id	sid	tid	token	lemma	pos
iajs-2430	1	1	microsoft	microsoft	PROPN
iajs-2430	1	2	word	word	NOUN
iajs-2430	1	3	95	95	NUM
iajs-2430	1	4	-	-	SYM
iajs-2430	1	5	106	106	NUM
iajs-2430	1	6	  	  	SPACE
iajs-2430	1	7	95	95	NUM
iajs-2430	1	8	  	  	SPACE
iajs-2430	1	9	ibn	ibn	PROPN
iajs-2430	1	10	al	al	PROPN
iajs-2430	1	11	-	-	PUNCT
iajs-2430	1	12	haitham	haitham	PROPN
iajs-2430	1	13	jour	jour	X
iajs-2430	1	14	.	.	PROPN
iajs-2430	2	1	for	for	ADP
iajs-2430	2	2	pure	pure	ADJ
iajs-2430	2	3	&	&	CCONJ
iajs-2430	2	4	appl	appl	PROPN
iajs-2430	2	5	.	.	PUNCT
iajs-2430	3	1	sci	sci	PROPN
iajs-2430	3	2	.	.	PROPN
iajs-2430	4	1	33	33	NUM
iajs-2430	4	2	(	(	PUNCT
iajs-2430	4	3	2	2	NUM
iajs-2430	4	4	)	)	PUNCT
iajs-2430	4	5	2020	2020	NUM
iajs-2430	4	6	      	      	SPACE
iajs-2430	4	7	interval	interval	NOUN
iajs-2430	4	8	value	value	NOUN
iajs-2430	4	9	fuzzy	fuzzy	ADJ
iajs-2430	4	10	k	k	NOUN
iajs-2430	4	11	-	-	NOUN
iajs-2430	4	12	ideals	ideal	NOUN
iajs-2430	4	13	of	of	ADP
iajs-2430	4	14	a	a	DET
iajs-2430	4	15	ku	ku	PROPN
iajs-2430	4	16	-	-	PUNCT
iajs-2430	4	17	semigroup	semigroup	NOUN
iajs-2430	4	18	         	         	SPACE
iajs-2430	4	19	article	article	NOUN
iajs-2430	4	20	history	history	NOUN
iajs-2430	4	21	:	:	PUNCT
iajs-2430	4	22	received	receive	VERB
iajs-2430	4	23	23	23	NUM
iajs-2430	4	24	june	june	PROPN
iajs-2430	4	25	2019	2019	NUM
iajs-2430	4	26	,	,	PUNCT
iajs-2430	4	27	accepted	accept	VERB
iajs-2430	4	28	6	6	NUM
iajs-2430	4	29	august	august	PROPN
iajs-2430	4	30	2019	2019	NUM
iajs-2430	4	31	,	,	PUNCT
iajs-2430	4	32	published	publish	VERB
iajs-2430	4	33	in	in	ADP
iajs-2430	4	34	april	april	PROPN
iajs-2430	4	35	2020	2020	NUM
iajs-2430	4	36	.	.	PUNCT
iajs-2430	5	1	abstract	abstract	ADJ
iajs-2430	5	2	the	the	DET
iajs-2430	5	3	notion	notion	NOUN
iajs-2430	5	4	of	of	ADP
iajs-2430	5	5	interval	interval	NOUN
iajs-2430	5	6	value	value	NOUN
iajs-2430	5	7	fuzzy	fuzzy	ADJ
iajs-2430	5	8	k	k	NOUN
iajs-2430	5	9	-	-	NOUN
iajs-2430	5	10	ideal	ideal	NOUN
iajs-2430	5	11	of	of	ADP
iajs-2430	5	12	ku	ku	PROPN
iajs-2430	5	13	-	-	PUNCT
iajs-2430	5	14	semigroup	semigroup	PROPN
iajs-2430	5	15	was	be	AUX
iajs-2430	5	16	studied	study	VERB
iajs-2430	5	17	as	as	ADP
iajs-2430	5	18	a	a	DET
iajs-2430	5	19	generalization	generalization	NOUN
iajs-2430	5	20	of	of	ADP
iajs-2430	5	21	afuzzy	afuzzy	ADJ
iajs-2430	5	22	k	k	NOUN
iajs-2430	5	23	-	-	NOUN
iajs-2430	5	24	ideal	ideal	NOUN
iajs-2430	5	25	of	of	ADP
iajs-2430	5	26	ku	ku	PROPN
iajs-2430	5	27	-	-	PUNCT
iajs-2430	5	28	semigroup	semigroup	PROPN
iajs-2430	5	29	.	.	PUNCT
iajs-2430	6	1	some	some	DET
iajs-2430	6	2	results	result	NOUN
iajs-2430	6	3	of	of	ADP
iajs-2430	6	4	this	this	DET
iajs-2430	6	5	idea	idea	NOUN
iajs-2430	6	6	under	under	ADP
iajs-2430	6	7	homomorphism	homomorphism	NOUN
iajs-2430	6	8	are	be	AUX
iajs-2430	6	9	discussed	discuss	VERB
iajs-2430	6	10	.	.	PUNCT
iajs-2430	7	1	also	also	ADV
iajs-2430	7	2	,	,	PUNCT
iajs-2430	7	3	we	we	PRON
iajs-2430	7	4	presented	present	VERB
iajs-2430	7	5	some	some	DET
iajs-2430	7	6	properties	property	NOUN
iajs-2430	7	7	about	about	ADP
iajs-2430	7	8	the	the	DET
iajs-2430	7	9	image	image	NOUN
iajs-2430	7	10	(	(	PUNCT
iajs-2430	7	11	preimage	preimage	NOUN
iajs-2430	7	12	)	)	PUNCT
iajs-2430	7	13	for	for	SCONJ
iajs-2430	7	14	interval~	interval~	PROPN
iajs-2430	7	15	valued	value	VERB
iajs-2430	7	16	fuzzy	fuzzy	NOUN
iajs-2430	7	17	~	~	SYM
iajs-2430	7	18	k	k	NOUN
iajs-2430	7	19	-	-	PUNCT
iajs-2430	7	20	ideals	ideal	NOUN
iajs-2430	7	21	of	of	ADP
iajs-2430	7	22	a	a	DET
iajs-2430	7	23	ku	ku	PROPN
iajs-2430	7	24	-	-	PUNCT
iajs-2430	7	25	semigroup	semigroup	PROPN
iajs-2430	7	26	.	.	PUNCT
iajs-2430	8	1	finally	finally	ADV
iajs-2430	8	2	,	,	PUNCT
iajs-2430	8	3	the~	the~	DET
iajs-2430	8	4	product	product	NOUN
iajs-2430	8	5	of~	of~	PROPN
iajs-2430	8	6	interval	interval	NOUN
iajs-2430	8	7	valued	value	VERB
iajs-2430	8	8	fuzzyk	fuzzyk	NOUN
iajs-2430	8	9	-	-	PUNCT
iajs-2430	8	10	ideals	ideal	NOUN
iajs-2430	8	11	is	be	AUX
iajs-2430	8	12	established	establish	VERB
iajs-2430	8	13	.	.	PUNCT
iajs-2430	9	1	keywords	keyword	NOUN
iajs-2430	9	2	:	:	PUNCT
iajs-2430	9	3	ku	ku	PROPN
iajs-2430	9	4	-	-	PUNCT
iajs-2430	9	5	algebra;ku	algebra;ku	PROPN
iajs-2430	9	6	-	-	PUNCT
iajs-2430	9	7	semigroup	semigroup	NOUN
iajs-2430	9	8	;	;	PUNCT
iajs-2430	9	9	interval	interval	NOUN
iajs-2430	9	10	valuefuzzy	valuefuzzy	NOUN
iajs-2430	9	11	s	s	NOUN
iajs-2430	9	12	-	-	NOUN
iajs-2430	9	13	ideal	ideal	ADJ
iajs-2430	9	14	;	;	PUNCT
iajs-2430	9	15	interval	interval	NOUN
iajs-2430	9	16	value	value	NOUN
iajs-2430	9	17	fuzzy	fuzzy	ADJ
iajs-2430	9	18	kideal	kideal	NOUN
iajs-2430	9	19	;	;	PUNCT
iajs-2430	9	20	interval	interval	NOUN
iajs-2430	9	21	value	value	NOUN
iajs-2430	9	22	fuzzy	fuzzy	ADJ
iajs-2430	9	23	p	p	NOUN
iajs-2430	9	24	-	-	PUNCT
iajs-2430	9	25	ideal	ideal	NOUN
iajs-2430	9	26	.	.	PUNCT
iajs-2430	10	1	1	1	X
iajs-2430	10	2	.	.	X
iajs-2430	10	3	introduction	introduction	NOUN
iajs-2430	10	4	prabpayaknandˑleerawat	prabpayaknandˑleerawat	VERB
iajs-2430	11	1	[	[	X
iajs-2430	11	2	1,2	1,2	NUM
iajs-2430	11	3	]	]	PUNCT
iajs-2430	11	4	.	.	PUNCT
iajs-2430	11	5	introduced	introduce	VERB
iajs-2430	11	6	the	the	DET
iajs-2430	11	7	ku	ku	PROPN
iajs-2430	11	8	-	-	PUNCT
iajs-2430	11	9	algebra	algebra	PROPN
iajs-2430	11	10	which	which	PRON
iajs-2430	11	11	is	be	AUX
iajs-2430	11	12	dual	dual	ADJ
iajs-2430	11	13	of	of	ADP
iajs-2430	11	14	bck	bck	NOUN
iajs-2430	11	15	-	-	PUNCT
iajs-2430	11	16	algebra	algebra	NOUN
iajs-2430	11	17	.	.	PUNCT
iajs-2430	12	1	inˑ[3	inˑ[3	NUM
iajs-2430	12	2	]	]	PUNCT
iajs-2430	12	3	.	.	PUNCT
iajs-2430	13	1	kareem	kareem	PROPN
iajs-2430	13	2	and	and	CCONJ
iajs-2430	13	3	hasan	hasan	PROPN
iajs-2430	13	4	introduced	introduce	VERB
iajs-2430	13	5	the	the	DET
iajs-2430	13	6	ku	ku	NOUN
iajs-2430	13	7	-	-	PUNCT
iajs-2430	13	8	semigroups	semigroup	NOUN
iajs-2430	13	9	and	and	CCONJ
iajs-2430	13	10	defined	define	VERB
iajs-2430	13	11	some	some	DET
iajs-2430	13	12	types	type	NOUN
iajs-2430	13	13	of	of	ADP
iajs-2430	13	14	ideals	ideal	NOUN
iajs-2430	13	15	in	in	ADP
iajs-2430	13	16	this	this	DET
iajs-2430	13	17	concept	concept	NOUN
iajs-2430	13	18	.	.	PUNCT
iajs-2430	14	1	the	the	DET
iajs-2430	14	2	fuzzy	fuzzy	ADJ
iajs-2430	14	3	set	set	NOUN
iajs-2430	14	4	was	be	AUX
iajs-2430	14	5	initiated	initiate	VERB
iajs-2430	14	6	by	by	ADP
iajs-2430	14	7	zadeh	zadeh	PROPN
iajs-2430	14	8	,	,	PUNCT
iajs-2430	14	9	in	in	ADP
iajs-2430	14	10	[	[	X
iajs-2430	14	11	4	4	NUM
iajs-2430	14	12	]	]	PUNCT
iajs-2430	14	13	.	.	PUNCT
iajs-2430	15	1	since	since	SCONJ
iajs-2430	15	2	then	then	ADV
iajs-2430	15	3	this	this	DET
iajs-2430	15	4	concept	concept	NOUN
iajs-2430	15	5	has	have	AUX
iajs-2430	15	6	been	be	AUX
iajs-2430	15	7	applied	apply	VERB
iajs-2430	15	8	in	in	ADP
iajs-2430	15	9	many	many	ADJ
iajs-2430	15	10	distinct	distinct	ADJ
iajs-2430	15	11	branches	branch	NOUN
iajs-2430	15	12	of	of	ADP
iajs-2430	15	13	mathematics	mathematic	NOUN
iajs-2430	15	14	such	such	ADJ
iajs-2430	15	15	as	as	ADP
iajs-2430	15	16	groups	group	NOUN
iajs-2430	15	17	,	,	PUNCT
iajs-2430	15	18	vector	vector	NOUN
iajs-2430	15	19	space	space	NOUN
iajs-2430	15	20	,	,	PUNCT
iajs-2430	15	21	topological	topological	ADJ
iajs-2430	15	22	space	space	NOUN
iajs-2430	15	23	and	and	CCONJ
iajs-2430	15	24	ring	ring	NOUN
iajs-2430	15	25	theory	theory	NOUN
iajs-2430	15	26	.	.	PUNCT
iajs-2430	16	1	in	in	ADP
iajs-2430	16	2	[	[	X
iajs-2430	16	3	5	5	NUM
iajs-2430	16	4	]	]	PUNCT
iajs-2430	16	5	.	.	PUNCT
iajs-2430	17	1	the	the	DET
iajs-2430	17	2	idea	idea	NOUN
iajs-2430	17	3	of	of	ADP
iajs-2430	17	4	fuzzy	fuzzy	ADJ
iajs-2430	17	5	ku	ku	PROPN
iajs-2430	17	6	-	-	PUNCT
iajs-2430	17	7	algebra	algebra	PROPN
iajs-2430	17	8	was	be	AUX
iajs-2430	17	9	introduced	introduce	VERB
iajs-2430	17	10	by	by	ADP
iajs-2430	17	11	mostafa	mostafa	PROPN
iajs-2430	17	12	et	et	PROPN
iajs-2430	17	13	al	al	PROPN
iajs-2430	17	14	.	.	PROPN
iajs-2430	17	15	and	and	CCONJ
iajs-2430	17	16	the	the	DET
iajs-2430	17	17	fuzzy	fuzzy	ADJ
iajs-2430	17	18	ku	ku	PROPN
iajs-2430	17	19	-	-	PUNCT
iajs-2430	17	20	semigroupwas	semigroupwas	PROPN
iajs-2430	17	21	studied	study	VERB
iajs-2430	17	22	by	by	ADP
iajs-2430	17	23	elaf	elaf	PROPN
iajs-2430	17	24	and	and	CCONJ
iajs-2430	17	25	kareem	kareem	PROPN
iajs-2430	17	26	in	in	ADP
iajs-2430	17	27	[	[	X
iajs-2430	17	28	6	6	NUM
iajs-2430	17	29	]	]	PUNCT
iajs-2430	17	30	.	.	PUNCT
iajs-2430	18	1	fuzzy	fuzzy	ADJ
iajs-2430	18	2	sets	set	NOUN
iajs-2430	18	3	extensions	extension	NOUN
iajs-2430	18	4	such	such	ADJ
iajs-2430	18	5	as	as	ADP
iajs-2430	18	6	intuitionistic	intuitionistic	ADJ
iajs-2430	18	7	fuzzy	fuzzy	ADJ
iajs-2430	18	8	sets	set	NOUN
iajs-2430	18	9	,	,	PUNCT
iajs-2430	18	10	bipolar	bipolar	ADJ
iajs-2430	18	11	-	-	PUNCT
iajs-2430	18	12	valued	value	VERB
iajs-2430	18	13	fuzzy	fuzzy	ADJ
iajs-2430	18	14	sets	set	NOUN
iajs-2430	18	15	,	,	PUNCT
iajs-2430	18	16	and	and	CCONJ
iajs-2430	18	17	interval	interval	NOUN
iajs-2430	18	18	valued	value	VERB
iajs-2430	18	19	fuzzy	fuzzy	ADJ
iajs-2430	18	20	sets	set	NOUN
iajs-2430	18	21	were	be	AUX
iajs-2430	18	22	studied	study	VERB
iajs-2430	18	23	by	by	ADP
iajs-2430	18	24	many	many	ADJ
iajs-2430	18	25	mathematicians	mathematician	NOUN
iajs-2430	18	26	see	see	VERB
iajs-2430	18	27	[	[	X
iajs-2430	18	28	7	7	NUM
iajs-2430	18	29	-	-	SYM
iajs-2430	18	30	12	12	NUM
iajs-2430	18	31	]	]	PUNCT
iajs-2430	18	32	.	.	PUNCT
iajs-2430	19	1	the	the	DET
iajs-2430	19	2	notion	notion	NOUN
iajs-2430	19	3	of	of	ADP
iajs-2430	19	4	interval	interval	NOUN
iajs-2430	19	5	value	value	NOUN
iajs-2430	19	6	fuzzy	fuzzy	ADJ
iajs-2430	19	7	k	k	NOUN
iajs-2430	19	8	-	-	NOUN
iajs-2430	19	9	ideal	ideal	NOUN
iajs-2430	19	10	of	of	ADP
iajs-2430	19	11	kusemigroup	kusemigroup	NOUN
iajs-2430	19	12	was	be	AUX
iajs-2430	19	13	studied	study	VERB
iajs-2430	19	14	in	in	ADP
iajs-2430	19	15	this	this	DET
iajs-2430	19	16	paper	paper	NOUN
iajs-2430	19	17	and	and	CCONJ
iajs-2430	19	18	few	few	ADJ
iajs-2430	19	19	properties	property	NOUN
iajs-2430	19	20	were	be	AUX
iajs-2430	19	21	investigated	investigate	VERB
iajs-2430	19	22	.	.	PUNCT
iajs-2430	20	1	some	some	DET
iajs-2430	20	2	results	result	NOUN
iajs-2430	20	3	of	of	ADP
iajs-2430	20	4	these	these	DET
iajs-2430	20	5	ideals	ideal	NOUN
iajs-2430	20	6	ina	ina	PROPN
iajs-2430	20	7	ku	ku	PROPN
iajs-2430	20	8	-	-	PUNCT
iajs-2430	20	9	semigroup´under	semigroup´under	NOUN
iajs-2430	20	10	homomorphism	homomorphism	NOUN
iajs-2430	20	11	are	be	AUX
iajs-2430	20	12	discussed	discuss	VERB
iajs-2430	20	13	.	.	PUNCT
iajs-2430	21	1	the	the	DET
iajs-2430	21	2	image	image	NOUN
iajs-2430	21	3	of	of	ADP
iajs-2430	21	4	these	these	DET
iajs-2430	21	5	ideals	ideal	NOUN
iajs-2430	21	6	in	in	ADP
iajs-2430	21	7	a	a	DET
iajs-2430	21	8	ku	ku	PROPN
iajs-2430	21	9	-	-	PUNCT
iajs-2430	21	10	semigroup	semigroup	PROPN
iajs-2430	21	11	was	be	AUX
iajs-2430	21	12	defined	define	VERB
iajs-2430	21	13	.	.	PUNCT
iajs-2430	22	1	finally	finally	ADV
iajs-2430	22	2	,	,	PUNCT
iajs-2430	22	3	the	the	DET
iajs-2430	22	4	product	product	NOUN
iajs-2430	22	5	of	of	ADP
iajs-2430	22	6	some	some	DET
iajs-2430	22	7	ideals	ideal	NOUN
iajs-2430	22	8	was	be	AUX
iajs-2430	22	9	established	establish	VERB
iajs-2430	22	10	.	.	PUNCT
iajs-2430	23	1	2	2	X
iajs-2430	23	2	.	.	NUM
iajs-2430	23	3	preliminaries	preliminary	NOUN
iajs-2430	23	4	in	in	ADP
iajs-2430	23	5	this	this	DET
iajs-2430	23	6	part	part	NOUN
iajs-2430	23	7	,	,	PUNCT
iajs-2430	23	8	we	we	PRON
iajs-2430	23	9	review	review	VERB
iajs-2430	23	10	some	some	DET
iajs-2430	23	11	concepts	concept	NOUN
iajs-2430	23	12	related	relate	VERB
iajs-2430	23	13	to	to	ADP
iajs-2430	23	14	ku	ku	PROPN
iajs-2430	23	15	-	-	PUNCT
iajs-2430	23	16	semigroup	semigroup	PROPN
iajs-2430	23	17	and	and	CCONJ
iajs-2430	23	18	interval	interval	NOUN
iajs-2430	23	19	valued	value	VERB
iajs-2430	23	20	fuzzy	fuzzy	ADJ
iajs-2430	23	21	logic	logic	NOUN
iajs-2430	23	22	.	.	PUNCT
iajs-2430	24	1	definition	definition	NOUN
iajs-2430	24	2	(	(	PUNCT
iajs-2430	24	3	1	1	X
iajs-2430	24	4	)	)	PUNCT
iajs-2430	25	1	[	[	X
iajs-2430	25	2	1	1	NUM
iajs-2430	25	3	-	-	SYM
iajs-2430	25	4	2	2	NUM
iajs-2430	25	5	]	]	PUNCT
iajs-2430	25	6	.	.	PUNCT
iajs-2430	26	1	algebra	algebra	PROPN
iajs-2430	26	2	ℵ,∗	ℵ,∗	PROPN
iajs-2430	26	3	,	,	PUNCT
iajs-2430	26	4	0	0	NUM
iajs-2430	26	5	is	be	AUX
iajs-2430	26	6	called	call	VERB
iajs-2430	26	7	a	a	DET
iajs-2430	26	8	ku	ku	NOUN
iajs-2430	26	9	-	-	PUNCT
iajs-2430	26	10	algebra	algebra	PROPN
iajs-2430	26	11	if	if	SCONJ
iajs-2430	26	12	,	,	PUNCT
iajs-2430	26	13	for	for	ADP
iajs-2430	26	14	all	all	DET
iajs-2430	26	15	𝜒	𝜒	NOUN
iajs-2430	26	16	,	,	PUNCT
iajs-2430	26	17	𝛾	𝛾	PROPN
iajs-2430	26	18	,	,	PUNCT
iajs-2430	26	19	τ	τ	PROPN
iajs-2430	26	20	∈	∈	PROPN
iajs-2430	26	21	ℵ	ℵ	NOUN
iajs-2430	26	22	,	,	PUNCT
iajs-2430	26	23	ibn	ibn	PROPN
iajs-2430	26	24	al	al	PROPN
iajs-2430	26	25	haitham	haitham	PROPN
iajs-2430	26	26	journal	journal	PROPN
iajs-2430	26	27	for	for	ADP
iajs-2430	26	28	pure	pure	ADJ
iajs-2430	26	29	and	and	CCONJ
iajs-2430	26	30	applied	apply	VERB
iajs-2430	26	31	science	science	NOUN
iajs-2430	26	32	journal	journal	PROPN
iajs-2430	26	33	homepage	homepage	NOUN
iajs-2430	26	34	:	:	PUNCT
iajs-2430	26	35	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2430	26	36	doi	doi	NOUN
iajs-2430	26	37	:	:	PUNCT
iajs-2430	26	38	10.30526/33.2.2430	10.30526/33.2.2430	NUM
iajs-2430	26	39	sally	sally	ADJ
iajs-2430	26	40	a.	a.	PROPN
iajs-2430	26	41	talib	talib	PROPN
iajs-2430	26	42	fatema	fatema	PROPN
iajs-2430	26	43	f.	f.	PROPN
iajs-2430	26	44	kareem	kareem	PROPN
iajs-2430	26	45	rim89@gmail.comsallyabdulka	rim89@gmail.comsallyabdulka	PROPN
iajs-2430	27	1	fa_sa20072000@yahoo.com	fa_sa20072000@yahoo.com	PROPN
iajs-2430	27	2	department	department	PROPN
iajs-2430	27	3	of	of	ADP
iajs-2430	27	4	mathematics	mathematics	PROPN
iajs-2430	27	5	,	,	PUNCT
iajs-2430	27	6	college	college	NOUN
iajs-2430	27	7	of	of	ADP
iajs-2430	27	8	education	education	NOUN
iajs-2430	27	9	for	for	ADP
iajs-2430	27	10	pure	pure	ADJ
iajs-2430	27	11	science	science	NOUN
iajs-2430	27	12	ibn	ibn	PROPN
iajs-2430	27	13	-	-	PUNCT
iajs-2430	27	14	al	al	PROPN
iajs-2430	27	15	-	-	PUNCT
iajs-2430	27	16	haitham	haitham	PROPN
iajs-2430	27	17	,	,	PUNCT
iajs-2430	27	18	university	university	PROPN
iajs-2430	27	19	of	of	ADP
iajs-2430	27	20	baghdad	baghdad	PROPN
iajs-2430	27	21	,	,	PUNCT
iajs-2430	27	22	baghdad	baghdad	PROPN
iajs-2430	27	23	,	,	PUNCT
iajs-2430	27	24	iraq	iraq	PROPN
iajs-2430	27	25	.	.	PUNCT
iajs-2430	27	26	    	    	SPACE
iajs-2430	28	1	96	96	NUM
iajs-2430	28	2	  	  	SPACE
iajs-2430	28	3	ibn	ibn	PROPN
iajs-2430	28	4	al	al	PROPN
iajs-2430	28	5	-	-	PUNCT
iajs-2430	28	6	haitham	haitham	PROPN
iajs-2430	28	7	jour	jour	X
iajs-2430	28	8	.	.	PROPN
iajs-2430	28	9	for	for	ADP
iajs-2430	28	10	pure	pure	ADJ
iajs-2430	28	11	&	&	CCONJ
iajs-2430	28	12	appl	appl	PROPN
iajs-2430	28	13	.	.	PUNCT
iajs-2430	29	1	sci	sci	PROPN
iajs-2430	29	2	.	.	PROPN
iajs-2430	30	1	33	33	NUM
iajs-2430	30	2	(	(	PUNCT
iajs-2430	30	3	2	2	NUM
iajs-2430	30	4	)	)	PUNCT
iajs-2430	30	5	2020	2020	NUM
iajs-2430	30	6	(	(	PUNCT
iajs-2430	30	7	ku	ku	PROPN
iajs-2430	30	8	)	)	PUNCT
iajs-2430	30	9	𝜒	𝜒	PROPN
iajs-2430	30	10	∗	∗	NOUN
iajs-2430	30	11	𝛾	𝛾	ADP
iajs-2430	30	12	∗	∗	NOUN
iajs-2430	30	13	𝛾	𝛾	ADP
iajs-2430	30	14	∗	∗	NOUN
iajs-2430	30	15	τ	τ	PROPN
iajs-2430	30	16	∗	∗	X
iajs-2430	30	17	𝜒	𝜒	PROPN
iajs-2430	30	18	∗	∗	NOUN
iajs-2430	30	19	τ	τ	X
iajs-2430	30	20	0	0	NUM
iajs-2430	30	21	,	,	PUNCT
iajs-2430	30	22	(	(	PUNCT
iajs-2430	30	23	ku	ku	PROPN
iajs-2430	30	24	)	)	PUNCT
iajs-2430	30	25	𝜒	𝜒	NOUN
iajs-2430	30	26	∗0	∗0	PROPN
iajs-2430	30	27	=	=	SYM
iajs-2430	30	28	0	0	NUM
iajs-2430	30	29	,	,	PUNCT
iajs-2430	30	30	(	(	PUNCT
iajs-2430	30	31	ku	ku	PROPN
iajs-2430	30	32	)	)	PUNCT
iajs-2430	30	33	0	0	PUNCT
iajs-2430	31	1	*	*	SYM
iajs-2430	31	2	𝜒	𝜒	X
iajs-2430	31	3	𝜒	𝜒	NOUN
iajs-2430	31	4	,	,	PUNCT
iajs-2430	31	5	(	(	PUNCT
iajs-2430	31	6	ku	ku	PROPN
iajs-2430	31	7	)	)	PUNCT
iajs-2430	31	8	𝜒*𝛾	𝜒*𝛾	NOUN
iajs-2430	31	9	0	0	NUM
iajs-2430	31	10	and	and	CCONJ
iajs-2430	31	11	𝛾	𝛾	PROPN
iajs-2430	31	12	∗	∗	NOUN
iajs-2430	31	13	𝜒	𝜒	X
iajs-2430	31	14	implies	imply	VERB
iajs-2430	31	15	𝜒	𝜒	NUM
iajs-2430	31	16	𝛾	𝛾	NOUN
iajs-2430	31	17	and	and	CCONJ
iajs-2430	31	18	,	,	PUNCT
iajs-2430	31	19	(	(	PUNCT
iajs-2430	31	20	ku	ku	PROPN
iajs-2430	31	21	)	)	PUNCT
iajs-2430	31	22	𝜒	𝜒	PROPN
iajs-2430	31	23	∗	∗	NOUN
iajs-2430	31	24	𝜒	𝜒	NOUN
iajs-2430	31	25	0	0	NUM
iajs-2430	31	26	.	.	PUNCT
iajs-2430	32	1	a	a	DET
iajs-2430	32	2	binary	binary	ADJ
iajs-2430	32	3	relation	relation	NOUN
iajs-2430	32	4	on	on	ADP
iajs-2430	32	5	ℵ	ℵ	NOUN
iajs-2430	32	6	is	be	AUX
iajs-2430	32	7	defined	define	VERB
iajs-2430	32	8	by	by	ADP
iajs-2430	32	9	𝜒	𝜒	X
iajs-2430	32	10	𝛾	𝛾	AUX
iajs-2430	32	11	⟺	⟺	PRON
iajs-2430	32	12	𝛾	𝛾	NOUN
iajs-2430	32	13	∗	∗	NOUN
iajs-2430	32	14	𝜒	𝜒	NOUN
iajs-2430	32	15	0	0	NUM
iajs-2430	32	16	.	.	PUNCT
iajs-2430	33	1	it	it	PRON
iajs-2430	33	2	follows	follow	VERB
iajs-2430	33	3	that	that	SCONJ
iajs-2430	33	4	(	(	PUNCT
iajs-2430	33	5	ℵ	ℵ	NOUN
iajs-2430	33	6	,	,	PUNCT
iajs-2430	33	7	is	be	AUX
iajs-2430	33	8	a~	a~	PROPN
iajs-2430	33	9	partially	partially	ADV
iajs-2430	33	10	ordered	order	VERB
iajs-2430	33	11	set~.	set~.	NOUN
iajs-2430	33	12	then	then	ADV
iajs-2430	33	13	,	,	PUNCT
iajs-2430	33	14	ℵ,∗	ℵ,∗	PROPN
iajs-2430	33	15	,	,	PUNCT
iajs-2430	33	16	0	0	NUM
iajs-2430	33	17	)	)	PUNCT
iajs-2430	33	18	~	~	PUNCT
iajs-2430	33	19	satisfies	satisfy	VERB
iajs-2430	33	20	the	the	DET
iajs-2430	33	21	following	following	ADJ
iajs-2430	33	22	statements	statement	NOUN
iajs-2430	33	23	.	.	PUNCT
iajs-2430	34	1	for	for	ADP
iajs-2430	34	2	all	all	DET
iajs-2430	34	3	𝜒	𝜒	NOUN
iajs-2430	34	4	,	,	PUNCT
iajs-2430	34	5	𝛾	𝛾	PROPN
iajs-2430	34	6	,	,	PUNCT
iajs-2430	34	7	τ	τ	PROPN
iajs-2430	34	8	∈	∈	PROPN
iajs-2430	34	9	ℵ	ℵ	NOUN
iajs-2430	34	10	,	,	PUNCT
iajs-2430	34	11	  	  	SPACE
iajs-2430	34	12	(	(	PUNCT
iajs-2430	34	13	ᴋu	ᴋu	NOUN
iajs-2430	34	14	)	)	PUNCT
iajs-2430	34	15	𝛾	𝛾	ADP
iajs-2430	34	16	∗	∗	NOUN
iajs-2430	34	17	τ	τ	PROPN
iajs-2430	34	18	∗	∗	X
iajs-2430	34	19	𝜒	𝜒	PROPN
iajs-2430	34	20	∗	∗	NOUN
iajs-2430	34	21	τ	τ	X
iajs-2430	34	22	𝜒	𝜒	X
iajs-2430	34	23	∗	∗	NOUN
iajs-2430	34	24	𝛾	𝛾	NOUN
iajs-2430	34	25	,	,	PUNCT
iajs-2430	34	26	(	(	PUNCT
iajs-2430	34	27	ᴋu	ᴋu	NOUN
iajs-2430	34	28	)	)	PUNCT
iajs-2430	34	29	0	0	NUM
iajs-2430	34	30	𝜒	𝜒	NUM
iajs-2430	34	31	,	,	PUNCT
iajs-2430	34	32	(	(	PUNCT
iajs-2430	34	33	ᴋu	ᴋu	NOUN
iajs-2430	34	34	)	)	PUNCT
iajs-2430	34	35	𝜒	𝜒	NOUN
iajs-2430	34	36	𝛾	𝛾	NOUN
iajs-2430	34	37	,	,	PUNCT
iajs-2430	34	38	𝛾	𝛾	ADP
iajs-2430	34	39	𝜒	𝜒	PRON
iajs-2430	34	40	implies	imply	VERB
iajs-2430	34	41			PROPN
iajs-2430	34	42			PROPN
iajs-2430	34	43	(	(	PUNCT
iajs-2430	34	44	ᴋu	ᴋu	NOUN
iajs-2430	34	45	)	)	PUNCT
iajs-2430	34	46	𝛾	𝛾	ADP
iajs-2430	34	47	∗	∗	NOUN
iajs-2430	34	48	𝜒	𝜒	X
iajs-2430	34	49	𝜒	𝜒	NOUN
iajs-2430	34	50	.	.	PUNCT
iajs-2430	35	1	example	example	NOUN
iajs-2430	35	2	(	(	PUNCT
iajs-2430	35	3	2)[1	2)[1	NUM
iajs-2430	35	4	]	]	PUNCT
iajs-2430	35	5	.	.	PUNCT
iajs-2430	36	1	let	let	VERB
iajs-2430	36	2	ℵ	ℵ	ADP
iajs-2430	36	3	0	0	NUM
iajs-2430	36	4	,	,	PUNCT
iajs-2430	36	5	𝑎	𝑎	NOUN
iajs-2430	36	6	,	,	PUNCT
iajs-2430	36	7	𝑏	𝑏	NOUN
iajs-2430	36	8	,	,	PUNCT
iajs-2430	36	9	𝑐	𝑐	PROPN
iajs-2430	36	10	be	be	AUX
iajs-2430	36	11	a	a	DET
iajs-2430	36	12	set	set	NOUN
iajs-2430	36	13	and	and	CCONJ
iajs-2430	36	14	∗	∗	NOUN
iajs-2430	36	15	a	a	DET
iajs-2430	36	16	binary	binary	ADJ
iajs-2430	36	17	operation	operation	NOUN
iajs-2430	36	18	defined	define	VERB
iajs-2430	36	19	in	in	ADP
iajs-2430	36	20	the	the	DET
iajs-2430	36	21	following	follow	VERB
iajs-2430	36	22	table	table	NOUN
iajs-2430	36	23	it	it	PRON
iajs-2430	36	24	is	be	AUX
iajs-2430	36	25	easy	easy	ADJ
iajs-2430	36	26	to	to	PART
iajs-2430	36	27	see	see	VERB
iajs-2430	36	28	that	that	SCONJ
iajs-2430	36	29	ℵ,∗	ℵ,∗	PROPN
iajs-2430	36	30	,	,	PUNCT
iajs-2430	36	31	0	0	NUM
iajs-2430	36	32	is	be	AUX
iajs-2430	36	33	a	a	DET
iajs-2430	36	34	ku	ku	NOUN
iajs-2430	36	35	-	-	PUNCT
iajs-2430	36	36	algebra	algebra	PROPN
iajs-2430	36	37	.	.	PUNCT
iajs-2430	37	1	ˑtheorem(ˑ3	ˑtheorem(ˑ3	NOUN
iajs-2430	37	2	)	)	PUNCT
iajs-2430	38	1	ˑ	ˑ	PROPN
iajs-2430	39	1	[	[	X
iajs-2430	39	2	2	2	NUM
iajs-2430	39	3	]	]	PUNCT
iajs-2430	39	4	.	.	PUNCT
iajs-2430	40	1	ˑlet	ˑlet	PROPN
iajs-2430	40	2	ℵ,∗	ℵ,∗	PROPN
iajs-2430	40	3	,	,	PUNCT
iajs-2430	40	4	0	0	NUM
iajs-2430	40	5	be	be	AUX
iajs-2430	40	6	a	a	DET
iajs-2430	40	7	ku	ku	NOUN
iajs-2430	40	8	-	-	PUNCT
iajs-2430	40	9	algebra	algebra	PROPN
iajs-2430	40	10	.	.	PUNCT
iajs-2430	41	1	then	then	ADV
iajs-2430	41	2	,	,	PUNCT
iajs-2430	41	3	for	for	ADP
iajs-2430	41	4	all	all	DET
iajs-2430	41	5	𝜒	𝜒	NOUN
iajs-2430	41	6	,	,	PUNCT
iajs-2430	41	7	𝛾	𝛾	PROPN
iajs-2430	41	8	,	,	PUNCT
iajs-2430	41	9	τ	τ	PROPN
iajs-2430	41	10	∈	∈	PROPN
iajs-2430	41	11	ℵ	ℵ	NOUN
iajs-2430	41	12	,	,	PUNCT
iajs-2430	41	13	(	(	PUNCT
iajs-2430	41	14	1	1	X
iajs-2430	41	15	)	)	PUNCT
iajs-2430	41	16	if	if	SCONJ
iajs-2430	41	17	𝜒	𝜒	PRON
iajs-2430	41	18	𝛾	𝛾	NOUN
iajs-2430	41	19	implies	imply	VERB
iajs-2430	41	20	𝛾	𝛾	AUX
iajs-2430	41	21	∗	∗	NOUN
iajs-2430	41	22	τ	τ	X
iajs-2430	41	23	𝜒	𝜒	X
iajs-2430	41	24	∗	∗	NOUN
iajs-2430	41	25	τ	τ	X
iajs-2430	41	26	,	,	PUNCT
iajs-2430	41	27	(	(	PUNCT
iajs-2430	41	28	2	2	NUM
iajs-2430	41	29	)	)	PUNCT
iajs-2430	41	30	𝜒	𝜒	NOUN
iajs-2430	41	31	∗	∗	NOUN
iajs-2430	41	32	𝛾	𝛾	ADP
iajs-2430	41	33	∗	∗	NOUN
iajs-2430	41	34	τ	τ	X
iajs-2430	41	35	𝛾	𝛾	PROPN
iajs-2430	41	36	∗	∗	NOUN
iajs-2430	41	37	𝜒	𝜒	NUM
iajs-2430	41	38	∗	∗	NOUN
iajs-2430	41	39	τ	τ	X
iajs-2430	41	40	,	,	PUNCT
iajs-2430	41	41	(	(	PUNCT
iajs-2430	41	42	3	3	NUM
iajs-2430	41	43	)	)	PUNCT
iajs-2430	41	44	(	(	PUNCT
iajs-2430	41	45	(	(	PUNCT
iajs-2430	41	46	𝛾	𝛾	NOUN
iajs-2430	41	47	∗	∗	X
iajs-2430	41	48	𝜒	𝜒	NUM
iajs-2430	41	49	∗	∗	NOUN
iajs-2430	41	50	𝜒	𝜒	X
iajs-2430	41	51	𝛾.	𝛾.	ADJ
iajs-2430	41	52	definition	definition	NOUN
iajs-2430	41	53	(	(	PUNCT
iajs-2430	41	54	4	4	NUM
iajs-2430	41	55	)	)	PUNCT
iajs-2430	42	1	[	[	X
iajs-2430	42	2	1	1	NUM
iajs-2430	42	3	-	-	SYM
iajs-2430	42	4	2	2	NUM
iajs-2430	42	5	]	]	PUNCT
iajs-2430	42	6	.	.	PUNCT
iajs-2430	43	1	let	let	VERB
iajs-2430	44	1	ℵ,∗	ℵ,∗	PROPN
iajs-2430	44	2	,	,	PUNCT
iajs-2430	44	3	0	0	NUM
iajs-2430	44	4	be	be	AUX
iajs-2430	44	5	a	a	DET
iajs-2430	44	6	ku	ku	NOUN
iajs-2430	44	7	-	-	PUNCT
iajs-2430	44	8	algebra	algebra	PROPN
iajs-2430	44	9	and	and	CCONJ
iajs-2430	44	10	𝛪	𝛪	PROPN
iajs-2430	44	11	be	be	VERB
iajs-2430	44	12	a	a	DET
iajs-2430	44	13	non-ˑempty	non-ˑempty	ADJ
iajs-2430	44	14	subset	subset	NOUN
iajs-2430	44	15	ofˑℵ.	ofˑℵ.	NOUN
iajs-2430	44	16	then	then	ADV
iajs-2430	44	17	𝛪	𝛪	PROPN
iajs-2430	44	18	is	be	AUX
iajs-2430	44	19	calledˑˑan	calledˑˑan	ADJ
iajs-2430	44	20	ˑideal	ˑideal	NOUN
iajs-2430	44	21	of	of	ADP
iajs-2430	44	22	ℵ	ℵ	NOUN
iajs-2430	44	23	if	if	SCONJ
iajs-2430	44	24	ˑfor	ˑfor	ADP
iajs-2430	44	25	any	any	DET
iajs-2430	44	26	𝜒	𝜒	NOUN
iajs-2430	44	27	,	,	PUNCT
iajs-2430	44	28	𝛾	𝛾	PROPN
iajs-2430	44	29	∈	∈	NOUN
iajs-2430	44	30	ℵ,ˑthen	ℵ,ˑthen	NOUN
iajs-2430	45	1	(	(	PUNCT
iajs-2430	45	2	i	i	NOUN
iajs-2430	45	3	)	)	PUNCT
iajs-2430	45	4	0	0	PUNCT
iajs-2430	46	1	∈	∈	PROPN
iajs-2430	46	2	𝛪and	𝛪and	PROPN
iajs-2430	46	3	(	(	PUNCT
iajs-2430	46	4	ii	ii	PROPN
iajs-2430	46	5	)	)	PUNCT
iajs-2430	46	6	if	if	SCONJ
iajs-2430	46	7	𝜒	𝜒	PRON
iajs-2430	46	8	∗	∗	NOUN
iajs-2430	46	9	𝛾	𝛾	ADP
iajs-2430	46	10	∈	∈	PROPN
iajs-2430	46	11	𝛪	𝛪	PROPN
iajs-2430	46	12	and	and	CCONJ
iajs-2430	46	13	𝜒	𝜒	NOUN
iajs-2430	46	14	∈	∈	PROPN
iajs-2430	46	15	𝛪	𝛪	PROPN
iajs-2430	46	16	imply	imply	VERB
iajs-2430	46	17	𝛾	𝛾	AUX
iajs-2430	46	18	∈	∈	NOUN
iajs-2430	46	19	𝛪.ˑˑ	𝛪.ˑˑ	X
iajs-2430	46	20	ˑdefinition	ˑdefinition	NOUN
iajs-2430	46	21	(	(	PUNCT
iajs-2430	46	22	5	5	NUM
iajs-2430	46	23	)	)	PUNCT
iajs-2430	46	24	[	[	X
iajs-2430	46	25	1	1	NUM
iajs-2430	46	26	-	-	SYM
iajs-2430	46	27	2].ˑ	2].ˑ	NUM
iajs-2430	46	28	let	let	VERB
iajs-2430	46	29	𝛪	𝛪	PRON
iajs-2430	46	30	be	be	AUX
iajs-2430	46	31	a	a	DET
iajs-2430	46	32	subset	subset	NOUN
iajs-2430	46	33	of	of	ADP
iajs-2430	46	34	a	a	DET
iajs-2430	46	35	ku	ku	PROPN
iajs-2430	46	36	-	-	PUNCT
iajs-2430	46	37	algebra	algebra	PROPN
iajs-2430	46	38	ℵ,∗	ℵ,∗	PROPN
iajs-2430	46	39	,	,	PUNCT
iajs-2430	46	40	0	0	NUM
iajs-2430	46	41	and	and	CCONJ
iajs-2430	46	42	𝜑.	𝜑.	VERB
iajs-2430	46	43	then	then	ADV
iajs-2430	46	44	𝛪	𝛪	PROPN
iajs-2430	46	45	is	be	AUX
iajs-2430	46	46	named	name	VERB
iajs-2430	46	47	a	a	DET
iajs-2430	46	48	ku	ku	NOUN
iajs-2430	46	49	-	-	PUNCT
iajs-2430	46	50	ideal	ideal	NOUN
iajs-2430	46	51	of	of	ADP
iajs-2430	46	52	ℵ	ℵ	NOUN
iajs-2430	46	53	,	,	PUNCT
iajs-2430	46	54	if	if	SCONJ
iajs-2430	46	55	(	(	PUNCT
iajs-2430	46	56	ι	ι	PART
iajs-2430	46	57	0	0	NUM
iajs-2430	46	58	∈	∈	PROPN
iajs-2430	46	59	𝛪	𝛪	PROPN
iajs-2430	46	60	and	and	CCONJ
iajs-2430	46	61	(	(	PUNCT
iajs-2430	46	62	ι	ι	X
iajs-2430	46	63	∀𝜒	∀𝜒	PROPN
iajs-2430	46	64	,	,	PUNCT
iajs-2430	46	65	𝛾	𝛾	PROPN
iajs-2430	46	66	,	,	PUNCT
iajs-2430	46	67	𝜏	𝜏	PROPN
iajs-2430	46	68	∈	∈	PROPN
iajs-2430	46	69	ℵ	ℵ	NOUN
iajs-2430	46	70	,	,	PUNCT
iajs-2430	46	71	𝜒	𝜒	PRON
iajs-2430	46	72	∗	∗	NOUN
iajs-2430	46	73	𝛾	𝛾	ADP
iajs-2430	46	74	∗	∗	NOUN
iajs-2430	46	75	𝜏	𝜏	PRON
iajs-2430	46	76	∈	∈	NOUN
iajs-2430	46	77	𝐼	𝐼	ADP
iajs-2430	46	78	an𝑑	an𝑑	PROPN
iajs-2430	46	79	𝛾	𝛾	ADP
iajs-2430	46	80	∈	∈	PROPN
iajs-2430	46	81	𝛪imply	𝛪imply	PROPN
iajs-2430	46	82	𝜒	𝜒	X
iajs-2430	46	83	∗	∗	NOUN
iajs-2430	46	84	𝜏	𝜏	PRON
iajs-2430	46	85	∈	∈	NOUN
iajs-2430	46	86	𝛪.	𝛪.	NOUN
iajs-2430	46	87	definition	definition	NOUN
iajs-2430	46	88	(	(	PUNCT
iajs-2430	46	89	6)[3	6)[3	ADJ
iajs-2430	46	90	]	]	PUNCT
iajs-2430	46	91	.	.	PUNCT
iajs-2430	47	1	a	a	DET
iajs-2430	47	2	ku	ku	PROPN
iajs-2430	47	3	-	-	PUNCT
iajs-2430	47	4	semigroup	semigroup	PROPN
iajs-2430	47	5	is	be	AUX
iajs-2430	47	6	a	a	DET
iajs-2430	47	7	nonempty	nonempty	ADJ
iajs-2430	47	8	set	set	VERB
iajs-2430	47	9	ℵ	ℵ	NOUN
iajs-2430	47	10	with	with	ADP
iajs-2430	47	11	two	two	NUM
iajs-2430	47	12	binary	binary	ADJ
iajs-2430	47	13	operations	operation	NOUN
iajs-2430	47	14	~∗,∘and	~∗,∘and	CCONJ
iajs-2430	47	15	a	a	DET
iajs-2430	47	16	constant	constant	ADJ
iajs-2430	47	17	~0	~0	NOUN
iajs-2430	47	18	satisfying~	satisfying~	VERB
iajs-2430	47	19	the	the	DET
iajs-2430	47	20	following~~	following~~	PROPN
iajs-2430	47	21	(	(	PUNCT
iajs-2430	47	22	i	i	NOUN
iajs-2430	47	23	)	)	PUNCT
iajs-2430	47	24	ℵ,∗	ℵ,∗	PROPN
iajs-2430	47	25	,	,	PUNCT
iajs-2430	47	26	0	0	NUM
iajs-2430	47	27	isˑa	isˑa	PROPN
iajs-2430	47	28	ku	ku	PROPN
iajs-2430	47	29	-	-	PUNCT
iajs-2430	47	30	algebra	algebra	PROPN
iajs-2430	47	31	,	,	PUNCT
iajs-2430	47	32	  	  	SPACE
iajs-2430	47	33	97	97	NUM
iajs-2430	47	34	  	  	SPACE
iajs-2430	47	35	ibn	ibn	PROPN
iajs-2430	47	36	al	al	PROPN
iajs-2430	47	37	-	-	PUNCT
iajs-2430	47	38	haitham	haitham	PROPN
iajs-2430	47	39	jour	jour	X
iajs-2430	47	40	.	.	PROPN
iajs-2430	48	1	for	for	ADP
iajs-2430	48	2	pure	pure	ADJ
iajs-2430	48	3	&	&	CCONJ
iajs-2430	48	4	appl	appl	PROPN
iajs-2430	48	5	.	.	PUNCT
iajs-2430	49	1	sci	sci	PROPN
iajs-2430	49	2	.	.	PROPN
iajs-2430	50	1	33	33	NUM
iajs-2430	50	2	(	(	PUNCT
iajs-2430	50	3	2	2	NUM
iajs-2430	50	4	)	)	PUNCT
iajs-2430	50	5	2020	2020	NUM
iajs-2430	50	6	(	(	PUNCT
iajs-2430	50	7	ii	ii	NOUN
iajs-2430	50	8	)	)	PUNCT
iajs-2430	50	9	(	(	PUNCT
iajs-2430	50	10	ℵ,∘	ℵ,∘	NOUN
iajs-2430	50	11	is	be	AUX
iajs-2430	50	12	aˑsemigroup	aˑsemigroup	NOUN
iajs-2430	50	13	,	,	PUNCT
iajs-2430	50	14	(	(	PUNCT
iajs-2430	50	15	iii	iii	X
iajs-2430	50	16	)	)	PUNCT
iajs-2430	50	17	the	the	DET
iajs-2430	50	18	operation	operation	NOUN
iajs-2430	50	19	∘	∘	PROPN
iajs-2430	50	20	is	be	AUX
iajs-2430	50	21	distributiveˑ	distributiveˑ	VERB
iajs-2430	50	22	(	(	PUNCT
iajs-2430	50	23	on	on	ADP
iajs-2430	50	24	both	both	DET
iajs-2430	50	25	sides	side	NOUN
iajs-2430	50	26	)	)	PUNCT
iajs-2430	50	27	over	over	ADP
iajs-2430	50	28	theoperation	theoperation	NOUN
iajs-2430	50	29	∗	∗	NOUN
iajs-2430	50	30	,	,	PUNCT
iajs-2430	50	31	i.e.	i.e.	X
iajs-2430	50	32	𝜒	𝜒	X
iajs-2430	50	33	∘	∘	NOUN
iajs-2430	50	34	𝛾	𝛾	ADP
iajs-2430	50	35	∗	∗	NOUN
iajs-2430	50	36	τ	τ	X
iajs-2430	50	37	𝜒	𝜒	PRON
iajs-2430	50	38	∘	∘	NOUN
iajs-2430	50	39	𝛾	𝛾	ADP
iajs-2430	50	40	∗	∗	NOUN
iajs-2430	50	41	𝜒	𝜒	X
iajs-2430	50	42	∘	∘	X
iajs-2430	50	43	τ	τ	X
iajs-2430	50	44	and	and	CCONJ
iajs-2430	50	45	𝜒	𝜒	NOUN
iajs-2430	50	46	∗	∗	NOUN
iajs-2430	50	47	𝛾	𝛾	ADP
iajs-2430	50	48	∘	∘	PROPN
iajs-2430	50	49	τ	τ	X
iajs-2430	50	50	𝜒	𝜒	X
iajs-2430	50	51	∘	∘	X
iajs-2430	50	52	τ	τ	PROPN
iajs-2430	50	53	∗	∗	NOUN
iajs-2430	50	54	𝛾	𝛾	ADP
iajs-2430	50	55	∘	∘	PROPN
iajs-2430	50	56	τ	τ	X
iajs-2430	50	57	,	,	PUNCT
iajs-2430	50	58	forall	forall	NOUN
iajs-2430	50	59	𝜒	𝜒	SYM
iajs-2430	50	60	,	,	PUNCT
iajs-2430	50	61	𝛾	𝛾	PROPN
iajs-2430	50	62	,	,	PUNCT
iajs-2430	50	63	τ	τ	PROPN
iajs-2430	50	64	∈	∈	PROPN
iajs-2430	50	65	𝛸.	𝛸.	PROPN
iajs-2430	50	66	example	example	NOUN
iajs-2430	50	67	(	(	PUNCT
iajs-2430	50	68	7)[3	7)[3	NOUN
iajs-2430	50	69	]	]	PUNCT
iajs-2430	50	70	.	.	PUNCT
iajs-2430	51	1	letˑℵ	letˑℵ	PROPN
iajs-2430	51	2	0,1,2,3	0,1,2,3	NUM
iajs-2430	51	3	beˑa	beˑa	NOUN
iajs-2430	51	4	set	set	NOUN
iajs-2430	51	5	.	.	PUNCT
iajs-2430	52	1	define∗ˑ-operation	define∗ˑ-operation	NOUN
iajs-2430	52	2	and	and	CCONJ
iajs-2430	52	3	∘-operationby	∘-operationby	NOUN
iajs-2430	52	4	the	the	DET
iajs-2430	52	5	following	follow	VERB
iajs-2430	52	6	tables	table	NOUN
iajs-2430	52	7	thenˑ	thenˑ	VERB
iajs-2430	52	8	ℵ,∗,∘	ℵ,∗,∘	PUNCT
iajs-2430	52	9	,	,	PUNCT
iajs-2430	52	10	0	0	NUM
iajs-2430	52	11	is	be	AUX
iajs-2430	52	12	aˑku	aˑku	NOUN
iajs-2430	52	13	-	-	PUNCT
iajs-2430	52	14	semigroup	semigroup	NOUN
iajs-2430	52	15	.	.	PUNCT
iajs-2430	52	16	"	"	PUNCT
iajs-2430	53	1	definition	definition	NOUN
iajs-2430	53	2	(	(	PUNCT
iajs-2430	53	3	8)[3	8)[3	NUM
iajs-2430	53	4	]	]	PUNCT
iajs-2430	53	5	.	.	PUNCT
iajs-2430	54	1	aˑsubku	aˑsubku	NOUN
iajs-2430	54	2	-	-	PUNCT
iajs-2430	54	3	semigroup	semigroup	PROPN
iajs-2430	54	4	is	be	AUX
iajs-2430	54	5	a	a	DET
iajs-2430	54	6	non	non	ADJ
iajs-2430	54	7	-	-	ADJ
iajs-2430	54	8	empty	empty	ADJ
iajs-2430	54	9	subseta	subseta	NOUN
iajs-2430	54	10	of	of	ADP
iajs-2430	54	11	aˑku	aˑku	NOUN
iajs-2430	54	12	-	-	PUNCT
iajs-2430	54	13	semigroup	semigroup	NOUN
iajs-2430	54	14	ℵ	ℵ	NOUN
iajs-2430	54	15	and	and	CCONJ
iajs-2430	54	16	it	it	PRON
iajs-2430	54	17	is	be	AUX
iajs-2430	54	18	satisfied	satisfied	ADJ
iajs-2430	54	19	𝜒	𝜒	X
iajs-2430	54	20	∗	∗	NOUN
iajs-2430	54	21	𝛾	𝛾	NOUN
iajs-2430	54	22	,	,	PUNCT
iajs-2430	54	23	𝜒	𝜒	X
iajs-2430	54	24	∘	∘	NOUN
iajs-2430	54	25	𝛾	𝛾	ADP
iajs-2430	54	26	∈	∈	PROPN
iajs-2430	54	27	α	α	NOUN
iajs-2430	54	28	,	,	PUNCT
iajs-2430	54	29	for	for	ADP
iajs-2430	54	30	all	all	DET
iajs-2430	54	31	𝜒	𝜒	NOUN
iajs-2430	54	32	,	,	PUNCT
iajs-2430	54	33	𝛾	𝛾	ADP
iajs-2430	54	34	∈	∈	PROPN
iajs-2430	55	1	α	α	NOUN
iajs-2430	55	2	.	.	PUNCT
iajs-2430	56	1	definition	definition	NOUN
iajs-2430	56	2	(	(	PUNCT
iajs-2430	56	3	9)[3	9)[3	NUM
iajs-2430	56	4	]	]	PUNCT
iajs-2430	56	5	.	.	PUNCT
iajs-2430	57	1	let	let	VERB
iajs-2430	57	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	57	3	,	,	PUNCT
iajs-2430	57	4	0	0	NUM
iajs-2430	57	5	be	be	AUX
iajs-2430	57	6	a	a	DET
iajs-2430	57	7	ku	ku	PROPN
iajs-2430	57	8	-	-	PUNCT
iajs-2430	57	9	semigroup	semigroup	PROPN
iajs-2430	57	10	and	and	CCONJ
iajs-2430	57	11	𝜑	𝜑	NOUN
iajs-2430	57	12	𝐼	𝐼	PROPN
iajs-2430	57	13	ℵ.	ℵ.	PROPN
iajs-2430	58	1	then	then	ADV
iajs-2430	58	2	,	,	PUNCT
iajs-2430	58	3	𝐼	𝐼	PROPN
iajs-2430	58	4	is	be	AUX
iajs-2430	58	5	named	name	VERB
iajs-2430	58	6	an	an	DET
iajs-2430	58	7	s	s	NOUN
iajs-2430	58	8	-	-	NOUN
iajs-2430	58	9	ideal	ideal	NOUN
iajs-2430	58	10	of	of	ADP
iajs-2430	58	11	ℵ	ℵ	NOUN
iajs-2430	58	12	,	,	PUNCT
iajs-2430	58	13	if	if	SCONJ
iajs-2430	58	14	i	i	PRON
iajs-2430	58	15	)	)	PUNCT
iajs-2430	59	1	𝐼	𝐼	PROPN
iajs-2430	59	2	is	be	AUX
iajs-2430	59	3	an	an	DET
iajs-2430	59	4	ideal	ideal	NOUN
iajs-2430	59	5	of	of	ADP
iajs-2430	59	6	a	a	DET
iajs-2430	59	7	ku	ku	PROPN
iajs-2430	59	8	-	-	PUNCT
iajs-2430	59	9	algebra	algebra	PROPN
iajs-2430	59	10	ℵ,∗	ℵ,∗	PROPN
iajs-2430	59	11	,	,	PUNCT
iajs-2430	59	12	0	0	NUM
iajs-2430	59	13	,	,	PUNCT
iajs-2430	59	14	ii	ii	PROPN
iajs-2430	59	15	)	)	PUNCT
iajs-2430	59	16	for	for	ADP
iajs-2430	59	17	all	all	DET
iajs-2430	59	18	𝜒	𝜒	NUM
iajs-2430	59	19	∈	∈	ADJ
iajs-2430	59	20	ℵ	ℵ	NOUN
iajs-2430	59	21	,	,	PUNCT
iajs-2430	59	22	𝑎	𝑎	PROPN
iajs-2430	59	23	∈	∈	PROPN
iajs-2430	59	24	𝐼	𝐼	NOUN
iajs-2430	59	25	,	,	PUNCT
iajs-2430	59	26	we	we	PRON
iajs-2430	59	27	have	have	VERB
iajs-2430	59	28	𝜒	𝜒	NUM
iajs-2430	59	29	∘	∘	NUM
iajs-2430	59	30	𝑎	𝑎	PRON
iajs-2430	59	31	∈	∈	NOUN
iajs-2430	59	32	𝐼	𝐼	ADP
iajs-2430	59	33	and𝑎	and𝑎	ADJ
iajs-2430	59	34	∘	∘	X
iajs-2430	59	35	𝜒	𝜒	PRON
iajs-2430	59	36	∈	∈	PROPN
iajs-2430	59	37	𝐼.	𝐼.	PROPN
iajs-2430	59	38	definition	definition	NOUN
iajs-2430	59	39	(	(	PUNCT
iajs-2430	59	40	10)[3	10)[3	NUM
iajs-2430	59	41	]	]	PUNCT
iajs-2430	59	42	.	.	PUNCT
iajs-2430	60	1	let	let	VERB
iajs-2430	60	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	60	3	,	,	PUNCT
iajs-2430	60	4	0	0	NUM
iajs-2430	60	5	~be	~be	X
iajs-2430	60	6	a	a	DET
iajs-2430	60	7	ku-~semigroup	ku-~semigroup	NUM
iajs-2430	60	8	and𝜑	and𝜑	ADJ
iajs-2430	60	9	𝐴	𝐴	PROPN
iajs-2430	60	10			PROPN
iajs-2430	61	1	ℵ.	ℵ.	PROPN
iajs-2430	61	2	then~	then~	PROPN
iajs-2430	61	3	α	α	PROPN
iajs-2430	61	4	is	be	AUX
iajs-2430	61	5	said	say	VERB
iajs-2430	61	6	to	to	PART
iajs-2430	61	7	be	be	AUX
iajs-2430	61	8	a	a	DET
iajs-2430	61	9	k	k	NOUN
iajs-2430	61	10	-	-	NOUN
iajs-2430	61	11	ideal	ideal	NOUN
iajs-2430	61	12	of	of	ADP
iajs-2430	61	13	ℵ~	ℵ~	PROPN
iajs-2430	61	14	,	,	PUNCT
iajs-2430	61	15	if~	if~	PROPN
iajs-2430	61	16	i	i	PRON
iajs-2430	61	17	)	)	PUNCT
iajs-2430	62	1	α	α	PROPN
iajs-2430	62	2	is	be	AUX
iajs-2430	62	3	an	an	DET
iajs-2430	62	4	ku~-ideal	ku~-ideal	NOUN
iajs-2430	62	5	~of	~of	PROPN
iajs-2430	62	6	a~	a~	PROPN
iajs-2430	62	7	ku-~algebra~	ku-~algebra~	PROPN
iajs-2430	62	8	ℵ,∗	ℵ,∗	PROPN
iajs-2430	62	9	,	,	PUNCT
iajs-2430	62	10	0	0	NUM
iajs-2430	62	11	,	,	PUNCT
iajs-2430	62	12	ii	ii	PROPN
iajs-2430	62	13	)	)	PUNCT
iajs-2430	62	14	for~	for~	PROPN
iajs-2430	62	15	all	all	PRON
iajs-2430	62	16	𝜒	𝜒	NOUN
iajs-2430	62	17	∈	∈	NOUN
iajs-2430	62	18	𝛸~	𝛸~	NOUN
iajs-2430	62	19	,	,	PUNCT
iajs-2430	62	20	𝑎	𝑎	PROPN
iajs-2430	62	21	∈	∈	PROPN
iajs-2430	62	22	α,~we	α,~we	PROPN
iajs-2430	62	23	have~	have~	PROPN
iajs-2430	62	24	𝜒	𝜒	NOUN
iajs-2430	62	25	∘	∘	NUM
iajs-2430	62	26	𝑎	𝑎	PRON
iajs-2430	62	27	∈	∈	PROPN
iajs-2430	62	28	α	α	NOUN
iajs-2430	62	29	and	and	CCONJ
iajs-2430	62	30	𝑎	𝑎	ADP
iajs-2430	62	31	∘	∘	X
iajs-2430	62	32	𝜒	𝜒	X
iajs-2430	62	33	∈	∈	NOUN
iajs-2430	62	34	α~.	α~.	NUM
iajs-2430	62	35	definition	definition	NOUN
iajs-2430	62	36	(	(	PUNCT
iajs-2430	62	37	11)[3	11)[3	NUM
iajs-2430	62	38	]	]	PUNCT
iajs-2430	62	39	.	.	PUNCT
iajs-2430	63	1	let	let	VERB
iajs-2430	63	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	63	3	,	,	PUNCT
iajs-2430	63	4	0	0	NUM
iajs-2430	63	5	,	,	PUNCT
iajs-2430	63	6	~~be	~~be	ADP
iajs-2430	63	7	a	a	DET
iajs-2430	63	8	~ku-~semigroup	~ku-~semigroup	NOUN
iajs-2430	63	9	and	and	CCONJ
iajs-2430	63	10	𝜑	𝜑	PROPN
iajs-2430	63	11	𝐴	𝐴	PROPN
iajs-2430	63	12			PROPN
iajs-2430	63	13	ℵ.	ℵ.	PROPN
iajs-2430	64	1	then	then	ADV
iajs-2430	64	2	,	,	PUNCT
iajs-2430	64	3	α	α	PROPN
iajs-2430	64	4	is~	is~	PROPN
iajs-2430	64	5	said	say	VERB
iajs-2430	64	6	to~	to~	PROPN
iajs-2430	64	7	be	be	AUX
iajs-2430	64	8	a	a	DET
iajs-2430	64	9	p~-idealof	p~-idealof	NOUN
iajs-2430	64	10	ℵ~	ℵ~	NOUN
iajs-2430	64	11	,	,	PUNCT
iajs-2430	64	12	if	if	SCONJ
iajs-2430	64	13	~	~	PUNCT
iajs-2430	64	14	(	(	PUNCT
iajs-2430	64	15	p1)~	p1)~	NOUN
iajs-2430	64	16	for	for	ADP
iajs-2430	64	17	any	any	DET
iajs-2430	64	18	𝜒	𝜒	NOUN
iajs-2430	64	19	,	,	PUNCT
iajs-2430	64	20	𝛾	𝛾	PROPN
iajs-2430	64	21	,	,	PUNCT
iajs-2430	64	22	τ	τ	PROPN
iajs-2430	64	23	∈	∈	PROPN
iajs-2430	64	24	ℵ	ℵ	NOUN
iajs-2430	64	25	,	,	PUNCT
iajs-2430	64	26	τ	τ	PROPN
iajs-2430	64	27	∗	∗	NOUN
iajs-2430	64	28	𝜒	𝜒	NOUN
iajs-2430	64	29	∗	∗	NOUN
iajs-2430	64	30	𝛾	𝛾	ADP
iajs-2430	64	31	∈	∈	NOUN
iajs-2430	64	32	α	α	NOUN
iajs-2430	64	33	and	and	CCONJ
iajs-2430	64	34	τ	τ	PROPN
iajs-2430	64	35	∗	∗	NOUN
iajs-2430	64	36	𝜒	𝜒	X
iajs-2430	64	37	∈	∈	PROPN
iajs-2430	64	38	α	α	NOUN
iajs-2430	64	39	⟹	⟹	PUNCT
iajs-2430	64	40	τ	τ	PROPN
iajs-2430	64	41	∗	∗	NOUN
iajs-2430	64	42	𝛾	𝛾	ADP
iajs-2430	64	43	∈	∈	PROPN
iajs-2430	64	44	α	α	NOUN
iajs-2430	64	45	.	.	PUNCT
iajs-2430	65	1	(	(	PUNCT
iajs-2430	65	2	p2	p2	PROPN
iajs-2430	65	3	)	)	PUNCT
iajs-2430	65	4	~for	~for	ADP
iajs-2430	65	5	all	all	DET
iajs-2430	65	6	𝜒	𝜒	NUM
iajs-2430	65	7	∈	∈	ADJ
iajs-2430	65	8	ℵ	ℵ	NOUN
iajs-2430	65	9	,	,	PUNCT
iajs-2430	65	10	𝑎	𝑎	PROPN
iajs-2430	65	11	∈	∈	NOUN
iajs-2430	65	12	α	α	NOUN
iajs-2430	65	13	,	,	PUNCT
iajs-2430	65	14	we	we	PRON
iajs-2430	65	15	have	have	VERB
iajs-2430	65	16	𝜒	𝜒	NUM
iajs-2430	65	17	∘	∘	NUM
iajs-2430	65	18	𝑎	𝑎	PRON
iajs-2430	65	19	∈	∈	PROPN
iajs-2430	65	20	α	α	NOUN
iajs-2430	65	21	and	and	CCONJ
iajs-2430	65	22	𝑎	𝑎	PRON
iajs-2430	65	23	∘	∘	X
iajs-2430	65	24	𝜒	𝜒	PRON
iajs-2430	65	25	∈	∈	PROPN
iajs-2430	65	26	α	α	NOUN
iajs-2430	65	27	.	.	PUNCT
iajs-2430	66	1	definition	definition	NOUN
iajs-2430	66	2	(	(	PUNCT
iajs-2430	66	3	12)[3	12)[3	NUM
iajs-2430	66	4	]	]	PUNCT
iajs-2430	66	5	.	.	PUNCT
iajs-2430	67	1	let	let	VERB
iajs-2430	67	2	ℵˑ	ℵˑ	PRON
iajs-2430	67	3	and	and	CCONJ
iajs-2430	67	4	ℵ	ℵ	NOUN
iajs-2430	67	5	be	be	VERB
iajs-2430	67	6	two	two	NUM
iajs-2430	67	7	ku	ku	NOUN
iajs-2430	67	8	-	-	PUNCT
iajs-2430	67	9	semigroups	semigroup	NOUN
iajs-2430	67	10	.	.	PUNCT
iajs-2430	68	1	a	a	DET
iajs-2430	68	2	mapping	mapping	NOUN
iajs-2430	68	3	ƒ	ƒ	X
iajs-2430	68	4	:	:	PUNCT
iajs-2430	68	5	ℵ	ℵ	PROPN
iajs-2430	68	6	⟶	⟶	NOUN
iajs-2430	68	7	ℵ	ℵ	PRON
iajs-2430	68	8	ˑis	ˑis	NOUN
iajs-2430	68	9	called	call	VERB
iajs-2430	68	10	a	a	DET
iajs-2430	68	11	ku	ku	PROPN
iajs-2430	68	12	-	-	PUNCT
iajs-2430	68	13	semigroup	semigroup	NOUN
iajs-2430	68	14	homomorphism	homomorphism	NOUN
iajs-2430	68	15	if	if	SCONJ
iajs-2430	68	16	ƒ	ƒ	X
iajs-2430	68	17	(	(	PUNCT
iajs-2430	68	18	𝜒	𝜒	NOUN
iajs-2430	68	19	∗	∗	NOUN
iajs-2430	68	20	𝛾	𝛾	NOUN
iajs-2430	68	21	)	)	PUNCT
iajs-2430	69	1	=	=	VERB
iajs-2430	69	2	ƒ	ƒ	X
iajs-2430	69	3	𝜒	𝜒	X
iajs-2430	69	4	∗	∗	NOUN
iajs-2430	69	5	ƒ	ƒ	ADP
iajs-2430	69	6	𝛾	𝛾	NOUN
iajs-2430	70	1	and	and	CCONJ
iajs-2430	70	2	ƒ	ƒ	X
iajs-2430	70	3	(	(	PUNCT
iajs-2430	70	4	𝜒	𝜒	NOUN
iajs-2430	70	5	∘	∘	NOUN
iajs-2430	70	6	𝛾	𝛾	NOUN
iajs-2430	70	7	)	)	PUNCT
iajs-2430	70	8	=	=	SYM
iajs-2430	71	1	ƒ	ƒ	PRON
iajs-2430	71	2	𝜒	𝜒	X
iajs-2430	71	3	∘	∘	NOUN
iajs-2430	71	4	ƒ	ƒ	ADP
iajs-2430	71	5	𝛾	𝛾	NOUN
iajs-2430	71	6	,	,	PUNCT
iajs-2430	71	7	for	for	ADP
iajs-2430	71	8	all	all	DET
iajs-2430	71	9	𝜒	𝜒	NOUN
iajs-2430	71	10	,	,	PUNCT
iajs-2430	71	11	𝛾	𝛾	PROPN
iajs-2430	71	12	∈	∈	PRON
iajs-2430	71	13	ℵ.	ℵ.	PUNCT
iajs-2430	72	1	the	the	DET
iajs-2430	72	2	kernel	kernel	NOUN
iajs-2430	72	3	of	of	ADP
iajs-2430	72	4	ƒ	ƒ	PROPN
iajs-2430	72	5	is	be	AUX
iajs-2430	72	6	denoted	denote	VERB
iajs-2430	72	7	by	by	ADP
iajs-2430	72	8	ker	ker	PROPN
iajs-2430	72	9	ƒ	ƒ	X
iajs-2430	72	10	and	and	CCONJ
iajs-2430	72	11	is	be	AUX
iajs-2430	72	12	defined	define	VERB
iajs-2430	72	13	by	by	ADP
iajs-2430	72	14	{	{	PUNCT
iajs-2430	72	15	𝜒	𝜒	PROPN
iajs-2430	72	16	∈	∈	NOUN
iajs-2430	72	17	ℵ	ℵ	NOUN
iajs-2430	72	18	:	:	PUNCT
iajs-2430	72	19	ƒ	ƒ	PRON
iajs-2430	72	20	𝜒	𝜒	X
iajs-2430	72	21	0	0	NUM
iajs-2430	72	22	.	.	PUNCT
iajs-2430	73	1	moreover	moreover	ADV
iajs-2430	73	2	,	,	PUNCT
iajs-2430	73	3	the	the	DET
iajs-2430	73	4	image	image	NOUN
iajs-2430	73	5	of	of	ADP
iajs-2430	73	6	ƒ	ƒ	PRON
iajs-2430	73	7	is	be	AUX
iajs-2430	73	8	denoted	denote	VERB
iajs-2430	73	9	by	by	ADP
iajs-2430	73	10	i	i	PRON
iajs-2430	73	11	m	m	VERB
iajs-2430	73	12	ƒ	ƒ	PRON
iajs-2430	73	13	and	and	CCONJ
iajs-2430	73	14	is	be	AUX
iajs-2430	73	15	defined	define	VERB
iajs-2430	73	16	by	by	ADP
iajs-2430	73	17	{	{	PUNCT
iajs-2430	73	18	ƒ	ƒ	X
iajs-2430	73	19	𝜒	𝜒	SYM
iajs-2430	73	20	∈	∈	ADJ
iajs-2430	73	21			NOUN
iajs-2430	73	22	:	:	PUNCT
iajs-2430	73	23	𝜒	𝜒	PROPN
iajs-2430	73	24	∈	∈	ADJ
iajs-2430	73	25	ℵ	ℵ	NOUN
iajs-2430	73	26	.	.	PUNCT
iajs-2430	74	1	we	we	PRON
iajs-2430	74	2	review	review	VERB
iajs-2430	74	3	some	some	DET
iajs-2430	74	4	concepts	concept	NOUN
iajs-2430	74	5	of	of	ADP
iajs-2430	74	6	fuzzy	fuzzy	ADJ
iajs-2430	74	7	logic	logic	NOUN
iajs-2430	74	8	.	.	PUNCT
iajs-2430	75	1	a	a	DET
iajs-2430	75	2	functionˑ𝜇	functionˑ𝜇	NOUN
iajs-2430	75	3	:	:	PUNCT
iajs-2430	75	4	ℵ	ℵ	PROPN
iajs-2430	75	5	⟶	⟶	NOUN
iajs-2430	75	6	0,1	0,1	NUM
iajs-2430	75	7	is	be	AUX
iajs-2430	75	8	said	say	VERB
iajs-2430	75	9	to	to	PART
iajs-2430	75	10	be	be	AUX
iajs-2430	75	11	a	a	DET
iajs-2430	75	12	fuzzy	fuzzy	ADJ
iajs-2430	75	13	set	set	NOUN
iajs-2430	75	14	of	of	ADP
iajs-2430	75	15	a	a	DET
iajs-2430	75	16	set	set	NOUN
iajs-2430	75	17	ℵ	ℵ	NOUN
iajs-2430	75	18	and	and	CCONJ
iajs-2430	75	19	the	the	DET
iajs-2430	75	20	set	set	NOUN
iajs-2430	75	21	𝑈	𝑈	PROPN
iajs-2430	75	22	𝜇	𝜇	PROPN
iajs-2430	75	23	,	,	PUNCT
iajs-2430	75	24	𝑡	𝑡	VERB
iajs-2430	75	25	𝜒	𝜒	NOUN
iajs-2430	75	26	∈	∈	ADJ
iajs-2430	75	27	ℵ	ℵ	NOUN
iajs-2430	75	28	:	:	PUNCT
iajs-2430	75	29	𝜇	𝜇	ADP
iajs-2430	75	30	𝜒	𝜒	X
iajs-2430	75	31	𝑡	𝑡	VERB
iajs-2430	75	32	ˑis	ˑis	NOUN
iajs-2430	75	33	said	say	VERB
iajs-2430	75	34	to	to	PART
iajs-2430	75	35	be	be	AUX
iajs-2430	75	36	a	a	DET
iajs-2430	75	37	level	level	NOUN
iajs-2430	75	38	set	set	NOUN
iajs-2430	75	39	of	of	ADP
iajs-2430	75	40	𝜇	𝜇	ADP
iajs-2430	75	41	,	,	PUNCT
iajs-2430	75	42	for	for	ADP
iajs-2430	75	43	𝑡	𝑡	PROPN
iajs-2430	75	44	in	in	ADP
iajs-2430	75	45	0,1	0,1	NUM
iajs-2430	75	46	.∞	.∞	NOUN
iajs-2430	75	47	  	  	SPACE
iajs-2430	75	48	98	98	NUM
iajs-2430	75	49	  	  	SPACE
iajs-2430	75	50	ibn	ibn	PROPN
iajs-2430	75	51	al	al	PROPN
iajs-2430	75	52	-	-	PUNCT
iajs-2430	75	53	haitham	haitham	PROPN
iajs-2430	75	54	jour	jour	X
iajs-2430	75	55	.	.	PROPN
iajs-2430	75	56	for	for	ADP
iajs-2430	75	57	pure	pure	ADJ
iajs-2430	75	58	&	&	CCONJ
iajs-2430	75	59	appl	appl	PROPN
iajs-2430	75	60	.	.	PUNCT
iajs-2430	76	1	sci	sci	PROPN
iajs-2430	76	2	.	.	PROPN
iajs-2430	77	1	33	33	NUM
iajs-2430	77	2	(	(	PUNCT
iajs-2430	77	3	2	2	NUM
iajs-2430	77	4	)	)	PUNCT
iajs-2430	77	5	2020	2020	NUM
iajs-2430	77	6	definition	definition	NOUN
iajs-2430	77	7	(	(	PUNCT
iajs-2430	77	8	13)[6	13)[6	NUM
iajs-2430	77	9	]	]	X
iajs-2430	77	10	.	.	PUNCT
iajs-2430	78	1	a	a	DET
iajs-2430	78	2	fuzzy	fuzzy	ADJ
iajs-2430	78	3	set	set	NOUN
iajs-2430	78	4	𝜇	𝜇	ADP
iajs-2430	78	5	of	of	ADP
iajs-2430	78	6	ℵ	ℵ	NOUN
iajs-2430	78	7	is	be	AUX
iajs-2430	78	8	called	call	VERB
iajs-2430	78	9	a	a	DET
iajs-2430	78	10	fuzzy	fuzzy	ADJ
iajs-2430	78	11	sub	sub	NOUN
iajs-2430	78	12	ku	ku	PROPN
iajs-2430	78	13	-	-	PUNCT
iajs-2430	78	14	semigroup	semigroup	PROPN
iajs-2430	78	15	if	if	SCONJ
iajs-2430	78	16	,	,	PUNCT
iajs-2430	78	17	for	for	ADP
iajs-2430	78	18	all	all	DET
iajs-2430	78	19	𝜒	𝜒	NOUN
iajs-2430	78	20	,	,	PUNCT
iajs-2430	78	21	𝛾	𝛾	ADP
iajs-2430	78	22	∈	∈	PROPN
iajs-2430	78	23	ℵ	ℵ	PROPN
iajs-2430	78	24	i	i	NOUN
iajs-2430	78	25	)	)	PUNCT
iajs-2430	78	26	ˑ	ˑ	PROPN
iajs-2430	78	27	µ	µ	PROPN
iajs-2430	78	28	ˑ𝜒	ˑ𝜒	ADP
iajs-2430	78	29	∗	∗	PROPN
iajs-2430	78	30	𝛾ˑ	𝛾ˑ	PROPN
iajs-2430	78	31	min	min	PROPN
iajs-2430	78	32	ˑ	ˑ	PROPN
iajs-2430	78	33	µ	µ	X
iajs-2430	78	34	𝜒	𝜒	X
iajs-2430	78	35	,	,	PUNCT
iajs-2430	78	36	ˑµ	ˑµ	NOUN
iajs-2430	78	37	𝛾	𝛾	PROPN
iajs-2430	78	38	,	,	PUNCT
iajs-2430	78	39	ii	ii	PROPN
iajs-2430	78	40	)	)	PUNCT
iajs-2430	78	41	ˑ𝜒	ˑ𝜒	ADP
iajs-2430	78	42	∘	∘	PROPN
iajs-2430	78	43	𝛾ˑ	𝛾ˑ	PROPN
iajs-2430	78	44	min	min	PROPN
iajs-2430	78	45	ˑ	ˑ	PROPN
iajs-2430	78	46	µ	µ	X
iajs-2430	78	47	𝜒	𝜒	X
iajs-2430	78	48	,	,	PUNCT
iajs-2430	78	49	ˑµ	ˑµ	NOUN
iajs-2430	78	50	𝛾	𝛾	NOUN
iajs-2430	78	51	.	.	PUNCT
iajs-2430	79	1	definition	definition	NOUN
iajs-2430	79	2	(	(	PUNCT
iajs-2430	79	3	14)[6	14)[6	NOUN
iajs-2430	79	4	]	]	X
iajs-2430	79	5	.	.	PUNCT
iajs-2430	80	1	a	a	DET
iajs-2430	80	2	fuzzy	fuzzy	ADJ
iajs-2430	80	3	set	set	NOUN
iajs-2430	80	4	µ	µ	DET
iajs-2430	80	5	inˑℵ	inˑℵ	NOUN
iajs-2430	80	6	is	be	AUX
iajs-2430	80	7	called	call	VERB
iajs-2430	80	8	a	a	DET
iajs-2430	80	9	fuzzyˑs	fuzzyˑs	NOUN
iajs-2430	80	10	-	-	PUNCT
iajs-2430	80	11	ideal	ideal	NOUN
iajs-2430	80	12	of	of	ADP
iajs-2430	80	13	ℵ	ℵ	ADJ
iajs-2430	80	14	if	if	SCONJ
iajs-2430	80	15	,	,	PUNCT
iajs-2430	80	16	for	for	ADP
iajs-2430	80	17	all𝜒	all𝜒	NOUN
iajs-2430	80	18	,	,	PUNCT
iajs-2430	80	19	𝛾	𝛾	PROPN
iajs-2430	80	20	∈	∈	PROPN
iajs-2430	80	21	ℵ.ˑ	ℵ.ˑ	PROPN
iajs-2430	80	22	(	(	PUNCT
iajs-2430	80	23	s1)ˑµ(0	s1)ˑµ(0	PROPN
iajs-2430	80	24	)	)	PUNCT
iajs-2430	80	25	µ	µ	X
iajs-2430	80	26	𝜒	𝜒	X
iajs-2430	80	27	,	,	PUNCT
iajs-2430	80	28	(	(	PUNCT
iajs-2430	80	29	s2	s2	PROPN
iajs-2430	80	30	)	)	PUNCT
iajs-2430	80	31	µ	µ	PROPN
iajs-2430	80	32	𝛾	𝛾	NOUN
iajs-2430	80	33	min	min	NOUN
iajs-2430	80	34	µ	µ	X
iajs-2430	80	35	𝜒	𝜒	NOUN
iajs-2430	80	36	∗	∗	NOUN
iajs-2430	80	37	𝛾	𝛾	NOUN
iajs-2430	80	38	,	,	PUNCT
iajs-2430	80	39	µ	µ	X
iajs-2430	80	40	𝜒	𝜒	X
iajs-2430	80	41	(	(	PUNCT
iajs-2430	80	42	s3	s3	PROPN
iajs-2430	80	43	)	)	PUNCT
iajs-2430	80	44	µ	µ	X
iajs-2430	80	45	𝜒	𝜒	X
iajs-2430	80	46	∘	∘	NOUN
iajs-2430	80	47	𝛾	𝛾	ADP
iajs-2430	80	48	min	min	NOUN
iajs-2430	80	49	µ	µ	X
iajs-2430	80	50	𝜒	𝜒	X
iajs-2430	80	51	ˑ	ˑ	NOUN
iajs-2430	80	52	,	,	PUNCT
iajs-2430	80	53	ˑµ	ˑµ	NOUN
iajs-2430	80	54	𝛾	𝛾	NOUN
iajs-2430	80	55	.	.	PUNCT
iajs-2430	81	1	definition	definition	NOUN
iajs-2430	81	2	(	(	PUNCT
iajs-2430	81	3	15)[6	15)[6	NUM
iajs-2430	81	4	]	]	PUNCT
iajs-2430	81	5	.	.	PUNCT
iajs-2430	82	1	a	a	DET
iajs-2430	82	2	fuzzy	fuzzy	ADJ
iajs-2430	82	3	setˑµ	setˑµ	NOUN
iajs-2430	82	4	in	in	ADP
iajs-2430	82	5	ℵˑis	ℵˑis	ADJ
iajs-2430	82	6	calledˑaˑfuzzyˑk	calledˑaˑfuzzyˑk	NOUN
iajs-2430	82	7	-	-	PUNCT
iajs-2430	82	8	idealˑof	idealˑof	NOUN
iajs-2430	82	9	ℵˑif	ℵˑif	NOUN
iajs-2430	82	10	it	it	PRON
iajs-2430	82	11	satisfies	satisfy	VERB
iajs-2430	82	12	the	the	DET
iajs-2430	82	13	following	follow	VERB
iajs-2430	82	14	conditions	condition	NOUN
iajs-2430	82	15	:	:	PUNCT
iajs-2430	82	16	for	for	ADP
iajs-2430	82	17	all	all	DET
iajs-2430	82	18	𝜒	𝜒	NOUN
iajs-2430	82	19	,	,	PUNCT
iajs-2430	82	20	𝛾	𝛾	PROPN
iajs-2430	82	21	,	,	PUNCT
iajs-2430	82	22	τ	τ	PROPN
iajs-2430	82	23	∈	∈	PROPN
iajs-2430	82	24	ℵ.	ℵ.	NOUN
iajs-2430	83	1	(	(	PUNCT
iajs-2430	83	2	k1	k1	NOUN
iajs-2430	83	3	)	)	PUNCT
iajs-2430	83	4	µ(0	µ(0	NOUN
iajs-2430	83	5	)	)	PUNCT
iajs-2430	83	6	µ	µ	NOUN
iajs-2430	83	7			NOUN
iajs-2430	83	8	,	,	PUNCT
iajs-2430	83	9	(	(	PUNCT
iajs-2430	83	10	k2)µ	k2)µ	ADJ
iajs-2430	83	11	𝜒	𝜒	X
iajs-2430	83	12	∗	∗	X
iajs-2430	83	13	τ	τ	X
iajs-2430	83	14	min	min	PROPN
iajs-2430	83	15	µ	µ	X
iajs-2430	83	16	𝜒	𝜒	NOUN
iajs-2430	83	17	∗	∗	NOUN
iajs-2430	83	18	𝛾	𝛾	ADP
iajs-2430	83	19	∗	∗	NOUN
iajs-2430	83	20	τ	τ	PROPN
iajs-2430	83	21	,	,	PUNCT
iajs-2430	83	22	ˑµ	ˑµ	PROPN
iajs-2430	83	23	𝛾	𝛾	PROPN
iajs-2430	83	24	(	(	PUNCT
iajs-2430	83	25	k3)µ	k3)µ	NOUN
iajs-2430	83	26	𝜒	𝜒	X
iajs-2430	83	27	∘	∘	PROPN
iajs-2430	83	28	𝛾	𝛾	ADP
iajs-2430	83	29	min	min	NOUN
iajs-2430	83	30	µ	µ	X
iajs-2430	83	31	𝜒	𝜒	X
iajs-2430	83	32	ˑ	ˑ	NOUN
iajs-2430	83	33	,	,	PUNCT
iajs-2430	83	34	ˑµ	ˑµ	NOUN
iajs-2430	83	35	𝛾	𝛾	NOUN
iajs-2430	83	36	.	.	PUNCT
iajs-2430	84	1	example	example	NOUN
iajs-2430	84	2	(	(	PUNCT
iajs-2430	84	3	16)[6	16)[6	NUM
iajs-2430	84	4	]	]	PUNCT
iajs-2430	84	5	.	.	PUNCT
iajs-2430	85	1	let	let	VERB
iajs-2430	85	2	ℵ	ℵ	ADP
iajs-2430	85	3	0	0	NUM
iajs-2430	85	4	,	,	PUNCT
iajs-2430	85	5	𝑎	𝑎	NOUN
iajs-2430	85	6	,	,	PUNCT
iajs-2430	85	7	𝑏	𝑏	NOUN
iajs-2430	85	8	,	,	PUNCT
iajs-2430	85	9	𝑐	𝑐	PROPN
iajs-2430	85	10	,	,	PUNCT
iajs-2430	85	11	𝑑	𝑑	PROPN
iajs-2430	85	12	ˑˑbe	ˑˑbe	VERB
iajs-2430	85	13	a	a	DET
iajs-2430	85	14	~set	~set	NOUN
iajs-2430	85	15	.	.	PUNCT
iajs-2430	86	1	define∗	define∗	PROPN
iajs-2430	86	2	-~operation	-~operation	PUNCT
iajs-2430	86	3	and	and	CCONJ
iajs-2430	86	4	∘-~operation~	∘-~operation~	NUM
iajs-2430	86	5	by	by	ADP
iajs-2430	86	6	the	the	DET
iajs-2430	86	7	~following	~followe	VERB
iajs-2430	86	8	~tablesˑ~	~tablesˑ~	PROPN
iajs-2430	86	9	thenˑ	thenˑ	NOUN
iajs-2430	86	10	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	86	11	,	,	PUNCT
iajs-2430	86	12	0	0	PUNCT
iajs-2430	86	13	ˑis	ˑi	VERB
iajs-2430	86	14	a	a	DET
iajs-2430	86	15	~	~	PROPN
iajs-2430	86	16	ku	ku	PROPN
iajs-2430	86	17	-	-	PUNCT
iajs-2430	86	18	semigroup	semigroup	PROPN
iajs-2430	86	19	.	.	PUNCT
iajs-2430	87	1	defineˑa	defineˑa	PROPN
iajs-2430	87	2	fuzzyˑsetˑ𝜇	fuzzyˑsetˑ𝜇	PROPN
iajs-2430	87	3	∶	∶	PROPN
iajs-2430	87	4	ℵ	ℵ	ADP
iajs-2430	87	5	⟶	⟶	NOUN
iajs-2430	87	6	0,1	0,1	NUM
iajs-2430	87	7	ˑˑby	ˑˑby	NOUN
iajs-2430	87	8	𝜇	𝜇	ADP
iajs-2430	87	9	0	0	NUM
iajs-2430	87	10	𝜇	𝜇	ADP
iajs-2430	87	11	𝑎	𝑎	PRON
iajs-2430	87	12	0.4	0.4	NUM
iajs-2430	87	13	,	,	PUNCT
iajs-2430	87	14	𝜇	𝜇	X
iajs-2430	87	15	(	(	PUNCT
iajs-2430	87	16	b)=	b)=	X
iajs-2430	87	17	𝜇	𝜇	ADP
iajs-2430	87	18	𝑐	𝑐	PROPN
iajs-2430	87	19	0.2ˑ	0.2ˑ	PROPN
iajs-2430	87	20	,	,	PUNCT
iajs-2430	87	21	𝜇	𝜇	ADP
iajs-2430	87	22	𝑑	𝑑	PROPN
iajs-2430	87	23	0.1	0.1	NUM
iajs-2430	87	24	.ˑ	.ˑ	NOUN
iajs-2430	87	25	.	.	PUNCT
iajs-2430	88	1	then	then	ADV
iajs-2430	88	2	by	by	ADP
iajs-2430	88	3	routine	routine	ADJ
iajs-2430	88	4	calculation	calculation	NOUN
iajs-2430	88	5	we	we	PRON
iajs-2430	88	6	can	can	AUX
iajs-2430	88	7	prove	prove	VERB
iajs-2430	88	8	thatˑ𝜇	thatˑ𝜇	PROPN
iajs-2430	88	9	is	be	AUX
iajs-2430	88	10	afuzzy	afuzzy	ADJ
iajs-2430	88	11	k	k	ADJ
iajs-2430	88	12	-	-	ADJ
iajs-2430	88	13	ideal	ideal	ADJ
iajs-2430	88	14	ofℵ.	ofℵ.	X
iajs-2430	88	15	definition	definition	NOUN
iajs-2430	88	16	(	(	PUNCT
iajs-2430	88	17	17	17	NUM
iajs-2430	88	18	)	)	PUNCT
iajs-2430	89	1	[	[	X
iajs-2430	89	2	6	6	NUM
iajs-2430	89	3	]	]	PUNCT
iajs-2430	89	4	.	.	PUNCT
iajs-2430	90	1	the	the	DET
iajs-2430	90	2	cartesian	cartesian	ADJ
iajs-2430	90	3	product	product	NOUN
iajs-2430	90	4	of	of	ADP
iajs-2430	90	5	two	two	NUM
iajs-2430	90	6	fuzzy	fuzzy	ADJ
iajs-2430	90	7	sets	set	NOUN
iajs-2430	90	8	𝜇	𝜇	ADP
iajs-2430	90	9	and	and	CCONJ
iajs-2430	90	10	𝛽of	𝛽of	PROPN
iajs-2430	90	11	ℵ	ℵ	NOUN
iajs-2430	90	12	is	be	AUX
iajs-2430	90	13	denoted	denote	VERB
iajs-2430	90	14	by	by	ADP
iajs-2430	90	15	𝜇	𝜇	ADP
iajs-2430	90	16	𝛽	𝛽	NOUN
iajs-2430	90	17	:	:	PUNCT
iajs-2430	90	18	ℵ	ℵ	NOUN
iajs-2430	90	19	ℵ	ℵ	NOUN
iajs-2430	90	20	→	→	SYM
iajs-2430	90	21	0,1	0,1	NUM
iajs-2430	90	22	and	and	CCONJ
iajs-2430	90	23	defined	define	VERB
iajs-2430	90	24	by	by	ADP
iajs-2430	90	25	𝜇	𝜇	ADP
iajs-2430	90	26	𝛽	𝛽	NOUN
iajs-2430	90	27	𝜒	𝜒	NOUN
iajs-2430	90	28	,	,	PUNCT
iajs-2430	90	29	𝛾	𝛾	VERB
iajs-2430	90	30	min	min	NOUN
iajs-2430	90	31	𝜇	𝜇	ADP
iajs-2430	90	32	𝜒	𝜒	NOUN
iajs-2430	90	33	,	,	PUNCT
iajs-2430	90	34	𝛽	𝛽	NOUN
iajs-2430	90	35	𝛾	𝛾	NOUN
iajs-2430	90	36	,	,	PUNCT
iajs-2430	90	37	∀𝜒	∀𝜒	PROPN
iajs-2430	90	38	,	,	PUNCT
iajs-2430	90	39	𝛾	𝛾	PROPN
iajs-2430	90	40	∈	∈	PRON
iajs-2430	91	1	ℵ.	ℵ.	NOUN
iajs-2430	91	2	definition	definition	NOUN
iajs-2430	91	3	(	(	PUNCT
iajs-2430	91	4	18)[6	18)[6	NUM
iajs-2430	91	5	]	]	PUNCT
iajs-2430	91	6	.	.	PUNCT
iajs-2430	92	1	let	let	AUX
iajs-2430	92	2	𝜇	𝜇	PART
iajs-2430	92	3	be	be	AUX
iajs-2430	92	4	a	a	DET
iajs-2430	92	5	fuzzy	fuzzy	ADJ
iajs-2430	92	6	set	set	NOUN
iajs-2430	92	7	in	in	ADP
iajs-2430	92	8	ℵ.	ℵ.	PROPN
iajs-2430	92	9	if	if	SCONJ
iajs-2430	92	10	𝜇	𝜇	PRON
iajs-2430	92	11	is	be	AUX
iajs-2430	92	12	defined	define	VERB
iajs-2430	92	13	by	by	ADP
iajs-2430	92	14	:	:	PUNCT
iajs-2430	92	15	ℵ	ℵ	ADJ
iajs-2430	92	16	ℵ	ℵ	NOUN
iajs-2430	92	17	→	→	SYM
iajs-2430	92	18	0,1	0,1	NUM
iajs-2430	92	19	,	,	PUNCT
iajs-2430	92	20	then	then	ADV
iajs-2430	92	21	𝜇	𝜇	ADP
iajs-2430	92	22	is	be	AUX
iajs-2430	92	23	said	say	VERB
iajs-2430	92	24	to	to	PART
iajs-2430	92	25	be	be	AUX
iajs-2430	92	26	a	a	DET
iajs-2430	92	27	fuzzy	fuzzy	ADJ
iajs-2430	92	28	relation	relation	NOUN
iajs-2430	92	29	on	on	ADP
iajs-2430	92	30	a	a	DET
iajs-2430	92	31	set	set	ADJ
iajs-2430	92	32	𝑆	𝑆	PROPN
iajs-2430	92	33	´	´	NOUN
iajs-2430	92	34	,	,	PUNCT
iajs-2430	92	35	where	where	SCONJ
iajs-2430	92	36	s⊆	s⊆	NOUN
iajs-2430	92	37	ℵ.	ℵ.	PROPN
iajs-2430	92	38	definition	definition	NOUN
iajs-2430	92	39	(	(	PUNCT
iajs-2430	92	40	19)[6	19)[6	NUM
iajs-2430	92	41	]	]	PUNCT
iajs-2430	92	42	.	.	PUNCT
iajs-2430	93	1	let	let	AUX
iajs-2430	93	2	𝜇	𝜇	PART
iajs-2430	93	3	be	be	AUX
iajs-2430	93	4	a	a	DET
iajs-2430	93	5	fuzzy	fuzzy	ADJ
iajs-2430	93	6	relation	relation	NOUN
iajs-2430	93	7	on	on	ADP
iajs-2430	93	8	ℵ	ℵ	NOUN
iajs-2430	93	9	and	and	NOUN
iajs-2430	93	10	be	be	AUX
iajs-2430	93	11	a	a	DET
iajs-2430	93	12	fuzzy	fuzzy	ADJ
iajs-2430	93	13	subset	subset	NOUN
iajs-2430	93	14	of	of	ADP
iajs-2430	93	15	ℵ.	ℵ.	PROPN
iajs-2430	93	16	then	then	ADV
iajs-2430	93	17	the	the	DET
iajs-2430	93	18	strongest	strong	ADJ
iajs-2430	93	19	fuzzy	fuzzy	ADJ
iajs-2430	93	20	relation	relation	NOUN
iajs-2430	93	21	on	on	ADP
iajs-2430	93	22	ℵ	ℵ	NOUN
iajs-2430	93	23	is	be	AUX
iajs-2430	93	24	denoted	denote	VERB
iajs-2430	93	25	by	by	ADP
iajs-2430	93	26	𝜇	𝜇	ADV
iajs-2430	93	27	and	and	CCONJ
iajs-2430	93	28	is	be	AUX
iajs-2430	93	29	defined	define	VERB
iajs-2430	93	30	as	as	SCONJ
iajs-2430	93	31	follows	follow	VERB
iajs-2430	93	32	𝜇	𝜇	ADP
iajs-2430	93	33	𝜒	𝜒	NOUN
iajs-2430	93	34	,	,	PUNCT
iajs-2430	93	35	𝛾	𝛾	NOUN
iajs-2430	93	36	min	min	NOUN
iajs-2430	93	37	𝛽	𝛽	NOUN
iajs-2430	93	38	𝜒	𝜒	NOUN
iajs-2430	93	39	,	,	PUNCT
iajs-2430	93	40	𝛽	𝛽	NOUN
iajs-2430	93	41	𝛾	𝛾	NOUN
iajs-2430	93	42	,	,	PUNCT
iajs-2430	93	43	∀𝜒	∀𝜒	PROPN
iajs-2430	93	44	,	,	PUNCT
iajs-2430	93	45	𝛾	𝛾	PROPN
iajs-2430	93	46	∈	∈	PRON
iajs-2430	93	47	ℵ.	ℵ.	NOUN
iajs-2430	94	1	3	3	X
iajs-2430	94	2	.	.	X
iajs-2430	94	3	interval	interval	NOUN
iajs-2430	94	4	value	value	NOUN
iajs-2430	94	5	fuzzy	fuzzy	ADJ
iajs-2430	94	6	k	k	NOUN
iajs-2430	94	7	-	-	NOUN
iajs-2430	94	8	ideals	ideal	NOUN
iajs-2430	94	9	in	in	ADP
iajs-2430	94	10	ku	ku	PROPN
iajs-2430	94	11	-	-	PUNCT
iajs-2430	94	12	semigroup	semigroup	PROPN
iajs-2430	94	13	in	in	ADP
iajs-2430	94	14	this	this	DET
iajs-2430	94	15	part	part	NOUN
iajs-2430	94	16	,	,	PUNCT
iajs-2430	94	17	we	we	PRON
iajs-2430	94	18	recall	recall	VERB
iajs-2430	94	19	the	the	DET
iajs-2430	94	20	definition	definition	NOUN
iajs-2430	94	21	of	of	ADP
iajs-2430	94	22	interval	interval	NOUN
iajs-2430	94	23	valued	value	VERB
iajs-2430	94	24	fuzzy	fuzzy	ADJ
iajs-2430	94	25	set	set	VERB
iajs-2430	94	26	𝜇	𝜇	ADP
iajs-2430	94	27	of	of	ADP
iajs-2430	94	28	ℵ	ℵ	PRON
iajs-2430	94	29	as	as	SCONJ
iajs-2430	94	30	follows	follow	VERB
iajs-2430	94	31	  	  	SPACE
iajs-2430	94	32	99	99	NUM
iajs-2430	94	33	  	  	SPACE
iajs-2430	94	34	ibn	ibn	PROPN
iajs-2430	94	35	al	al	PROPN
iajs-2430	94	36	-	-	PUNCT
iajs-2430	94	37	haitham	haitham	PROPN
iajs-2430	94	38	jour	jour	X
iajs-2430	94	39	.	.	PROPN
iajs-2430	95	1	for	for	ADP
iajs-2430	95	2	pure	pure	ADJ
iajs-2430	95	3	&	&	CCONJ
iajs-2430	95	4	appl	appl	PROPN
iajs-2430	95	5	.	.	PUNCT
iajs-2430	96	1	sci	sci	PROPN
iajs-2430	96	2	.	.	PROPN
iajs-2430	97	1	33	33	NUM
iajs-2430	97	2	(	(	PUNCT
iajs-2430	97	3	2	2	NUM
iajs-2430	97	4	)	)	PUNCT
iajs-2430	97	5	2020	2020	NUM
iajs-2430	97	6	𝜇	𝜇	ADP
iajs-2430	97	7	𝜒	𝜒	NOUN
iajs-2430	97	8	,	,	PUNCT
iajs-2430	97	9	𝜇	𝜇	X
iajs-2430	97	10	𝜒	𝜒	X
iajs-2430	97	11	,	,	PUNCT
iajs-2430	97	12	𝜇	𝜇	X
iajs-2430	97	13	𝜒	𝜒	X
iajs-2430	97	14	:	:	PUNCT
iajs-2430	97	15	𝜒	𝜒	NUM
iajs-2430	97	16	∈	∈	ADJ
iajs-2430	97	17	ℵ	ℵ	NOUN
iajs-2430	97	18	,	,	PUNCT
iajs-2430	97	19	by	by	ADP
iajs-2430	97	20	briefly	briefly	NOUN
iajs-2430	97	21	𝜇	𝜇	ADP
iajs-2430	97	22	𝜇	𝜇	X
iajs-2430	97	23	,	,	PUNCT
iajs-2430	97	24	𝜇	𝜇	X
iajs-2430	97	25	,	,	PUNCT
iajs-2430	97	26	where	where	SCONJ
iajs-2430	97	27	𝜇	𝜇	ADP
iajs-2430	97	28	and	and	CCONJ
iajs-2430	97	29	𝜇	𝜇	X
iajs-2430	97	30	are	be	AUX
iajs-2430	97	31	two	two	NUM
iajs-2430	97	32	fuzzy	fuzzy	ADJ
iajs-2430	97	33	sets	set	NOUN
iajs-2430	97	34	in	in	ADP
iajs-2430	97	35	ℵ	ℵ	DET
iajs-2430	97	36	such	such	ADJ
iajs-2430	97	37	that	that	SCONJ
iajs-2430	97	38	𝜇	𝜇	ADP
iajs-2430	97	39	𝜒	𝜒	ADP
iajs-2430	97	40	𝜇	𝜇	X
iajs-2430	97	41	𝜒	𝜒	NOUN
iajs-2430	97	42	,	,	PUNCT
iajs-2430	97	43	for	for	ADP
iajs-2430	97	44	all	all	DET
iajs-2430	97	45	𝜒	𝜒	NUM
iajs-2430	97	46	∈	∈	ADJ
iajs-2430	97	47	ℵ	ℵ	NOUN
iajs-2430	97	48	and	and	CCONJ
iajs-2430	97	49	the	the	DET
iajs-2430	97	50	closed~	closed~	PROPN
iajs-2430	98	1	sub	sub	NOUN
iajs-2430	98	2	-	-	NOUN
iajs-2430	98	3	intervals	interval	NOUN
iajs-2430	98	4	̂of	̂of	NOUN
iajs-2430	98	5	̂[0	̂[0	VERB
iajs-2430	98	6	,	,	PUNCT
iajs-2430	98	7	1	1	NUM
iajs-2430	98	8	]	]	PUNCT
iajs-2430	98	9	is	be	AUX
iajs-2430	98	10	denoted	denote	VERB
iajs-2430	98	11	by	by	ADP
iajs-2430	98	12	d	d	PROPN
iajs-2430	98	13	[	[	X
iajs-2430	98	14	0	0	NUM
iajs-2430	98	15	,	,	PUNCT
iajs-2430	98	16	1]̂~.	1]̂~.	NUM
iajs-2430	98	17	note	note	VERB
iajs-2430	98	18	that	that	SCONJ
iajs-2430	98	19	,	,	PUNCT
iajs-2430	98	20	if	if	SCONJ
iajs-2430	98	21	𝜇	𝜇	ADP
iajs-2430	98	22	𝜒	𝜒	X
iajs-2430	98	23	𝜇	𝜇	X
iajs-2430	98	24	𝜒	𝜒	X
iajs-2430	98	25	𝑐	𝑐	NOUN
iajs-2430	98	26	,	,	PUNCT
iajs-2430	98	27	~where	~where	NUM
iajs-2430	98	28	~0	~0	NUM
iajs-2430	98	29	𝑐	𝑐	PROPN
iajs-2430	98	30	1	1	NUM
iajs-2430	98	31	,	,	PUNCT
iajs-2430	98	32	~then	~then	X
iajs-2430	98	33	𝜇	𝜇	ADP
iajs-2430	98	34	𝜒	𝜒	PROPN
iajs-2430	98	35	𝑐	𝑐	NOUN
iajs-2430	98	36	,	,	PUNCT
iajs-2430	98	37	𝑐	𝑐	NOUN
iajs-2430	98	38	~	~	PUNCT
iajs-2430	98	39	,	,	PUNCT
iajs-2430	98	40	~it	~it	PROPN
iajs-2430	98	41	follows	follow	VERB
iajs-2430	98	42	that	that	SCONJ
iajs-2430	98	43	𝜇	𝜇	SCONJ
iajs-2430	98	44	𝜒	𝜒	X
iajs-2430	98	45	∈	∈	NOUN
iajs-2430	98	46	d	d	NOUN
iajs-2430	98	47	[	[	X
iajs-2430	98	48	0	0	NUM
iajs-2430	98	49	,	,	PUNCT
iajs-2430	98	50	1]~and	1]~and	NUM
iajs-2430	98	51	it	it	PRON
iajs-2430	98	52	is	be	AUX
iajs-2430	98	53	given	give	VERB
iajs-2430	98	54	by	by	ADP
iajs-2430	98	55	𝜇	𝜇	ADP
iajs-2430	98	56	:	:	PUNCT
iajs-2430	98	57	ℵ	ℵ	ADJ
iajs-2430	98	58	→	→	SYM
iajs-2430	98	59	𝐷	𝐷	NOUN
iajs-2430	98	60	0,1	0,1	NUM
iajs-2430	98	61	,	,	PUNCT
iajs-2430	98	62	for	for	ADP
iajs-2430	98	63	all	all	DET
iajs-2430	98	64	χ∈ℵ	χ∈ℵ	NOUN
iajs-2430	98	65	and	and	CCONJ
iajs-2430	98	66	d	d	X
iajs-2430	99	1	[	[	X
iajs-2430	99	2	0	0	NUM
iajs-2430	99	3	,	,	PUNCT
iajs-2430	99	4	1]={[𝑎	1]={[𝑎	NUM
iajs-2430	99	5	,	,	PUNCT
iajs-2430	99	6	𝑎	𝑎	X
iajs-2430	99	7	:	:	PUNCT
iajs-2430	99	8	𝑎	𝑎	NUM
iajs-2430	99	9	𝑎	𝑎	PRON
iajs-2430	99	10	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
iajs-2430	99	11	𝑎	𝑎	NOUN
iajs-2430	99	12	,	,	PUNCT
iajs-2430	99	13	𝑎	𝑎	PROPN
iajs-2430	99	14	∈	∈	NOUN
iajs-2430	99	15	0	0	NUM
iajs-2430	99	16	,	,	PUNCT
iajs-2430	99	17	1	1	NUM
iajs-2430	99	18	.	.	PUNCT
iajs-2430	100	1	consider	consider	VERB
iajs-2430	100	2	,	,	PUNCT
iajs-2430	100	3	two	two	NUM
iajs-2430	100	4	elements	element	NOUN
iajs-2430	100	5	]	]	PUNCT
iajs-2430	100	6	,	,	PUNCT
iajs-2430	100	7	[	[	X
iajs-2430	100	8	1	1	NUM
iajs-2430	100	9	ul	ul	INTJ
iajs-2430	100	10	aad	aad	INTJ
iajs-2430	100	11			PROPN
iajs-2430	101	1	and	and	CCONJ
iajs-2430	101	2	]	]	X
iajs-2430	101	3	,	,	PUNCT
iajs-2430	101	4	[	[	X
iajs-2430	101	5	2	2	NUM
iajs-2430	101	6	ul	ul	NOUN
iajs-2430	101	7	bbd	bbd	PROPN
iajs-2430	101	8			PROPN
iajs-2430	101	9	in	in	ADP
iajs-2430	101	10	d	d	PROPN
iajs-2430	101	11	[	[	X
iajs-2430	101	12	0	0	NUM
iajs-2430	101	13	,	,	PUNCT
iajs-2430	101	14	1	1	NUM
iajs-2430	101	15	]	]	PUNCT
iajs-2430	101	16	are	be	AUX
iajs-2430	101	17	defined	define	VERB
iajs-2430	101	18	by	by	ADP
iajs-2430	101	19	)	)	PUNCT
iajs-2430	101	20	]	]	PUNCT
iajs-2430	101	21	,	,	PUNCT
iajs-2430	101	22	min(,),[min(),min	min(,),[min(),min	PROPN
iajs-2430	101	23	(	(	PUNCT
iajs-2430	101	24	21	21	NUM
iajs-2430	101	25	uull	uull	ADJ
iajs-2430	101	26	babaddr	babaddr	NOUN
iajs-2430	101	27			NUM
iajs-2430	101	28	and	and	CCONJ
iajs-2430	101	29			ADJ
iajs-2430	101	30	),max(,),max(),max	),max(,),max(),max	PRON
iajs-2430	101	31	(	(	PUNCT
iajs-2430	101	32	21	21	NUM
iajs-2430	101	33	uull	uull	ADJ
iajs-2430	101	34	babaddr	babaddr	NOUN
iajs-2430	101	35			PROPN
iajs-2430	101	36	.	.	PUNCT
iajs-2430	102	1	definition	definition	NOUN
iajs-2430	102	2	(	(	PUNCT
iajs-2430	102	3	20	20	NUM
iajs-2430	102	4	)	)	PUNCT
iajs-2430	102	5	.	.	PUNCT
iajs-2430	103	1	let	let	VERB
iajs-2430	103	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	103	3	,	,	PUNCT
iajs-2430	103	4	0	0	NUM
iajs-2430	103	5	~be	~be	X
iajs-2430	103	6	a	a	DET
iajs-2430	103	7	ku-~semigroup	ku-~semigroup	NOUN
iajs-2430	103	8	.	.	PUNCT
iajs-2430	104	1	if	if	SCONJ
iajs-2430	104	2	𝜇	𝜇	X
iajs-2430	104	3	:	:	PUNCT
iajs-2430	104	4	ℵ	ℵ	ADJ
iajs-2430	104	5	→	→	SYM
iajs-2430	104	6	𝐷	𝐷	NOUN
iajs-2430	104	7	0,1	0,1	NUM
iajs-2430	104	8	,	,	PUNCT
iajs-2430	104	9	then	then	ADV
iajs-2430	104	10	the	the	DET
iajs-2430	104	11	level	level	NOUN
iajs-2430	104	12	subset	subset	NOUN
iajs-2430	104	13	of	of	ADP
iajs-2430	104	14	𝜇~is	𝜇~is	PROPN
iajs-2430	104	15	~denoted	~denote	VERB
iajs-2430	104	16	~by	~by	PROPN
iajs-2430	104	17	𝜇	𝜇	NOUN
iajs-2430	104	18	and	and	CCONJ
iajs-2430	104	19	it	it	PRON
iajs-2430	104	20	is	be	AUX
iajs-2430	104	21	~defined	~define	VERB
iajs-2430	104	22	~by	~by	PROPN
iajs-2430	104	23	𝜇	𝜇	X
iajs-2430	104	24	{	{	PUNCT
iajs-2430	104	25	𝜒	𝜒	NOUN
iajs-2430	104	26	∈	∈	ADJ
iajs-2430	104	27	ℵ	ℵ	NOUN
iajs-2430	104	28	:	:	PUNCT
iajs-2430	104	29	𝜇	𝜇	ADP
iajs-2430	104	30	𝜒	𝜒	X
iajs-2430	104	31	�	�	PROPN
iajs-2430	104	32	̃	̃	NOUN
iajs-2430	104	33	�	�	PROPN
iajs-2430	104	34	,	,	PUNCT
iajs-2430	104	35	for	for	ADP
iajs-2430	104	36	every	every	DET
iajs-2430	104	37	~[0,0	~[0,0	NOUN
iajs-2430	104	38	]	]	SYM
iajs-2430	104	39	�	�	PROPN
iajs-2430	104	40	̃	̃	PROPN
iajs-2430	104	41	�	�	PROPN
iajs-2430	104	42	1,1	1,1	NUM
iajs-2430	104	43	.	.	PUNCT
iajs-2430	105	1	definition	definition	NOUN
iajs-2430	105	2	(	(	PUNCT
iajs-2430	105	3	?	?	NOUN
iajs-2430	105	4	21	21	NUM
iajs-2430	105	5	)	)	PUNCT
iajs-2430	105	6	.	.	PUNCT
iajs-2430	106	1	~	~	PUNCT
iajs-2430	106	2	let	let	VERB
iajs-2430	106	3	~	~	PUNCT
iajs-2430	106	4	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	106	5	,	,	PUNCT
iajs-2430	106	6	0	0	NUM
iajs-2430	106	7	be	be	AUX
iajs-2430	106	8	a	a	DET
iajs-2430	106	9	ku-~semigroup	ku-~semigroup	NUM
iajs-2430	106	10	and	and	CCONJ
iajs-2430	106	11	𝜇	𝜇	ADP
iajs-2430	106	12	:	:	PUNCT
iajs-2430	106	13	ℵ	ℵ	ADJ
iajs-2430	106	14	→	→	SYM
iajs-2430	106	15	𝐷	𝐷	NOUN
iajs-2430	106	16	0,1	0,1	NUM
iajs-2430	106	17	.	.	PUNCT
iajs-2430	107	1	then	then	ADV
iajs-2430	107	2	~𝜇	~𝜇	NUM
iajs-2430	107	3	is	be	AUX
iajs-2430	107	4	called	call	VERB
iajs-2430	107	5	an	an	DET
iajs-2430	107	6	~interval	~interval	NOUN
iajs-2430	107	7	~valued	~valued	NUM
iajs-2430	107	8	~	~	SYM
iajs-2430	107	9	fuzzy	fuzzy	ADJ
iajs-2430	107	10	sub	sub	NOUN
iajs-2430	107	11	ku-~semigroup	ku-~semigroup	PROPN
iajs-2430	107	12	ℵ	ℵ	NOUN
iajs-2430	107	13	,	,	PUNCT
iajs-2430	107	14	if	if	SCONJ
iajs-2430	107	15	it	it	PRON
iajs-2430	107	16	satisfies	satisfy	VERB
iajs-2430	107	17	the	the	DET
iajs-2430	107	18	following	follow	VERB
iajs-2430	107	19	conditions	condition	NOUN
iajs-2430	107	20	𝜇	𝜇	ADP
iajs-2430	107	21	𝜒	𝜒	NOUN
iajs-2430	107	22	∗	∗	NOUN
iajs-2430	107	23	𝛾	𝛾	ADP
iajs-2430	107	24	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	107	25	𝜇	𝜇	ADP
iajs-2430	107	26	𝜒	𝜒	NOUN
iajs-2430	107	27	ˑ	ˑ	NOUN
iajs-2430	107	28	,	,	PUNCT
iajs-2430	107	29	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	107	30	𝛾	𝛾	ADP
iajs-2430	107	31	~,𝜇	~,𝜇	PROPN
iajs-2430	107	32	𝜒	𝜒	X
iajs-2430	107	33	∘	∘	NOUN
iajs-2430	107	34	𝛾	𝛾	ADP
iajs-2430	107	35	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	107	36	𝜇	𝜇	ADP
iajs-2430	107	37	𝜒	𝜒	NOUN
iajs-2430	107	38	ˑ	ˑ	NOUN
iajs-2430	107	39	,	,	PUNCT
iajs-2430	107	40	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	107	41	𝛾	𝛾	NOUN
iajs-2430	107	42	,	,	PUNCT
iajs-2430	107	43	∀𝜒	∀𝜒	X
iajs-2430	107	44	,	,	PUNCT
iajs-2430	107	45	𝛾	𝛾	PROPN
iajs-2430	107	46	∈	∈	PRON
iajs-2430	107	47	ℵ.	ℵ.	NOUN
iajs-2430	107	48	example	example	NOUN
iajs-2430	107	49	(	(	PUNCT
iajs-2430	107	50	22	22	NUM
iajs-2430	107	51	)	)	PUNCT
iajs-2430	107	52	.	.	PUNCT
iajs-2430	108	1	let	let	VERB
iajs-2430	108	2	ℵ	ℵ	PRON
iajs-2430	108	3	0,1,2,3	0,1,2,3	NUM
iajs-2430	108	4	~be	~be	X
iajs-2430	108	5	a	a	DET
iajs-2430	108	6	set	set	NOUN
iajs-2430	108	7	.	.	PUNCT
iajs-2430	109	1	we	we	PRON
iajs-2430	109	2	define	define	VERB
iajs-2430	109	3	two	two	NUM
iajs-2430	109	4	operations	operation	NOUN
iajs-2430	109	5	by	by	ADP
iajs-2430	109	6	the	the	DET
iajs-2430	109	7	following	follow	VERB
iajs-2430	109	8	tables	table	NOUN
iajs-2430	109	9	then	then	ADV
iajs-2430	109	10	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	109	11	,	,	PUNCT
iajs-2430	109	12	0	0	PUNCT
iajs-2430	109	13	ˑis	ˑi	VERB
iajs-2430	109	14	a	a	DET
iajs-2430	109	15	ku	ku	PROPN
iajs-2430	109	16	-	-	PUNCT
iajs-2430	109	17	semigroup	semigroup	PROPN
iajs-2430	109	18	.	.	PUNCT
iajs-2430	110	1	define	define	VERB
iajs-2430	110	2	𝜇	𝜇	SCONJ
iajs-2430	110	3	𝜒	𝜒	X
iajs-2430	110	4	ˑas	ˑas	NOUN
iajs-2430	110	5	follows	follow	VERB
iajs-2430	110	6	𝜇	𝜇	ADP
iajs-2430	110	7	𝜒	𝜒	NOUN
iajs-2430	110	8	0.2,0.7	0.2,0.7	NOUN
iajs-2430	110	9	𝑖𝑓	𝑖𝑓	NUM
iajs-2430	110	10	𝜒	𝜒	NOUN
iajs-2430	110	11	0,1,2	0,1,2	NOUN
iajs-2430	110	12	0.1,0.3	0.1,0.3	NUM
iajs-2430	110	13	𝑖𝑓	𝑖𝑓	NUM
iajs-2430	110	14	𝜒	𝜒	NUM
iajs-2430	110	15	3	3	NUM
iajs-2430	110	16	and	and	CCONJ
iajs-2430	110	17	by	by	ADP
iajs-2430	110	18	applying	apply	VERB
iajs-2430	110	19	definition21	definition21	NOUN
iajs-2430	110	20	,	,	PUNCT
iajs-2430	110	21	we	we	PRON
iajs-2430	110	22	can	can	AUX
iajs-2430	110	23	prove	prove	VERB
iajs-2430	110	24	that	that	SCONJ
iajs-2430	110	25	𝜇	𝜇	ADP
iajs-2430	110	26	𝜒	𝜒	NOUN
iajs-2430	110	27	is	be	AUX
iajs-2430	110	28	an	an	DET
iajs-2430	110	29	interval	interval	NOUN
iajs-2430	110	30	valued	value	VERB
iajs-2430	110	31	fuzzysub	fuzzysub	PROPN
iajs-2430	110	32	kusemi	kusemi	PROPN
iajs-2430	110	33	group	group	NOUN
iajs-2430	110	34	of	of	ADP
iajs-2430	110	35	ℵˑ.	ℵˑ.	PROPN
iajs-2430	110	36	definition	definition	NOUN
iajs-2430	110	37	(	(	PUNCT
iajs-2430	110	38	23	23	NUM
iajs-2430	110	39	)	)	PUNCT
iajs-2430	110	40	.	.	PUNCT
iajs-2430	111	1	let	let	VERB
iajs-2430	111	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	111	3	,	,	PUNCT
iajs-2430	111	4	0	0	NUM
iajs-2430	111	5	be	be	AUX
iajs-2430	111	6	a	a	DET
iajs-2430	111	7	ku	ku	PROPN
iajs-2430	111	8	-	-	PUNCT
iajs-2430	111	9	semigroup	semigroup	PROPN
iajs-2430	111	10	and	and	CCONJ
iajs-2430	111	11	𝜇	𝜇	ADP
iajs-2430	111	12	:	:	PUNCT
iajs-2430	111	13	ℵ	ℵ	ADJ
iajs-2430	111	14	→	→	SYM
iajs-2430	111	15	𝐷	𝐷	NOUN
iajs-2430	111	16	0,1	0,1	NUM
iajs-2430	111	17	.	.	PUNCT
iajs-2430	112	1	then	then	ADV
iajs-2430	112	2	𝜇	𝜇	X
iajs-2430	112	3	is	be	AUX
iajs-2430	112	4	named	name	VERB
iajs-2430	112	5	an	an	DET
iajs-2430	112	6	interval	interval	NOUN
iajs-2430	112	7	valuedfuzzy	valuedfuzzy	NOUN
iajs-2430	112	8	s	s	NOUN
iajs-2430	112	9	-	-	NOUN
iajs-2430	112	10	ideal	ideal	NOUN
iajs-2430	112	11	of	of	ADP
iajs-2430	112	12	ℵ	ℵ	ADJ
iajs-2430	112	13	if	if	SCONJ
iajs-2430	112	14	𝑖	𝑖	X
iajs-2430	112	15	𝜇	𝜇	X
iajs-2430	112	16	(	(	PUNCT
iajs-2430	112	17	0	0	NUM
iajs-2430	112	18	)	)	PUNCT
iajs-2430	112	19	𝜇	𝜇	ADP
iajs-2430	112	20	𝜒	𝜒	NOUN
iajs-2430	112	21	,	,	PUNCT
iajs-2430	112	22	∀𝜒	∀𝜒	NOUN
iajs-2430	112	23	∈	∈	PROPN
iajs-2430	112	24	ℵ	ℵ	NOUN
iajs-2430	112	25	,	,	PUNCT
iajs-2430	112	26	𝑖	𝑖	VERB
iajs-2430	112	27	for	for	ADP
iajs-2430	112	28	all	all	PRON
iajs-2430	112	29	,	,	PUNCT
iajs-2430	112	30			VERB
iajs-2430	112	31	,	,	PUNCT
iajs-2430	112	32	𝛾	𝛾	PROPN
iajs-2430	112	33	∈	∈	NOUN
iajs-2430	112	34			NOUN
iajs-2430	112	35	,	,	PUNCT
iajs-2430	112	36	𝜇	𝜇	ADP
iajs-2430	112	37	𝛾	𝛾	ADP
iajs-2430	112	38	𝑟	𝑟	X
iajs-2430	112	39	min	min	NOUN
iajs-2430	112	40	𝜇	𝜇	ADP
iajs-2430	112	41	𝜒	𝜒	X
iajs-2430	112	42	∗	∗	NOUN
iajs-2430	112	43	𝛾	𝛾	NOUN
iajs-2430	112	44	,	,	PUNCT
iajs-2430	112	45	𝜇	𝜇	ADP
iajs-2430	112	46	𝜒	𝜒	X
iajs-2430	112	47	,	,	PUNCT
iajs-2430	112	48	𝑖	𝑖	X
iajs-2430	112	49	𝜇	𝜇	X
iajs-2430	112	50	𝜒	𝜒	X
iajs-2430	112	51	∘	∘	NOUN
iajs-2430	112	52	𝛾	𝛾	ADP
iajs-2430	112	53	𝑟min	𝑟min	NOUN
iajs-2430	112	54	𝜇	𝜇	ADP
iajs-2430	112	55	𝜒	𝜒	NOUN
iajs-2430	112	56	ˑ	ˑ	NOUN
iajs-2430	112	57	,	,	PUNCT
iajs-2430	112	58	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	112	59	𝛾	𝛾	NOUN
iajs-2430	112	60	.	.	PUNCT
iajs-2430	113	1	definition	definition	NOUN
iajs-2430	113	2	(	(	PUNCT
iajs-2430	113	3	24	24	NUM
iajs-2430	113	4	)	)	PUNCT
iajs-2430	113	5	.	.	PUNCT
iajs-2430	114	1	let	let	VERB
iajs-2430	114	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	114	3	,	,	PUNCT
iajs-2430	114	4	0	0	NUM
iajs-2430	114	5	be	be	AUX
iajs-2430	114	6	a	a	DET
iajs-2430	114	7	ku	ku	PROPN
iajs-2430	114	8	-	-	PUNCT
iajs-2430	114	9	semigroup	semigroup	PROPN
iajs-2430	114	10	and	and	CCONJ
iajs-2430	114	11	𝜇	𝜇	ADP
iajs-2430	114	12	:	:	PUNCT
iajs-2430	114	13	ℵ	ℵ	ADJ
iajs-2430	114	14	→	→	SYM
iajs-2430	114	15	𝐷	𝐷	NOUN
iajs-2430	114	16	0,1	0,1	NUM
iajs-2430	114	17	.	.	PUNCT
iajs-2430	115	1	then	then	ADV
iajs-2430	115	2	𝜇~is	𝜇~is	PROPN
iajs-2430	115	3	named	name	VERB
iajs-2430	115	4	~an	~an	PROPN
iajs-2430	115	5	interval	interval	NOUN
iajs-2430	115	6	valued	value	VERB
iajs-2430	115	7	fuzzy	fuzzy	ADJ
iajs-2430	115	8	k-~´ideal	k-~´ideal	NOUN
iajs-2430	115	9	of´ℵ~	of´ℵ~	X
iajs-2430	115	10	if	if	SCONJ
iajs-2430	115	11	(	(	PUNCT
iajs-2430	115	12	,	,	PUNCT
iajs-2430	115	13	ƒ	ƒ	X
iajs-2430	115	14	𝜇	𝜇	X
iajs-2430	115	15	(	(	PUNCT
iajs-2430	115	16	0	0	NUM
iajs-2430	115	17	)	)	PUNCT
iajs-2430	115	18	𝜇	𝜇	ADP
iajs-2430	115	19	𝜒	𝜒	NOUN
iajs-2430	115	20	,	,	PUNCT
iajs-2430	115	21	∀𝜒	∀𝜒	NOUN
iajs-2430	115	22	∈	∈	PROPN
iajs-2430	115	23	ℵ	ℵ	NOUN
iajs-2430	115	24	,	,	PUNCT
iajs-2430	115	25	(	(	PUNCT
iajs-2430	115	26	ƒ	ƒ	NOUN
iajs-2430	115	27	for	for	ADP
iajs-2430	115	28	all	all	DET
iajs-2430	115	29	𝜒	𝜒	NOUN
iajs-2430	115	30	,	,	PUNCT
iajs-2430	115	31	𝛾	𝛾	PROPN
iajs-2430	115	32	,	,	PUNCT
iajs-2430	115	33	τ	τ	PROPN
iajs-2430	115	34	∈	∈	PROPN
iajs-2430	115	35	ℵ,𝜇	ℵ,𝜇	VERB
iajs-2430	115	36	𝜒	𝜒	NOUN
iajs-2430	115	37	∗	∗	X
iajs-2430	115	38	τ	τ	PROPN
iajs-2430	115	39	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	115	40	𝜇	𝜇	ADP
iajs-2430	115	41	𝜒	𝜒	PROPN
iajs-2430	115	42	∗	∗	NOUN
iajs-2430	115	43	𝛾	𝛾	ADP
iajs-2430	115	44	∗	∗	NOUN
iajs-2430	115	45	τ	τ	X
iajs-2430	115	46	,	,	PUNCT
iajs-2430	115	47	𝜇	𝜇	ADP
iajs-2430	115	48	𝛾	𝛾	PROPN
iajs-2430	115	49	,	,	PUNCT
iajs-2430	115	50	(	(	PUNCT
iajs-2430	115	51	ƒ	ƒ	X
iajs-2430	115	52	𝜇	𝜇	X
iajs-2430	115	53	𝜒	𝜒	X
iajs-2430	115	54	∘	∘	NOUN
iajs-2430	115	55	𝛾	𝛾	ADP
iajs-2430	115	56	𝑟min	𝑟min	NOUN
iajs-2430	115	57	𝜇	𝜇	ADP
iajs-2430	115	58	𝜒	𝜒	NOUN
iajs-2430	115	59	ˑ	ˑ	NOUN
iajs-2430	115	60	,	,	PUNCT
iajs-2430	115	61	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	115	62	𝛾	𝛾	NOUN
iajs-2430	115	63	.	.	PUNCT
iajs-2430	115	64	  	  	SPACE
iajs-2430	116	1	100	100	NUM
iajs-2430	116	2	  	  	SPACE
iajs-2430	116	3	ibn	ibn	PROPN
iajs-2430	116	4	al	al	PROPN
iajs-2430	116	5	-	-	PUNCT
iajs-2430	116	6	haitham	haitham	PROPN
iajs-2430	116	7	jour	jour	X
iajs-2430	116	8	.	.	PROPN
iajs-2430	116	9	for	for	ADP
iajs-2430	116	10	pure	pure	ADJ
iajs-2430	116	11	&	&	CCONJ
iajs-2430	116	12	appl	appl	PROPN
iajs-2430	116	13	.	.	PUNCT
iajs-2430	117	1	sci	sci	PROPN
iajs-2430	117	2	.	.	PROPN
iajs-2430	118	1	33	33	NUM
iajs-2430	118	2	(	(	PUNCT
iajs-2430	118	3	2	2	NUM
iajs-2430	118	4	)	)	PUNCT
iajs-2430	118	5	2020	2020	NUM
iajs-2430	118	6	example	example	NOUN
iajs-2430	118	7	(	(	PUNCT
iajs-2430	118	8	25	25	NUM
iajs-2430	118	9	)	)	PUNCT
iajs-2430	118	10	.	.	PUNCT
iajs-2430	119	1	from	from	ADP
iajs-2430	119	2	example16	example16	PROPN
iajs-2430	119	3	,	,	PUNCT
iajs-2430	119	4	we	we	PRON
iajs-2430	119	5	define	define	VERB
iajs-2430	119	6	𝜇	𝜇	DET
iajs-2430	119	7	𝜒	𝜒	NOUN
iajs-2430	119	8	as	as	SCONJ
iajs-2430	119	9	follows	follow	VERB
iajs-2430	119	10	:	:	PUNCT
iajs-2430	120	1	[	[	X
iajs-2430	120	2	0.3,0.9	0.3,0.9	X
iajs-2430	120	3	]	]	X
iajs-2430	120	4	if	if	SCONJ
iajs-2430	120	5	{	{	PUNCT
iajs-2430	120	6	0,a	0,a	PROPN
iajs-2430	120	7	,	,	PUNCT
iajs-2430	120	8	b	b	NOUN
iajs-2430	120	9	,	,	PUNCT
iajs-2430	120	10	c	c	NOUN
iajs-2430	120	11	}	}	PUNCT
iajs-2430	120	12	(	(	PUNCT
iajs-2430	120	13	)	)	PUNCT
iajs-2430	120	14	[	[	X
iajs-2430	120	15	0.1,0.6	0.1,0.6	X
iajs-2430	120	16	]	]	X
iajs-2430	120	17	if	if	SCONJ
iajs-2430	120	18	d	d	NOUN
iajs-2430	120	19			VERB
iajs-2430	120	20			NOUN
iajs-2430	120	21			NOUN
iajs-2430	120	22			VERB
iajs-2430	120	23			PROPN
iajs-2430	120	24			NUM
iajs-2430	120	25			NOUN
iajs-2430	120	26			NOUN
iajs-2430	120	27			NOUN
iajs-2430	120	28	.	.	PUNCT
iajs-2430	121	1	by	by	ADP
iajs-2430	121	2	this	this	DET
iajs-2430	121	3	definition	definition	NOUN
iajs-2430	121	4	of	of	ADP
iajs-2430	121	5	𝜇	𝜇	ADP
iajs-2430	121	6	,	,	PUNCT
iajs-2430	121	7	we	we	PRON
iajs-2430	121	8	can	can	AUX
iajs-2430	121	9	prove	prove	VERB
iajs-2430	121	10	it	it	PRON
iajs-2430	121	11	is	be	AUX
iajs-2430	121	12	an	an	DET
iajs-2430	121	13	interval	interval	NOUN
iajs-2430	121	14	valued~	valued~	VERB
iajs-2430	121	15	fuzzy	fuzzy	ADJ
iajs-2430	121	16	k-~ideal	k-~ideal	NOUN
iajs-2430	121	17	.	.	PUNCT
iajs-2430	122	1	theorem	theorem	NOUN
iajs-2430	122	2	(	(	PUNCT
iajs-2430	122	3	26).~let	26).~let	NUM
iajs-2430	122	4	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	122	5	,	,	PUNCT
iajs-2430	122	6	0	0	X
iajs-2430	122	7	be	be	SYM
iajs-2430	122	8	a	a	DET
iajs-2430	122	9	ku	ku	PROPN
iajs-2430	122	10	-	-	PUNCT
iajs-2430	122	11	semigroup	semigroup	PROPN
iajs-2430	122	12	.	.	PUNCT
iajs-2430	123	1	then	then	ADV
iajs-2430	123	2	𝜇in~ℵ	𝜇in~ℵ	NOUN
iajs-2430	123	3	is	be	AUX
iajs-2430	123	4	an	an	DET
iajs-2430	123	5	interval	interval	NOUN
iajs-2430	123	6	valuedfuzzy	valuedfuzzy	NOUN
iajs-2430	123	7	~	~	PUNCT
iajs-2430	123	8	kidealif~and	kidealif~and	PUNCT
iajs-2430	123	9	only	only	ADJ
iajs-2430	123	10	ifit	ifit	PROPN
iajs-2430	123	11	is	be	AUX
iajs-2430	123	12	an	an	DET
iajs-2430	123	13	interval	interval	NOUN
iajs-2430	123	14	valued	value	VERB
iajs-2430	123	15	fuzzy	fuzzy	ADJ
iajs-2430	123	16	s-~ideal	s-~ideal	NOUN
iajs-2430	123	17	ofˑℵ~.	ofˑℵ~.	NOUN
iajs-2430	123	18	proof	proof	NOUN
iajs-2430	123	19	.	.	PUNCT
iajs-2430	124	1	(	(	PUNCT
iajs-2430	124	2	)~by	)~by	X
iajs-2430	124	3	taking	take	VERB
iajs-2430	124	4	𝜒	𝜒	X
iajs-2430	124	5	0	0	NUM
iajs-2430	124	6	in(ƒ	in(ƒ	NOUN
iajs-2430	124	7	,	,	PUNCT
iajs-2430	124	8	(	(	PUNCT
iajs-2430	124	9	ƒ	ƒ	X
iajs-2430	124	10	and	and	CCONJ
iajs-2430	124	11	using	use	VERB
iajs-2430	124	12	(	(	PUNCT
iajs-2430	124	13	ku	ku	PROPN
iajs-2430	124	14	)	)	PUNCT
iajs-2430	124	15	,	,	PUNCT
iajs-2430	124	16	we	we	PRON
iajs-2430	124	17	obtain~~	obtain~~	VERB
iajs-2430	124	18	∀𝛾	∀𝛾	PROPN
iajs-2430	124	19	,	,	PUNCT
iajs-2430	124	20	τ	τ	PROPN
iajs-2430	124	21	∈	∈	PROPN
iajs-2430	124	22	ℵ	ℵ	X
iajs-2430	124	23	𝜇	𝜇	X
iajs-2430	124	24	τ	τ	X
iajs-2430	124	25	𝜇	𝜇	ADP
iajs-2430	124	26	0	0	PROPN
iajs-2430	124	27	∗	∗	NOUN
iajs-2430	124	28	τ	τ	PROPN
iajs-2430	124	29	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	124	30	𝜇	𝜇	ADP
iajs-2430	124	31	0	0	NUM
iajs-2430	124	32	∗	∗	NOUN
iajs-2430	124	33	𝛾	𝛾	ADP
iajs-2430	124	34	∗	∗	NOUN
iajs-2430	124	35	τ	τ	X
iajs-2430	124	36	,	,	PUNCT
iajs-2430	124	37	𝜇	𝜇	ADP
iajs-2430	124	38	𝛾	𝛾	ADP
iajs-2430	124	39	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	124	40	𝜇	𝜇	ADP
iajs-2430	124	41	𝛾	𝛾	PROPN
iajs-2430	124	42	∗	∗	X
iajs-2430	124	43	τ	τ	X
iajs-2430	124	44	,	,	PUNCT
iajs-2430	124	45	𝜇	𝜇	ADP
iajs-2430	124	46	𝛾	𝛾	NOUN
iajs-2430	124	47	and	and	CCONJ
iajs-2430	124	48	𝜇	𝜇	X
iajs-2430	124	49	𝜒	𝜒	X
iajs-2430	124	50	∘	∘	NOUN
iajs-2430	124	51	𝛾	𝛾	ADP
iajs-2430	124	52	𝑟min	𝑟min	NOUN
iajs-2430	124	53	𝜇	𝜇	ADP
iajs-2430	124	54	𝜒	𝜒	NOUN
iajs-2430	124	55	ˑ	ˑ	NOUN
iajs-2430	124	56	,	,	PUNCT
iajs-2430	124	57	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	124	58	𝛾	𝛾	NOUN
iajs-2430	124	59	are	be	AUX
iajs-2430	124	60	satisfied	satisfied	ADJ
iajs-2430	124	61	.	.	PUNCT
iajs-2430	125	1	(	(	PUNCT
iajs-2430	125	2	?	?	PUNCT
iajs-2430	125	3			NOUN
iajs-2430	125	4	)	)	PUNCT
iajs-2430	125	5	we	we	PRON
iajs-2430	125	6	have	have	VERB
iajs-2430	125	7	𝜇	𝜇	ADP
iajs-2430	125	8	𝜒	𝜒	NOUN
iajs-2430	125	9	∗	∗	NOUN
iajs-2430	125	10	τ	τ	X
iajs-2430	125	11	𝑟	𝑟	NOUN
iajs-2430	125	12	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	125	13	𝜇	𝜇	ADP
iajs-2430	125	14	𝛾	𝛾	PROPN
iajs-2430	125	15	∗	∗	X
iajs-2430	125	16	𝜒	𝜒	NUM
iajs-2430	125	17	∗	∗	NOUN
iajs-2430	125	18	τ	τ	X
iajs-2430	125	19	,	,	PUNCT
iajs-2430	125	20	𝜇	𝜇	ADP
iajs-2430	125	21	𝛾	𝛾	NOUN
iajs-2430	125	22	and	and	CCONJ
iajs-2430	125	23	by	by	ADP
iajs-2430	125	24	apply	apply	NOUN
iajs-2430	125	25	theorem	theorem	NOUN
iajs-2430	125	26	3	3	NUM
iajs-2430	125	27	,	,	PUNCT
iajs-2430	125	28	we	we	PRON
iajs-2430	125	29	get	get	VERB
iajs-2430	125	30	𝜇	𝜇	ADP
iajs-2430	125	31	𝜒	𝜒	NOUN
iajs-2430	125	32	∗	∗	NOUN
iajs-2430	125	33	τ	τ	PROPN
iajs-2430	125	34	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	125	35	𝜇	𝜇	ADP
iajs-2430	125	36	𝜒	𝜒	PROPN
iajs-2430	125	37	∗	∗	NOUN
iajs-2430	125	38	𝛾	𝛾	ADP
iajs-2430	125	39	∗	∗	NOUN
iajs-2430	125	40	τ	τ	X
iajs-2430	125	41	,	,	PUNCT
iajs-2430	125	42	𝜇	𝜇	ADP
iajs-2430	125	43	𝛾	𝛾	NOUN
iajs-2430	125	44	and	and	CCONJ
iajs-2430	125	45	by	by	ADP
iajs-2430	125	46	definition	definition	NOUN
iajs-2430	125	47	24	24	NUM
iajs-2430	125	48	,	,	PUNCT
iajs-2430	125	49	we	we	PRON
iajs-2430	125	50	have	have	VERB
iajs-2430	125	51	𝜇	𝜇	ADP
iajs-2430	125	52	𝜒	𝜒	X
iajs-2430	125	53	∘	∘	NOUN
iajs-2430	125	54	𝛾	𝛾	ADP
iajs-2430	125	55	𝑟min	𝑟min	NOUN
iajs-2430	125	56	𝜇	𝜇	ADP
iajs-2430	125	57	𝜒	𝜒	NOUN
iajs-2430	125	58	ˑ	ˑ	NOUN
iajs-2430	125	59	,	,	PUNCT
iajs-2430	125	60	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	125	61	𝛾	𝛾	NOUN
iajs-2430	125	62	.	.	PUNCT
iajs-2430	126	1	thus	thus	ADV
iajs-2430	126	2	𝜇	𝜇	PRON
iajs-2430	126	3	is	be	AUX
iajs-2430	126	4	an	an	DET
iajs-2430	126	5	interval	interval	NOUN
iajs-2430	126	6	valuedfuzzy	valuedfuzzy	NOUN
iajs-2430	126	7	k	k	NOUN
iajs-2430	126	8	-	-	NOUN
iajs-2430	126	9	ideal	ideal	NOUN
iajs-2430	126	10	of	of	ADP
iajs-2430	126	11	ℵ.	ℵ.	PROPN
iajs-2430	126	12	theorem	theorem	PROPN
iajs-2430	126	13	(	(	PUNCT
iajs-2430	126	14	27	27	NUM
iajs-2430	126	15	)	)	PUNCT
iajs-2430	126	16	.	.	PUNCT
iajs-2430	127	1	let	let	PROPN
iajs-2430	127	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	127	3	,	,	PUNCT
iajs-2430	127	4	0	0	NUM
iajs-2430	127	5	be	be	VERB
iajs-2430	127	6	a	a	DET
iajs-2430	127	7	ku	ku	PROPN
iajs-2430	127	8	-	-	PUNCT
iajs-2430	127	9	semigroup	semigroup	PROPN
iajs-2430	127	10	,	,	PUNCT
iajs-2430	127	11	𝛢	𝛢	NOUN
iajs-2430	127	12	be	be	VERB
iajs-2430	127	13	a	a	DET
iajs-2430	127	14	nonempty	nonempty	NOUN
iajs-2430	127	15	subset	subset	NOUN
iajs-2430	127	16	of	of	ADP
iajs-2430	127	17	?	?	PUNCT
iajs-2430	128	1	ℵ	ℵ	NOUN
iajs-2430	128	2	and	and	CCONJ
iajs-2430	128	3	?	?	PUNCT
iajs-2430	129	1	𝜇	𝜇	X
iajs-2430	129	2	be	be	AUX
iajs-2430	129	3	an	an	DET
iajs-2430	129	4	interval	interval	NOUN
iajs-2430	129	5	valued	value	VERB
iajs-2430	129	6	fuzzy	fuzzy	ADJ
iajs-2430	129	7	set	set	VERB
iajs-2430	129	8	in	in	ADP
iajs-2430	129	9	?	?	PUNCT
iajs-2430	130	1	ℵ.	ℵ.	PROPN
iajs-2430	131	1	we	we	PRON
iajs-2430	131	2	define	define	VERB
iajs-2430	131	3	𝜇	𝜇	ADP
iajs-2430	131	4	as	as	SCONJ
iajs-2430	131	5	follows	follow	VERB
iajs-2430	131	6	𝜇	𝜇	ADP
iajs-2430	131	7	𝜒	𝜒	X
iajs-2430	131	8	𝑡	𝑡	NOUN
iajs-2430	131	9	,	,	PUNCT
iajs-2430	131	10	𝑡	𝑡	PROPN
iajs-2430	131	11	𝜒	𝜒	NOUN
iajs-2430	131	12	∈	∈	PROPN
iajs-2430	131	13	𝛢	𝛢	NOUN
iajs-2430	131	14	𝛼1	𝛼1	NOUN
iajs-2430	131	15	,	,	PUNCT
iajs-2430	131	16	𝛼1	𝛼1	NOUN
iajs-2430	131	17	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	VERB
iajs-2430	131	18	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	PROPN
iajs-2430	131	19	𝑡	𝑡	PROPN
iajs-2430	131	20	𝛼1	𝛼1	NOUN
iajs-2430	131	21	,	,	PUNCT
iajs-2430	131	22	𝑡	𝑡	PROPN
iajs-2430	131	23	𝛼2	𝛼2	VERB
iajs-2430	131	24	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2430	131	25	𝛼1	𝛼1	NOUN
iajs-2430	131	26	,	,	PUNCT
iajs-2430	131	27	𝛼1,𝑡	𝛼1,𝑡	PROPN
iajs-2430	131	28	,	,	PUNCT
iajs-2430	131	29	𝑡	𝑡	PROPN
iajs-2430	131	30	∈	∈	PROPN
iajs-2430	131	31	d	d	SYM
iajs-2430	131	32	0,1	0,1	NUM
iajs-2430	131	33	.	.	PUNCT
iajs-2430	132	1	then	then	ADV
iajs-2430	132	2	𝛢	𝛢	PROPN
iajs-2430	132	3	is	be	AUX
iajs-2430	132	4	'	'	PUNCT
iajs-2430	132	5	a	a	DET
iajs-2430	132	6	k-´´ideal	k-´´ideal	NOUN
iajs-2430	132	7	of	of	ADP
iajs-2430	132	8	ℵ´if	ℵ´if	PROPN
iajs-2430	132	9	´	´	PROPN
iajs-2430	132	10	and	and	CCONJ
iajs-2430	132	11	only	only	ADV
iajs-2430	132	12	´	´	NOUN
iajs-2430	132	13	if	if	SCONJ
iajs-2430	132	14	𝜇	𝜇	ADV
iajs-2430	132	15	is	be	AUX
iajs-2430	132	16	´	´	VERB
iajs-2430	132	17	an	an	DET
iajs-2430	132	18	´	´	PROPN
iajs-2430	132	19	interval	interval	NOUN
iajs-2430	132	20	´	´	NOUN
iajs-2430	132	21	valued	value	VERB
iajs-2430	132	22	'	'	PUNCT
iajs-2430	132	23	fuzzy	fuzzy	ADJ
iajs-2430	132	24	k´-´ideal´´of	k´-´ideal´´of	PROPN
iajs-2430	132	25	´	´	PROPN
iajs-2430	132	26	ℵ.	ℵ.	PUNCT
iajs-2430	133	1	moreover,´ℵ	moreover,´ℵ	PROPN
iajs-2430	133	2	α	α	PROPN
iajs-2430	133	3	,	,	PUNCT
iajs-2430	133	4	such	such	ADJ
iajs-2430	133	5	that´ℵ	that´ℵ	NOUN
iajs-2430	133	6	𝜒	𝜒	PROPN
iajs-2430	133	7	∈	∈	ADJ
iajs-2430	133	8	ℵ	ℵ	NOUN
iajs-2430	133	9	:	:	PUNCT
iajs-2430	133	10	𝜇	𝜇	X
iajs-2430	133	11	(	(	PUNCT
iajs-2430	133	12	𝜒	𝜒	NOUN
iajs-2430	133	13	)	)	PUNCT
iajs-2430	133	14	𝜇	𝜇	ADP
iajs-2430	133	15	0	0	NUM
iajs-2430	133	16	.	.	PUNCT
iajs-2430	133	17	  	  	SPACE
iajs-2430	134	1	proof	proof	NOUN
iajs-2430	134	2	.	.	PUNCT
iajs-2430	135	1	(	(	PUNCT
iajs-2430	135	2			NOUN
iajs-2430	135	3	)	)	PUNCT
iajs-2430	135	4	since	since	SCONJ
iajs-2430	135	5	𝜇	𝜇	ADP
iajs-2430	135	6	0	0	NUM
iajs-2430	135	7	𝜇	𝜇	PRON
iajs-2430	135	8	𝜒	𝜒	NOUN
iajs-2430	135	9	,	,	PUNCT
iajs-2430	135	10	∀𝜒	∀𝜒	NOUN
iajs-2430	135	11	∈	∈	PROPN
iajs-2430	135	12	ℵ	ℵ	NOUN
iajs-2430	135	13	,	,	PUNCT
iajs-2430	135	14	we	we	PRON
iajs-2430	135	15	get	get	VERB
iajs-2430	135	16	𝜇	𝜇	ADP
iajs-2430	135	17	0	0	NUM
iajs-2430	135	18	𝑡	𝑡	PROPN
iajs-2430	135	19	,	,	PUNCT
iajs-2430	135	20	𝑡	𝑡	PROPN
iajs-2430	135	21	,	,	PUNCT
iajs-2430	135	22	then	then	ADV
iajs-2430	135	23	0	0	NUM
iajs-2430	135	24	∈	∈	NOUN
iajs-2430	135	25	α	α	NOUN
iajs-2430	135	26	.	.	PUNCT
iajs-2430	136	1	let	let	VERB
iajs-2430	136	2	𝜒	𝜒	NUM
iajs-2430	136	3	∗	∗	NOUN
iajs-2430	136	4	𝛾	𝛾	ADP
iajs-2430	136	5	∗	∗	NOUN
iajs-2430	136	6	τ	τ	X
iajs-2430	136	7	∈	∈	PROPN
iajs-2430	136	8	α	α	NOUN
iajs-2430	136	9	and	and	CCONJ
iajs-2430	136	10	𝛾	𝛾	ADP
iajs-2430	136	11	∈	∈	PROPN
iajs-2430	136	12	α	α	NOUN
iajs-2430	136	13	,	,	PUNCT
iajs-2430	136	14	for	for	ADP
iajs-2430	136	15	any	any	DET
iajs-2430	136	16	𝜒	𝜒	NOUN
iajs-2430	136	17	,	,	PUNCT
iajs-2430	136	18	𝛾	𝛾	PROPN
iajs-2430	136	19	,	,	PUNCT
iajs-2430	136	20	τ	τ	PROPN
iajs-2430	136	21	∈	∈	PROPN
iajs-2430	136	22	ℵ.	ℵ.	NOUN
iajs-2430	136	23	using	use	VERB
iajs-2430	136	24	)	)	PUNCT
iajs-2430	136	25	(	(	PUNCT
iajs-2430	136	26	2f	2f	NOUN
iajs-2430	136	27	,	,	PUNCT
iajs-2430	136	28	we	we	PRON
iajs-2430	136	29	have	have	AUX
iajs-2430	136	30	𝜇	𝜇	ADP
iajs-2430	136	31	𝜒	𝜒	NOUN
iajs-2430	136	32	∗	∗	NOUN
iajs-2430	136	33	τ	τ	PROPN
iajs-2430	136	34	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	136	35	𝜇	𝜇	ADP
iajs-2430	136	36	𝜒	𝜒	PROPN
iajs-2430	136	37	∗	∗	NOUN
iajs-2430	136	38	𝛾	𝛾	ADP
iajs-2430	136	39	∗	∗	NOUN
iajs-2430	136	40	τ	τ	X
iajs-2430	136	41	,	,	PUNCT
iajs-2430	136	42	𝜇	𝜇	ADP
iajs-2430	136	43	𝛾	𝛾	NOUN
iajs-2430	136	44	=	=	SYM
iajs-2430	136	45	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	136	46	{	{	PUNCT
iajs-2430	137	1	[	[	X
iajs-2430	137	2	𝑡	𝑡	X
iajs-2430	137	3	,	,	PUNCT
iajs-2430	137	4	𝑡	𝑡	PROPN
iajs-2430	137	5	,	,	PUNCT
iajs-2430	137	6	𝑡	𝑡	PROPN
iajs-2430	137	7	,	,	PUNCT
iajs-2430	137	8	𝑡	𝑡	PROPN
iajs-2430	137	9	𝑡	𝑡	PROPN
iajs-2430	137	10	,	,	PUNCT
iajs-2430	137	11	𝑡	𝑡	PROPN
iajs-2430	137	12	and	and	CCONJ
iajs-2430	137	13	thus	thus	ADV
iajs-2430	137	14	𝜇	𝜇	ADP
iajs-2430	137	15	𝜒	𝜒	X
iajs-2430	137	16	∗	∗	NOUN
iajs-2430	137	17	τ	τ	X
iajs-2430	137	18	𝑡	𝑡	PROPN
iajs-2430	137	19	,	,	PUNCT
iajs-2430	137	20	𝑡	𝑡	PROPN
iajs-2430	137	21	,	,	PUNCT
iajs-2430	137	22	it	it	PRON
iajs-2430	137	23	follows	follow	VERB
iajs-2430	137	24	that	that	SCONJ
iajs-2430	137	25	𝜒	𝜒	X
iajs-2430	137	26	∗	∗	NOUN
iajs-2430	137	27	τ	τ	X
iajs-2430	137	28	∈	∈	PROPN
iajs-2430	137	29	α	α	NOUN
iajs-2430	137	30	.	.	PUNCT
iajs-2430	138	1	now	now	ADV
iajs-2430	138	2	,	,	PUNCT
iajs-2430	138	3	let	let	VERB
iajs-2430	138	4	𝜒	𝜒	NUM
iajs-2430	138	5	∈	∈	NOUN
iajs-2430	138	6	𝛢	𝛢	NOUN
iajs-2430	138	7	and	and	CCONJ
iajs-2430	138	8	𝛾	𝛾	ADP
iajs-2430	138	9	∈	∈	NOUN
iajs-2430	138	10	𝛢	𝛢	NOUN
iajs-2430	138	11	by	by	ADP
iajs-2430	138	12	using	use	VERB
iajs-2430	138	13	)	)	PUNCT
iajs-2430	138	14	(	(	PUNCT
iajs-2430	138	15	3f	3f	PROPN
iajs-2430	138	16	,	,	PUNCT
iajs-2430	138	17	we	we	PRON
iajs-2430	138	18	know	know	VERB
iajs-2430	138	19	that	that	SCONJ
iajs-2430	138	20	𝜇	𝜇	ADP
iajs-2430	138	21	𝜒	𝜒	X
iajs-2430	138	22	∘	∘	NOUN
iajs-2430	138	23	𝛾	𝛾	ADP
iajs-2430	138	24	𝑟	𝑟	X
iajs-2430	138	25	min	min	NOUN
iajs-2430	138	26	𝜇	𝜇	ADP
iajs-2430	138	27	𝜒	𝜒	X
iajs-2430	138	28	ˑ	ˑ	NOUN
iajs-2430	138	29	,	,	PUNCT
iajs-2430	138	30	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	138	31	𝛾	𝛾	ADP
iajs-2430	138	32	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	138	33	𝑡	𝑡	PROPN
iajs-2430	138	34	,	,	PUNCT
iajs-2430	138	35	𝑡	𝑡	PROPN
iajs-2430	138	36	,	,	PUNCT
iajs-2430	138	37	𝑡	𝑡	PROPN
iajs-2430	138	38	,	,	PUNCT
iajs-2430	138	39	𝑡	𝑡	PROPN
iajs-2430	138	40	𝑡	𝑡	PROPN
iajs-2430	138	41	,	,	PUNCT
iajs-2430	138	42	𝑡	𝑡	PROPN
iajs-2430	138	43	,	,	PUNCT
iajs-2430	138	44	and	and	CCONJ
iajs-2430	138	45	thus	thus	ADV
iajs-2430	138	46	𝜇	𝜇	X
iajs-2430	138	47	𝜒	𝜒	X
iajs-2430	138	48	∘	∘	NOUN
iajs-2430	138	49	𝛾	𝛾	ADP
iajs-2430	138	50	𝑡	𝑡	PROPN
iajs-2430	138	51	,	,	PUNCT
iajs-2430	138	52	𝑡	𝑡	PROPN
iajs-2430	138	53	.	.	PUNCT
iajs-2430	139	1	hence	hence	ADV
iajs-2430	139	2	𝜒	𝜒	NUM
iajs-2430	139	3	∘	∘	NOUN
iajs-2430	139	4	𝛾	𝛾	ADP
iajs-2430	139	5	∈	∈	NOUN
iajs-2430	139	6	𝛢	𝛢	NOUN
iajs-2430	140	1	and	and	CCONJ
iajs-2430	140	2	it	it	PRON
iajs-2430	140	3	follows	follow	VERB
iajs-2430	140	4	that	that	SCONJ
iajs-2430	140	5	𝛢	𝛢	NOUN
iajs-2430	140	6	is	be	AUX
iajs-2430	140	7	a	a	DET
iajs-2430	140	8	k	k	NOUN
iajs-2430	140	9	-	-	NOUN
iajs-2430	140	10	ideal	ideal	NOUN
iajs-2430	140	11	of	of	ADP
iajs-2430	140	12	ℵ.	ℵ.	PROPN
iajs-2430	140	13	(	(	PUNCT
iajs-2430	140	14			NOUN
iajs-2430	140	15	)	)	PUNCT
iajs-2430	140	16	since	since	SCONJ
iajs-2430	140	17	0	0	NUM
iajs-2430	140	18	∈	∈	PROPN
iajs-2430	140	19	𝛢	𝛢	NOUN
iajs-2430	140	20	,	,	PUNCT
iajs-2430	140	21	it	it	PRON
iajs-2430	140	22	follows	follow	VERB
iajs-2430	140	23	that	that	SCONJ
iajs-2430	140	24	𝜇	𝜇	ADP
iajs-2430	140	25	(	(	PUNCT
iajs-2430	140	26	0	0	NUM
iajs-2430	140	27	)	)	PUNCT
iajs-2430	140	28	𝑡	𝑡	NOUN
iajs-2430	140	29	,	,	PUNCT
iajs-2430	140	30	𝑡	𝑡	X
iajs-2430	140	31	𝜇	𝜇	ADP
iajs-2430	140	32	𝜒	𝜒	NOUN
iajs-2430	140	33	for	for	ADP
iajs-2430	140	34	all	all	DET
iajs-2430	140	35	𝜒	𝜒	NOUN
iajs-2430	140	36	∈	∈	NOUN
iajs-2430	140	37	ℵ.	ℵ.	NOUN
iajs-2430	140	38	let	let	VERB
iajs-2430	140	39	𝜒	𝜒	NUM
iajs-2430	140	40	,	,	PUNCT
iajs-2430	140	41	𝛾	𝛾	PROPN
iajs-2430	140	42	,	,	PUNCT
iajs-2430	140	43	τ	τ	PROPN
iajs-2430	140	44	∈	∈	PROPN
iajs-2430	140	45	ℵ.	ℵ.	NOUN
iajs-2430	141	1	if	if	SCONJ
iajs-2430	141	2	𝛾	𝛾	PROPN
iajs-2430	141	3	∉	∉	PROPN
iajs-2430	141	4	α	α	NOUN
iajs-2430	141	5	and	and	CCONJ
iajs-2430	141	6	𝜒	𝜒	NUM
iajs-2430	141	7	∗	∗	NOUN
iajs-2430	141	8	τ	τ	X
iajs-2430	141	9	∈	∈	PROPN
iajs-2430	141	10	α	α	NOUN
iajs-2430	141	11	,	,	PUNCT
iajs-2430	141	12	then	then	ADV
iajs-2430	141	13	clearly	clearly	ADV
iajs-2430	141	14	𝜇	𝜇	ADP
iajs-2430	141	15	𝜒	𝜒	X
iajs-2430	141	16	∗	∗	NOUN
iajs-2430	141	17	τ	τ	PROPN
iajs-2430	141	18	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	141	19	𝜇	𝜇	ADP
iajs-2430	141	20	𝜒	𝜒	PROPN
iajs-2430	141	21	∗	∗	NOUN
iajs-2430	141	22	𝛾	𝛾	ADP
iajs-2430	141	23	∗	∗	NOUN
iajs-2430	141	24	τ	τ	X
iajs-2430	141	25	,	,	PUNCT
iajs-2430	141	26	𝜇	𝜇	ADP
iajs-2430	141	27	𝛾	𝛾	NOUN
iajs-2430	141	28	.	.	PUNCT
iajs-2430	142	1	assume	assume	VERB
iajs-2430	142	2	that	that	SCONJ
iajs-2430	142	3	𝛾	𝛾	PROPN
iajs-2430	142	4	∈	∈	NOUN
iajs-2430	142	5	𝛢	𝛢	NOUN
iajs-2430	142	6	and	and	CCONJ
iajs-2430	142	7	𝜒	𝜒	NUM
iajs-2430	142	8	∗	∗	NOUN
iajs-2430	142	9	τ	τ	PROPN
iajs-2430	142	10	∉	∉	PROPN
iajs-2430	142	11	𝛢.	𝛢.	PROPN
iajs-2430	142	12	then	then	ADV
iajs-2430	142	13	by	by	ADP
iajs-2430	142	14	definition9	definition9	PROPN
iajs-2430	142	15	,	,	PUNCT
iajs-2430	142	16	we	we	PRON
iajs-2430	142	17	have	have	VERB
iajs-2430	142	18	χ	χ	PROPN
iajs-2430	142	19	∗	∗	NOUN
iajs-2430	142	20	γ	γ	PROPN
iajs-2430	142	21	∗	∗	PROPN
iajs-2430	142	22	τ	τ	PROPN
iajs-2430	142	23	∉	∉	ADJ
iajs-2430	142	24	𝛢	𝛢	NOUN
iajs-2430	142	25	.	.	PUNCT
iajs-2430	143	1	therefore	therefore	ADV
iajs-2430	143	2	𝜇	𝜇	ADP
iajs-2430	143	3	𝜒	𝜒	X
iajs-2430	143	4	∗	∗	X
iajs-2430	143	5	τ	τ	PROPN
iajs-2430	143	6	𝛼1	𝛼1	NOUN
iajs-2430	143	7	,	,	PUNCT
iajs-2430	143	8	𝛼1	𝛼1	PROPN
iajs-2430	143	9	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	143	10	𝜇	𝜇	ADP
iajs-2430	143	11	𝜒	𝜒	PROPN
iajs-2430	143	12	∗	∗	NOUN
iajs-2430	143	13	𝛾	𝛾	ADP
iajs-2430	143	14	∗	∗	NOUN
iajs-2430	143	15	τ	τ	X
iajs-2430	143	16	,	,	PUNCT
iajs-2430	143	17	𝜇	𝜇	ADP
iajs-2430	143	18	𝛾	𝛾	NOUN
iajs-2430	143	19	.	.	PUNCT
iajs-2430	144	1	also	also	ADV
iajs-2430	144	2	,	,	PUNCT
iajs-2430	144	3	let	let	VERB
iajs-2430	144	4	𝜒	𝜒	NUM
iajs-2430	144	5	∈	∈	NOUN
iajs-2430	144	6	𝛢	𝛢	NOUN
iajs-2430	144	7	,	,	PUNCT
iajs-2430	144	8	γ	γ	PROPN
iajs-2430	144	9	∉	∉	PROPN
iajs-2430	144	10	𝛢	𝛢	NOUN
iajs-2430	144	11	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
iajs-2430	144	12	𝜒	𝜒	NUM
iajs-2430	144	13	∘	∘	NOUN
iajs-2430	144	14	𝛾	𝛾	ADP
iajs-2430	144	15	∈	∈	PROPN
iajs-2430	144	16	𝛢	𝛢	NOUN
iajs-2430	144	17	,	,	PUNCT
iajs-2430	144	18	then	then	ADV
iajs-2430	144	19	clearly	clearly	ADV
iajs-2430	144	20	:	:	PUNCT
iajs-2430	144	21	𝜇	𝜇	X
iajs-2430	144	22	𝜒	𝜒	X
iajs-2430	144	23	∘	∘	X
iajs-2430	144	24	τ	τ	X
iajs-2430	144	25	𝑡	𝑡	PROPN
iajs-2430	144	26	,	,	PUNCT
iajs-2430	144	27	𝑡	𝑡	PROPN
iajs-2430	144	28	𝑟	𝑟	NOUN
iajs-2430	144	29	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	144	30	𝜇	𝜇	ADP
iajs-2430	144	31	𝜒	𝜒	X
iajs-2430	144	32	,	,	PUNCT
iajs-2430	144	33	𝜇	𝜇	ADP
iajs-2430	144	34	𝛾	𝛾	NOUN
iajs-2430	144	35	.	.	PUNCT
iajs-2430	145	1	assume	assume	VERB
iajs-2430	145	2	that	that	SCONJ
iajs-2430	145	3	𝜒	𝜒	X
iajs-2430	145	4	∉	∉	ADJ
iajs-2430	145	5	𝛢	𝛢	NOUN
iajs-2430	145	6	,	,	PUNCT
iajs-2430	145	7	𝛾	𝛾	ADP
iajs-2430	145	8	∈	∈	PROPN
iajs-2430	145	9	α	α	NOUN
iajs-2430	145	10	and	and	CCONJ
iajs-2430	145	11	𝜒	𝜒	ADP
iajs-2430	145	12	∘	∘	PROPN
iajs-2430	145	13	𝛾	𝛾	ADP
iajs-2430	145	14	∉	∉	PROPN
iajs-2430	145	15	𝛢.	𝛢.	PROPN
iajs-2430	145	16	then	then	ADV
iajs-2430	145	17	𝜇	𝜇	ADP
iajs-2430	145	18	𝜒	𝜒	X
iajs-2430	145	19	∘	∘	NOUN
iajs-2430	145	20	𝛾	𝛾	ADP
iajs-2430	145	21	𝛼	𝛼	PRON
iajs-2430	145	22	,	,	PUNCT
iajs-2430	145	23	𝛼	𝛼	PROPN
iajs-2430	145	24	=	=	NOUN
iajs-2430	145	25	𝑟	𝑟	NOUN
iajs-2430	145	26	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	145	27	𝜇	𝜇	ADP
iajs-2430	145	28	𝜒	𝜒	X
iajs-2430	145	29	,	,	PUNCT
iajs-2430	145	30	𝜇	𝜇	ADP
iajs-2430	145	31	𝛾	𝛾	NOUN
iajs-2430	145	32	.	.	PUNCT
iajs-2430	146	1	hence	hence	ADV
iajs-2430	146	2	𝜇	𝜇	ADV
iajs-2430	146	3	is	be	AUX
iajs-2430	146	4	a	a	DET
iajs-2430	146	5	fuzzy	fuzzy	ADJ
iajs-2430	146	6	k	k	NOUN
iajs-2430	146	7	-	-	NOUN
iajs-2430	146	8	ideal	ideal	NOUN
iajs-2430	146	9	in	in	ADP
iajs-2430	146	10	ℵ.	ℵ.	PROPN
iajs-2430	146	11	  	  	SPACE
iajs-2430	147	1	101	101	NUM
iajs-2430	147	2	  	  	SPACE
iajs-2430	147	3	ibn	ibn	PROPN
iajs-2430	147	4	al	al	PROPN
iajs-2430	147	5	-	-	PUNCT
iajs-2430	147	6	haitham	haitham	PROPN
iajs-2430	147	7	jour	jour	X
iajs-2430	147	8	.	.	PROPN
iajs-2430	147	9	for	for	ADP
iajs-2430	147	10	pure	pure	ADJ
iajs-2430	147	11	&	&	CCONJ
iajs-2430	147	12	appl	appl	PROPN
iajs-2430	147	13	.	.	PUNCT
iajs-2430	148	1	sci	sci	PROPN
iajs-2430	148	2	.	.	PROPN
iajs-2430	149	1	33	33	NUM
iajs-2430	149	2	(	(	PUNCT
iajs-2430	149	3	2	2	NUM
iajs-2430	149	4	)	)	PUNCT
iajs-2430	149	5	2020	2020	NUM
iajs-2430	149	6	finally	finally	ADV
iajs-2430	149	7	,	,	PUNCT
iajs-2430	149	8	we	we	PRON
iajs-2430	149	9	have	have	VERB
iajs-2430	149	10	´	´	NOUN
iajs-2430	149	11	ℵ	ℵ	ADP
iajs-2430	149	12	𝜒	𝜒	NOUN
iajs-2430	149	13	∈	∈	ADJ
iajs-2430	149	14	ℵ	ℵ	NOUN
iajs-2430	149	15	:	:	PUNCT
iajs-2430	149	16	𝜇	𝜇	X
iajs-2430	149	17	(	(	PUNCT
iajs-2430	149	18	𝜒	𝜒	NOUN
iajs-2430	149	19	)	)	PUNCT
iajs-2430	149	20	𝜇	𝜇	ADP
iajs-2430	149	21	0	0	NUM
iajs-2430	149	22	𝜒	𝜒	SYM
iajs-2430	149	23	∈	∈	NOUN
iajs-2430	149	24	ℵ:𝜇	ℵ:𝜇	NOUN
iajs-2430	149	25	(	(	PUNCT
iajs-2430	149	26	𝜒	𝜒	NOUN
iajs-2430	149	27	)	)	PUNCT
iajs-2430	149	28	𝑡	𝑡	NOUN
iajs-2430	149	29	,	,	PUNCT
iajs-2430	149	30	𝑡	𝑡	PROPN
iajs-2430	149	31	α	α	PROPN
iajs-2430	149	32	.	.	PUNCT
iajs-2430	150	1	theorem	theorem	PROPN
iajs-2430	150	2	(	(	PUNCT
iajs-2430	150	3	28	28	NUM
iajs-2430	150	4	)	)	PUNCT
iajs-2430	150	5	.	.	PUNCT
iajs-2430	151	1	let	let	VERB
iajs-2430	151	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	151	3	,	,	PUNCT
iajs-2430	151	4	0	0	NUM
iajs-2430	151	5	be	be	AUX
iajs-2430	151	6	a	a	DET
iajs-2430	151	7	ku	ku	PROPN
iajs-2430	151	8	-	-	PUNCT
iajs-2430	151	9	semigroup	semigroup	PROPN
iajs-2430	151	10	and	and	CCONJ
iajs-2430	151	11	𝜇	𝜇	ADP
iajs-2430	151	12	:	:	PUNCT
iajs-2430	151	13	ℵ	ℵ	ADJ
iajs-2430	151	14	→	→	SYM
iajs-2430	151	15	𝐷	𝐷	NOUN
iajs-2430	151	16	0,1	0,1	NUM
iajs-2430	151	17	.	.	PUNCT
iajs-2430	152	1	then	then	ADV
iajs-2430	152	2	the	the	DET
iajs-2430	152	3	level	level	NOUN
iajs-2430	152	4	set	set	VERB
iajs-2430	152	5	𝜇	𝜇	ADP
iajs-2430	152	6	of	of	ADP
iajs-2430	152	7	𝜇	𝜇	PRON
iajs-2430	152	8	is	be	AUX
iajs-2430	152	9	a	a	DET
iajs-2430	152	10	k	k	NOUN
iajs-2430	152	11	-	-	NOUN
iajs-2430	152	12	ideal	ideal	NOUN
iajs-2430	152	13	in	in	ADP
iajs-2430	152	14	ℵ	ℵ	DET
iajs-2430	152	15	iff	iff	PROPN
iajs-2430	152	16	𝜇	𝜇	ADP
iajs-2430	152	17	is	be	AUX
iajs-2430	152	18	an	an	DET
iajs-2430	152	19	interval	interval	NOUN
iajs-2430	152	20	valued	value	VERB
iajs-2430	152	21	fuzzy	fuzzy	ADJ
iajs-2430	152	22	k	k	NOUN
iajs-2430	152	23	-	-	NOUN
iajs-2430	152	24	ideal	ideal	ADJ
iajs-2430	152	25	.	.	PUNCT
iajs-2430	153	1	©	©	ADJ
iajs-2430	153	2	?	?	PUNCT
iajs-2430	153	3	?	?	PUNCT
iajs-2430	154	1	proof	proof	NOUN
iajs-2430	154	2	.	.	PUNCT
iajs-2430	155	1	(	(	PUNCT
iajs-2430	155	2			NOUN
iajs-2430	155	3	)	)	PUNCT
iajs-2430	155	4	for	for	ADP
iajs-2430	155	5	any	any	DET
iajs-2430	155	6	�	�	PROPN
iajs-2430	155	7	̃	̃	PROPN
iajs-2430	155	8	�	�	PROPN
iajs-2430	155	9	𝑡	𝑡	PROPN
iajs-2430	155	10	,	,	PUNCT
iajs-2430	155	11	𝑡	𝑡	PROPN
iajs-2430	155	12	∈	∈	PROPN
iajs-2430	155	13	d	d	SYM
iajs-2430	155	14	0,1	0,1	NUM
iajs-2430	155	15	,	,	PUNCT
iajs-2430	155	16	assume	assume	VERB
iajs-2430	155	17	𝜇	𝜇	ADP
iajs-2430	155	18	is	be	AUX
iajs-2430	155	19	a	a	DET
iajs-2430	155	20	non	non	X
iajs-2430	155	21	empty	empty	ADJ
iajs-2430	155	22	,	,	PUNCT
iajs-2430	155	23	then	then	ADV
iajs-2430	155	24	there	there	PRON
iajs-2430	155	25	exists	exist	VERB
iajs-2430	155	26	𝜒	𝜒	PRON
iajs-2430	155	27	∈	∈	PROPN
iajs-2430	155	28	𝜇	𝜇	ADP
iajs-2430	155	29	and	and	CCONJ
iajs-2430	155	30	𝜇	𝜇	X
iajs-2430	155	31	𝜒	𝜒	X
iajs-2430	155	32	�	�	PROPN
iajs-2430	155	33	̃	̃	PROPN
iajs-2430	155	34	�	�	PROPN
iajs-2430	155	35	.	.	PUNCT
iajs-2430	156	1	itˑ	itˑ	PROPN
iajs-2430	156	2	followsˑ	followsˑ	NOUN
iajs-2430	156	3	from	from	ADP
iajs-2430	156	4	definition	definition	NOUN
iajs-2430	156	5	24	24	NUM
iajs-2430	156	6	that	that	PRON
iajs-2430	156	7	𝜇	𝜇	ADP
iajs-2430	156	8	0	0	PUNCT
iajs-2430	156	9	𝜇	𝜇	ADP
iajs-2430	156	10	𝜒	𝜒	X
iajs-2430	156	11	�	�	PROPN
iajs-2430	156	12	̃	̃	PROPN
iajs-2430	156	13	�	�	PROPN
iajs-2430	156	14	,	,	PUNCT
iajs-2430	156	15	so	so	SCONJ
iajs-2430	156	16	that	that	SCONJ
iajs-2430	156	17	0	0	NUM
iajs-2430	156	18	∈	∈	PROPN
iajs-2430	156	19	𝜇	𝜇	X
iajs-2430	156	20	.	.	PUNCT
iajs-2430	157	1	let	let	VERB
iajs-2430	157	2	𝜒	𝜒	NUM
iajs-2430	157	3	,	,	PUNCT
iajs-2430	157	4	𝛾	𝛾	PROPN
iajs-2430	157	5	,	,	PUNCT
iajs-2430	157	6	τ	τ	PROPN
iajs-2430	157	7	∈	∈	PROPN
iajs-2430	157	8	ℵ	ℵ	ADP
iajs-2430	157	9	such	such	ADJ
iajs-2430	157	10	that	that	SCONJ
iajs-2430	157	11	𝜒	𝜒	NUM
iajs-2430	157	12	∗	∗	NOUN
iajs-2430	157	13	𝛾	𝛾	ADP
iajs-2430	157	14	∗	∗	NOUN
iajs-2430	157	15	τ	τ	X
iajs-2430	157	16	∈	∈	PROPN
iajs-2430	157	17	𝜇	𝜇	ADP
iajs-2430	157	18	and	and	CCONJ
iajs-2430	157	19	t	t	NOUN
iajs-2430	157	20			NOUN
iajs-2430	157	21			PROPN
iajs-2430	157	22	.	.	PUNCT
iajs-2430	158	1	we	we	PRON
iajs-2430	158	2	have	have	VERB
iajs-2430	158	3	(	(	PUNCT
iajs-2430	158	4	(	(	PUNCT
iajs-2430	158	5	)	)	PUNCT
iajs-2430	158	6	)	)	PUNCT
iajs-2430	159	1	t	t	PROPN
iajs-2430	159	2			VERB
iajs-2430	159	3			NUM
iajs-2430	159	4			PROPN
iajs-2430	159	5			PROPN
iajs-2430	159	6			NUM
iajs-2430	159	7			PROPN
iajs-2430	159	8	and	and	CCONJ
iajs-2430	159	9	𝜇	𝜇	ADP
iajs-2430	159	10	𝛾	𝛾	PROPN
iajs-2430	159	11	�	�	PROPN
iajs-2430	159	12	̃	̃	PROPN
iajs-2430	159	13	�	�	PROPN
iajs-2430	159	14	.	.	PUNCT
iajs-2430	160	1	from	from	ADP
iajs-2430	160	2	definition	definition	NOUN
iajs-2430	160	3	24	24	NUM
iajs-2430	160	4	,	,	PUNCT
iajs-2430	160	5	we	we	PRON
iajs-2430	160	6	get	get	VERB
iajs-2430	160	7	the	the	DET
iajs-2430	160	8	following	following	NOUN
iajs-2430	160	9	𝜇	𝜇	ADP
iajs-2430	160	10	𝜒	𝜒	NOUN
iajs-2430	160	11	∗	∗	NOUN
iajs-2430	160	12	τ	τ	PROPN
iajs-2430	160	13	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	160	14	𝜇	𝜇	ADP
iajs-2430	160	15	𝜒	𝜒	PROPN
iajs-2430	160	16	∗	∗	NOUN
iajs-2430	160	17	𝛾	𝛾	ADP
iajs-2430	160	18	∗	∗	NOUN
iajs-2430	160	19	τ	τ	X
iajs-2430	160	20	,	,	PUNCT
iajs-2430	160	21	𝜇	𝜇	ADP
iajs-2430	160	22	𝛾	𝛾	ADP
iajs-2430	160	23	𝑟	𝑟	PRON
iajs-2430	160	24	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
iajs-2430	160	25	�	�	PROPN
iajs-2430	160	26	̃	̃	PROPN
iajs-2430	160	27	�	�	PROPN
iajs-2430	160	28	,	,	PUNCT
iajs-2430	160	29	�	�	PROPN
iajs-2430	160	30	̃	̃	PROPN
iajs-2430	160	31	�	�	PROPN
iajs-2430	160	32	�	�	PROPN
iajs-2430	160	33	̃	̃	PROPN
iajs-2430	160	34	�	�	PROPN
iajs-2430	160	35	,	,	PUNCT
iajs-2430	160	36	thus	thus	ADV
iajs-2430	160	37	𝜒	𝜒	NUM
iajs-2430	160	38	∗	∗	NOUN
iajs-2430	160	39	τ	τ	X
iajs-2430	160	40	∈	∈	PROPN
iajs-2430	160	41	𝜇	𝜇	X
iajs-2430	160	42	.	.	PUNCT
iajs-2430	161	1	now	now	ADV
iajs-2430	161	2	,	,	PUNCT
iajs-2430	161	3	let	let	VERB
iajs-2430	161	4	𝑎	𝑎	PRON
iajs-2430	161	5	∈	∈	NOUN
iajs-2430	161	6	𝜇	𝜇	ADP
iajs-2430	161	7	and	and	CCONJ
iajs-2430	161	8	𝜒	𝜒	NOUN
iajs-2430	161	9	∈	∈	ADJ
iajs-2430	161	10	ℵ	ℵ	NOUN
iajs-2430	161	11	,	,	PUNCT
iajs-2430	161	12	then	then	ADV
iajs-2430	161	13	𝜇	𝜇	ADP
iajs-2430	161	14	𝜒	𝜒	X
iajs-2430	161	15	�	�	PROPN
iajs-2430	161	16	̃	̃	PROPN
iajs-2430	161	17	�	�	PROPN
iajs-2430	161	18	and	and	CCONJ
iajs-2430	161	19	𝜇	𝜇	ADP
iajs-2430	161	20	𝑎	𝑎	PROPN
iajs-2430	161	21	�	�	PROPN
iajs-2430	161	22	̃	̃	NOUN
iajs-2430	161	23	�	�	PROPN
iajs-2430	161	24	.	.	PUNCT
iajs-2430	162	1	we	we	PRON
iajs-2430	162	2	get	get	VERB
iajs-2430	162	3	𝜇	𝜇	PRON
iajs-2430	162	4	𝜒	𝜒	NOUN
iajs-2430	162	5	∘	∘	NOUN
iajs-2430	162	6	𝑎	𝑎	NUM
iajs-2430	162	7	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	162	8	𝜇	𝜇	ADP
iajs-2430	162	9	𝜒	𝜒	NOUN
iajs-2430	162	10	,	,	PUNCT
iajs-2430	162	11	𝜇	𝜇	ADP
iajs-2430	162	12	𝑎	𝑎	PROPN
iajs-2430	162	13	�	�	PROPN
iajs-2430	162	14	̃	̃	PROPN
iajs-2430	162	15	�	�	PROPN
iajs-2430	162	16	,	,	PUNCT
iajs-2430	162	17	�	�	PROPN
iajs-2430	162	18	̃	̃	PROPN
iajs-2430	162	19	�	�	PROPN
iajs-2430	162	20	�	�	PROPN
iajs-2430	162	21	̃	̃	PROPN
iajs-2430	162	22	�	�	PROPN
iajs-2430	162	23	,	,	PUNCT
iajs-2430	162	24	whichimpliesthat	whichimpliesthat	PRON
iajs-2430	162	25	𝜒	𝜒	X
iajs-2430	162	26	∘	∘	X
iajs-2430	162	27	𝑎	𝑎	PRON
iajs-2430	162	28	∈	∈	PROPN
iajs-2430	162	29	𝜇	𝜇	ADP
iajs-2430	162	30	.	.	PUNCT
iajs-2430	163	1	similarly	similarly	ADV
iajs-2430	163	2	,	,	PUNCT
iajs-2430	163	3	𝑎	𝑎	X
iajs-2430	163	4	∘	∘	X
iajs-2430	163	5	𝜒	𝜒	PRON
iajs-2430	163	6	∈	∈	NOUN
iajs-2430	163	7	𝜇	𝜇	ADP
iajs-2430	163	8	.	.	PUNCT
iajs-2430	164	1	therefore~𝜇	therefore~𝜇	X
iajs-2430	164	2	is	be	AUX
iajs-2430	164	3	a	a	DET
iajs-2430	164	4	k-~ideal	k-~ideal	NOUN
iajs-2430	164	5	of	of	ADP
iajs-2430	164	6	ℵ~	ℵ~	PROPN
iajs-2430	164	7	?	?	PUNCT
iajs-2430	164	8	?	?	PUNCT
iajs-2430	164	9	?	?	PUNCT
iajs-2430	165	1	(	(	PUNCT
iajs-2430	165	2	⟹	⟹	PROPN
iajs-2430	165	3	let	let	VERB
iajs-2430	165	4	μ	μ	NOUN
iajs-2430	165	5	be	be	AUX
iajs-2430	165	6	a	a	DET
iajs-2430	165	7	non	non	ADJ
iajs-2430	165	8	-	-	ADJ
iajs-2430	165	9	empty	empty	ADJ
iajs-2430	165	10	and	and	CCONJ
iajs-2430	165	11	a	a	DET
iajs-2430	165	12	k	k	NOUN
iajs-2430	165	13	-	-	NOUN
iajs-2430	165	14	ideal	ideal	NOUN
iajs-2430	165	15	of	of	ADP
iajs-2430	165	16	ℵ	ℵ	NOUN
iajs-2430	165	17	,	,	PUNCT
iajs-2430	165	18	we	we	PRON
iajs-2430	165	19	have	have	VERB
iajs-2430	165	20	𝜇	𝜇	ADP
iajs-2430	165	21	𝜒	𝜒	PRON
iajs-2430	165	22	�	�	PROPN
iajs-2430	165	23	̃	̃	NOUN
iajs-2430	165	24	�	�	PROPN
iajs-2430	165	25	,	,	PUNCT
iajs-2430	165	26	for	for	ADP
iajs-2430	165	27	every	every	DET
iajs-2430	165	28	�	�	PROPN
iajs-2430	165	29	̃	̃	PROPN
iajs-2430	165	30	�	�	NOUN
iajs-2430	165	31	∈	∈	PROPN
iajs-2430	165	32	d	d	SYM
iajs-2430	165	33	0,1	0,1	NUM
iajs-2430	165	34	and	and	CCONJ
iajs-2430	165	35	for	for	ADP
iajs-2430	165	36	any	any	DET
iajs-2430	165	37	𝜒	𝜒	NUM
iajs-2430	165	38	∈	∈	NOUN
iajs-2430	165	39	ℵ.	ℵ.	NOUN
iajs-2430	166	1	this	this	PRON
iajs-2430	166	2	implies	imply	VERB
iajs-2430	166	3	that	that	SCONJ
iajs-2430	166	4	𝜒	𝜒	PROPN
iajs-2430	166	5	∈	∈	PROPN
iajs-2430	166	6	μ	μ	NOUN
iajs-2430	166	7	.	.	PUNCT
iajs-2430	167	1	and	and	CCONJ
iajs-2430	167	2	since	since	SCONJ
iajs-2430	167	3	0	0	NUM
iajs-2430	167	4	∈	∈	PROPN
iajs-2430	167	5	μ	μ	PROPN
iajs-2430	167	6	,	,	PUNCT
iajs-2430	167	7	then	then	ADV
iajs-2430	167	8	𝜇	𝜇	ADP
iajs-2430	167	9	0	0	PROPN
iajs-2430	167	10	�	�	PROPN
iajs-2430	167	11	̃	̃	PROPN
iajs-2430	167	12	�	�	PROPN
iajs-2430	167	13	𝜇	𝜇	ADP
iajs-2430	167	14	𝜒	𝜒	NOUN
iajs-2430	167	15	.	.	PUNCT
iajs-2430	168	1	now	now	ADV
iajs-2430	168	2	,	,	PUNCT
iajs-2430	168	3	we	we	PRON
iajs-2430	168	4	show	show	VERB
iajs-2430	168	5	that	that	SCONJ
iajs-2430	168	6	𝜇	𝜇	ADP
iajs-2430	168	7	satisifies	satisifie	NOUN
iajs-2430	168	8	𝑘	𝑘	PRON
iajs-2430	168	9	and	and	CCONJ
iajs-2430	168	10	𝑘	𝑘	X
iajs-2430	168	11	.	.	PUNCT
iajs-2430	169	1	if	if	SCONJ
iajs-2430	169	2	not	not	PART
iajs-2430	169	3	,	,	PUNCT
iajs-2430	169	4	suppose	suppose	VERB
iajs-2430	169	5	that	that	SCONJ
iajs-2430	169	6	∃𝑙	∃𝑙	PROPN
iajs-2430	169	7	,	,	PUNCT
iajs-2430	169	8	𝑚	𝑚	PROPN
iajs-2430	169	9	,	,	PUNCT
iajs-2430	169	10	𝑛	𝑛	DET
iajs-2430	169	11	∈	∈	NOUN
iajs-2430	169	12	ℵ	ℵ	ADP
iajs-2430	169	13	such	such	ADJ
iajs-2430	169	14	that	that	SCONJ
iajs-2430	169	15	𝜇	𝜇	ADP
iajs-2430	169	16	𝑙	𝑙	X
iajs-2430	169	17	∗	∗	ADP
iajs-2430	169	18	𝑛	𝑛	PROPN
iajs-2430	169	19	𝑟	𝑟	NOUN
iajs-2430	169	20	min	min	NOUN
iajs-2430	169	21	𝜇	𝜇	ADP
iajs-2430	169	22	𝑙	𝑙	PROPN
iajs-2430	169	23	∗	∗	X
iajs-2430	169	24	𝑚	𝑚	ADP
iajs-2430	169	25	∗	∗	NOUN
iajs-2430	169	26	𝑛	𝑛	PRON
iajs-2430	169	27	,	,	PUNCT
iajs-2430	169	28	𝜇	𝜇	ADP
iajs-2430	169	29	𝑚	𝑚	X
iajs-2430	169	30	.	.	PUNCT
iajs-2430	170	1	put	put	VERB
iajs-2430	170	2	�	�	PROPN
iajs-2430	170	3	̃	̃	PROPN
iajs-2430	170	4	�	�	PROPN
iajs-2430	170	5	𝜇	𝜇	ADP
iajs-2430	170	6	𝑙	𝑙	PROPN
iajs-2430	170	7	∗	∗	X
iajs-2430	170	8	𝑛	𝑛	PROPN
iajs-2430	170	9	𝑟	𝑟	NOUN
iajs-2430	170	10	min	min	NOUN
iajs-2430	170	11	𝜇	𝜇	ADP
iajs-2430	170	12	𝑙	𝑙	PROPN
iajs-2430	170	13	∗	∗	X
iajs-2430	170	14	𝑚	𝑚	ADP
iajs-2430	170	15	∗	∗	NOUN
iajs-2430	170	16	𝑛	𝑛	PRON
iajs-2430	170	17	,	,	PUNCT
iajs-2430	170	18	𝜇	𝜇	ADP
iajs-2430	170	19	𝑚	𝑚	PROPN
iajs-2430	170	20	,	,	PUNCT
iajs-2430	170	21	for	for	ADP
iajs-2430	170	22	n	n	PRON
iajs-2430	170	23	any	any	DET
iajs-2430	170	24	integer	integer	NOUN
iajs-2430	170	25	number	number	NOUN
iajs-2430	170	26	,	,	PUNCT
iajs-2430	170	27	so	so	SCONJ
iajs-2430	170	28	𝜇	𝜇	ADP
iajs-2430	170	29	𝑙	𝑙	X
iajs-2430	170	30	∗	∗	X
iajs-2430	170	31	𝑛	𝑛	PRON
iajs-2430	170	32	�	�	PROPN
iajs-2430	170	33	̃	̃	PROPN
iajs-2430	170	34	�	�	PROPN
iajs-2430	170	35	𝑟	𝑟	NOUN
iajs-2430	170	36	min	min	PROPN
iajs-2430	170	37	𝜇	𝜇	ADP
iajs-2430	170	38	𝑙	𝑙	PROPN
iajs-2430	170	39	∗	∗	X
iajs-2430	170	40	𝑚	𝑚	ADP
iajs-2430	170	41	∗	∗	NOUN
iajs-2430	170	42	𝑛	𝑛	PRON
iajs-2430	170	43	,	,	PUNCT
iajs-2430	170	44	𝜇	𝜇	ADP
iajs-2430	170	45	𝑚	𝑚	PROPN
iajs-2430	170	46	.	.	PUNCT
iajs-2430	171	1	implies	imply	VERB
iajs-2430	171	2	that	that	SCONJ
iajs-2430	171	3	𝑙	𝑙	DET
iajs-2430	171	4	∗	∗	NOUN
iajs-2430	171	5	𝑚	𝑚	ADP
iajs-2430	171	6	∗	∗	NOUN
iajs-2430	171	7	𝑛	𝑛	DET
iajs-2430	171	8	∈	∈	PROPN
iajs-2430	171	9	𝜇	𝜇	ADP
iajs-2430	171	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2430	171	11	𝑚	𝑚	PROPN
iajs-2430	171	12	∈	∈	PROPN
iajs-2430	171	13	𝜇	𝜇	ADP
iajs-2430	171	14	,	,	PUNCT
iajs-2430	171	15	but	but	CCONJ
iajs-2430	171	16	𝑙	𝑙	X
iajs-2430	171	17	∗	∗	NOUN
iajs-2430	171	18	𝑛	𝑛	DET
iajs-2430	171	19	∉	∉	PROPN
iajs-2430	171	20	𝜇	𝜇	ADP
iajs-2430	171	21	,	,	PUNCT
iajs-2430	171	22	which	which	PRON
iajs-2430	171	23	implies	imply	VERB
iajs-2430	171	24	𝜇	𝜇	ADP
iajs-2430	171	25	is	be	AUX
iajs-2430	171	26	not	not	PART
iajs-2430	171	27	a	a	DET
iajs-2430	171	28	kideal	kideal	NOUN
iajs-2430	171	29	of	of	ADP
iajs-2430	171	30	ℵ.	ℵ.	PROPN
iajs-2430	172	1	then	then	ADV
iajs-2430	172	2	,	,	PUNCT
iajs-2430	172	3	it	it	PRON
iajs-2430	172	4	is	be	AUX
iajs-2430	172	5	a	a	DET
iajs-2430	172	6	contradiction	contradiction	NOUN
iajs-2430	172	7	.	.	PUNCT
iajs-2430	173	1	let	let	VERB
iajs-2430	173	2	l	l	NOUN
iajs-2430	173	3	,	,	PUNCT
iajs-2430	173	4	m∈	m∈	PROPN
iajs-2430	173	5	𝜇	𝜇	ADP
iajs-2430	173	6	such	such	ADJ
iajs-2430	173	7	that	that	SCONJ
iajs-2430	173	8	𝜇	𝜇	ADP
iajs-2430	173	9	𝑙	𝑙	PRON
iajs-2430	173	10	∘	∘	NOUN
iajs-2430	173	11	𝑚	𝑚	ADP
iajs-2430	173	12	𝑟	𝑟	NOUN
iajs-2430	173	13	min	min	PROPN
iajs-2430	173	14	𝜇	𝜇	ADP
iajs-2430	173	15	𝑙	𝑙	X
iajs-2430	173	16	,	,	PUNCT
iajs-2430	173	17	𝜇	𝜇	ADP
iajs-2430	173	18	𝑚	𝑚	PROPN
iajs-2430	173	19	.	.	PUNCT
iajs-2430	174	1	then	then	ADV
iajs-2430	174	2	by	by	ADP
iajs-2430	174	3	taking	take	VERB
iajs-2430	174	4	�	�	PROPN
iajs-2430	174	5	̃	̃	PROPN
iajs-2430	174	6	�	�	PROPN
iajs-2430	174	7	𝜇	𝜇	ADP
iajs-2430	174	8	𝑙	𝑙	PRON
iajs-2430	174	9	∘	∘	PROPN
iajs-2430	174	10	𝑚	𝑚	ADP
iajs-2430	174	11	𝑟	𝑟	NOUN
iajs-2430	174	12	min	min	PROPN
iajs-2430	174	13	𝜇	𝜇	ADP
iajs-2430	174	14	𝑙	𝑙	X
iajs-2430	174	15	,	,	PUNCT
iajs-2430	174	16	𝜇	𝜇	ADP
iajs-2430	174	17	𝑚	𝑚	NOUN
iajs-2430	174	18	.	.	PUNCT
iajs-2430	175	1	we	we	PRON
iajs-2430	175	2	have	have	VERB
iajs-2430	175	3	𝜇	𝜇	ADP
iajs-2430	175	4	𝑙	𝑙	DET
iajs-2430	175	5	∘	∘	PROPN
iajs-2430	175	6	𝑚	𝑚	ADP
iajs-2430	175	7	�	�	PROPN
iajs-2430	175	8	̃	̃	PROPN
iajs-2430	175	9	�	�	PROPN
iajs-2430	175	10	𝑟	𝑟	NOUN
iajs-2430	175	11	min	min	PROPN
iajs-2430	175	12	𝜇	𝜇	ADP
iajs-2430	175	13	𝑙	𝑙	X
iajs-2430	175	14	,	,	PUNCT
iajs-2430	175	15	𝜇	𝜇	ADP
iajs-2430	175	16	𝑚	𝑚	PROPN
iajs-2430	175	17	.	.	PUNCT
iajs-2430	176	1	then	then	ADV
iajs-2430	176	2	,	,	PUNCT
iajs-2430	176	3	𝑙	𝑙	X
iajs-2430	176	4	,	,	PUNCT
iajs-2430	176	5	𝑚	𝑚	PROPN
iajs-2430	176	6	∈	∈	PROPN
iajs-2430	176	7	𝜇	𝜇	ADP
iajs-2430	176	8	but	but	CCONJ
iajs-2430	176	9	𝑙	𝑙	PROPN
iajs-2430	176	10	∘	∘	PROPN
iajs-2430	176	11	𝑚	𝑚	X
iajs-2430	176	12	∉	∉	PROPN
iajs-2430	176	13	𝜇	𝜇	ADP
iajs-2430	176	14	.	.	PUNCT
iajs-2430	177	1	it	it	PRON
iajs-2430	177	2	means	mean	VERB
iajs-2430	177	3	that	that	SCONJ
iajs-2430	177	4	𝜇	𝜇	ADP
iajs-2430	177	5	is	be	AUX
iajs-2430	177	6	not	not	PART
iajs-2430	177	7	k	k	NOUN
iajs-2430	177	8	-	-	NOUN
iajs-2430	177	9	ideal	ideal	NOUN
iajs-2430	177	10	of	of	ADP
iajs-2430	177	11	ℵ	ℵ	NOUN
iajs-2430	177	12	and	and	CCONJ
iajs-2430	177	13	~this	~this	PUNCT
iajs-2430	177	14	'	'	PUNCT
iajs-2430	177	15	is~	is~	PROPN
iajs-2430	177	16	a	a	DET
iajs-2430	177	17	´	´	NOUN
iajs-2430	177	18	~contradiction	~contradiction	NUM
iajs-2430	177	19	.	.	PUNCT
iajs-2430	177	20	'	'	PUNCT
iajs-2430	178	1	the	the	DET
iajs-2430	178	2	proof	proof	NOUN
iajs-2430	178	3	is	be	AUX
iajs-2430	178	4	completed	complete	VERB
iajs-2430	178	5	.	.	PUNCT
iajs-2430	179	1	definition	definition	NOUN
iajs-2430	179	2	(	(	PUNCT
iajs-2430	179	3	29	29	NUM
iajs-2430	179	4	)	)	PUNCT
iajs-2430	179	5	.	.	PUNCT
iajs-2430	180	1	a	a	DET
iajs-2430	180	2	fuzzy	fuzzy	ADJ
iajs-2430	180	3	set	set	NOUN
iajs-2430	180	4	𝜇	𝜇	ADP
iajs-2430	180	5	in	in	ADP
iajs-2430	180	6	ℵ	ℵ	DET
iajs-2430	180	7	isˑ	isˑ	NOUN
iajs-2430	180	8	calledˑ	calledˑ	VERB
iajs-2430	180	9	an	an	DET
iajs-2430	180	10	interval	interval	NOUN
iajs-2430	180	11	valued	value	VERB
iajs-2430	180	12	fuzzy	fuzzy	ADJ
iajs-2430	180	13	p	p	NOUN
iajs-2430	180	14	-	-	PUNCT
iajs-2430	180	15	ideal	ideal	NOUN
iajs-2430	180	16	ofˑℵ	ofˑℵ	NOUN
iajs-2430	181	1	if	if	SCONJ
iajs-2430	181	2	,	,	PUNCT
iajs-2430	181	3	for	for	ADP
iajs-2430	181	4	all	all	DET
iajs-2430	181	5	𝜒	𝜒	NOUN
iajs-2430	181	6	,	,	PUNCT
iajs-2430	181	7	𝛾	𝛾	PROPN
iajs-2430	181	8	,	,	PUNCT
iajs-2430	181	9	τ	τ	PROPN
iajs-2430	181	10	∈	∈	PROPN
iajs-2430	181	11	ℵ	ℵ	ADP
iajs-2430	181	12	𝑃	𝑃	NOUN
iajs-2430	181	13	𝜇	𝜇	X
iajs-2430	181	14	(	(	PUNCT
iajs-2430	181	15	0	0	NUM
iajs-2430	181	16	)	)	PUNCT
iajs-2430	181	17	𝜇	𝜇	ADP
iajs-2430	181	18	𝜒	𝜒	X
iajs-2430	181	19	𝑃	𝑃	NOUN
iajs-2430	181	20	𝜇	𝜇	ADP
iajs-2430	181	21	τ	τ	PROPN
iajs-2430	181	22	∗	∗	NOUN
iajs-2430	181	23	𝛾	𝛾	ADP
iajs-2430	181	24	𝑟	𝑟	PRON
iajs-2430	181	25	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	181	26	𝜇	𝜇	ADP
iajs-2430	181	27	τ	τ	PROPN
iajs-2430	181	28	∗	∗	X
iajs-2430	181	29	𝜒	𝜒	PROPN
iajs-2430	181	30	∗	∗	NOUN
iajs-2430	181	31	𝛾	𝛾	NOUN
iajs-2430	181	32	,	,	PUNCT
iajs-2430	181	33	𝜇	𝜇	ADP
iajs-2430	181	34	τ	τ	PROPN
iajs-2430	181	35	∗	∗	X
iajs-2430	181	36	𝜒	𝜒	NOUN
iajs-2430	181	37	.	.	PUNCT
iajs-2430	182	1	𝑃	𝑃	NOUN
iajs-2430	182	2	𝜇	𝜇	DET
iajs-2430	182	3	𝜒	𝜒	NOUN
iajs-2430	182	4	∘	∘	NOUN
iajs-2430	182	5	𝛾	𝛾	ADP
iajs-2430	182	6	𝑟	𝑟	X
iajs-2430	182	7	min	min	NOUN
iajs-2430	182	8	𝜇	𝜇	ADP
iajs-2430	182	9	𝜒	𝜒	X
iajs-2430	182	10	ˑ	ˑ	NOUN
iajs-2430	182	11	,	,	PUNCT
iajs-2430	182	12	ˑ𝜇	ˑ𝜇	NOUN
iajs-2430	182	13	𝛾	𝛾	NOUN
iajs-2430	182	14	.	.	PUNCT
iajs-2430	183	1	example30	example30	NOUN
iajs-2430	183	2	.	.	PUNCT
iajs-2430	184	1	let	let	VERB
iajs-2430	184	2	ℵ	ℵ	ADP
iajs-2430	184	3	0	0	NUM
iajs-2430	184	4	,	,	PUNCT
iajs-2430	184	5	𝑎	𝑎	NOUN
iajs-2430	184	6	,	,	PUNCT
iajs-2430	184	7	𝑏	𝑏	PROPN
iajs-2430	184	8	be	be	VERB
iajs-2430	184	9	a	a	DET
iajs-2430	184	10	set	set	NOUN
iajs-2430	184	11	.	.	PUNCT
iajs-2430	185	1	define	define	VERB
iajs-2430	185	2	∗-ˑoperation	∗-ˑoperation	NOUN
iajs-2430	185	3	and∘-operation	and∘-operation	NOUN
iajs-2430	185	4	byˑtheˑfollowing	byˑtheˑfollowe	VERB
iajs-2430	185	5	tables	table	NOUN
iajs-2430	185	6	  	  	SPACE
iajs-2430	185	7	102	102	NUM
iajs-2430	185	8	  	  	SPACE
iajs-2430	185	9	ibn	ibn	PROPN
iajs-2430	185	10	al	al	PROPN
iajs-2430	185	11	-	-	PUNCT
iajs-2430	185	12	haitham	haitham	PROPN
iajs-2430	185	13	jour	jour	X
iajs-2430	185	14	.	.	PROPN
iajs-2430	186	1	for	for	ADP
iajs-2430	186	2	pure	pure	ADJ
iajs-2430	186	3	&	&	CCONJ
iajs-2430	186	4	appl	appl	PROPN
iajs-2430	186	5	.	.	PUNCT
iajs-2430	187	1	sci	sci	PROPN
iajs-2430	187	2	.	.	PROPN
iajs-2430	188	1	33	33	NUM
iajs-2430	188	2	(	(	PUNCT
iajs-2430	188	3	2	2	NUM
iajs-2430	188	4	)	)	PUNCT
iajs-2430	188	5	2020	2020	NUM
iajs-2430	189	1	then	then	ADV
iajs-2430	189	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	189	3	,	,	PUNCT
iajs-2430	189	4	0	0	NUM
iajs-2430	189	5	is	be	AUX
iajs-2430	189	6	a	a	DET
iajs-2430	189	7	ku	ku	PROPN
iajs-2430	189	8	-	-	PUNCT
iajs-2430	189	9	semigroup	semigroup	PROPN
iajs-2430	189	10	.	.	PUNCT
iajs-2430	190	1	define	define	VERB
iajs-2430	190	2	𝜇	𝜇	DET
iajs-2430	190	3	𝜒	𝜒	NOUN
iajs-2430	190	4	as	as	SCONJ
iajs-2430	190	5	follows	follow	VERB
iajs-2430	190	6	𝜇	𝜇	ADP
iajs-2430	190	7	𝜒	𝜒	DET
iajs-2430	190	8	0.3,0.8	0.3,0.8	NUM
iajs-2430	190	9	𝜒	𝜒	NUM
iajs-2430	190	10	0	0	NUM
iajs-2430	190	11	0.2,0.4	0.2,0.4	NOUN
iajs-2430	191	1	𝜒	𝜒	SYM
iajs-2430	191	2	0	0	NUM
iajs-2430	191	3	.	.	PUNCT
iajs-2430	192	1	then	then	ADV
iajs-2430	192	2	𝜇	𝜇	X
iajs-2430	192	3	𝜒	𝜒	X
iajs-2430	192	4	~is	~is	NOUN
iajs-2430	192	5	an	an	DET
iajs-2430	192	6	interval	interval	NOUN
iajs-2430	192	7	valued	value	VERB
iajs-2430	192	8	fuzzy	fuzzy	ADJ
iajs-2430	192	9	p	p	NOUN
iajs-2430	192	10	-	-	PUNCT
iajs-2430	192	11	ideal	ideal	NOUN
iajs-2430	192	12	of	of	ADP
iajs-2430	192	13	ℵˑ~.	ℵˑ~.	PROPN
iajs-2430	192	14	theorem31	theorem31	NOUN
iajs-2430	192	15	.	.	PUNCT
iajs-2430	193	1	let	let	VERB
iajs-2430	193	2	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	193	3	,	,	PUNCT
iajs-2430	193	4	0	0	NUM
iajs-2430	193	5	be	be	AUX
iajs-2430	193	6	a	a	DET
iajs-2430	193	7	ku	ku	PROPN
iajs-2430	193	8	-	-	PUNCT
iajs-2430	193	9	semigroup	semigroup	PROPN
iajs-2430	193	10	and	and	CCONJ
iajs-2430	193	11	𝜇	𝜇	ADP
iajs-2430	193	12	:	:	PUNCT
iajs-2430	193	13	ℵ	ℵ	ADJ
iajs-2430	193	14	→	→	SYM
iajs-2430	193	15	𝐷	𝐷	NOUN
iajs-2430	193	16	0,1	0,1	NUM
iajs-2430	193	17	.	.	PUNCT
iajs-2430	194	1	if	if	SCONJ
iajs-2430	194	2	𝜇	𝜇	VERB
iajs-2430	194	3	´	´	NOUN
iajs-2430	194	4	is	be	AUX
iajs-2430	194	5	~an	~an	ADP
iajs-2430	194	6	interval	interval	NOUN
iajs-2430	194	7	~	~	SYM
iajs-2430	194	8	valued	value	VERB
iajs-2430	194	9	´	´	NOUN
iajs-2430	194	10	fuzzy	fuzzy	ADJ
iajs-2430	194	11	~p-~ideal	~p-~ideal	NOUN
iajs-2430	194	12	,	,	PUNCT
iajs-2430	194	13	then~	then~	PROPN
iajs-2430	194	14	it	it	PRON
iajs-2430	194	15	is	be	AUX
iajs-2430	194	16	an	an	DET
iajs-2430	194	17	interval~	interval~	PROPN
iajs-2430	194	18	valued	value	VERB
iajs-2430	194	19	´	´	NOUN
iajs-2430	194	20	fuzzy	fuzzy	ADJ
iajs-2430	194	21	s-´ideal	s-´ideal	ADV
iajs-2430	194	22	.	.	PUNCT
iajs-2430	195	1	µ	µ	PRON
iajs-2430	195	2	proof~.	proof~.	NUM
iajs-2430	195	3	by	by	ADP
iajs-2430	195	4	(	(	PUNCT
iajs-2430	195	5	p2	p2	NOUN
iajs-2430	195	6	)	)	PUNCT
iajs-2430	195	7	weget	weget	NOUN
iajs-2430	195	8	:	:	PUNCT
iajs-2430	195	9	𝜇	𝜇	ADP
iajs-2430	195	10	τ	τ	PROPN
iajs-2430	195	11	∗	∗	X
iajs-2430	195	12	𝛾	𝛾	PROPN
iajs-2430	195	13	𝑟	𝑟	PRON
iajs-2430	195	14	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	195	15	𝜇	𝜇	ADP
iajs-2430	195	16	τ	τ	PROPN
iajs-2430	195	17	∗	∗	X
iajs-2430	195	18	𝜒	𝜒	PROPN
iajs-2430	195	19	∗	∗	NOUN
iajs-2430	195	20	𝛾	𝛾	NOUN
iajs-2430	195	21	,	,	PUNCT
iajs-2430	195	22	𝜇	𝜇	ADP
iajs-2430	195	23	τ	τ	PROPN
iajs-2430	195	24	∗	∗	X
iajs-2430	195	25	𝜒	𝜒	NOUN
iajs-2430	195	26	,	,	PUNCT
iajs-2430	195	27	put	put	VERB
iajs-2430	195	28	τ	τ	PROPN
iajs-2430	195	29	0	0	NUM
iajs-2430	195	30	,	,	PUNCT
iajs-2430	195	31	we	we	PRON
iajs-2430	195	32	get	get	VERB
iajs-2430	195	33	:	:	PUNCT
iajs-2430	195	34	𝜇	𝜇	NOUN
iajs-2430	195	35	0	0	NUM
iajs-2430	195	36	∗	∗	NOUN
iajs-2430	195	37	𝛾	𝛾	ADP
iajs-2430	195	38	𝑟	𝑟	PRON
iajs-2430	195	39	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	195	40	𝜇	𝜇	ADP
iajs-2430	195	41	0	0	NUM
iajs-2430	195	42	∗	∗	NOUN
iajs-2430	195	43	𝜒	𝜒	NUM
iajs-2430	195	44	∗	∗	NOUN
iajs-2430	195	45	𝛾	𝛾	NOUN
iajs-2430	195	46	,	,	PUNCT
iajs-2430	195	47	𝜇	𝜇	ADP
iajs-2430	195	48	0	0	NUM
iajs-2430	195	49	∗	∗	NOUN
iajs-2430	195	50	𝜒	𝜒	NOUN
iajs-2430	195	51	thus	thus	ADV
iajs-2430	195	52	𝜇	𝜇	ADP
iajs-2430	195	53	𝛾	𝛾	NOUN
iajs-2430	195	54	𝑟	𝑟	PRON
iajs-2430	195	55	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2430	195	56	𝜇	𝜇	ADP
iajs-2430	195	57	𝜒	𝜒	NOUN
iajs-2430	195	58	∗	∗	NOUN
iajs-2430	195	59	𝛾	𝛾	NOUN
iajs-2430	195	60	,	,	PUNCT
iajs-2430	195	61	𝜇	𝜇	ADP
iajs-2430	195	62	𝜒	𝜒	NOUN
iajs-2430	195	63	.	.	PUNCT
iajs-2430	196	1	the	the	DET
iajs-2430	196	2	reverse	reverse	NOUN
iajs-2430	196	3	of	of	ADP
iajs-2430	196	4	theorem~31is	theorem~31i	NOUN
iajs-2430	196	5	incorrect	incorrect	ADJ
iajs-2430	196	6	.	.	PUNCT
iajs-2430	197	1	the~	the~	DET
iajs-2430	197	2	example32~	example32~	NOUN
iajs-2430	197	3	shows	show	VERB
iajs-2430	197	4	~the	~the	DET
iajs-2430	197	5	reverse	reverse	NOUN
iajs-2430	197	6	.	.	PUNCT
iajs-2430	197	7	example	example	NOUN
iajs-2430	198	1	32	32	NUM
iajs-2430	198	2	.	.	PUNCT
iajs-2430	199	1	let	let	VERB
iajs-2430	199	2	ℵ	ℵ	ADP
iajs-2430	199	3	0	0	NUM
iajs-2430	199	4	,	,	PUNCT
iajs-2430	199	5	𝑎	𝑎	NOUN
iajs-2430	199	6	,	,	PUNCT
iajs-2430	199	7	𝑏	𝑏	DET
iajs-2430	199	8	beˑaˑset	beˑaˑset	NOUN
iajs-2430	199	9	.	.	PUNCT
iajs-2430	200	1	define	define	NOUN
iajs-2430	200	2	(	(	PUNCT
iajs-2430	200	3	∗-operation	∗-operation	NOUN
iajs-2430	200	4	)	)	PUNCT
iajs-2430	200	5	and	and	CCONJ
iajs-2430	200	6	(	(	PUNCT
iajs-2430	200	7	∘-operation	∘-operation	NOUN
iajs-2430	200	8	)	)	PUNCT
iajs-2430	200	9	by	by	ADP
iajs-2430	200	10	the	the	DET
iajs-2430	200	11	following	follow	VERB
iajs-2430	200	12	tables	table	NOUN
iajs-2430	200	13	then	then	ADV
iajs-2430	200	14	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2430	200	15	,	,	PUNCT
iajs-2430	200	16	0	0	PUNCT
iajs-2430	200	17	ˑis	ˑi	VERB
iajs-2430	200	18	a	a	DET
iajs-2430	200	19	ku	ku	PROPN
iajs-2430	200	20	-	-	PUNCT
iajs-2430	200	21	semigroup	semigroup	PROPN
iajs-2430	200	22	.	.	PUNCT
iajs-2430	201	1	define	define	VERB
iajs-2430	201	2	𝜇	𝜇	DET
iajs-2430	201	3	𝜒	𝜒	NOUN
iajs-2430	201	4	as	as	SCONJ
iajs-2430	201	5	follows	follow	VERB
iajs-2430	201	6	:	:	PUNCT
iajs-2430	201	7	𝜇	𝜇	ADP
iajs-2430	201	8	𝜒	𝜒	NOUN
iajs-2430	201	9	0.4,0.8	0.4,0.8	NUM
iajs-2430	201	10	𝑖𝑓	𝑖𝑓	NOUN
iajs-2430	201	11	𝜒	𝜒	NOUN
iajs-2430	201	12	0	0	NUM
iajs-2430	201	13	,	,	PUNCT
iajs-2430	201	14	𝑏	𝑏	PROPN
iajs-2430	201	15	0.1,0.3	0.1,0.3	PROPN
iajs-2430	201	16	𝑖𝑓	𝑖𝑓	VERB
iajs-2430	201	17	𝜒	𝜒	NUM
iajs-2430	201	18	𝑎	𝑎	NOUN
iajs-2430	201	19	.	.	PUNCT
iajs-2430	202	1	we	we	PRON
iajs-2430	202	2	can	can	AUX
iajs-2430	202	3	easily	easily	ADV
iajs-2430	202	4	prove	prove	VERB
iajs-2430	202	5	that~𝜇	that~𝜇	PRON
iajs-2430	202	6	is	be	AUX
iajs-2430	202	7	~an	~an	PUNCT
iajs-2430	202	8	interval~	interval~	PROPN
iajs-2430	202	9	valued	value	VERB
iajs-2430	202	10	fuzzy	fuzzy	ADJ
iajs-2430	202	11	~s	~s	NUM
iajs-2430	202	12	-	-	PUNCT
iajs-2430	202	13	idealof~ℵ	idealof~ℵ	NOUN
iajs-2430	202	14	,	,	PUNCT
iajs-2430	202	15	butˑ~it	butˑ~it	VERB
iajs-2430	202	16	~is	~is	NOUN
iajs-2430	202	17	not	not	PART
iajs-2430	202	18	an	an	DET
iajs-2430	202	19	interval~	interval~	PROPN
iajs-2430	202	20	valued	value	VERB
iajs-2430	202	21	fuzzy	fuzzy	ADJ
iajs-2430	202	22	p	p	NOUN
iajs-2430	202	23	-	-	PUNCT
iajs-2430	202	24	ideal	ideal	NOUN
iajs-2430	202	25	,	,	PUNCT
iajs-2430	202	26	since	since	SCONJ
iajs-2430	202	27	𝜇	𝜇	ADP
iajs-2430	202	28	0	0	NUM
iajs-2430	202	29	∗	∗	NOUN
iajs-2430	203	1	𝑎	𝑎	NUM
iajs-2430	203	2	0.1,0.3	0.1,0.3	PROPN
iajs-2430	203	3	𝑟	𝑟	NOUN
iajs-2430	203	4	min	min	NOUN
iajs-2430	203	5	𝜇	𝜇	ADP
iajs-2430	203	6	0	0	NUM
iajs-2430	203	7	∗	∗	NOUN
iajs-2430	203	8	𝑏	𝑏	PROPN
iajs-2430	203	9	∗	∗	NOUN
iajs-2430	203	10	𝑎	𝑎	NOUN
iajs-2430	203	11	,	,	PUNCT
iajs-2430	203	12	𝜇	𝜇	ADP
iajs-2430	203	13	0	0	NUM
iajs-2430	203	14	∗	∗	NOUN
iajs-2430	203	15	𝑏	𝑏	NOUN
iajs-2430	203	16	0.4,0.8	0.4,0.8	PROPN
iajs-2430	203	17	.	.	PUNCT
iajs-2430	204	1	4	4	X
iajs-2430	204	2	.	.	X
iajs-2430	204	3	study	study	NOUN
iajs-2430	204	4	of	of	ADP
iajs-2430	204	5	image	image	NOUN
iajs-2430	204	6	(	(	PUNCT
iajs-2430	204	7	pre-´image	pre-´image	NOUN
iajs-2430	204	8	)	)	PUNCT
iajs-2430	204	9	for	for	ADP
iajs-2430	204	10	interval	interval	NOUN
iajs-2430	204	11	valued	value	VERB
iajs-2430	204	12	´	´	NOUN
iajs-2430	205	1	fuzzy	fuzzy	ADJ
iajs-2430	205	2	k-´ideal	k-´ideal	NOUN
iajs-2430	205	3	the	the	DET
iajs-2430	205	4	image	image	NOUN
iajs-2430	205	5	and	and	CCONJ
iajs-2430	205	6	thepre-image	thepre-image	NOUN
iajs-2430	205	7	are	be	AUX
iajs-2430	205	8	important	important	ADJ
iajs-2430	205	9	topics	topic	NOUN
iajs-2430	205	10	in	in	ADP
iajs-2430	205	11	modern	modern	ADJ
iajs-2430	205	12	algebra	algebra	NOUN
iajs-2430	205	13	so	so	SCONJ
iajs-2430	205	14	we	we	PRON
iajs-2430	205	15	will	will	AUX
iajs-2430	205	16	focus	focus	VERB
iajs-2430	205	17	on	on	ADP
iajs-2430	205	18	these	these	DET
iajs-2430	205	19	two	two	NUM
iajs-2430	205	20	concepts	concept	NOUN
iajs-2430	205	21	in	in	ADP
iajs-2430	205	22	this	this	DET
iajs-2430	205	23	part	part	NOUN
iajs-2430	205	24	of	of	ADP
iajs-2430	205	25	our	our	PRON
iajs-2430	205	26	paper	paper	NOUN
iajs-2430	205	27	.	.	PUNCT
iajs-2430	206	1	we	we	PRON
iajs-2430	206	2	will	will	AUX
iajs-2430	206	3	study	study	VERB
iajs-2430	206	4	these	these	DET
iajs-2430	206	5	concepts	concept	NOUN
iajs-2430	206	6	with~	with~	VERB
iajs-2430	206	7	the	the	DET
iajs-2430	206	8	interval~	interval~	PROPN
iajs-2430	206	9	  	  	SPACE
iajs-2430	206	10	103	103	NUM
iajs-2430	206	11	  	  	SPACE
iajs-2430	206	12	ibn	ibn	PROPN
iajs-2430	206	13	al	al	PROPN
iajs-2430	206	14	-	-	PUNCT
iajs-2430	206	15	haitham	haitham	PROPN
iajs-2430	206	16	jour	jour	X
iajs-2430	206	17	.	.	PROPN
iajs-2430	207	1	for	for	ADP
iajs-2430	207	2	pure	pure	ADJ
iajs-2430	207	3	&	&	CCONJ
iajs-2430	207	4	appl	appl	PROPN
iajs-2430	207	5	.	.	PUNCT
iajs-2430	208	1	sci	sci	PROPN
iajs-2430	208	2	.	.	PROPN
iajs-2430	209	1	33	33	NUM
iajs-2430	209	2	(	(	PUNCT
iajs-2430	209	3	2	2	NUM
iajs-2430	209	4	)	)	PUNCT
iajs-2430	209	5	2020	2020	NUM
iajs-2430	209	6	valued	value	VERB
iajs-2430	209	7	fuzzy~k	fuzzy~k	NOUN
iajs-2430	209	8	-	-	PUNCT
iajs-2430	209	9	ideals	ideal	NOUN
iajs-2430	209	10	ina	ina	X
iajs-2430	209	11	ku-~semigroup~ℵˑunder~	ku-~semigroup~ℵˑunder~	PROPN
iajs-2430	209	12	homomorphism	homomorphism	PROPN
iajs-2430	209	13	.	.	PUNCT
iajs-2430	210	1	also	also	ADV
iajs-2430	210	2	,	,	PUNCT
iajs-2430	210	3	wewill	wewill	ADV
iajs-2430	210	4	prove	prove	VERB
iajs-2430	210	5	that	that	SCONJ
iajs-2430	210	6	the	the	DET
iajs-2430	210	7	products	product	NOUN
iajs-2430	210	8	of	of	ADP
iajs-2430	210	9	~interval~	~interval~	PUNCT
iajs-2430	210	10	valued~	valued~	VERB
iajs-2430	210	11	fuzzy	fuzzy	ADJ
iajs-2430	210	12	k-~ideals~	k-~ideals~	NOUN
iajs-2430	210	13	are	be	AUX
iajs-2430	210	14	a	a	DET
iajs-2430	210	15	fuzzy	fuzzy	ADJ
iajs-2430	210	16	~k-~ideal	~k-~ideal	NOUN
iajs-2430	210	17	of~	of~	PROPN
iajs-2430	210	18	a	a	DET
iajs-2430	210	19	ku	ku	PROPN
iajs-2430	210	20	-	-	PUNCT
iajs-2430	210	21	semigroup	semigroup	PROPN
iajs-2430	210	22	ℵˑ.	ℵˑ.	NOUN
iajs-2430	210	23	definition33	definition33	NOUN
iajs-2430	210	24	.	.	PUNCT
iajs-2430	211	1	let	let	VERB
iajs-2430	211	2	𝑓	𝑓	PRON
iajs-2430	211	3	:	:	PUNCT
iajs-2430	211	4	ℵ	ℵ	PROPN
iajs-2430	211	5	→	→	SYM
iajs-2430	211	6	𝑌	𝑌	PROPN
iajs-2430	211	7	be	be	VERB
iajs-2430	211	8	a	a	DET
iajs-2430	211	9	mapping	mapping	NOUN
iajs-2430	211	10	from	from	ADP
iajs-2430	211	11	ku	ku	PROPN
iajs-2430	211	12	-	-	PUNCT
iajs-2430	211	13	semigroup	semigroup	PROPN
iajs-2430	211	14	ℵ	ℵ	NOUN
iajs-2430	211	15	into	into	ADP
iajs-2430	211	16	ku	ku	PROPN
iajs-2430	211	17	-	-	PUNCT
iajs-2430	211	18	semigroup	semigroup	PROPN
iajs-2430	211	19	𝑌	𝑌	PROPN
iajs-2430	211	20	and	and	CCONJ
iajs-2430	211	21	𝜇	𝜇	SCONJ
iajs-2430	211	22	be	be	VERB
iajs-2430	211	23	an	an	DET
iajs-2430	211	24	interval	interval	NOUN
iajs-2430	211	25	valued	value	VERB
iajs-2430	211	26	fuzzy	fuzzy	ADJ
iajs-2430	211	27	subset	subset	NOUN
iajs-2430	211	28	of	of	ADP
iajs-2430	211	29	ℵ.	ℵ.	PROPN
iajs-2430	212	1	we	we	PRON
iajs-2430	212	2	define	define	VERB
iajs-2430	212	3	the	the	DET
iajs-2430	212	4	image	image	NOUN
iajs-2430	212	5	for	for	ADP
iajs-2430	212	6	𝜇	𝜇	ADP
iajs-2430	212	7	under𝑓	under𝑓	NOUN
iajs-2430	212	8	,	,	PUNCT
iajs-2430	212	9	denoted	denote	VERB
iajs-2430	212	10	by	by	ADP
iajs-2430	212	11	𝑓	𝑓	PRON
iajs-2430	212	12	(	(	PUNCT
iajs-2430	212	13	𝜇	𝜇	ADP
iajs-2430	212	14	as	as	SCONJ
iajs-2430	212	15	follows	follow	VERB
iajs-2430	212	16	𝑓	𝑓	PRON
iajs-2430	212	17	𝜇	𝜇	ADP
iajs-2430	212	18	𝛾	𝛾	ADP
iajs-2430	212	19	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	ADJ
iajs-2430	212	20	𝜒	𝜒	NOUN
iajs-2430	212	21	∈	∈	NOUN
iajs-2430	212	22	,	,	PUNCT
iajs-2430	212	23	𝑖𝑓	𝑖𝑓	ADP
iajs-2430	212	24	𝑓	𝑓	PRON
iajs-2430	212	25	𝛾	𝛾	NOUN
iajs-2430	212	26	𝜒	𝜒	PRON
iajs-2430	212	27	∈	∈	ADJ
iajs-2430	212	28	ℵ	ℵ	NOUN
iajs-2430	212	29	:	:	PUNCT
iajs-2430	212	30	𝑓	𝑓	DET
iajs-2430	212	31	𝜒	𝜒	NOUN
iajs-2430	212	32	𝛾	𝛾	NOUN
iajs-2430	212	33	∅	∅	NOUN
iajs-2430	212	34	0	0	NUM
iajs-2430	212	35	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2430	212	36	and	and	CCONJ
iajs-2430	212	37	the	the	DET
iajs-2430	212	38	pre	pre	NOUN
iajs-2430	212	39	-	-	NOUN
iajs-2430	212	40	image	image	NOUN
iajs-2430	212	41	for	for	ADP
iajs-2430	212	42	𝛽	𝛽	NOUN
iajs-2430	212	43	under	under	ADP
iajs-2430	212	44	f	f	NOUN
iajs-2430	212	45	,	,	PUNCT
iajs-2430	212	46	where	where	SCONJ
iajs-2430	212	47	𝛽	𝛽	NOUN
iajs-2430	212	48	is	be	AUX
iajs-2430	212	49	´	´	VERB
iajs-2430	212	50	an	an	DET
iajs-2430	212	51	interval	interval	NOUN
iajs-2430	212	52	´	´	NOUN
iajs-2430	212	53	valued	value	VERB
iajs-2430	212	54	fuzzy	fuzzy	ADJ
iajs-2430	212	55	´	´	NOUN
iajs-2430	212	56	subset	subset	NOUN
iajs-2430	212	57	of	of	ADP
iajs-2430	212	58	𝑌	𝑌	PROPN
iajs-2430	212	59	´	´	NOUN
iajs-2430	212	60	,	,	PUNCT
iajs-2430	212	61	denoted	denote	VERB
iajs-2430	212	62	´	´	NOUN
iajs-2430	212	63	by	by	ADP
iajs-2430	212	64	𝑓	𝑓	DET
iajs-2430	212	65	𝟏	𝟏	PROPN
iajs-2430	212	66	𝛽	𝛽	NOUN
iajs-2430	212	67	in	in	ADP
iajs-2430	212	68	ℵ	ℵ	NOUN
iajs-2430	212	69	by	by	ADP
iajs-2430	212	70	𝜇	𝜇	ADP
iajs-2430	212	71	𝜒	𝜒	NOUN
iajs-2430	212	72	𝑓	𝑓	PRON
iajs-2430	212	73	𝟏	𝟏	NUM
iajs-2430	212	74	𝛽	𝛽	NOUN
iajs-2430	212	75	𝛽	𝛽	NOUN
iajs-2430	212	76	𝑓	𝑓	ADV
iajs-2430	212	77	𝜒	𝜒	NOUN
iajs-2430	212	78	,	,	PUNCT
iajs-2430	212	79	∀	∀	X
iajs-2430	212	80	𝜒	𝜒	SYM
iajs-2430	212	81	∈	∈	NOUN
iajs-2430	212	82	ℵ.	ℵ.	NOUN
iajs-2430	212	83	lemma34.´let	lemma34.´let	VERB
iajs-2430	212	84	ƒ	ƒ	PRON
iajs-2430	212	85	be	be	AUX
iajs-2430	212	86	a	a	DET
iajs-2430	212	87	homomorphism	homomorphism	NOUN
iajs-2430	212	88	mapping	mapping	NOUN
iajs-2430	212	89	from	from	ADP
iajs-2430	212	90	a	a	DET
iajs-2430	212	91	ku	ku	PROPN
iajs-2430	212	92	-	-	PUNCT
iajs-2430	212	93	semigroup	semigroup	PROPN
iajs-2430	212	94	ℵ,∗,∘	ℵ,∗,∘	X
iajs-2430	212	95	,	,	PUNCT
iajs-2430	212	96	0	0	NUM
iajs-2430	212	97	into	into	ADP
iajs-2430	212	98	a	a	DET
iajs-2430	212	99	kusemigroup	kusemigroup	NOUN
iajs-2430	212	100	ℵ′,∗′,∘′	ℵ′,∗′,∘′	PROPN
iajs-2430	212	101	,	,	PUNCT
iajs-2430	212	102	0′	0′	NUM
iajs-2430	212	103	.	.	PUNCT
iajs-2430	213	1	then	then	ADV
iajs-2430	213	2	ƒ	ƒ	X
iajs-2430	213	3	𝟏	𝟏	PROPN
iajs-2430	213	4	𝛽	𝛽	NOUN
iajs-2430	213	5	is	be	AUX
iajs-2430	213	6	an	an	DET
iajs-2430	213	7	interval	interval	NOUN
iajs-2430	213	8	valued	value	VERB
iajs-2430	213	9	fuzzy	fuzzy	ADJ
iajs-2430	213	10	k´-ideal	k´-ideal	NOUN
iajs-2430	213	11	of	of	ADP
iajs-2430	213	12	ℵ	ℵ	NOUN
iajs-2430	213	13	,	,	PUNCT
iajs-2430	213	14	if	if	SCONJ
iajs-2430	213	15	the	the	DET
iajs-2430	213	16	mapping	mapping	NOUN
iajs-2430	213	17	𝛽	𝛽	NOUN
iajs-2430	213	18	is	be	AUX
iajs-2430	213	19	an	an	DET
iajs-2430	213	20	interval	interval	NOUN
iajs-2430	213	21	valued	value	VERB
iajs-2430	214	1	fuzzy	fuzzy	ADJ
iajs-2430	214	2	k	k	NOUN
iajs-2430	214	3	-	-	NOUN
iajs-2430	214	4	ideal	ideal	NOUN
iajs-2430	214	5	of	of	ADP
iajs-2430	214	6	ℵ′.~	ℵ′.~	PROPN
iajs-2430	214	7	?	?	PUNCT
iajs-2430	214	8	?	?	PUNCT
iajs-2430	215	1	proof	proof	NOUN
iajs-2430	215	2	.	.	PUNCT
iajs-2430	216	1	for	for	ADP
iajs-2430	216	2	all	all	DET
iajs-2430	216	3	𝜒	𝜒	NUM
iajs-2430	216	4	∈	∈	ADJ
iajs-2430	216	5	ℵ	ℵ	NOUN
iajs-2430	216	6	,	,	PUNCT
iajs-2430	216	7	we	we	PRON
iajs-2430	216	8	have	have	VERB
iajs-2430	216	9	𝜇	𝜇	ADP
iajs-2430	216	10	0	0	NUM
iajs-2430	216	11	=	=	SYM
iajs-2430	216	12	𝛽	𝛽	NOUN
iajs-2430	216	13	ƒ	ƒ	NUM
iajs-2430	216	14	0	0	PUNCT
iajs-2430	216	15	𝛽	𝛽	NOUN
iajs-2430	216	16	ƒ	ƒ	PRON
iajs-2430	216	17	𝜒	𝜒	NOUN
iajs-2430	216	18	𝜇	𝜇	X
iajs-2430	216	19	𝜒	𝜒	NOUN
iajs-2430	216	20	.	.	PUNCT
iajs-2430	217	1	let	let	AUX
iajs-2430	217	2	,	,	PUNCT
iajs-2430	217	3	𝛾	𝛾	PROPN
iajs-2430	217	4	,	,	PUNCT
iajs-2430	217	5	τ	τ	PROPN
iajs-2430	217	6	∈	∈	PROPN
iajs-2430	217	7	ℵ	ℵ	NOUN
iajs-2430	217	8	,	,	PUNCT
iajs-2430	217	9	then	then	ADV
iajs-2430	217	10	we	we	PRON
iajs-2430	217	11	have	have	VERB
iajs-2430	217	12	𝜇	𝜇	ADP
iajs-2430	217	13	𝜒	𝜒	NOUN
iajs-2430	217	14	∗	∗	NOUN
iajs-2430	217	15	τ	τ	X
iajs-2430	218	1	𝛽	𝛽	NOUN
iajs-2430	218	2	ƒ	ƒ	PRON
iajs-2430	218	3	𝜒	𝜒	X
iajs-2430	218	4	∗	∗	NOUN
iajs-2430	218	5	τ	τ	X
iajs-2430	218	6	𝛽	𝛽	NOUN
iajs-2430	218	7	ƒ	ƒ	PRON
iajs-2430	218	8	𝜒	𝜒	X
iajs-2430	218	9	∗	∗	NOUN
iajs-2430	218	10	𝑓	𝑓	SYM
iajs-2430	218	11	τ	τ	PROPN
iajs-2430	218	12	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	218	13	𝛽	𝛽	PROPN
iajs-2430	218	14	ƒ	ƒ	ADP
iajs-2430	218	15	𝜒	𝜒	X
iajs-2430	218	16	∗	∗	NOUN
iajs-2430	218	17	ƒ	ƒ	ADP
iajs-2430	218	18	𝛾	𝛾	NOUN
iajs-2430	218	19	∗	∗	NOUN
iajs-2430	218	20	ƒ	ƒ	X
iajs-2430	218	21	τ	τ	X
iajs-2430	218	22	,	,	PUNCT
iajs-2430	218	23	𝛽ƒ	𝛽ƒ	PROPN
iajs-2430	218	24	𝛾	𝛾	PROPN
iajs-2430	218	25	=	=	SYM
iajs-2430	218	26	r	r	NOUN
iajs-2430	218	27	min	min	NOUN
iajs-2430	218	28	𝛽	𝛽	NOUN
iajs-2430	218	29	ƒ	ƒ	PRON
iajs-2430	218	30	𝜒	𝜒	PRON
iajs-2430	218	31	∗	∗	NOUN
iajs-2430	218	32	𝛾	𝛾	ADP
iajs-2430	218	33	∗	∗	NOUN
iajs-2430	218	34	τ	τ	X
iajs-2430	218	35	,	,	PUNCT
iajs-2430	218	36	𝛽ƒ	𝛽ƒ	PROPN
iajs-2430	218	37	𝛾	𝛾	PROPN
iajs-2430	218	38	=	=	SYM
iajs-2430	218	39	r	r	NOUN
iajs-2430	218	40	min	min	NOUN
iajs-2430	218	41	𝜇	𝜇	ADP
iajs-2430	218	42	𝜒	𝜒	NOUN
iajs-2430	218	43	∗	∗	NOUN
iajs-2430	218	44	𝛾	𝛾	ADP
iajs-2430	218	45	∗	∗	NOUN
iajs-2430	218	46	τ	τ	X
iajs-2430	218	47	,	,	PUNCT
iajs-2430	218	48	𝜇	𝜇	ADP
iajs-2430	218	49	𝛾	𝛾	VERB
iajs-2430	218	50	also	also	ADV
iajs-2430	218	51	,	,	PUNCT
iajs-2430	218	52	we	we	PRON
iajs-2430	218	53	have	have	VERB
iajs-2430	218	54	𝜇	𝜇	ADP
iajs-2430	218	55	𝜒	𝜒	X
iajs-2430	218	56	∘	∘	NOUN
iajs-2430	218	57	𝛾	𝛾	ADP
iajs-2430	218	58	𝛽	𝛽	NOUN
iajs-2430	218	59	ƒ	ƒ	PRON
iajs-2430	218	60	𝜒	𝜒	DET
iajs-2430	218	61	∘	∘	NOUN
iajs-2430	218	62	𝛾	𝛾	ADP
iajs-2430	218	63	𝛽	𝛽	NOUN
iajs-2430	218	64	𝑓	𝑓	DET
iajs-2430	218	65	𝜒	𝜒	NOUN
iajs-2430	218	66	∘′	∘′	NOUN
iajs-2430	218	67	ƒ	ƒ	X
iajs-2430	218	68	τ	τ	PROPN
iajs-2430	218	69	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	218	70	𝛽	𝛽	PROPN
iajs-2430	218	71	ƒ	ƒ	X
iajs-2430	218	72	𝜒	𝜒	X
iajs-2430	218	73	,	,	PUNCT
iajs-2430	218	74	𝛽	𝛽	PROPN
iajs-2430	218	75	ƒ	ƒ	ADP
iajs-2430	218	76	𝛾	𝛾	ADP
iajs-2430	218	77	=	=	NOUN
iajs-2430	218	78	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	218	79	𝜇	𝜇	ADP
iajs-2430	218	80	𝜒	𝜒	NOUN
iajs-2430	218	81	,	,	PUNCT
iajs-2430	218	82	𝜇	𝜇	ADP
iajs-2430	218	83	𝛾	𝛾	NOUN
iajs-2430	218	84	.	.	PUNCT
iajs-2430	219	1	hence´the	hence´the	PROPN
iajs-2430	219	2	proof´is	proof´is	NOUN
iajs-2430	219	3	completed	complete	VERB
iajs-2430	219	4	.	.	PUNCT
iajs-2430	220	1	theorem(35	theorem(35	NOUN
iajs-2430	220	2	)	)	PUNCT
iajs-2430	220	3	.	.	PUNCT
iajs-2430	221	1	letƒ	letƒ	VERB
iajs-2430	221	2	:	:	PUNCT
iajs-2430	221	3	ℵ	ℵ	PROPN
iajs-2430	221	4	→	→	SYM
iajs-2430	221	5	ℵ′	ℵ′	PUNCT
iajs-2430	221	6	be	be	AUX
iajs-2430	221	7	an	an	DET
iajs-2430	221	8	epimorphism	epimorphism	NOUN
iajs-2430	221	9	between	between	ADP
iajs-2430	221	10	two	two	NUM
iajs-2430	221	11	ku	ku	PROPN
iajs-2430	221	12	-	-	PUNCT
iajs-2430	221	13	semigroups	semigroup	NOUN
iajs-2430	221	14	ℵ	ℵ	NOUN
iajs-2430	221	15	and	and	CCONJ
iajs-2430	221	16	ℵ'.ƒ	ℵ'.ƒ	PROPN
iajs-2430	221	17	𝜇	𝜇	ADP
iajs-2430	221	18	~is	~is	PROPN
iajs-2430	221	19	´	´	PROPN
iajs-2430	221	20	an~	an~	PROPN
iajs-2430	221	21	interval	interval	NOUN
iajs-2430	221	22	´	´	NOUN
iajs-2430	221	23	valued	value	VERB
iajs-2430	221	24	~fuzzy	~fuzzy	NUM
iajs-2430	221	25	´	´	NOUN
iajs-2430	221	26	k-~ideal	k-~ideal	NOUN
iajs-2430	221	27	of	of	ADP
iajs-2430	221	28	´	´	NOUN
iajs-2430	221	29	ℵ	ℵ	NOUN
iajs-2430	221	30	,	,	PUNCT
iajs-2430	221	31	if	if	SCONJ
iajs-2430	221	32	𝜇~is	𝜇~is	PROPN
iajs-2430	221	33	´	´	VERB
iajs-2430	221	34	an	an	DET
iajs-2430	221	35	~	~	NOUN
iajs-2430	221	36	interval	interval	NOUN
iajs-2430	221	37	´	´	NOUN
iajs-2430	221	38	valued	value	VERB
iajs-2430	221	39	~fuzzy´k-~ideal	~fuzzy´k-~ideal	PROPN
iajs-2430	221	40	ofℵ´~.µ	ofℵ´~.µ	PROPN
iajs-2430	221	41	proof	proof	NOUN
iajs-2430	221	42	.	.	PUNCT
iajs-2430	221	43	'	'	PUNCT
iajs-2430	222	1	µlet	µlet	NOUN
iajs-2430	222	2	𝜒′	𝜒′	NOUN
iajs-2430	222	3	,	,	PUNCT
iajs-2430	222	4	𝛾	𝛾	VERB
iajs-2430	222	5	′	′	NUM
iajs-2430	222	6	∈	∈	PROPN
iajs-2430	222	7	ℵ′	ℵ′	PUNCT
iajs-2430	222	8	,	,	PUNCT
iajs-2430	222	9	~then	~then	ADJ
iajs-2430	222	10	∃𝜒	∃𝜒	PROPN
iajs-2430	222	11	,	,	PUNCT
iajs-2430	222	12	𝛾	𝛾	ADP
iajs-2430	222	13	∈	∈	PROPN
iajs-2430	222	14	ℵ	ℵ	ADP
iajs-2430	222	15	such	such	ADJ
iajs-2430	222	16	that	that	SCONJ
iajs-2430	222	17	ƒ	ƒ	NUM
iajs-2430	222	18	𝜒	𝜒	SYM
iajs-2430	222	19	𝜒	𝜒	NOUN
iajs-2430	222	20	and	and	CCONJ
iajs-2430	222	21	ƒ	ƒ	NOUN
iajs-2430	222	22	𝛾	𝛾	NOUN
iajs-2430	222	23	𝛾	𝛾	NOUN
iajs-2430	222	24	.	.	PUNCT
iajs-2430	223	1	by	by	ADP
iajs-2430	223	2	definition	definition	NOUN
iajs-2430	223	3	of	of	ADP
iajs-2430	223	4	image	image	NOUN
iajs-2430	223	5	,	,	PUNCT
iajs-2430	223	6	we	we	PRON
iajs-2430	223	7	have	have	VERB
iajs-2430	223	8	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	223	9	𝜒′	𝜒′	NUM
iajs-2430	223	10	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	223	11	𝜒	𝜒	X
iajs-2430	223	12	∈ƒ	∈ƒ	NOUN
iajs-2430	223	13	𝟏	𝟏	NUM
iajs-2430	223	14	′	′	NOUN
iajs-2430	223	15	,	,	PUNCT
iajs-2430	223	16	for	for	ADP
iajs-2430	223	17	some	some	DET
iajs-2430	223	18	𝜒	𝜒	NUM
iajs-2430	223	19	∈	∈	ADJ
iajs-2430	223	20	ℵ	ℵ	NOUN
iajs-2430	223	21	,	,	PUNCT
iajs-2430	223	22	and	and	CCONJ
iajs-2430	223	23	,	,	PUNCT
iajs-2430	223	24	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	223	25	𝛾	𝛾	ADP
iajs-2430	223	26	′	′	NUM
iajs-2430	223	27	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	ADJ
iajs-2430	223	28	𝛾	𝛾	PART
iajs-2430	223	29	∈ƒ	∈ƒ	NOUN
iajs-2430	223	30	𝟏	𝟏	NUM
iajs-2430	223	31	′	′	NOUN
iajs-2430	223	32	,	,	PUNCT
iajs-2430	223	33	for	for	ADP
iajs-2430	223	34	some	some	PRON
iajs-2430	223	35	𝛾	𝛾	PROPN
iajs-2430	223	36	∈	∈	PRON
iajs-2430	223	37	ℵ.	ℵ.	PROPN
iajs-2430	223	38	weˑ	weˑ	PROPN
iajs-2430	223	39	have𝜇	have𝜇	PROPN
iajs-2430	223	40	(	(	PUNCT
iajs-2430	223	41	0	0	NUM
iajs-2430	223	42	)	)	PUNCT
iajs-2430	223	43	𝜇	𝜇	ADP
iajs-2430	223	44	𝜒	𝜒	NOUN
iajs-2430	223	45	,	,	PUNCT
iajs-2430	223	46	∀𝜒	∀𝜒	X
iajs-2430	223	47	∈	∈	PROPN
iajs-2430	223	48	ℵ.	ℵ.	PROPN
iajs-2430	223	49	then	then	ADV
iajs-2430	223	50	,	,	PUNCT
iajs-2430	223	51	(	(	PUNCT
iajs-2430	223	52	i	i	NOUN
iajs-2430	223	53	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	223	54	0′	0′	X
iajs-2430	224	1	=	=	NUM
iajs-2430	224	2	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	ADJ
iajs-2430	224	3	0	0	NUM
iajs-2430	224	4	∈ƒ	∈ƒ	NOUN
iajs-2430	224	5	𝟏	𝟏	NUM
iajs-2430	224	6	′	′	NUM
iajs-2430	224	7	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	224	8	𝜒	𝜒	X
iajs-2430	224	9	∈ƒ	∈ƒ	NOUN
iajs-2430	224	10	𝟏	𝟏	NUM
iajs-2430	224	11	′	′	NUM
iajs-2430	224	12	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	224	13	𝜒′	𝜒′	PROPN
iajs-2430	224	14	,	,	PUNCT
iajs-2430	224	15	for	for	ADP
iajs-2430	224	16	any	any	DET
iajs-2430	224	17	𝜒′	𝜒′	NUM
iajs-2430	224	18	∈	∈	NOUN
iajs-2430	224	19	ℵ.	ℵ.	NOUN
iajs-2430	224	20	(	(	PUNCT
iajs-2430	224	21	ii	ii	NOUN
iajs-2430	224	22	)	)	PUNCT
iajs-2430	224	23	for	for	ADP
iajs-2430	224	24	any𝜒′	any𝜒′	NOUN
iajs-2430	224	25	,	,	PUNCT
iajs-2430	224	26	𝛾	𝛾	NOUN
iajs-2430	224	27	′	′	NUM
iajs-2430	224	28	,	,	PUNCT
iajs-2430	224	29	τ′	τ′	X
iajs-2430	224	30	∈	∈	PROPN
iajs-2430	224	31	ℵ′	ℵ′	NUM
iajs-2430	224	32	,	,	PUNCT
iajs-2430	224	33	let	let	VERB
iajs-2430	224	34	𝜒	𝜒	NOUN
iajs-2430	224	35	∈	∈	NOUN
iajs-2430	224	36	ƒ	ƒ	PRON
iajs-2430	224	37	𝟏	𝟏	NUM
iajs-2430	224	38	𝜒′	𝜒′	NOUN
iajs-2430	224	39	,	,	PUNCT
iajs-2430	224	40	𝛾	𝛾	PROPN
iajs-2430	224	41	∈	∈	PROPN
iajs-2430	224	42	ƒ	ƒ	PRON
iajs-2430	224	43	𝟏	𝟏	NUM
iajs-2430	224	44	𝛾	𝛾	NOUN
iajs-2430	224	45	′	′	NUM
iajs-2430	224	46	,	,	PUNCT
iajs-2430	224	47	τ	τ	PROPN
iajs-2430	224	48	∈	∈	PROPN
iajs-2430	224	49	ƒ	ƒ	X
iajs-2430	224	50	𝟏	𝟏	NUM
iajs-2430	224	51	τ′	τ′	PUNCT
iajs-2430	224	52	,	,	PUNCT
iajs-2430	224	53	and	and	CCONJ
iajs-2430	224	54	since	since	SCONJ
iajs-2430	224	55	f	f	PROPN
iajs-2430	224	56	is	be	AUX
iajs-2430	224	57	ahomomorphism	ahomomorphism	ADJ
iajs-2430	224	58	,	,	PUNCT
iajs-2430	224	59	thenƒ(𝜇	thenƒ(𝜇	NOUN
iajs-2430	224	60	𝜒′	𝜒′	NOUN
iajs-2430	224	61	∗	∗	NOUN
iajs-2430	224	62	τ′	τ′	PUNCT
iajs-2430	225	1	=	=	AUX
iajs-2430	225	2	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	225	3	𝜒	𝜒	X
iajs-2430	225	4	∗	∗	NOUN
iajs-2430	225	5	τ	τ	NOUN
iajs-2430	226	1	∗τ	∗τ	PROPN
iajs-2430	226	2	∈ƒ	∈ƒ	NOUN
iajs-2430	226	3	𝟏	𝟏	NUM
iajs-2430	226	4	′∗	′∗	NUM
iajs-2430	226	5	τ′	τ′	PUNCT
iajs-2430	226	6	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	226	7	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	226	8	𝜒	𝜒	X
iajs-2430	226	9	∗	∗	NOUN
iajs-2430	226	10	𝛾	𝛾	ADP
iajs-2430	226	11	∗	∗	NOUN
iajs-2430	226	12	τ	τ	PROPN
iajs-2430	226	13	∗	∗	NOUN
iajs-2430	226	14	∗τ	∗τ	PROPN
iajs-2430	226	15	∈ƒ	∈ƒ	NOUN
iajs-2430	226	16	𝟏	𝟏	NUM
iajs-2430	226	17	′∗	′∗	NUM
iajs-2430	226	18	′∗	′∗	NUM
iajs-2430	226	19	τ′	τ′	X
iajs-2430	226	20	,	,	PUNCT
iajs-2430	226	21	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	226	22	𝛾	𝛾	PART
iajs-2430	226	23	∈ƒ	∈ƒ	NOUN
iajs-2430	226	24	𝟏	𝟏	NUM
iajs-2430	226	25	′	′	NUM
iajs-2430	227	1	=	=	NUM
iajs-2430	227	2	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	227	3	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	227	4	𝜒′	𝜒′	PROPN
iajs-2430	227	5	∗	∗	NOUN
iajs-2430	227	6	𝛾	𝛾	ADP
iajs-2430	227	7	′	′	NUM
iajs-2430	227	8	∗	∗	NOUN
iajs-2430	227	9	τ′	τ′	PUNCT
iajs-2430	227	10	,	,	PUNCT
iajs-2430	227	11	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	227	12	𝛾	𝛾	ADP
iajs-2430	227	13	′	′	NUM
iajs-2430	227	14	(	(	PUNCT
iajs-2430	227	15	iii	iii	NOUN
iajs-2430	227	16	)	)	PUNCT
iajs-2430	227	17	ˑforany	ˑforany	NOUN
iajs-2430	227	18	𝜒′	𝜒′	NOUN
iajs-2430	227	19	,	,	PUNCT
iajs-2430	227	20	𝛾	𝛾	VERB
iajs-2430	227	21	′	′	NUM
iajs-2430	227	22	∈	∈	PROPN
iajs-2430	227	23	ℵ′	ℵ′	ADV
iajs-2430	227	24	,	,	PUNCT
iajs-2430	227	25	let	let	VERB
iajs-2430	227	26	𝜒	𝜒	NOUN
iajs-2430	227	27	∈	∈	NOUN
iajs-2430	227	28	ƒ	ƒ	PRON
iajs-2430	227	29	𝟏	𝟏	NUM
iajs-2430	227	30	𝜒′	𝜒′	NOUN
iajs-2430	227	31	,	,	PUNCT
iajs-2430	228	1	𝛾	𝛾	AUX
iajs-2430	228	2	∈	∈	PROPN
iajs-2430	228	3	ƒ	ƒ	PRON
iajs-2430	228	4	𝟏	𝟏	NUM
iajs-2430	228	5	𝛾	𝛾	NOUN
iajs-2430	228	6	′	′	NOUN
iajs-2430	228	7	be	be	AUX
iajs-2430	228	8	such	such	ADJ
iajs-2430	228	9	that	that	SCONJ
iajs-2430	228	10	:	:	PUNCT
iajs-2430	228	11	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	228	12	𝜒′	𝜒′	NOUN
iajs-2430	228	13	∘	∘	NOUN
iajs-2430	228	14	𝛾	𝛾	ADP
iajs-2430	228	15	′	′	NOUN
iajs-2430	229	1	=	=	VERB
iajs-2430	229	2	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	229	3	𝜒	𝜒	X
iajs-2430	229	4	∘	∘	NOUN
iajs-2430	229	5	𝛾	𝛾	ADP
iajs-2430	229	6	∘	∘	PROPN
iajs-2430	229	7	τ	τ	X
iajs-2430	229	8	∈ƒ	∈ƒ	NOUN
iajs-2430	229	9	𝟏	𝟏	PUNCT
iajs-2430	229	10	′∘	′∘	PROPN
iajs-2430	229	11	τ′	τ′	PUNCT
iajs-2430	229	12	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	229	13	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	229	14	𝜒	𝜒	X
iajs-2430	229	15	∈ƒ	∈ƒ	NOUN
iajs-2430	229	16	𝟏	𝟏	NUM
iajs-2430	229	17	′	′	NUM
iajs-2430	229	18	,	,	PUNCT
iajs-2430	229	19	𝑠𝑢𝑝𝜇	𝑠𝑢𝑝𝜇	VERB
iajs-2430	229	20	𝛾	𝛾	PART
iajs-2430	229	21	∈ƒ	∈ƒ	NOUN
iajs-2430	229	22	𝟏	𝟏	NUM
iajs-2430	229	23	′	′	NUM
iajs-2430	229	24	  	  	SPACE
iajs-2430	229	25	104	104	NUM
iajs-2430	229	26	  	  	SPACE
iajs-2430	229	27	ibn	ibn	PROPN
iajs-2430	229	28	al	al	PROPN
iajs-2430	229	29	-	-	PUNCT
iajs-2430	229	30	haitham	haitham	PROPN
iajs-2430	229	31	jour	jour	X
iajs-2430	229	32	.	.	PROPN
iajs-2430	230	1	for	for	ADP
iajs-2430	230	2	pure	pure	ADJ
iajs-2430	230	3	&	&	CCONJ
iajs-2430	230	4	appl	appl	PROPN
iajs-2430	230	5	.	.	PUNCT
iajs-2430	231	1	sci	sci	PROPN
iajs-2430	231	2	.	.	PROPN
iajs-2430	232	1	33	33	NUM
iajs-2430	232	2	(	(	PUNCT
iajs-2430	232	3	2	2	NUM
iajs-2430	232	4	)	)	PUNCT
iajs-2430	232	5	2020	2020	NUM
iajs-2430	233	1	=	=	SYM
iajs-2430	233	2	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	233	3	{	{	PUNCT
iajs-2430	233	4	(	(	PUNCT
iajs-2430	233	5	𝜇	𝜇	ADP
iajs-2430	233	6	𝜒′	𝜒′	NOUN
iajs-2430	233	7	,	,	PUNCT
iajs-2430	233	8	ƒ(𝜇	ƒ(𝜇	PROPN
iajs-2430	233	9	𝛾	𝛾	ADP
iajs-2430	233	10	′	′	NOUN
iajs-2430	233	11	hence	hence	ADV
iajs-2430	233	12	´	´	VERB
iajs-2430	233	13	the	the	DET
iajs-2430	233	14	~proof	~proof	NUM
iajs-2430	233	15	´	´	NOUN
iajs-2430	233	16	is~	is~	PROPN
iajs-2430	233	17	completed	complete	VERB
iajs-2430	233	18	.	.	PUNCT
iajs-2430	234	1	definition	definition	NOUN
iajs-2430	234	2	(	(	PUNCT
iajs-2430	234	3	36	36	NUM
iajs-2430	234	4	)	)	PUNCT
iajs-2430	234	5	.	.	PUNCT
iajs-2430	235	1	if	if	SCONJ
iajs-2430	235	2	~𝜇	~𝜇	NUM
iajs-2430	235	3	and	and	CCONJ
iajs-2430	235	4	𝛽	𝛽	NOUN
iajs-2430	235	5	are	be	AUX
iajs-2430	235	6	´	´	NOUN
iajs-2430	235	7	two	two	NUM
iajs-2430	235	8	~interval	~interval	NOUN
iajs-2430	235	9	´	´	NOUN
iajs-2430	235	10	valued	value	VERB
iajs-2430	235	11	~fuzzy´subsets	~fuzzy´subset	NOUN
iajs-2430	235	12	of	of	ADP
iajs-2430	235	13	a	a	DET
iajs-2430	235	14	~set	~set	NOUN
iajs-2430	235	15	ℵ.	ℵ.	PROPN
iajs-2430	235	16	´	´	VERB
iajs-2430	236	1	then~	then~	NOUN
iajs-2430	236	2	the	the	DET
iajs-2430	236	3	product	product	NOUN
iajs-2430	236	4	of	of	ADP
iajs-2430	236	5	𝜇	𝜇	ADP
iajs-2430	236	6	and	and	CCONJ
iajs-2430	236	7	𝛽	𝛽	NOUN
iajs-2430	236	8	denoted	denote	VERB
iajs-2430	236	9	by	by	ADP
iajs-2430	236	10	𝜇	𝜇	ADP
iajs-2430	236	11	𝛽	𝛽	NOUN
iajs-2430	236	12	is	be	AUX
iajs-2430	236	13	defined	define	VERB
iajs-2430	236	14	by	by	ADP
iajs-2430	236	15	:	:	PUNCT
iajs-2430	236	16	𝜇	𝜇	ADP
iajs-2430	236	17	𝛽	𝛽	NOUN
iajs-2430	236	18	𝜒	𝜒	NOUN
iajs-2430	236	19	,	,	PUNCT
iajs-2430	236	20	𝛾	𝛾	ADP
iajs-2430	236	21	𝑟	𝑟	PRON
iajs-2430	236	22	min	min	NOUN
iajs-2430	236	23	𝜇	𝜇	ADP
iajs-2430	236	24	𝜒	𝜒	NOUN
iajs-2430	236	25	,	,	PUNCT
iajs-2430	236	26	𝛽	𝛽	NOUN
iajs-2430	236	27	𝛾	𝛾	NOUN
iajs-2430	236	28	,	,	PUNCT
iajs-2430	236	29	for	for	ADP
iajs-2430	236	30	all	all	DET
iajs-2430	236	31	𝜒	𝜒	NOUN
iajs-2430	236	32	,	,	PUNCT
iajs-2430	236	33	𝛾	𝛾	AUX
iajs-2430	236	34	∈	∈	PROPN
iajs-2430	236	35	ℵ	ℵ	NOUN
iajs-2430	236	36	ℵ.	ℵ.	PROPN
iajs-2430	236	37	theorem	theorem	ADJ
iajs-2430	236	38	(	(	PUNCT
iajs-2430	236	39	37	37	NUM
iajs-2430	236	40	)	)	PUNCT
iajs-2430	236	41	.	.	PUNCT
iajs-2430	237	1	the	the	DET
iajs-2430	237	2	product	product	NOUN
iajs-2430	237	3	𝜇	𝜇	ADP
iajs-2430	237	4	𝛽	𝛽	NOUN
iajs-2430	237	5	is~	is~	PROPN
iajs-2430	237	6	an	an	DET
iajs-2430	237	7	interval	interval	NOUN
iajs-2430	237	8	~valued	~value	VERB
iajs-2430	237	9	fuzzy	fuzzy	ADJ
iajs-2430	237	10	~k	~k	NOUN
iajs-2430	237	11	-	-	PUNCT
iajs-2430	237	12	ideal~	ideal~	PROPN
iajs-2430	237	13	of	of	ADP
iajs-2430	237	14	ℵ	ℵ	ADJ
iajs-2430	237	15	ℵ	ℵ	NOUN
iajs-2430	237	16	,	,	PUNCT
iajs-2430	237	17	if	if	SCONJ
iajs-2430	237	18	𝜇	𝜇	ADP
iajs-2430	237	19	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2430	237	20	𝛽~are~	𝛽~are~	PROPN
iajs-2430	237	21	interval	interval	NOUN
iajs-2430	237	22	valued	value	VERB
iajs-2430	237	23	~fuzzy	~fuzzy	NUM
iajs-2430	237	24	~k-´ideals	~k-´ideal	NOUN
iajs-2430	237	25	of	of	ADP
iajs-2430	237	26	~a	~a	PUNCT
iajs-2430	237	27	ku	ku	PROPN
iajs-2430	237	28	-	-	PUNCT
iajs-2430	237	29	semigroup	semigroup	PROPN
iajs-2430	237	30	ℵ.	ℵ.	PROPN
iajs-2430	237	31	proof	proof	NOUN
iajs-2430	237	32	.	.	PUNCT
iajs-2430	238	1	let	let	VERB
iajs-2430	238	2	𝜒	𝜒	NOUN
iajs-2430	238	3	,	,	PUNCT
iajs-2430	238	4	𝛾	𝛾	ADP
iajs-2430	238	5	∈	∈	NOUN
iajs-2430	238	6	ℵ	ℵ	NOUN
iajs-2430	238	7	ℵ	ℵ	NOUN
iajs-2430	238	8	,	,	PUNCT
iajs-2430	238	9	we	we	PRON
iajs-2430	238	10	have	have	VERB
iajs-2430	238	11	𝜇	𝜇	ADP
iajs-2430	238	12	𝛽	𝛽	NOUN
iajs-2430	238	13	0	0	NUM
iajs-2430	238	14	,	,	PUNCT
iajs-2430	238	15	0	0	NUM
iajs-2430	238	16	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	238	17	𝜇	𝜇	ADP
iajs-2430	238	18	0	0	NUM
iajs-2430	238	19	,	,	PUNCT
iajs-2430	238	20	𝛽	𝛽	PROPN
iajs-2430	238	21	0	0	NUM
iajs-2430	238	22	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	238	23	𝜇	𝜇	ADP
iajs-2430	238	24	𝜒	𝜒	NOUN
iajs-2430	238	25	,	,	PUNCT
iajs-2430	238	26	𝛽	𝛽	PROPN
iajs-2430	238	27	𝛾	𝛾	NOUN
iajs-2430	238	28	𝜇	𝜇	ADP
iajs-2430	238	29	𝛽	𝛽	NOUN
iajs-2430	238	30	𝜒	𝜒	NOUN
iajs-2430	238	31	,	,	PUNCT
iajs-2430	238	32	𝛾	𝛾	NOUN
iajs-2430	238	33	.	.	PUNCT
iajs-2430	239	1	now	now	ADV
iajs-2430	239	2	,	,	PUNCT
iajs-2430	239	3	let	let	VERB
iajs-2430	239	4	(	(	PUNCT
iajs-2430	239	5	𝜒	𝜒	X
iajs-2430	239	6	,	,	PUNCT
iajs-2430	239	7	,	,	PUNCT
iajs-2430	239	8	𝜒	𝜒	X
iajs-2430	239	9	,	,	PUNCT
iajs-2430	239	10	𝛾	𝛾	PROPN
iajs-2430	239	11	,	,	PUNCT
iajs-2430	239	12	𝛾	𝛾	PROPN
iajs-2430	239	13	,	,	PUNCT
iajs-2430	239	14	τ	τ	PROPN
iajs-2430	239	15	,	,	PUNCT
iajs-2430	239	16	τ	τ	PROPN
iajs-2430	239	17	∈	∈	PROPN
iajs-2430	239	18	ℵ	ℵ	ADP
iajs-2430	239	19	ℵ	ℵ	NOUN
iajs-2430	239	20	,	,	PUNCT
iajs-2430	239	21	then	then	ADV
iajs-2430	239	22	𝜇	𝜇	ADP
iajs-2430	239	23	𝛽	𝛽	ADP
iajs-2430	239	24	𝜒	𝜒	X
iajs-2430	239	25	∗	∗	X
iajs-2430	239	26	τ	τ	X
iajs-2430	239	27	,	,	PUNCT
iajs-2430	239	28	𝜒	𝜒	PROPN
iajs-2430	239	29	∗	∗	NOUN
iajs-2430	239	30	τ	τ	PROPN
iajs-2430	239	31	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	32	𝜇	𝜇	ADP
iajs-2430	239	33	𝜒	𝜒	X
iajs-2430	239	34	∗	∗	NOUN
iajs-2430	239	35	τ	τ	X
iajs-2430	239	36	,	,	PUNCT
iajs-2430	239	37	𝛽	𝛽	NOUN
iajs-2430	239	38	𝜒	𝜒	X
iajs-2430	239	39	∗	∗	X
iajs-2430	239	40	τ	τ	PROPN
iajs-2430	239	41	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	42	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	43	𝜇	𝜇	ADP
iajs-2430	239	44	𝜒	𝜒	X
iajs-2430	239	45	∗	∗	NOUN
iajs-2430	239	46	𝛾	𝛾	ADP
iajs-2430	239	47	∗	∗	NOUN
iajs-2430	239	48	τ	τ	X
iajs-2430	239	49	,	,	PUNCT
iajs-2430	239	50	𝜇	𝜇	ADP
iajs-2430	239	51	𝛾	𝛾	PROPN
iajs-2430	239	52	,	,	PUNCT
iajs-2430	239	53	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	54	𝛽	𝛽	NOUN
iajs-2430	239	55	𝜒	𝜒	NOUN
iajs-2430	239	56	∗	∗	NOUN
iajs-2430	239	57	𝛾	𝛾	ADP
iajs-2430	239	58	∗	∗	NOUN
iajs-2430	239	59	τ	τ	X
iajs-2430	239	60	,	,	PUNCT
iajs-2430	239	61	𝛽	𝛽	NOUN
iajs-2430	239	62	𝛾	𝛾	NOUN
iajs-2430	239	63	=	=	SYM
iajs-2430	239	64	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	239	65	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	66	𝜇	𝜇	ADP
iajs-2430	239	67	𝜒	𝜒	X
iajs-2430	239	68	∗	∗	NOUN
iajs-2430	239	69	𝛾	𝛾	ADP
iajs-2430	239	70	∗	∗	NOUN
iajs-2430	239	71	τ	τ	X
iajs-2430	239	72	,	,	PUNCT
iajs-2430	239	73	𝛽	𝛽	NOUN
iajs-2430	239	74	𝜒	𝜒	NOUN
iajs-2430	239	75	∗	∗	NOUN
iajs-2430	239	76	𝛾	𝛾	ADP
iajs-2430	239	77	∗	∗	NOUN
iajs-2430	239	78	τ	τ	X
iajs-2430	239	79	,	,	PUNCT
iajs-2430	239	80	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	81	𝜇	𝜇	ADP
iajs-2430	239	82	𝛾	𝛾	PROPN
iajs-2430	239	83	,	,	PUNCT
iajs-2430	239	84	𝛽	𝛽	NOUN
iajs-2430	239	85	𝛾	𝛾	NOUN
iajs-2430	239	86	=	=	NOUN
iajs-2430	239	87	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	239	88	𝜇	𝜇	ADP
iajs-2430	239	89	𝛽	𝛽	NOUN
iajs-2430	239	90	𝜒	𝜒	NOUN
iajs-2430	239	91	∗	∗	NOUN
iajs-2430	239	92	𝛾	𝛾	ADP
iajs-2430	239	93	∗	∗	NOUN
iajs-2430	239	94	τ	τ	X
iajs-2430	239	95	,	,	PUNCT
iajs-2430	239	96	𝜒	𝜒	PROPN
iajs-2430	239	97	∗	∗	NOUN
iajs-2430	239	98	𝛾	𝛾	ADP
iajs-2430	239	99	∗	∗	NOUN
iajs-2430	239	100	τ	τ	X
iajs-2430	239	101	,	,	PUNCT
iajs-2430	239	102	𝜇	𝜇	ADP
iajs-2430	239	103	𝛽	𝛽	NOUN
iajs-2430	239	104	𝛾	𝛾	NOUN
iajs-2430	239	105	,	,	PUNCT
iajs-2430	239	106	𝛾	𝛾	PROPN
iajs-2430	239	107	.	.	PUNCT
iajs-2430	240	1	and	and	CCONJ
iajs-2430	240	2	,	,	PUNCT
iajs-2430	240	3	𝜇	𝜇	ADP
iajs-2430	240	4	𝛽	𝛽	NOUN
iajs-2430	240	5	𝜒	𝜒	X
iajs-2430	240	6	∘	∘	NOUN
iajs-2430	240	7	𝜒	𝜒	NOUN
iajs-2430	240	8	𝛾	𝛾	NOUN
iajs-2430	240	9	∘	∘	NOUN
iajs-2430	240	10	𝛾	𝛾	ADP
iajs-2430	240	11	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	12	𝜇	𝜇	ADP
iajs-2430	240	13	𝜒	𝜒	X
iajs-2430	240	14	∘	∘	X
iajs-2430	240	15	𝜒	𝜒	NOUN
iajs-2430	240	16	,	,	PUNCT
iajs-2430	240	17	𝛽	𝛽	NOUN
iajs-2430	240	18	𝛾	𝛾	ADP
iajs-2430	240	19	∘	∘	NOUN
iajs-2430	240	20	𝛾	𝛾	ADP
iajs-2430	240	21	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	22	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	23	𝜇	𝜇	ADP
iajs-2430	240	24	𝜒	𝜒	NOUN
iajs-2430	240	25	,	,	PUNCT
iajs-2430	240	26	𝜇	𝜇	X
iajs-2430	240	27	𝜒	𝜒	NOUN
iajs-2430	240	28	,	,	PUNCT
iajs-2430	240	29	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	30	𝛽	𝛽	NOUN
iajs-2430	240	31	𝛾	𝛾	PROPN
iajs-2430	240	32	,	,	PUNCT
iajs-2430	240	33	𝛽	𝛽	NOUN
iajs-2430	240	34	𝛾	𝛾	NOUN
iajs-2430	240	35	=	=	SYM
iajs-2430	240	36	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	240	37	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	38	𝜇	𝜇	ADP
iajs-2430	240	39	𝜒	𝜒	NOUN
iajs-2430	240	40	,	,	PUNCT
iajs-2430	240	41	𝛽	𝛽	NOUN
iajs-2430	240	42	𝛾	𝛾	NOUN
iajs-2430	240	43	,	,	PUNCT
iajs-2430	240	44	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	45	𝜇	𝜇	ADP
iajs-2430	240	46	𝜒	𝜒	NOUN
iajs-2430	240	47	,	,	PUNCT
iajs-2430	240	48	𝛽	𝛽	NOUN
iajs-2430	240	49	𝛾	𝛾	NOUN
iajs-2430	240	50	=	=	NOUN
iajs-2430	240	51	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	240	52	𝜇	𝜇	ADP
iajs-2430	240	53	𝛽	𝛽	NOUN
iajs-2430	240	54	𝜒	𝜒	NOUN
iajs-2430	240	55	,	,	PUNCT
iajs-2430	240	56	𝛾	𝛾	NOUN
iajs-2430	240	57	,	,	PUNCT
iajs-2430	240	58	𝜇	𝜇	ADP
iajs-2430	240	59	𝛽	𝛽	NOUN
iajs-2430	240	60	𝜒	𝜒	NOUN
iajs-2430	240	61	,	,	PUNCT
iajs-2430	240	62	𝛾	𝛾	NOUN
iajs-2430	240	63	.	.	PUNCT
iajs-2430	241	1	definition´(38	definition´(38	PROPN
iajs-2430	241	2	)	)	PUNCT
iajs-2430	241	3	.	.	PUNCT
iajs-2430	242	1	let	let	VERB
iajs-2430	242	2	𝜇	𝜇	PART
iajs-2430	242	3	be	be	AUX
iajs-2430	242	4	an	an	DET
iajs-2430	242	5	interval~	interval~	PROPN
iajs-2430	242	6	valued	value	VERB
iajs-2430	242	7	fuzzy	fuzzy	ADJ
iajs-2430	242	8	set	set	VERB
iajs-2430	242	9	in	in	ADP
iajs-2430	242	10	ℵ.	ℵ.	PROPN
iajs-2430	242	11	if	if	SCONJ
iajs-2430	242	12	𝜇	𝜇	PRON
iajs-2430	242	13	´	´	PROPN
iajs-2430	242	14	is	be	AUX
iajs-2430	242	15	defined	define	VERB
iajs-2430	242	16	by	by	ADP
iajs-2430	242	17	:	:	PUNCT
iajs-2430	242	18	𝜇	𝜇	ADP
iajs-2430	242	19	:	:	PUNCT
iajs-2430	242	20	𝑆	𝑆	PROPN
iajs-2430	242	21	𝑆	𝑆	PROPN
iajs-2430	242	22	→	→	SYM
iajs-2430	242	23	𝐷	𝐷	NOUN
iajs-2430	242	24	0,1	0,1	NUM
iajs-2430	242	25	,	,	PUNCT
iajs-2430	242	26	then	then	ADV
iajs-2430	242	27	𝜇	𝜇	SCONJ
iajs-2430	242	28	´	´	NOUN
iajs-2430	242	29	is	be	AUX
iajs-2430	242	30	named	name	VERB
iajs-2430	242	31	an	an	DET
iajs-2430	242	32	interval~	interval~	PROPN
iajs-2430	242	33	valued	value	VERB
iajs-2430	242	34	fuzzy	fuzzy	ADJ
iajs-2430	242	35	relation	relation	NOUN
iajs-2430	242	36	on	on	ADP
iajs-2430	242	37	a	a	DET
iajs-2430	242	38	set	set	ADJ
iajs-2430	242	39	𝑆	𝑆	PROPN
iajs-2430	242	40	´	´	NOUN
iajs-2430	242	41	,	,	PUNCT
iajs-2430	242	42	where	where	SCONJ
iajs-2430	242	43	s⊆	s⊆	NOUN
iajs-2430	242	44	ℵ.	ℵ.	PROPN
iajs-2430	242	45	definition	definition	NOUN
iajs-2430	242	46	(	(	PUNCT
iajs-2430	242	47	39	39	NUM
iajs-2430	242	48	)	)	PUNCT
iajs-2430	242	49	.	.	PUNCT
iajs-2430	243	1	let	let	VERB
iajs-2430	243	2	𝛽	𝛽	NOUN
iajs-2430	243	3	be	be	AUX
iajs-2430	243	4	an	an	DET
iajs-2430	243	5	interval~	interval~	PROPN
iajs-2430	243	6	valued	value	VERB
iajs-2430	243	7	fuzzy	fuzzy	ADJ
iajs-2430	243	8	set	set	VERB
iajs-2430	243	9	in	in	ADP
iajs-2430	243	10	ℵ.	ℵ.	PROPN
iajs-2430	243	11	then	then	ADV
iajs-2430	243	12	the	the	DET
iajs-2430	243	13	strongest	strong	ADJ
iajs-2430	243	14	interval~	interval~	PROPN
iajs-2430	243	15	valued	value	VERB
iajs-2430	243	16	fuzzy	fuzzy	ADJ
iajs-2430	243	17	relation	relation	NOUN
iajs-2430	243	18	on	on	ADP
iajs-2430	243	19	ℵ	ℵ	NOUN
iajs-2430	243	20	by	by	ADP
iajs-2430	243	21	𝛽	𝛽	NOUN
iajs-2430	243	22	is	be	AUX
iajs-2430	243	23	denoted	denote	VERB
iajs-2430	243	24	by	by	ADP
iajs-2430	243	25	𝜇	𝜇	X
iajs-2430	243	26	and	and	CCONJ
iajs-2430	243	27	defined	define	VERB
iajs-2430	243	28	as	as	SCONJ
iajs-2430	243	29	follows	follow	VERB
iajs-2430	243	30	𝜇	𝜇	ADP
iajs-2430	243	31	𝜒	𝜒	NOUN
iajs-2430	243	32	,	,	PUNCT
iajs-2430	243	33	𝛾	𝛾	ADP
iajs-2430	243	34	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	243	35	𝛽	𝛽	NOUN
iajs-2430	243	36	𝜒	𝜒	X
iajs-2430	243	37	,	,	PUNCT
iajs-2430	243	38	𝛽	𝛽	NOUN
iajs-2430	243	39	𝛾	𝛾	NOUN
iajs-2430	243	40	,	,	PUNCT
iajs-2430	243	41	for	for	ADP
iajs-2430	243	42	all	all	DET
iajs-2430	243	43	𝜒	𝜒	NOUN
iajs-2430	243	44	,	,	PUNCT
iajs-2430	243	45	𝛾	𝛾	PROPN
iajs-2430	243	46	∈	∈	PRON
iajs-2430	244	1	ℵ.	ℵ.	PROPN
iajs-2430	244	2	lemma	lemma	PROPN
iajs-2430	244	3	(	(	PUNCT
iajs-2430	244	4	40	40	NUM
iajs-2430	244	5	)	)	PUNCT
iajs-2430	244	6	.	.	PUNCT
iajs-2430	245	1	if	if	SCONJ
iajs-2430	245	2	the	the	DET
iajs-2430	245	3	strongest	strong	ADJ
iajs-2430	245	4	interval~	interval~	PROPN
iajs-2430	245	5	valued	value	VERB
iajs-2430	245	6	fuzzy	fuzzy	ADJ
iajs-2430	245	7	relation	relation	NOUN
iajs-2430	245	8	on~ℵ	on~ℵ	VERB
iajs-2430	245	9	is	be	AUX
iajs-2430	245	10	´	´	NOUN
iajs-2430	245	11	an	an	DET
iajs-2430	245	12	interval	interval	NOUN
iajs-2430	245	13	''	''	PUNCT
iajs-2430	245	14	valued	value	VERB
iajs-2430	245	15	fuzzy	fuzzy	ADJ
iajs-2430	245	16	'	'	PUNCT
iajs-2430	245	17	k	k	NOUN
iajs-2430	245	18	-	-	NOUN
iajs-2430	245	19	ideal	ideal	NOUN
iajs-2430	245	20	of	of	ADP
iajs-2430	245	21	ℵ	ℵ	DET
iajs-2430	245	22	ℵ,´~then𝛽	ℵ,´~then𝛽	NUM
iajs-2430	245	23	𝜒	𝜒	PRON
iajs-2430	245	24	𝛽	𝛽	NOUN
iajs-2430	245	25	0	0	NUM
iajs-2430	245	26	,	,	PUNCT
iajs-2430	245	27	for	for	ADP
iajs-2430	245	28	all	all	DET
iajs-2430	245	29	𝜒	𝜒	NUM
iajs-2430	245	30	∈	∈	ADJ
iajs-2430	245	31	ℵ	ℵ	NOUN
iajs-2430	245	32	and	and	CCONJ
iajs-2430	245	33	𝛽	𝛽	NOUN
iajs-2430	245	34	is	be	AUX
iajs-2430	245	35	an	an	DET
iajs-2430	245	36	interval	interval	NOUN
iajs-2430	245	37	valued	value	VERB
iajs-2430	245	38	fuzzy	fuzzy	ADJ
iajs-2430	245	39	set	set	NOUN
iajs-2430	245	40	of	of	ADP
iajs-2430	245	41	a	a	DET
iajs-2430	245	42	ku	ku	NOUN
iajs-2430	245	43	-	-	PUNCT
iajs-2430	245	44	semigroupℵ.	semigroupℵ.	NOUN
iajs-2430	245	45	proof	proof	NOUN
iajs-2430	245	46	.	.	PUNCT
iajs-2430	246	1	´	´	NOUN
iajs-2430	246	2	let	let	VERB
iajs-2430	246	3	𝜇	𝜇	SCONJ
iajs-2430	246	4	´	´	NOUN
iajs-2430	246	5	´	´	NOUN
iajs-2430	246	6	be	be	VERB
iajs-2430	246	7	an	an	DET
iajs-2430	246	8	interval	interval	NOUN
iajs-2430	246	9	valued	value	VERB
iajs-2430	246	10	´	´	NOUN
iajs-2430	246	11	fuzzy	fuzzy	ADJ
iajs-2430	246	12	k-´ideal	k-´ideal	NOUN
iajs-2430	246	13	of	of	ADP
iajs-2430	246	14	ℵ	ℵ	NOUN
iajs-2430	246	15	ℵ	ℵ	NOUN
iajs-2430	246	16	´	´	NOUN
iajs-2430	246	17	,	,	PUNCT
iajs-2430	246	18	it	it	PRON
iajs-2430	246	19	follows	follow	VERB
iajs-2430	246	20	that	that	SCONJ
iajs-2430	246	21	:	:	PUNCT
iajs-2430	246	22	𝛽	𝛽	NOUN
iajs-2430	246	23	𝜒	𝜒	NUM
iajs-2430	246	24	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	246	25	𝛽	𝛽	NOUN
iajs-2430	246	26	𝜒	𝜒	NOUN
iajs-2430	246	27	,	,	PUNCT
iajs-2430	246	28	𝛽	𝛽	NOUN
iajs-2430	246	29	𝜒	𝜒	PRON
iajs-2430	246	30	𝜇	𝜇	X
iajs-2430	246	31	𝜒	𝜒	NOUN
iajs-2430	246	32	,	,	PUNCT
iajs-2430	246	33	𝜒	𝜒	SYM
iajs-2430	246	34	𝜇	𝜇	ADP
iajs-2430	246	35	0,0	0,0	NUM
iajs-2430	246	36	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	246	37	𝛽	𝛽	NOUN
iajs-2430	246	38	0	0	NUM
iajs-2430	246	39	,	,	PUNCT
iajs-2430	246	40	𝛽	𝛽	NOUN
iajs-2430	246	41	0	0	NUM
iajs-2430	246	42	𝛽	𝛽	NOUN
iajs-2430	246	43	0	0	NUM
iajs-2430	246	44	.	.	PUNCT
iajs-2430	247	1	hence	hence	ADV
iajs-2430	247	2	,	,	PUNCT
iajs-2430	247	3	𝛽	𝛽	PROPN
iajs-2430	247	4	𝜒	𝜒	NOUN
iajs-2430	247	5	𝛽	𝛽	NOUN
iajs-2430	247	6	0	0	NUM
iajs-2430	247	7	.	.	PUNCT
iajs-2430	247	8	  	  	SPACE
iajs-2430	248	1	105	105	NUM
iajs-2430	248	2	  	  	SPACE
iajs-2430	248	3	ibn	ibn	PROPN
iajs-2430	248	4	al	al	PROPN
iajs-2430	248	5	-	-	PUNCT
iajs-2430	248	6	haitham	haitham	PROPN
iajs-2430	248	7	jour	jour	X
iajs-2430	248	8	.	.	PROPN
iajs-2430	248	9	for	for	ADP
iajs-2430	248	10	pure	pure	ADJ
iajs-2430	248	11	&	&	CCONJ
iajs-2430	248	12	appl	appl	PROPN
iajs-2430	248	13	.	.	PUNCT
iajs-2430	249	1	sci	sci	PROPN
iajs-2430	249	2	.	.	PROPN
iajs-2430	250	1	33	33	NUM
iajs-2430	250	2	(	(	PUNCT
iajs-2430	250	3	2	2	NUM
iajs-2430	250	4	)	)	PUNCT
iajs-2430	250	5	2020	2020	NUM
iajs-2430	250	6	theorem(41	theorem(41	NOUN
iajs-2430	250	7	)	)	PUNCT
iajs-2430	250	8	.	.	PUNCT
iajs-2430	251	1	the	the	DET
iajs-2430	251	2	strongest´~	strongest´~	PROPN
iajs-2430	251	3	interval	interval	NOUN
iajs-2430	251	4	valued	value	VERB
iajs-2430	251	5	fuzzy	fuzzy	ADJ
iajs-2430	251	6	relation~𝜇	relation~𝜇	NOUN
iajs-2430	251	7	on~ℵ	on~ℵ	NUM
iajs-2430	251	8	is	be	AUX
iajs-2430	251	9	an	an	DET
iajs-2430	251	10	interval	interval	NOUN
iajs-2430	251	11	'	'	PUNCT
iajs-2430	251	12	valued	value	VERB
iajs-2430	251	13	fuzzy	fuzzy	ADJ
iajs-2430	251	14	k''-ideal	k''-ideal	PROPN
iajs-2430	251	15	of´~ℵ	of´~ℵ	ADP
iajs-2430	251	16	ℵ	ℵ	PRON
iajs-2430	251	17	iff	iff	PROPN
iajs-2430	251	18	the	the	DET
iajs-2430	251	19	mapping	mapping	NOUN
iajs-2430	252	1	𝛽~is	𝛽~is	PROPN
iajs-2430	252	2	´	´	NOUN
iajs-2430	252	3	an	an	DET
iajs-2430	252	4	~	~	PUNCT
iajs-2430	252	5	interval~´valued	interval~´value	VERB
iajs-2430	252	6	~	~	SYM
iajs-2430	252	7	fuzzy''k	fuzzy''k	PROPN
iajs-2430	252	8	-	-	PUNCT
iajs-2430	252	9	ideal	ideal	NOUN
iajs-2430	252	10	of´~ℵ.	of´~ℵ.	PROPN
iajs-2430	252	11	?	?	PUNCT
iajs-2430	252	12	?	?	PUNCT
iajs-2430	252	13	?	?	PUNCT
iajs-2430	252	14	?	?	PUNCT
iajs-2430	253	1	proof	proof	NOUN
iajs-2430	253	2	.	.	PUNCT
iajs-2430	254	1	(	(	PUNCT
iajs-2430	254	2	)~since	)~since	NOUN
iajs-2430	254	3	𝛽~is	𝛽~is	PROPN
iajs-2430	254	4	an	an	DET
iajs-2430	254	5	interval	interval	NOUN
iajs-2430	254	6	valued	value	VERB
iajs-2430	254	7	fuzzy~´k	fuzzy~´k	ADJ
iajs-2430	254	8	-	-	PUNCT
iajs-2430	254	9	ideal	ideal	ADJ
iajs-2430	254	10	of~ℵ~	of~ℵ~	ADJ
iajs-2430	254	11	´	´	NOUN
iajs-2430	254	12	,	,	PUNCT
iajs-2430	254	13	~then	~then	ADP
iajs-2430	254	14	𝜇	𝜇	ADP
iajs-2430	254	15	0,0	0,0	NUM
iajs-2430	254	16	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	254	17	𝛽	𝛽	NOUN
iajs-2430	254	18	0	0	NUM
iajs-2430	254	19	,	,	PUNCT
iajs-2430	254	20	𝛽	𝛽	PROPN
iajs-2430	254	21	0	0	NUM
iajs-2430	254	22	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	254	23	𝛽	𝛽	NOUN
iajs-2430	254	24	𝜒	𝜒	X
iajs-2430	254	25	,	,	PUNCT
iajs-2430	254	26	𝛽	𝛽	NOUN
iajs-2430	254	27	𝛾	𝛾	NOUN
iajs-2430	254	28	𝜇	𝜇	ADP
iajs-2430	254	29	𝜒	𝜒	NOUN
iajs-2430	254	30	,	,	PUNCT
iajs-2430	254	31	𝛾	𝛾	NOUN
iajs-2430	254	32	.	.	PUNCT
iajs-2430	255	1	now	now	ADV
iajs-2430	255	2	,	,	PUNCT
iajs-2430	255	3	for	for	ADP
iajs-2430	255	4	any(𝜒	any(𝜒	PROPN
iajs-2430	255	5	,	,	PUNCT
iajs-2430	255	6	,	,	PUNCT
iajs-2430	255	7	𝜒	𝜒	NOUN
iajs-2430	255	8	𝛾	𝛾	NOUN
iajs-2430	255	9	,	,	PUNCT
iajs-2430	255	10	𝛾	𝛾	PROPN
iajs-2430	255	11	τ	τ	PROPN
iajs-2430	255	12	,	,	PUNCT
iajs-2430	255	13	τ	τ	PROPN
iajs-2430	255	14	∈	∈	PROPN
iajs-2430	255	15	ℵ	ℵ	ADP
iajs-2430	255	16	ℵ	ℵ	NOUN
iajs-2430	255	17	´	´	NOUN
iajs-2430	255	18	,	,	PUNCT
iajs-2430	255	19	´	´	NOUN
iajs-2430	255	20	we	we	PRON
iajs-2430	255	21	have	have	VERB
iajs-2430	255	22	´	´	NOUN
iajs-2430	255	23	:	:	PUNCT
iajs-2430	255	24	𝜇	𝜇	ADP
iajs-2430	255	25	𝜒	𝜒	X
iajs-2430	255	26	,	,	PUNCT
iajs-2430	255	27	∗	∗	X
iajs-2430	255	28	τ	τ	PROPN
iajs-2430	255	29	,	,	PUNCT
iajs-2430	255	30	𝜒	𝜒	PROPN
iajs-2430	255	31	∗	∗	NOUN
iajs-2430	255	32	τ	τ	PROPN
iajs-2430	255	33	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	255	34	𝛽	𝛽	NOUN
iajs-2430	255	35	𝜒	𝜒	X
iajs-2430	255	36	,	,	PUNCT
iajs-2430	255	37	∗	∗	X
iajs-2430	255	38	τ	τ	X
iajs-2430	255	39	,	,	PUNCT
iajs-2430	255	40	𝛽	𝛽	NOUN
iajs-2430	255	41	𝜒	𝜒	X
iajs-2430	255	42	∗	∗	X
iajs-2430	255	43	τ	τ	PROPN
iajs-2430	255	44	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	255	45	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	255	46	𝛽	𝛽	NOUN
iajs-2430	255	47	𝜒	𝜒	X
iajs-2430	255	48	,	,	PUNCT
iajs-2430	255	49	∗	∗	NOUN
iajs-2430	255	50	𝛾	𝛾	ADP
iajs-2430	255	51	∗	∗	NOUN
iajs-2430	255	52	τ	τ	X
iajs-2430	255	53	,	,	PUNCT
iajs-2430	255	54	𝛽	𝛽	PROPN
iajs-2430	255	55	𝛾	𝛾	NOUN
iajs-2430	255	56	,	,	PUNCT
iajs-2430	255	57	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	255	58	𝛽	𝛽	NOUN
iajs-2430	255	59	𝜒	𝜒	X
iajs-2430	255	60	,	,	PUNCT
iajs-2430	255	61	∗	∗	NOUN
iajs-2430	255	62	𝛾	𝛾	ADP
iajs-2430	255	63	∗	∗	NOUN
iajs-2430	255	64	τ	τ	X
iajs-2430	255	65	,	,	PUNCT
iajs-2430	256	1	𝛽	𝛽	NOUN
iajs-2430	256	2	𝛾	𝛾	NOUN
iajs-2430	256	3	=	=	SYM
iajs-2430	256	4	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	256	5	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	256	6	𝛽	𝛽	NOUN
iajs-2430	256	7	𝜒	𝜒	X
iajs-2430	256	8	,	,	PUNCT
iajs-2430	256	9	∗	∗	NOUN
iajs-2430	256	10	𝛾	𝛾	ADP
iajs-2430	256	11	∗	∗	NOUN
iajs-2430	256	12	τ	τ	X
iajs-2430	256	13	,	,	PUNCT
iajs-2430	256	14	𝛽	𝛽	NOUN
iajs-2430	256	15	𝜒	𝜒	NOUN
iajs-2430	256	16	,	,	PUNCT
iajs-2430	256	17	∗	∗	NOUN
iajs-2430	256	18	𝛾	𝛾	ADP
iajs-2430	256	19	∗	∗	NOUN
iajs-2430	256	20	τ	τ	PROPN
iajs-2430	256	21	,	,	PUNCT
iajs-2430	256	22	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	256	23	𝛽	𝛽	NOUN
iajs-2430	256	24	𝛾	𝛾	PROPN
iajs-2430	256	25	,	,	PUNCT
iajs-2430	256	26	𝛽	𝛽	NOUN
iajs-2430	256	27	𝛾	𝛾	NOUN
iajs-2430	256	28	=	=	NOUN
iajs-2430	256	29	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	256	30	𝜇	𝜇	ADP
iajs-2430	256	31	𝜒	𝜒	X
iajs-2430	256	32	,	,	PUNCT
iajs-2430	256	33	∗	∗	NOUN
iajs-2430	256	34	𝛾	𝛾	ADP
iajs-2430	256	35	∗	∗	NOUN
iajs-2430	256	36	τ	τ	PROPN
iajs-2430	256	37	,	,	PUNCT
iajs-2430	256	38	𝜒	𝜒	X
iajs-2430	256	39	,	,	PUNCT
iajs-2430	256	40	∗	∗	NOUN
iajs-2430	256	41	𝛾	𝛾	ADP
iajs-2430	256	42	∗	∗	NOUN
iajs-2430	256	43	τ	τ	X
iajs-2430	256	44	,	,	PUNCT
iajs-2430	256	45	𝜇	𝜇	ADP
iajs-2430	256	46	𝛾	𝛾	NOUN
iajs-2430	256	47	,	,	PUNCT
iajs-2430	256	48	𝛾	𝛾	PROPN
iajs-2430	256	49	.	.	PUNCT
iajs-2430	257	1	and	and	CCONJ
iajs-2430	257	2	,	,	PUNCT
iajs-2430	257	3	𝜇	𝜇	ADP
iajs-2430	257	4	𝜒	𝜒	X
iajs-2430	257	5	,	,	PUNCT
iajs-2430	257	6	∘	∘	X
iajs-2430	257	7	𝜒	𝜒	NOUN
iajs-2430	257	8	,	,	PUNCT
iajs-2430	257	9	𝛾	𝛾	ADP
iajs-2430	257	10	∘	∘	PROPN
iajs-2430	257	11	𝛾	𝛾	ADP
iajs-2430	257	12	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	257	13	𝛽	𝛽	NOUN
iajs-2430	257	14	𝜒	𝜒	PRON
iajs-2430	257	15	∘	∘	X
iajs-2430	257	16	𝜒	𝜒	NOUN
iajs-2430	257	17	,	,	PUNCT
iajs-2430	257	18	𝛽	𝛽	NOUN
iajs-2430	257	19	𝛾	𝛾	ADP
iajs-2430	257	20	∘	∘	NOUN
iajs-2430	257	21	𝛾	𝛾	ADP
iajs-2430	257	22	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	257	23	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	257	24	𝛽	𝛽	NOUN
iajs-2430	257	25	𝜒	𝜒	X
iajs-2430	257	26	,	,	PUNCT
iajs-2430	257	27	,	,	PUNCT
iajs-2430	257	28	𝛽	𝛽	NOUN
iajs-2430	257	29	𝜒	𝜒	NOUN
iajs-2430	257	30	,	,	PUNCT
iajs-2430	257	31	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	257	32	𝛽	𝛽	NOUN
iajs-2430	257	33	𝛾	𝛾	PROPN
iajs-2430	257	34	,	,	PUNCT
iajs-2430	257	35	,	,	PUNCT
iajs-2430	257	36	𝛽	𝛽	NOUN
iajs-2430	257	37	𝛾	𝛾	NOUN
iajs-2430	257	38	=	=	NOUN
iajs-2430	257	39	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	257	40	𝜇	𝜇	ADP
iajs-2430	257	41	𝜒	𝜒	X
iajs-2430	257	42	,	,	PUNCT
iajs-2430	257	43	𝜒	𝜒	X
iajs-2430	257	44	,	,	PUNCT
iajs-2430	257	45	𝜇	𝜇	ADP
iajs-2430	257	46	𝛾	𝛾	NOUN
iajs-2430	257	47	,	,	PUNCT
iajs-2430	257	48	𝛾	𝛾	PROPN
iajs-2430	257	49	.	.	PUNCT
iajs-2430	258	1	(	(	PUNCT
iajs-2430	258	2			NOUN
iajs-2430	258	3	)	)	PUNCT
iajs-2430	258	4	for	for	ADP
iajs-2430	258	5	all	all	PRON
iajs-2430	258	6	,	,	PUNCT
iajs-2430	258	7	𝜒	𝜒	X
iajs-2430	258	8	,	,	PUNCT
iajs-2430	258	9	𝛾	𝛾	PROPN
iajs-2430	258	10	∈	∈	PROPN
iajs-2430	258	11	ℵ	ℵ	PRON
iajs-2430	258	12	ℵ~	ℵ~	PROPN
iajs-2430	258	13	,	,	PUNCT
iajs-2430	258	14	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	258	15	𝛽	𝛽	NOUN
iajs-2430	258	16	0	0	NUM
iajs-2430	258	17	,	,	PUNCT
iajs-2430	258	18	𝛽	𝛽	PROPN
iajs-2430	258	19	0	0	NUM
iajs-2430	258	20	𝜇	𝜇	ADP
iajs-2430	258	21	0,0	0,0	NUM
iajs-2430	258	22	𝜇	𝜇	ADP
iajs-2430	258	23	𝜒	𝜒	NOUN
iajs-2430	258	24	,	,	PUNCT
iajs-2430	258	25	𝛾	𝛾	ADP
iajs-2430	258	26	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	258	27	𝛽	𝛽	NOUN
iajs-2430	258	28	𝜒	𝜒	X
iajs-2430	258	29	,	,	PUNCT
iajs-2430	258	30	𝛽	𝛽	NOUN
iajs-2430	258	31	𝛾	𝛾	NOUN
iajs-2430	258	32	.~	.~	NOUN
iajs-2430	258	33	then	then	ADV
iajs-2430	258	34	𝛽	𝛽	NOUN
iajs-2430	258	35	0	0	PUNCT
iajs-2430	258	36	𝛽	𝛽	NOUN
iajs-2430	258	37	𝜒	𝜒	PRON
iajs-2430	258	38	´	´	NOUN
iajs-2430	258	39	,	,	PUNCT
iajs-2430	258	40	∀𝜒	∀𝜒	X
iajs-2430	258	41	∈	∈	PROPN
iajs-2430	258	42	ℵ.	ℵ.	PROPN
iajs-2430	259	1	now	now	ADV
iajs-2430	259	2	,	,	PUNCT
iajs-2430	259	3	let	let	VERB
iajs-2430	259	4	𝜒	𝜒	NOUN
iajs-2430	259	5	,	,	PUNCT
iajs-2430	259	6	𝜒	𝜒	NOUN
iajs-2430	259	7	𝛾	𝛾	NOUN
iajs-2430	259	8	,	,	PUNCT
iajs-2430	259	9	𝛾	𝛾	PROPN
iajs-2430	259	10	τ	τ	PROPN
iajs-2430	259	11	,	,	PUNCT
iajs-2430	259	12	τ	τ	PROPN
iajs-2430	259	13	∈	∈	PROPN
iajs-2430	259	14	ℵ	ℵ	NOUN
iajs-2430	259	15	ℵ.	ℵ.	PROPN
iajs-2430	259	16	then	then	ADV
iajs-2430	259	17	´	´	VERB
iajs-2430	259	18	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	259	19	𝛽	𝛽	NOUN
iajs-2430	259	20	𝜒	𝜒	X
iajs-2430	259	21	∗	∗	X
iajs-2430	259	22	τ	τ	X
iajs-2430	259	23	,	,	PUNCT
iajs-2430	260	1	𝛽	𝛽	NOUN
iajs-2430	260	2	𝜒	𝜒	X
iajs-2430	260	3	∗	∗	NOUN
iajs-2430	260	4	τ	τ	X
iajs-2430	260	5	𝜇	𝜇	ADP
iajs-2430	260	6	𝜒	𝜒	X
iajs-2430	260	7	∗	∗	NOUN
iajs-2430	260	8	τ	τ	X
iajs-2430	260	9	,	,	PUNCT
iajs-2430	260	10	𝜒	𝜒	PROPN
iajs-2430	260	11	∗	∗	NOUN
iajs-2430	260	12	τ	τ	PROPN
iajs-2430	260	13	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	14	𝜇	𝜇	ADP
iajs-2430	260	15	𝜒	𝜒	X
iajs-2430	260	16	∗	∗	NOUN
iajs-2430	260	17	τ	τ	X
iajs-2430	260	18	,	,	PUNCT
iajs-2430	260	19	𝜇	𝜇	ADP
iajs-2430	260	20	𝜒	𝜒	X
iajs-2430	260	21	∗	∗	X
iajs-2430	260	22	τ	τ	X
iajs-2430	260	23	=	=	SYM
iajs-2430	260	24	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	260	25	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	26	𝜇	𝜇	ADP
iajs-2430	260	27	𝜒	𝜒	X
iajs-2430	260	28	∗	∗	NOUN
iajs-2430	260	29	𝛾	𝛾	ADP
iajs-2430	260	30	∗	∗	NOUN
iajs-2430	260	31	τ	τ	X
iajs-2430	260	32	,	,	PUNCT
iajs-2430	260	33	𝜇	𝜇	ADP
iajs-2430	260	34	𝛾	𝛾	PROPN
iajs-2430	260	35	,	,	PUNCT
iajs-2430	260	36	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	37	𝜇	𝜇	ADP
iajs-2430	260	38	𝜒	𝜒	X
iajs-2430	260	39	∗	∗	NOUN
iajs-2430	260	40	𝛾	𝛾	ADP
iajs-2430	260	41	∗	∗	NOUN
iajs-2430	260	42	τ	τ	X
iajs-2430	260	43	,	,	PUNCT
iajs-2430	260	44	𝜇	𝜇	ADP
iajs-2430	260	45	𝛾	𝛾	ADP
iajs-2430	260	46	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	47	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	48	𝜇	𝜇	ADP
iajs-2430	260	49	𝜒	𝜒	NOUN
iajs-2430	260	50	∗	∗	NOUN
iajs-2430	260	51	𝛾	𝛾	ADP
iajs-2430	260	52	∗	∗	NOUN
iajs-2430	260	53	τ	τ	X
iajs-2430	260	54	,	,	PUNCT
iajs-2430	260	55	𝜇	𝜇	ADP
iajs-2430	260	56	𝜒	𝜒	X
iajs-2430	260	57	,	,	PUNCT
iajs-2430	260	58	∗	∗	NOUN
iajs-2430	260	59	𝛾	𝛾	ADP
iajs-2430	260	60	∗	∗	NOUN
iajs-2430	260	61	τ	τ	X
iajs-2430	260	62	,	,	PUNCT
iajs-2430	260	63	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	64	𝜇	𝜇	ADP
iajs-2430	260	65	𝛾	𝛾	PROPN
iajs-2430	260	66	,	,	PUNCT
iajs-2430	260	67	𝜇	𝜇	ADP
iajs-2430	260	68	𝛾	𝛾	ADP
iajs-2430	260	69	=	=	NOUN
iajs-2430	260	70	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	71	𝑟𝑚𝑖𝑛𝜇	𝑟𝑚𝑖𝑛𝜇	NOUN
iajs-2430	260	72	𝜒	𝜒	NUM
iajs-2430	260	73	∗	∗	NOUN
iajs-2430	260	74	𝛾	𝛾	ADP
iajs-2430	260	75	∗	∗	NOUN
iajs-2430	260	76	τ	τ	X
iajs-2430	260	77	,	,	PUNCT
iajs-2430	260	78	𝜒	𝜒	PROPN
iajs-2430	260	79	∗	∗	NOUN
iajs-2430	260	80	𝛾	𝛾	ADP
iajs-2430	260	81	∗	∗	NOUN
iajs-2430	260	82	τ	τ	X
iajs-2430	260	83	,	,	PUNCT
iajs-2430	260	84	𝜇	𝜇	ADP
iajs-2430	260	85	𝛾	𝛾	NOUN
iajs-2430	260	86	,	,	PUNCT
iajs-2430	260	87	𝛾	𝛾	PROPN
iajs-2430	260	88	=	=	NOUN
iajs-2430	260	89	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	260	90	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	91	𝛽	𝛽	NOUN
iajs-2430	260	92	𝜒	𝜒	NOUN
iajs-2430	260	93	∗	∗	NOUN
iajs-2430	260	94	𝛾	𝛾	ADP
iajs-2430	260	95	∗	∗	NOUN
iajs-2430	260	96	τ	τ	X
iajs-2430	260	97	,	,	PUNCT
iajs-2430	260	98	𝛽	𝛽	NOUN
iajs-2430	260	99	𝜒	𝜒	NOUN
iajs-2430	260	100	∗	∗	NOUN
iajs-2430	260	101	𝛾	𝛾	ADP
iajs-2430	260	102	∗	∗	NOUN
iajs-2430	260	103	τ	τ	PROPN
iajs-2430	260	104	,	,	PUNCT
iajs-2430	260	105	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	106	𝛽	𝛽	NOUN
iajs-2430	260	107	𝛾	𝛾	PROPN
iajs-2430	260	108	,	,	PUNCT
iajs-2430	260	109	𝛽	𝛽	NOUN
iajs-2430	260	110	𝛾	𝛾	NOUN
iajs-2430	260	111	=	=	SYM
iajs-2430	260	112	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	PROPN
iajs-2430	260	113	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	114	𝛽	𝛽	NOUN
iajs-2430	260	115	𝜒	𝜒	NOUN
iajs-2430	260	116	∗	∗	NOUN
iajs-2430	260	117	𝛾	𝛾	ADP
iajs-2430	260	118	∗	∗	NOUN
iajs-2430	260	119	τ	τ	X
iajs-2430	260	120	,	,	PUNCT
iajs-2430	260	121	𝛽	𝛽	PROPN
iajs-2430	260	122	𝛾	𝛾	NOUN
iajs-2430	260	123	,	,	PUNCT
iajs-2430	260	124	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2430	260	125	𝛽	𝛽	NOUN
iajs-2430	260	126	𝜒	𝜒	NOUN
iajs-2430	260	127	∗	∗	NOUN
iajs-2430	260	128	𝛾	𝛾	ADP
iajs-2430	260	129	∗	∗	NOUN
iajs-2430	260	130	τ	τ	X
iajs-2430	260	131	,	,	PUNCT
iajs-2430	260	132	𝛽	𝛽	NOUN
iajs-2430	260	133	𝛾	𝛾	NOUN
iajs-2430	260	134	in	in	ADP
iajs-2430	260	135	particular	particular	ADJ
iajs-2430	260	136	,	,	PUNCT
iajs-2430	260	137	if	if	SCONJ
iajs-2430	260	138	we	we	PRON
iajs-2430	260	139	take	take	VERB
iajs-2430	260	140	𝜒	𝜒	NUM
iajs-2430	260	141	𝛾	𝛾	NOUN
iajs-2430	260	142	τ	τ	X
iajs-2430	260	143	0	0	NUM
iajs-2430	260	144	,	,	PUNCT
iajs-2430	260	145	then	then	ADV
iajs-2430	260	146	𝛽	𝛽	NOUN
iajs-2430	260	147	𝜒	𝜒	X
iajs-2430	260	148	∗	∗	NOUN
iajs-2430	260	149	τ	τ	PROPN
iajs-2430	260	150	𝑟	𝑟	NOUN
iajs-2430	260	151	min	min	NOUN
iajs-2430	260	152	𝛽	𝛽	NOUN
iajs-2430	260	153	𝜒	𝜒	NOUN
iajs-2430	260	154	∗	∗	NOUN
iajs-2430	260	155	𝛾	𝛾	ADP
iajs-2430	260	156	∗	∗	NOUN
iajs-2430	260	157	τ	τ	X
iajs-2430	260	158	,	,	PUNCT
iajs-2430	260	159	𝛽	𝛽	NOUN
iajs-2430	260	160	𝛾	𝛾	NOUN
iajs-2430	260	161	.	.	PUNCT
iajs-2430	260	162	 	 	SPACE
iajs-2430	260	163	hence	hence	ADV
iajs-2430	260	164	,	,	PUNCT
iajs-2430	260	165	 	 	SPACE
iajs-2430	260	166	´	´	VERB
iajs-2430	260	167	the	the	DET
iajs-2430	260	168	proof	proof	NOUN
iajs-2430	260	169	´	´	NOUN
iajs-2430	260	170	is	be	AUX
iajs-2430	260	171	completed	complete	VERB
iajs-2430	260	172	´	´	NOUN
iajs-2430	260	173	.	.	PUNCT
iajs-2430	261	1	5	5	X
iajs-2430	261	2	.	.	X
iajs-2430	261	3	conclusion	conclusion	NOUN
iajs-2430	261	4	the	the	DET
iajs-2430	261	5	concept	concept	NOUN
iajs-2430	261	6	of	of	ADP
iajs-2430	261	7	an	an	DET
iajs-2430	261	8	interval	interval	NOUN
iajs-2430	261	9	value	value	NOUN
iajs-2430	261	10	fuzzy	fuzzy	ADJ
iajs-2430	261	11	set	set	NOUN
iajs-2430	261	12	of	of	ADP
iajs-2430	261	13	ku	ku	PROPN
iajs-2430	261	14	-	-	PUNCT
iajs-2430	261	15	semigroup	semigroup	PROPN
iajs-2430	261	16	is	be	AUX
iajs-2430	261	17	introduced	introduce	VERB
iajs-2430	261	18	and	and	CCONJ
iajs-2430	261	19	some	some	DET
iajs-2430	261	20	related	related	ADJ
iajs-2430	261	21	properties	property	NOUN
iajs-2430	261	22	are	be	AUX
iajs-2430	261	23	investigated	investigate	VERB
iajs-2430	261	24	.	.	PUNCT
iajs-2430	262	1	also	also	ADV
iajs-2430	262	2	,	,	PUNCT
iajs-2430	262	3	some	some	DET
iajs-2430	262	4	types	type	NOUN
iajs-2430	262	5	of	of	ADP
iajs-2430	262	6	interval	interval	NOUN
iajs-2430	262	7	value	value	NOUN
iajs-2430	262	8	fuzzy	fuzzy	ADJ
iajs-2430	262	9	ideals	ideal	NOUN
iajs-2430	262	10	are	be	AUX
iajs-2430	262	11	studied	study	VERB
iajs-2430	262	12	and	and	CCONJ
iajs-2430	262	13	the	the	DET
iajs-2430	262	14	relationship	relationship	NOUN
iajs-2430	262	15	between	between	ADP
iajs-2430	262	16	them	they	PRON
iajs-2430	262	17	is	be	AUX
iajs-2430	262	18	stated	state	VERB
iajs-2430	262	19	.	.	PUNCT
iajs-2430	263	1	then	then	ADV
iajs-2430	263	2	,	,	PUNCT
iajs-2430	263	3	an	an	DET
iajs-2430	263	4	interval	interval	NOUN
iajs-2430	263	5	value	value	NOUN
iajs-2430	263	6	fuzzy	fuzzy	ADJ
iajs-2430	263	7	k	k	NOUN
iajs-2430	263	8	-	-	NOUN
iajs-2430	263	9	ideal	ideal	NOUN
iajs-2430	263	10	of	of	ADP
iajs-2430	263	11	kusemigroup	kusemigroup	NOUN
iajs-2430	263	12	is	be	AUX
iajs-2430	263	13	studied	study	VERB
iajs-2430	263	14	and	and	CCONJ
iajs-2430	263	15	a	a	DET
iajs-2430	263	16	few	few	ADJ
iajs-2430	263	17	properties	property	NOUN
iajs-2430	263	18	are	be	AUX
iajs-2430	263	19	obtained	obtain	VERB
iajs-2430	263	20	.	.	PUNCT
iajs-2430	264	1	furthermore	furthermore	ADV
iajs-2430	264	2	,	,	PUNCT
iajs-2430	264	3	the	the	DET
iajs-2430	264	4	notion	notion	NOUN
iajs-2430	264	5	of	of	ADP
iajs-2430	264	6	a	a	DET
iajs-2430	264	7	  	  	SPACE
iajs-2430	264	8	106	106	NUM
iajs-2430	264	9	  	  	SPACE
iajs-2430	264	10	ibn	ibn	PROPN
iajs-2430	264	11	al	al	PROPN
iajs-2430	264	12	-	-	PUNCT
iajs-2430	264	13	haitham	haitham	PROPN
iajs-2430	264	14	jour	jour	X
iajs-2430	264	15	.	.	PROPN
iajs-2430	264	16	for	for	ADP
iajs-2430	264	17	pure	pure	ADJ
iajs-2430	264	18	&	&	CCONJ
iajs-2430	264	19	appl	appl	PROPN
iajs-2430	264	20	.	.	PUNCT
iajs-2430	265	1	sci	sci	PROPN
iajs-2430	265	2	.	.	PROPN
iajs-2430	266	1	33	33	NUM
iajs-2430	266	2	(	(	PUNCT
iajs-2430	266	3	2	2	NUM
iajs-2430	266	4	)	)	PUNCT
iajs-2430	266	5	2020	2020	NUM
iajs-2430	267	1	homomorphism	homomorphism	NOUN
iajs-2430	267	2	is	be	AUX
iajs-2430	267	3	discussed	discuss	VERB
iajs-2430	267	4	.	.	PUNCT
iajs-2430	268	1	main	main	ADJ
iajs-2430	268	2	purpose	purpose	NOUN
iajs-2430	268	3	of	of	ADP
iajs-2430	268	4	our	our	PRON
iajs-2430	268	5	future	future	ADJ
iajs-2430	268	6	work	work	NOUN
iajs-2430	268	7	is	be	AUX
iajs-2430	268	8	to	to	PART
iajs-2430	268	9	investigate	investigate	VERB
iajs-2430	268	10	fuzzy	fuzzy	ADJ
iajs-2430	268	11	of	of	ADP
iajs-2430	268	12	several	several	ADJ
iajs-2430	268	13	types	type	NOUN
iajs-2430	268	14	of	of	ADP
iajs-2430	268	15	ideals	ideal	NOUN
iajs-2430	268	16	with	with	ADP
iajs-2430	268	17	special	special	ADJ
iajs-2430	268	18	properties	property	NOUN
iajs-2430	268	19	such	such	ADJ
iajs-2430	268	20	as	as	ADP
iajs-2430	268	21	an	an	DET
iajs-2430	268	22	intuitionistic	intuitionistic	ADJ
iajs-2430	268	23	fuzzy	fuzzy	ADJ
iajs-2430	268	24	and	and	CCONJ
iajs-2430	268	25	hyper	hyper	NOUN
iajs-2430	268	26	of	of	ADP
iajs-2430	268	27	kusemigroup	kusemigroup	NOUN
iajs-2430	268	28	.	.	PUNCT
iajs-2430	269	1	references	reference	NOUN
iajs-2430	269	2	1	1	NUM
iajs-2430	269	3	.	.	PUNCT
iajs-2430	270	1	prabpayak	prabpayak	NOUN
iajs-2430	270	2	,	,	PUNCT
iajs-2430	270	3	c.	c.	PROPN
iajs-2430	270	4	;	;	PUNCT
iajs-2430	270	5	leerawat	leerawat	NOUN
iajs-2430	270	6	,	,	PUNCT
iajs-2430	270	7	u.	u.	VERB
iajs-2430	270	8	on	on	ADP
iajs-2430	270	9	ideals	ideal	NOUN
iajs-2430	270	10	and	and	CCONJ
iajs-2430	270	11	congruence	congruence	NOUN
iajs-2430	270	12	in	in	ADP
iajs-2430	270	13	ku-algebras.scientia	ku-algebras.scientia	PROPN
iajs-2430	270	14	magna	magna	PROPN
iajs-2430	270	15	journal.2009	journal.2009	PROPN
iajs-2430	270	16	,	,	PUNCT
iajs-2430	270	17	5	5	NUM
iajs-2430	270	18	,	,	PUNCT
iajs-2430	270	19	1	1	NUM
iajs-2430	270	20	,	,	PUNCT
iajs-2430	270	21	54	54	NUM
iajs-2430	270	22	-	-	SYM
iajs-2430	270	23	57	57	NUM
iajs-2430	270	24	.	.	X
iajs-2430	271	1	2	2	NUM
iajs-2430	271	2	.	.	X
iajs-2430	271	3	prabpayak	prabpayak	NOUN
iajs-2430	271	4	,	,	PUNCT
iajs-2430	271	5	c.	c.	PROPN
iajs-2430	271	6	;	;	PUNCT
iajs-2430	271	7	leerawat	leerawat	NOUN
iajs-2430	271	8	,	,	PUNCT
iajs-2430	271	9	u.	u.	PROPN
iajs-2430	271	10	on	on	ADP
iajs-2430	271	11	isomorphisms	isomorphism	NOUN
iajs-2430	271	12	of	of	ADP
iajs-2430	271	13	ku-algebras.scientia	ku-algebras.scientia	PROPN
iajs-2430	271	14	magna	magna	PROPN
iajs-2430	271	15	journal.2009	journal.2009	PROPN
iajs-2430	271	16	,	,	PUNCT
iajs-2430	271	17	5	5	NUM
iajs-2430	271	18	,	,	PUNCT
iajs-2430	271	19	3	3	NUM
iajs-2430	271	20	,	,	PUNCT
iajs-2430	271	21	25	25	NUM
iajs-2430	271	22	-	-	SYM
iajs-2430	271	23	31	31	NUM
iajs-2430	271	24	.	.	PUNCT
iajs-2430	272	1	3	3	X
iajs-2430	272	2	.	.	X
iajs-2430	272	3	kareem	kareem	PROPN
iajs-2430	272	4	,	,	PUNCT
iajs-2430	272	5	f.f.;hasan	f.f.;hasan	PROPN
iajs-2430	272	6	,	,	PUNCT
iajs-2430	272	7	e.	e.	PROPN
iajs-2430	272	8	r.on	r.on	PROPN
iajs-2430	272	9	ku-semigroups.international	ku-semigroups.international	PROPN
iajs-2430	272	10	journal	journal	NOUN
iajs-2430	272	11	of	of	ADP
iajs-2430	272	12	science	science	NOUN
iajs-2430	272	13	and	and	CCONJ
iajs-2430	272	14	nature.2018	nature.2018	NOUN
iajs-2430	272	15	,	,	PUNCT
iajs-2430	272	16	9	9	NUM
iajs-2430	272	17	,	,	PUNCT
iajs-2430	272	18	1	1	NUM
iajs-2430	272	19	,	,	PUNCT
iajs-2430	272	20	79	79	NUM
iajs-2430	272	21	-	-	SYM
iajs-2430	272	22	84	84	NUM
iajs-2430	272	23	.	.	PUNCT
iajs-2430	273	1	4	4	NUM
iajs-2430	273	2	.	.	X
iajs-2430	273	3	zadeh	zadeh	PROPN
iajs-2430	273	4	,	,	PUNCT
iajs-2430	273	5	l.a	l.a	PROPN
iajs-2430	273	6	.	.	PROPN
iajs-2430	273	7	fuzzy	fuzzy	ADJ
iajs-2430	273	8	sets.inform	sets.inform	NOUN
iajs-2430	273	9	and	and	CCONJ
iajs-2430	273	10	control.1965	control.1965	PROPN
iajs-2430	273	11	,	,	PUNCT
iajs-2430	273	12	8	8	NUM
iajs-2430	273	13	,	,	PUNCT
iajs-2430	273	14	338	338	NUM
iajs-2430	273	15	-	-	SYM
iajs-2430	273	16	353	353	NUM
iajs-2430	273	17	.	.	NOUN
iajs-2430	273	18	5	5	NUM
iajs-2430	273	19	.	.	X
iajs-2430	273	20	mostafa	mostafa	PROPN
iajs-2430	273	21	,	,	PUNCT
iajs-2430	273	22	s.m	s.m	PROPN
iajs-2430	273	23	.	.	PROPN
iajs-2430	273	24	;	;	PUNCT
iajs-2430	273	25	abd	abd	PROPN
iajs-2430	273	26	-	-	PUNCT
iajs-2430	273	27	elnaby	elnaby	PROPN
iajs-2430	273	28	,	,	PUNCT
iajs-2430	273	29	m.a	m.a	PROPN
iajs-2430	273	30	.	.	PROPN
iajs-2430	273	31	;	;	PUNCT
iajs-2430	273	32	yousef	yousef	PROPN
iajs-2430	273	33	,	,	PUNCT
iajs-2430	273	34	m.m.m	m.m.m	INTJ
iajs-2430	273	35	.	.	PUNCT
iajs-2430	273	36	fuzzy	fuzzy	ADJ
iajs-2430	273	37	ideals	ideal	NOUN
iajs-2430	273	38	of	of	ADP
iajs-2430	273	39	ku	ku	PROPN
iajs-2430	273	40	-	-	PUNCT
iajs-2430	273	41	algebras	algebras	PROPN
iajs-2430	273	42	.	.	PUNCT
iajs-2430	274	1	international	international	ADJ
iajs-2430	274	2	mathematical	mathematical	ADJ
iajs-2430	274	3	forum.2011	forum.2011	PROPN
iajs-2430	274	4	,	,	PUNCT
iajs-2430	274	5	6	6	NUM
iajs-2430	274	6	,	,	PUNCT
iajs-2430	274	7	63	63	NUM
iajs-2430	274	8	,	,	PUNCT
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iajs-2430	274	10	-	-	SYM
iajs-2430	274	11	3149	3149	NUM
iajs-2430	274	12	.	.	PUNCT
iajs-2430	275	1	6	6	NUM
iajs-2430	275	2	.	.	X
iajs-2430	275	3	hasan	hasan	PROPN
iajs-2430	275	4	,	,	PUNCT
iajs-2430	275	5	e.r	e.r	AUX
iajs-2430	275	6	.	.	PROPN
iajs-2430	275	7	;	;	PUNCT
iajs-2430	275	8	kareem	kareem	PROPN
iajs-2430	275	9	,	,	PUNCT
iajs-2430	275	10	f.f	f.f	PROPN
iajs-2430	275	11	.	.	PROPN
iajs-2430	275	12	fuzzy	fuzzy	PROPN
iajs-2430	275	13	ku	ku	PROPN
iajs-2430	275	14	-	-	PUNCT
iajs-2430	275	15	semi	semi	NOUN
iajs-2430	275	16	-	-	NOUN
iajs-2430	275	17	groups	group	NOUN
iajs-2430	275	18	and	and	CCONJ
iajs-2430	275	19	investigate	investigate	VERB
iajs-2430	275	20	some	some	DET
iajs-2430	275	21	basic	basic	ADJ
iajs-2430	275	22	properties.journal	properties.journal	PROPN
iajs-2430	275	23	of	of	ADP
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iajs-2430	275	25	and	and	CCONJ
iajs-2430	275	26	applied	apply	VERB
iajs-2430	275	27	science.2018	science.2018	PROPN
iajs-2430	275	28	,	,	PUNCT
iajs-2430	275	29	13	13	NUM
iajs-2430	275	30	,	,	PUNCT
iajs-2430	275	31	18	18	NUM
iajs-2430	275	32	,	,	PUNCT
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iajs-2430	275	34	-	-	SYM
iajs-2430	275	35	7744	7744	NUM
iajs-2430	275	36	.	.	PUNCT
iajs-2430	276	1	7	7	X
iajs-2430	276	2	.	.	X
iajs-2430	276	3	akram	akram	PROPN
iajs-2430	276	4	,	,	PUNCT
iajs-2430	276	5	m.	m.	NOUN
iajs-2430	276	6	;	;	PUNCT
iajs-2430	276	7	yaqoob	yaqoob	NOUN
iajs-2430	276	8	,	,	PUNCT
iajs-2430	276	9	n.	n.	NOUN
iajs-2430	276	10	;	;	PUNCT
iajs-2430	276	11	kavikumar	kavikumar	PROPN
iajs-2430	276	12	,	,	PUNCT
iajs-2430	276	13	j.	j.	PROPN
iajs-2430	276	14	interval	interval	PROPN
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iajs-2430	276	16	)	)	PUNCT
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iajs-2430	276	18	,	,	PUNCT
iajs-2430	276	19	~	~	PUNCT
iajs-2430	276	20	(	(	PUNCT
iajs-2430	276	21			NOUN
iajs-2430	276	22	-fuzzy	-fuzzy	PROPN
iajs-2430	276	23	ku	ku	NOUN
iajs-2430	276	24	-	-	PUNCT
iajs-2430	276	25	ideals	ideal	NOUN
iajs-2430	276	26	of	of	ADP
iajs-2430	276	27	kualgebras.int	kualgebras.int	PROPN
iajs-2430	276	28	.	.	PUNCT
iajs-2430	277	1	j.	j.	PROPN
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iajs-2430	277	4	.	.	PUNCT
iajs-2430	278	1	math.2014	math.2014	PROPN
iajs-2430	278	2	,	,	PUNCT
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iajs-2430	278	4	,	,	PUNCT
iajs-2430	278	5	3	3	NUM
iajs-2430	278	6	,	,	PUNCT
iajs-2430	278	7	335	335	NUM
iajs-2430	278	8	-	-	SYM
iajs-2430	278	9	349	349	NUM
iajs-2430	278	10	.	.	NOUN
iajs-2430	279	1	8	8	NUM
iajs-2430	279	2	.	.	X
iajs-2430	279	3	lee	lee	PROPN
iajs-2430	279	4	,	,	PUNCT
iajs-2430	279	5	k.m	k.m	PROPN
iajs-2430	279	6	.	.	PROPN
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iajs-2430	279	8	-	-	PUNCT
iajs-2430	279	9	valued	value	VERB
iajs-2430	279	10	fuzzy	fuzzy	ADJ
iajs-2430	279	11	sets	set	NOUN
iajs-2430	279	12	and	and	CCONJ
iajs-2430	279	13	their	their	PRON
iajs-2430	279	14	operations	operation	NOUN
iajs-2430	279	15	.	.	PUNCT
iajs-2430	280	1	proc	proc	NOUN
iajs-2430	280	2	.	.	PUNCT
iajs-2430	281	1	int	int	NOUN
iajs-2430	281	2	.	.	PUNCT
iajs-2430	281	3	conf	conf	PROPN
iajs-2430	281	4	.	.	PUNCT
iajs-2430	282	1	onintelligent	onintelligent	PROPN
iajs-2430	282	2	technologies	technologies	PROPN
iajs-2430	282	3	,	,	PUNCT
iajs-2430	282	4	bangkok	bangkok	PROPN
iajs-2430	282	5	,	,	PUNCT
iajs-2430	282	6	thailand.2000	thailand.2000	PROPN
iajs-2430	282	7	,	,	PUNCT
iajs-2430	282	8	307	307	NUM
iajs-2430	282	9	-	-	SYM
iajs-2430	282	10	312	312	NUM
iajs-2430	282	11	.	.	NOUN
iajs-2430	282	12	9	9	NUM
iajs-2430	282	13	.	.	X
iajs-2430	282	14	lee	lee	PROPN
iajs-2430	282	15	,	,	PUNCT
iajs-2430	282	16	k.m	k.m	PROPN
iajs-2430	282	17	.	.	PROPN
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iajs-2430	282	19	of	of	ADP
iajs-2430	282	20	interval	interval	NOUN
iajs-2430	282	21	-	-	PUNCT
iajs-2430	282	22	valued	value	VERB
iajs-2430	282	23	fuzzy	fuzzy	ADJ
iajs-2430	282	24	sets	set	NOUN
iajs-2430	282	25	,	,	PUNCT
iajs-2430	282	26	intuitionistic	intuitionistic	ADJ
iajs-2430	282	27	fuzzy	fuzzy	ADJ
iajs-2430	282	28	sets	set	NOUN
iajs-2430	282	29	,	,	PUNCT
iajs-2430	282	30	and	and	CCONJ
iajs-2430	282	31	bipolar	bipolar	ADV
iajs-2430	282	32	-	-	PUNCT
iajs-2430	282	33	valued	value	VERB
iajs-2430	282	34	fuzzy	fuzzy	ADJ
iajs-2430	282	35	sets.j	sets.j	PROPN
iajs-2430	282	36	.	.	PUNCT
iajs-2430	283	1	fuzzy	fuzzy	ADJ
iajs-2430	283	2	logic	logic	NOUN
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iajs-2430	283	4	systems.2004	systems.2004	PROPN
iajs-2430	283	5	,	,	PUNCT
iajs-2430	283	6	14	14	NUM
iajs-2430	283	7	,	,	PUNCT
iajs-2430	283	8	125	125	NUM
iajs-2430	283	9	-	-	SYM
iajs-2430	283	10	129	129	NUM
iajs-2430	283	11	.	.	PUNCT
iajs-2430	284	1	10	10	NUM
iajs-2430	284	2	.	.	X
iajs-2430	285	1	mostafa	mostafa	PROPN
iajs-2430	285	2	,	,	PUNCT
iajs-2430	285	3	s.m	s.m	PROPN
iajs-2430	285	4	.	.	PROPN
iajs-2430	285	5	;	;	PUNCT
iajs-2430	285	6	abd	abd	PROPN
iajs-2430	285	7	-	-	PUNCT
iajs-2430	285	8	elnaby	elnaby	PROPN
iajs-2430	285	9	,	,	PUNCT
iajs-2430	285	10	m.a	m.a	PROPN
iajs-2430	285	11	.	.	PROPN
iajs-2430	285	12	;	;	PUNCT
iajs-2430	285	13	elgendy	elgendy	ADJ
iajs-2430	285	14	,	,	PUNCT
iajs-2430	285	15	o.	o.	PROPN
iajs-2430	285	16	r.	r.	PROPN
iajs-2430	285	17	interval	interval	PROPN
iajs-2430	285	18	-	-	PUNCT
iajs-2430	285	19	valued	value	VERB
iajs-2430	285	20	fuzzy	fuzzy	ADJ
iajs-2430	285	21	ku	ku	PROPN
iajs-2430	285	22	ideals	ideal	NOUN
iajs-2430	285	23	in	in	ADP
iajs-2430	285	24	ku-algebras.int	ku-algebras.int	PROPN
iajs-2430	285	25	.	.	PUNCT
iajs-2430	286	1	math	math	NOUN
iajs-2430	286	2	.	.	PUNCT
iajs-2430	287	1	forum.2011	forum.2011	NOUN
iajs-2430	287	2	,	,	PUNCT
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iajs-2430	287	4	,	,	PUNCT
iajs-2430	287	5	64	64	NUM
iajs-2430	287	6	,	,	PUNCT
iajs-2430	287	7	3151	3151	NUM
iajs-2430	287	8	-	-	SYM
iajs-2430	287	9	3159	3159	NUM
iajs-2430	287	10	.	.	PUNCT
iajs-2430	288	1	11	11	NUM
iajs-2430	288	2	.	.	X
iajs-2430	289	1	mostafa	mostafa	PROPN
iajs-2430	289	2	,	,	PUNCT
iajs-2430	289	3	s.m	s.m	PROPN
iajs-2430	289	4	.	.	PROPN
iajs-2430	289	5	;	;	PUNCT
iajs-2430	289	6	radwan	radwan	NOUN
iajs-2430	289	7	,	,	PUNCT
iajs-2430	289	8	a.e	a.e	PROPN
iajs-2430	289	9	.	.	PROPN
iajs-2430	289	10	;	;	PUNCT
iajs-2430	289	11	ibrahem	ibrahem	NOUN
iajs-2430	289	12	,	,	PUNCT
iajs-2430	289	13	f.a	f.a	PROPN
iajs-2430	289	14	.	.	PROPN
iajs-2430	289	15	;	;	PUNCT
iajs-2430	289	16	kareem	kareem	PROPN
iajs-2430	289	17	,	,	PUNCT
iajs-2430	289	18	f.f	f.f	PROPN
iajs-2430	289	19	.	.	PROPN
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iajs-2430	289	21	value	value	NOUN
iajs-2430	289	22	fuzzy	fuzzy	ADJ
iajs-2430	289	23	n	n	CCONJ
iajs-2430	289	24	-	-	ADJ
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iajs-2430	289	26	ku	ku	NOUN
iajs-2430	289	27	-	-	PUNCT
iajs-2430	289	28	ideals	ideal	NOUN
iajs-2430	289	29	of	of	ADP
iajs-2430	289	30	ku	ku	PROPN
iajs-2430	289	31	-	-	PUNCT
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iajs-2430	289	33	.	.	PUNCT
iajs-2430	290	1	math.comput.sci.2015	math.comput.sci.2015	PROPN
iajs-2430	290	2	,	,	PUNCT
iajs-2430	290	3	5	5	NUM
iajs-2430	290	4	,	,	PUNCT
iajs-2430	290	5	2	2	NUM
iajs-2430	290	6	,	,	PUNCT
iajs-2430	290	7	246	246	NUM
iajs-2430	290	8	-	-	SYM
iajs-2430	290	9	264	264	NUM
iajs-2430	290	10	.	.	PUNCT
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iajs-2430	290	12	.	.	PUNCT
iajs-2430	291	1	saeid	saeid	PROPN
iajs-2430	291	2	,	,	PUNCT
iajs-2430	291	3	a.b	a.b	PROPN
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iajs-2430	291	6	-	-	PUNCT
iajs-2430	291	7	valued	value	VERB
iajs-2430	291	8	fuzzy	fuzzy	ADJ
iajs-2430	291	9	bck\bci	bck\bci	NOUN
iajs-2430	291	10	-	-	PUNCT
iajs-2430	291	11	algebras	algebras	PROPN
iajs-2430	291	12	.	.	PUNCT
iajs-2430	292	1	world	world	PROPN
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iajs-2430	292	3	sciences	science	NOUN
iajs-2430	292	4	journal.2009	journal.2009	PROPN
iajs-2430	292	5	,	,	PUNCT
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iajs-2430	292	7	,	,	PUNCT
iajs-2430	292	8	11	11	NUM
iajs-2430	292	9	,	,	PUNCT
iajs-2430	292	10	1404	1404	NUM
iajs-2430	292	11	-	-	SYM
iajs-2430	292	12	1411	1411	NUM
iajs-2430	292	13	.	.	PUNCT