id	sid	tid	token	lemma	pos
iajs-905	1	1	ibn	ibn	PROPN
iajs-905	1	2	alhaitham	alhaitham	NOUN
iajs-905	1	3	j.	j.	PROPN
iajs-905	1	4	for	for	ADP
iajs-905	1	5	pure	pure	ADJ
iajs-905	1	6	&	&	CCONJ
iajs-905	1	7	appl	appl	PROPN
iajs-905	1	8	.	.	PUNCT
iajs-905	2	1	sci	sci	PROPN
iajs-905	2	2	.	.	PUNCT
iajs-905	3	1	vol.23	vol.23	PROPN
iajs-905	3	2	(	(	PUNCT
iajs-905	3	3	3	3	NUM
iajs-905	3	4	)	)	PUNCT
iajs-905	3	5	2010	2010	NUM
iajs-905	3	6	on	on	ADP
iajs-905	3	7	fuzzy	fuzzy	ADJ
iajs-905	3	8	internal	internal	ADJ
iajs-905	3	9	direct	direct	ADJ
iajs-905	3	10	product	product	NOUN
iajs-905	3	11	nn	nn	PROPN
iajs-905	3	12	..	..	PROPN
iajs-905	3	13	mm	mm	PROPN
iajs-905	3	14	..	..	PUNCT
iajs-905	3	15	nnaammaa	nnaammaa	NOUN
iajs-905	3	16	..	..	PUNCT
iajs-905	3	17	ddee	ddee	NOUN
iajs-905	3	18	ppaarrttmmee	ppaarrttmmee	NOUN
iajs-905	3	19	nntt	nntt	VERB
iajs-905	3	20	ooff	ooff	ADJ
iajs-905	3	21	mmaatthhee	mmaatthhee	NOUN
iajs-905	3	22	mmaattiiccss	mmaattiiccss	PROPN
iajs-905	3	23	,	,	PUNCT
iajs-905	3	24	,	,	PUNCT
iajs-905	3	25	ccoollllee	ccoollllee	PROPN
iajs-905	3	26	ggee	ggee	PROPN
iajs-905	3	27	ooff	ooff	PROPN
iajs-905	3	28	sscciiee	sscciiee	VERB
iajs-905	3	29	nnccee	nnccee	ADJ
iajs-905	3	30	ooff	ooff	NOUN
iajs-905	3	31	wwoommeenn	wwoommeenn	NOUN
iajs-905	3	32	,	,	PUNCT
iajs-905	3	33	,	,	PUNCT
iajs-905	3	34	uunniivvee	uunniivvee	PROPN
iajs-905	3	35	rrss	rrss	VERB
iajs-905	3	36	iittyy	iittyy	ADJ
iajs-905	3	37	ooff	ooff	ADJ
iajs-905	3	38	bbaagghhddaa	bbaagghhddaa	NOUN
iajs-905	3	39	dd	dd	NOUN
iajs-905	3	40	..	..	PUNCT
iajs-905	3	41	abstract	abstract	ADJ
iajs-905	3	42	the	the	DET
iajs-905	3	43	main	main	ADJ
iajs-905	3	44	aim	aim	NOUN
iajs-905	3	45	of	of	ADP
iajs-905	3	46	this	this	DET
iajs-905	3	47	paper	paper	NOUN
iajs-905	3	48	is	be	AUX
iajs-905	3	49	to	to	PART
iajs-905	3	50	introduce	introduce	VERB
iajs-905	3	51	the	the	DET
iajs-905	3	52	concept	concept	NOUN
iajs-905	3	53	of	of	ADP
iajs-905	3	54	a	a	DET
iajs-905	3	55	fuzzy	fuzzy	ADJ
iajs-905	3	56	internal	internal	ADJ
iajs-905	3	57	direct	direct	ADJ
iajs-905	3	58	product	product	NOUN
iajs-905	3	59	of	of	ADP
iajs-905	3	60	fuzzy	fuzzy	ADJ
iajs-905	3	61	subgroups	subgroup	NOUN
iajs-905	3	62	of	of	ADP
iajs-905	3	63	group	group	NOUN
iajs-905	3	64	.	.	PUNCT
iajs-905	4	1	we	we	PRON
iajs-905	4	2	study	study	VERB
iajs-905	4	3	some	some	DET
iajs-905	4	4	properties	property	NOUN
iajs-905	4	5	and	and	CCONJ
iajs-905	4	6	prove	prove	VERB
iajs-905	4	7	some	some	DET
iajs-905	4	8	theorems	theorem	NOUN
iajs-905	4	9	about	about	ADP
iajs-905	4	10	this	this	DET
iajs-905	4	11	concept	concept	NOUN
iajs-905	4	12	,	,	PUNCT
iajs-905	4	13	which	which	PRON
iajs-905	4	14	is	be	AUX
iajs-905	4	15	very	very	ADV
iajs-905	4	16	important	important	ADJ
iajs-905	4	17	and	and	CCONJ
iajs-905	4	18	interesting	interesting	ADJ
iajs-905	4	19	of	of	ADP
iajs-905	4	20	fuzzy	fuzzy	ADJ
iajs-905	4	21	groups	group	NOUN
iajs-905	4	22	and	and	CCONJ
iajs-905	4	23	very	very	ADV
iajs-905	4	24	useful	useful	ADJ
iajs-905	4	25	in	in	ADP
iajs-905	4	26	applications	application	NOUN
iajs-905	4	27	of	of	ADP
iajs-905	4	28	fuzzy	fuzzy	ADJ
iajs-905	4	29	mathematics	mathematic	NOUN
iajs-905	4	30	in	in	ADP
iajs-905	4	31	general	general	ADJ
iajs-905	4	32	and	and	CCONJ
iajs-905	4	33	especially	especially	ADV
iajs-905	4	34	in	in	ADP
iajs-905	4	35	fuzzy	fuzzy	ADJ
iajs-905	4	36	groups	group	NOUN
iajs-905	4	37	.	.	PUNCT
iajs-905	5	1	introduction	introduction	NOUN
iajs-905	5	2	applying	apply	VERB
iajs-905	5	3	the	the	DET
iajs-905	5	4	concept	concept	NOUN
iajs-905	5	5	of	of	ADP
iajs-905	5	6	fuzzy	fuzzy	ADJ
iajs-905	5	7	sets	set	NOUN
iajs-905	5	8	of	of	ADP
iajs-905	5	9	zadeh	zadeh	PROPN
iajs-905	5	10	to	to	ADP
iajs-905	5	11	the	the	DET
iajs-905	5	12	group	group	NOUN
iajs-905	5	13	theory	theory	NOUN
iajs-905	5	14	,	,	PUNCT
iajs-905	5	15	rosenfeld	rosenfeld	PROPN
iajs-905	5	16	introduced	introduce	VERB
iajs-905	5	17	the	the	DET
iajs-905	5	18	notion	notion	NOUN
iajs-905	5	19	of	of	ADP
iajs-905	5	20	a	a	DET
iajs-905	5	21	fuzzy	fuzzy	ADJ
iajs-905	5	22	group	group	NOUN
iajs-905	5	23	as	as	ADV
iajs-905	5	24	early	early	ADV
iajs-905	5	25	as	as	ADP
iajs-905	5	26	1971	1971	NUM
iajs-905	5	27	.	.	PUNCT
iajs-905	6	1	the	the	DET
iajs-905	6	2	technique	technique	NOUN
iajs-905	6	3	of	of	ADP
iajs-905	6	4	generating	generate	VERB
iajs-905	6	5	a	a	DET
iajs-905	6	6	fuzzy	fuzzy	ADJ
iajs-905	6	7	group	group	NOUN
iajs-905	6	8	(	(	PUNCT
iajs-905	6	9	the	the	DET
iajs-905	6	10	smallest	small	ADJ
iajs-905	6	11	fuzzy	fuzzy	ADJ
iajs-905	6	12	group	group	NOUN
iajs-905	6	13	)	)	PUNCT
iajs-905	6	14	containing	contain	VERB
iajs-905	6	15	an	an	DET
iajs-905	6	16	arbitrarily	arbitrarily	ADV
iajs-905	6	17	chosen	choose	VERB
iajs-905	6	18	fuzzy	fuzzy	ADJ
iajs-905	6	19	set	set	NOUN
iajs-905	6	20	was	be	AUX
iajs-905	6	21	developed	develop	VERB
iajs-905	6	22	only	only	ADV
iajs-905	6	23	in	in	ADP
iajs-905	6	24	1992	1992	NUM
iajs-905	6	25	by	by	ADP
iajs-905	6	26	malik	malik	PROPN
iajs-905	6	27	,	,	PUNCT
iajs-905	6	28	mordeson	mordeson	NOUN
iajs-905	6	29	and	and	CCONJ
iajs-905	6	30	nair	nair	NOUN
iajs-905	6	31	,	,	PUNCT
iajs-905	6	32	[	[	X
iajs-905	6	33	1	1	NUM
iajs-905	6	34	]	]	PUNCT
iajs-905	6	35	.	.	PUNCT
iajs-905	7	1	in	in	ADP
iajs-905	7	2	this	this	DET
iajs-905	7	3	paper	paper	NOUN
iajs-905	7	4	,	,	PUNCT
iajs-905	7	5	we	we	PRON
iajs-905	7	6	use	use	VERB
iajs-905	7	7	our	our	PRON
iajs-905	7	8	notion	notion	NOUN
iajs-905	7	9	of	of	ADP
iajs-905	7	10	fuzzy	fuzzy	ADJ
iajs-905	7	11	inner	inner	ADJ
iajs-905	7	12	product	product	NOUN
iajs-905	7	13	to	to	PART
iajs-905	7	14	generate	generate	VERB
iajs-905	7	15	fuzzy	fuzzy	ADJ
iajs-905	7	16	internal	internal	ADJ
iajs-905	7	17	direct	direct	ADJ
iajs-905	7	18	product	product	NOUN
iajs-905	7	19	of	of	ADP
iajs-905	7	20	fuzzy	fuzzy	ADJ
iajs-905	7	21	subgroups	subgroup	NOUN
iajs-905	7	22	of	of	ADP
iajs-905	7	23	group	group	NOUN
iajs-905	7	24	.	.	PUNCT
iajs-905	8	1	now	now	ADV
iajs-905	8	2	we	we	PRON
iajs-905	8	3	introduce	introduce	VERB
iajs-905	8	4	the	the	DET
iajs-905	8	5	following	follow	VERB
iajs-905	8	6	definitions	definition	NOUN
iajs-905	8	7	which	which	PRON
iajs-905	8	8	is	be	AUX
iajs-905	8	9	necessary	necessary	ADJ
iajs-905	8	10	and	and	CCONJ
iajs-905	8	11	needed	need	VERB
iajs-905	8	12	in	in	ADP
iajs-905	8	13	the	the	DET
iajs-905	8	14	next	next	ADJ
iajs-905	8	15	section	section	NOUN
iajs-905	8	16	:	:	PUNCT
iajs-905	8	17	definition	definition	NOUN
iajs-905	8	18	1.1	1.1	NUM
iajs-905	8	19	[	[	X
iajs-905	8	20	1	1	NUM
iajs-905	8	21	]	]	PUNCT
iajs-905	8	22	,	,	PUNCT
iajs-905	8	23	[	[	X
iajs-905	8	24	2	2	NUM
iajs-905	8	25	]	]	PUNCT
iajs-905	8	26	:	:	PUNCT
iajs-905	8	27	a	a	DET
iajs-905	8	28	mapping	mapping	NOUN
iajs-905	8	29	from	from	ADP
iajs-905	8	30	a	a	DET
iajs-905	8	31	nonempty	nonempty	ADV
iajs-905	8	32	set	set	VERB
iajs-905	8	33	x	x	INTJ
iajs-905	8	34	to	to	ADP
iajs-905	8	35	the	the	DET
iajs-905	8	36	interval	interval	NOUN
iajs-905	8	37	[	[	X
iajs-905	8	38	0	0	NUM
iajs-905	8	39	,	,	PUNCT
iajs-905	8	40	1	1	NUM
iajs-905	8	41	]	]	PUNCT
iajs-905	8	42	is	be	AUX
iajs-905	8	43	called	call	VERB
iajs-905	8	44	a	a	DET
iajs-905	8	45	fuzzy	fuzzy	ADJ
iajs-905	8	46	subset	subset	NOUN
iajs-905	8	47	of	of	ADP
iajs-905	8	48	x	x	X
iajs-905	8	49	.	.	PUNCT
iajs-905	9	1	next	next	ADV
iajs-905	9	2	,	,	PUNCT
iajs-905	9	3	we	we	PRON
iajs-905	9	4	shall	shall	AUX
iajs-905	9	5	give	give	VERB
iajs-905	9	6	some	some	DET
iajs-905	9	7	definitions	definition	NOUN
iajs-905	9	8	and	and	CCONJ
iajs-905	9	9	concepts	concept	NOUN
iajs-905	9	10	related	relate	VERB
iajs-905	9	11	to	to	ADP
iajs-905	9	12	fuzzy	fuzzy	ADJ
iajs-905	9	13	subsets	subset	NOUN
iajs-905	9	14	of	of	ADP
iajs-905	9	15	g.	g.	PROPN
iajs-905	9	16	definition	definition	NOUN
iajs-905	9	17	1.2	1.2	NUM
iajs-905	9	18	:	:	PUNCT
iajs-905	9	19	let	let	VERB
iajs-905	9	20	v,	v,	PART
iajs-905	9	21	be	be	AUX
iajs-905	9	22	fuzzy	fuzzy	ADJ
iajs-905	9	23	subsets	subset	NOUN
iajs-905	9	24	of	of	ADP
iajs-905	9	25	g	g	NOUN
iajs-905	9	26	,	,	PUNCT
iajs-905	9	27	if	if	SCONJ
iajs-905	9	28			NOUN
iajs-905	9	29			SYM
iajs-905	9	30			NOUN
iajs-905	10	1	xvx	xvx	NOUN
iajs-905	10	2			NOUN
iajs-905	10	3	for	for	ADP
iajs-905	10	4	every	every	DET
iajs-905	10	5	gx	gx	NOUN
iajs-905	10	6	,	,	PUNCT
iajs-905	10	7	then	then	ADV
iajs-905	10	8	we	we	PRON
iajs-905	10	9	say	say	VERB
iajs-905	10	10	that	that	DET
iajs-905	10	11			NOUN
iajs-905	10	12	is	be	AUX
iajs-905	10	13	contained	contain	VERB
iajs-905	10	14	in	in	ADP
iajs-905	10	15	v	v	NOUN
iajs-905	10	16	(	(	PUNCT
iajs-905	10	17	or	or	CCONJ
iajs-905	10	18	v	v	NOUN
iajs-905	10	19	contains	contain	VERB
iajs-905	10	20			NOUN
iajs-905	10	21	)	)	PUNCT
iajs-905	10	22	and	and	CCONJ
iajs-905	10	23	we	we	PRON
iajs-905	10	24	write	write	VERB
iajs-905	10	25	v	v	PROPN
iajs-905	10	26	(	(	PUNCT
iajs-905	10	27	or	or	CCONJ
iajs-905	10	28			PROPN
iajs-905	10	29			PROPN
iajs-905	10	30	)	)	PUNCT
iajs-905	10	31	.	.	PUNCT
iajs-905	11	1	if	if	SCONJ
iajs-905	11	2	v	v	PROPN
iajs-905	11	3	and	and	CCONJ
iajs-905	11	4	v	v	PROPN
iajs-905	11	5	,	,	PUNCT
iajs-905	11	6	then	then	ADV
iajs-905	11	7			NOUN
iajs-905	11	8	is	be	AUX
iajs-905	11	9	said	say	VERB
iajs-905	11	10	to	to	PART
iajs-905	11	11	be	be	AUX
iajs-905	11	12	properly	properly	ADV
iajs-905	11	13	contained	contain	VERB
iajs-905	11	14	in	in	ADP
iajs-905	11	15	v	v	NOUN
iajs-905	11	16	(	(	PUNCT
iajs-905	11	17	or	or	CCONJ
iajs-905	11	18	vproperly	vproperly	ADV
iajs-905	11	19	contains	contain	VERB
iajs-905	11	20			NOUN
iajs-905	11	21	)	)	PUNCT
iajs-905	11	22	and	and	CCONJ
iajs-905	11	23	we	we	PRON
iajs-905	11	24	write	write	VERB
iajs-905	11	25	v	v	NOUN
iajs-905	11	26	(	(	PUNCT
iajs-905	11	27	or	or	CCONJ
iajs-905	11	28			PROPN
iajs-905	11	29			PROPN
iajs-905	11	30	)	)	PUNCT
iajs-905	12	1	.[3	.[3	SYM
iajs-905	12	2	]	]	PUNCT
iajs-905	12	3	note	note	VERB
iajs-905	12	4	that	that	SCONJ
iajs-905	12	5	:	:	PUNCT
iajs-905	12	6	v	v	X
iajs-905	12	7	if	if	SCONJ
iajs-905	12	8	and	and	CCONJ
iajs-905	12	9	only	only	ADV
iajs-905	12	10	if	if	SCONJ
iajs-905	12	11			NOUN
iajs-905	12	12			SYM
iajs-905	12	13			NOUN
iajs-905	12	14	xvx	xvx	NOUN
iajs-905	12	15			ADJ
iajs-905	12	16	for	for	ADP
iajs-905	12	17	all	all	DET
iajs-905	12	18	gx	gx	VERB
iajs-905	12	19	.[4	.[4	PROPN
iajs-905	12	20	]	]	PUNCT
iajs-905	12	21	definition	definition	NOUN
iajs-905	12	22	1.3	1.3	NUM
iajs-905	13	1	[	[	X
iajs-905	13	2	3	3	NUM
iajs-905	13	3	]	]	PUNCT
iajs-905	13	4	:	:	PUNCT
iajs-905	13	5	let	let	VERB
iajs-905	13	6	v,	v,	PART
iajs-905	13	7	be	be	AUX
iajs-905	13	8	two	two	NUM
iajs-905	13	9	fuzzy	fuzzy	ADJ
iajs-905	13	10	subsets	subset	NOUN
iajs-905	13	11	of	of	ADP
iajs-905	13	12	g.	g.	PROPN
iajs-905	13	13	then	then	ADV
iajs-905	13	14	v	v	NUM
iajs-905	13	15	and	and	CCONJ
iajs-905	13	16	v	v	NUM
iajs-905	13	17			NOUN
iajs-905	13	18			NOUN
iajs-905	13	19	are	be	AUX
iajs-905	13	20	fuzzy	fuzzy	ADJ
iajs-905	13	21	subsets	subset	NOUN
iajs-905	13	22	as	as	SCONJ
iajs-905	13	23	follows	follow	VERB
iajs-905	13	24	:	:	PUNCT
iajs-905	13	25	(	(	PUNCT
iajs-905	13	26	i	i	NOUN
iajs-905	13	27	)	)	PUNCT
iajs-905	13	28			NOUN
iajs-905	13	29			PUNCT
iajs-905	13	30			PROPN
iajs-905	13	31	)(),(max	)(),(max	NUM
iajs-905	13	32	)	)	PUNCT
iajs-905	13	33	(	(	PUNCT
iajs-905	13	34	xvxxv	xvxxv	PROPN
iajs-905	13	35			PROPN
iajs-905	13	36			PROPN
iajs-905	13	37	(	(	PUNCT
iajs-905	13	38	ii	ii	NOUN
iajs-905	13	39	)	)	PUNCT
iajs-905	13	40			NOUN
iajs-905	13	41			PUNCT
iajs-905	13	42			NUM
iajs-905	13	43	)(),(min	)(),(min	NOUN
iajs-905	13	44	)	)	PUNCT
iajs-905	13	45	(	(	PUNCT
iajs-905	13	46	xvxxv	xvxxv	PROPN
iajs-905	13	47			PROPN
iajs-905	13	48			PROPN
iajs-905	13	49	,	,	PUNCT
iajs-905	13	50	for	for	ADP
iajs-905	13	51	all	all	PRON
iajs-905	13	52	gx	gx	VERB
iajs-905	13	53	then	then	ADV
iajs-905	13	54	vandv	vandv	NOUN
iajs-905	13	55			PROPN
iajs-905	13	56			PROPN
iajs-905	13	57	are	be	AUX
iajs-905	13	58	called	call	VERB
iajs-905	13	59	the	the	DET
iajs-905	13	60	union	union	NOUN
iajs-905	13	61	and	and	CCONJ
iajs-905	13	62	intersection	intersection	NOUN
iajs-905	13	63	of	of	ADP
iajs-905	13	64			NOUN
iajs-905	13	65	and	and	CCONJ
iajs-905	13	66	v	v	NOUN
iajs-905	13	67	,	,	PUNCT
iajs-905	13	68	respectively	respectively	ADV
iajs-905	13	69	.	.	PUNCT
iajs-905	14	1	now	now	ADV
iajs-905	14	2	,	,	PUNCT
iajs-905	14	3	we	we	PRON
iajs-905	14	4	are	be	AUX
iajs-905	14	5	ready	ready	ADJ
iajs-905	14	6	to	to	PART
iajs-905	14	7	give	give	VERB
iajs-905	14	8	the	the	DET
iajs-905	14	9	definition	definition	NOUN
iajs-905	14	10	of	of	ADP
iajs-905	14	11	a	a	DET
iajs-905	14	12	fuzzy	fuzzy	ADJ
iajs-905	14	13	subgroup	subgroup	NOUN
iajs-905	14	14	of	of	ADP
iajs-905	14	15	a	a	DET
iajs-905	14	16	group	group	NOUN
iajs-905	14	17	.	.	PUNCT
iajs-905	15	1	definition	definition	NOUN
iajs-905	15	2	1.4[1	1.4[1	NUM
iajs-905	15	3	]	]	PUNCT
iajs-905	15	4	,	,	PUNCT
iajs-905	15	5	[	[	X
iajs-905	15	6	5	5	NUM
iajs-905	15	7	]	]	PUNCT
iajs-905	15	8	:	:	PUNCT
iajs-905	15	9	a	a	DET
iajs-905	15	10	fuzzy	fuzzy	ADJ
iajs-905	15	11	subset	subset	NOUN
iajs-905	15	12			NOUN
iajs-905	15	13	of	of	ADP
iajs-905	15	14	a	a	DET
iajs-905	15	15	group	group	NOUN
iajs-905	15	16	g	g	NOUN
iajs-905	15	17	is	be	AUX
iajs-905	15	18	a	a	DET
iajs-905	15	19	fuzzy	fuzzy	ADJ
iajs-905	15	20	subgroup	subgroup	NOUN
iajs-905	15	21	of	of	ADP
iajs-905	15	22	g	g	PROPN
iajs-905	16	1	if	if	SCONJ
iajs-905	16	2	:	:	PUNCT
iajs-905	16	3	(	(	PUNCT
iajs-905	16	4	i	i	NOUN
iajs-905	16	5	)	)	PUNCT
iajs-905	16	6			NOUN
iajs-905	16	7			SYM
iajs-905	16	8			NOUN
iajs-905	16	9			ADJ
iajs-905	16	10			PROPN
iajs-905	16	11			PROPN
iajs-905	16	12	min	min	NOUN
iajs-905	16	13	a	a	PRON
iajs-905	16	14	,	,	PUNCT
iajs-905	16	15	b	b	PROPN
iajs-905	16	16	a	a	PROPN
iajs-905	16	17	*	*	NOUN
iajs-905	16	18	b	b	NUM
iajs-905	16	19			NOUN
iajs-905	16	20			NOUN
iajs-905	16	21	(	(	PUNCT
iajs-905	16	22	ii	ii	NOUN
iajs-905	16	23	)	)	PUNCT
iajs-905	16	24			NOUN
iajs-905	16	25			PROPN
iajs-905	16	26			NOUN
iajs-905	16	27	aa	aa	PUNCT
iajs-905	17	1			PROPN
iajs-905	17	2	1	1	NOUN
iajs-905	17	3	,	,	PUNCT
iajs-905	17	4	for	for	ADP
iajs-905	17	5	all	all	DET
iajs-905	17	6	gba	gba	PROPN
iajs-905	17	7			NOUN
iajs-905	17	8	,	,	PUNCT
iajs-905	17	9	.	.	PUNCT
iajs-905	18	1	ihjpas	ihjpa	VERB
iajs-905	18	2	ibn	ibn	PROPN
iajs-905	18	3	alhaitham	alhaitham	PROPN
iajs-905	18	4	j.	j.	PROPN
iajs-905	18	5	for	for	ADP
iajs-905	18	6	pure	pure	ADJ
iajs-905	18	7	&	&	CCONJ
iajs-905	18	8	appl	appl	PROPN
iajs-905	18	9	.	.	PUNCT
iajs-905	19	1	sci	sci	PROPN
iajs-905	19	2	.	.	PUNCT
iajs-905	20	1	vol.23	vol.23	PROPN
iajs-905	20	2	(	(	PUNCT
iajs-905	20	3	3	3	NUM
iajs-905	20	4	)	)	PUNCT
iajs-905	20	5	2010	2010	NUM
iajs-905	20	6	proposition	proposition	NOUN
iajs-905	20	7	1.5	1.5	NUM
iajs-905	21	1	[	[	SYM
iajs-905	21	2	6	6	NUM
iajs-905	21	3	]	]	PUNCT
iajs-905	21	4	:	:	PUNCT
iajs-905	21	5	let	let	VERB
iajs-905	21	6			NOUN
iajs-905	21	7	be	be	AUX
iajs-905	21	8	a	a	DET
iajs-905	21	9	fuzzy	fuzzy	ADJ
iajs-905	21	10	group	group	NOUN
iajs-905	21	11	.	.	PUNCT
iajs-905	22	1	then	then	ADV
iajs-905	22	2			NOUN
iajs-905	22	3			SYM
iajs-905	22	4			NOUN
iajs-905	22	5			PROPN
iajs-905	22	6	gaea	gaea	NOUN
iajs-905	22	7			X
iajs-905	22	8			PROPN
iajs-905	22	9	.	.	PUNCT
iajs-905	23	1	definition	definition	NOUN
iajs-905	23	2	1.6	1.6	NUM
iajs-905	24	1	[	[	X
iajs-905	24	2	7	7	NUM
iajs-905	24	3	]	]	X
iajs-905	24	4	:	:	PUNCT
iajs-905	24	5	if	if	SCONJ
iajs-905	24	6			NOUN
iajs-905	24	7	is	be	AUX
iajs-905	24	8	a	a	DET
iajs-905	24	9	fuzzy	fuzzy	ADJ
iajs-905	24	10	subgroup	subgroup	NOUN
iajs-905	24	11	of	of	ADP
iajs-905	24	12	g	g	PROPN
iajs-905	24	13	,	,	PUNCT
iajs-905	24	14	then	then	ADV
iajs-905	24	15			NOUN
iajs-905	24	16	is	be	AUX
iajs-905	24	17	said	say	VERB
iajs-905	24	18	to	to	PART
iajs-905	24	19	be	be	AUX
iajs-905	24	20	abelian	abelian	ADJ
iajs-905	24	21	if	if	SCONJ
iajs-905	24	22	gyx	gyx	VERB
iajs-905	24	23			PROPN
iajs-905	24	24	,	,	PUNCT
iajs-905	24	25	,	,	PUNCT
iajs-905	24	26			NOUN
iajs-905	24	27			SYM
iajs-905	24	28			NOUN
iajs-905	24	29			NUM
iajs-905	24	30	0,0	0,0	NUM
iajs-905	24	31			NOUN
iajs-905	24	32	yx	yx	NOUN
iajs-905	24	33			PROPN
iajs-905	24	34	,	,	PUNCT
iajs-905	24	35	then	then	ADV
iajs-905	24	36			NOUN
iajs-905	24	37			SYM
iajs-905	24	38			NOUN
iajs-905	24	39	yxxy	yxxy	NOUN
iajs-905	24	40			PROPN
iajs-905	24	41			PRON
iajs-905	24	42	.	.	PUNCT
iajs-905	25	1	definition	definition	NOUN
iajs-905	25	2	1.7	1.7	NUM
iajs-905	26	1	[	[	X
iajs-905	26	2	8	8	NUM
iajs-905	26	3	]	]	PUNCT
iajs-905	26	4	,	,	PUNCT
iajs-905	26	5	[	[	X
iajs-905	26	6	9	9	NUM
iajs-905	26	7	]	]	X
iajs-905	26	8	:	:	PUNCT
iajs-905	26	9	a	a	DET
iajs-905	26	10	fuzzy	fuzzy	ADJ
iajs-905	26	11	subgroup	subgroup	NOUN
iajs-905	26	12			NOUN
iajs-905	26	13	of	of	ADP
iajs-905	26	14	g	g	PROPN
iajs-905	26	15	is	be	AUX
iajs-905	26	16	said	say	VERB
iajs-905	26	17	to	to	PART
iajs-905	26	18	be	be	AUX
iajs-905	26	19	normal	normal	ADJ
iajs-905	26	20	fuzzy	fuzzy	ADJ
iajs-905	26	21	subgroup	subgroup	NOUN
iajs-905	26	22	if	if	SCONJ
iajs-905	26	23			PROPN
iajs-905	26	24			PROPN
iajs-905	26	25			PROPN
iajs-905	26	26	x	x	PROPN
iajs-905	26	27	*	*	PUNCT
iajs-905	26	28	y	y	PROPN
iajs-905	26	29	y*x	y*x	NOUN
iajs-905	26	30	,	,	PUNCT
iajs-905	26	31	x	x	X
iajs-905	26	32	,	,	PUNCT
iajs-905	26	33	y	y	PROPN
iajs-905	26	34	g	g	PROPN
iajs-905	26	35			NOUN
iajs-905	26	36			VERB
iajs-905	26	37			NOUN
iajs-905	26	38	.	.	PUNCT
iajs-905	27	1	fuzzy	fuzzy	ADJ
iajs-905	27	2	internal	internal	ADJ
iajs-905	27	3	direct	direct	ADJ
iajs-905	27	4	product	product	NOUN
iajs-905	27	5	now	now	ADV
iajs-905	27	6	we	we	PRON
iajs-905	27	7	are	be	AUX
iajs-905	27	8	ready	ready	ADJ
iajs-905	27	9	to	to	PART
iajs-905	27	10	introduce	introduce	VERB
iajs-905	27	11	the	the	DET
iajs-905	27	12	definition	definition	NOUN
iajs-905	27	13	and	and	CCONJ
iajs-905	27	14	some	some	DET
iajs-905	27	15	theorems	theorem	NOUN
iajs-905	27	16	about	about	ADP
iajs-905	27	17	fuzzy	fuzzy	ADJ
iajs-905	27	18	internal	internal	ADJ
iajs-905	27	19	direct	direct	ADJ
iajs-905	27	20	p	p	NOUN
iajs-905	27	21	roduct	roduct	NOUN
iajs-905	27	22	of	of	ADP
iajs-905	27	23	fuzzy	fuzzy	ADJ
iajs-905	27	24	sub	sub	NOUN
iajs-905	27	25	groups	group	NOUN
iajs-905	27	26	of	of	ADP
iajs-905	27	27	groups	group	NOUN
iajs-905	27	28	:	:	PUNCT
iajs-905	27	29	definition	definition	NOUN
iajs-905	27	30	2.1	2.1	NUM
iajs-905	27	31	let	let	VERB
iajs-905	27	32	a	a	PRON
iajs-905	27	33	be	be	AUX
iajs-905	27	34	a	a	DET
iajs-905	27	35	fuzzy	fuzzy	ADJ
iajs-905	27	36	subgroup	subgroup	NOUN
iajs-905	27	37	of	of	ADP
iajs-905	27	38	a	a	DET
iajs-905	27	39	group	group	NOUN
iajs-905	27	40	(	(	PUNCT
iajs-905	27	41	g	g	NOUN
iajs-905	27	42	,	,	PUNCT
iajs-905	27	43	.	.	PUNCT
iajs-905	27	44	)	)	PUNCT
iajs-905	28	1	and	and	CCONJ
iajs-905	28	2	n1,n2,	n1,n2,	PROPN
iajs-905	28	3	…	…	PUNCT
iajs-905	28	4	,nn	,nn	PUNCT
iajs-905	28	5	be	be	VERB
iajs-905	28	6	fuzzy	fuzzy	ADJ
iajs-905	28	7	normal	normal	ADJ
iajs-905	28	8	subgroups	subgroup	NOUN
iajs-905	28	9	in	in	ADP
iajs-905	28	10	a	a	DET
iajs-905	28	11	such	such	ADJ
iajs-905	28	12	that	that	PRON
iajs-905	28	13	:	:	PUNCT
iajs-905	28	14	1a	1a	X
iajs-905	28	15	=	=	SYM
iajs-905	28	16	n1n2	n1n2	PROPN
iajs-905	28	17	…	…	PUNCT
iajs-905	28	18	nn	nn	NOUN
iajs-905	28	19	2let	2let	PROPN
iajs-905	28	20	xt	xt	PROPN
iajs-905	28	21			PROPN
iajs-905	28	22	a	a	PRON
iajs-905	28	23	,	,	PUNCT
iajs-905	28	24	xg	xg	NOUN
iajs-905	28	25	,	,	PUNCT
iajs-905	28	26	t[0	t[0	NUM
iajs-905	28	27	,	,	PUNCT
iajs-905	28	28	1	1	NUM
iajs-905	28	29	]	]	PUNCT
iajs-905	28	30	,	,	PUNCT
iajs-905	28	31	then	then	ADV
iajs-905	28	32	:	:	PUNCT
iajs-905	28	33	xt(y)=	xt(y)=	PROPN
iajs-905	28	34	ni	ni	PROPN
iajs-905	28	35			PROPN
iajs-905	28	36	ni	ni	PROPN
iajs-905	28	37	in	in	ADP
iajs-905	28	38	unique	unique	ADJ
iajs-905	28	39	way	way	NOUN
iajs-905	28	40	.	.	PUNCT
iajs-905	29	1	then	then	ADV
iajs-905	29	2	a	a	PRON
iajs-905	29	3	is	be	AUX
iajs-905	29	4	said	say	VERB
iajs-905	29	5	to	to	PART
iajs-905	29	6	be	be	AUX
iajs-905	29	7	fuzzy	fuzzy	ADJ
iajs-905	29	8	internal	internal	ADJ
iajs-905	29	9	direct	direct	ADJ
iajs-905	29	10	p	p	NOUN
iajs-905	29	11	roduct	roduct	NOUN
iajs-905	29	12	of	of	ADP
iajs-905	29	13	n1,n2,	n1,n2,	PROPN
iajs-905	29	14	…	…	SYM
iajs-905	29	15	,nn	,nn	PROPN
iajs-905	29	16	.	.	PUNCT
iajs-905	30	1	now	now	ADV
iajs-905	30	2	we	we	PRON
iajs-905	30	3	introduce	introduce	VERB
iajs-905	30	4	the	the	DET
iajs-905	30	5	following	follow	VERB
iajs-905	30	6	theorem	theorem	NOUN
iajs-905	30	7	:	:	PUNCT
iajs-905	30	8	theorem	theorem	VERB
iajs-905	30	9	2.2	2.2	NUM
iajs-905	30	10	if	if	SCONJ
iajs-905	30	11	a	a	DET
iajs-905	30	12	fuzzy	fuzzy	ADJ
iajs-905	30	13	subgroup	subgroup	NOUN
iajs-905	30	14	a	a	PRON
iajs-905	30	15	of	of	ADP
iajs-905	30	16	a	a	DET
iajs-905	30	17	group	group	NOUN
iajs-905	30	18	(	(	PUNCT
iajs-905	30	19	g	g	NOUN
iajs-905	30	20	,	,	PUNCT
iajs-905	30	21	.	.	PUNCT
iajs-905	30	22	)	)	PUNCT
iajs-905	31	1	is	be	AUX
iajs-905	31	2	the	the	DET
iajs-905	31	3	fuzzy	fuzzy	ADJ
iajs-905	31	4	internal	internal	ADJ
iajs-905	31	5	direct	direct	ADJ
iajs-905	31	6	product	product	NOUN
iajs-905	31	7	of	of	ADP
iajs-905	31	8	fuzzy	fuzzy	ADJ
iajs-905	31	9	normal	normal	ADJ
iajs-905	31	10	subgroups	subgroup	NOUN
iajs-905	31	11	:	:	PUNCT
iajs-905	31	12	n1,n2	n1,n2	PROPN
iajs-905	31	13	,	,	PUNCT
iajs-905	31	14	…	…	PUNCT
iajs-905	31	15	,	,	PUNCT
iajs-905	31	16	nn	nn	INTJ
iajs-905	31	17	,	,	PUNCT
iajs-905	31	18	then	then	ADV
iajs-905	31	19	for	for	ADP
iajs-905	31	20	i	i	PROPN
iajs-905	31	21			PROPN
iajs-905	31	22	j	j	PROPN
iajs-905	31	23	,	,	PUNCT
iajs-905	31	24	ni	ni	PROPN
iajs-905	31	25			PUNCT
iajs-905	31	26	nj	nj	PROPN
iajs-905	31	27	=	=	PUNCT
iajs-905	31	28	{	{	PUNCT
iajs-905	31	29	et	et	NOUN
iajs-905	31	30	}	}	PUNCT
iajs-905	31	31	,	,	PUNCT
iajs-905	31	32	and	and	CCONJ
iajs-905	31	33	if	if	SCONJ
iajs-905	31	34	ni	ni	PROPN
iajs-905	31	35			PROPN
iajs-905	31	36	ni	ni	PROPN
iajs-905	31	37	,	,	PUNCT
iajs-905	31	38	nj	nj	PROPN
iajs-905	31	39			PROPN
iajs-905	31	40	nj	nj	PROPN
iajs-905	31	41	then	then	ADV
iajs-905	31	42	ni	ni	PROPN
iajs-905	31	43	nj	nj	PROPN
iajs-905	32	1	=	=	PROPN
iajs-905	32	2	nj	nj	PROPN
iajs-905	32	3	ni	ni	PROPN
iajs-905	32	4	.	.	PUNCT
iajs-905	33	1	proof	proof	NOUN
iajs-905	33	2	:	:	PUNCT
iajs-905	33	3	suppose	suppose	VERB
iajs-905	33	4	that	that	SCONJ
iajs-905	33	5	ns	ns	ADJ
iajs-905	33	6			PROPN
iajs-905	33	7	ni	ni	PROPN
iajs-905	33	8			PROPN
iajs-905	33	9	nj	nj	PROPN
iajs-905	33	10	then	then	ADV
iajs-905	33	11	we	we	PRON
iajs-905	33	12	can	can	AUX
iajs-905	33	13	write	write	VERB
iajs-905	33	14	ns	ns	NOUN
iajs-905	33	15	as	as	ADP
iajs-905	33	16	viewing	view	VERB
iajs-905	33	17	ns	ns	ADP
iajs-905	33	18	a	a	DET
iajs-905	33	19	an	an	DET
iajs-905	33	20	fuzzy	fuzzy	ADJ
iajs-905	33	21	singleton	singleton	NOUN
iajs-905	33	22	in	in	ADP
iajs-905	33	23	ni	ni	PROPN
iajs-905	33	24	.	.	PUNCT
iajs-905	34	1	similarly	similarly	ADV
iajs-905	34	2	,	,	PUNCT
iajs-905	34	3	we	we	PRON
iajs-905	34	4	can	can	AUX
iajs-905	34	5	write	write	VERB
iajs-905	34	6	ns	ns	NOUN
iajs-905	34	7	as	as	ADP
iajs-905	34	8			NUM
iajs-905	34	9			PROPN
iajs-905	34	10			NUM
iajs-905	34	11			NUM
iajs-905	34	12			ADP
iajs-905	35	1			PROPN
iajs-905	35	2	otherwise0	otherwise0	NOUN
iajs-905	35	3	xyifgy,	xyifgy,	ADJ
iajs-905	35	4	...	...	PUNCT
iajs-905	35	5	,y	,y	NOUN
iajs-905	35	6	,	,	PUNCT
iajs-905	35	7	yandy	yandy	NOUN
iajs-905	35	8	...	...	PUNCT
iajs-905	35	9	yyy	yyy	PROPN
iajs-905	35	10	)	)	PUNCT
iajs-905	35	11	}	}	PUNCT
iajs-905	35	12	y(n),	y(n),	NUM
iajs-905	35	13	...	...	PUNCT
iajs-905	35	14	,y(n),y(nsup{min	,y(n),y(nsup{min	PUNCT
iajs-905	35	15	{	{	PUNCT
iajs-905	35	16	n21n21	n21n21	ADV
iajs-905	35	17	nn2211	nn2211	PROPN
iajs-905	35	18			VERB
iajs-905	36	1			NUM
iajs-905	36	2			ADP
iajs-905	36	3			NOUN
iajs-905	36	4			PROPN
iajs-905	36	5			PROPN
iajs-905	36	6	wiseother0	wiseother0	PROPN
iajs-905	36	7	n	n	CCONJ
iajs-905	36	8	...	...	PUNCT
iajs-905	36	9	,,1inn	,,1inn	PROPN
iajs-905	36	10	,	,	PUNCT
iajs-905	36	11	nyif}}e,	nyif}}e,	PROPN
iajs-905	36	12	...	...	PUNCT
iajs-905	36	13	,e	,e	PROPN
iajs-905	36	14	...	...	PUNCT
iajs-905	36	15	,e),y(n	,e),y(n	X
iajs-905	36	16	,	,	PUNCT
iajs-905	36	17	e,	e,	NOUN
iajs-905	36	18	...	...	NOUN
iajs-905	36	19	,e{{minsup	,e{{minsup	PUNCT
iajs-905	36	20	(	(	PUNCT
iajs-905	36	21	y)n	y)n	PROPN
iajs-905	36	22	iinj1is1i1	iinj1is1i1	PROPN
iajs-905	36	23	s	s	NOUN
iajs-905	36	24			PROPN
iajs-905	36	25			NOUN
iajs-905	36	26			ADP
iajs-905	36	27			NOUN
iajs-905	36	28			PROPN
iajs-905	36	29			PROPN
iajs-905	36	30	wiseother0	wiseother0	PROPN
iajs-905	36	31	n,	n,	NUM
iajs-905	36	32	...	...	PUNCT
iajs-905	36	33	1inn	1inn	NUM
iajs-905	36	34	,	,	PUNCT
iajs-905	36	35	nyif}}e,	nyif}}e,	PROPN
iajs-905	36	36	...	...	PUNCT
iajs-905	36	37	,e),y(n	,e),y(n	X
iajs-905	36	38	,	,	PUNCT
iajs-905	36	39	e,	e,	X
iajs-905	36	40	...	...	PUNCT
iajs-905	36	41	,e,	,e,	PUNCT
iajs-905	36	42	...	...	PUNCT
iajs-905	36	43	,e{{minsup	,e{{minsup	PUNCT
iajs-905	36	44	(	(	PUNCT
iajs-905	36	45	y)n	y)n	ADJ
iajs-905	36	46	iin1js1ji1	iin1js1ji1	PROPN
iajs-905	36	47	s	s	PART
iajs-905	36	48	ihjpas	ihjpas	NOUN
iajs-905	36	49	ibn	ibn	PROPN
iajs-905	36	50	alhaitham	alhaitham	PROPN
iajs-905	36	51	j.	j.	PROPN
iajs-905	36	52	for	for	ADP
iajs-905	36	53	pure	pure	ADJ
iajs-905	36	54	&	&	CCONJ
iajs-905	36	55	appl	appl	PROPN
iajs-905	36	56	.	.	PUNCT
iajs-905	37	1	sci	sci	PROPN
iajs-905	37	2	.	.	PUNCT
iajs-905	38	1	vol.23	vol.23	PROPN
iajs-905	38	2	(	(	PUNCT
iajs-905	38	3	3	3	NUM
iajs-905	38	4	)	)	PUNCT
iajs-905	38	5	2010	2010	NUM
iajs-905	38	6	since	since	SCONJ
iajs-905	38	7	the	the	DET
iajs-905	38	8	two	two	NUM
iajs-905	38	9	decompositions	decomposition	NOUN
iajs-905	38	10	in	in	ADP
iajs-905	38	11	this	this	DET
iajs-905	38	12	form	form	NOUN
iajs-905	38	13	for	for	ADP
iajs-905	38	14	ns	ns	NUM
iajs-905	38	15	must	must	AUX
iajs-905	38	16	coincide	coincide	VERB
iajs-905	38	17	,	,	PUNCT
iajs-905	38	18	the	the	DET
iajs-905	38	19	entry	entry	NOUN
iajs-905	38	20	from	from	ADP
iajs-905	38	21	ni	ni	PROPN
iajs-905	38	22	in	in	ADP
iajs-905	38	23	each	each	PRON
iajs-905	38	24	must	must	AUX
iajs-905	38	25	be	be	AUX
iajs-905	38	26	equal	equal	ADJ
iajs-905	38	27	.	.	PUNCT
iajs-905	39	1	in	in	ADP
iajs-905	39	2	the	the	DET
iajs-905	39	3	first	first	ADJ
iajs-905	39	4	decomposition	decomposition	NOUN
iajs-905	39	5	this	this	DET
iajs-905	39	6	entry	entry	NOUN
iajs-905	39	7	is	be	AUX
iajs-905	39	8	ns	ns	ADJ
iajs-905	39	9	,	,	PUNCT
iajs-905	39	10	in	in	ADP
iajs-905	39	11	the	the	DET
iajs-905	39	12	other	other	ADJ
iajs-905	39	13	it	it	PRON
iajs-905	39	14	is	be	AUX
iajs-905	39	15	et	et	NOUN
iajs-905	39	16	;	;	PUNCT
iajs-905	39	17	hence	hence	ADV
iajs-905	39	18	ns=	ns=	VERB
iajs-905	39	19	et	et	NOUN
iajs-905	39	20	.	.	PUNCT
iajs-905	40	1	thus	thus	ADV
iajs-905	40	2	ni	ni	PROPN
iajs-905	40	3			PUNCT
iajs-905	40	4	nj=	nj=	PROPN
iajs-905	40	5	{	{	PUNCT
iajs-905	40	6	et	et	NOUN
iajs-905	40	7	}	}	PUNCT
iajs-905	40	8	for	for	ADP
iajs-905	40	9	i	i	PRON
iajs-905	40	10			PROPN
iajs-905	40	11	j.	j.	PROPN
iajs-905	40	12	suppose	suppose	VERB
iajs-905	40	13	ni	ni	PROPN
iajs-905	40	14	ni	ni	PROPN
iajs-905	40	15	,	,	PUNCT
iajs-905	40	16	nj	nj	PROPN
iajs-905	40	17	nj	nj	NUM
iajs-905	40	18	,	,	PUNCT
iajs-905	40	19	and	and	CCONJ
iajs-905	40	20	ij	ij	PROPN
iajs-905	40	21	then	then	ADV
iajs-905	40	22	ninjni	ninjni	VERB
iajs-905	40	23	-1nj	-1nj	NOUN
iajs-905	40	24	since	since	SCONJ
iajs-905	40	25	nj	nj	PROPN
iajs-905	40	26	is	be	AUX
iajs-905	40	27	normal	normal	ADJ
iajs-905	40	28	fuzzy	fuzzy	ADJ
iajs-905	40	29	;	;	PUNCT
iajs-905	40	30	thus	thus	ADV
iajs-905	40	31	ninjni	ninjni	VERB
iajs-905	40	32	1	1	NUM
iajs-905	40	33	nj	nj	NOUN
iajs-905	40	34	-1	-1	ADP
iajs-905	40	35	nj	nj	PROPN
iajs-905	40	36	.	.	PUNCT
iajs-905	41	1	similarly	similarly	ADV
iajs-905	41	2	,	,	PUNCT
iajs-905	41	3	since	since	SCONJ
iajs-905	41	4	ni	ni	PROPN
iajs-905	41	5	-1	-1	PROPN
iajs-905	41	6	ni	ni	PROPN
iajs-905	41	7	,	,	PUNCT
iajs-905	41	8	njni	njni	PROPN
iajs-905	41	9	-1	-1	PROPN
iajs-905	41	10	nj	nj	PROPN
iajs-905	41	11	-1	-1	PUNCT
iajs-905	42	1	ni	ni	PROPN
iajs-905	42	2	,	,	PUNCT
iajs-905	42	3	whence	whence	ADP
iajs-905	42	4	ninjni	ninjni	PROPN
iajs-905	42	5	-1	-1	PROPN
iajs-905	42	6	nj	nj	PROPN
iajs-905	42	7	-1	-1	PROPN
iajs-905	42	8			PROPN
iajs-905	42	9	ni	ni	PROPN
iajs-905	42	10	then	then	ADV
iajs-905	42	11	ninjni	ninjni	VERB
iajs-905	42	12	-1	-1	PROPN
iajs-905	42	13	nj	nj	PROPN
iajs-905	43	1	-1	-1	PROPN
iajs-905	43	2			PROPN
iajs-905	43	3	ni	ni	PROPN
iajs-905	43	4	nj	nj	PROPN
iajs-905	43	5	=	=	SYM
iajs-905	43	6	{	{	PUNCT
iajs-905	43	7	et	et	NOUN
iajs-905	43	8	}	}	PUNCT
iajs-905	43	9	.	.	PUNCT
iajs-905	44	1	thus	thus	ADV
iajs-905	44	2	ninjni	ninjni	VERB
iajs-905	44	3	-1	-1	ADP
iajs-905	44	4	nj	nj	PROPN
iajs-905	44	5	-1	-1	PUNCT
iajs-905	45	1	=	=	SYM
iajs-905	45	2	et	et	NOUN
iajs-905	45	3	;	;	PUNCT
iajs-905	45	4	this	this	PRON
iajs-905	45	5	gives	give	VERB
iajs-905	45	6	the	the	DET
iajs-905	45	7	desired	desire	VERB
iajs-905	45	8	result	result	NOUN
iajs-905	45	9	ninj	ninj	NOUN
iajs-905	45	10	=	=	SYM
iajs-905	45	11	njni	njni	NOUN
iajs-905	45	12	.	.	PUNCT
iajs-905	46	1	remark	remark	VERB
iajs-905	46	2	one	one	PRON
iajs-905	46	3	should	should	AUX
iajs-905	46	4	point	point	VERB
iajs-905	46	5	out	out	ADP
iajs-905	46	6	that	that	SCONJ
iajs-905	46	7	if	if	SCONJ
iajs-905	46	8	k1,	k1,	NOUN
iajs-905	46	9	…	…	SYM
iajs-905	46	10	,kn	,kn	NUM
iajs-905	46	11	are	be	AUX
iajs-905	46	12	normal	normal	ADJ
iajs-905	46	13	fuzzy	fuzzy	ADJ
iajs-905	46	14	subgroups	subgroup	NOUN
iajs-905	46	15	of	of	ADP
iajs-905	46	16	a	a	PRON
iajs-905	46	17	,	,	PUNCT
iajs-905	46	18	such	such	ADJ
iajs-905	46	19	that	that	SCONJ
iajs-905	46	20	a=	a=	ADV
iajs-905	46	21	k1k2	k1k2	NOUN
iajs-905	46	22	…	…	SYM
iajs-905	46	23	kn	kn	X
iajs-905	46	24	and	and	CCONJ
iajs-905	46	25	ki	ki	PROPN
iajs-905	46	26	kj={et	kj={et	VERB
iajs-905	46	27	}	}	PUNCT
iajs-905	46	28	for	for	ADP
iajs-905	46	29	ij	ij	PROPN
iajs-905	46	30	it	it	PRON
iajs-905	46	31	need	need	AUX
iajs-905	46	32	not	not	PART
iajs-905	46	33	be	be	AUX
iajs-905	46	34	true	true	ADJ
iajs-905	46	35	that	that	SCONJ
iajs-905	46	36	a	a	PRON
iajs-905	46	37	is	be	AUX
iajs-905	46	38	the	the	DET
iajs-905	46	39	fuzzy	fuzzy	ADJ
iajs-905	46	40	internal	internal	ADJ
iajs-905	46	41	direct	direct	ADJ
iajs-905	46	42	product	product	NOUN
iajs-905	46	43	of	of	ADP
iajs-905	46	44	k1,	k1,	NOUN
iajs-905	46	45	…	…	SYM
iajs-905	46	46	,kn	,kn	NOUN
iajs-905	46	47	.	.	PUNCT
iajs-905	47	1	a	a	DET
iajs-905	47	2	more	more	ADV
iajs-905	47	3	stringent	stringent	ADJ
iajs-905	47	4	condition	condition	NOUN
iajs-905	47	5	is	be	AUX
iajs-905	47	6	needed	need	VERB
iajs-905	47	7	.	.	PUNCT
iajs-905	48	1	clearly	clearly	ADV
iajs-905	48	2	from	from	ADP
iajs-905	48	3	the	the	DET
iajs-905	48	4	above	above	ADJ
iajs-905	48	5	theorem	theorem	NOUN
iajs-905	48	6	we	we	PRON
iajs-905	48	7	can	can	AUX
iajs-905	48	8	obtain	obtain	VERB
iajs-905	48	9	the	the	DET
iajs-905	48	10	following	follow	VERB
iajs-905	48	11	corollary	corollary	NOUN
iajs-905	48	12	:	:	PUNCT
iajs-905	48	13	corollary	corollary	ADJ
iajs-905	48	14	2.3	2.3	NUM
iajs-905	48	15	a	a	DET
iajs-905	48	16	fuzzy	fuzzy	ADJ
iajs-905	48	17	subgroup	subgroup	NOUN
iajs-905	48	18	a	a	PRON
iajs-905	48	19	of	of	ADP
iajs-905	48	20	a	a	DET
iajs-905	48	21	group	group	NOUN
iajs-905	48	22	(	(	PUNCT
iajs-905	48	23	g	g	NOUN
iajs-905	48	24	,	,	PUNCT
iajs-905	48	25	.	.	PUNCT
iajs-905	48	26	)	)	PUNCT
iajs-905	48	27	is	be	AUX
iajs-905	48	28	the	the	DET
iajs-905	48	29	fuzzy	fuzzy	ADJ
iajs-905	48	30	internal	internal	ADJ
iajs-905	48	31	direct	direct	ADJ
iajs-905	48	32	p	p	NOUN
iajs-905	48	33	roduct	roduct	NOUN
iajs-905	48	34	of	of	ADP
iajs-905	48	35	the	the	DET
iajs-905	48	36	normal	normal	ADJ
iajs-905	48	37	fuzzy	fuzzy	ADJ
iajs-905	48	38	subgroups	subgroup	NOUN
iajs-905	48	39	n1,	n1,	NOUN
iajs-905	48	40	…	…	SYM
iajs-905	48	41	,nn	,nn	PUNCT
iajs-905	48	42	if	if	SCONJ
iajs-905	48	43	and	and	CCONJ
iajs-905	48	44	only	only	ADV
iajs-905	48	45	if	if	SCONJ
iajs-905	48	46	:	:	PUNCT
iajs-905	48	47	(	(	PUNCT
iajs-905	48	48	1	1	X
iajs-905	48	49	)	)	PUNCT
iajs-905	48	50	a	a	DET
iajs-905	48	51	=	=	PUNCT
iajs-905	48	52	n1n2	n1n2	NOUN
iajs-905	48	53	…	…	SYM
iajs-905	48	54	nn	nn	X
iajs-905	48	55	(	(	PUNCT
iajs-905	48	56	2	2	NUM
iajs-905	48	57	)	)	PUNCT
iajs-905	48	58	ni	ni	PROPN
iajs-905	48	59	(n1n2	(n1n2	PROPN
iajs-905	48	60	…	…	X
iajs-905	48	61	ni-1ni+1	ni-1ni+1	NOUN
iajs-905	48	62	…	…	SYM
iajs-905	48	63	nn	nn	NOUN
iajs-905	48	64	)	)	PUNCT
iajs-905	48	65	=	=	PRON
iajs-905	48	66	{	{	PUNCT
iajs-905	48	67	e	e	NOUN
iajs-905	48	68	}	}	PUNCT
iajs-905	48	69	for	for	ADP
iajs-905	48	70	i=	i=	PROPN
iajs-905	48	71	1,	1,	NUM
iajs-905	48	72	…	…	PUNCT
iajs-905	48	73	,n	,n	PUNCT
iajs-905	48	74	definition	definition	NOUN
iajs-905	48	75	2.4	2.4	NUM
iajs-905	48	76	let	let	VERB
iajs-905	48	77	h	h	NOUN
iajs-905	48	78	and	and	CCONJ
iajs-905	48	79	k	k	PROPN
iajs-905	48	80	are	be	AUX
iajs-905	48	81	fuzzy	fuzzy	ADJ
iajs-905	48	82	subgroups	subgroup	NOUN
iajs-905	48	83	of	of	ADP
iajs-905	48	84	a	a	DET
iajs-905	48	85	group	group	NOUN
iajs-905	48	86	(	(	PUNCT
iajs-905	48	87	g	g	NOUN
iajs-905	48	88	,	,	PUNCT
iajs-905	48	89	.	.	PUNCT
iajs-905	48	90	)	)	PUNCT
iajs-905	49	1	.the	.the	NOUN
iajs-905	49	2	join	join	VERB
iajs-905	49	3	hk	hk	ADV
iajs-905	49	4	of	of	ADP
iajs-905	49	5	h	h	PROPN
iajs-905	49	6	and	and	CCONJ
iajs-905	49	7	k	k	PROPN
iajs-905	49	8	is	be	AUX
iajs-905	49	9	the	the	DET
iajs-905	49	10	intersection	intersection	NOUN
iajs-905	49	11	of	of	ADP
iajs-905	49	12	all	all	DET
iajs-905	49	13	fuzzy	fuzzy	ADJ
iajs-905	49	14	subgroups	subgroup	NOUN
iajs-905	49	15	of	of	ADP
iajs-905	49	16	g	g	NOUN
iajs-905	49	17	containing	contain	VERB
iajs-905	49	18	h.k	h.k	NOUN
iajs-905	49	19	.	.	PUNCT
iajs-905	50	1	clearly	clearly	ADV
iajs-905	50	2	,	,	PUNCT
iajs-905	50	3	this	this	DET
iajs-905	50	4	intersection	intersection	NOUN
iajs-905	50	5	will	will	AUX
iajs-905	50	6	be	be	AUX
iajs-905	50	7	the	the	DET
iajs-905	50	8	smallest	small	ADJ
iajs-905	50	9	possible	possible	ADJ
iajs-905	50	10	fuzzy	fuzzy	ADJ
iajs-905	50	11	subgroup	subgroup	NOUN
iajs-905	50	12	of	of	ADP
iajs-905	50	13	g	g	PROPN
iajs-905	50	14	containing	contain	VERB
iajs-905	50	15	h.k	h.k	NOUN
iajs-905	50	16	,	,	PUNCT
iajs-905	50	17	and	and	CCONJ
iajs-905	50	18	if	if	SCONJ
iajs-905	50	19	elements	element	NOUN
iajs-905	50	20	in	in	ADP
iajs-905	50	21	h	h	PROPN
iajs-905	50	22	and	and	CCONJ
iajs-905	50	23	k	k	PROPN
iajs-905	50	24	commute	commute	NOUN
iajs-905	50	25	,	,	PUNCT
iajs-905	50	26	in	in	ADP
iajs-905	50	27	particular	particular	ADJ
iajs-905	50	28	,	,	PUNCT
iajs-905	50	29	if	if	SCONJ
iajs-905	50	30	g	g	PROPN
iajs-905	50	31	is	be	AUX
iajs-905	50	32	abelian	abelian	ADJ
iajs-905	50	33	,	,	PUNCT
iajs-905	50	34	we	we	PRON
iajs-905	50	35	have	have	VERB
iajs-905	50	36	h	h	NOUN
iajs-905	50	37			NOUN
iajs-905	50	38	k	k	X
iajs-905	50	39	=	=	PUNCT
iajs-905	50	40	h.k	h.k	PROPN
iajs-905	50	41	,	,	PUNCT
iajs-905	50	42	since	since	SCONJ
iajs-905	50	43	h	h	PRON
iajs-905	50	44	.	.	PUNCT
iajs-905	51	1	k	k	PROPN
iajs-905	51	2			PROPN
iajs-905	51	3	h	h	PROPN
iajs-905	51	4			NOUN
iajs-905	51	5	k	k	PROPN
iajs-905	51	6	(	(	PUNCT
iajs-905	51	7	by	by	ADP
iajs-905	51	8	definition	definition	NOUN
iajs-905	51	9	2.4	2.4	NUM
iajs-905	51	10	)	)	PUNCT
iajs-905	51	11	and	and	CCONJ
iajs-905	51	12	since	since	SCONJ
iajs-905	51	13	:	:	PUNCT
iajs-905	51	14	h(x1	h(x1	ADJ
iajs-905	51	15	)	)	PUNCT
iajs-905	51	16	=	=	SYM
iajs-905	51	17	sup{min	sup{min	NOUN
iajs-905	51	18	{	{	PUNCT
iajs-905	51	19	h(x1	h(x1	NOUN
iajs-905	51	20	)	)	PUNCT
iajs-905	51	21	,	,	PUNCT
iajs-905	51	22	k(e	k(e	PROPN
iajs-905	51	23	)	)	PUNCT
iajs-905	51	24	}	}	PUNCT
iajs-905	52	1	x1	x1	NUM
iajs-905	52	2			NOUN
iajs-905	52	3	g	g	NOUN
iajs-905	52	4	}	}	PUNCT
iajs-905	52	5	and	and	CCONJ
iajs-905	52	6	k(x2	k(x2	NOUN
iajs-905	52	7	)	)	PUNCT
iajs-905	53	1	=	=	SYM
iajs-905	53	2	sup	sup	NOUN
iajs-905	53	3	{	{	PUNCT
iajs-905	53	4	min	min	NOUN
iajs-905	53	5	{	{	PUNCT
iajs-905	53	6	h(e	h(e	PROPN
iajs-905	53	7	)	)	PUNCT
iajs-905	53	8	,	,	PUNCT
iajs-905	53	9	k(x2	k(x2	NOUN
iajs-905	53	10	)	)	PUNCT
iajs-905	53	11	}	}	PUNCT
iajs-905	54	1	x2	x2	PRON
iajs-905	54	2			NOUN
iajs-905	54	3	g	g	PROPN
iajs-905	54	4	}	}	PUNCT
iajs-905	54	5	,	,	PUNCT
iajs-905	54	6	then	then	ADV
iajs-905	54	7	:	:	PUNCT
iajs-905	54	8	h	h	PROPN
iajs-905	54	9			PROPN
iajs-905	54	10	h	h	PROPN
iajs-905	54	11	.	.	PUNCT
iajs-905	55	1	k	k	PROPN
iajs-905	55	2	and	and	CCONJ
iajs-905	55	3	k	k	PROPN
iajs-905	55	4			PROPN
iajs-905	55	5	h	h	PROPN
iajs-905	55	6	.	.	PUNCT
iajs-905	56	1	k	k	X
iajs-905	56	2	,	,	PUNCT
iajs-905	56	3	so	so	ADV
iajs-905	57	1	h	h	NOUN
iajs-905	57	2			NOUN
iajs-905	57	3	k	k	PROPN
iajs-905	57	4			PROPN
iajs-905	57	5	h	h	NOUN
iajs-905	57	6	.	.	PUNCT
iajs-905	58	1	k	k	NOUN
iajs-905	58	2	,	,	PUNCT
iajs-905	58	3	thus	thus	ADV
iajs-905	58	4	h	h	NOUN
iajs-905	58	5			NOUN
iajs-905	58	6	k	k	X
iajs-905	58	7	=	=	PUNCT
iajs-905	58	8	h.k	h.k	PROPN
iajs-905	58	9	.	.	PUNCT
iajs-905	59	1	note	note	VERB
iajs-905	59	2	that	that	SCONJ
iajs-905	59	3	clearly	clearly	ADV
iajs-905	59	4	,	,	PUNCT
iajs-905	59	5	h	h	NOUN
iajs-905	59	6			NOUN
iajs-905	59	7	k	k	PROPN
iajs-905	59	8	would	would	AUX
iajs-905	59	9	be	be	AUX
iajs-905	59	10	contained	contain	VERB
iajs-905	59	11	in	in	ADP
iajs-905	59	12	any	any	DET
iajs-905	59	13	fuzzy	fuzzy	ADJ
iajs-905	59	14	subgroup	subgroup	NOUN
iajs-905	59	15	containing	contain	VERB
iajs-905	59	16	both	both	DET
iajs-905	59	17	h	h	NOUN
iajs-905	59	18	and	and	CCONJ
iajs-905	59	19	k	k	PROPN
iajs-905	59	20	.	.	PUNCT
iajs-905	60	1	thus	thus	ADV
iajs-905	60	2	we	we	PRON
iajs-905	60	3	see	see	VERB
iajs-905	60	4	that	that	SCONJ
iajs-905	60	5	h	h	PROPN
iajs-905	60	6	k	k	PROPN
iajs-905	60	7	is	be	AUX
iajs-905	60	8	the	the	DET
iajs-905	60	9	smallest	small	ADJ
iajs-905	60	10	fuzzy	fuzzy	ADJ
iajs-905	60	11	subgroup	subgroup	NOUN
iajs-905	60	12	of	of	ADP
iajs-905	60	13	g	g	NOUN
iajs-905	60	14	containing	contain	VERB
iajs-905	60	15	both	both	DET
iajs-905	60	16	h	h	NOUN
iajs-905	60	17	and	and	CCONJ
iajs-905	60	18	k	k	PROPN
iajs-905	60	19	.	.	PUNCT
iajs-905	61	1	theorem	theorem	VERB
iajs-905	61	2	2.5	2.5	NUM
iajs-905	61	3	a	a	DET
iajs-905	61	4	fuzzy	fuzzy	ADJ
iajs-905	61	5	subgroup	subgroup	NOUN
iajs-905	61	6	a	a	PRON
iajs-905	61	7	of	of	ADP
iajs-905	61	8	a	a	DET
iajs-905	61	9	group	group	NOUN
iajs-905	61	10	(	(	PUNCT
iajs-905	61	11	g	g	NOUN
iajs-905	61	12	,	,	PUNCT
iajs-905	61	13	.	.	PUNCT
iajs-905	61	14	)	)	PUNCT
iajs-905	62	1	is	be	AUX
iajs-905	62	2	the	the	DET
iajs-905	62	3	fuzzy	fuzzy	ADJ
iajs-905	62	4	internal	internal	ADJ
iajs-905	62	5	direct	direct	ADJ
iajs-905	62	6	product	product	NOUN
iajs-905	62	7	of	of	ADP
iajs-905	62	8	fuzzy	fuzzy	ADJ
iajs-905	62	9	subgroups	subgroup	NOUN
iajs-905	62	10	h	h	PROPN
iajs-905	62	11	and	and	CCONJ
iajs-905	62	12	k	k	PROPN
iajs-905	63	1	if	if	SCONJ
iajs-905	63	2	and	and	CCONJ
iajs-905	63	3	only	only	ADV
iajs-905	63	4	if	if	SCONJ
iajs-905	63	5	:	:	PUNCT
iajs-905	63	6			NOUN
iajs-905	63	7			NOUN
iajs-905	63	8			ADP
iajs-905	63	9			NUM
iajs-905	63	10			NUM
iajs-905	63	11	wiseother	wiseother	NOUN
iajs-905	63	12	xifgxxxxxxkxh	xifgxxxxxxkxh	VERB
iajs-905	63	13	k	k	PROPN
iajs-905	63	14	0	0	PUNCT
iajs-905	63	15	)	)	PUNCT
iajs-905	63	16	im(},,)}(),({{minsup	im(},,)}(),({{minsup	PROPN
iajs-905	63	17	.h	.h	NOUN
iajs-905	63	18	212121	212121	NUM
iajs-905	63	19	ihjpas	ihjpa	VERB
iajs-905	63	20	ibn	ibn	PROPN
iajs-905	63	21	alhaitham	alhaitham	PROPN
iajs-905	63	22	j.	j.	PROPN
iajs-905	64	1	fo	fo	ADP
iajs-905	64	2	r	r	NOUN
iajs-905	64	3	pure	pure	ADJ
iajs-905	64	4	&	&	CCONJ
iajs-905	64	5	appl	appl	PROPN
iajs-905	64	6	.	.	PUNCT
iajs-905	65	1	sc	sc	PROPN
iajs-905	65	2	i.	i.	PROPN
iajs-905	65	3	vo	vo	PROPN
iajs-905	65	4	l.23	l.23	PROPN
iajs-905	65	5	(	(	PUNCT
iajs-905	65	6	3	3	NUM
iajs-905	65	7	)	)	PUNCT
iajs-905	65	8	2010	2010	NUM
iajs-905	65	9	1	1	NUM
iajs-905	65	10	)	)	PUNCT
iajs-905	65	11	a	a	PRON
iajs-905	65	12	=	=	NOUN
iajs-905	65	13	h	h	NOUN
iajs-905	65	14			NOUN
iajs-905	65	15	k	k	PROPN
iajs-905	65	16	2	2	X
iajs-905	65	17	)	)	PUNCT
iajs-905	65	18	xt	xt	PUNCT
iajs-905	65	19	.	.	PUNCT
iajs-905	66	1	ys	ys	NOUN
iajs-905	66	2	=	=	PUNCT
iajs-905	66	3	y	y	PROPN
iajs-905	66	4	s	s	PROPN
iajs-905	66	5	.	.	PUNCT
iajs-905	67	1	xt	xt	X
iajs-905	68	1	for	for	ADP
iajs-905	68	2	all	all	DET
iajs-905	68	3	xt	xt	PROPN
iajs-905	68	4			PROPN
iajs-905	68	5	h	h	PROPN
iajs-905	68	6	and	and	CCONJ
iajs-905	68	7	ys	ys	PROPN
iajs-905	68	8			PROPN
iajs-905	68	9	k	k	PROPN
iajs-905	68	10	,	,	PUNCT
iajs-905	68	11	t	t	PROPN
iajs-905	68	12	,	,	PUNCT
iajs-905	68	13	s	s	PART
iajs-905	68	14			NOUN
iajs-905	69	1	[	[	X
iajs-905	69	2	0,1	0,1	NUM
iajs-905	69	3	]	]	PUNCT
iajs-905	69	4	,	,	PUNCT
iajs-905	69	5	x	x	PRON
iajs-905	69	6	,	,	PUNCT
iajs-905	69	7	y	y	PROPN
iajs-905	69	8			PROPN
iajs-905	69	9	g.	g.	PROPN
iajs-905	69	10	3	3	NUM
iajs-905	69	11	)	)	PUNCT
iajs-905	69	12	h	h	NOUN
iajs-905	69	13			PUNCT
iajs-905	69	14	k	k	NOUN
iajs-905	69	15	=	=	PUNCT
iajs-905	69	16	{	{	PUNCT
iajs-905	69	17	e	e	NOUN
iajs-905	69	18	}	}	PUNCT
iajs-905	69	19	proof	proof	NOUN
iajs-905	69	20	:	:	PUNCT
iajs-905	69	21	let	let	VERB
iajs-905	69	22	a	a	PRON
iajs-905	69	23	be	be	AUX
iajs-905	69	24	the	the	DET
iajs-905	69	25	fuzzy	fuzzy	ADJ
iajs-905	69	26	internal	internal	ADJ
iajs-905	69	27	direct	direct	ADJ
iajs-905	69	28	p	p	NOUN
iajs-905	69	29	roduct	roduct	NOUN
iajs-905	69	30	of	of	ADP
iajs-905	69	31	h	h	NOUN
iajs-905	69	32	and	and	CCONJ
iajs-905	69	33	k.	k.	NOUN
iajs-905	70	1	we	we	PRON
iajs-905	70	2	claim	claim	VERB
iajs-905	70	3	that	that	SCONJ
iajs-905	70	4	(	(	PUNCT
iajs-905	70	5	1	1	NUM
iajs-905	70	6	)	)	PUNCT
iajs-905	70	7	,	,	PUNCT
iajs-905	70	8	(	(	PUNCT
iajs-905	70	9	2	2	X
iajs-905	70	10	)	)	PUNCT
iajs-905	70	11	and	and	CCONJ
iajs-905	70	12	(	(	PUNCT
iajs-905	70	13	3	3	X
iajs-905	70	14	)	)	PUNCT
iajs-905	70	15	are	be	AUX
iajs-905	70	16	obvious	obvious	ADJ
iajs-905	70	17	if	if	SCONJ
iajs-905	70	18	one	one	PRON
iajs-905	70	19	will	will	AUX
iajs-905	70	20	regard	regard	VERB
iajs-905	70	21	a	a	DET
iajs-905	70	22	as	as	ADV
iajs-905	70	23	isomorphic	isomorphic	ADJ
iajs-905	70	24	to	to	ADP
iajs-905	70	25	the	the	DET
iajs-905	70	26	fuzzy	fuzzy	ADJ
iajs-905	70	27	internal	internal	ADJ
iajs-905	70	28	direct	direct	ADJ
iajs-905	70	29	p	p	NOUN
iajs-905	70	30	roduct	roduct	NOUN
iajs-905	70	31	of	of	ADP
iajs-905	70	32	h	h	NOUN
iajs-905	70	33	and	and	CCONJ
iajs-905	70	34	k	k	PROPN
iajs-905	70	35	under	under	ADP
iajs-905	70	36	the	the	DET
iajs-905	70	37	map	map	NOUN
iajs-905	70	38			NOUN
iajs-905	70	39	,	,	PUNCT
iajs-905	70	40	with	with	ADP
iajs-905	70	41	(xt	(xt	NOUN
iajs-905	70	42	,	,	PUNCT
iajs-905	70	43	ys	ys	NOUN
iajs-905	70	44	)	)	PUNCT
iajs-905	71	1	=	=	SYM
iajs-905	71	2	xt	xt	PROPN
iajs-905	71	3	.	.	PUNCT
iajs-905	72	1	ys	ys	PROPN
iajs-905	72	2	=	=	PUNCT
iajs-905	73	1	(	(	PUNCT
iajs-905	73	2	x	x	INTJ
iajs-905	73	3	.	.	PUNCT
iajs-905	73	4	y)r	y)r	X
iajs-905	74	1	where	where	SCONJ
iajs-905	74	2	r	r	NOUN
iajs-905	74	3	=	=	SYM
iajs-905	74	4	min	min	PROPN
iajs-905	74	5	{	{	PUNCT
iajs-905	74	6	t	t	PROPN
iajs-905	74	7	,	,	PUNCT
iajs-905	74	8	s	s	AUX
iajs-905	74	9	}	}	PUNCT
iajs-905	74	10	,	,	PUNCT
iajs-905	74	11	x	x	X
iajs-905	74	12	,	,	PUNCT
iajs-905	74	13	y	y	PROPN
iajs-905	74	14			PROPN
iajs-905	74	15	g	g	PROPN
iajs-905	74	16	,	,	PUNCT
iajs-905	74	17	r	r	NOUN
iajs-905	74	18	,	,	PUNCT
iajs-905	74	19	s	s	AUX
iajs-905	74	20			NOUN
iajs-905	74	21	[	[	X
iajs-905	74	22	0	0	NUM
iajs-905	74	23	,	,	PUNCT
iajs-905	74	24	1	1	NUM
iajs-905	74	25	]	]	PUNCT
iajs-905	74	26	.	.	PUNCT
iajs-905	75	1	under	under	ADP
iajs-905	75	2	this	this	DET
iajs-905	75	3	map	map	NOUN
iajs-905	75	4	corresponds	correspond	VERB
iajs-905	75	5	to	to	ADP
iajs-905	75	6	h	h	NOUN
iajs-905	75	7	,	,	PUNCT
iajs-905	75	8	and	and	CCONJ
iajs-905	75	9	corresponds	correspond	VERB
iajs-905	75	10	to	to	ADP
iajs-905	75	11	k	k	PROPN
iajs-905	75	12	.	.	PUNCT
iajs-905	76	1	then	then	ADV
iajs-905	76	2	(	(	PUNCT
iajs-905	76	3	1	1	NUM
iajs-905	76	4	)	)	PUNCT
iajs-905	76	5	,	,	PUNCT
iajs-905	76	6	(	(	PUNCT
iajs-905	76	7	2	2	X
iajs-905	76	8	)	)	PUNCT
iajs-905	76	9	and	and	CCONJ
iajs-905	76	10	(	(	PUNCT
iajs-905	76	11	3	3	X
iajs-905	76	12	)	)	PUNCT
iajs-905	76	13	follow	follow	VERB
iajs-905	76	14	immediately	immediately	ADV
iajs-905	76	15	from	from	ADP
iajs-905	76	16	the	the	DET
iajs-905	76	17	corresponding	corresponding	ADJ
iajs-905	76	18	assertions	assertion	NOUN
iajs-905	76	19	regarding	regard	VERB
iajs-905	76	20	in	in	ADP
iajs-905	76	21	h	h	PROPN
iajs-905	76	22			PROPN
iajs-905	76	23	k	k	NOUN
iajs-905	76	24	,	,	PUNCT
iajs-905	76	25	which	which	PRON
iajs-905	76	26	are	be	AUX
iajs-905	76	27	obvious	obvious	ADJ
iajs-905	76	28	.	.	PUNCT
iajs-905	77	1	conversely	conversely	ADV
iajs-905	77	2	,	,	PUNCT
iajs-905	77	3	let	let	VERB
iajs-905	77	4	(	(	PUNCT
iajs-905	77	5	1	1	NUM
iajs-905	77	6	)	)	PUNCT
iajs-905	77	7	,	,	PUNCT
iajs-905	77	8	(	(	PUNCT
iajs-905	77	9	2	2	X
iajs-905	77	10	)	)	PUNCT
iajs-905	77	11	and	and	CCONJ
iajs-905	77	12	(	(	PUNCT
iajs-905	77	13	3	3	X
iajs-905	77	14	)	)	PUNCT
iajs-905	77	15	hold	hold	NOUN
iajs-905	77	16	.	.	PUNCT
iajs-905	78	1	we	we	PRON
iajs-905	78	2	must	must	AUX
iajs-905	78	3	show	show	VERB
iajs-905	78	4	that	that	SCONJ
iajs-905	78	5	the	the	DET
iajs-905	78	6	map	map	NOUN
iajs-905	78	7			NOUN
iajs-905	78	8	of	of	ADP
iajs-905	78	9	the	the	DET
iajs-905	78	10	fuzzy	fuzzy	ADJ
iajs-905	78	11	internal	internal	ADJ
iajs-905	78	12	direct	direct	ADJ
iajs-905	78	13	product	product	NOUN
iajs-905	78	14	h	h	NOUN
iajs-905	78	15			NOUN
iajs-905	78	16	k	k	PROPN
iajs-905	78	17	in	in	ADP
iajs-905	78	18	to	to	ADP
iajs-905	78	19	a	a	PRON
iajs-905	78	20	,	,	PUNCT
iajs-905	78	21	given	give	VERB
iajs-905	78	22	by	by	ADP
iajs-905	78	23	:	:	PUNCT
iajs-905	78	24	(xt	(xt	NOUN
iajs-905	78	25	,	,	PUNCT
iajs-905	78	26	ys	ys	NOUN
iajs-905	78	27	)	)	PUNCT
iajs-905	78	28	=	=	SYM
iajs-905	78	29	xt	xt	PROPN
iajs-905	78	30	.	.	PUNCT
iajs-905	79	1	ys	ys	PROPN
iajs-905	79	2	=	=	PUNCT
iajs-905	80	1	(	(	PUNCT
iajs-905	80	2	x.y)r	x.y)r	X
iajs-905	80	3	where	where	SCONJ
iajs-905	80	4	r	r	NOUN
iajs-905	80	5	=	=	SYM
iajs-905	80	6	min{t	min{t	PROPN
iajs-905	80	7	,	,	PUNCT
iajs-905	80	8	s	s	PART
iajs-905	80	9	}	}	PUNCT
iajs-905	80	10	,	,	PUNCT
iajs-905	80	11	x	x	X
iajs-905	80	12	,	,	PUNCT
iajs-905	80	13	y	y	PROPN
iajs-905	80	14			PROPN
iajs-905	80	15	g	g	PROPN
iajs-905	80	16	,	,	PUNCT
iajs-905	80	17	r	r	NOUN
iajs-905	80	18	,	,	PUNCT
iajs-905	80	19	s	s	PART
iajs-905	80	20			NOUN
iajs-905	81	1	[	[	X
iajs-905	81	2	0,1	0,1	NUM
iajs-905	81	3	]	]	PUNCT
iajs-905	81	4	,	,	PUNCT
iajs-905	81	5	is	be	AUX
iajs-905	81	6	an	an	DET
iajs-905	81	7	isomorphism	isomorphism	NOUN
iajs-905	81	8	.	.	PUNCT
iajs-905	82	1	the	the	DET
iajs-905	82	2	map	map	NOUN
iajs-905	82	3			NOUN
iajs-905	82	4	has	have	AUX
iajs-905	82	5	already	already	ADV
iajs-905	82	6	been	be	AUX
iajs-905	82	7	defined	define	VERB
iajs-905	82	8	suppose	suppose	VERB
iajs-905	82	9	(xt1	(xt1	NOUN
iajs-905	82	10	,	,	PUNCT
iajs-905	82	11	ys1	ys1	NUM
iajs-905	82	12	)	)	PUNCT
iajs-905	82	13	=	=	SYM
iajs-905	82	14	(xt2	(xt2	NOUN
iajs-905	82	15	,	,	PUNCT
iajs-905	82	16	ys2	ys2	NOUN
iajs-905	82	17	)	)	PUNCT
iajs-905	82	18	.	.	PUNCT
iajs-905	83	1	then	then	ADV
iajs-905	83	2	xt1	xt1	PROPN
iajs-905	83	3	.	.	PUNCT
iajs-905	84	1	ys1	ys1	X
iajs-905	84	2	=	=	PUNCT
iajs-905	84	3	xt2	xt2	PROPN
iajs-905	84	4	.	.	PUNCT
iajs-905	85	1	ys2	ys2	PROPN
iajs-905	85	2	;	;	PUNCT
iajs-905	85	3	consequently	consequently	ADV
iajs-905	85	4	but	but	CCONJ
iajs-905	85	5	x	x	X
iajs-905	85	6	-1	-1	PUNCT
iajs-905	85	7	t2	t2	PROPN
iajs-905	85	8	.	.	PUNCT
iajs-905	86	1	xt1	xt1	PROPN
iajs-905	86	2			PROPN
iajs-905	86	3	h	h	PROPN
iajs-905	86	4	and	and	CCONJ
iajs-905	86	5	ys2	ys2	NOUN
iajs-905	86	6	.	.	PUNCT
iajs-905	87	1	y	y	PROPN
iajs-905	87	2	-1	-1	PROPN
iajs-905	87	3	s1	s1	PROPN
iajs-905	87	4			PROPN
iajs-905	87	5	k	k	PROPN
iajs-905	87	6	and	and	CCONJ
iajs-905	87	7	they	they	PRON
iajs-905	87	8	are	be	AUX
iajs-905	87	9	the	the	DET
iajs-905	87	10	same	same	ADJ
iajs-905	87	11	element	element	NOUN
iajs-905	87	12	and	and	CCONJ
iajs-905	87	13	thus	thus	ADV
iajs-905	87	14	in	in	ADP
iajs-905	87	15	h	h	NOUN
iajs-905	87	16			PUNCT
iajs-905	88	1	k	k	NOUN
iajs-905	88	2	=	=	PUNCT
iajs-905	88	3	{	{	PUNCT
iajs-905	88	4	e	e	NOUN
iajs-905	88	5	}	}	PUNCT
iajs-905	88	6	by	by	ADP
iajs-905	88	7	(	(	PUNCT
iajs-905	88	8	3	3	NUM
iajs-905	88	9	)	)	PUNCT
iajs-905	88	10	.	.	PUNCT
iajs-905	89	1	therefore	therefore	ADV
iajs-905	89	2	,	,	PUNCT
iajs-905	89	3	x-1	x-1	PROPN
iajs-905	89	4	t2	t2	PROPN
iajs-905	89	5	.	.	PUNCT
iajs-905	89	6	xt1	xt1	X
iajs-905	90	1	=	=	PUNCT
iajs-905	90	2	e	e	PROPN
iajs-905	90	3	and	and	CCONJ
iajs-905	90	4	xt1	xt1	X
iajs-905	91	1	=	=	PROPN
iajs-905	91	2	xt2	xt2	PROPN
iajs-905	91	3	.	.	PUNCT
iajs-905	92	1	likewise	likewise	ADV
iajs-905	92	2	,	,	PUNCT
iajs-905	92	3	ys1	ys1	PROPN
iajs-905	92	4	=	=	SYM
iajs-905	92	5	y	y	PROPN
iajs-905	92	6	s2	s2	PROPN
iajs-905	92	7	,	,	PUNCT
iajs-905	92	8	so	so	CCONJ
iajs-905	92	9	(	(	PUNCT
iajs-905	92	10	xt1	xt1	X
iajs-905	92	11	,	,	PUNCT
iajs-905	92	12	ys1)=	ys1)=	X
iajs-905	92	13	(	(	PUNCT
iajs-905	92	14	xt2	xt2	PROPN
iajs-905	92	15	,	,	PUNCT
iajs-905	92	16	ys2	ys2	PROPN
iajs-905	92	17	)	)	PUNCT
iajs-905	92	18	.	.	PUNCT
iajs-905	93	1	this	this	PRON
iajs-905	93	2	shows	show	VERB
iajs-905	93	3	that	that	SCONJ
iajs-905	93	4			NOUN
iajs-905	93	5	is	be	AUX
iajs-905	93	6	one	one	NUM
iajs-905	93	7	to	to	ADP
iajs-905	93	8	one	one	NUM
iajs-905	93	9	.	.	PUNCT
iajs-905	94	1	the	the	DET
iajs-905	94	2	fact	fact	NOUN
iajs-905	94	3	that	that	SCONJ
iajs-905	94	4	xt	xt	VERB
iajs-905	94	5	.	.	PUNCT
iajs-905	95	1	ys	ys	PROPN
iajs-905	95	2	=	=	PUNCT
iajs-905	95	3	y	y	PROPN
iajs-905	95	4	s	s	PROPN
iajs-905	95	5	.	.	PUNCT
iajs-905	95	6	xt	xt	PUNCT
iajs-905	96	1	by	by	ADP
iajs-905	96	2	(	(	PUNCT
iajs-905	96	3	2	2	NUM
iajs-905	96	4	)	)	PUNCT
iajs-905	96	5	for	for	ADP
iajs-905	96	6	all	all	DET
iajs-905	96	7	xt	xt	PROPN
iajs-905	96	8			PROPN
iajs-905	96	9	h	h	PROPN
iajs-905	96	10	and	and	CCONJ
iajs-905	96	11	ys	ys	PROPN
iajs-905	96	12			PROPN
iajs-905	96	13	k	k	PROPN
iajs-905	96	14	means	mean	VERB
iajs-905	96	15	that	that	SCONJ
iajs-905	96	16	h.k(x	h.k(x	VERB
iajs-905	96	17	)	)	PUNCT
iajs-905	96	18	=	=	PRON
iajs-905	96	19	{	{	PUNCT
iajs-905	96	20	sup	sup	NOUN
iajs-905	96	21	{	{	PUNCT
iajs-905	96	22	min{h(x1	min{h(x1	NOUN
iajs-905	96	23	)	)	PUNCT
iajs-905	96	24	,	,	PUNCT
iajs-905	96	25	k(x2	k(x2	NOUN
iajs-905	96	26	)	)	PUNCT
iajs-905	96	27	}	}	PUNCT
iajs-905	96	28	}	}	PUNCT
iajs-905	96	29	;	;	PUNCT
iajs-905	97	1	x	x	SYM
iajs-905	97	2	=	=	SYM
iajs-905	97	3	x1	x1	PROPN
iajs-905	97	4	.	.	PUNCT
iajs-905	98	1	x2	x2	INTJ
iajs-905	98	2	,	,	PUNCT
iajs-905	98	3	x1	x1	PROPN
iajs-905	98	4	,	,	PUNCT
iajs-905	98	5	x2	x2	PROPN
iajs-905	98	6			NOUN
iajs-905	98	7	g	g	NOUN
iajs-905	98	8	}	}	PUNCT
iajs-905	98	9	is	be	AUX
iajs-905	98	10	a	a	DET
iajs-905	98	11	fuzzy	fuzzy	ADJ
iajs-905	98	12	subgroup	subgroup	NOUN
iajs-905	98	13	,	,	PUNCT
iajs-905	98	14	for	for	SCONJ
iajs-905	98	15	we	we	PRON
iajs-905	98	16	have	have	AUX
iajs-905	98	17	seen	see	VERB
iajs-905	98	18	that	that	SCONJ
iajs-905	98	19	this	this	PRON
iajs-905	98	20	is	be	AUX
iajs-905	98	21	the	the	DET
iajs-905	98	22	case	case	NOUN
iajs-905	98	23	if	if	SCONJ
iajs-905	98	24	fuzzy	fuzzy	ADJ
iajs-905	98	25	singletons	singleton	NOUN
iajs-905	98	26	of	of	ADP
iajs-905	98	27	h	h	NOUN
iajs-905	98	28	commute	commute	NOUN
iajs-905	98	29	with	with	ADP
iajs-905	98	30	those	those	PRON
iajs-905	98	31	of	of	ADP
iajs-905	98	32	k	k	PROPN
iajs-905	98	33	.	.	PUNCT
iajs-905	99	1	thus	thus	ADV
iajs-905	99	2	by	by	ADP
iajs-905	99	3	(	(	PUNCT
iajs-905	99	4	1	1	NUM
iajs-905	99	5	)	)	PUNCT
iajs-905	99	6	,	,	PUNCT
iajs-905	99	7	h.	h.	PROPN
iajs-905	99	8	k	k	PROPN
iajs-905	100	1	=	=	PUNCT
iajs-905	100	2	h	h	PROPN
iajs-905	100	3			NOUN
iajs-905	101	1	k	k	X
iajs-905	101	2	=	=	PUNCT
iajs-905	101	3	a	a	NOUN
iajs-905	101	4	,	,	PUNCT
iajs-905	101	5	so	so	ADV
iajs-905	101	6			NOUN
iajs-905	101	7	is	be	AUX
iajs-905	101	8	on	on	ADP
iajs-905	101	9	to	to	ADP
iajs-905	101	10	a	a	PRON
iajs-905	101	11	.	.	PUNCT
iajs-905	102	1	finally	finally	ADV
iajs-905	102	2	,	,	PUNCT
iajs-905	102	3	[(xt1	[(xt1	NOUN
iajs-905	102	4	,	,	PUNCT
iajs-905	102	5	ys1)(xt2	ys1)(xt2	NOUN
iajs-905	102	6	,	,	PUNCT
iajs-905	102	7	ys2	ys2	NOUN
iajs-905	102	8	)	)	PUNCT
iajs-905	102	9	]	]	PUNCT
iajs-905	103	1	=	=	SYM
iajs-905	103	2	(xt1.xt2	(xt1.xt2	X
iajs-905	103	3	,	,	PUNCT
iajs-905	103	4	ys1.ys2	ys1.ys2	PROPN
iajs-905	103	5	)	)	PUNCT
iajs-905	103	6	=	=	SYM
iajs-905	103	7	xt1.xt2.ys1.ys2	xt1.xt2.ys1.ys2	PROPN
iajs-905	103	8	,	,	PUNCT
iajs-905	103	9	while	while	SCONJ
iajs-905	103	10	[	[	X
iajs-905	103	11	(xt1,ys1)][(xt2,ys2	(xt1,ys1)][(xt2,ys2	NUM
iajs-905	103	12	)	)	PUNCT
iajs-905	103	13	]	]	PUNCT
iajs-905	104	1	=	=	SYM
iajs-905	104	2	xt1.ys1.xt2.ys2	xt1.ys1.xt2.ys2	PROPN
iajs-905	104	3	.	.	PUNCT
iajs-905	105	1	but	but	CCONJ
iajs-905	105	2	by	by	ADP
iajs-905	105	3	(	(	PUNCT
iajs-905	105	4	2	2	X
iajs-905	105	5	)	)	PUNCT
iajs-905	105	6	we	we	PRON
iajs-905	105	7	have	have	VERB
iajs-905	105	8	ys1	ys1	X
iajs-905	105	9	.	.	PUNCT
iajs-905	106	1	xt2	xt2	X
iajs-905	106	2	=	=	PUNCT
iajs-905	106	3	xt2	xt2	PROPN
iajs-905	106	4	.	.	PUNCT
iajs-905	107	1	ys1	ys1	INTJ
iajs-905	107	2	.	.	PUNCT
iajs-905	108	1	thus	thus	ADV
iajs-905	108	2	:	:	PUNCT
iajs-905	108	3	[(xt1,ys1)(xt2,ys2	[(xt1,ys1)(xt2,ys2	NOUN
iajs-905	108	4	)	)	PUNCT
iajs-905	108	5	]	]	PUNCT
iajs-905	109	1	=	=	PUNCT
iajs-905	110	1	[	[	X
iajs-905	110	2	(xt1,ys1)][(xt2	(xt1,ys1)][(xt2	VERB
iajs-905	110	3	,	,	PUNCT
iajs-905	110	4	ys2	ys2	NOUN
iajs-905	110	5	)	)	PUNCT
iajs-905	110	6	]	]	PUNCT
iajs-905	110	7	.	.	PUNCT
iajs-905	111	1	remark	remark	NOUN
iajs-905	111	2	:	:	PUNCT
iajs-905	111	3	not	not	PART
iajs-905	111	4	every	every	DET
iajs-905	111	5	fuzzy	fuzzy	ADJ
iajs-905	111	6	subgroup	subgroup	NOUN
iajs-905	111	7	of	of	ADP
iajs-905	111	8	abelian	abelian	PROPN
iajs-905	111	9	group	group	NOUN
iajs-905	111	10	is	be	AUX
iajs-905	111	11	the	the	DET
iajs-905	111	12	fuzzy	fuzzy	ADJ
iajs-905	111	13	internal	internal	ADJ
iajs-905	111	14	direct	direct	ADJ
iajs-905	111	15	p	p	NOUN
iajs-905	111	16	roduct	roduct	NOUN
iajs-905	111	17	of	of	ADP
iajs-905	111	18	two	two	NUM
iajs-905	111	19	proper	proper	ADJ
iajs-905	111	20	fuzzy	fuzzy	ADJ
iajs-905	111	21	subgroups	subgroup	NOUN
iajs-905	111	22	.	.	PUNCT
iajs-905	112	1	the	the	DET
iajs-905	112	2	following	follow	VERB
iajs-905	112	3	corollary	corollary	NOUN
iajs-905	112	4	is	be	AUX
iajs-905	112	5	immediate	immediate	ADJ
iajs-905	112	6	consequences	consequence	NOUN
iajs-905	112	7	of	of	ADP
iajs-905	112	8	the	the	DET
iajs-905	112	9	above	above	ADJ
iajs-905	112	10	theorem	theorem	NOUN
iajs-905	112	11	.	.	PUNCT
iajs-905	112	12	}	}	PUNCT
iajs-905	112	13	hx|)e	hx|)e	PROPN
iajs-905	112	14	,	,	PUNCT
iajs-905	112	15	x{(h	x{(h	PROPN
iajs-905	112	16	tt	tt	PROPN
iajs-905	112	17			NOUN
iajs-905	112	18	}	}	PUNCT
iajs-905	112	19	|	|	ADV
iajs-905	112	20	)	)	PUNCT
iajs-905	112	21	,	,	PUNCT
iajs-905	112	22	{	{	PUNCT
iajs-905	112	23	(	(	PUNCT
iajs-905	112	24	kyyek	kyyek	NOUN
iajs-905	112	25	ss	ss	NOUN
iajs-905	112	26			PROPN
iajs-905	112	27	kandh	kandh	PROPN
iajs-905	112	28	1	1	NUM
iajs-905	112	29	1s2s1	1s2s1	NUM
iajs-905	112	30	t	t	NOUN
iajs-905	112	31	1	1	NUM
iajs-905	112	32	2	2	NUM
iajs-905	112	33	t	t	NOUN
iajs-905	112	34	yyxx	yyxx	NOUN
iajs-905	112	35			NOUN
iajs-905	112	36			NOUN
iajs-905	112	37	ihjpas	ihjpa	VERB
iajs-905	112	38	ibn	ibn	PROPN
iajs-905	112	39	alhaitham	alhaitham	NOUN
iajs-905	113	1	j.	j.	PROPN
iajs-905	114	1	fo	fo	ADP
iajs-905	114	2	r	r	NOUN
iajs-905	114	3	pure	pure	ADJ
iajs-905	114	4	&	&	CCONJ
iajs-905	114	5	appl	appl	PROPN
iajs-905	114	6	.	.	PUNCT
iajs-905	115	1	sc	sc	PROPN
iajs-905	115	2	i.	i.	PROPN
iajs-905	115	3	vo	vo	PROPN
iajs-905	115	4	l.23	l.23	PROPN
iajs-905	115	5	(	(	PUNCT
iajs-905	115	6	3	3	NUM
iajs-905	115	7	)	)	PUNCT
iajs-905	115	8	2010	2010	NUM
iajs-905	115	9	corollary	corollary	NOUN
iajs-905	115	10	2.6	2.6	NUM
iajs-905	115	11	let	let	VERB
iajs-905	115	12	a	a	PRON
iajs-905	115	13	be	be	AUX
iajs-905	115	14	fuzzy	fuzzy	ADJ
iajs-905	115	15	subgroup	subgroup	NOUN
iajs-905	115	16	of	of	ADP
iajs-905	115	17	a	a	DET
iajs-905	115	18	group	group	NOUN
iajs-905	115	19	(	(	PUNCT
iajs-905	115	20	g	g	NOUN
iajs-905	115	21	,	,	PUNCT
iajs-905	115	22	.	.	PUNCT
iajs-905	115	23	)	)	PUNCT
iajs-905	115	24	.	.	PUNCT
iajs-905	116	1	let	let	VERB
iajs-905	116	2	xt	xt	X
iajs-905	117	1	and	and	CCONJ
iajs-905	117	2	ys	ys	PROPN
iajs-905	117	3	be	be	AUX
iajs-905	117	4	fuzzy	fuzzy	ADJ
iajs-905	117	5	singletons	singleton	NOUN
iajs-905	117	6	of	of	ADP
iajs-905	117	7	a	a	DET
iajs-905	117	8	which	which	DET
iajs-905	117	9	commute	commute	NOUN
iajs-905	117	10	and	and	CCONJ
iajs-905	117	11	are	be	AUX
iajs-905	117	12	of	of	ADP
iajs-905	117	13	relatively	relatively	ADV
iajs-905	117	14	prime	prime	ADJ
iajs-905	117	15	orders	order	NOUN
iajs-905	117	16	r	r	NOUN
iajs-905	117	17	and	and	CCONJ
iajs-905	117	18	s	s	PROPN
iajs-905	117	19	and	and	CCONJ
iajs-905	117	20	<	<	X
iajs-905	117	21	xt	xt	X
iajs-905	117	22	>	>	X
iajs-905	117	23	,	,	PUNCT
iajs-905	117	24	<	<	X
iajs-905	117	25	ys	ys	X
iajs-905	117	26	>	>	X
iajs-905	117	27	are	be	AUX
iajs-905	117	28	fuzzy	fuzzy	ADJ
iajs-905	117	29	subgroups	subgroup	NOUN
iajs-905	117	30	of	of	ADP
iajs-905	117	31	<	<	X
iajs-905	117	32	xt	xt	X
iajs-905	117	33	>	>	X
iajs-905	117	34			NOUN
iajs-905	117	35	<	<	X
iajs-905	117	36	ys	ys	X
iajs-905	117	37	>	>	X
iajs-905	117	38	.	.	PUNCT
iajs-905	118	1	then	then	ADV
iajs-905	118	2	xt.ys	xt.ys	PROPN
iajs-905	118	3	is	be	AUX
iajs-905	118	4	of	of	ADP
iajs-905	118	5	order	order	NOUN
iajs-905	119	1	r.s	r.s	PROPN
iajs-905	119	2	.	.	PUNCT
iajs-905	120	1	fuzzy	fuzzy	ADJ
iajs-905	120	2	invariants	invariant	NOUN
iajs-905	120	3	of	of	ADP
iajs-905	120	4	fuzzy	fuzzy	ADJ
iajs-905	120	5	subgroup	subgroup	NOUN
iajs-905	120	6	in	in	ADP
iajs-905	120	7	this	this	DET
iajs-905	120	8	section	section	NOUN
iajs-905	120	9	we	we	PRON
iajs-905	120	10	introduce	introduce	VERB
iajs-905	120	11	the	the	DET
iajs-905	120	12	following	follow	VERB
iajs-905	120	13	definition	definition	NOUN
iajs-905	120	14	and	and	CCONJ
iajs-905	120	15	lemma	lemma	PROPN
iajs-905	120	16	about	about	ADP
iajs-905	120	17	fuzzy	fuzzy	ADJ
iajs-905	120	18	invariants	invariant	NOUN
iajs-905	120	19	of	of	ADP
iajs-905	120	20	fuzzy	fuzzy	ADJ
iajs-905	120	21	subgroup	subgroup	NOUN
iajs-905	120	22	:	:	PUNCT
iajs-905	120	23	definition	definition	NOUN
iajs-905	120	24	3.1	3.1	NUM
iajs-905	120	25	let	let	VERB
iajs-905	120	26	a	a	PRON
iajs-905	120	27	be	be	AUX
iajs-905	120	28	fuzzy	fuzzy	ADJ
iajs-905	120	29	subgroup	subgroup	NOUN
iajs-905	120	30	of	of	ADP
iajs-905	120	31	abelian	abelian	PROPN
iajs-905	120	32	group	group	NOUN
iajs-905	120	33	(	(	PUNCT
iajs-905	120	34	g	g	NOUN
iajs-905	120	35	,	,	PUNCT
iajs-905	120	36	.	.	PUNCT
iajs-905	120	37	)	)	PUNCT
iajs-905	121	1	of	of	ADP
iajs-905	121	2	order	order	NOUN
iajs-905	121	3	pn	pn	NOUN
iajs-905	121	4	,	,	PUNCT
iajs-905	121	5	p	p	X
iajs-905	121	6	a	a	DET
iajs-905	121	7	prime	prime	NOUN
iajs-905	121	8	,	,	PUNCT
iajs-905	121	9	and	and	CCONJ
iajs-905	121	10	a	a	DET
iajs-905	121	11	=	=	NOUN
iajs-905	121	12	a1	a1	NOUN
iajs-905	121	13			PROPN
iajs-905	121	14	a2	a2	PROPN
iajs-905	121	15			PROPN
iajs-905	121	16	…	…	PROPN
iajs-905	121	17	.	.	PROPN
iajs-905	121	18	ak	ak	PROPN
iajs-905	121	19	where	where	SCONJ
iajs-905	121	20	each	each	PRON
iajs-905	121	21	ai	ai	VERB
iajs-905	121	22	is	be	AUX
iajs-905	121	23	fuzzy	fuzzy	ADJ
iajs-905	121	24	generating	generate	VERB
iajs-905	121	25	set	set	NOUN
iajs-905	121	26	of	of	ADP
iajs-905	121	27	order	order	NOUN
iajs-905	121	28	pni	pni	NOUN
iajs-905	121	29	with	with	ADP
iajs-905	121	30	n1	n1	PROPN
iajs-905	121	31			NUM
iajs-905	121	32	n2	n2	NOUN
iajs-905	121	33			NUM
iajs-905	121	34	…	…	SYM
iajs-905	121	35			X
iajs-905	121	36	nk	nk	PROPN
iajs-905	121	37	>	>	X
iajs-905	121	38	0	0	PROPN
iajs-905	121	39	,	,	PUNCT
iajs-905	121	40	then	then	ADV
iajs-905	121	41	the	the	DET
iajs-905	121	42	integers	integer	NOUN
iajs-905	121	43	n1	n1	NOUN
iajs-905	121	44	,	,	PUNCT
iajs-905	121	45	n2	n2	NOUN
iajs-905	121	46	,	,	PUNCT
iajs-905	121	47	…	…	PUNCT
iajs-905	121	48	,	,	PUNCT
iajs-905	121	49	nk	nk	PROPN
iajs-905	121	50	are	be	AUX
iajs-905	121	51	called	call	VERB
iajs-905	121	52	the	the	DET
iajs-905	121	53	fuzzy	fuzzy	ADJ
iajs-905	121	54	invariants	invariant	NOUN
iajs-905	121	55	of	of	ADP
iajs-905	121	56	a	a	PRON
iajs-905	121	57	just	just	ADV
iajs-905	121	58	because	because	SCONJ
iajs-905	121	59	we	we	PRON
iajs-905	121	60	called	call	VERB
iajs-905	121	61	the	the	DET
iajs-905	121	62	integers	integer	NOUN
iajs-905	121	63	above	above	ADP
iajs-905	121	64	the	the	DET
iajs-905	121	65	fuzzy	fuzzy	ADJ
iajs-905	121	66	invariants	invariant	NOUN
iajs-905	121	67	of	of	ADP
iajs-905	121	68	a	a	PRON
iajs-905	121	69	does	do	AUX
iajs-905	121	70	not	not	PART
iajs-905	121	71	mean	mean	VERB
iajs-905	121	72	that	that	SCONJ
iajs-905	121	73	they	they	PRON
iajs-905	121	74	are	be	AUX
iajs-905	121	75	really	really	ADV
iajs-905	121	76	the	the	DET
iajs-905	121	77	fuzzy	fuzzy	ADJ
iajs-905	121	78	invariants	invariant	NOUN
iajs-905	121	79	of	of	ADP
iajs-905	121	80	a.	a.	NOUN
iajs-905	121	81	that	that	PRON
iajs-905	121	82	is	be	AUX
iajs-905	121	83	,	,	PUNCT
iajs-905	121	84	it	it	PRON
iajs-905	121	85	is	be	AUX
iajs-905	121	86	possible	possible	ADJ
iajs-905	121	87	to	to	PART
iajs-905	121	88	assign	assign	VERB
iajs-905	121	89	different	different	ADJ
iajs-905	121	90	sets	set	NOUN
iajs-905	121	91	of	of	ADP
iajs-905	121	92	fuzzy	fuzzy	ADJ
iajs-905	121	93	invariants	invariant	NOUN
iajs-905	121	94	to	to	ADP
iajs-905	121	95	a	a	PRON
iajs-905	121	96	.	.	PUNCT
iajs-905	122	1	we	we	PRON
iajs-905	122	2	shall	shall	AUX
iajs-905	122	3	soon	soon	ADV
iajs-905	122	4	show	show	VERB
iajs-905	122	5	that	that	SCONJ
iajs-905	122	6	the	the	DET
iajs-905	122	7	fuzzy	fuzzy	ADJ
iajs-905	122	8	invariants	invariant	NOUN
iajs-905	122	9	of	of	ADP
iajs-905	122	10	a	a	PRON
iajs-905	122	11	are	be	AUX
iajs-905	122	12	indeed	indeed	ADV
iajs-905	122	13	unique	unique	ADJ
iajs-905	122	14	and	and	CCONJ
iajs-905	122	15	completely	completely	ADV
iajs-905	122	16	describe	describe	VERB
iajs-905	122	17	a.	a.	NOUN
iajs-905	122	18	note	note	NOUN
iajs-905	122	19	one	one	NUM
iajs-905	122	20	other	other	ADJ
iajs-905	122	21	thing	thing	NOUN
iajs-905	122	22	about	about	ADP
iajs-905	122	23	the	the	DET
iajs-905	122	24	fuzzy	fuzzy	ADJ
iajs-905	122	25	invariants	invariant	NOUN
iajs-905	122	26	of	of	ADP
iajs-905	122	27	a.	a.	NOUN
iajs-905	122	28	if	if	SCONJ
iajs-905	122	29	a	a	DET
iajs-905	122	30	=	=	NOUN
iajs-905	122	31	a1	a1	NOUN
iajs-905	122	32			NOUN
iajs-905	122	33	…	…	PUNCT
iajs-905	122	34			PROPN
iajs-905	122	35	ak	ak	PROPN
iajs-905	122	36	,	,	PUNCT
iajs-905	122	37	where	where	SCONJ
iajs-905	122	38	ai	ai	NOUN
iajs-905	122	39	is	be	AUX
iajs-905	122	40	fuzzy	fuzzy	ADJ
iajs-905	122	41	generating	generate	VERB
iajs-905	122	42	set	set	NOUN
iajs-905	122	43	of	of	ADP
iajs-905	122	44	order	order	NOUN
iajs-905	122	45	p	p	PROPN
iajs-905	122	46	ni	ni	PROPN
iajs-905	122	47	,	,	PUNCT
iajs-905	122	48	n1	n1	PROPN
iajs-905	122	49			NUM
iajs-905	122	50	n2	n2	PROPN
iajs-905	122	51			NUM
iajs-905	122	52	…	…	SYM
iajs-905	122	53			NUM
iajs-905	122	54	nk	nk	PROPN
iajs-905	122	55	>	>	X
iajs-905	122	56	0	0	PROPN
iajs-905	122	57	,	,	PUNCT
iajs-905	122	58	then	then	ADV
iajs-905	122	59	o(a	o(a	NUM
iajs-905	122	60	)	)	PUNCT
iajs-905	122	61	=	=	SYM
iajs-905	122	62	o(a1)o(a2	o(a1)o(a2	PROPN
iajs-905	122	63	)	)	PUNCT
iajs-905	122	64	…	…	PUNCT
iajs-905	122	65	o(ak	o(ak	PROPN
iajs-905	122	66	)	)	PUNCT
iajs-905	122	67	,	,	PUNCT
iajs-905	122	68	hence	hence	ADV
iajs-905	122	69	p	p	NOUN
iajs-905	122	70	n	n	NOUN
iajs-905	122	71	=	=	PUNCT
iajs-905	122	72	pn1pn2	pn1pn2	VERB
iajs-905	122	73	…	…	PUNCT
iajs-905	122	74	pnk	pnk	NOUN
iajs-905	122	75	=	=	SYM
iajs-905	122	76	pn1+n2+	pn1+n2+	NOUN
iajs-905	122	77	…	…	SYM
iajs-905	122	78	+nk	+nk	ADJ
iajs-905	122	79	,	,	PUNCT
iajs-905	122	80	whence	whence	NOUN
iajs-905	122	81	n	n	PROPN
iajs-905	122	82	=	=	SYM
iajs-905	122	83	n1+n2	n1+n2	PROPN
iajs-905	122	84	+	+	NUM
iajs-905	122	85	…	…	PUNCT
iajs-905	122	86	+	+	CCONJ
iajs-905	122	87	nk	nk	PROPN
iajs-905	122	88	in	in	ADP
iajs-905	122	89	other	other	ADJ
iajs-905	122	90	words	word	NOUN
iajs-905	122	91	,	,	PUNCT
iajs-905	122	92	n1	n1	NOUN
iajs-905	122	93	,	,	PUNCT
iajs-905	122	94	n2	n2	NOUN
iajs-905	122	95	,	,	PUNCT
iajs-905	122	96	…	…	PUNCT
iajs-905	122	97	,	,	PUNCT
iajs-905	122	98	nk	nk	PROPN
iajs-905	122	99	give	give	VERB
iajs-905	122	100	us	we	PRON
iajs-905	122	101	a	a	DET
iajs-905	122	102	partition	partition	NOUN
iajs-905	122	103	of	of	ADP
iajs-905	122	104	n	n	PROPN
iajs-905	122	105	.	.	PUNCT
iajs-905	123	1	before	before	ADP
iajs-905	123	2	discussing	discuss	VERB
iajs-905	123	3	the	the	DET
iajs-905	123	4	uniqueness	uniqueness	NOUN
iajs-905	123	5	of	of	ADP
iajs-905	123	6	the	the	DET
iajs-905	123	7	fuzzy	fuzzy	ADJ
iajs-905	123	8	invariants	invariant	NOUN
iajs-905	123	9	of	of	ADP
iajs-905	123	10	a	a	PRON
iajs-905	123	11	,	,	PUNCT
iajs-905	123	12	one	one	NUM
iajs-905	123	13	thing	thing	NOUN
iajs-905	123	14	should	should	AUX
iajs-905	123	15	be	be	AUX
iajs-905	123	16	made	make	VERB
iajs-905	123	17	absolutely	absolutely	ADV
iajs-905	123	18	clear	clear	ADJ
iajs-905	123	19	the	the	DET
iajs-905	123	20	singleton	singleton	PROPN
iajs-905	123	21	fuzzy	fuzzy	ADJ
iajs-905	123	22	a1	a1	NOUN
iajs-905	123	23	,	,	PUNCT
iajs-905	123	24	…	…	NUM
iajs-905	123	25	,	,	PUNCT
iajs-905	123	26	ak	ak	PROPN
iajs-905	123	27	and	and	CCONJ
iajs-905	123	28	the	the	DET
iajs-905	123	29	fuzzy	fuzzy	ADJ
iajs-905	123	30	subgroups	subgroup	NOUN
iajs-905	123	31	a1	a1	VERB
iajs-905	123	32	,	,	PUNCT
iajs-905	123	33	…	…	PUNCT
iajs-905	123	34	,	,	PUNCT
iajs-905	123	35	ak	ak	PROPN
iajs-905	123	36	which	which	PRON
iajs-905	123	37	they	they	PRON
iajs-905	123	38	generate	generate	VERB
iajs-905	123	39	,	,	PUNCT
iajs-905	123	40	which	which	PRON
iajs-905	123	41	a	a	DET
iajs-905	123	42	rose	rose	NOUN
iajs-905	123	43	above	above	ADV
iajs-905	123	44	to	to	PART
iajs-905	123	45	give	give	VERB
iajs-905	123	46	the	the	DET
iajs-905	123	47	decomposition	decomposition	NOUN
iajs-905	123	48	of	of	ADP
iajs-905	123	49	a	a	DET
iajs-905	123	50	in	in	ADP
iajs-905	123	51	to	to	ADP
iajs-905	123	52	a	a	DET
iajs-905	123	53	fuzzy	fuzzy	ADJ
iajs-905	123	54	internal	internal	ADJ
iajs-905	123	55	direct	direct	ADJ
iajs-905	123	56	product	product	NOUN
iajs-905	123	57	of	of	ADP
iajs-905	123	58	fuzzy	fuzzy	ADJ
iajs-905	123	59	generating	generating	NOUN
iajs-905	123	60	subgroups	subgroup	NOUN
iajs-905	123	61	,	,	PUNCT
iajs-905	123	62	are	be	AUX
iajs-905	123	63	not	not	PART
iajs-905	123	64	unique	unique	ADJ
iajs-905	123	65	.	.	PUNCT
iajs-905	124	1	let	let	VERB
iajs-905	124	2	’s	’s	NOUN
iajs-905	124	3	see	see	VERB
iajs-905	124	4	this	this	PRON
iajs-905	124	5	in	in	ADP
iajs-905	124	6	a	a	DET
iajs-905	124	7	very	very	ADV
iajs-905	124	8	simple	simple	ADJ
iajs-905	124	9	example	example	NOUN
iajs-905	124	10	let	let	VERB
iajs-905	124	11	g	g	NOUN
iajs-905	124	12	=	=	SYM
iajs-905	124	13	{	{	PUNCT
iajs-905	124	14	e	e	NOUN
iajs-905	124	15	,	,	PUNCT
iajs-905	124	16	a	a	DET
iajs-905	124	17	,	,	PUNCT
iajs-905	124	18	b	b	NOUN
iajs-905	124	19	,	,	PUNCT
iajs-905	124	20	a.b	a.b	AUX
iajs-905	124	21	}	}	PUNCT
iajs-905	124	22	be	be	AUX
iajs-905	124	23	an	an	DET
iajs-905	124	24	abelian	abelian	ADJ
iajs-905	124	25	group	group	NOUN
iajs-905	124	26	of	of	ADP
iajs-905	124	27	order	order	NOUN
iajs-905	124	28	4	4	NUM
iajs-905	125	1	where	where	SCONJ
iajs-905	125	2	a	a	DET
iajs-905	125	3	2	2	NUM
iajs-905	125	4	=	=	SYM
iajs-905	125	5	b	b	SYM
iajs-905	125	6	2	2	NUM
iajs-905	125	7	=	=	SYM
iajs-905	125	8	e	e	NOUN
iajs-905	125	9	,	,	PUNCT
iajs-905	125	10	ab	ab	PROPN
iajs-905	125	11	=	=	SYM
iajs-905	125	12	ba	ba	PROPN
iajs-905	125	13	and	and	CCONJ
iajs-905	125	14	a(e	a(e	PROPN
iajs-905	125	15	)	)	PUNCT
iajs-905	125	16	=	=	SYM
iajs-905	125	17	a(a	a(a	PROPN
iajs-905	125	18	)	)	PUNCT
iajs-905	125	19	=	=	SYM
iajs-905	125	20	1	1	X
iajs-905	125	21	,	,	PUNCT
iajs-905	125	22	a(b	a(b	ADJ
iajs-905	125	23	)	)	PUNCT
iajs-905	125	24	=	=	PUNCT
iajs-905	125	25	a(a.b	a(a.b	NOUN
iajs-905	125	26	)	)	PUNCT
iajs-905	125	27	=	=	SYM
iajs-905	125	28	3/4	3/4	NUM
iajs-905	125	29	.	.	PUNCT
iajs-905	126	1	then	then	ADV
iajs-905	126	2	a=	a=	PROPN
iajs-905	126	3	h	h	PROPN
iajs-905	126	4			PROPN
iajs-905	127	1	k	k	PROPN
iajs-905	128	1	where	where	SCONJ
iajs-905	128	2	h	h	NOUN
iajs-905	129	1	=	=	PUNCT
iajs-905	129	2	<	<	X
iajs-905	129	3	at	at	ADP
iajs-905	129	4	>	>	X
iajs-905	129	5	,	,	PUNCT
iajs-905	129	6	k	k	X
iajs-905	129	7	=	=	PUNCT
iajs-905	129	8	<	<	X
iajs-905	129	9	bt	bt	X
iajs-905	129	10	>	>	X
iajs-905	129	11	are	be	AUX
iajs-905	129	12	fuzzy	fuzzy	ADJ
iajs-905	129	13	generating	generating	NOUN
iajs-905	129	14	subgroups	subgroup	NOUN
iajs-905	129	15	of	of	ADP
iajs-905	129	16	order	order	NOUN
iajs-905	129	17	2	2	NUM
iajs-905	129	18	.	.	PUNCT
iajs-905	130	1	but	but	CCONJ
iajs-905	130	2	we	we	PRON
iajs-905	130	3	have	have	VERB
iajs-905	130	4	another	another	DET
iajs-905	130	5	decomposition	decomposition	NOUN
iajs-905	130	6	of	of	ADP
iajs-905	130	7	a	a	PRON
iajs-905	130	8	as	as	ADP
iajs-905	130	9	a	a	DET
iajs-905	130	10	fuzzy	fuzzy	ADJ
iajs-905	130	11	internal	internal	ADJ
iajs-905	130	12	direct	direct	ADJ
iajs-905	130	13	product	product	NOUN
iajs-905	130	14	,	,	PUNCT
iajs-905	130	15	namely	namely	ADV
iajs-905	130	16	a	a	DET
iajs-905	130	17	=	=	SYM
iajs-905	130	18	n	n	NOUN
iajs-905	130	19			NOUN
iajs-905	130	20	k	k	PROPN
iajs-905	130	21	where	where	SCONJ
iajs-905	130	22	n	n	ADV
iajs-905	130	23	=	=	SYM
iajs-905	130	24	<	<	X
iajs-905	130	25	ab	ab	X
iajs-905	130	26	>	>	X
iajs-905	130	27	and	and	CCONJ
iajs-905	130	28	k	k	PROPN
iajs-905	130	29	=	=	PUNCT
iajs-905	131	1	<	<	X
iajs-905	131	2	b	b	X
iajs-905	131	3	>	>	X
iajs-905	131	4	.	.	PUNCT
iajs-905	132	1	so	so	ADV
iajs-905	132	2	,	,	PUNCT
iajs-905	132	3	even	even	ADV
iajs-905	132	4	in	in	ADP
iajs-905	132	5	this	this	DET
iajs-905	132	6	fuzzy	fuzzy	ADJ
iajs-905	132	7	subgroup	subgroup	NOUN
iajs-905	132	8	of	of	ADP
iajs-905	132	9	very	very	ADV
iajs-905	132	10	small	small	ADJ
iajs-905	132	11	order	order	NOUN
iajs-905	132	12	,	,	PUNCT
iajs-905	132	13	we	we	PRON
iajs-905	132	14	can	can	AUX
iajs-905	132	15	get	get	VERB
iajs-905	132	16	distinct	distinct	ADJ
iajs-905	132	17	decompositions	decomposition	NOUN
iajs-905	132	18	of	of	ADP
iajs-905	132	19	the	the	DET
iajs-905	132	20	fuzzy	fuzzy	ADJ
iajs-905	132	21	subgroup	subgroup	NOUN
iajs-905	132	22	as	as	ADP
iajs-905	132	23	the	the	DET
iajs-905	132	24	internal	internal	ADJ
iajs-905	132	25	direct	direct	ADJ
iajs-905	132	26	product	product	NOUN
iajs-905	132	27	of	of	ADP
iajs-905	132	28	fuzzy	fuzzy	ADJ
iajs-905	132	29	generating	generating	NOUN
iajs-905	132	30	subgroups	subgroup	NOUN
iajs-905	132	31	.	.	PUNCT
iajs-905	133	1	lemma	lemma	PROPN
iajs-905	133	2	3.2	3.2	NUM
iajs-905	133	3	let	let	VERB
iajs-905	133	4	a	a	PRON
iajs-905	133	5	be	be	AUX
iajs-905	133	6	fuzzy	fuzzy	ADJ
iajs-905	133	7	subgroup	subgroup	NOUN
iajs-905	133	8	of	of	ADP
iajs-905	133	9	abelian	abelian	PROPN
iajs-905	133	10	group	group	NOUN
iajs-905	133	11	(	(	PUNCT
iajs-905	133	12	g	g	NOUN
iajs-905	133	13	,	,	PUNCT
iajs-905	133	14	.	.	PUNCT
iajs-905	133	15	)	)	PUNCT
iajs-905	134	1	of	of	ADP
iajs-905	134	2	order	order	NOUN
iajs-905	134	3	pn	pn	NOUN
iajs-905	134	4	,	,	PUNCT
iajs-905	134	5	p	p	X
iajs-905	134	6	a	a	DET
iajs-905	134	7	prime	prime	NOUN
iajs-905	134	8	.	.	PUNCT
iajs-905	134	9	suppose	suppose	VERB
iajs-905	134	10	that	that	SCONJ
iajs-905	134	11	a	a	DET
iajs-905	134	12	=	=	NOUN
iajs-905	134	13	a1	a1	NOUN
iajs-905	134	14			PROPN
iajs-905	134	15	a2	a2	PROPN
iajs-905	134	16			PROPN
iajs-905	134	17	…	…	PROPN
iajs-905	134	18	.	.	PROPN
iajs-905	134	19	ak	ak	PROPN
iajs-905	134	20	,	,	PUNCT
iajs-905	134	21	where	where	SCONJ
iajs-905	134	22	each	each	PRON
iajs-905	134	23	ai	ai	VERB
iajs-905	134	24	=	=	PUNCT
iajs-905	134	25	<	<	X
iajs-905	134	26	asi	asi	X
iajs-905	134	27	>	>	X
iajs-905	134	28	is	be	AUX
iajs-905	134	29	fuzzy	fuzzy	ADJ
iajs-905	134	30	generating	generating	NOUN
iajs-905	134	31	of	of	ADP
iajs-905	134	32	order	order	NOUN
iajs-905	134	33	pni	pni	NOUN
iajs-905	134	34	,	,	PUNCT
iajs-905	134	35	and	and	CCONJ
iajs-905	134	36	n1	n1	PROPN
iajs-905	134	37			NUM
iajs-905	134	38	n2	n2	PROPN
iajs-905	134	39			NUM
iajs-905	134	40	…	…	PUNCT
iajs-905	134	41			NUM
iajs-905	134	42	nk	nk	PROPN
iajs-905	134	43	>	>	X
iajs-905	134	44	0	0	PROPN
iajs-905	134	45	.	.	PUNCT
iajs-905	135	1	if	if	SCONJ
iajs-905	135	2	m	m	NOUN
iajs-905	135	3	is	be	AUX
iajs-905	135	4	an	an	DET
iajs-905	135	5	integer	integer	NOUN
iajs-905	135	6	such	such	ADJ
iajs-905	135	7	that	that	SCONJ
iajs-905	136	1	nt	not	PART
iajs-905	136	2	>	>	X
iajs-905	136	3	m	m	PROPN
iajs-905	136	4			NUM
iajs-905	136	5	nt+1	nt+1	X
iajs-905	136	6	then	then	ADV
iajs-905	136	7	:	:	PUNCT
iajs-905	136	8	a(p	a(p	PROPN
iajs-905	136	9	m	m	NOUN
iajs-905	136	10	)	)	PUNCT
iajs-905	136	11	=	=	SYM
iajs-905	136	12	b1	b1	NOUN
iajs-905	136	13			NOUN
iajs-905	136	14	…	…	PUNCT
iajs-905	136	15			PROPN
iajs-905	136	16	bt	bt	PROPN
iajs-905	136	17			PROPN
iajs-905	136	18	at+1	at+1	PROPN
iajs-905	136	19			PROPN
iajs-905	136	20	…	…	PUNCT
iajs-905	136	21			PROPN
iajs-905	136	22	ak	ak	PROPN
iajs-905	136	23	where	where	SCONJ
iajs-905	136	24	bi	bi	NOUN
iajs-905	136	25	is	be	AUX
iajs-905	136	26	fuzzy	fuzzy	ADJ
iajs-905	136	27	generating	generating	NOUN
iajs-905	136	28	of	of	ADP
iajs-905	136	29	order	order	NOUN
iajs-905	136	30	pm	pm	NOUN
iajs-905	136	31	,	,	PUNCT
iajs-905	136	32	generated	generate	VERB
iajs-905	136	33	by	by	ADP
iajs-905	136	34	,	,	PUNCT
iajs-905	136	35	for	for	ADP
iajs-905	136	36	i	i	PRON
iajs-905	136	37			NOUN
iajs-905	136	38	t.	t.	NOUN
iajs-905	136	39	the	the	DET
iajs-905	136	40	order	order	NOUN
iajs-905	136	41	of	of	ADP
iajs-905	136	42	a(p	a(p	PROPN
iajs-905	136	43	m	m	NOUN
iajs-905	136	44	)	)	PUNCT
iajs-905	136	45	is	be	AUX
iajs-905	136	46	p	p	PROPN
iajs-905	136	47	u	u	NOUN
iajs-905	136	48	,	,	PUNCT
iajs-905	136	49	where	where	SCONJ
iajs-905	136	50	mni	mni	PROPN
iajs-905	136	51	p	p	X
iajs-905	136	52	si	si	PROPN
iajs-905	136	53	a	a	DET
iajs-905	136	54			NOUN
iajs-905	136	55	ihjpas	ihjpa	VERB
iajs-905	136	56	ibn	ibn	PROPN
iajs-905	136	57	alhaitham	alhaitham	PROPN
iajs-905	136	58	j.	j.	PROPN
iajs-905	136	59	for	for	ADP
iajs-905	136	60	pure	pure	ADJ
iajs-905	136	61	&	&	CCONJ
iajs-905	136	62	appl	appl	PROPN
iajs-905	136	63	.	.	PUNCT
iajs-905	137	1	sci	sci	PROPN
iajs-905	137	2	.	.	PUNCT
iajs-905	138	1	vol.23	vol.23	PROPN
iajs-905	138	2	(	(	PUNCT
iajs-905	138	3	3	3	NUM
iajs-905	138	4	)	)	PUNCT
iajs-905	138	5	2010	2010	NUM
iajs-905	138	6	proof	proof	NOUN
iajs-905	138	7	:	:	PUNCT
iajs-905	138	8	first	first	ADV
iajs-905	138	9	of	of	ADP
iajs-905	138	10	all	all	PRON
iajs-905	138	11	,	,	PUNCT
iajs-905	138	12	we	we	PRON
iajs-905	138	13	claim	claim	VERB
iajs-905	138	14	that	that	SCONJ
iajs-905	138	15	at+1	at+1	X
iajs-905	138	16	,	,	PUNCT
iajs-905	138	17	…	…	PUNCT
iajs-905	138	18	.	.	PROPN
iajs-905	138	19	,	,	PUNCT
iajs-905	138	20	ak	ak	PROPN
iajs-905	138	21	are	be	AUX
iajs-905	138	22	all	all	PRON
iajs-905	138	23	in	in	ADP
iajs-905	138	24	a(p	a(p	PROPN
iajs-905	138	25	m	m	NOUN
iajs-905	138	26	)	)	PUNCT
iajs-905	138	27	,	,	PUNCT
iajs-905	138	28	since	since	SCONJ
iajs-905	138	29	m	m	PROPN
iajs-905	138	30			NUM
iajs-905	138	31	nt+1	nt+1	ADP
iajs-905	138	32			NUM
iajs-905	138	33	…	…	SYM
iajs-905	138	34			NUM
iajs-905	138	35	nk	nk	PROPN
iajs-905	138	36	>	>	X
iajs-905	138	37	0	0	PROPN
iajs-905	138	38	,	,	PUNCT
iajs-905	138	39	if	if	SCONJ
iajs-905	138	40	j	j	PROPN
iajs-905	138	41			NUM
iajs-905	138	42	t+1	t+1	NUM
iajs-905	138	43	,	,	PUNCT
iajs-905	138	44	hence	hence	ADV
iajs-905	138	45	aj	aj	PROPN
iajs-905	138	46	,	,	PUNCT
iajs-905	138	47	for	for	ADP
iajs-905	138	48	j	j	PROPN
iajs-905	138	49			PROPN
iajs-905	138	50	t+1	t+1	PROPN
iajs-905	138	51	lies	lie	VERB
iajs-905	138	52	in	in	ADP
iajs-905	138	53	a(p	a(p	PROPN
iajs-905	138	54	m	m	NOUN
iajs-905	138	55	)	)	PUNCT
iajs-905	138	56	.	.	PUNCT
iajs-905	139	1	secondly	secondly	ADV
iajs-905	139	2	,	,	PUNCT
iajs-905	139	3	if	if	SCONJ
iajs-905	139	4	i	i	PRON
iajs-905	139	5			VERB
iajs-905	139	6	t	t	NOUN
iajs-905	139	7	then	then	ADV
iajs-905	139	8	ni	ni	PROPN
iajs-905	139	9	>	>	X
iajs-905	139	10	m	m	PROPN
iajs-905	139	11	and	and	CCONJ
iajs-905	139	12	whence	whence	ADP
iajs-905	139	13	each	each	DET
iajs-905	139	14	such	such	ADJ
iajs-905	139	15	is	be	AUX
iajs-905	139	16	in	in	ADP
iajs-905	139	17	a(pm	a(pm	PROPN
iajs-905	139	18	)	)	PUNCT
iajs-905	139	19	and	and	CCONJ
iajs-905	139	20	so	so	ADV
iajs-905	139	21	the	the	DET
iajs-905	139	22	fuzzy	fuzzy	ADJ
iajs-905	139	23	subgroup	subgroup	NOUN
iajs-905	139	24	it	it	PRON
iajs-905	139	25	generates	generate	VERB
iajs-905	139	26	,	,	PUNCT
iajs-905	139	27	bi	bi	NOUN
iajs-905	139	28	,	,	PUNCT
iajs-905	139	29	is	be	AUX
iajs-905	139	30	also	also	ADV
iajs-905	139	31	in	in	ADP
iajs-905	139	32	a(p	a(p	PROPN
iajs-905	139	33	m	m	NOUN
iajs-905	139	34	)	)	PUNCT
iajs-905	139	35	.	.	PUNCT
iajs-905	140	1	since	since	SCONJ
iajs-905	140	2	b1	b1	NOUN
iajs-905	140	3	,	,	PUNCT
iajs-905	140	4	…	…	PUNCT
iajs-905	140	5	,	,	PUNCT
iajs-905	140	6	bt	bt	PROPN
iajs-905	140	7	,	,	PUNCT
iajs-905	140	8	at+1	at+1	X
iajs-905	140	9	,	,	PUNCT
iajs-905	140	10	…	…	PUNCT
iajs-905	140	11	,	,	PUNCT
iajs-905	140	12	ak	ak	PROPN
iajs-905	140	13	all	all	PRON
iajs-905	140	14	in	in	ADP
iajs-905	140	15	a(p	a(p	PROPN
iajs-905	140	16	m	m	NOUN
iajs-905	140	17	)	)	PUNCT
iajs-905	140	18	,	,	PUNCT
iajs-905	140	19	their	their	PRON
iajs-905	140	20	product	product	NOUN
iajs-905	140	21	(	(	PUNCT
iajs-905	140	22	which	which	PRON
iajs-905	140	23	is	be	AUX
iajs-905	140	24	fuzzy	fuzzy	ADJ
iajs-905	140	25	direct	direct	ADJ
iajs-905	140	26	,	,	PUNCT
iajs-905	140	27	since	since	SCONJ
iajs-905	140	28	the	the	DET
iajs-905	140	29	product	product	NOUN
iajs-905	140	30	a1	a1	NOUN
iajs-905	140	31			PROPN
iajs-905	140	32	a2	a2	PROPN
iajs-905	140	33			PROPN
iajs-905	140	34	…	…	PUNCT
iajs-905	140	35			PROPN
iajs-905	140	36	ak	ak	PROPN
iajs-905	140	37	is	be	AUX
iajs-905	140	38	fuzzy	fuzzy	ADJ
iajs-905	140	39	direct	direct	ADJ
iajs-905	140	40	)	)	PUNCT
iajs-905	140	41	is	be	AUX
iajs-905	140	42	in	in	ADP
iajs-905	140	43	a(p	a(p	PROPN
iajs-905	140	44	m	m	NOUN
iajs-905	140	45	)	)	PUNCT
iajs-905	140	46	.	.	PUNCT
iajs-905	141	1	hence	hence	ADV
iajs-905	141	2	a(p	a(p	PROPN
iajs-905	141	3	m	m	NOUN
iajs-905	141	4	)	)	PUNCT
iajs-905	141	5			PROPN
iajs-905	141	6	b1	b1	PROPN
iajs-905	141	7			PROPN
iajs-905	141	8	…	…	PUNCT
iajs-905	141	9			PROPN
iajs-905	141	10	bt	bt	PROPN
iajs-905	141	11			PROPN
iajs-905	141	12	at+1	at+1	PROPN
iajs-905	141	13			PROPN
iajs-905	141	14	…	…	PUNCT
iajs-905	141	15			PROPN
iajs-905	141	16	ak	ak	PROPN
iajs-905	141	17	.	.	PROPN
iajs-905	141	18	on	on	ADP
iajs-905	141	19	the	the	DET
iajs-905	141	20	other	other	ADJ
iajs-905	141	21	hand	hand	NOUN
iajs-905	141	22	,	,	PUNCT
iajs-905	141	23	if	if	SCONJ
iajs-905	141	24	:	:	PUNCT
iajs-905	141	25	is	be	AUX
iajs-905	141	26	in	in	ADP
iajs-905	141	27	a(p	a(p	PROPN
iajs-905	141	28	m	m	NOUN
iajs-905	141	29	)	)	PUNCT
iajs-905	141	30	,	,	PUNCT
iajs-905	141	31	since	since	SCONJ
iajs-905	141	32	it	it	PRON
iajs-905	141	33	then	then	ADV
iajs-905	141	34	satisfies	satisfy	VERB
iajs-905	141	35	we	we	PRON
iajs-905	141	36	set	set	VERB
iajs-905	141	37	:	:	PUNCT
iajs-905	141	38	however	however	ADV
iajs-905	141	39	,	,	PUNCT
iajs-905	141	40	the	the	DET
iajs-905	141	41	product	product	NOUN
iajs-905	141	42	of	of	ADP
iajs-905	141	43	the	the	DET
iajs-905	141	44	fuzzy	fuzzy	ADJ
iajs-905	141	45	subgroups	subgroup	NOUN
iajs-905	141	46	a1	a1	VERB
iajs-905	141	47	,	,	PUNCT
iajs-905	141	48	…	…	PUNCT
iajs-905	141	49	,	,	PUNCT
iajs-905	141	50	ak	ak	PROPN
iajs-905	141	51	is	be	AUX
iajs-905	141	52	fuzzy	fuzzy	ADJ
iajs-905	141	53	direct	direct	ADJ
iajs-905	141	54	,	,	PUNCT
iajs-905	141	55	so	so	SCONJ
iajs-905	141	56	we	we	PRON
iajs-905	141	57	get	get	VERB
iajs-905	141	58	:	:	PUNCT
iajs-905	141	59	thus	thus	ADV
iajs-905	141	60	the	the	DET
iajs-905	141	61	order	order	NOUN
iajs-905	141	62	of	of	ADP
iajs-905	141	63	ai	ai	NOUN
iajs-905	141	64	,	,	PUNCT
iajs-905	141	65	that	that	ADV
iajs-905	141	66	is	is	ADV
iajs-905	141	67	,	,	PUNCT
iajs-905	141	68	p	p	PROPN
iajs-905	141	69	ni	ni	PROPN
iajs-905	141	70	must	must	AUX
iajs-905	141	71	divide	divide	VERB
iajs-905	141	72	ip	ip	PROPN
iajs-905	141	73	m	m	NOUN
iajs-905	141	74	for	for	ADP
iajs-905	141	75	i	i	PRON
iajs-905	141	76	=	=	SYM
iajs-905	141	77	1	1	NUM
iajs-905	141	78	,	,	PUNCT
iajs-905	141	79	2	2	NUM
iajs-905	141	80	,	,	PUNCT
iajs-905	141	81	…	…	PUNCT
iajs-905	141	82	,	,	PUNCT
iajs-905	141	83	k.	k.	PROPN
iajs-905	142	1	if	if	SCONJ
iajs-905	142	2	i	i	PRON
iajs-905	142	3			NUM
iajs-905	142	4	t+	t+	PUNCT
iajs-905	142	5	1	1	NUM
iajs-905	142	6	this	this	PRON
iajs-905	142	7	is	be	AUX
iajs-905	142	8	automatically	automatically	ADV
iajs-905	142	9	true	true	ADJ
iajs-905	142	10	whatever	whatever	PRON
iajs-905	142	11	be	be	VERB
iajs-905	142	12	the	the	DET
iajs-905	142	13	choice	choice	NOUN
iajs-905	142	14	of	of	ADP
iajs-905	142	15	t+1	t+1	PROPN
iajs-905	142	16	,	,	PUNCT
iajs-905	142	17	...	...	PUNCT
iajs-905	142	18	,	,	PUNCT
iajs-905	142	19	k	k	X
iajs-905	142	20	since	since	SCONJ
iajs-905	142	21	m	m	PROPN
iajs-905	142	22			NUM
iajs-905	142	23	nt+1	nt+1	ADP
iajs-905	142	24			NUM
iajs-905	142	25	…	…	SYM
iajs-905	142	26			NUM
iajs-905	142	27	nk	nk	PROPN
iajs-905	142	28	,	,	PUNCT
iajs-905	142	29	hence	hence	ADV
iajs-905	142	30	p	p	PROPN
iajs-905	142	31	ni	ni	PROPN
iajs-905	142	32	|	|	ADV
iajs-905	142	33	pm	pm	NOUN
iajs-905	142	34	,	,	PUNCT
iajs-905	142	35	i	i	PRON
iajs-905	142	36			PROPN
iajs-905	142	37	t+1	t+1	NUM
iajs-905	142	38	.	.	PUNCT
iajs-905	143	1	however	however	ADV
iajs-905	143	2	,	,	PUNCT
iajs-905	143	3	for	for	ADP
iajs-905	143	4	i	i	PROPN
iajs-905	143	5			NUM
iajs-905	143	6	t	t	NOUN
iajs-905	143	7	,	,	PUNCT
iajs-905	143	8	we	we	PRON
iajs-905	143	9	get	get	VERB
iajs-905	143	10	from	from	ADP
iajs-905	143	11	pni	pni	NOUN
iajs-905	144	1	|	|	ADV
iajs-905	144	2	ip	ip	PROPN
iajs-905	144	3	m	m	VERB
iajs-905	144	4	that	that	SCONJ
iajs-905	144	5	p	p	PROPN
iajs-905	144	6	ni	ni	PROPN
iajs-905	144	7	-	-	PUNCT
iajs-905	144	8	m|	m|	PROPN
iajs-905	144	9	i	i	PROPN
iajs-905	144	10	,	,	PUNCT
iajs-905	144	11	therefore	therefore	ADV
iajs-905	144	12	i	i	PROPN
iajs-905	144	13	=	=	SYM
iajs-905	144	14	vi	vi	NOUN
iajs-905	144	15	pni	pni	NOUN
iajs-905	144	16	-	-	NOUN
iajs-905	144	17	m	m	NOUN
iajs-905	144	18	for	for	ADP
iajs-905	144	19	some	some	DET
iajs-905	144	20	integer	integer	PROPN
iajs-905	144	21	vi	vi	PROPN
iajs-905	144	22	.	.	PUNCT
iajs-905	145	1	putting	put	VERB
iajs-905	145	2	all	all	DET
iajs-905	145	3	this	this	PRON
iajs-905	145	4	in	in	ADP
iajs-905	145	5	formation	formation	NOUN
iajs-905	145	6	in	in	ADP
iajs-905	145	7	to	to	ADP
iajs-905	145	8	the	the	DET
iajs-905	145	9	values	value	NOUN
iajs-905	145	10	of	of	ADP
iajs-905	145	11	the	the	DET
iajs-905	145	12	i	i	NOUN
iajs-905	145	13	's	's	PART
iajs-905	145	14	in	in	ADP
iajs-905	145	15	the	the	DET
iajs-905	145	16	expression	expression	NOUN
iajs-905	145	17	for	for	ADP
iajs-905	145	18	yr	yr	NOUN
iajs-905	145	19	as	as	ADP
iajs-905	145	20	:	:	PUNCT
iajs-905	145	21	we	we	PRON
iajs-905	145	22	see	see	VERB
iajs-905	145	23	that	that	SCONJ
iajs-905	145	24	this	this	PRON
iajs-905	145	25	says	say	VERB
iajs-905	145	26	that	that	SCONJ
iajs-905	145	27	y	y	PROPN
iajs-905	145	28	r	r	PROPN
iajs-905	145	29			PROPN
iajs-905	145	30	b1	b1	NOUN
iajs-905	145	31			NOUN
iajs-905	146	1	…	…	PUNCT
iajs-905	146	2			NOUN
iajs-905	146	3	bt	bt	PROPN
iajs-905	146	4			PROPN
iajs-905	146	5	at+1	at+1	PROPN
iajs-905	146	6			PROPN
iajs-905	146	7	…	…	PUNCT
iajs-905	146	8			PROPN
iajs-905	146	9	ak	ak	PROPN
iajs-905	146	10	.	.	PROPN
iajs-905	146	11	now	now	ADV
iajs-905	146	12	since	since	SCONJ
iajs-905	146	13	each	each	DET
iajs-905	146	14	bi	bi	NOUN
iajs-905	146	15	is	be	AUX
iajs-905	146	16	of	of	ADP
iajs-905	146	17	order	order	NOUN
iajs-905	146	18	pm	pm	NOUN
iajs-905	146	19	and	and	CCONJ
iajs-905	146	20	since	since	SCONJ
iajs-905	146	21	o(ai	o(ai	PROPN
iajs-905	146	22	)	)	PUNCT
iajs-905	146	23	=	=	VERB
iajs-905	146	24	pni	pni	NOUN
iajs-905	146	25	and	and	CCONJ
iajs-905	146	26	since	since	SCONJ
iajs-905	146	27	a	a	DET
iajs-905	146	28	=	=	PUNCT
iajs-905	146	29	b1	b1	NOUN
iajs-905	146	30			NOUN
iajs-905	146	31	…	…	PUNCT
iajs-905	146	32			NOUN
iajs-905	146	33	bt	bt	PROPN
iajs-905	146	34			PROPN
iajs-905	146	35	at+1	at+1	PROPN
iajs-905	146	36			PROPN
iajs-905	146	37	…	…	PUNCT
iajs-905	146	38			PROPN
iajs-905	146	39	ak	ak	PROPN
iajs-905	146	40	,	,	PUNCT
iajs-905	146	41	o(a	o(a	PROPN
iajs-905	146	42	)	)	PUNCT
iajs-905	146	43	=	=	SYM
iajs-905	146	44	o(b1	o(b1	VERB
iajs-905	146	45	)	)	PUNCT
iajs-905	146	46	o	o	NOUN
iajs-905	146	47	(	(	PUNCT
iajs-905	146	48	b2	b2	NOUN
iajs-905	146	49	)	)	PUNCT
iajs-905	146	50	…	…	PUNCT
iajs-905	146	51	.	.	PUNCT
iajs-905	147	1	o(bt	o(bt	NUM
iajs-905	147	2	)	)	PUNCT
iajs-905	147	3	o	o	NOUN
iajs-905	147	4	(	(	PUNCT
iajs-905	147	5	at+1	at+1	NOUN
iajs-905	147	6	)	)	PUNCT
iajs-905	147	7	…	…	PUNCT
iajs-905	147	8	o	o	NOUN
iajs-905	147	9	(	(	PUNCT
iajs-905	147	10	ak	ak	PROPN
iajs-905	147	11	)	)	PUNCT
iajs-905	147	12	=	=	NOUN
iajs-905	147	13	pmpm	pmpm	NOUN
iajs-905	147	14	…	…	PUNCT
iajs-905	147	15	pm	pm	NOUN
iajs-905	147	16	pnt+1	pnt+1	NOUN
iajs-905	147	17	…	…	PUNCT
iajs-905	147	18	pnk	pnk	NOUN
iajs-905	147	19	thus	thus	ADV
iajs-905	147	20	,	,	PUNCT
iajs-905	147	21	if	if	SCONJ
iajs-905	147	22	we	we	PRON
iajs-905	147	23	write	write	VERB
iajs-905	147	24	o	o	PROPN
iajs-905	147	25	(	(	PUNCT
iajs-905	147	26	a	a	X
iajs-905	147	27	)	)	PUNCT
iajs-905	147	28	=	=	SYM
iajs-905	148	1	p	p	NOUN
iajs-905	148	2	u	u	NOUN
iajs-905	148	3	,	,	PUNCT
iajs-905	148	4	then	then	ADV
iajs-905	148	5	:	:	PUNCT
iajs-905	148	6	the	the	DET
iajs-905	148	7	lemma	lemma	PROPN
iajs-905	148	8	is	be	AUX
iajs-905	148	9	proved	prove	VERB
iajs-905	148	10	.	.	PUNCT
iajs-905	149	1	corollary	corollary	ADJ
iajs-905	149	2	3.3	3.3	NUM
iajs-905	149	3	if	if	SCONJ
iajs-905	149	4	a	a	PRON
iajs-905	149	5	is	be	AUX
iajs-905	149	6	a	a	DET
iajs-905	149	7	fuzzy	fuzzy	ADJ
iajs-905	149	8	subgroup	subgroup	NOUN
iajs-905	149	9	of	of	ADP
iajs-905	149	10	a	a	DET
iajs-905	149	11	group	group	NOUN
iajs-905	149	12	in	in	ADP
iajs-905	149	13	lemma	lemma	PROPN
iajs-905	149	14	(	(	PUNCT
iajs-905	149	15	3.2	3.2	NUM
iajs-905	149	16	)	)	PUNCT
iajs-905	149	17	,	,	PUNCT
iajs-905	149	18	then	then	ADV
iajs-905	149	19	o(a(p	o(a(p	ADJ
iajs-905	149	20	)	)	PUNCT
iajs-905	149	21	)	)	PUNCT
iajs-905	150	1	=	=	PUNCT
iajs-905	151	1	p	p	PROPN
iajs-905	151	2	k	k	PROPN
iajs-905	151	3	.	.	PUNCT
iajs-905	152	1	proof	proof	NOUN
iajs-905	152	2	:	:	PUNCT
iajs-905	152	3	apply	apply	VERB
iajs-905	152	4	the	the	DET
iajs-905	152	5	above	above	ADJ
iajs-905	152	6	lemma	lemma	PROPN
iajs-905	152	7	to	to	ADP
iajs-905	152	8	the	the	DET
iajs-905	152	9	case	case	NOUN
iajs-905	152	10	m	m	NOUN
iajs-905	152	11	=	=	NOUN
iajs-905	153	1	1	1	X
iajs-905	153	2	.	.	PUNCT
iajs-905	153	3	then	then	ADV
iajs-905	153	4	t	t	PROPN
iajs-905	153	5	=	=	SYM
iajs-905	153	6	k	k	NOUN
iajs-905	153	7	,	,	PUNCT
iajs-905	153	8	hence	hence	ADV
iajs-905	153	9	u	u	NOUN
iajs-905	153	10	=	=	PROPN
iajs-905	153	11	l.	l.	PROPN
iajs-905	153	12	k	k	PROPN
iajs-905	153	13	=	=	PUNCT
iajs-905	153	14	k	k	PROPN
iajs-905	153	15	and	and	CCONJ
iajs-905	153	16	so	so	ADV
iajs-905	153	17	o	o	X
iajs-905	153	18	(	(	PUNCT
iajs-905	153	19	a	a	X
iajs-905	153	20	)	)	PUNCT
iajs-905	153	21	=	=	SYM
iajs-905	153	22	p	p	PROPN
iajs-905	153	23	k.	k.	PROPN
iajs-905	153	24			PROPN
iajs-905	153	25			PROPN
iajs-905	153	26			ADJ
iajs-905	153	27	k	k	PROPN
iajs-905	153	28	1ti	1ti	PROPN
iajs-905	154	1	i	i	PRON
iajs-905	154	2	nmtu	nmtu	NOUN
iajs-905	155	1	e)a(a	e)a(a	PROPN
iajs-905	155	2	njmnjm	njmnjm	PROPN
iajs-905	156	1	pp	pp	PROPN
iajs-905	156	2	j	j	PROPN
iajs-905	157	1	p	p	X
iajs-905	157	2	j	j	PROPN
iajs-905	157	3			NUM
iajs-905	157	4			PROPN
iajs-905	157	5	ea)a	ea)a	NUM
iajs-905	157	6	(	(	PUNCT
iajs-905	157	7	nimni	nimni	NOUN
iajs-905	158	1	p	p	NOUN
iajs-905	159	1	i	i	PRON
iajs-905	159	2	p	p	NOUN
iajs-905	159	3	i	i	PRON
iajs-905	159	4			NUM
iajs-905	159	5			NOUN
iajs-905	159	6	mnip	mnip	NOUN
iajs-905	159	7	i	i	PRON
iajs-905	159	8	a	a	DET
iajs-905	159	9			PROPN
iajs-905	159	10	k	k	PROPN
iajs-905	159	11	k	k	PROPN
iajs-905	159	12	2	2	NUM
iajs-905	159	13	2	2	NUM
iajs-905	159	14	1	1	NUM
iajs-905	159	15	1r	1r	NUM
iajs-905	159	16	a	a	PRON
iajs-905	159	17	....	....	PUNCT
iajs-905	159	18	a.ay	a.ay	PROPN
iajs-905	159	19			PROPN
iajs-905	159	20	,	,	PUNCT
iajs-905	159	21	ey	ey	INTJ
iajs-905	159	22	mp	mp	NOUN
iajs-905	159	23	r	r	NOUN
iajs-905	159	24			NOUN
iajs-905	159	25	mmm	mmm	INTJ
iajs-905	159	26	kp	kp	PROPN
iajs-905	159	27	k	k	PROPN
iajs-905	159	28	p1	p1	PROPN
iajs-905	159	29	1	1	NUM
iajs-905	159	30	p	p	NOUN
iajs-905	159	31	r	r	NOUN
iajs-905	160	1	a	a	PRON
iajs-905	160	2	....	....	PUNCT
iajs-905	160	3	aye	aye	NOUN
iajs-905	160	4			X
iajs-905	160	5	ea,	ea,	PROPN
iajs-905	160	6	....	....	PROPN
iajs-905	160	7	,ea	,ea	PUNCT
iajs-905	160	8	mm	mm	INTJ
iajs-905	160	9	kp	kp	PROPN
iajs-905	160	10	k	k	PROPN
iajs-905	160	11	p1	p1	PROPN
iajs-905	160	12	1	1	NUM
iajs-905	160	13			NUM
iajs-905	160	14			NOUN
iajs-905	160	15	k	k	PROPN
iajs-905	160	16	k	k	PROPN
iajs-905	160	17	1	1	NUM
iajs-905	160	18	1r	1r	NUM
iajs-905	160	19	a	a	PRON
iajs-905	160	20	...	...	PUNCT
iajs-905	161	1	ay	ay	NOUN
iajs-905	161	2			PROPN
iajs-905	161	3	k	k	PROPN
iajs-905	162	1	k	k	PROPN
iajs-905	162	2	1	1	NUM
iajs-905	162	3	t	t	PROPN
iajs-905	162	4	1	1	NUM
iajs-905	162	5	t	t	NOUN
iajs-905	162	6	vtp	vtp	NOUN
iajs-905	162	7	t	t	PROPN
iajs-905	163	1	p1v	p1v	PROPN
iajs-905	163	2	1r	1r	PROPN
iajs-905	163	3	a	a	DET
iajs-905	163	4	....	....	NOUN
iajs-905	163	5	aa	aa	NOUN
iajs-905	163	6	...	...	PUNCT
iajs-905	163	7	ay	ay	PROPN
iajs-905	163	8	mntm1n	mntm1n	PROPN
iajs-905	163	9			NOUN
iajs-905	163	10			ADV
iajs-905	163	11			X
iajs-905	163	12			NUM
iajs-905	163	13			PROPN
iajs-905	163	14			ADJ
iajs-905	163	15			ADJ
iajs-905	163	16	k	k	PROPN
iajs-905	163	17	1ti	1ti	NOUN
iajs-905	164	1	i	i	PRON
iajs-905	164	2	nmtu	nmtu	NOUN
iajs-905	164	3	ihjpas	ihjpa	VERB
iajs-905	164	4	ibn	ibn	PROPN
iajs-905	164	5	alhaitham	alhaitham	PROPN
iajs-905	165	1	j.	j.	PROPN
iajs-905	166	1	fo	fo	ADP
iajs-905	166	2	r	r	NOUN
iajs-905	166	3	pure	pure	ADJ
iajs-905	166	4	&	&	CCONJ
iajs-905	166	5	appl	appl	PROPN
iajs-905	166	6	.	.	PUNCT
iajs-905	167	1	sc	sc	PROPN
iajs-905	167	2	i.	i.	PROPN
iajs-905	167	3	vo	vo	PROPN
iajs-905	167	4	l.23	l.23	PROPN
iajs-905	167	5	(	(	PUNCT
iajs-905	167	6	3	3	NUM
iajs-905	167	7	)	)	PUNCT
iajs-905	167	8	2010	2010	NUM
iajs-905	167	9	references	reference	NOUN
iajs-905	167	10	1	1	NUM
iajs-905	167	11	.	.	PUNCT
iajs-905	167	12	malik	malik	PROPN
iajs-905	167	13	.	.	PUNCT
iajs-905	168	1	d.	d.	PROPN
iajs-905	168	2	s.	s.	PROPN
iajs-905	168	3	,	,	PUNCT
iajs-905	168	4	mordeson	mordeson	NOUN
iajs-905	168	5	.	.	PUNCT
iajs-905	169	1	j.	j.	PROPN
iajs-905	169	2	n.	n.	PROPN
iajs-905	169	3	and	and	CCONJ
iajs-905	169	4	nair	nair	NOUN
iajs-905	169	5	.	.	PUNCT
iajs-905	170	1	p.	p.	PROPN
iajs-905	170	2	s.	s.	PROPN
iajs-905	170	3	,	,	PUNCT
iajs-905	170	4	(	(	PUNCT
iajs-905	170	5	1992	1992	NUM
iajs-905	170	6	)	)	PUNCT
iajs-905	170	7	”	"	PUNCT
iajs-905	170	8	fuzzy	fuzzy	ADJ
iajs-905	170	9	generators	generator	NOUN
iajs-905	170	10	and	and	CCONJ
iajs-905	170	11	fuzzy	fuzzy	ADJ
iajs-905	170	12	direct	direct	ADJ
iajs-905	170	13	sums	sum	NOUN
iajs-905	170	14	of	of	ADP
iajs-905	170	15	abelian	abelian	ADJ
iajs-905	170	16	groups	group	NOUN
iajs-905	170	17	”	"	PUNCT
iajs-905	170	18	,	,	PUNCT
iajs-905	170	19	fuzzy	fuzzy	ADJ
iajs-905	170	20	sets	set	NOUN
iajs-905	170	21	and	and	CCONJ
iajs-905	170	22	systems,.50,.193	systems,.50,.193	NOUN
iajs-905	170	23	-	-	SYM
iajs-905	170	24	199	199	NUM
iajs-905	170	25	,	,	PUNCT
iajs-905	170	26	.	.	PUNCT
iajs-905	171	1	2	2	X
iajs-905	171	2	.	.	NUM
iajs-905	171	3	majeed.s.n	majeed.s.n	NOUN
iajs-905	171	4	.	.	NOUN
iajs-905	171	5	,	,	PUNCT
iajs-905	171	6	(	(	PUNCT
iajs-905	171	7	1999)”on	1999)”on	NUM
iajs-905	171	8	fuzzy	fuzzy	ADJ
iajs-905	171	9	subgroups	subgroup	NOUN
iajs-905	171	10	of	of	ADP
iajs-905	171	11	abelian	abelian	ADJ
iajs-905	171	12	groups”,m.sc	groups”,m.sc	PROPN
iajs-905	171	13	.	.	PUNCT
iajs-905	172	1	thesis	thesis	NOUN
iajs-905	172	2	,	,	PUNCT
iajs-905	172	3	university	university	NOUN
iajs-905	172	4	of	of	ADP
iajs-905	172	5	baghdad	baghdad	PROPN
iajs-905	172	6	,	,	PUNCT
iajs-905	172	7	.	.	PUNCT
iajs-905	173	1	3	3	X
iajs-905	173	2	.	.	X
iajs-905	173	3	mordesn	mordesn	PROPN
iajs-905	173	4	j.n	j.n	PROPN
iajs-905	173	5	.	.	PROPN
iajs-905	173	6	,	,	PUNCT
iajs-905	173	7	(	(	PUNCT
iajs-905	173	8	1996	1996	NUM
iajs-905	173	9	)	)	PUNCT
iajs-905	173	10	”	"	PUNCT
iajs-905	173	11	l	l	NOUN
iajs-905	173	12	-	-	NOUN
iajs-905	173	13	subspaces	subspace	NOUN
iajs-905	173	14	and	and	CCONJ
iajs-905	173	15	l	l	NOUN
iajs-905	173	16	-	-	PUNCT
iajs-905	173	17	subfields	subfield	NOUN
iajs-905	173	18	”	"	PUNCT
iajs-905	173	19	.	.	PUNCT
iajs-905	174	1	4	4	X
iajs-905	174	2	.	.	X
iajs-905	174	3	hussein	hussein	PROPN
iajs-905	174	4	.	.	PUNCT
iajs-905	175	1	r.	r.	PROPN
iajs-905	175	2	w.	w.	PROPN
iajs-905	175	3	,	,	PUNCT
iajs-905	175	4	(	(	PUNCT
iajs-905	175	5	1999	1999	NUM
iajs-905	175	6	)	)	PUNCT
iajs-905	175	7	”	"	PUNCT
iajs-905	175	8	some	some	DET
iajs-905	175	9	results	result	NOUN
iajs-905	175	10	of	of	ADP
iajs-905	175	11	fuzzy	fuzzy	ADJ
iajs-905	175	12	rings	ring	NOUN
iajs-905	175	13	”	"	PUNCT
iajs-905	175	14	,	,	PUNCT
iajs-905	175	15	m.sc	m.sc	PROPN
iajs-905	175	16	.	.	PUNCT
iajs-905	176	1	thesis	thesis	NOUN
iajs-905	176	2	,	,	PUNCT
iajs-905	176	3	university	university	NOUN
iajs-905	176	4	of	of	ADP
iajs-905	176	5	baghdad	baghdad	PROPN
iajs-905	176	6	,	,	PUNCT
iajs-905	176	7	.	.	PUNCT
iajs-905	177	1	5	5	X
iajs-905	177	2	.	.	X
iajs-905	177	3	abou	abou	PROPN
iajs-905	177	4	-	-	PUNCT
iajs-905	177	5	zaid	zaid	PROPN
iajs-905	177	6	.	.	PROPN
iajs-905	177	7	,	,	PUNCT
iajs-905	177	8	(	(	PUNCT
iajs-905	177	9	1988	1988	NUM
iajs-905	177	10	)	)	PUNCT
iajs-905	177	11	.	.	PUNCT
iajs-905	177	12	”	"	PUNCT
iajs-905	178	1	on	on	ADP
iajs-905	178	2	normal	normal	ADJ
iajs-905	178	3	fuzzy	fuzzy	ADJ
iajs-905	178	4	subgroups	subgroup	NOUN
iajs-905	178	5	”	"	PUNCT
iajs-905	178	6	,	,	PUNCT
iajs-905	178	7	j.facu.edu	j.facu.edu	PROPN
iajs-905	178	8	.	.	PUNCT
iajs-905	178	9	,.13	,.13	PROPN
iajs-905	178	10	.	.	PUNCT
iajs-905	179	1	6	6	X
iajs-905	179	2	.	.	X
iajs-905	179	3	seselja	seselja	PROPN
iajs-905	179	4	.b	.b	PROPN
iajs-905	180	1	and	and	CCONJ
iajs-905	180	2	tepavcevic	tepavcevic	ADJ
iajs-905	180	3	a.	a.	NOUN
iajs-905	180	4	,(1997).“anote	,(1997).“anote	PUNCT
iajs-905	180	5	on	on	ADP
iajs-905	180	6	fuzzy	fuzzy	ADJ
iajs-905	180	7	groups	group	NOUN
iajs-905	180	8	”	"	PUNCT
iajs-905	180	9	,	,	PUNCT
iajs-905	180	10	j.yugoslav.oper.rese	j.yugoslav.oper.rese	ADJ
iajs-905	180	11	,	,	PUNCT
iajs-905	180	12	.7,.1,.49	.7,.1,.49	PROPN
iajs-905	180	13	-	-	PUNCT
iajs-905	180	14	54	54	NUM
iajs-905	180	15	,	,	PUNCT
iajs-905	180	16	.	.	PUNCT
iajs-905	181	1	7	7	X
iajs-905	181	2	.	.	X
iajs-905	181	3	gupta	gupta	PROPN
iajs-905	181	4	k.c	k.c	PROPN
iajs-905	181	5	and	and	CCONJ
iajs-905	181	6	sarma	sarma	PROPN
iajs-905	181	7	b.k	b.k	PROPN
iajs-905	181	8	.	.	PROPN
iajs-905	181	9	,	,	PUNCT
iajs-905	181	10	(	(	PUNCT
iajs-905	181	11	1999	1999	NUM
iajs-905	181	12	)	)	PUNCT
iajs-905	181	13	.	.	PUNCT
iajs-905	182	1	“	"	PUNCT
iajs-905	182	2	nilpotent	nilpotent	ADJ
iajs-905	182	3	fuzzy	fuzzy	ADJ
iajs-905	182	4	groups	group	NOUN
iajs-905	182	5	”	"	PUNCT
iajs-905	182	6	,	,	PUNCT
iajs-905	182	7	fuzzy	fuzzy	ADJ
iajs-905	182	8	set	set	NOUN
iajs-905	182	9	and	and	CCONJ
iajs-905	182	10	systems	system	NOUN
iajs-905	182	11	,	,	PUNCT
iajs-905	182	12	.101,.167	.101,.167	NOUN
iajs-905	182	13	-	-	X
iajs-905	182	14	176	176	NUM
iajs-905	182	15	,	,	PUNCT
iajs-905	182	16	.	.	PUNCT
iajs-905	183	1	8	8	X
iajs-905	183	2	.	.	X
iajs-905	183	3	seselja	seselja	ADJ
iajs-905	183	4	.	.	PUNCT
iajs-905	184	1	b	b	X
iajs-905	184	2	.	.	PUNCT
iajs-905	185	1	and	and	CCONJ
iajs-905	185	2	tepavcevic	tepavcevic	ADJ
iajs-905	185	3	a.	a.	NOUN
iajs-905	185	4	,	,	PUNCT
iajs-905	185	5	(	(	PUNCT
iajs-905	185	6	1996	1996	NUM
iajs-905	185	7	)	)	PUNCT
iajs-905	185	8	“	"	PUNCT
iajs-905	185	9	fuzzy	fuzzy	ADJ
iajs-905	185	10	groups	group	NOUN
iajs-905	185	11	and	and	CCONJ
iajs-905	185	12	collections	collection	NOUN
iajs-905	185	13	of	of	ADP
iajs-905	185	14	subgroups	subgroup	NOUN
iajs-905	185	15	”	"	PUNCT
iajs-905	185	16	,	,	PUNCT
iajs-905	185	17	fuzzy	fuzzy	ADJ
iajs-905	185	18	sets	set	NOUN
iajs-905	185	19	and	and	CCONJ
iajs-905	185	20	systems	system	NOUN
iajs-905	185	21	,	,	PUNCT
iajs-905	185	22	.83	.83	NUM
iajs-905	185	23	,	,	PUNCT
iajs-905	185	24	.85	.85	NUM
iajs-905	185	25	-	-	SYM
iajs-905	185	26	91	91	NUM
iajs-905	185	27	,	,	PUNCT
iajs-905	185	28	.	.	PUNCT
iajs-905	186	1	9	9	X
iajs-905	186	2	.	.	X
iajs-905	186	3	bandler	bandler	NOUN
iajs-905	186	4	.	.	PUNCT
iajs-905	187	1	w.	w.	PROPN
iajs-905	187	2	and	and	CCONJ
iajs-905	187	3	kohout	kohout	PROPN
iajs-905	187	4	.	.	PUNCT
iajs-905	188	1	l.	l.	PROPN
iajs-905	188	2	,	,	PUNCT
iajs-905	188	3	(	(	PUNCT
iajs-905	188	4	2000	2000	NUM
iajs-905	188	5	)	)	PUNCT
iajs-905	188	6	semantics	semantic	NOUN
iajs-905	188	7	of	of	ADP
iajs-905	188	8	implication	implication	NOUN
iajs-905	188	9	operators	operator	NOUN
iajs-905	188	10	and	and	CCONJ
iajs-905	188	11	fuzzy	fuzzy	ADJ
iajs-905	188	12	relational	relational	ADJ
iajs-905	188	13	products	product	NOUN
iajs-905	188	14	,	,	PUNCT
iajs-905	188	15	internat	internat	PROPN
iajs-905	188	16	.	.	PUNCT
iajs-905	189	1	j.	j.	PROPN
iajs-905	189	2	manmachine	manmachine	PROPN
iajs-905	189	3	studies	study	NOUN
iajs-905	189	4	12	12	NUM
iajs-905	189	5	.	.	PUNCT
iajs-905	189	6	89	89	NUM
iajs-905	189	7	-	-	SYM
iajs-905	189	8	116	116	NUM
iajs-905	189	9	.	.	PUNCT
iajs-905	190	1	ihjpas	ihjpas	PROPN
iajs-905	190	2	للعلوم	للعلوم	NOUN
iajs-905	190	3	الصر	الصر	ADV
iajs-905	190	4	2010	2010	NUM
iajs-905	190	5	)	)	PUNCT
iajs-905	191	1	3	3	NUM
iajs-905	191	2	(	(	PUNCT
iajs-905	191	3	23فة	23فة	NUM
iajs-905	191	4	والتطبیقیة	والتطبیقیة	NOUN
iajs-905	191	5	المجلدمجلة	المجلدمجلة	VERB
iajs-905	191	6	ابن	ابن	PROPN
iajs-905	191	7	الھیثم	الھیثم	PROPN
iajs-905	191	8	الجداء	الجداء	PROPN
iajs-905	191	9	المباشر	المباشر	PROPN
iajs-905	191	10	الداخلي	الداخلي	PROPN
iajs-905	191	11	الضبابيحول	الضبابيحول	PROPN
iajs-905	191	12	نغم	نغم	PROPN
iajs-905	191	13	موسى	موسى	PROPN
iajs-905	191	14	نعمة	نعمة	PROPN
iajs-905	191	15	جامعة	جامعة	PROPN
iajs-905	191	16	بغداد	بغداد	PROPN
iajs-905	191	17	،	،	PROPN
iajs-905	191	18	العلوم	العلوم	PROPN
iajs-905	191	19	للبناتكلیة	للبناتكلیة	PROPN
iajs-905	191	20	،	،	PROPN
iajs-905	192	1	قسم	قسم	PROPN
iajs-905	192	2	الریاضیات	الریاضیات	PROPN
iajs-905	192	3	الخالصة	الخالصة	PROPN
iajs-905	192	4	یتضـــمن	یتضـــمن	PROPN
iajs-905	192	5	البحـــث	البحـــث	PROPN
iajs-905	192	6	تقــــدیم	تقــــدیم	PROPN
iajs-905	192	7	تعریـــف	تعریـــف	NOUN
iajs-905	192	8	الجــــداء	الجــــداء	PROPN
iajs-905	192	9	المباشـــر	المباشـــر	PROPN
iajs-905	192	10	الــــداخلي	الــــداخلي	PROPN
iajs-905	192	11	الضـــبابي	الضـــبابي	PROPN
iajs-905	192	12	وبعــــض	وبعــــض	NOUN
iajs-905	192	13	الخـــواص	الخـــواص	PROPN
iajs-905	192	14	والمبرهنــــات	والمبرهنــــات	PROPN
iajs-905	192	15	فـي	فـي	PROPN
iajs-905	192	16	مجـال	مجـال	PROPN
iajs-905	192	17	الریاضـیات	الریاضـیات	VERB
iajs-905	192	18	الضـبابیة	الضـبابیة	NOUN
iajs-905	192	19	بشـكل	بشـكل	NOUN
iajs-905	192	20	تطبیقـات	تطبیقـات	PROPN
iajs-905	192	21	واسـعة	واسـعة	PROPN
iajs-905	192	22	اوذ	اوذ	ADV
iajs-905	192	23	اً	اً	PUNCT
iajs-905	192	24	ضـروریو	ضـروریو	NOUN
iajs-905	192	25	اً	اً	PUNCT
iajs-905	192	26	مـمهـ	مـمهـ	X
iajs-905	192	27	اً	اً	PUNCT
iajs-905	192	28	موضوع	موضوع	PROPN
iajs-905	192	29	دالذي	دالذي	PROPN
iajs-905	192	30	یع	یع	VERB
iajs-905	192	31	المتعلقة	المتعلقة	PROPN
iajs-905	192	32	به	به	VERB
iajs-905	192	33	.عام	.عام	PUNCT
iajs-905	193	1	وفي	وفي	INTJ
iajs-905	193	2	مجال	مجال	ADJ
iajs-905	193	3	الزمرة	الزمرة	VERB
iajs-905	193	4	الضبابیة	الضبابیة	NOUN
iajs-905	193	5	بشكل	بشكل	NOUN
iajs-905	193	6	خاص	خاص	ADJ
iajs-905	193	7	ihjpas	ihjpa	VERB