
iajs-1241	300	2	a	a	PRON
iajs-1241	300	3	is	be	AUX
iajs-1241	300	4	a	a	DET
iajs-1241	300	5	fuzzy	fuzzy	ADJ
iajs-1241	300	6	semimaximal	semimaximal	ADJ
iajs-1241	300	7	ideal	ideal	NOUN
iajs-1241	300	8	of	of	ADP
iajs-1241	300	9	r	r	NOUN
iajs-1241	300	10	,	,	PUNCT
iajs-1241	300	11	then	then	ADV
iajs-1241	300	12	a	a	PRON
iajs-1241	300	13	=	=	SYM
iajs-1241	300	14	1	1	NUM
iajs-1241	300	15			PROPN
iajs-1241	300	16	n	n	NUM
iajs-1241	300	17	i	i	PRON
iajs-1241	300	18	i	i	PRON
iajs-1241	300	19	,	,	PUNCT
iajs-1241	300	20	where	where	SCONJ
iajs-1241	300	21	ai	ai	NOUN
iajs-1241	300	22	is	be	AUX
iajs-1241	300	23	a	a	DET
iajs-1241	300	24	fuzzy	fuzzy	ADJ
iajs-1241	300	25	maximal	maximal	ADJ
iajs-1241	300	26	ideal	ideal	NOUN
iajs-1241	300	27	of	of	ADP
iajs-1241	300	28	r.	r.	PROPN
iajs-1241	300	29	by	by	ADP
iajs-1241	300	30	lemma	lemma	PROPN
iajs-1241	300	31	4.4	4.4	NUM
iajs-1241	300	32	,	,	PUNCT
iajs-1241	300	33	for	for	ADP
iajs-1241	300	34	each	each	DET
iajs-1241	300	35	i	i	NOUN
iajs-1241	300	36	=	=	NOUN
iajs-1241	300	37	1	1	NUM
iajs-1241	300	38	,	,	PUNCT
iajs-1241	300	39	2	2	NUM
iajs-1241	300	40	,	,	PUNCT
iajs-1241	300	41			PROPN
iajs-1241	300	42	,	,	PUNCT
iajs-1241	300	43	n	n	CCONJ
iajs-1241	300	44	,	,	PUNCT
iajs-1241	300	45	either	either	CCONJ
iajs-1241	300	46	ai	ai	VERB
iajs-1241	300	47	=	=	ADJ
iajs-1241	300	48	bi	bi	ADJ
iajs-1241	300	49			PROPN
iajs-1241	300	50	r2	r2	PROPN
iajs-1241	300	51			ADJ
iajs-1241	300	52	or	or	CCONJ
iajs-1241	300	53	ai	ai	VERB
iajs-1241	300	54	=	=	PROPN
iajs-1241	300	55	r2	r2	PROPN
iajs-1241	300	56			X
iajs-1241	300	57			PROPN
iajs-1241	300	58	ci	ci	PROPN
iajs-1241	300	59	,	,	PUNCT
iajs-1241	300	60	where	where	SCONJ
iajs-1241	300	61	bi	bi	NOUN
iajs-1241	300	62	,	,	PUNCT
iajs-1241	300	63	ci	ci	PROPN
iajs-1241	300	64	are	be	AUX
iajs-1241	300	65	fuzzy	fuzzy	ADJ
iajs-1241	300	66	maximal	maximal	ADJ
iajs-1241	300	67	ideals	ideal	NOUN
iajs-1241	300	68	of	of	ADP
iajs-1241	300	69	r1	r1	NOUN
iajs-1241	300	70	,	,	PUNCT
iajs-1241	300	71	r2	r2	PROPN
iajs-1241	300	72	respectively	respectively	ADV
iajs-1241	300	73	.	.	PUNCT
iajs-1241	301	1	if	if	SCONJ
iajs-1241	301	2	ai	ai	VERB
iajs-1241	301	3	=	=	ADJ
iajs-1241	301	4	bi	bi	ADJ
iajs-1241	301	5			PROPN
iajs-1241	301	6	r2	r2	PROPN
iajs-1241	301	7			NOUN
iajs-1241	301	8	,	,	PUNCT
iajs-1241	301	9	for	for	ADP
iajs-1241	301	10	all	all	DET
iajs-1241	301	11	i	i	PRON
iajs-1241	301	12	=	=	NOUN
iajs-1241	301	13	1	1	NUM
iajs-1241	301	14	,	,	PUNCT
iajs-1241	301	15	2	2	NUM
iajs-1241	301	16	,	,	PUNCT
iajs-1241	301	17			PROPN
iajs-1241	301	18	,	,	PUNCT
iajs-1241	301	19	n	n	CCONJ
iajs-1241	301	20	,	,	PUNCT
iajs-1241	301	21	then	then	ADV
iajs-1241	301	22	a	a	PRON
iajs-1241	301	23	=	=	SYM
iajs-1241	301	24	1	1	NUM
iajs-1241	301	25			PROPN
iajs-1241	302	1	n	n	NUM
iajs-1241	302	2	i	i	PRON
iajs-1241	303	1	i	i	PRON
iajs-1241	303	2	=	=	PUNCT
iajs-1241	304	1	1	1	NUM
iajs-1241	304	2			NOUN
iajs-1241	304	3	n	n	CCONJ
iajs-1241	305	1	i	i	PRON
iajs-1241	306	1	i	i	PRON
iajs-1241	306	2			PROPN
iajs-1241	306	3	r2	r2	PROPN
iajs-1241	306	4			NOUN
iajs-1241	306	5	,	,	PUNCT
iajs-1241	306	6	putting	put	VERB
iajs-1241	306	7	1	1	NUM
iajs-1241	306	8			NOUN
iajs-1241	306	9	n	n	CCONJ
iajs-1241	307	1	i	i	PRON
iajs-1241	308	1	i	i	PRON
iajs-1241	308	2	=	=	SYM
iajs-1241	308	3	c	c	X
iajs-1241	308	4	,	,	PUNCT
iajs-1241	308	5	we	we	PRON
iajs-1241	308	6	get	get	VERB
iajs-1241	308	7	a	a	DET
iajs-1241	308	8	=	=	SYM
iajs-1241	308	9	b	b	PROPN
iajs-1241	308	10			ADJ
iajs-1241	308	11	r2	r2	PROPN
iajs-1241	308	12			ADJ
iajs-1241	308	13	and	and	CCONJ
iajs-1241	308	14	b	b	NOUN
iajs-1241	308	15	is	be	AUX
iajs-1241	308	16	a	a	DET
iajs-1241	308	17	fuzzy	fuzzy	ADJ
iajs-1241	308	18	semimaximal	semimaximal	ADJ
iajs-1241	308	19	ideal	ideal	NOUN
iajs-1241	308	20	of	of	ADP
iajs-1241	308	21	r1	r1	PROPN
iajs-1241	308	22	.	.	PUNCT
iajs-1241	309	1	if	if	SCONJ
iajs-1241	309	2	ai	ai	VERB
iajs-1241	309	3	=	=	VERB
iajs-1241	309	4	r1	r1	PROPN
iajs-1241	309	5			PROPN
iajs-1241	309	6			PROPN
iajs-1241	309	7	ci	ci	PROPN
iajs-1241	309	8	,	,	PUNCT
iajs-1241	309	9	for	for	ADP
iajs-1241	309	10	all	all	DET
iajs-1241	309	11	i	i	PRON
iajs-1241	309	12	=	=	NOUN
iajs-1241	309	13	1	1	NUM
iajs-1241	309	14	,	,	PUNCT
iajs-1241	309	15	2	2	NUM
iajs-1241	309	16	,	,	PUNCT
iajs-1241	309	17			PROPN
iajs-1241	309	18	,	,	PUNCT
iajs-1241	309	19	n	n	CCONJ
iajs-1241	309	20	,	,	PUNCT
iajs-1241	309	21	then	then	ADV
iajs-1241	309	22	a	a	DET
iajs-1241	309	23	=	=	PROPN
iajs-1241	309	24	r1	r1	PROPN
iajs-1241	309	25			NOUN
iajs-1241	309	26			PROPN
iajs-1241	309	27	c	c	NOUN
iajs-1241	309	28	,	,	PUNCT
iajs-1241	309	29	where	where	SCONJ
iajs-1241	309	30	c	c	NOUN
iajs-1241	309	31	=	=	SYM
iajs-1241	309	32	1	1	NUM
iajs-1241	309	33	c	c	X
iajs-1241	309	34			VERB
iajs-1241	309	35	n	n	CCONJ
iajs-1241	310	1	i	i	PRON
iajs-1241	310	2	i	i	PRON
iajs-1241	310	3	and	and	CCONJ
iajs-1241	310	4	c	c	PROPN
iajs-1241	310	5	is	be	AUX
iajs-1241	310	6	a	a	DET
iajs-1241	310	7	fuzzy	fuzzy	ADJ
iajs-1241	310	8	semimaximal	semimaximal	ADJ
iajs-1241	310	9	ideal	ideal	NOUN
iajs-1241	310	10	of	of	ADP
iajs-1241	310	11	r2	r2	PROPN
iajs-1241	310	12	.	.	PUNCT
iajs-1241	311	1	now	now	ADV
iajs-1241	311	2	if	if	SCONJ
iajs-1241	311	3	ai	ai	VERB
iajs-1241	311	4	=	=	ADJ
iajs-1241	311	5	bi	bi	ADJ
iajs-1241	311	6			PROPN
iajs-1241	311	7	r2	r2	PROPN
iajs-1241	311	8			NOUN
iajs-1241	311	9	,	,	PUNCT
iajs-1241	311	10	for	for	ADP
iajs-1241	311	11	some	some	DET
iajs-1241	311	12	i	i	NOUN
iajs-1241	311	13	=	=	NOUN
iajs-1241	311	14	1	1	NUM
iajs-1241	311	15	,	,	PUNCT
iajs-1241	311	16	2	2	NUM
iajs-1241	311	17	,	,	PUNCT
iajs-1241	311	18			PROPN
iajs-1241	311	19	,	,	PUNCT
iajs-1241	311	20	n.	n.	PROPN
iajs-1241	311	21	then	then	ADV
iajs-1241	311	22	without	without	ADP
iajs-1241	311	23	loss	loss	NOUN
iajs-1241	311	24	of	of	ADP
iajs-1241	311	25	generality	generality	NOUN
iajs-1241	311	26	,	,	PUNCT
iajs-1241	311	27	we	we	PRON
iajs-1241	311	28	can	can	AUX
iajs-1241	311	29	assume	assume	VERB
iajs-1241	311	30	that	that	SCONJ
iajs-1241	311	31	ai	ai	VERB
iajs-1241	311	32	=	=	ADJ
iajs-1241	311	33	bi	bi	ADJ
iajs-1241	311	34			PROPN
iajs-1241	311	35	r2	r2	PROPN
iajs-1241	311	36			NOUN
iajs-1241	311	37	,	,	PUNCT
iajs-1241	311	38	for	for	ADP
iajs-1241	311	39	some	some	DET
iajs-1241	311	40	i	i	NOUN
iajs-1241	311	41	=	=	NOUN
iajs-1241	311	42	1	1	NUM
iajs-1241	311	43	,	,	PUNCT
iajs-1241	311	44	2,	2,	NUM
iajs-1241	311	45	,	,	PUNCT
iajs-1241	311	46	k	k	PROPN
iajs-1241	311	47	,	,	PUNCT
iajs-1241	311	48	k	k	X
iajs-1241	311	49	<	<	X
iajs-1241	311	50	n	n	PROPN
iajs-1241	311	51	and	and	CCONJ
iajs-1241	311	52	ai=	ai=	PROPN
iajs-1241	311	53	r1	r1	PROPN
iajs-1241	311	54			ADJ
iajs-1241	311	55	ci	ci	NOUN
iajs-1241	311	56	,	,	PUNCT
iajs-1241	311	57	for	for	ADP
iajs-1241	311	58	all	all	DET
iajs-1241	311	59	i	i	PRON
iajs-1241	311	60	=	=	SYM
iajs-1241	311	61	k+1	k+1	X
iajs-1241	311	62	,	,	PUNCT
iajs-1241	311	63			PROPN
iajs-1241	311	64	,	,	PUNCT
iajs-1241	311	65	n.	n.	NOUN
iajs-1241	311	66	hence	hence	ADV
iajs-1241	312	1	1	1	NUM
iajs-1241	312	2			PROPN
iajs-1241	313	1	n	n	NUM
iajs-1241	314	1	i	i	PRON
iajs-1241	315	1	i	i	PRON
iajs-1241	316	1	=	=	PUNCT
iajs-1241	317	1	1	1	NUM
iajs-1241	318	1			PROPN
iajs-1241	318	2	k	k	PROPN
iajs-1241	319	1	i	i	PRON
iajs-1241	319	2	i	i	PRON
iajs-1241	319	3			PUNCT
iajs-1241	319	4	1	1	NUM
iajs-1241	319	5			ADJ
iajs-1241	319	6			NOUN
iajs-1241	319	7			PROPN
iajs-1241	319	8			PROPN
iajs-1241	319	9			PROPN
iajs-1241	320	1			ADJ
iajs-1241	320	2			NOUN
iajs-1241	320	3	n	n	VERB
iajs-1241	321	1	i	i	PRON
iajs-1241	321	2	i	i	VERB
iajs-1241	321	3	k	k	PROPN
iajs-1241	321	4	1	1	NUM
iajs-1241	322	1			PROPN
iajs-1241	322	2	n	n	NUM
iajs-1241	323	1	i	i	PRON
iajs-1241	323	2	i	i	PRON
iajs-1241	323	3	=	=	PUNCT
iajs-1241	323	4	r2	r2	PROPN
iajs-1241	323	5	1	1	NUM
iajs-1241	323	6	(	(	PUNCT
iajs-1241	323	7	)	)	PUNCT
iajs-1241	323	8			NUM
iajs-1241	323	9			NOUN
iajs-1241	323	10			PROPN
iajs-1241	324	1	k	k	PROPN
iajs-1241	324	2	i	i	PRON
iajs-1241	324	3	i	i	PRON
iajs-1241	324	4			VERB
iajs-1241	324	5	1	1	NUM
iajs-1241	324	6	1	1	NUM
iajs-1241	324	7	(	(	PUNCT
iajs-1241	324	8	c	c	NOUN
iajs-1241	324	9	)	)	PUNCT
iajs-1241	325	1			PROPN
iajs-1241	325	2			ADV
iajs-1241	325	3			PROPN
iajs-1241	325	4	n	n	PRON
iajs-1241	325	5	r	r	NOUN
iajs-1241	326	1	i	i	PRON
iajs-1241	326	2	i	i	PROPN
iajs-1241	326	3	k	k	PROPN
iajs-1241	326	4			X
iajs-1241	326	5	=	=	PROPN
iajs-1241	326	6	r2	r2	PROPN
iajs-1241	326	7	1	1	PROPN
iajs-1241	326	8			NOUN
iajs-1241	326	9			PROPN
iajs-1241	326	10			PROPN
iajs-1241	326	11			PROPN
iajs-1241	326	12			PROPN
iajs-1241	327	1			PROPN
iajs-1241	327	2			NOUN
iajs-1241	328	1	k	k	X
iajs-1241	328	2	i	i	PRON
iajs-1241	328	3	i	i	PRON
iajs-1241	328	4			VERB
iajs-1241	328	5			X
iajs-1241	328	6	1	1	NUM
iajs-1241	328	7	1	1	NUM
iajs-1241	328	8	c	c	X
iajs-1241	328	9			PROPN
iajs-1241	328	10			ADV
iajs-1241	328	11			NOUN
iajs-1241	328	12			PROPN
iajs-1241	328	13			VERB
iajs-1241	328	14			DET
iajs-1241	328	15			PROPN
iajs-1241	328	16			NOUN
iajs-1241	328	17	n	n	VERB
iajs-1241	328	18	r	r	NOUN
iajs-1241	329	1	i	i	PRON
iajs-1241	329	2	i	i	PROPN
iajs-1241	329	3	k	k	PROPN
iajs-1241	329	4			PROPN
iajs-1241	329	5	,	,	PUNCT
iajs-1241	329	6	by	by	ADP
iajs-1241	329	7	(	(	PUNCT
iajs-1241	329	8	lemma	lemma	PROPN
iajs-1241	329	9	2.4(16	2.4(16	NOUN
iajs-1241	329	10	)	)	PUNCT
iajs-1241	329	11	)	)	PUNCT
iajs-1241	330	1	=	=	PUNCT
iajs-1241	331	1	1	1	NUM
iajs-1241	331	2			NOUN
iajs-1241	331	3	k	k	PROPN
iajs-1241	332	1	i	i	PRON
iajs-1241	332	2	i	i	PRON
iajs-1241	332	3			VERB
iajs-1241	332	4	1	1	NUM
iajs-1241	332	5	c	c	PROPN
iajs-1241	332	6			PROPN
iajs-1241	332	7			PUNCT
iajs-1241	332	8	n	n	CCONJ
iajs-1241	333	1	i	i	PRON
iajs-1241	333	2	i	i	VERB
iajs-1241	334	1	k	k	VERB
iajs-1241	334	2	letting	let	VERB
iajs-1241	334	3	b	b	PROPN
iajs-1241	334	4	=	=	SYM
iajs-1241	334	5	1	1	NUM
iajs-1241	334	6			NOUN
iajs-1241	334	7	k	k	PROPN
iajs-1241	335	1	i	i	PRON
iajs-1241	335	2	i	i	PRON
iajs-1241	335	3	,	,	PUNCT
iajs-1241	335	4	c	c	NOUN
iajs-1241	335	5	=	=	SYM
iajs-1241	335	6	1	1	NUM
iajs-1241	335	7	c	c	X
iajs-1241	335	8			NOUN
iajs-1241	335	9			PUNCT
iajs-1241	335	10	n	n	CCONJ
iajs-1241	336	1	i	i	PRON
iajs-1241	336	2	i	i	VERB
iajs-1241	337	1	k	k	NOUN
iajs-1241	337	2	,	,	PUNCT
iajs-1241	337	3	then	then	ADV
iajs-1241	337	4	b	b	X
iajs-1241	337	5	,	,	PUNCT
iajs-1241	337	6	c	c	PROPN
iajs-1241	337	7	are	be	AUX
iajs-1241	337	8	fuzzy	fuzzy	ADJ
iajs-1241	337	9	semimaximal	semimaximal	NOUN
iajs-1241	337	10	of	of	ADP
iajs-1241	337	11	r1	r1	PROPN
iajs-1241	337	12	,	,	PUNCT
iajs-1241	337	13	r2	r2	PROPN
iajs-1241	337	14	respectively	respectively	ADV
iajs-1241	337	15	.	.	PUNCT
iajs-1241	338	1	thus	thus	ADV
iajs-1241	338	2	a	a	DET
iajs-1241	338	3	=	=	SYM
iajs-1241	338	4	b	b	PROPN
iajs-1241	338	5			PROPN
iajs-1241	338	6	c.	c.	NOUN
iajs-1241	338	7	remark	remark	NOUN
iajs-1241	338	8	4.7	4.7	NUM
iajs-1241	338	9	:	:	PUNCT
iajs-1241	338	10	the	the	DET
iajs-1241	338	11	converse	converse	NOUN
iajs-1241	338	12	of	of	ADP
iajs-1241	338	13	theorem	theorem	ADJ
iajs-1241	338	14	4.6	4.6	NUM
iajs-1241	338	15	is	be	AUX
iajs-1241	338	16	not	not	PART
iajs-1241	338	17	necessary	necessary	ADJ
iajs-1241	338	18	true	true	ADJ
iajs-1241	338	19	in	in	ADP
iajs-1241	338	20	general	general	ADJ
iajs-1241	338	21	,	,	PUNCT
iajs-1241	338	22	in	in	ADP
iajs-1241	338	23	fact	fact	NOUN
iajs-1241	338	24	when	when	SCONJ
iajs-1241	338	25	b	b	X
iajs-1241	338	26	,	,	PUNCT
iajs-1241	338	27	c	c	PROPN
iajs-1241	338	28	are	be	AUX
iajs-1241	338	29	fuzzy	fuzzy	ADJ
iajs-1241	338	30	semimaximal	semimaximal	ADJ
iajs-1241	338	31	ideals	ideal	NOUN
iajs-1241	338	32	of	of	ADP
iajs-1241	338	33	r1	r1	PROPN
iajs-1241	338	34			ADJ
iajs-1241	338	35	r2	r2	PROPN
iajs-1241	338	36	,	,	PUNCT
iajs-1241	338	37	then	then	ADV
iajs-1241	338	38	b	b	PROPN
iajs-1241	338	39			PROPN
iajs-1241	338	40	c	c	X
iajs-1241	338	41	=	=	PUNCT
iajs-1241	338	42	a	a	DET
iajs-1241	338	43	need	need	NOUN
iajs-1241	338	44	not	not	PART
iajs-1241	338	45	a	a	DET
iajs-1241	338	46	fuzzy	fuzzy	ADJ
iajs-1241	338	47	semimaximal	semimaximal	ADJ
iajs-1241	338	48	ideal	ideal	NOUN
iajs-1241	338	49	of	of	ADP
iajs-1241	338	50	r1	r1	PROPN
iajs-1241	338	51			ADJ
iajs-1241	338	52	r2	r2	PROPN
iajs-1241	338	53	.	.	PUNCT
iajs-1241	339	1	we	we	PRON
iajs-1241	339	2	can	can	AUX
iajs-1241	339	3	give	give	VERB
iajs-1241	339	4	the	the	DET
iajs-1241	339	5	following	follow	VERB
iajs-1241	339	6	example	example	NOUN
iajs-1241	339	7	:	:	PUNCT
iajs-1241	339	8	example	example	NOUN
iajs-1241	339	9	:	:	PUNCT
iajs-1241	339	10	let	let	VERB
iajs-1241	339	11	b	b	X
iajs-1241	339	12	,	,	PUNCT
iajs-1241	339	13	c	c	NOUN
iajs-1241	339	14	:	:	PUNCT
iajs-1241	339	15	z	z	NOUN
iajs-1241	339	16			X
iajs-1241	340	1	[	[	X
iajs-1241	340	2	0,1	0,1	NUM
iajs-1241	340	3	]	]	PUNCT
iajs-1241	340	4	defined	define	VERB
iajs-1241	340	5	by	by	ADP
iajs-1241	340	6	ibn	ibn	PROPN
iajs-1241	340	7	alhaitham	alhaitham	NOUN
iajs-1241	340	8	j.	j.	PROPN
iajs-1241	340	9	for	for	ADP
iajs-1241	340	10	pure	pure	ADJ
iajs-1241	340	11	&	&	CCONJ
iajs-1241	340	12	appl	appl	PROPN
iajs-1241	340	13	.	.	PUNCT
iajs-1241	341	1	sci	sci	PROPN
iajs-1241	341	2	vol.22	vol.22	PROPN
iajs-1241	341	3	(	(	PUNCT
iajs-1241	341	4	2	2	NUM
iajs-1241	341	5	)	)	PUNCT
iajs-1241	341	6	2009	2009	NUM
iajs-1241	341	7	1	1	NUM
iajs-1241	341	8	2	2	NUM
iajs-1241	341	9	,	,	PUNCT
iajs-1241	341	10	(	(	PUNCT
iajs-1241	341	11	)	)	PUNCT
iajs-1241	341	12	1	1	NUM
iajs-1241	341	13	otherwise	otherwise	ADV
iajs-1241	341	14	3	3	NUM
iajs-1241	341	15			NOUN
iajs-1241	341	16			NOUN
iajs-1241	341	17			NUM
iajs-1241	341	18			NOUN
iajs-1241	341	19			PROPN
iajs-1241	342	1			NUM
iajs-1241	342	2			NOUN
iajs-1241	342	3	x	x	PUNCT
iajs-1241	342	4	x	x	SYM
iajs-1241	342	5	1	1	NUM
iajs-1241	342	6	3	3	NUM
iajs-1241	342	7	,	,	PUNCT
iajs-1241	342	8	c	c	X
iajs-1241	342	9	(	(	PUNCT
iajs-1241	342	10	)	)	PUNCT
iajs-1241	342	11	1	1	NUM
iajs-1241	342	12	otherwise	otherwise	ADV
iajs-1241	342	13	2	2	NUM
iajs-1241	342	14			NOUN
iajs-1241	342	15			PUNCT
iajs-1241	342	16			NUM
iajs-1241	342	17			NUM
iajs-1241	342	18			NUM
iajs-1241	342	19			NOUN
iajs-1241	342	20	x	x	X
iajs-1241	342	21	x	x	SYM
iajs-1241	342	22	b	b	PROPN
iajs-1241	342	23	,	,	PUNCT
iajs-1241	342	24	c	c	PROPN
iajs-1241	342	25	are	be	AUX
iajs-1241	342	26	fuzzy	fuzzy	ADJ
iajs-1241	342	27	semimaximal	semimaximal	ADJ
iajs-1241	342	28	ideals	ideal	NOUN
iajs-1241	342	29	of	of	ADP
iajs-1241	342	30	z	z	NOUN
iajs-1241	342	31	since	since	SCONJ
iajs-1241	342	32	b	b	PROPN
iajs-1241	342	33	,	,	PUNCT
iajs-1241	342	34	c	c	PROPN
iajs-1241	342	35	are	be	AUX
iajs-1241	342	36	fuzzy	fuzzy	ADJ
iajs-1241	342	37	maximal	maximal	ADJ
iajs-1241	342	38	ideals	ideal	NOUN
iajs-1241	342	39	of	of	ADP
iajs-1241	342	40	z.	z.	PROPN
iajs-1241	342	41	on	on	ADP
iajs-1241	342	42	the	the	DET
iajs-1241	342	43	other	other	ADJ
iajs-1241	342	44	hand	hand	NOUN
iajs-1241	342	45	,	,	PUNCT
iajs-1241	342	46	b	b	PROPN
iajs-1241	342	47			ADJ
iajs-1241	342	48	c	c	NOUN
iajs-1241	342	49	:	:	PUNCT
iajs-1241	342	50	z	z	PROPN
iajs-1241	342	51			PROPN
iajs-1241	342	52	z	z	PROPN
iajs-1241	342	53			PROPN
iajs-1241	343	1	[	[	X
iajs-1241	343	2	0,1	0,1	NUM
iajs-1241	343	3	]	]	PUNCT
iajs-1241	343	4	and	and	CCONJ
iajs-1241	343	5	1	1	NUM
iajs-1241	343	6	(	(	PUNCT
iajs-1241	343	7	,	,	PUNCT
iajs-1241	343	8	)	)	PUNCT
iajs-1241	343	9	2	2	NUM
iajs-1241	343	10	3	3	NUM
iajs-1241	343	11	,	,	PUNCT
iajs-1241	343	12	1	1	NUM
iajs-1241	343	13	(	(	PUNCT
iajs-1241	343	14	c	c	NOUN
iajs-1241	343	15	)	)	PUNCT
iajs-1241	343	16	(	(	PUNCT
iajs-1241	343	17	)	)	PUNCT
iajs-1241	343	18	(	(	PUNCT
iajs-1241	343	19	,	,	PUNCT
iajs-1241	343	20	)	)	PUNCT
iajs-1241	343	21	2	2	NUM
iajs-1241	343	22	(	(	PUNCT
iajs-1241	343	23	3	3	NUM
iajs-1241	343	24	)	)	PUNCT
iajs-1241	343	25	,	,	PUNCT
iajs-1241	343	26	2	2	NUM
iajs-1241	343	27	1	1	NUM
iajs-1241	343	28	otherwise	otherwise	ADV
iajs-1241	343	29	3	3	NUM
iajs-1241	343	30			ADV
iajs-1241	343	31			NOUN
iajs-1241	343	32			NUM
iajs-1241	343	33			NOUN
iajs-1241	343	34			NUM
iajs-1241	343	35			PRON
iajs-1241	343	36			PROPN
iajs-1241	343	37			PROPN
iajs-1241	343	38			PROPN
iajs-1241	343	39			NOUN
iajs-1241	343	40			NUM
iajs-1241	343	41			PROPN
iajs-1241	343	42			PROPN
iajs-1241	343	43			NOUN
iajs-1241	343	44			NUM
iajs-1241	343	45			NUM
iajs-1241	343	46			PROPN
iajs-1241	343	47	a	a	PROPN
iajs-1241	343	48	b	b	PROPN
iajs-1241	343	49	a	a	DET
iajs-1241	343	50	b	b	NOUN
iajs-1241	343	51	a	a	DET
iajs-1241	343	52	b	b	NOUN
iajs-1241	343	53	b	b	X
iajs-1241	343	54			PROPN
iajs-1241	343	55	c	c	NOUN
iajs-1241	343	56	is	be	AUX
iajs-1241	343	57	not	not	PART
iajs-1241	343	58	a	a	DET
iajs-1241	343	59	fuzzy	fuzzy	ADJ
iajs-1241	343	60	semimaximal	semimaximal	NOUN
iajs-1241	343	61	ideal	ideal	NOUN
iajs-1241	343	62	,	,	PUNCT
iajs-1241	343	63	since	since	SCONJ
iajs-1241	343	64	im	im	ADV
iajs-1241	343	65	(	(	PUNCT
iajs-1241	343	66	bc)	bc)	NOUN
iajs-1241	343	67	=	=	SYM
iajs-1241	343	68	3	3	X
iajs-1241	343	69	.	.	NUM
iajs-1241	343	70	references	reference	NOUN
iajs-1241	343	71	1	1	NUM
iajs-1241	343	72	.	.	PUNCT
iajs-1241	343	73	malik	malik	PROPN
iajs-1241	343	74	,	,	PUNCT
iajs-1241	343	75	d.s	d.s	PROPN
iajs-1241	343	76	.	.	PROPN
iajs-1241	343	77	and	and	CCONJ
iajs-1241	343	78	mordeson	mordeson	NOUN
iajs-1241	343	79	,	,	PUNCT
iajs-1241	343	80	j.n	j.n	PROPN
iajs-1241	343	81	.	.	PROPN
iajs-1241	343	82	,	,	PUNCT
iajs-1241	343	83	(	(	PUNCT
iajs-1241	343	84	1991	1991	NUM
iajs-1241	343	85	)	)	PUNCT
iajs-1241	343	86	,	,	PUNCT
iajs-1241	343	87	"	"	PUNCT
iajs-1241	343	88	fuzzy	fuzzy	ADJ
iajs-1241	343	89	maximal	maximal	ADJ
iajs-1241	343	90	,	,	PUNCT
iajs-1241	343	91	radical	radical	ADJ
iajs-1241	343	92	and	and	CCONJ
iajs-1241	343	93	primary	primary	ADJ
iajs-1241	343	94	ideals	ideal	NOUN
iajs-1241	343	95	of	of	ADP
iajs-1241	343	96	a	a	DET
iajs-1241	343	97	ring	ring	NOUN
iajs-1241	343	98	"	"	PUNCT
iajs-1241	343	99	,	,	PUNCT
iajs-1241	343	100	information	information	NOUN
iajs-1241	343	101	sciences	science	NOUN
iajs-1241	343	102	,	,	PUNCT
iajs-1241	343	103	53	53	NUM
iajs-1241	343	104	:	:	SYM
iajs-1241	343	105	237	237	NUM
iajs-1241	343	106	-	-	SYM
iajs-1241	343	107	250	250	NUM
iajs-1241	343	108	.	.	PUNCT
iajs-1241	344	1	2	2	X
iajs-1241	344	2	.	.	X
iajs-1241	344	3	goodreal	goodreal	NOUN
iajs-1241	344	4	,	,	PUNCT
iajs-1241	344	5	k.r	k.r	PROPN
iajs-1241	344	6	.	.	PROPN
iajs-1241	344	7	,	,	PUNCT
iajs-1241	344	8	(	(	PUNCT
iajs-1241	344	9	1979	1979	NUM
iajs-1241	344	10	)	)	PUNCT
iajs-1241	344	11	,	,	PUNCT
iajs-1241	344	12	"	"	PUNCT
iajs-1241	344	13	ring	ring	NOUN
iajs-1241	344	14	theorey	theorey	VERB
iajs-1241	344	15	-	-	PUNCT
iajs-1241	344	16	non	non	X
iajs-1241	344	17	singular	singular	NOUN
iajs-1241	344	18	ring	ring	NOUN
iajs-1241	344	19	and	and	CCONJ
iajs-1241	344	20	modules	module	NOUN
iajs-1241	344	21	"	"	PUNCT
iajs-1241	344	22	,	,	PUNCT
iajs-1241	344	23	marceidekker	marceidekker	PROPN
iajs-1241	344	24	,	,	PUNCT
iajs-1241	344	25	new	new	PROPN
iajs-1241	344	26	york	york	PROPN
iajs-1241	344	27	and	and	CCONJ
iajs-1241	344	28	basel	basel	PROPN
iajs-1241	344	29	.	.	PUNCT
iajs-1241	345	1	3	3	X
iajs-1241	345	2	.	.	X
iajs-1241	345	3	k.y.hatem	k.y.hatem	PROPN
iajs-1241	345	4	,	,	PUNCT
iajs-1241	345	5	(	(	PUNCT
iajs-1241	345	6	2007	2007	NUM
iajs-1241	345	7	)	)	PUNCT
iajs-1241	345	8	,	,	PUNCT
iajs-1241	345	9	"	"	PUNCT
iajs-1241	345	10	semimaximal	semimaximal	ADJ
iajs-1241	345	11	submodules	submodules	NOUN
iajs-1241	345	12	"	"	PUNCT
iajs-1241	345	13	,	,	PUNCT
iajs-1241	345	14	ph.d.thesis	ph.d.thesis	PROPN
iajs-1241	345	15	,	,	PUNCT
iajs-1241	345	16	university	university	NOUN
iajs-1241	345	17	of	of	ADP
iajs-1241	345	18	baghdad	baghdad	PROPN
iajs-1241	345	19	.	.	PUNCT
iajs-1241	346	1	4	4	NUM
iajs-1241	346	2	.	.	X
iajs-1241	347	1	zaheb	zaheb	PROPN
iajs-1241	347	2	,	,	PUNCT
iajs-1241	347	3	l.a	l.a	PROPN
iajs-1241	347	4	.	.	PROPN
iajs-1241	347	5	(	(	PUNCT
iajs-1241	347	6	1965	1965	NUM
iajs-1241	347	7	)	)	PUNCT
iajs-1241	347	8	,	,	PUNCT
iajs-1241	347	9	"	"	PUNCT
iajs-1241	347	10	fuzzy	fuzzy	ADJ
iajs-1241	347	11	sets	set	NOUN
iajs-1241	347	12	,	,	PUNCT
iajs-1241	347	13	information	information	NOUN
iajs-1241	347	14	and	and	CCONJ
iajs-1241	347	15	control	control	NOUN
iajs-1241	347	16	"	"	PUNCT
iajs-1241	347	17	,	,	PUNCT
iajs-1241	347	18	8:338	8:338	NUM
iajs-1241	347	19	-	-	SYM
iajs-1241	347	20	353	353	NUM
iajs-1241	347	21	.	.	NOUN
iajs-1241	347	22	5	5	NUM
iajs-1241	347	23	.	.	X
iajs-1241	347	24	zahedi	zahedi	PROPN
iajs-1241	347	25	,	,	PUNCT
iajs-1241	347	26	m.m	m.m	PROPN
iajs-1241	347	27	.	.	PROPN
iajs-1241	347	28	(	(	PUNCT
iajs-1241	347	29	1992	1992	NUM
iajs-1241	347	30	)	)	PUNCT
iajs-1241	347	31	,	,	PUNCT
iajs-1241	347	32	"	"	PUNCT
iajs-1241	347	33	on	on	ADP
iajs-1241	347	34	l	l	ADJ
iajs-1241	347	35	-	-	ADJ
iajs-1241	347	36	fuzzy	fuzzy	ADJ
iajs-1241	347	37	residual	residual	ADJ
iajs-1241	347	38	quotient	quotient	NOUN
iajs-1241	347	39	modules	module	NOUN
iajs-1241	347	40	and	and	CCONJ
iajs-1241	347	41	p.primary	p.primary	ADJ
iajs-1241	347	42	submodules	submodule	NOUN
iajs-1241	347	43	"	"	PUNCT
iajs-1241	347	44	,	,	PUNCT
iajs-1241	347	45	fuzzy	fuzzy	ADJ
iajs-1241	347	46	sets	set	NOUN
iajs-1241	347	47	and	and	CCONJ
iajs-1241	347	48	systems,51:333	systems,51:333	NOUN
iajs-1241	347	49	-	-	PUNCT
iajs-1241	347	50	344	344	NUM
iajs-1241	347	51	.	.	NOUN
iajs-1241	347	52	6	6	NUM
iajs-1241	347	53	.	.	X
iajs-1241	347	54	zahedi	zahedi	PROPN
iajs-1241	347	55	,	,	PUNCT
iajs-1241	347	56	m.m	m.m	PROPN
iajs-1241	347	57	,	,	PUNCT
iajs-1241	347	58	(	(	PUNCT
iajs-1241	347	59	1991	1991	NUM
iajs-1241	347	60	)	)	PUNCT
iajs-1241	347	61	,	,	PUNCT
iajs-1241	347	62	"	"	PUNCT
iajs-1241	347	63	a	a	DET
iajs-1241	347	64	characterization	characterization	NOUN
iajs-1241	347	65	of	of	ADP
iajs-1241	347	66	l	l	ADJ
iajs-1241	347	67	-	-	ADJ
iajs-1241	347	68	fuzzy	fuzzy	ADJ
iajs-1241	347	69	prime	prime	ADJ
iajs-1241	347	70	ideals	ideal	NOUN
iajs-1241	347	71	"	"	PUNCT
iajs-1241	347	72	,	,	PUNCT
iajs-1241	347	73	fuzzy	fuzzy	ADJ
iajs-1241	347	74	sets	set	NOUN
iajs-1241	347	75	and	and	CCONJ
iajs-1241	347	76	systems	system	NOUN
iajs-1241	347	77	,	,	PUNCT
iajs-1241	347	78	vol.44	vol.44	NOUN
iajs-1241	347	79	,	,	PUNCT
iajs-1241	347	80	pp.147	pp.147	PROPN
iajs-1241	347	81	-	-	PUNCT
iajs-1241	347	82	160	160	NUM
iajs-1241	347	83	.	.	PUNCT
iajs-1241	348	1	7	7	NUM
iajs-1241	348	2	.	.	X
iajs-1241	348	3	kumar	kumar	PROPN
iajs-1241	348	4	,	,	PUNCT
iajs-1241	348	5	r.	r.	PROPN
iajs-1241	348	6	(	(	PUNCT
iajs-1241	348	7	1991	1991	NUM
iajs-1241	348	8	)	)	PUNCT
iajs-1241	348	9	,	,	PUNCT
iajs-1241	348	10	"	"	PUNCT
iajs-1241	348	11	fuzzy	fuzzy	ADJ
iajs-1241	348	12	semiprimary	semiprimary	ADJ
iajs-1241	348	13	ideals	ideal	NOUN
iajs-1241	348	14	of	of	ADP
iajs-1241	348	15	ring	ring	NOUN
iajs-1241	348	16	"	"	PUNCT
iajs-1241	348	17	,	,	PUNCT
iajs-1241	348	18	fuzzy	fuzzy	ADJ
iajs-1241	348	19	sets	set	NOUN
iajs-1241	348	20	and	and	CCONJ
iajs-1241	348	21	systems	system	NOUN
iajs-1241	348	22	,	,	PUNCT
iajs-1241	348	23	42:263	42:263	NUM
iajs-1241	348	24	-	-	SYM
iajs-1241	348	25	272	272	NUM
iajs-1241	348	26	.	.	NOUN
iajs-1241	348	27	8	8	NUM
iajs-1241	348	28	.	.	X
iajs-1241	349	1	zhao	zhao	PROPN
iajs-1241	349	2	jiandi	jiandi	PROPN
iajs-1241	349	3	,	,	PUNCT
iajs-1241	349	4	shik	shik	PROPN
iajs-1241	349	5	.	.	PUNCT
iajs-1241	350	1	yue	yue	PROPN
iajs-1241	350	2	m.	m.	PROPN
iajs-1241	350	3	,	,	PUNCT
iajs-1241	350	4	(	(	PUNCT
iajs-1241	350	5	1993	1993	NUM
iajs-1241	350	6	)	)	PUNCT
iajs-1241	350	7	,	,	PUNCT
iajs-1241	350	8	"	"	PUNCT
iajs-1241	350	9	fuzzy	fuzzy	ADJ
iajs-1241	350	10	modules	module	NOUN
iajs-1241	350	11	over	over	ADP
iajs-1241	350	12	fuzzy	fuzzy	ADJ
iajs-1241	350	13	rings	ring	NOUN
iajs-1241	350	14	"	"	PUNCT
iajs-1241	350	15	,	,	PUNCT
iajs-1241	350	16	the	the	DET
iajs-1241	350	17	j.	j.	NOUN
iajs-1241	350	18	of	of	ADP
iajs-1241	350	19	fuzzy	fuzzy	ADJ
iajs-1241	350	20	math	math	NOUN
iajs-1241	350	21	.	.	PUNCT
iajs-1241	351	1	3	3	NUM
iajs-1241	351	2	:	:	PUNCT
iajs-1241	351	3	531	531	NUM
iajs-1241	351	4	-	-	SYM
iajs-1241	351	5	540	540	NUM
iajs-1241	351	6	.	.	PUNCT
iajs-1241	352	1	9	9	NUM
iajs-1241	352	2	.	.	X
iajs-1241	352	3	kumar	kumar	PROPN
iajs-1241	352	4	,	,	PUNCT
iajs-1241	352	5	r.	r.	PROPN
iajs-1241	352	6	,	,	PUNCT
iajs-1241	352	7	(	(	PUNCT
iajs-1241	352	8	1992	1992	NUM
iajs-1241	352	9	)	)	PUNCT
iajs-1241	352	10	,	,	PUNCT
iajs-1241	352	11	"	"	PUNCT
iajs-1241	352	12	fuzzy	fuzzy	ADJ
iajs-1241	352	13	cosets	coset	NOUN
iajs-1241	352	14	and	and	CCONJ
iajs-1241	352	15	some	some	DET
iajs-1241	352	16	fuzzy	fuzzy	ADJ
iajs-1241	352	17	radicals	radical	NOUN
iajs-1241	352	18	"	"	PUNCT
iajs-1241	352	19	,	,	PUNCT
iajs-1241	352	20	fuzzy	fuzzy	ADJ
iajs-1241	352	21	sets	set	NOUN
iajs-1241	352	22	and	and	CCONJ
iajs-1241	352	23	systems	system	NOUN
iajs-1241	352	24	,	,	PUNCT
iajs-1241	352	25	vol.46	vol.46	NOUN
iajs-1241	352	26	,	,	PUNCT
iajs-1241	352	27	pp.261	pp.261	PROPN
iajs-1241	352	28	-	-	PUNCT
iajs-1241	352	29	265	265	NUM
iajs-1241	352	30	.	.	PUNCT
iajs-1241	353	1	10	10	NUM
iajs-1241	353	2	.	.	PUNCT
iajs-1241	354	1	liu	liu	PROPN
iajs-1241	354	2	,	,	PUNCT
iajs-1241	354	3	w.j	w.j	PROPN
iajs-1241	354	4	,	,	PUNCT
iajs-1241	354	5	(	(	PUNCT
iajs-1241	354	6	1982	1982	NUM
iajs-1241	354	7	)	)	PUNCT
iajs-1241	354	8	,	,	PUNCT
iajs-1241	354	9	"	"	PUNCT
iajs-1241	354	10	fuzzy	fuzzy	ADJ
iajs-1241	354	11	invariant	invariant	ADJ
iajs-1241	354	12	subgroups	subgroup	NOUN
iajs-1241	354	13	and	and	CCONJ
iajs-1241	354	14	fuzzy	fuzzy	ADJ
iajs-1241	354	15	ideals	ideal	NOUN
iajs-1241	354	16	"	"	PUNCT
iajs-1241	354	17	,	,	PUNCT
iajs-1241	354	18	fuzzy	fuzzy	ADJ
iajs-1241	354	19	sets	set	NOUN
iajs-1241	354	20	and	and	CCONJ
iajs-1241	354	21	systems	system	NOUN
iajs-1241	354	22	,	,	PUNCT
iajs-1241	354	23	8:133	8:133	NUM
iajs-1241	354	24	-	-	SYM
iajs-1241	354	25	139	139	NUM
iajs-1241	354	26	.	.	PUNCT
iajs-1241	354	27	11	11	NUM
iajs-1241	354	28	.	.	X
iajs-1241	354	29	mordeson	mordeson	NOUN
iajs-1241	354	30	,	,	PUNCT
iajs-1241	354	31	j.n	j.n	PROPN
iajs-1241	354	32	.	.	PROPN
iajs-1241	354	33	(	(	PUNCT
iajs-1241	354	34	1996	1996	NUM
iajs-1241	354	35	)	)	PUNCT
iajs-1241	354	36	,	,	PUNCT
iajs-1241	354	37	"	"	PUNCT
iajs-1241	354	38	fuzzy	fuzzy	ADJ
iajs-1241	354	39	intersection	intersection	NOUN
iajs-1241	354	40	equations	equation	NOUN
iajs-1241	354	41	and	and	CCONJ
iajs-1241	354	42	primary	primary	ADJ
iajs-1241	354	43	representations	representation	NOUN
iajs-1241	354	44	"	"	PUNCT
iajs-1241	354	45	,	,	PUNCT
iajs-1241	354	46	fuzzy	fuzzy	ADJ
iajs-1241	354	47	sets	set	NOUN
iajs-1241	354	48	and	and	CCONJ
iajs-1241	354	49	systems	system	NOUN
iajs-1241	354	50	,	,	PUNCT
iajs-1241	354	51	83	83	NUM
iajs-1241	354	52	:	:	SYM
iajs-1241	354	53	93	93	NUM
iajs-1241	354	54	-	-	SYM
iajs-1241	354	55	98	98	NUM
iajs-1241	354	56	.	.	PUNCT
iajs-1241	355	1	ibn	ibn	PROPN
iajs-1241	355	2	alhaitham	alhaitham	PROPN
iajs-1241	355	3	j.	j.	PROPN
iajs-1241	355	4	for	for	ADP
iajs-1241	355	5	pure	pure	ADJ
iajs-1241	355	6	&	&	CCONJ
iajs-1241	355	7	appl	appl	PROPN
iajs-1241	355	8	.	.	PUNCT
iajs-1241	356	1	sci	sci	PROPN
iajs-1241	356	2	vol.22	vol.22	PROPN
iajs-1241	356	3	(	(	PUNCT
iajs-1241	356	4	2	2	NUM
iajs-1241	356	5	)	)	PUNCT
iajs-1241	356	6	2009	2009	NUM
iajs-1241	356	7	12	12	NUM
iajs-1241	356	8	.	.	PUNCT
iajs-1241	357	1	swamy	swamy	PROPN
iajs-1241	357	2	,	,	PUNCT
iajs-1241	357	3	k.l.n	k.l.n	NOUN
iajs-1241	357	4	.	.	PUNCT
iajs-1241	357	5	and	and	CCONJ
iajs-1241	357	6	swamy	swamy	PROPN
iajs-1241	357	7	,	,	PUNCT
iajs-1241	357	8	v.m	v.m	PROPN
iajs-1241	357	9	.	.	PROPN
iajs-1241	357	10	,	,	PUNCT
iajs-1241	357	11	(	(	PUNCT
iajs-1241	357	12	1988	1988	NUM
iajs-1241	357	13	)	)	PUNCT
iajs-1241	357	14	,	,	PUNCT
iajs-1241	357	15	"	"	PUNCT
iajs-1241	357	16	fuzzy	fuzzy	ADJ
iajs-1241	357	17	prime	prime	ADJ
iajs-1241	357	18	ideals	ideal	NOUN
iajs-1241	357	19	of	of	ADP
iajs-1241	357	20	rings	ring	NOUN
iajs-1241	357	21	,	,	PUNCT
iajs-1241	357	22	j.math	j.math	NOUN
iajs-1241	357	23	.	.	PUNCT
iajs-1241	358	1	anal.appl	anal.appl	NOUN
iajs-1241	358	2	,	,	PUNCT
iajs-1241	358	3	134:94	134:94	PROPN
iajs-1241	358	4	-	-	SYM
iajs-1241	358	5	103	103	NUM
iajs-1241	358	6	.	.	NOUN
iajs-1241	358	7	13	13	NUM
iajs-1241	358	8	.	.	PUNCT
iajs-1241	359	1	hadi	hadi	PROPN
iajs-1241	359	2	,	,	PUNCT
iajs-1241	359	3	i.m.a	i.m.a	NOUN
iajs-1241	359	4	.	.	PROPN
iajs-1241	359	5	,	,	PUNCT
iajs-1241	359	6	(	(	PUNCT
iajs-1241	359	7	2001	2001	NUM
iajs-1241	359	8	)	)	PUNCT
iajs-1241	359	9	,	,	PUNCT
iajs-1241	359	10	"	"	PUNCT
iajs-1241	359	11	on	on	ADP
iajs-1241	359	12	fuzzy	fuzzy	ADJ
iajs-1241	359	13	ideals	ideal	NOUN
iajs-1241	359	14	of	of	ADP
iajs-1241	359	15	fuzzy	fuzzy	ADJ
iajs-1241	359	16	rings	ring	NOUN
iajs-1241	359	17	"	"	PUNCT
iajs-1241	359	18	,	,	PUNCT
iajs-1241	359	19	math	math	NOUN
iajs-1241	359	20	.	.	PUNCT
iajs-1241	360	1	and	and	CCONJ
iajs-1241	360	2	physics	physics	PROPN
iajs-1241	360	3	j.	j.	PROPN
iajs-1241	360	4	16(4	16(4	PROPN
iajs-1241	360	5	):	):	PUNCT
iajs-1241	360	6	14	14	NUM
iajs-1241	360	7	.	.	PUNCT
iajs-1241	361	1	abu	abu	PROPN
iajs-1241	361	2	-	-	PUNCT
iajs-1241	361	3	dareb	dareb	PROPN
iajs-1241	361	4	,	,	PUNCT
iajs-1241	361	5	a.t.h	a.t.h	ADJ
iajs-1241	361	6	.	.	PROPN
iajs-1241	361	7	,	,	PUNCT
iajs-1241	361	8	(	(	PUNCT
iajs-1241	361	9	2000	2000	NUM
iajs-1241	361	10	)	)	PUNCT
iajs-1241	361	11	,	,	PUNCT
iajs-1241	361	12	"	"	PUNCT
iajs-1241	361	13	on	on	ADP
iajs-1241	361	14	qusi	qusi	ADJ
iajs-1241	361	15	-	-	PUNCT
iajs-1241	361	16	frobenius	frobenius	ADJ
iajs-1241	361	17	fuzzy	fuzzy	ADJ
iajs-1241	361	18	rings	ring	NOUN
iajs-1241	361	19	"	"	PUNCT
iajs-1241	361	20	,	,	PUNCT
iajs-1241	361	21	m.sc	m.sc	PROPN
iajs-1241	361	22	.	.	PUNCT
iajs-1241	362	1	thesis	thesis	NOUN
iajs-1241	362	2	,	,	PUNCT
iajs-1241	362	3	university	university	NOUN
iajs-1241	362	4	of	of	ADP
iajs-1241	362	5	baghdad	baghdad	PROPN
iajs-1241	362	6	.	.	PUNCT
iajs-1241	363	1	15	15	NUM
iajs-1241	363	2	.	.	X
iajs-1241	364	1	larsen	larsen	PROPN
iajs-1241	364	2	,	,	PUNCT
iajs-1241	364	3	m.d	m.d	PROPN
iajs-1241	364	4	.	.	PROPN
iajs-1241	364	5	and	and	CCONJ
iajs-1241	364	6	carthy	carthy	PROPN
iajs-1241	364	7	,	,	PUNCT
iajs-1241	364	8	p.j.mc	p.j.mc	PROPN
iajs-1241	364	9	,	,	PUNCT
iajs-1241	364	10	(	(	PUNCT
iajs-1241	364	11	1971	1971	NUM
iajs-1241	364	12	)	)	PUNCT
iajs-1241	364	13	,	,	PUNCT
iajs-1241	364	14	"	"	PUNCT
iajs-1241	364	15	multiplicative	multiplicative	ADJ
iajs-1241	364	16	theory	theory	NOUN
iajs-1241	364	17	of	of	ADP
iajs-1241	364	18	ideals	ideal	NOUN
iajs-1241	364	19	"	"	PUNCT
iajs-1241	364	20	,	,	PUNCT
iajs-1241	364	21	academic	academic	ADJ
iajs-1241	364	22	press	press	NOUN
iajs-1241	364	23	,	,	PUNCT
iajs-1241	364	24	new	new	PROPN
iajs-1241	364	25	york	york	PROPN
iajs-1241	364	26	.	.	PUNCT
iajs-1241	365	1	16	16	NUM
iajs-1241	365	2	.	.	PUNCT
iajs-1241	366	1	hadi	hadi	PROPN
iajs-1241	366	2	,	,	PUNCT
iajs-1241	366	3	i.m.a	i.m.a	NOUN
iajs-1241	366	4	.	.	PUNCT
iajs-1241	366	5	and	and	CCONJ
iajs-1241	366	6	abu	abu	PROPN
iajs-1241	366	7	-	-	PUNCT
iajs-1241	366	8	dareb	dareb	PROPN
iajs-1241	366	9	,	,	PUNCT
iajs-1241	366	10	a.t.h	a.t.h	ADJ
iajs-1241	366	11	.	.	PROPN
iajs-1241	366	12	,	,	PUNCT
iajs-1241	366	13	(	(	PUNCT
iajs-1241	366	14	2004	2004	NUM
iajs-1241	366	15	)	)	PUNCT
iajs-1241	366	16	,	,	PUNCT
iajs-1241	366	17	"	"	PUNCT
iajs-1241	366	18	p	p	X
iajs-1241	366	19	-	-	PUNCT
iajs-1241	366	20	f	f	NOUN
iajs-1241	366	21	fuzzy	fuzzy	ADJ
iajs-1241	366	22	rings	ring	NOUN
iajs-1241	366	23	and	and	CCONJ
iajs-1241	366	24	normal	normal	ADJ
iajs-1241	366	25	fuzzy	fuzzy	ADJ
iajs-1241	366	26	ring	ring	NOUN
iajs-1241	366	27	"	"	PUNCT
iajs-1241	366	28	,	,	PUNCT
iajs-1241	366	29	ibn	ibn	PROPN
iajs-1241	366	30	-	-	PUNCT
iajs-1241	366	31	al	al	PROPN
iajs-1241	366	32	-	-	PUNCT
iajs-1241	366	33	haitham	haitham	PROPN
iajs-1241	366	34	j.	j.	PROPN
iajs-1241	366	35	of	of	ADP
iajs-1241	366	36	pure	pure	ADJ
iajs-1241	366	37	and	and	CCONJ
iajs-1241	366	38	applies	apply	VERB
iajs-1241	366	39	sciences	science	NOUN
iajs-1241	366	40	,	,	PUNCT
iajs-1241	366	41	17(1	17(1	NUM
iajs-1241	366	42	):	):	PUNCT
iajs-1241	366	43	2002	2002	NUM
iajs-1241	366	44	(	(	PUNCT
iajs-1241	366	45	2	2	NUM
iajs-1241	366	46	)	)	PUNCT
iajs-1241	366	47	22مجلة	22مجلة	NUM
iajs-1241	366	48	ابن	ابن	VERB
iajs-1241	366	49	الهيثم	الهيثم	ADJ
iajs-1241	366	50	للعلوم	للعلوم	PROPN
iajs-1241	366	51	الصرفة	الصرفة	PROPN
iajs-1241	366	52	والتطبيقية	والتطبيقية	NOUN
iajs-1241	367	1	المجلد	المجلد	PROPN
iajs-1241	367	2	المـثاليات	المـثاليات	PROPN
iajs-1241	367	3	الضبابية	الضبابية	PROPN
iajs-1241	367	4	شبه	شبه	PROPN
iajs-1241	367	5	األعظمية	األعظمية	PROPN
iajs-1241	367	6	أنعام	أنعام	PROPN
iajs-1241	367	7	محمد	محمد	PROPN
iajs-1241	367	8	علي	علي	NOUN
iajs-1241	367	9	هادي	هادي	NOUN
iajs-1241	367	10	،	،	NOUN
iajs-1241	367	11	ميسون	ميسون	NOUN
iajs-1241	367	12	عبد	عبد	VERB
iajs-1241	367	13	هامل	هامل	NOUN
iajs-1241	367	14	ابن	ابن	PROPN
iajs-1241	367	15	الهيثم	الهيثم	PROPN
iajs-1241	367	16	،	،	PROPN
iajs-1241	367	17	جامعة	جامعة	PROPN
iajs-1241	367	18	بغداد	بغداد	PROPN
iajs-1241	367	19	-قسم	-قسم	SYM
iajs-1241	367	20	الرياضيات	الرياضيات	PROPN
iajs-1241	367	21	،	،	PROPN
iajs-1241	367	22	كلية	كلية	PROPN
iajs-1241	367	23	التربية	التربية	PROPN
iajs-1241	367	24	الخالصة	الخالصة	PROPN
iajs-1241	367	25	تقاطع	تقاطع	PROPN
iajs-1241	367	26	عدد	عدد	VERB
iajs-1241	367	27	منته	منته	PROPN
iajs-1241	367	28	من	من	NUM
iajs-1241	367	29	مثاليات	مثاليات	PROPN
iajs-1241	368	1	iيسمى	iيسمى	PROPN
iajs-1241	368	2	شبه	شبه	VERB
iajs-1241	368	3	أعظمي	أعظمي	VERB
iajs-1241	368	4	إذا	إذا	NUM
iajs-1241	368	5	كان	كان	NOUN
iajs-1241	368	6	rمثالي	rمثالي	ADJ
iajs-1241	368	7	فعلي	فعلي	NOUN
iajs-1241	368	8	في	في	ADP
iajs-1241	368	9	iالية	iالية	NOUN
iajs-1241	368	10	ذا	ذا	PRON
iajs-1241	368	11	محايد.حلقة	محايد.حلقة	NUM
iajs-1241	368	12	ابد	ابد	PRON
iajs-1241	368	13	rلتكن	rلتكن	NOUN
iajs-1241	368	14	rعلى	rعلى	PROPN
iajs-1241	368	15	aيكون	aيكون	PROPN
iajs-1241	368	16	المثالي	المثالي	PROPN
iajs-1241	368	17	الضبابي	الضبابي	PROPN
iajs-1241	368	18	اذ	اذ	PROPN
iajs-1241	368	19	،	،	PROPN
iajs-1241	368	20	rعظمى	rعظمى	PROPN
iajs-1241	368	21	.	.	PUNCT
iajs-1241	369	1	في	في	PRON
iajs-1241	369	2	هذا	هذا	NOUN
iajs-1241	369	3	البحث	البحث	NOUN
iajs-1241	369	4	قمنا	قمنا	ADV
iajs-1241	369	5	بتنصيب	بتنصيب	ADJ
iajs-1241	369	6	هذا	هذا	ADJ
iajs-1241	369	7	المفهوم	المفهوم	NOUN
iajs-1241	369	8	إلى	إلى	NOUN
iajs-1241	369	9	المثاليات	المثاليات	PROPN
iajs-1241	369	10	الضبابية	الضبابية	PROPN
iajs-1241	369	11	على	على	NOUN
iajs-1241	369	12	فضال	فضال	NOUN
iajs-1241	369	13	عن	عن	PROPN
iajs-1241	369	14	مثاليا	مثاليا	ADV
iajs-1241	369	15	شبه	شبه	VERB
iajs-1241	369	16	أعظمي	أعظمي	VERB
iajs-1241	369	17	إذا	إذا	NOUN
iajs-1241	369	18	كان	كان	NOUN
iajs-1241	369	19	تقاطع	تقاطع	ADV
iajs-1241	369	20	عدد	عدد	VERB
iajs-1241	369	21	منته	منته	ADJ
iajs-1241	369	22	من	من	PRON
iajs-1241	369	23	المثاليات	المثاليات	PROPN
iajs-1241	369	24	الضبابية	الضبابية	PROPN
iajs-1241	369	25	العظمى	العظمى	PROPN
iajs-1241	369	26	.	.	PUNCT
iajs-1241	370	1	خواص	خواص	ADV
iajs-1241	370	2	أساسية	أساسية	VERB
iajs-1241	370	3	مختلفة	مختلفة	PROPN
iajs-1241	370	4	قد	قد	INTJ
iajs-1241	370	5	أعطيت	أعطيت	PROPN
iajs-1241	370	6	قد	قد	PROPN
iajs-1241	370	7	أعطيت	أعطيت	PROPN
iajs-1241	370	8	بعض	بعض	NOUN
iajs-1241	370	9	األمثلة	األمثلة	PROPN
iajs-1241	370	10	لتوضيح	لتوضيح	NOUN
iajs-1241	370	11	هذا	هذا	NOUN
iajs-1241	370	12	المفهوم	المفهوم	NOUN
iajs-1241	370	13	.	.	PUNCT
iajs-1241	371	1	هذا	هذا	NOUN