id	sid	tid	token	lemma	pos
iajs-2767	1	1	37	37	NUM
iajs-2767	1	2	this	this	DET
iajs-2767	1	3	work	work	NOUN
iajs-2767	1	4	is	be	AUX
iajs-2767	1	5	licensed	license	VERB
iajs-2767	1	6	under	under	ADP
iajs-2767	1	7	a	a	DET
iajs-2767	1	8	creative	creative	ADJ
iajs-2767	1	9	commons	common	NOUN
iajs-2767	1	10	attribution	attribution	NOUN
iajs-2767	1	11	4.0	4.0	NUM
iajs-2767	1	12	international	international	ADJ
iajs-2767	1	13	license	license	NOUN
iajs-2767	1	14	.	.	PUNCT
iajs-2767	2	1	study	study	NOUN
iajs-2767	2	2	of	of	ADP
iajs-2767	2	3	fuzzy	fuzzy	ADJ
iajs-2767	2	4	𝛔	𝛔	NOUN
iajs-2767	2	5	–	–	PUNCT
iajs-2767	2	6	𝐑𝐢𝐧𝐠	𝐑𝐢𝐧𝐠	PROPN
iajs-2767	2	7	and	and	CCONJ
iajs-2767	2	8	some	some	DET
iajs-2767	2	9	related	relate	VERB
iajs-2767	2	10	concepts	concept	NOUN
iajs-2767	2	11	abstract	abstract	ADP
iajs-2767	2	12	this	this	DET
iajs-2767	2	13	paper	paper	NOUN
iajs-2767	2	14	introduces	introduce	VERB
iajs-2767	2	15	the	the	DET
iajs-2767	2	16	concept	concept	NOUN
iajs-2767	2	17	of	of	ADP
iajs-2767	2	18	fuzzy	fuzzy	ADJ
iajs-2767	2	19	σ	σ	PROPN
iajs-2767	2	20	–	–	PUNCT
iajs-2767	2	21	ring	ring	NOUN
iajs-2767	2	22	as	as	ADP
iajs-2767	2	23	a	a	DET
iajs-2767	2	24	generalization	generalization	NOUN
iajs-2767	2	25	of	of	ADP
iajs-2767	2	26	fuzzy	fuzzy	ADJ
iajs-2767	2	27	σ	σ	PROPN
iajs-2767	2	28	−algebra	−algebra	PROPN
iajs-2767	2	29	and	and	CCONJ
iajs-2767	2	30	basic	basic	ADJ
iajs-2767	2	31	properties	property	NOUN
iajs-2767	2	32	;	;	PUNCT
iajs-2767	2	33	examples	example	NOUN
iajs-2767	2	34	of	of	ADP
iajs-2767	2	35	this	this	DET
iajs-2767	2	36	concept	concept	NOUN
iajs-2767	2	37	have	have	AUX
iajs-2767	2	38	been	be	AUX
iajs-2767	2	39	given	give	VERB
iajs-2767	2	40	.	.	PUNCT
iajs-2767	3	1	as	as	ADP
iajs-2767	3	2	the	the	DET
iajs-2767	3	3	first	first	ADJ
iajs-2767	3	4	result	result	NOUN
iajs-2767	3	5	,	,	PUNCT
iajs-2767	3	6	it	it	PRON
iajs-2767	3	7	has	have	AUX
iajs-2767	3	8	been	be	AUX
iajs-2767	3	9	proved	prove	VERB
iajs-2767	3	10	that	that	SCONJ
iajs-2767	3	11	every	every	DET
iajs-2767	3	12	fuzzy	fuzzy	ADJ
iajs-2767	3	13	σ	σ	NOUN
iajs-2767	3	14	–	–	PUNCT
iajs-2767	3	15	algebra	algebra	NOUN
iajs-2767	3	16	over	over	ADP
iajs-2767	3	17	a	a	DET
iajs-2767	3	18	fuzzy	fuzzy	ADJ
iajs-2767	3	19	set	set	NOUN
iajs-2767	3	20	𝒳∗	𝒳∗	NOUN
iajs-2767	3	21	is	be	AUX
iajs-2767	3	22	a	a	DET
iajs-2767	3	23	fuzzy	fuzzy	ADJ
iajs-2767	3	24	σ	σ	NOUN
iajs-2767	3	25	–	–	PUNCT
iajs-2767	3	26	ring	ring	NOUN
iajs-2767	3	27	over	over	ADP
iajs-2767	3	28	a	a	DET
iajs-2767	3	29	fuzzy	fuzzy	ADJ
iajs-2767	3	30	set	set	VERB
iajs-2767	3	31	𝒳∗	𝒳∗	X
iajs-2767	3	32	and	and	CCONJ
iajs-2767	3	33	construct	construct	VERB
iajs-2767	3	34	their	their	PRON
iajs-2767	3	35	converse	converse	NOUN
iajs-2767	3	36	by	by	ADP
iajs-2767	3	37	example	example	NOUN
iajs-2767	3	38	.	.	PUNCT
iajs-2767	4	1	furthermore	furthermore	ADV
iajs-2767	4	2	,	,	PUNCT
iajs-2767	4	3	the	the	DET
iajs-2767	4	4	fuzzy	fuzzy	ADJ
iajs-2767	4	5	ring	ring	NOUN
iajs-2767	4	6	concept	concept	NOUN
iajs-2767	4	7	has	have	AUX
iajs-2767	4	8	been	be	AUX
iajs-2767	4	9	studied	study	VERB
iajs-2767	4	10	to	to	PART
iajs-2767	4	11	generalize	generalize	VERB
iajs-2767	4	12	fuzzy	fuzzy	ADJ
iajs-2767	4	13	algebra	algebra	NOUN
iajs-2767	4	14	and	and	CCONJ
iajs-2767	4	15	its	its	PRON
iajs-2767	4	16	relation	relation	NOUN
iajs-2767	4	17	.	.	PUNCT
iajs-2767	5	1	investigating	investigate	VERB
iajs-2767	5	2	that	that	SCONJ
iajs-2767	5	3	the	the	DET
iajs-2767	5	4	concept	concept	NOUN
iajs-2767	5	5	of	of	ADP
iajs-2767	5	6	fuzzy	fuzzy	ADJ
iajs-2767	5	7	σ	σ	PROPN
iajs-2767	5	8	–	–	PUNCT
iajs-2767	5	9	ring	ring	NOUN
iajs-2767	5	10	is	be	AUX
iajs-2767	5	11	a	a	DET
iajs-2767	5	12	stronger	strong	ADJ
iajs-2767	5	13	form	form	NOUN
iajs-2767	5	14	of	of	ADP
iajs-2767	5	15	a	a	DET
iajs-2767	5	16	fuzzy	fuzzy	ADJ
iajs-2767	5	17	ring	ring	NOUN
iajs-2767	5	18	that	that	PRON
iajs-2767	5	19	is	be	AUX
iajs-2767	5	20	every	every	DET
iajs-2767	5	21	fuzzy	fuzzy	ADJ
iajs-2767	5	22	σ	σ	PROPN
iajs-2767	5	23	–	–	PUNCT
iajs-2767	5	24	ring	ring	NOUN
iajs-2767	5	25	over	over	ADP
iajs-2767	5	26	a	a	DET
iajs-2767	5	27	fuzzy	fuzzy	ADJ
iajs-2767	5	28	set	set	NOUN
iajs-2767	5	29	𝒳∗	𝒳∗	NOUN
iajs-2767	5	30	is	be	AUX
iajs-2767	5	31	a	a	DET
iajs-2767	5	32	fuzzy	fuzzy	ADJ
iajs-2767	5	33	ring	ring	NOUN
iajs-2767	5	34	over	over	ADP
iajs-2767	5	35	a	a	DET
iajs-2767	5	36	fuzzy	fuzzy	ADJ
iajs-2767	5	37	set	set	VERB
iajs-2767	5	38	𝒳∗	𝒳∗	X
iajs-2767	5	39	and	and	CCONJ
iajs-2767	5	40	construct	construct	VERB
iajs-2767	5	41	their	their	PRON
iajs-2767	5	42	converse	converse	NOUN
iajs-2767	5	43	by	by	ADP
iajs-2767	5	44	example	example	NOUN
iajs-2767	5	45	.	.	PUNCT
iajs-2767	6	1	in	in	ADP
iajs-2767	6	2	addition	addition	NOUN
iajs-2767	6	3	,	,	PUNCT
iajs-2767	6	4	the	the	DET
iajs-2767	6	5	idea	idea	NOUN
iajs-2767	6	6	of	of	ADP
iajs-2767	6	7	the	the	DET
iajs-2767	6	8	smallest	small	ADJ
iajs-2767	6	9	,	,	PUNCT
iajs-2767	6	10	as	as	ADP
iajs-2767	6	11	an	an	DET
iajs-2767	6	12	important	important	ADJ
iajs-2767	6	13	property	property	NOUN
iajs-2767	6	14	in	in	ADP
iajs-2767	6	15	the	the	DET
iajs-2767	6	16	study	study	NOUN
iajs-2767	6	17	of	of	ADP
iajs-2767	6	18	real	real	ADJ
iajs-2767	6	19	analysis	analysis	NOUN
iajs-2767	6	20	,	,	PUNCT
iajs-2767	6	21	is	be	AUX
iajs-2767	6	22	studied	study	VERB
iajs-2767	6	23	as	as	ADV
iajs-2767	6	24	well	well	ADV
iajs-2767	6	25	.	.	PUNCT
iajs-2767	7	1	finally	finally	ADV
iajs-2767	7	2	,	,	PUNCT
iajs-2767	7	3	the	the	DET
iajs-2767	7	4	main	main	ADJ
iajs-2767	7	5	goal	goal	NOUN
iajs-2767	7	6	of	of	ADP
iajs-2767	7	7	this	this	DET
iajs-2767	7	8	paper	paper	NOUN
iajs-2767	7	9	is	be	AUX
iajs-2767	7	10	to	to	PART
iajs-2767	7	11	study	study	VERB
iajs-2767	7	12	these	these	DET
iajs-2767	7	13	concepts	concept	NOUN
iajs-2767	7	14	and	and	CCONJ
iajs-2767	7	15	give	give	VERB
iajs-2767	7	16	basic	basic	ADJ
iajs-2767	7	17	properties	property	NOUN
iajs-2767	7	18	,	,	PUNCT
iajs-2767	7	19	examples	example	NOUN
iajs-2767	7	20	,	,	PUNCT
iajs-2767	7	21	characterizations	characterization	NOUN
iajs-2767	7	22	and	and	CCONJ
iajs-2767	7	23	relationships	relationship	NOUN
iajs-2767	7	24	between	between	ADP
iajs-2767	7	25	them	they	PRON
iajs-2767	7	26	.	.	PUNCT
iajs-2767	8	1	keywords	keyword	NOUN
iajs-2767	8	2	:	:	PUNCT
iajs-2767	8	3	σ	σ	PROPN
iajs-2767	8	4	−algebra	−algebra	PROPN
iajs-2767	8	5	,	,	PUNCT
iajs-2767	8	6	σ	σ	PROPN
iajs-2767	8	7	–	–	PUNCT
iajs-2767	8	8	ring	ring	NOUN
iajs-2767	8	9	,	,	PUNCT
iajs-2767	8	10	fuzzy	fuzzy	ADJ
iajs-2767	8	11	σ	σ	PROPN
iajs-2767	8	12	−algebra	−algebra	PROPN
iajs-2767	8	13	,	,	PUNCT
iajs-2767	8	14	fuzzy	fuzzy	ADJ
iajs-2767	8	15	algebra	algebra	NOUN
iajs-2767	8	16	,	,	PUNCT
iajs-2767	8	17	measure	measure	NOUN
iajs-2767	8	18	.	.	PUNCT
iajs-2767	9	1	1	1	X
iajs-2767	9	2	.	.	X
iajs-2767	9	3	introduction	introduction	NOUN
iajs-2767	9	4	the	the	DET
iajs-2767	9	5	generalized	generalized	ADJ
iajs-2767	9	6	measure	measure	NOUN
iajs-2767	9	7	theory	theory	NOUN
iajs-2767	9	8	,	,	PUNCT
iajs-2767	9	9	which	which	PRON
iajs-2767	9	10	is	be	AUX
iajs-2767	9	11	the	the	DET
iajs-2767	9	12	subject	subject	NOUN
iajs-2767	9	13	of	of	ADP
iajs-2767	9	14	this	this	DET
iajs-2767	9	15	thesis	thesis	NOUN
iajs-2767	9	16	emerged	emerge	VERB
iajs-2767	9	17	from	from	ADP
iajs-2767	9	18	the	the	DET
iajs-2767	9	19	wellestablished	wellestablished	ADJ
iajs-2767	9	20	classical	classical	ADJ
iajs-2767	9	21	measure	measure	NOUN
iajs-2767	9	22	theory	theory	NOUN
iajs-2767	9	23	by	by	ADP
iajs-2767	9	24	the	the	DET
iajs-2767	9	25	process	process	NOUN
iajs-2767	9	26	of	of	ADP
iajs-2767	9	27	generalization	generalization	NOUN
iajs-2767	9	28	.	.	PUNCT
iajs-2767	10	1	as	as	SCONJ
iajs-2767	10	2	is	be	AUX
iajs-2767	10	3	well	well	ADV
iajs-2767	10	4	known	know	VERB
iajs-2767	10	5	,	,	PUNCT
iajs-2767	10	6	classical	classical	ADJ
iajs-2767	10	7	measures	measure	NOUN
iajs-2767	10	8	are	be	AUX
iajs-2767	10	9	nonnegative	nonnegative	ADJ
iajs-2767	10	10	real	real	ADV
iajs-2767	10	11	-	-	PUNCT
iajs-2767	10	12	valued	value	VERB
iajs-2767	10	13	set	set	NOUN
iajs-2767	10	14	functions	function	NOUN
iajs-2767	10	15	,	,	PUNCT
iajs-2767	10	16	each	each	PRON
iajs-2767	10	17	defined	define	VERB
iajs-2767	10	18	on	on	ADP
iajs-2767	10	19	a	a	DET
iajs-2767	10	20	specific	specific	ADJ
iajs-2767	10	21	class	class	NOUN
iajs-2767	10	22	of	of	ADP
iajs-2767	10	23	subsets	subset	NOUN
iajs-2767	10	24	of	of	ADP
iajs-2767	10	25	a	a	DET
iajs-2767	10	26	given	give	VERB
iajs-2767	10	27	universal	universal	ADJ
iajs-2767	10	28	set	set	NOUN
iajs-2767	10	29	,	,	PUNCT
iajs-2767	10	30	that	that	PRON
iajs-2767	10	31	satisfies	satisfy	VERB
iajs-2767	10	32	certain	certain	ADJ
iajs-2767	10	33	axiomatic	axiomatic	ADJ
iajs-2767	10	34	requirements	requirement	NOUN
iajs-2767	10	35	.	.	PUNCT
iajs-2767	11	1	one	one	NUM
iajs-2767	11	2	of	of	ADP
iajs-2767	11	3	these	these	DET
iajs-2767	11	4	requirements	requirement	NOUN
iajs-2767	11	5	,	,	PUNCT
iajs-2767	11	6	crucial	crucial	ADJ
iajs-2767	11	7	to	to	ADP
iajs-2767	11	8	classical	classical	ADJ
iajs-2767	11	9	measures	measure	NOUN
iajs-2767	11	10	,	,	PUNCT
iajs-2767	11	11	is	be	AUX
iajs-2767	11	12	known	know	VERB
iajs-2767	11	13	as	as	ADP
iajs-2767	11	14	the	the	DET
iajs-2767	11	15	requirement	requirement	NOUN
iajs-2767	11	16	of	of	ADP
iajs-2767	11	17	additively	additively	ADV
iajs-2767	11	18	.	.	PUNCT
iajs-2767	12	1	measure	measure	NOUN
iajs-2767	12	2	theory	theory	NOUN
iajs-2767	12	3	plays	play	VERB
iajs-2767	12	4	a	a	DET
iajs-2767	12	5	vital	vital	ADJ
iajs-2767	12	6	role	role	NOUN
iajs-2767	12	7	in	in	ADP
iajs-2767	12	8	mathematics	mathematic	NOUN
iajs-2767	12	9	,	,	PUNCT
iajs-2767	12	10	particularly	particularly	ADV
iajs-2767	12	11	in	in	ADP
iajs-2767	12	12	probability	probability	NOUN
iajs-2767	12	13	theory	theory	NOUN
iajs-2767	12	14	's	's	PART
iajs-2767	12	15	foundation	foundation	NOUN
iajs-2767	12	16	.	.	PUNCT
iajs-2767	13	1	the	the	DET
iajs-2767	13	2	theory	theory	NOUN
iajs-2767	13	3	of	of	ADP
iajs-2767	13	4	measure	measure	NOUN
iajs-2767	13	5	has	have	AUX
iajs-2767	13	6	been	be	AUX
iajs-2767	13	7	extensively	extensively	ADV
iajs-2767	13	8	studied	study	VERB
iajs-2767	13	9	and	and	CCONJ
iajs-2767	13	10	is	be	AUX
iajs-2767	13	11	used	use	VERB
iajs-2767	13	12	in	in	ADP
iajs-2767	13	13	modeling	model	VERB
iajs-2767	13	14	the	the	DET
iajs-2767	13	15	physical	physical	ADJ
iajs-2767	13	16	world	world	NOUN
iajs-2767	13	17	.	.	PUNCT
iajs-2767	14	1	the	the	DET
iajs-2767	14	2	notion	notion	NOUN
iajs-2767	14	3	of	of	ADP
iajs-2767	14	4	σ	σ	PROPN
iajs-2767	14	5	–	–	PUNCT
iajs-2767	14	6	field	field	NOUN
iajs-2767	14	7	is	be	AUX
iajs-2767	14	8	essential	essential	ADJ
iajs-2767	14	9	in	in	ADP
iajs-2767	14	10	measure	measure	NOUN
iajs-2767	14	11	theory	theory	NOUN
iajs-2767	14	12	and	and	CCONJ
iajs-2767	14	13	probability	probability	NOUN
iajs-2767	14	14	theory	theory	NOUN
iajs-2767	14	15	.	.	PUNCT
iajs-2767	15	1	in	in	ADP
iajs-2767	15	2	2019	2019	NUM
iajs-2767	15	3	ahmed	ahme	VERB
iajs-2767	15	4	and	and	CCONJ
iajs-2767	15	5	ebrahim	ebrahim	PROPN
iajs-2767	16	1	[	[	X
iajs-2767	16	2	1	1	X
iajs-2767	16	3	]	]	PUNCT
iajs-2767	16	4	studied	study	VERB
iajs-2767	16	5	the	the	DET
iajs-2767	16	6	concept	concept	NOUN
iajs-2767	16	7	of	of	ADP
iajs-2767	16	8	σ	σ	PROPN
iajs-2767	16	9	–	–	PUNCT
iajs-2767	16	10	field	field	NOUN
iajs-2767	16	11	and	and	CCONJ
iajs-2767	16	12	discussed	discuss	VERB
iajs-2767	16	13	many	many	ADJ
iajs-2767	16	14	details	detail	NOUN
iajs-2767	16	15	about	about	ADP
iajs-2767	16	16	some	some	DET
iajs-2767	16	17	generalizations	generalization	NOUN
iajs-2767	16	18	of	of	ADP
iajs-2767	16	19	this	this	DET
iajs-2767	16	20	concept	concept	NOUN
iajs-2767	16	21	.	.	PUNCT
iajs-2767	17	1	they	they	PRON
iajs-2767	17	2	proved	prove	VERB
iajs-2767	17	3	some	some	DET
iajs-2767	17	4	important	important	ADJ
iajs-2767	17	5	results	result	NOUN
iajs-2767	17	6	in	in	ADP
iajs-2767	17	7	measure	measure	NOUN
iajs-2767	17	8	theory	theory	NOUN
iajs-2767	17	9	.	.	PUNCT
iajs-2767	18	1	many	many	ADJ
iajs-2767	18	2	authors	author	NOUN
iajs-2767	18	3	were	be	AUX
iajs-2767	18	4	interested	interested	ADJ
iajs-2767	18	5	in	in	ADP
iajs-2767	18	6	studying	study	VERB
iajs-2767	18	7	σ	σ	PROPN
iajs-2767	18	8	–	–	PUNCT
iajs-2767	18	9	field	field	NOUN
iajs-2767	18	10	and	and	CCONJ
iajs-2767	18	11	σ	σ	PROPN
iajs-2767	18	12	–	–	PUNCT
iajs-2767	18	13	ring	ring	NOUN
iajs-2767	18	14	,	,	PUNCT
iajs-2767	18	15	ibn	ibn	PROPN
iajs-2767	18	16	al	al	PROPN
iajs-2767	18	17	haitham	haitham	PROPN
iajs-2767	18	18	journal	journal	PROPN
iajs-2767	18	19	for	for	ADP
iajs-2767	18	20	pure	pure	ADJ
iajs-2767	18	21	and	and	CCONJ
iajs-2767	18	22	applied	applied	ADJ
iajs-2767	18	23	sciences	sciences	PROPN
iajs-2767	18	24	journal	journal	PROPN
iajs-2767	18	25	homepage	homepage	NOUN
iajs-2767	18	26	:	:	PUNCT
iajs-2767	18	27	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2767	18	28	doi	doi	NOUN
iajs-2767	18	29	:	:	PUNCT
iajs-2767	18	30	10.30526/35.2.2767	10.30526/35.2.2767	PROPN
iajs-2767	18	31	article	article	NOUN
iajs-2767	18	32	history	history	NOUN
iajs-2767	18	33	:	:	PUNCT
iajs-2767	18	34	received	receive	VERB
iajs-2767	18	35	5	5	NUM
iajs-2767	18	36	,	,	PUNCT
iajs-2767	18	37	january	january	PROPN
iajs-2767	18	38	,	,	PUNCT
iajs-2767	18	39	2022	2022	NUM
iajs-2767	18	40	,	,	PUNCT
iajs-2767	18	41	accepted,25	accepted,25	PROPN
iajs-2767	18	42	,	,	PUNCT
iajs-2767	18	43	january	january	PROPN
iajs-2767	18	44	,	,	PUNCT
iajs-2767	18	45	2022	2022	NUM
iajs-2767	18	46	,	,	PUNCT
iajs-2767	18	47	published	publish	VERB
iajs-2767	18	48	in	in	ADP
iajs-2767	18	49	april	april	PROPN
iajs-2767	18	50	2022	2022	NUM
iajs-2767	18	51	.	.	PUNCT
iajs-2767	19	1	ibrahim	ibrahim	PROPN
iajs-2767	19	2	s.	s.	PROPN
iajs-2767	19	3	ahmed	ahmed	PROPN
iajs-2767	19	4	mathematical	mathematical	PROPN
iajs-2767	19	5	department	department	PROPN
iajs-2767	19	6	/college	/college	PROPN
iajs-2767	19	7	of	of	ADP
iajs-2767	19	8	computer	computer	NOUN
iajs-2767	19	9	science	science	NOUN
iajs-2767	19	10	and	and	CCONJ
iajs-2767	19	11	mathematics	mathematics	PROPN
iajs-2767	19	12	/tikrit	/tikrit	VERB
iajs-2767	19	13	university	university	NOUN
iajs-2767	19	14	ibrahim1992@tu.edu.iq	ibrahim1992@tu.edu.iq	PROPN
iajs-2767	19	15	hassan	hassan	PROPN
iajs-2767	19	16	h.	h.	PROPN
iajs-2767	19	17	ebrahim	ebrahim	PROPN
iajs-2767	19	18	mathematical	mathematical	PROPN
iajs-2767	19	19	department	department	PROPN
iajs-2767	19	20	/college	/college	PROPN
iajs-2767	19	21	of	of	ADP
iajs-2767	19	22	computer	computer	NOUN
iajs-2767	19	23	science	science	NOUN
iajs-2767	19	24	and	and	CCONJ
iajs-2767	19	25	mathematics	mathematics	PROPN
iajs-2767	19	26	/tikrit	/tikrit	PROPN
iajs-2767	19	27	university	university	PROPN
iajs-2767	19	28	hassan1962pl@tu.edu.iq	hassan1962pl@tu.edu.iq	PROPN
iajs-2767	19	29	ali	ali	PROPN
iajs-2767	19	30	al	al	PROPN
iajs-2767	19	31	-	-	PUNCT
iajs-2767	19	32	fayadh	fayadh	PROPN
iajs-2767	19	33	department	department	NOUN
iajs-2767	19	34	of	of	ADP
iajs-2767	19	35	mathematics	mathematics	PROPN
iajs-2767	19	36	and	and	CCONJ
iajs-2767	19	37	computer	computer	NOUN
iajs-2767	19	38	applications	application	NOUN
iajs-2767	19	39	/	/	SYM
iajs-2767	19	40	college	college	NOUN
iajs-2767	19	41	of	of	ADP
iajs-2767	19	42	science	science	PROPN
iajs-2767	19	43	/	/	SYM
iajs-2767	19	44	al	al	PROPN
iajs-2767	19	45	–	–	PUNCT
iajs-2767	19	46	nahrain	nahrain	PROPN
iajs-2767	19	47	university	university	NOUN
iajs-2767	19	48	aalfayadh@yahoo.com	aalfayadh@yahoo.com	X
iajs-2767	19	49	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2767	19	50	mailto:ibrahim1992@tu.edu.iq	mailto:ibrahim1992@tu.edu.iq	PROPN
iajs-2767	19	51	mailto:hassan1962pl@tu.edu.iq	mailto:hassan1962pl@tu.edu.iq	PROPN
iajs-2767	19	52	mailto:aalfayadh@yahoo.com	mailto:aalfayadh@yahoo.com	X
iajs-2767	19	53	ibn	ibn	PROPN
iajs-2767	19	54	al	al	PROPN
iajs-2767	19	55	-	-	PUNCT
iajs-2767	19	56	haitham	haitham	PROPN
iajs-2767	19	57	jour	jour	X
iajs-2767	19	58	.	.	PROPN
iajs-2767	20	1	for	for	ADP
iajs-2767	20	2	pure	pure	ADJ
iajs-2767	20	3	&	&	CCONJ
iajs-2767	20	4	appl	appl	PROPN
iajs-2767	20	5	.	.	PUNCT
iajs-2767	21	1	sci	sci	PROPN
iajs-2767	21	2	.	.	PROPN
iajs-2767	22	1	53	53	NUM
iajs-2767	22	2	(	(	PUNCT
iajs-2767	22	3	2)2022	2)2022	VERB
iajs-2767	22	4	38	38	NUM
iajs-2767	22	5	for	for	ADP
iajs-2767	22	6	example	example	NOUN
iajs-2767	22	7	see	see	VERB
iajs-2767	22	8	[	[	X
iajs-2767	22	9	2	2	NUM
iajs-2767	22	10	-	-	SYM
iajs-2767	22	11	6	6	NUM
iajs-2767	22	12	]	]	PUNCT
iajs-2767	22	13	.	.	PUNCT
iajs-2767	23	1	zadeh	zadeh	PROPN
iajs-2767	23	2	in	in	ADP
iajs-2767	23	3	1965	1965	NUM
iajs-2767	23	4	[	[	X
iajs-2767	23	5	7	7	NUM
iajs-2767	23	6	]	]	PUNCT
iajs-2767	23	7	first	first	ADV
iajs-2767	23	8	introduced	introduce	VERB
iajs-2767	23	9	the	the	DET
iajs-2767	23	10	concept	concept	NOUN
iajs-2767	23	11	of	of	ADP
iajs-2767	23	12	the	the	DET
iajs-2767	23	13	fuzzy	fuzzy	ADJ
iajs-2767	23	14	set	set	NOUN
iajs-2767	23	15	where	where	SCONJ
iajs-2767	23	16	𝒳	𝒳	PROPN
iajs-2767	23	17	is	be	AUX
iajs-2767	23	18	a	a	DET
iajs-2767	23	19	nonempty	nonempty	ADJ
iajs-2767	23	20	set	set	NOUN
iajs-2767	23	21	,	,	PUNCT
iajs-2767	23	22	then	then	ADV
iajs-2767	23	23	a	a	DET
iajs-2767	23	24	fuzzy	fuzzy	ADJ
iajs-2767	23	25	set	set	VERB
iajs-2767	23	26	f	f	PROPN
iajs-2767	23	27	in	in	ADP
iajs-2767	23	28	𝒳	𝒳	PROPN
iajs-2767	23	29	was	be	AUX
iajs-2767	23	30	defined	define	VERB
iajs-2767	23	31	as	as	ADP
iajs-2767	23	32	a	a	DET
iajs-2767	23	33	set	set	NOUN
iajs-2767	23	34	of	of	ADP
iajs-2767	23	35	ordered	order	VERB
iajs-2767	23	36	pairs	pair	NOUN
iajs-2767	23	37	{	{	PUNCT
iajs-2767	23	38	(	(	PUNCT
iajs-2767	23	39	ω	ω	NOUN
iajs-2767	23	40	,	,	PUNCT
iajs-2767	23	41	𝓋f(𝜔	𝓋f(𝜔	NUM
iajs-2767	23	42	)	)	PUNCT
iajs-2767	23	43	)	)	PUNCT
iajs-2767	23	44	:	:	PUNCT
iajs-2767	24	1	𝜔	𝜔	X
iajs-2767	24	2	∈	∈	PROPN
iajs-2767	24	3	𝒳	𝒳	NOUN
iajs-2767	24	4	}	}	PUNCT
iajs-2767	24	5	where	where	SCONJ
iajs-2767	24	6	𝓋f	𝓋f	ADV
iajs-2767	24	7	:	:	PUNCT
iajs-2767	24	8	𝒳	𝒳	PROPN
iajs-2767	24	9	→	→	SYM
iajs-2767	24	10	[	[	X
iajs-2767	24	11	0	0	NUM
iajs-2767	24	12	,	,	PUNCT
iajs-2767	24	13	1	1	NUM
iajs-2767	24	14	]	]	PUNCT
iajs-2767	24	15	was	be	AUX
iajs-2767	24	16	a	a	DET
iajs-2767	24	17	function	function	NOUN
iajs-2767	24	18	such	such	ADJ
iajs-2767	24	19	that	that	PRON
iajs-2767	24	20	for	for	ADP
iajs-2767	24	21	every	every	DET
iajs-2767	24	22	𝜔	𝜔	PRON
iajs-2767	24	23	∈	∈	PROPN
iajs-2767	24	24	𝒳	𝒳	PROPN
iajs-2767	24	25	,	,	PUNCT
iajs-2767	24	26	𝓋f(𝜔	𝓋f(𝜔	NUM
iajs-2767	24	27	)	)	PUNCT
iajs-2767	24	28	represented	represent	VERB
iajs-2767	24	29	the	the	DET
iajs-2767	24	30	degree	degree	NOUN
iajs-2767	24	31	of	of	ADP
iajs-2767	24	32	membership	membership	NOUN
iajs-2767	24	33	of	of	ADP
iajs-2767	24	34	𝜔	𝜔	PRON
iajs-2767	24	35	in	in	ADP
iajs-2767	24	36	f.	f.	PROPN
iajs-2767	24	37	brown	brown	PROPN
iajs-2767	25	1	[	[	X
iajs-2767	25	2	8	8	NUM
iajs-2767	25	3	]	]	PUNCT
iajs-2767	25	4	and	and	CCONJ
iajs-2767	25	5	wang	wang	PROPN
iajs-2767	25	6	[	[	X
iajs-2767	25	7	9	9	NUM
iajs-2767	25	8	]	]	PUNCT
iajs-2767	25	9	studied	study	VERB
iajs-2767	25	10	some	some	DET
iajs-2767	25	11	types	type	NOUN
iajs-2767	25	12	of	of	ADP
iajs-2767	25	13	fuzzy	fuzzy	ADJ
iajs-2767	25	14	sets	set	NOUN
iajs-2767	25	15	such	such	ADJ
iajs-2767	25	16	as	as	ADP
iajs-2767	25	17	fuzzy	fuzzy	ADJ
iajs-2767	25	18	power	power	NOUN
iajs-2767	25	19	set	set	NOUN
iajs-2767	25	20	,	,	PUNCT
iajs-2767	25	21	empty	empty	ADJ
iajs-2767	25	22	fuzzy	fuzzy	ADJ
iajs-2767	25	23	set	set	NOUN
iajs-2767	25	24	,	,	PUNCT
iajs-2767	25	25	universal	universal	ADJ
iajs-2767	25	26	fuzzy	fuzzy	ADJ
iajs-2767	25	27	set	set	NOUN
iajs-2767	25	28	,	,	PUNCT
iajs-2767	25	29	the	the	DET
iajs-2767	25	30	complement	complement	NOUN
iajs-2767	25	31	of	of	ADP
iajs-2767	25	32	a	a	DET
iajs-2767	25	33	fuzzy	fuzzy	ADJ
iajs-2767	25	34	set	set	NOUN
iajs-2767	25	35	,	,	PUNCT
iajs-2767	25	36	the	the	DET
iajs-2767	25	37	union	union	NOUN
iajs-2767	25	38	of	of	ADP
iajs-2767	25	39	two	two	NUM
iajs-2767	25	40	fuzzy	fuzzy	ADJ
iajs-2767	25	41	sets	set	NOUN
iajs-2767	25	42	and	and	CCONJ
iajs-2767	25	43	the	the	DET
iajs-2767	25	44	intersection	intersection	NOUN
iajs-2767	25	45	of	of	ADP
iajs-2767	25	46	two	two	NUM
iajs-2767	25	47	fuzzy	fuzzy	ADJ
iajs-2767	25	48	sets	set	NOUN
iajs-2767	25	49	.	.	PUNCT
iajs-2767	26	1	ahmed	ahmed	PROPN
iajs-2767	26	2	et	et	PROPN
iajs-2767	26	3	al	al	PROPN
iajs-2767	27	1	[	[	X
iajs-2767	27	2	10	10	NUM
iajs-2767	27	3	]	]	PUNCT
iajs-2767	27	4	first	first	ADV
iajs-2767	27	5	introduced	introduce	VERB
iajs-2767	27	6	the	the	DET
iajs-2767	27	7	concept	concept	NOUN
iajs-2767	27	8	of	of	ADP
iajs-2767	27	9	fuzzy	fuzzy	ADJ
iajs-2767	27	10	σ	σ	NOUN
iajs-2767	27	11	−algebra	−algebra	NOUN
iajs-2767	27	12	and	and	CCONJ
iajs-2767	27	13	fuzzy	fuzzy	ADJ
iajs-2767	27	14	algebra	algebra	NOUN
iajs-2767	27	15	,	,	PUNCT
iajs-2767	27	16	where	where	SCONJ
iajs-2767	27	17	𝒳	𝒳	PROPN
iajs-2767	27	18	≠	≠	PROPN
iajs-2767	27	19	∅.	∅.	VERB
iajs-2767	27	20	a	a	DET
iajs-2767	27	21	nonempty	nonempty	ADJ
iajs-2767	27	22	class	class	NOUN
iajs-2767	27	23	ℋ∗	ℋ∗	NOUN
iajs-2767	27	24	⊆	⊆	NUM
iajs-2767	27	25	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	27	26	)	)	PUNCT
iajs-2767	27	27	is	be	AUX
iajs-2767	27	28	called	call	VERB
iajs-2767	27	29	a	a	DET
iajs-2767	27	30	fuzzy	fuzzy	ADJ
iajs-2767	27	31	σ	σ	PROPN
iajs-2767	27	32	–	–	PUNCT
iajs-2767	27	33	algebra	algebra	NOUN
iajs-2767	27	34	over	over	ADP
iajs-2767	27	35	a	a	DET
iajs-2767	27	36	fuzzy	fuzzy	ADJ
iajs-2767	27	37	set	set	VERB
iajs-2767	27	38	𝒳∗	𝒳∗	INTJ
iajs-2767	27	39	,	,	PUNCT
iajs-2767	27	40	if	if	SCONJ
iajs-2767	27	41	1	1	NUM
iajs-2767	27	42	.	.	PUNCT
iajs-2767	27	43	∅∗	∅∗	PROPN
iajs-2767	27	44	∈	∈	PROPN
iajs-2767	27	45	ℋ∗	ℋ∗	NUM
iajs-2767	27	46	,	,	PUNCT
iajs-2767	27	47	where	where	SCONJ
iajs-2767	27	48	∅∗	∅∗	VERB
iajs-2767	27	49	=	=	PRON
iajs-2767	27	50	{	{	PUNCT
iajs-2767	27	51	(	(	PUNCT
iajs-2767	27	52	ω	ω	PROPN
iajs-2767	27	53	,	,	PUNCT
iajs-2767	27	54	0	0	NUM
iajs-2767	27	55	)	)	PUNCT
iajs-2767	27	56	:	:	PUNCT
iajs-2767	27	57	∀𝜔	∀𝜔	NUM
iajs-2767	27	58	∈	∈	PROPN
iajs-2767	27	59	𝒳	𝒳	PROPN
iajs-2767	27	60	}	}	PUNCT
iajs-2767	27	61	.	.	PUNCT
iajs-2767	28	1	2.if	2.if	NUM
iajs-2767	28	2	e	e	X
iajs-2767	28	3	∈	∈	PROPN
iajs-2767	28	4	ℋ∗	ℋ∗	NUM
iajs-2767	28	5	,	,	PUNCT
iajs-2767	28	6	then	then	ADV
iajs-2767	28	7	e𝑐	e𝑐	PROPN
iajs-2767	29	1	∈	∈	PROPN
iajs-2767	29	2	ℋ∗.	ℋ∗.	PROPN
iajs-2767	29	3	3.if	3.if	NUM
iajs-2767	29	4	e1	e1	PROPN
iajs-2767	29	5	,	,	PUNCT
iajs-2767	29	6	e2	e2	PROPN
iajs-2767	29	7	,	,	PUNCT
iajs-2767	29	8	…	…	PUNCT
iajs-2767	29	9	∈	∈	PROPN
iajs-2767	29	10	ℋ∗	ℋ∗	NUM
iajs-2767	29	11	,	,	PUNCT
iajs-2767	29	12	then	then	ADV
iajs-2767	29	13	⋂	⋂	PROPN
iajs-2767	29	14	ek	ek	PROPN
iajs-2767	29	15	∞	∞	PROPN
iajs-2767	29	16	𝑘=1	𝑘=1	SYM
iajs-2767	29	17	∈	∈	PROPN
iajs-2767	29	18	ℋ∗.	ℋ∗.	NOUN
iajs-2767	29	19	if	if	SCONJ
iajs-2767	29	20	condition	condition	NOUN
iajs-2767	29	21	3	3	NUM
iajs-2767	29	22	is	be	AUX
iajs-2767	29	23	satisfied	satisfied	ADJ
iajs-2767	29	24	only	only	ADV
iajs-2767	29	25	for	for	ADP
iajs-2767	29	26	finite	finite	ADJ
iajs-2767	29	27	sets	set	NOUN
iajs-2767	29	28	,	,	PUNCT
iajs-2767	29	29	then	then	ADV
iajs-2767	29	30	ℋ∗	ℋ∗	NUM
iajs-2767	29	31	is	be	AUX
iajs-2767	29	32	said	say	VERB
iajs-2767	29	33	to	to	PART
iajs-2767	29	34	be	be	AUX
iajs-2767	29	35	a	a	DET
iajs-2767	29	36	fuzzy	fuzzy	ADJ
iajs-2767	29	37	algebra	algebra	NOUN
iajs-2767	29	38	over	over	ADP
iajs-2767	29	39	a	a	DET
iajs-2767	29	40	fuzzy	fuzzy	ADJ
iajs-2767	29	41	set	set	VERB
iajs-2767	29	42	𝒳∗.	𝒳∗.	PROPN
iajs-2767	29	43	in	in	ADP
iajs-2767	29	44	this	this	DET
iajs-2767	29	45	work	work	NOUN
iajs-2767	29	46	we	we	PRON
iajs-2767	29	47	introduce	introduce	VERB
iajs-2767	29	48	the	the	DET
iajs-2767	29	49	concept	concept	NOUN
iajs-2767	29	50	of	of	ADP
iajs-2767	29	51	fuzzy	fuzzy	ADJ
iajs-2767	29	52	σ	σ	PROPN
iajs-2767	29	53	–	–	PUNCT
iajs-2767	29	54	ring	ring	NOUN
iajs-2767	29	55	and	and	CCONJ
iajs-2767	29	56	fuzzy	fuzzy	ADJ
iajs-2767	29	57	ring	ring	NOUN
iajs-2767	29	58	which	which	PRON
iajs-2767	29	59	are	be	AUX
iajs-2767	29	60	generalizations	generalization	NOUN
iajs-2767	29	61	of	of	ADP
iajs-2767	29	62	fuzzy	fuzzy	ADJ
iajs-2767	29	63	σ	σ	PROPN
iajs-2767	29	64	−algebra	−algebra	PROPN
iajs-2767	29	65	.	.	PUNCT
iajs-2767	30	1	the	the	DET
iajs-2767	30	2	main	main	ADJ
iajs-2767	30	3	goal	goal	NOUN
iajs-2767	30	4	of	of	ADP
iajs-2767	30	5	this	this	DET
iajs-2767	30	6	paper	paper	NOUN
iajs-2767	30	7	is	be	AUX
iajs-2767	30	8	to	to	PART
iajs-2767	30	9	study	study	VERB
iajs-2767	30	10	these	these	DET
iajs-2767	30	11	concepts	concept	NOUN
iajs-2767	30	12	,	,	PUNCT
iajs-2767	30	13	give	give	VERB
iajs-2767	30	14	basic	basic	ADJ
iajs-2767	30	15	properties	property	NOUN
iajs-2767	30	16	,	,	PUNCT
iajs-2767	30	17	examples	example	NOUN
iajs-2767	30	18	,	,	PUNCT
iajs-2767	30	19	characterizations	characterization	NOUN
iajs-2767	30	20	and	and	CCONJ
iajs-2767	30	21	studied	study	VERB
iajs-2767	30	22	relationships	relationship	NOUN
iajs-2767	30	23	between	between	ADP
iajs-2767	30	24	them	they	PRON
iajs-2767	30	25	.	.	PUNCT
iajs-2767	31	1	2	2	X
iajs-2767	31	2	.	.	X
iajs-2767	31	3	preliminaries	preliminary	NOUN
iajs-2767	31	4	this	this	DET
iajs-2767	31	5	section	section	NOUN
iajs-2767	31	6	is	be	AUX
iajs-2767	31	7	going	go	VERB
iajs-2767	31	8	to	to	PART
iajs-2767	31	9	review	review	VERB
iajs-2767	31	10	some	some	DET
iajs-2767	31	11	well	well	ADV
iajs-2767	31	12	-	-	PUNCT
iajs-2767	31	13	known	know	VERB
iajs-2767	31	14	definitions	definition	NOUN
iajs-2767	31	15	in	in	ADP
iajs-2767	31	16	measure	measure	NOUN
iajs-2767	31	17	theory	theory	NOUN
iajs-2767	31	18	.	.	PUNCT
iajs-2767	32	1	definition	definition	NOUN
iajs-2767	32	2	2.1	2.1	NUM
iajs-2767	32	3	[	[	X
iajs-2767	32	4	3	3	NUM
iajs-2767	32	5	]	]	PUNCT
iajs-2767	32	6	let	let	VERB
iajs-2767	32	7	𝒳	𝒳	PRON
iajs-2767	32	8	≠	≠	NOUN
iajs-2767	32	9	∅.	∅.	VERB
iajs-2767	32	10	a	a	DET
iajs-2767	32	11	collection	collection	NOUN
iajs-2767	32	12	ℋ	ℋ	PROPN
iajs-2767	32	13	is	be	AUX
iajs-2767	32	14	called	call	VERB
iajs-2767	32	15	σ	σ	PROPN
iajs-2767	32	16	–	–	PUNCT
iajs-2767	32	17	ring	ring	NOUN
iajs-2767	32	18	iff	iff	NOUN
iajs-2767	32	19	:	:	PUNCT
iajs-2767	32	20	1.if	1.if	NUM
iajs-2767	32	21	f	f	NOUN
iajs-2767	32	22	,	,	PUNCT
iajs-2767	32	23	e	e	PROPN
iajs-2767	32	24	∈	∈	PROPN
iajs-2767	32	25	ℋ	ℋ	PROPN
iajs-2767	32	26	,	,	PUNCT
iajs-2767	32	27	then	then	ADV
iajs-2767	32	28	f	f	X
iajs-2767	32	29	∖	∖	X
iajs-2767	32	30	e	e	PROPN
iajs-2767	32	31	∈	∈	PROPN
iajs-2767	32	32	ℋ	ℋ	PROPN
iajs-2767	32	33	.	.	PUNCT
iajs-2767	33	1	2.if	2.if	NUM
iajs-2767	33	2	e1	e1	PROPN
iajs-2767	33	3	,	,	PUNCT
iajs-2767	33	4	e2	e2	PROPN
iajs-2767	33	5	,	,	PUNCT
iajs-2767	33	6	…	…	PUNCT
iajs-2767	33	7	∈	∈	PROPN
iajs-2767	33	8	ℋ	ℋ	PROPN
iajs-2767	33	9	,	,	PUNCT
iajs-2767	33	10	then	then	ADV
iajs-2767	33	11	⋃	⋃	ADP
iajs-2767	33	12	ek	ek	PROPN
iajs-2767	33	13	∞	∞	PROPN
iajs-2767	33	14	𝑘=1	𝑘=1	SYM
iajs-2767	33	15	∈	∈	PROPN
iajs-2767	33	16	ℋ	ℋ	PROPN
iajs-2767	33	17	.	.	PUNCT
iajs-2767	34	1	definition	definition	NOUN
iajs-2767	34	2	2.2	2.2	NUM
iajs-2767	34	3	[	[	X
iajs-2767	34	4	1	1	NUM
iajs-2767	34	5	]	]	PUNCT
iajs-2767	34	6	let	let	VERB
iajs-2767	34	7	𝒳	𝒳	PRON
iajs-2767	34	8	≠	≠	NOUN
iajs-2767	34	9	∅.	∅.	VERB
iajs-2767	34	10	a	a	DET
iajs-2767	34	11	collection	collection	NOUN
iajs-2767	34	12	ℋ	ℋ	PROPN
iajs-2767	34	13	is	be	AUX
iajs-2767	34	14	called	call	VERB
iajs-2767	34	15	σ	σ	PROPN
iajs-2767	34	16	–	–	PUNCT
iajs-2767	34	17	field	field	NOUN
iajs-2767	34	18	iff	iff	NOUN
iajs-2767	34	19	:	:	PUNCT
iajs-2767	34	20	1.𝒳	1.𝒳	NUM
iajs-2767	34	21	∈	∈	PROPN
iajs-2767	34	22	ℋ	ℋ	PROPN
iajs-2767	34	23	.	.	PUNCT
iajs-2767	35	1	2.if	2.if	NUM
iajs-2767	35	2	f	f	X
iajs-2767	35	3	∈	∈	PROPN
iajs-2767	35	4	ℋ	ℋ	PROPN
iajs-2767	35	5	,	,	PUNCT
iajs-2767	35	6	then	then	ADV
iajs-2767	35	7	f𝑐	f𝑐	ADP
iajs-2767	35	8	∈	∈	PROPN
iajs-2767	35	9	ℋ	ℋ	PROPN
iajs-2767	35	10	.	.	PUNCT
iajs-2767	36	1	3.if	3.if	NUM
iajs-2767	36	2	e1	e1	PROPN
iajs-2767	36	3	,	,	PUNCT
iajs-2767	36	4	e2	e2	PROPN
iajs-2767	36	5	,	,	PUNCT
iajs-2767	36	6	…	…	PUNCT
iajs-2767	36	7	∈	∈	PROPN
iajs-2767	36	8	ℋ	ℋ	PROPN
iajs-2767	36	9	,	,	PUNCT
iajs-2767	36	10	then	then	ADV
iajs-2767	36	11	⋃	⋃	ADP
iajs-2767	36	12	ek	ek	PROPN
iajs-2767	36	13	∞	∞	PROPN
iajs-2767	36	14	𝑘=1	𝑘=1	PUNCT
iajs-2767	36	15	∈	∈	PROPN
iajs-2767	36	16	ℋ	ℋ	PROPN
iajs-2767	36	17	.	.	PUNCT
iajs-2767	36	18	proposition	proposition	NOUN
iajs-2767	36	19	2.3	2.3	NUM
iajs-2767	37	1	[	[	X
iajs-2767	37	2	5	5	NUM
iajs-2767	37	3	]	]	PUNCT
iajs-2767	37	4	every	every	DET
iajs-2767	37	5	σ	σ	PROPN
iajs-2767	37	6	–	–	PUNCT
iajs-2767	37	7	field	field	NOUN
iajs-2767	37	8	is	be	AUX
iajs-2767	37	9	a	a	DET
iajs-2767	37	10	σ	σ	PROPN
iajs-2767	37	11	–	–	PUNCT
iajs-2767	37	12	ring	ring	NOUN
iajs-2767	37	13	.	.	PUNCT
iajs-2767	38	1	definition	definition	NOUN
iajs-2767	38	2	2.4	2.4	NUM
iajs-2767	38	3	[	[	SYM
iajs-2767	38	4	8	8	NUM
iajs-2767	38	5	,	,	PUNCT
iajs-2767	38	6	9	9	NUM
iajs-2767	38	7	]	]	PUNCT
iajs-2767	38	8	let	let	VERB
iajs-2767	38	9	𝒳	𝒳	PRON
iajs-2767	38	10	be	be	AUX
iajs-2767	38	11	a	a	DET
iajs-2767	38	12	nonempty	nonempty	ADV
iajs-2767	38	13	set	set	VERB
iajs-2767	38	14	.	.	PUNCT
iajs-2767	39	1	then	then	ADV
iajs-2767	39	2	:	:	PUNCT
iajs-2767	39	3	1.the	1.the	DET
iajs-2767	39	4	collection	collection	NOUN
iajs-2767	39	5	of	of	ADP
iajs-2767	39	6	all	all	DET
iajs-2767	39	7	fuzzy	fuzzy	ADJ
iajs-2767	39	8	sets	set	NOUN
iajs-2767	39	9	in	in	ADP
iajs-2767	39	10	𝒳	𝒳	PROPN
iajs-2767	39	11	is	be	AUX
iajs-2767	39	12	called	call	VERB
iajs-2767	39	13	a	a	DET
iajs-2767	39	14	fuzzy	fuzzy	ADJ
iajs-2767	39	15	power	power	NOUN
iajs-2767	39	16	set	set	NOUN
iajs-2767	39	17	and	and	CCONJ
iajs-2767	39	18	is	be	AUX
iajs-2767	39	19	denoted	denote	VERB
iajs-2767	39	20	by	by	ADP
iajs-2767	39	21	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	39	22	)	)	PUNCT
iajs-2767	39	23	,	,	PUNCT
iajs-2767	39	24	in	in	ADP
iajs-2767	39	25	symbols	symbol	NOUN
iajs-2767	39	26	:	:	PUNCT
iajs-2767	39	27	𝒫∗(𝒳	𝒫∗(𝒳	X
iajs-2767	39	28	)	)	PUNCT
iajs-2767	39	29	=	=	PRON
iajs-2767	39	30	{	{	PUNCT
iajs-2767	39	31	f	f	PROPN
iajs-2767	39	32	∶	∶	NOUN
iajs-2767	39	33	f	f	PROPN
iajs-2767	39	34	is	be	AUX
iajs-2767	39	35	a	a	DET
iajs-2767	39	36	fuzzy	fuzzy	ADJ
iajs-2767	39	37	set	set	NOUN
iajs-2767	39	38	in	in	ADP
iajs-2767	39	39	𝒳	𝒳	PROPN
iajs-2767	39	40	}	}	PUNCT
iajs-2767	39	41	.	.	PUNCT
iajs-2767	40	1	2.the	2.the	DET
iajs-2767	40	2	empty	empty	ADJ
iajs-2767	40	3	fuzzy	fuzzy	ADJ
iajs-2767	40	4	set	set	NOUN
iajs-2767	40	5	in	in	ADP
iajs-2767	40	6	𝒳	𝒳	PROPN
iajs-2767	40	7	is	be	AUX
iajs-2767	40	8	denoted	denote	VERB
iajs-2767	40	9	by	by	ADP
iajs-2767	40	10	∅∗	∅∗	PROPN
iajs-2767	40	11	and	and	CCONJ
iajs-2767	40	12	defined	define	VERB
iajs-2767	40	13	as	as	ADP
iajs-2767	40	14	:	:	PUNCT
iajs-2767	40	15	∅∗	∅∗	PART
iajs-2767	40	16	=	=	PRON
iajs-2767	40	17	{	{	PUNCT
iajs-2767	40	18	(	(	PUNCT
iajs-2767	40	19	ω	ω	PROPN
iajs-2767	40	20	,	,	PUNCT
iajs-2767	40	21	0	0	NUM
iajs-2767	40	22	)	)	PUNCT
iajs-2767	40	23	:	:	PUNCT
iajs-2767	41	1	∀𝜔	∀𝜔	NUM
iajs-2767	41	2	∈	∈	PROPN
iajs-2767	41	3	𝒳	𝒳	PROPN
iajs-2767	41	4	}	}	PUNCT
iajs-2767	41	5	.	.	PUNCT
iajs-2767	42	1	3.the	3.the	DET
iajs-2767	42	2	fuzzy	fuzzy	ADJ
iajs-2767	42	3	set	set	VERB
iajs-2767	42	4	𝒳∗	𝒳∗	X
iajs-2767	42	5	in	in	ADP
iajs-2767	42	6	𝒳	𝒳	PROPN
iajs-2767	42	7	is	be	AUX
iajs-2767	42	8	defined	define	VERB
iajs-2767	42	9	as	as	ADP
iajs-2767	42	10	:	:	PUNCT
iajs-2767	42	11	𝒳∗	𝒳∗	X
iajs-2767	43	1	=	=	PRON
iajs-2767	43	2	{	{	PUNCT
iajs-2767	43	3	(	(	PUNCT
iajs-2767	43	4	ω	ω	PROPN
iajs-2767	43	5	,	,	PUNCT
iajs-2767	43	6	1	1	NUM
iajs-2767	43	7	)	)	PUNCT
iajs-2767	43	8	:	:	PUNCT
iajs-2767	43	9	∀𝜔	∀𝜔	NUM
iajs-2767	43	10	∈	∈	PROPN
iajs-2767	43	11	𝒳	𝒳	PROPN
iajs-2767	43	12	}	}	PUNCT
iajs-2767	43	13	.	.	PUNCT
iajs-2767	44	1	definition	definition	NOUN
iajs-2767	44	2	2.5	2.5	NUM
iajs-2767	44	3	[	[	X
iajs-2767	44	4	7	7	NUM
iajs-2767	44	5	]	]	PUNCT
iajs-2767	44	6	let	let	VERB
iajs-2767	44	7	𝒳	𝒳	PRON
iajs-2767	44	8	be	be	AUX
iajs-2767	44	9	a	a	DET
iajs-2767	44	10	nonempty	nonempty	ADJ
iajs-2767	44	11	set	set	VERB
iajs-2767	44	12	.	.	PUNCT
iajs-2767	45	1	then	then	ADV
iajs-2767	45	2	the	the	DET
iajs-2767	45	3	union	union	NOUN
iajs-2767	45	4	of	of	ADP
iajs-2767	45	5	the	the	DET
iajs-2767	45	6	two	two	NUM
iajs-2767	45	7	fuzzy	fuzzy	ADJ
iajs-2767	45	8	sets	set	NOUN
iajs-2767	45	9	f	f	NOUN
iajs-2767	45	10	and	and	CCONJ
iajs-2767	45	11	e	e	PROPN
iajs-2767	45	12	in	in	ADP
iajs-2767	45	13	𝒳	𝒳	PROPN
iajs-2767	45	14	with	with	ADP
iajs-2767	45	15	respective	respective	ADJ
iajs-2767	45	16	membership	membership	NOUN
iajs-2767	45	17	functions	function	NOUN
iajs-2767	45	18	𝓋f(𝜔	𝓋f(𝜔	PUNCT
iajs-2767	45	19	)	)	PUNCT
iajs-2767	45	20	and	and	CCONJ
iajs-2767	45	21	𝓋e(𝜔	𝓋e(𝜔	NUM
iajs-2767	45	22	)	)	PUNCT
iajs-2767	45	23	is	be	AUX
iajs-2767	45	24	a	a	DET
iajs-2767	45	25	fuzzy	fuzzy	ADJ
iajs-2767	45	26	set	set	VERB
iajs-2767	45	27	g	g	NOUN
iajs-2767	45	28	in	in	ADP
iajs-2767	45	29	𝒳	𝒳	ADP
iajs-2767	45	30	whose	whose	DET
iajs-2767	45	31	membership	membership	NOUN
iajs-2767	45	32	function	function	NOUN
iajs-2767	45	33	is	be	AUX
iajs-2767	45	34	related	relate	VERB
iajs-2767	45	35	to	to	ADP
iajs-2767	45	36	those	those	PRON
iajs-2767	45	37	of	of	ADP
iajs-2767	45	38	f	f	PROPN
iajs-2767	45	39	and	and	CCONJ
iajs-2767	45	40	e	e	X
iajs-2767	45	41	by	by	ADP
iajs-2767	45	42	𝓋g(𝜔	𝓋g(𝜔	NOUN
iajs-2767	45	43	)	)	PUNCT
iajs-2767	45	44	=	=	SYM
iajs-2767	45	45	max	max	PROPN
iajs-2767	45	46	𝜔∈𝒳	𝜔∈𝒳	PROPN
iajs-2767	45	47	{	{	PUNCT
iajs-2767	45	48	𝓋f(𝜔	𝓋f(𝜔	PROPN
iajs-2767	45	49	)	)	PUNCT
iajs-2767	45	50	,	,	PUNCT
iajs-2767	45	51	𝓋e(𝜔	𝓋e(𝜔	X
iajs-2767	45	52	)	)	PUNCT
iajs-2767	45	53	}	}	PUNCT
iajs-2767	45	54	,	,	PUNCT
iajs-2767	45	55	in	in	ADP
iajs-2767	45	56	symbols	symbol	NOUN
iajs-2767	45	57	:	:	PUNCT
iajs-2767	45	58	ibn	ibn	PROPN
iajs-2767	45	59	al	al	PROPN
iajs-2767	45	60	-	-	PUNCT
iajs-2767	45	61	haitham	haitham	PROPN
iajs-2767	45	62	jour	jour	X
iajs-2767	45	63	.	.	PROPN
iajs-2767	46	1	for	for	ADP
iajs-2767	46	2	pure	pure	ADJ
iajs-2767	46	3	&	&	CCONJ
iajs-2767	46	4	appl	appl	PROPN
iajs-2767	46	5	.	.	PUNCT
iajs-2767	47	1	sci	sci	PROPN
iajs-2767	47	2	.	.	PROPN
iajs-2767	48	1	53	53	NUM
iajs-2767	48	2	(	(	PUNCT
iajs-2767	48	3	2)2022	2)2022	VERB
iajs-2767	48	4	39	39	NUM
iajs-2767	48	5	g	g	NOUN
iajs-2767	48	6	=	=	PUNCT
iajs-2767	48	7	f⋃e	f⋃e	PROPN
iajs-2767	48	8	⇔	⇔	PROPN
iajs-2767	48	9	g	g	PROPN
iajs-2767	48	10	=	=	PRON
iajs-2767	48	11	{	{	PUNCT
iajs-2767	48	12	(	(	PUNCT
iajs-2767	48	13	ω	ω	PROPN
iajs-2767	48	14	,	,	PUNCT
iajs-2767	48	15	max	max	PROPN
iajs-2767	48	16	𝜔∈𝒳	𝜔∈𝒳	PROPN
iajs-2767	48	17	{	{	PUNCT
iajs-2767	48	18	𝓋f(𝜔	𝓋f(𝜔	PROPN
iajs-2767	48	19	)	)	PUNCT
iajs-2767	48	20	,	,	PUNCT
iajs-2767	48	21	𝓋e(𝜔	𝓋e(𝜔	X
iajs-2767	48	22	)	)	PUNCT
iajs-2767	48	23	}	}	PUNCT
iajs-2767	48	24	)	)	PUNCT
iajs-2767	49	1	∶	∶	VERB
iajs-2767	49	2	𝜔	𝜔	ADP
iajs-2767	49	3	∈	∈	NOUN
iajs-2767	49	4	𝒳	𝒳	PROPN
iajs-2767	49	5	}	}	PUNCT
iajs-2767	49	6	.	.	PUNCT
iajs-2767	50	1	definition	definition	NOUN
iajs-2767	50	2	2.6	2.6	NUM
iajs-2767	51	1	[	[	SYM
iajs-2767	51	2	8	8	NUM
iajs-2767	51	3	]	]	PUNCT
iajs-2767	51	4	let	let	VERB
iajs-2767	51	5	𝒳	𝒳	PRON
iajs-2767	51	6	be	be	AUX
iajs-2767	51	7	a	a	DET
iajs-2767	51	8	nonempty	nonempty	ADJ
iajs-2767	51	9	set	set	VERB
iajs-2767	51	10	.	.	PUNCT
iajs-2767	52	1	then	then	ADV
iajs-2767	52	2	the	the	DET
iajs-2767	52	3	intersection	intersection	NOUN
iajs-2767	52	4	of	of	ADP
iajs-2767	52	5	two	two	NUM
iajs-2767	52	6	fuzzy	fuzzy	ADJ
iajs-2767	52	7	sets	set	NOUN
iajs-2767	52	8	f	f	NOUN
iajs-2767	52	9	and	and	CCONJ
iajs-2767	52	10	e	e	PROPN
iajs-2767	52	11	in	in	ADP
iajs-2767	52	12	𝒳	𝒳	PROPN
iajs-2767	52	13	with	with	ADP
iajs-2767	52	14	respective	respective	ADJ
iajs-2767	52	15	membership	membership	NOUN
iajs-2767	52	16	functions	function	NOUN
iajs-2767	52	17	𝓋f(𝜔	𝓋f(𝜔	PUNCT
iajs-2767	52	18	)	)	PUNCT
iajs-2767	52	19	and	and	CCONJ
iajs-2767	52	20	𝓋e(𝜔	𝓋e(𝜔	NUM
iajs-2767	52	21	)	)	PUNCT
iajs-2767	52	22	is	be	AUX
iajs-2767	52	23	a	a	DET
iajs-2767	52	24	fuzzy	fuzzy	ADJ
iajs-2767	52	25	set	set	VERB
iajs-2767	52	26	g	g	NOUN
iajs-2767	52	27	in	in	ADP
iajs-2767	52	28	𝒳	𝒳	ADP
iajs-2767	52	29	whose	whose	DET
iajs-2767	52	30	membership	membership	NOUN
iajs-2767	52	31	function	function	NOUN
iajs-2767	52	32	is	be	AUX
iajs-2767	52	33	related	relate	VERB
iajs-2767	52	34	to	to	ADP
iajs-2767	52	35	those	those	PRON
iajs-2767	52	36	of	of	ADP
iajs-2767	52	37	f	f	PROPN
iajs-2767	52	38	and	and	CCONJ
iajs-2767	52	39	e	e	X
iajs-2767	52	40	by	by	ADP
iajs-2767	52	41	𝓋g(𝜔	𝓋g(𝜔	NOUN
iajs-2767	52	42	)	)	PUNCT
iajs-2767	52	43	=	=	SYM
iajs-2767	52	44	min	min	NOUN
iajs-2767	52	45	𝜔∈𝒳	𝜔∈𝒳	X
iajs-2767	52	46	{	{	PUNCT
iajs-2767	52	47	𝓋f(𝜔	𝓋f(𝜔	PROPN
iajs-2767	52	48	)	)	PUNCT
iajs-2767	52	49	,	,	PUNCT
iajs-2767	52	50	𝓋e(𝜔	𝓋e(𝜔	X
iajs-2767	52	51	)	)	PUNCT
iajs-2767	52	52	}	}	PUNCT
iajs-2767	52	53	,	,	PUNCT
iajs-2767	52	54	in	in	ADP
iajs-2767	52	55	symbols	symbol	NOUN
iajs-2767	52	56	:	:	PUNCT
iajs-2767	52	57	g	g	PROPN
iajs-2767	52	58	=	=	SYM
iajs-2767	52	59	f⋂e	f⋂e	PROPN
iajs-2767	52	60	⇔	⇔	PROPN
iajs-2767	52	61	g	g	PROPN
iajs-2767	52	62	=	=	PRON
iajs-2767	52	63	{	{	PUNCT
iajs-2767	52	64	(	(	PUNCT
iajs-2767	52	65	ω	ω	PROPN
iajs-2767	52	66	,	,	PUNCT
iajs-2767	52	67	min	min	PROPN
iajs-2767	52	68	𝜔∈𝒳	𝜔∈𝒳	X
iajs-2767	52	69	{	{	PUNCT
iajs-2767	52	70	𝓋f(𝜔	𝓋f(𝜔	PROPN
iajs-2767	52	71	)	)	PUNCT
iajs-2767	52	72	,	,	PUNCT
iajs-2767	52	73	𝓋e(𝜔	𝓋e(𝜔	X
iajs-2767	52	74	)	)	PUNCT
iajs-2767	52	75	}	}	PUNCT
iajs-2767	52	76	)	)	PUNCT
iajs-2767	53	1	∶	∶	VERB
iajs-2767	53	2	𝜔	𝜔	ADP
iajs-2767	53	3	∈	∈	NOUN
iajs-2767	53	4	𝒳	𝒳	PROPN
iajs-2767	53	5	}	}	PUNCT
iajs-2767	53	6	.	.	PUNCT
iajs-2767	54	1	definition	definition	NOUN
iajs-2767	54	2	2.7	2.7	NUM
iajs-2767	54	3	[	[	X
iajs-2767	54	4	9	9	NUM
iajs-2767	54	5	]	]	PUNCT
iajs-2767	54	6	let	let	VERB
iajs-2767	54	7	𝒳	𝒳	PRON
iajs-2767	54	8	be	be	AUX
iajs-2767	54	9	a	a	DET
iajs-2767	54	10	nonempty	nonempty	ADV
iajs-2767	54	11	set	set	VERB
iajs-2767	54	12	and	and	CCONJ
iajs-2767	54	13	f	f	PROPN
iajs-2767	54	14	is	be	AUX
iajs-2767	54	15	a	a	DET
iajs-2767	54	16	fuzzy	fuzzy	ADJ
iajs-2767	54	17	set	set	NOUN
iajs-2767	54	18	in	in	ADP
iajs-2767	54	19	𝒳.	𝒳.	PROPN
iajs-2767	54	20	then	then	ADV
iajs-2767	54	21	the	the	DET
iajs-2767	54	22	complement	complement	NOUN
iajs-2767	54	23	of	of	ADP
iajs-2767	54	24	a	a	DET
iajs-2767	54	25	fuzzy	fuzzy	ADJ
iajs-2767	54	26	set	set	NOUN
iajs-2767	54	27	f	f	PROPN
iajs-2767	54	28	is	be	AUX
iajs-2767	54	29	denoted	denote	VERB
iajs-2767	54	30	by	by	ADP
iajs-2767	54	31	fc	fc	PROPN
iajs-2767	54	32	and	and	CCONJ
iajs-2767	54	33	defined	define	VERB
iajs-2767	54	34	as	as	ADP
iajs-2767	54	35	:	:	PUNCT
iajs-2767	54	36	fc	fc	PROPN
iajs-2767	54	37	=	=	PRON
iajs-2767	54	38	{	{	PUNCT
iajs-2767	54	39	(	(	PUNCT
iajs-2767	54	40	ω	ω	PROPN
iajs-2767	54	41	,	,	PUNCT
iajs-2767	54	42	1	1	NUM
iajs-2767	54	43	−	−	NOUN
iajs-2767	54	44	𝓋f(𝜔	𝓋f(𝜔	NUM
iajs-2767	54	45	)	)	PUNCT
iajs-2767	54	46	)	)	PUNCT
iajs-2767	54	47	:	:	PUNCT
iajs-2767	55	1	𝜔	𝜔	X
iajs-2767	55	2	∈	∈	PROPN
iajs-2767	55	3	𝒳	𝒳	PROPN
iajs-2767	55	4	}	}	PUNCT
iajs-2767	55	5	.	.	PUNCT
iajs-2767	56	1	proposition	proposition	NOUN
iajs-2767	56	2	2.8	2.8	NUM
iajs-2767	56	3	[	[	SYM
iajs-2767	56	4	10	10	NUM
iajs-2767	56	5	]	]	PUNCT
iajs-2767	56	6	every	every	DET
iajs-2767	56	7	fuzzy	fuzzy	ADJ
iajs-2767	56	8	σ	σ	PROPN
iajs-2767	56	9	–	–	PUNCT
iajs-2767	56	10	algebra	algebra	NOUN
iajs-2767	56	11	is	be	AUX
iajs-2767	56	12	a	a	DET
iajs-2767	56	13	fuzzy	fuzzy	ADJ
iajs-2767	56	14	algebra	algebra	NOUN
iajs-2767	56	15	.	.	PUNCT
iajs-2767	57	1	3	3	X
iajs-2767	57	2	.	.	X
iajs-2767	57	3	the	the	DET
iajs-2767	57	4	main	main	ADJ
iajs-2767	57	5	results	result	NOUN
iajs-2767	57	6	:	:	PUNCT
iajs-2767	57	7	in	in	ADP
iajs-2767	57	8	this	this	DET
iajs-2767	57	9	section	section	NOUN
iajs-2767	57	10	,	,	PUNCT
iajs-2767	57	11	the	the	DET
iajs-2767	57	12	basic	basic	ADJ
iajs-2767	57	13	definitions	definition	NOUN
iajs-2767	57	14	and	and	CCONJ
iajs-2767	57	15	facts	fact	NOUN
iajs-2767	57	16	related	relate	VERB
iajs-2767	57	17	to	to	ADP
iajs-2767	57	18	this	this	DET
iajs-2767	57	19	work	work	NOUN
iajs-2767	57	20	are	be	AUX
iajs-2767	57	21	recalled	recall	VERB
iajs-2767	57	22	,	,	PUNCT
iajs-2767	57	23	which	which	PRON
iajs-2767	57	24	starts	start	VERB
iajs-2767	57	25	with	with	ADP
iajs-2767	57	26	the	the	DET
iajs-2767	57	27	following	follow	VERB
iajs-2767	57	28	definition	definition	NOUN
iajs-2767	57	29	.	.	PUNCT
iajs-2767	58	1	definition	definition	NOUN
iajs-2767	58	2	3.1	3.1	NUM
iajs-2767	58	3	let	let	VERB
iajs-2767	58	4	𝒳	𝒳	PRON
iajs-2767	58	5	≠	≠	NOUN
iajs-2767	58	6	∅.	∅.	VERB
iajs-2767	58	7	a	a	DET
iajs-2767	58	8	collection	collection	NOUN
iajs-2767	58	9	ℋ∗	ℋ∗	VERB
iajs-2767	58	10	⊆	⊆	NUM
iajs-2767	58	11	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	58	12	)	)	PUNCT
iajs-2767	58	13	is	be	AUX
iajs-2767	58	14	a	a	DET
iajs-2767	58	15	fuzzy	fuzzy	ADJ
iajs-2767	58	16	σ	σ	NOUN
iajs-2767	58	17	–	–	PUNCT
iajs-2767	58	18	ring	ring	NOUN
iajs-2767	58	19	over	over	ADP
iajs-2767	58	20	a	a	DET
iajs-2767	58	21	fuzzy	fuzzy	ADJ
iajs-2767	58	22	set	set	VERB
iajs-2767	58	23	𝒳∗	𝒳∗	PROPN
iajs-2767	58	24	,	,	PUNCT
iajs-2767	58	25	iff	iff	PROPN
iajs-2767	58	26	4	4	NUM
iajs-2767	58	27	.	.	PUNCT
iajs-2767	59	1	∅∗	∅∗	PROPN
iajs-2767	59	2	∈	∈	PROPN
iajs-2767	59	3	ℋ∗.	ℋ∗.	PROPN
iajs-2767	59	4	5.if	5.if	NUM
iajs-2767	59	5	f	f	NOUN
iajs-2767	59	6	,	,	PUNCT
iajs-2767	59	7	e	e	PROPN
iajs-2767	59	8	∈	∈	PROPN
iajs-2767	59	9	ℋ∗	ℋ∗	NUM
iajs-2767	59	10	,	,	PUNCT
iajs-2767	59	11	then	then	ADV
iajs-2767	59	12	f	f	X
iajs-2767	59	13	∖	∖	X
iajs-2767	59	14	e	e	PROPN
iajs-2767	59	15	∈	∈	PROPN
iajs-2767	59	16	ℋ∗.	ℋ∗.	PROPN
iajs-2767	59	17	6.if	6.if	NUM
iajs-2767	59	18	e1	e1	PROPN
iajs-2767	59	19	,	,	PUNCT
iajs-2767	59	20	e2	e2	PROPN
iajs-2767	59	21	,	,	PUNCT
iajs-2767	59	22	…	…	PUNCT
iajs-2767	59	23	∈	∈	PROPN
iajs-2767	59	24	ℋ∗	ℋ∗	NUM
iajs-2767	59	25	,	,	PUNCT
iajs-2767	59	26	then	then	ADV
iajs-2767	59	27	⋃	⋃	ADP
iajs-2767	59	28	ek	ek	PROPN
iajs-2767	59	29	∞	∞	PROPN
iajs-2767	59	30	𝑘=1	𝑘=1	SYM
iajs-2767	59	31	∈	∈	PROPN
iajs-2767	59	32	ℋ∗.	ℋ∗.	PROPN
iajs-2767	59	33	definition	definition	NOUN
iajs-2767	59	34	3.2	3.2	NUM
iajs-2767	59	35	a	a	DET
iajs-2767	59	36	fuzzy	fuzzy	ADJ
iajs-2767	59	37	measurable	measurable	ADJ
iajs-2767	59	38	space	space	NOUN
iajs-2767	59	39	relatively	relatively	ADV
iajs-2767	59	40	to	to	ADP
iajs-2767	59	41	a	a	DET
iajs-2767	59	42	fuzzy	fuzzy	ADJ
iajs-2767	59	43	σ	σ	PROPN
iajs-2767	59	44	–	–	PUNCT
iajs-2767	59	45	ring	ring	NOUN
iajs-2767	59	46	is	be	AUX
iajs-2767	59	47	an	an	DET
iajs-2767	59	48	ordered	order	VERB
iajs-2767	59	49	pair	pair	NOUN
iajs-2767	59	50	(	(	PUNCT
iajs-2767	59	51	𝒳∗	𝒳∗	X
iajs-2767	59	52	,	,	PUNCT
iajs-2767	59	53	ℋ∗	ℋ∗	NUM
iajs-2767	59	54	)	)	PUNCT
iajs-2767	59	55	,	,	PUNCT
iajs-2767	59	56	where	where	SCONJ
iajs-2767	59	57	𝒳	𝒳	PROPN
iajs-2767	59	58	≠	≠	PROPN
iajs-2767	59	59	∅and	∅and	NUM
iajs-2767	59	60	ℋ∗	ℋ∗	NOUN
iajs-2767	60	1	⊆	⊆	NUM
iajs-2767	60	2	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	60	3	)	)	PUNCT
iajs-2767	60	4	be	be	VERB
iajs-2767	60	5	a	a	DET
iajs-2767	60	6	fuzzy	fuzzy	ADJ
iajs-2767	60	7	σ	σ	NOUN
iajs-2767	60	8	–	–	PUNCT
iajs-2767	60	9	ring	ring	NOUN
iajs-2767	60	10	over	over	ADP
iajs-2767	60	11	a	a	DET
iajs-2767	60	12	fuzzy	fuzzy	ADJ
iajs-2767	60	13	set	set	VERB
iajs-2767	60	14	𝒳∗	𝒳∗	PROPN
iajs-2767	60	15	and	and	CCONJ
iajs-2767	60	16	an	an	DET
iajs-2767	60	17	element	element	NOUN
iajs-2767	60	18	of	of	ADP
iajs-2767	60	19	ℋ∗	ℋ∗	NUM
iajs-2767	60	20	is	be	AUX
iajs-2767	60	21	called	call	VERB
iajs-2767	60	22	a	a	DET
iajs-2767	60	23	measurable	measurable	ADJ
iajs-2767	60	24	set	set	VERB
iajs-2767	60	25	relatively	relatively	ADV
iajs-2767	60	26	to	to	ADP
iajs-2767	60	27	fuzzy	fuzzy	ADJ
iajs-2767	60	28	σ	σ	PROPN
iajs-2767	60	29	–	–	PUNCT
iajs-2767	60	30	ring	ring	NOUN
iajs-2767	60	31	.	.	PUNCT
iajs-2767	61	1	example	example	NOUN
iajs-2767	61	2	3.1	3.1	NUM
iajs-2767	61	3	let	let	VERB
iajs-2767	61	4	𝒳	𝒳	PRON
iajs-2767	61	5	≠	≠	NOUN
iajs-2767	61	6	∅.	∅.	VERB
iajs-2767	61	7	then	then	ADV
iajs-2767	61	8	each	each	PRON
iajs-2767	61	9	of	of	ADP
iajs-2767	61	10	∅∗	∅∗	PROPN
iajs-2767	61	11	and	and	CCONJ
iajs-2767	61	12	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	61	13	)	)	PUNCT
iajs-2767	61	14	is	be	AUX
iajs-2767	61	15	a	a	DET
iajs-2767	61	16	fuzzy	fuzzy	ADJ
iajs-2767	61	17	σ	σ	NOUN
iajs-2767	61	18	–	–	PUNCT
iajs-2767	61	19	ring	ring	NOUN
iajs-2767	61	20	over	over	ADP
iajs-2767	61	21	a	a	DET
iajs-2767	61	22	fuzzy	fuzzy	ADJ
iajs-2767	61	23	set	set	NOUN
iajs-2767	61	24	𝒳∗	𝒳∗	PROPN
iajs-2767	61	25	.	.	PUNCT
iajs-2767	62	1	example	example	NOUN
iajs-2767	62	2	3.2	3.2	NUM
iajs-2767	62	3	assume	assume	VERB
iajs-2767	62	4	𝒳	𝒳	PRON
iajs-2767	62	5	=	=	PRON
iajs-2767	62	6	{	{	PUNCT
iajs-2767	62	7	𝑎	𝑎	PROPN
iajs-2767	62	8	,	,	PUNCT
iajs-2767	62	9	b	b	NOUN
iajs-2767	62	10	}	}	PUNCT
iajs-2767	62	11	and	and	CCONJ
iajs-2767	62	12	ℋ∗	ℋ∗	NUM
iajs-2767	63	1	=	=	SYM
iajs-2767	63	2	{	{	PUNCT
iajs-2767	63	3	ek	ek	X
iajs-2767	63	4	,	,	PUNCT
iajs-2767	63	5	ek	ek	PROPN
iajs-2767	63	6	c	c	PROPN
iajs-2767	63	7	⊂	⊂	PROPN
iajs-2767	63	8	𝒳∗	𝒳∗	VERB
iajs-2767	63	9	:	:	PUNCT
iajs-2767	63	10	ek	ek	PROPN
iajs-2767	63	11	⊃	⊃	PROPN
iajs-2767	63	12	ek+1	ek+1	NUM
iajs-2767	63	13	,	,	PUNCT
iajs-2767	63	14	for	for	ADP
iajs-2767	63	15	every	every	DET
iajs-2767	63	16	k	k	NOUN
iajs-2767	63	17	=	=	SYM
iajs-2767	63	18	1,2	1,2	NUM
iajs-2767	63	19	,	,	PUNCT
iajs-2767	63	20	…	…	PUNCT
iajs-2767	63	21	}	}	PUNCT
iajs-2767	63	22	.	.	PUNCT
iajs-2767	64	1	then	then	ADV
iajs-2767	64	2	∅∗	∅∗	PROPN
iajs-2767	64	3	,	,	PUNCT
iajs-2767	64	4	𝒳∗	𝒳∗	PUNCT
iajs-2767	65	1	∈	∈	PROPN
iajs-2767	65	2	ℋ∗.	ℋ∗.	PROPN
iajs-2767	65	3	if	if	SCONJ
iajs-2767	65	4	f	f	X
iajs-2767	65	5	,	,	PUNCT
iajs-2767	65	6	e	e	PROPN
iajs-2767	65	7	∈	∈	PROPN
iajs-2767	65	8	ℋ∗	ℋ∗	NUM
iajs-2767	65	9	,	,	PUNCT
iajs-2767	65	10	then	then	ADV
iajs-2767	65	11	f	f	PROPN
iajs-2767	65	12	c	c	PROPN
iajs-2767	65	13	,	,	PUNCT
iajs-2767	65	14	e	e	PROPN
iajs-2767	65	15	c	c	NOUN
iajs-2767	65	16	∈	∈	PROPN
iajs-2767	65	17	ℋ∗	ℋ∗	NOUN
iajs-2767	65	18	and	and	CCONJ
iajs-2767	66	1	either	either	CCONJ
iajs-2767	66	2	e	e	PROPN
iajs-2767	66	3	⊂	⊂	PROPN
iajs-2767	66	4	f	f	PROPN
iajs-2767	66	5	c	c	PROPN
iajs-2767	66	6	or	or	CCONJ
iajs-2767	66	7	e	e	PROPN
iajs-2767	66	8	⊃	⊃	PROPN
iajs-2767	66	9	f	f	PROPN
iajs-2767	66	10	c.	c.	PROPN
iajs-2767	67	1	if	if	SCONJ
iajs-2767	67	2	e	e	PROPN
iajs-2767	67	3	⊂	⊂	PROPN
iajs-2767	67	4	f	f	PROPN
iajs-2767	67	5	c	c	X
iajs-2767	67	6	,	,	PUNCT
iajs-2767	67	7	then	then	ADV
iajs-2767	67	8	f	f	PROPN
iajs-2767	67	9	⊂	⊂	PROPN
iajs-2767	67	10	e	e	PROPN
iajs-2767	67	11	c	c	X
iajs-2767	67	12	,	,	PUNCT
iajs-2767	67	13	hence	hence	ADV
iajs-2767	67	14	f⋂e	f⋂e	PROPN
iajs-2767	67	15	c	c	PROPN
iajs-2767	67	16	∈	∈	PROPN
iajs-2767	67	17	ℋ∗	ℋ∗	PROPN
iajs-2767	67	18	,	,	PUNCT
iajs-2767	67	19	that	that	PRON
iajs-2767	67	20	is	be	AUX
iajs-2767	67	21	f\e	f\e	NOUN
iajs-2767	67	22	∈	∈	PROPN
iajs-2767	67	23	ℋ∗.	ℋ∗.	NOUN
iajs-2767	67	24	if	if	SCONJ
iajs-2767	67	25	e	e	PROPN
iajs-2767	67	26	⊃	⊃	PROPN
iajs-2767	67	27	f	f	PROPN
iajs-2767	67	28	c	c	PROPN
iajs-2767	67	29	,	,	PUNCT
iajs-2767	67	30	then	then	ADV
iajs-2767	67	31	e	e	PROPN
iajs-2767	67	32	c	c	PROPN
iajs-2767	67	33	⊂	⊂	PROPN
iajs-2767	67	34	f	f	X
iajs-2767	67	35	,	,	PUNCT
iajs-2767	67	36	hence	hence	ADV
iajs-2767	67	37	f⋂e	f⋂e	PROPN
iajs-2767	67	38	c	c	PROPN
iajs-2767	67	39	∈	∈	PROPN
iajs-2767	67	40	ℋ∗	ℋ∗	PROPN
iajs-2767	67	41	,	,	PUNCT
iajs-2767	67	42	that	that	PRON
iajs-2767	67	43	is	be	AUX
iajs-2767	67	44	f\e	f\e	NOUN
iajs-2767	67	45	∈	∈	PROPN
iajs-2767	67	46	ℋ∗.	ℋ∗.	PROPN
iajs-2767	67	47	now	now	ADV
iajs-2767	67	48	,	,	PUNCT
iajs-2767	67	49	if	if	SCONJ
iajs-2767	67	50	e1	e1	PROPN
iajs-2767	67	51	,	,	PUNCT
iajs-2767	67	52	e2	e2	PROPN
iajs-2767	67	53	,	,	PUNCT
iajs-2767	67	54	…	…	PUNCT
iajs-2767	67	55	∈	∈	PROPN
iajs-2767	67	56	ℋ∗	ℋ∗	NUM
iajs-2767	67	57	,	,	PUNCT
iajs-2767	67	58	then	then	ADV
iajs-2767	67	59	ek	ek	PROPN
iajs-2767	67	60	⊃	⊃	PROPN
iajs-2767	67	61	ek+1	ek+1	VERB
iajs-2767	67	62	for	for	ADP
iajs-2767	67	63	every	every	DET
iajs-2767	67	64	(	(	PUNCT
iajs-2767	67	65	k	k	NOUN
iajs-2767	67	66	=	=	SYM
iajs-2767	67	67	1,2	1,2	NUM
iajs-2767	67	68	,	,	PUNCT
iajs-2767	67	69	…	…	PUNCT
iajs-2767	67	70	)	)	PUNCT
iajs-2767	67	71	and	and	CCONJ
iajs-2767	67	72	hence	hence	ADV
iajs-2767	67	73	𝓋ek	𝓋ek	NOUN
iajs-2767	67	74	(	(	PUNCT
iajs-2767	67	75	ω	ω	NOUN
iajs-2767	67	76	)	)	PUNCT
iajs-2767	67	77	>	>	X
iajs-2767	67	78	𝓋ek+1	𝓋ek+1	X
iajs-2767	67	79	(	(	PUNCT
iajs-2767	67	80	ω	ω	NOUN
iajs-2767	67	81	)	)	PUNCT
iajs-2767	67	82	for	for	ADP
iajs-2767	67	83	all	all	DET
iajs-2767	67	84	ω	ω	NUM
iajs-2767	67	85	∈	∈	PROPN
iajs-2767	67	86	𝒳	𝒳	PROPN
iajs-2767	67	87	and	and	CCONJ
iajs-2767	67	88	hence	hence	ADV
iajs-2767	67	89	⋃	⋃	ADP
iajs-2767	67	90	ek	ek	NOUN
iajs-2767	67	91	∞	∞	PROPN
iajs-2767	67	92	𝑘=1	𝑘=1	PUNCT
iajs-2767	68	1	=	=	SYM
iajs-2767	68	2	{	{	PUNCT
iajs-2767	68	3	(	(	PUNCT
iajs-2767	68	4	ω	ω	NOUN
iajs-2767	68	5	,	,	PUNCT
iajs-2767	68	6	sup	sup	NOUN
iajs-2767	68	7	{	{	PUNCT
iajs-2767	68	8	𝓋e1	𝓋e1	PROPN
iajs-2767	68	9	(	(	PUNCT
iajs-2767	68	10	ω	ω	NOUN
iajs-2767	68	11	)	)	PUNCT
iajs-2767	68	12	,	,	PUNCT
iajs-2767	68	13	𝓋e2	𝓋e2	NOUN
iajs-2767	68	14	(	(	PUNCT
iajs-2767	68	15	ω	ω	NOUN
iajs-2767	68	16	)	)	PUNCT
iajs-2767	68	17	,	,	PUNCT
iajs-2767	68	18	𝓋e3	𝓋e3	NOUN
iajs-2767	68	19	(	(	PUNCT
iajs-2767	68	20	ω	ω	NOUN
iajs-2767	68	21	)	)	PUNCT
iajs-2767	68	22	,	,	PUNCT
iajs-2767	68	23	…	…	PUNCT
iajs-2767	68	24	}	}	PUNCT
iajs-2767	68	25	:	:	PUNCT
iajs-2767	68	26	∀ω	∀ω	PUNCT
iajs-2767	68	27	∈	∈	PROPN
iajs-2767	68	28	𝒳	𝒳	PROPN
iajs-2767	68	29	)	)	PUNCT
iajs-2767	68	30	}	}	PUNCT
iajs-2767	68	31	=	=	SYM
iajs-2767	68	32	{	{	PUNCT
iajs-2767	68	33	(	(	PUNCT
iajs-2767	68	34	ω	ω	PROPN
iajs-2767	68	35	,	,	PUNCT
iajs-2767	68	36	𝓋e1	𝓋e1	PROPN
iajs-2767	68	37	(	(	PUNCT
iajs-2767	68	38	ω	ω	NOUN
iajs-2767	68	39	)	)	PUNCT
iajs-2767	68	40	)	)	PUNCT
iajs-2767	68	41	∶	∶	NOUN
iajs-2767	68	42	∀ω	∀ω	NOUN
iajs-2767	68	43	∈	∈	PROPN
iajs-2767	68	44	𝒳	𝒳	PROPN
iajs-2767	68	45	}	}	PUNCT
iajs-2767	68	46	=	=	SYM
iajs-2767	68	47	e1	e1	NOUN
iajs-2767	68	48	thus	thus	ADV
iajs-2767	68	49	,	,	PUNCT
iajs-2767	68	50	⋃	⋃	ADP
iajs-2767	68	51	ek	ek	NOUN
iajs-2767	68	52	∞	∞	PROPN
iajs-2767	68	53	𝑘=1	𝑘=1	PUNCT
iajs-2767	68	54	∈	∈	PROPN
iajs-2767	68	55	ℋ	ℋ	PROPN
iajs-2767	68	56	∗.	∗.	PROPN
iajs-2767	68	57	therefore	therefore	ADV
iajs-2767	68	58	,	,	PUNCT
iajs-2767	68	59	ℋ∗	ℋ∗	NUM
iajs-2767	68	60	is	be	AUX
iajs-2767	68	61	a	a	DET
iajs-2767	68	62	fuzzy	fuzzy	ADJ
iajs-2767	68	63	σ	σ	NOUN
iajs-2767	68	64	–	–	PUNCT
iajs-2767	68	65	ring	ring	NOUN
iajs-2767	68	66	over	over	ADP
iajs-2767	68	67	a	a	DET
iajs-2767	68	68	fuzzy	fuzzy	ADJ
iajs-2767	68	69	set	set	VERB
iajs-2767	68	70	𝒳∗	𝒳∗	PROPN
iajs-2767	68	71	and	and	CCONJ
iajs-2767	68	72	hence	hence	ADV
iajs-2767	68	73	(	(	PUNCT
iajs-2767	68	74	𝒳∗	𝒳∗	INTJ
iajs-2767	68	75	,	,	PUNCT
iajs-2767	68	76	ℋ∗	ℋ∗	NUM
iajs-2767	68	77	)	)	PUNCT
iajs-2767	68	78	is	be	AUX
iajs-2767	68	79	a	a	DET
iajs-2767	68	80	fuzzy	fuzzy	ADJ
iajs-2767	68	81	measurable	measurable	ADJ
iajs-2767	68	82	space	space	NOUN
iajs-2767	68	83	relatively	relatively	ADV
iajs-2767	68	84	to	to	ADP
iajs-2767	68	85	the	the	DET
iajs-2767	68	86	fuzzy	fuzzy	ADJ
iajs-2767	68	87	σ	σ	PROPN
iajs-2767	68	88	–	–	PUNCT
iajs-2767	68	89	ring	ring	NOUN
iajs-2767	68	90	.	.	PUNCT
iajs-2767	69	1	ibn	ibn	PROPN
iajs-2767	69	2	al	al	PROPN
iajs-2767	69	3	-	-	PUNCT
iajs-2767	69	4	haitham	haitham	PROPN
iajs-2767	69	5	jour	jour	X
iajs-2767	69	6	.	.	PROPN
iajs-2767	69	7	for	for	ADP
iajs-2767	69	8	pure	pure	ADJ
iajs-2767	69	9	&	&	CCONJ
iajs-2767	69	10	appl	appl	PROPN
iajs-2767	69	11	.	.	PUNCT
iajs-2767	70	1	sci	sci	PROPN
iajs-2767	70	2	.	.	PROPN
iajs-2767	71	1	53	53	NUM
iajs-2767	71	2	(	(	PUNCT
iajs-2767	71	3	2)2022	2)2022	NUM
iajs-2767	71	4	40	40	NUM
iajs-2767	71	5	example	example	NOUN
iajs-2767	71	6	3.3	3.3	NUM
iajs-2767	71	7	let	let	VERB
iajs-2767	71	8	𝒳	𝒳	PRON
iajs-2767	71	9	=	=	PRON
iajs-2767	71	10	{	{	PUNCT
iajs-2767	71	11	𝑎	𝑎	PROPN
iajs-2767	71	12	,	,	PUNCT
iajs-2767	71	13	b	b	NOUN
iajs-2767	71	14	,	,	PUNCT
iajs-2767	71	15	c	c	NOUN
iajs-2767	71	16	}	}	PUNCT
iajs-2767	71	17	and	and	CCONJ
iajs-2767	71	18	ℋ∗	ℋ∗	NUM
iajs-2767	72	1	=	=	NOUN
iajs-2767	72	2	{	{	PUNCT
iajs-2767	72	3	∅∗	∅∗	PROPN
iajs-2767	72	4	,	,	PUNCT
iajs-2767	72	5	{	{	PUNCT
iajs-2767	72	6	(	(	PUNCT
iajs-2767	72	7	𝑎,0.3	𝑎,0.3	PROPN
iajs-2767	72	8	)	)	PUNCT
iajs-2767	72	9	,	,	PUNCT
iajs-2767	72	10	(	(	PUNCT
iajs-2767	72	11	𝑏,0.3),(c,0.4	𝑏,0.3),(c,0.4	NOUN
iajs-2767	72	12	)	)	PUNCT
iajs-2767	72	13	}	}	PUNCT
iajs-2767	72	14	,	,	PUNCT
iajs-2767	72	15	{	{	PUNCT
iajs-2767	72	16	(	(	PUNCT
iajs-2767	72	17	𝑎,0.4),(𝑏,0.4),(c,0.2	𝑎,0.4),(𝑏,0.4),(c,0.2	PROPN
iajs-2767	72	18	)	)	PUNCT
iajs-2767	72	19	}	}	PUNCT
iajs-2767	72	20	}	}	PUNCT
iajs-2767	72	21	.	.	PUNCT
iajs-2767	73	1	then	then	ADV
iajs-2767	73	2	ℋ∗	ℋ∗	NUM
iajs-2767	73	3	is	be	AUX
iajs-2767	73	4	not	not	PART
iajs-2767	73	5	a	a	DET
iajs-2767	73	6	fuzzy	fuzzy	ADJ
iajs-2767	73	7	σ	σ	NOUN
iajs-2767	73	8	–	–	PUNCT
iajs-2767	73	9	ring	ring	NOUN
iajs-2767	73	10	over	over	ADP
iajs-2767	73	11	a	a	DET
iajs-2767	73	12	fuzzy	fuzzy	ADJ
iajs-2767	73	13	set	set	VERB
iajs-2767	73	14	𝒳∗	𝒳∗	PROPN
iajs-2767	73	15	,	,	PUNCT
iajs-2767	73	16	because	because	SCONJ
iajs-2767	73	17	{	{	PUNCT
iajs-2767	73	18	(	(	PUNCT
iajs-2767	73	19	𝑎,0.3	𝑎,0.3	PROPN
iajs-2767	73	20	)	)	PUNCT
iajs-2767	73	21	,	,	PUNCT
iajs-2767	73	22	(	(	PUNCT
iajs-2767	73	23	𝑏,0.3),(c,0.4	𝑏,0.3),(c,0.4	NOUN
iajs-2767	73	24	)	)	PUNCT
iajs-2767	73	25	}	}	PUNCT
iajs-2767	73	26	,	,	PUNCT
iajs-2767	73	27	{	{	PUNCT
iajs-2767	73	28	(	(	PUNCT
iajs-2767	73	29	𝑎,0.4),(𝑏,0.4),(c,0.2	𝑎,0.4),(𝑏,0.4),(c,0.2	NOUN
iajs-2767	73	30	)	)	PUNCT
iajs-2767	73	31	}	}	PUNCT
iajs-2767	73	32	∈	∈	PROPN
iajs-2767	73	33	ℋ∗	ℋ∗	NUM
iajs-2767	73	34	,	,	PUNCT
iajs-2767	73	35	but	but	CCONJ
iajs-2767	73	36	{	{	PUNCT
iajs-2767	73	37	(	(	PUNCT
iajs-2767	73	38	𝑎	𝑎	X
iajs-2767	73	39	,	,	PUNCT
iajs-2767	73	40	0.3	0.3	NUM
iajs-2767	73	41	)	)	PUNCT
iajs-2767	73	42	,	,	PUNCT
iajs-2767	73	43	(	(	PUNCT
iajs-2767	73	44	𝑏	𝑏	NOUN
iajs-2767	73	45	,	,	PUNCT
iajs-2767	73	46	0.3	0.3	NUM
iajs-2767	73	47	)	)	PUNCT
iajs-2767	73	48	,	,	PUNCT
iajs-2767	73	49	(	(	PUNCT
iajs-2767	73	50	𝑐	𝑐	NOUN
iajs-2767	73	51	,	,	PUNCT
iajs-2767	73	52	0.4	0.4	NUM
iajs-2767	73	53	)	)	PUNCT
iajs-2767	73	54	}	}	PUNCT
iajs-2767	73	55	⋃	⋃	VERB
iajs-2767	73	56	{	{	PUNCT
iajs-2767	73	57	(	(	PUNCT
iajs-2767	73	58	𝑎	𝑎	X
iajs-2767	73	59	,	,	PUNCT
iajs-2767	73	60	0.4	0.4	NUM
iajs-2767	73	61	)	)	PUNCT
iajs-2767	73	62	,	,	PUNCT
iajs-2767	73	63	(	(	PUNCT
iajs-2767	73	64	𝑏	𝑏	NOUN
iajs-2767	73	65	,	,	PUNCT
iajs-2767	73	66	0.4	0.4	NUM
iajs-2767	73	67	)	)	PUNCT
iajs-2767	73	68	,	,	PUNCT
iajs-2767	73	69	(	(	PUNCT
iajs-2767	73	70	𝑐	𝑐	NOUN
iajs-2767	73	71	,	,	PUNCT
iajs-2767	73	72	0.2	0.2	NUM
iajs-2767	73	73	)	)	PUNCT
iajs-2767	73	74	}	}	PUNCT
iajs-2767	74	1	=	=	PRON
iajs-2767	74	2	{	{	PUNCT
iajs-2767	74	3	(	(	PUNCT
iajs-2767	74	4	𝑎,0.4),(𝑏,0.4),(c,0.4)}∉	𝑎,0.4),(𝑏,0.4),(c,0.4)}∉	NOUN
iajs-2767	74	5	ℋ∗.	ℋ∗.	PROPN
iajs-2767	74	6	lemma	lemma	PROPN
iajs-2767	74	7	3.1	3.1	NUM
iajs-2767	74	8	let	let	VERB
iajs-2767	74	9	{	{	PUNCT
iajs-2767	74	10	ℋi	ℋi	PROPN
iajs-2767	74	11	∗}i∈ι	∗}i∈ι	PROPN
iajs-2767	74	12	be	be	AUX
iajs-2767	74	13	a	a	DET
iajs-2767	74	14	nonempty	nonempty	ADJ
iajs-2767	74	15	collection	collection	NOUN
iajs-2767	74	16	of	of	ADP
iajs-2767	74	17	fuzzy	fuzzy	ADJ
iajs-2767	74	18	σ	σ	PROPN
iajs-2767	74	19	–	–	PUNCT
iajs-2767	74	20	ring	ring	NOUN
iajs-2767	74	21	over	over	ADP
iajs-2767	74	22	a	a	DET
iajs-2767	74	23	fuzzy	fuzzy	ADJ
iajs-2767	74	24	set	set	NOUN
iajs-2767	74	25	𝒳∗	𝒳∗	PROPN
iajs-2767	74	26	.	.	PUNCT
iajs-2767	75	1	then	then	ADV
iajs-2767	75	2	⋂	⋂	PROPN
iajs-2767	75	3	ℋi	ℋi	PROPN
iajs-2767	75	4	∗	∗	NOUN
iajs-2767	75	5	i∈ι	i∈ι	NOUN
iajs-2767	75	6	is	be	AUX
iajs-2767	75	7	a	a	DET
iajs-2767	75	8	fuzzy	fuzzy	ADJ
iajs-2767	75	9	σ	σ	NOUN
iajs-2767	75	10	–	–	PUNCT
iajs-2767	75	11	ring	ring	NOUN
iajs-2767	75	12	over	over	ADP
iajs-2767	75	13	a	a	DET
iajs-2767	75	14	fuzzy	fuzzy	ADJ
iajs-2767	75	15	set	set	NOUN
iajs-2767	75	16	𝒳∗.	𝒳∗.	PROPN
iajs-2767	75	17	proof	proof	NOUN
iajs-2767	75	18	since	since	SCONJ
iajs-2767	75	19	ℋi	ℋi	PROPN
iajs-2767	75	20	∗	∗	NOUN
iajs-2767	75	21	is	be	AUX
iajs-2767	75	22	a	a	DET
iajs-2767	75	23	nonempty	nonempty	ADJ
iajs-2767	75	24	collection	collection	NOUN
iajs-2767	75	25	of	of	ADP
iajs-2767	75	26	a	a	DET
iajs-2767	75	27	fuzzy	fuzzy	ADJ
iajs-2767	75	28	σ	σ	NOUN
iajs-2767	75	29	–	–	PUNCT
iajs-2767	75	30	ring	ring	NOUN
iajs-2767	75	31	over	over	ADP
iajs-2767	75	32	a	a	DET
iajs-2767	75	33	fuzzy	fuzzy	ADJ
iajs-2767	75	34	set	set	VERB
iajs-2767	75	35	𝒳∗	𝒳∗	PRON
iajs-2767	75	36	∀i	∀i	NOUN
iajs-2767	75	37	∈	∈	PROPN
iajs-2767	75	38	ι	ι	NOUN
iajs-2767	75	39	,	,	PUNCT
iajs-2767	75	40	then	then	ADV
iajs-2767	75	41	there	there	PRON
iajs-2767	75	42	is	be	VERB
iajs-2767	75	43	∅∗	∅∗	PROPN
iajs-2767	75	44	≠	≠	PROPN
iajs-2767	75	45	e	e	X
iajs-2767	75	46	∈	∈	NOUN
iajs-2767	75	47	ℋi	ℋi	PROPN
iajs-2767	75	48	∗	∗	NOUN
iajs-2767	75	49	∀i	∀i	NOUN
iajs-2767	75	50	∈	∈	PROPN
iajs-2767	75	51	ι	ι	X
iajs-2767	75	52	,	,	PUNCT
iajs-2767	75	53	which	which	PRON
iajs-2767	75	54	implies	imply	VERB
iajs-2767	75	55	that	that	SCONJ
iajs-2767	75	56	e	e	PROPN
iajs-2767	75	57	∈	∈	PROPN
iajs-2767	75	58	⋂	⋂	PROPN
iajs-2767	75	59	ℋi	ℋi	PROPN
iajs-2767	75	60	∗	∗	NOUN
iajs-2767	75	61	iϵι	iϵι	VERB
iajs-2767	75	62	and	and	CCONJ
iajs-2767	75	63	hence	hence	ADV
iajs-2767	75	64	⋂	⋂	PROPN
iajs-2767	75	65	ℋi	ℋi	PROPN
iajs-2767	75	66	∗	∗	NOUN
iajs-2767	75	67	iϵι	iϵι	VERB
iajs-2767	75	68	≠	≠	PROPN
iajs-2767	75	69	∅∗.	∅∗.	NUM
iajs-2767	75	70	it	it	PRON
iajs-2767	75	71	is	be	AUX
iajs-2767	75	72	clear	clear	ADJ
iajs-2767	75	73	that	that	SCONJ
iajs-2767	75	74	∅∗	∅∗	PROPN
iajs-2767	75	75	∈	∈	PROPN
iajs-2767	75	76	⋂	⋂	PROPN
iajs-2767	75	77	ℋi	ℋi	PROPN
iajs-2767	75	78	∗	∗	NOUN
iajs-2767	75	79	iϵι	iϵι	VERB
iajs-2767	75	80	.	.	PUNCT
iajs-2767	76	1	let	let	VERB
iajs-2767	76	2	f	f	X
iajs-2767	76	3	,	,	PUNCT
iajs-2767	76	4	e	e	PROPN
iajs-2767	76	5	∈	∈	PROPN
iajs-2767	76	6	⋂	⋂	PROPN
iajs-2767	76	7	ℋi	ℋi	PROPN
iajs-2767	76	8	∗	∗	NOUN
iajs-2767	76	9	iϵι	iϵι	VERB
iajs-2767	76	10	.	.	PUNCT
iajs-2767	77	1	then	then	ADV
iajs-2767	77	2	f	f	X
iajs-2767	77	3	,	,	PUNCT
iajs-2767	77	4	e	e	PROPN
iajs-2767	77	5	∈	∈	PROPN
iajs-2767	77	6	ℋi	ℋi	PROPN
iajs-2767	77	7	∗	∗	NOUN
iajs-2767	77	8	∀i	∀i	NOUN
iajs-2767	77	9	∈	∈	PROPN
iajs-2767	77	10	ι	ι	NOUN
iajs-2767	77	11	,	,	PUNCT
iajs-2767	77	12	hence	hence	ADV
iajs-2767	77	13	f\e	f\e	VERB
iajs-2767	77	14	∈	∈	PROPN
iajs-2767	77	15	ℋi	ℋi	PROPN
iajs-2767	77	16	∗	∗	NOUN
iajs-2767	77	17	∀i	∀i	NOUN
iajs-2767	77	18	∈	∈	PROPN
iajs-2767	77	19	ι	ι	X
iajs-2767	77	20	.	.	PUNCT
iajs-2767	78	1	therefore	therefore	ADV
iajs-2767	78	2	f\e	f\e	VERB
iajs-2767	78	3	∈	∈	PROPN
iajs-2767	78	4	⋂	⋂	PROPN
iajs-2767	78	5	ℋi	ℋi	PROPN
iajs-2767	78	6	∗	∗	NOUN
iajs-2767	78	7	iϵι	iϵι	VERB
iajs-2767	78	8	.	.	PUNCT
iajs-2767	79	1	let	let	VERB
iajs-2767	79	2	e1	e1	PROPN
iajs-2767	79	3	,	,	PUNCT
iajs-2767	79	4	e2	e2	PROPN
iajs-2767	79	5	,	,	PUNCT
iajs-2767	79	6	…	…	PUNCT
iajs-2767	79	7	∈	∈	PROPN
iajs-2767	79	8	⋂	⋂	PROPN
iajs-2767	79	9	ℋi	ℋi	PROPN
iajs-2767	79	10	∗	∗	NOUN
iajs-2767	79	11	iϵι	iϵι	VERB
iajs-2767	79	12	,	,	PUNCT
iajs-2767	79	13	then	then	ADV
iajs-2767	79	14	e1	e1	PROPN
iajs-2767	79	15	,	,	PUNCT
iajs-2767	79	16	e2	e2	PROPN
iajs-2767	79	17	,	,	PUNCT
iajs-2767	79	18	…	…	PUNCT
iajs-2767	79	19	∈	∈	PROPN
iajs-2767	80	1	ℋi	ℋi	NOUN
iajs-2767	80	2	∗	∗	NOUN
iajs-2767	80	3	∀i	∀i	X
iajs-2767	80	4	∈	∈	PROPN
iajs-2767	80	5	ι	ι	X
iajs-2767	80	6	and	and	CCONJ
iajs-2767	80	7	hence	hence	ADV
iajs-2767	80	8	⋃	⋃	ADV
iajs-2767	80	9	ek	ek	NOUN
iajs-2767	80	10	∞	∞	PROPN
iajs-2767	80	11	k=1	k=1	PUNCT
iajs-2767	81	1	∈	∈	PROPN
iajs-2767	82	1	ℋi	ℋi	PROPN
iajs-2767	82	2	∗	∗	NOUN
iajs-2767	82	3	∀i	∀i	NOUN
iajs-2767	82	4	∈	∈	PROPN
iajs-2767	82	5	ι	ι	NOUN
iajs-2767	82	6	,	,	PUNCT
iajs-2767	82	7	thus	thus	ADV
iajs-2767	82	8	⋃	⋃	PUNCT
iajs-2767	82	9	ek	ek	X
iajs-2767	82	10	∞	∞	PROPN
iajs-2767	82	11	k=1	k=1	PUNCT
iajs-2767	83	1	∈	∈	PROPN
iajs-2767	83	2	⋂	⋂	PROPN
iajs-2767	83	3	ℋi	ℋi	PROPN
iajs-2767	83	4	∗	∗	NOUN
iajs-2767	83	5	iϵι	iϵι	VERB
iajs-2767	83	6	.	.	PUNCT
iajs-2767	84	1	therefore	therefore	ADV
iajs-2767	84	2	,	,	PUNCT
iajs-2767	84	3	⋂	⋂	PROPN
iajs-2767	84	4	ℋi	ℋi	PROPN
iajs-2767	84	5	∗	∗	NOUN
iajs-2767	84	6	i∈ι	i∈ι	NOUN
iajs-2767	84	7	is	be	AUX
iajs-2767	84	8	a	a	DET
iajs-2767	84	9	fuzzy	fuzzy	ADJ
iajs-2767	84	10	σ	σ	NOUN
iajs-2767	84	11	–	–	PUNCT
iajs-2767	84	12	ring	ring	NOUN
iajs-2767	84	13	over	over	ADP
iajs-2767	84	14	a	a	DET
iajs-2767	84	15	fuzzy	fuzzy	ADJ
iajs-2767	84	16	set	set	VERB
iajs-2767	84	17	𝒳∗.	𝒳∗.	PROPN
iajs-2767	84	18	in	in	ADP
iajs-2767	84	19	the	the	DET
iajs-2767	84	20	following	follow	VERB
iajs-2767	84	21	example	example	NOUN
iajs-2767	84	22	shows	show	VERB
iajs-2767	84	23	the	the	DET
iajs-2767	84	24	union	union	NOUN
iajs-2767	84	25	for	for	ADP
iajs-2767	84	26	two	two	NUM
iajs-2767	84	27	fuzzy	fuzzy	ADJ
iajs-2767	84	28	σ	σ	NOUN
iajs-2767	84	29	–	–	PUNCT
iajs-2767	84	30	ring	ring	NOUN
iajs-2767	84	31	over	over	ADP
iajs-2767	84	32	a	a	DET
iajs-2767	84	33	fuzzy	fuzzy	ADJ
iajs-2767	84	34	set	set	NOUN
iajs-2767	84	35	𝒳∗	𝒳∗	PRON
iajs-2767	84	36	needs	need	VERB
iajs-2767	84	37	not	not	PART
iajs-2767	84	38	be	be	AUX
iajs-2767	84	39	a	a	DET
iajs-2767	84	40	fuzzy	fuzzy	ADJ
iajs-2767	84	41	σ	σ	NOUN
iajs-2767	84	42	–	–	PUNCT
iajs-2767	84	43	ring	ring	NOUN
iajs-2767	84	44	over	over	ADP
iajs-2767	84	45	a	a	DET
iajs-2767	84	46	fuzzy	fuzzy	ADJ
iajs-2767	84	47	set	set	NOUN
iajs-2767	84	48	𝒳∗.	𝒳∗.	PROPN
iajs-2767	84	49	example	example	NOUN
iajs-2767	84	50	3.4	3.4	NUM
iajs-2767	84	51	let	let	VERB
iajs-2767	84	52	𝒳	𝒳	PRON
iajs-2767	84	53	=	=	PRON
iajs-2767	84	54	{	{	PUNCT
iajs-2767	84	55	𝜔1	𝜔1	ADJ
iajs-2767	84	56	,	,	PUNCT
iajs-2767	84	57	𝜔2	𝜔2	PROPN
iajs-2767	84	58	}	}	PUNCT
iajs-2767	84	59	and	and	CCONJ
iajs-2767	84	60	ℋ1	ℋ1	PROPN
iajs-2767	84	61	∗	∗	NOUN
iajs-2767	84	62	=	=	SYM
iajs-2767	84	63	{	{	PUNCT
iajs-2767	84	64	∅∗	∅∗	PROPN
iajs-2767	84	65	,	,	PUNCT
iajs-2767	84	66	{	{	PUNCT
iajs-2767	84	67	(	(	PUNCT
iajs-2767	84	68	𝜔1,0.1	𝜔1,0.1	NOUN
iajs-2767	84	69	)	)	PUNCT
iajs-2767	84	70	,	,	PUNCT
iajs-2767	84	71	(	(	PUNCT
iajs-2767	84	72	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	84	73	)	)	PUNCT
iajs-2767	84	74	}	}	PUNCT
iajs-2767	84	75	,	,	PUNCT
iajs-2767	84	76	{	{	PUNCT
iajs-2767	84	77	(	(	PUNCT
iajs-2767	84	78	𝜔1,0.9	𝜔1,0.9	NUM
iajs-2767	84	79	)	)	PUNCT
iajs-2767	84	80	,	,	PUNCT
iajs-2767	84	81	(	(	PUNCT
iajs-2767	84	82	𝜔2	𝜔2	ADV
iajs-2767	84	83	,	,	PUNCT
iajs-2767	84	84	0.5	0.5	NUM
iajs-2767	84	85	)	)	PUNCT
iajs-2767	84	86	}	}	PUNCT
iajs-2767	84	87	,	,	PUNCT
iajs-2767	84	88	𝒳∗	𝒳∗	PROPN
iajs-2767	84	89	}	}	PUNCT
iajs-2767	84	90	,	,	PUNCT
iajs-2767	84	91	ℋ2	ℋ2	PROPN
iajs-2767	84	92	∗	∗	NOUN
iajs-2767	84	93	=	=	PROPN
iajs-2767	84	94	{	{	PUNCT
iajs-2767	84	95	∅∗	∅∗	PROPN
iajs-2767	84	96	,	,	PUNCT
iajs-2767	84	97	{	{	PUNCT
iajs-2767	84	98	(	(	PUNCT
iajs-2767	84	99	𝜔1,0.2),(𝜔2,0.4	𝜔1,0.2),(𝜔2,0.4	NOUN
iajs-2767	84	100	)	)	PUNCT
iajs-2767	84	101	}	}	PUNCT
iajs-2767	84	102	,	,	PUNCT
iajs-2767	84	103	{	{	PUNCT
iajs-2767	84	104	(	(	PUNCT
iajs-2767	84	105	𝜔1,0.8	𝜔1,0.8	NOUN
iajs-2767	84	106	)	)	PUNCT
iajs-2767	84	107	,	,	PUNCT
iajs-2767	84	108	(	(	PUNCT
iajs-2767	84	109	𝜔2,0.6	𝜔2,0.6	X
iajs-2767	84	110	)	)	PUNCT
iajs-2767	84	111	}	}	PUNCT
iajs-2767	84	112	,	,	PUNCT
iajs-2767	84	113	𝒳∗	𝒳∗	PROPN
iajs-2767	84	114	}	}	PUNCT
iajs-2767	84	115	.	.	PUNCT
iajs-2767	85	1	then	then	ADV
iajs-2767	85	2	ℋ1	ℋ1	PROPN
iajs-2767	85	3	∗	∗	NOUN
iajs-2767	85	4	and	and	CCONJ
iajs-2767	85	5	ℋ2	ℋ2	PROPN
iajs-2767	85	6	∗	∗	NOUN
iajs-2767	85	7	are	be	AUX
iajs-2767	85	8	fuzzy	fuzzy	ADJ
iajs-2767	85	9	σ	σ	NOUN
iajs-2767	85	10	–	–	PUNCT
iajs-2767	85	11	ring	ring	NOUN
iajs-2767	85	12	over	over	ADP
iajs-2767	85	13	a	a	DET
iajs-2767	85	14	fuzzy	fuzzy	ADJ
iajs-2767	85	15	set	set	NOUN
iajs-2767	85	16	𝒳∗.	𝒳∗.	PROPN
iajs-2767	85	17	now	now	ADV
iajs-2767	85	18	,	,	PUNCT
iajs-2767	85	19	ℋ1	ℋ1	NOUN
iajs-2767	85	20	∗	∗	NOUN
iajs-2767	85	21	⋃	⋃	NOUN
iajs-2767	85	22	ℋ2	ℋ2	NOUN
iajs-2767	85	23	∗	∗	NOUN
iajs-2767	85	24	=	=	SYM
iajs-2767	85	25	{	{	PUNCT
iajs-2767	85	26	∅∗	∅∗	PROPN
iajs-2767	85	27	,	,	PUNCT
iajs-2767	85	28	{	{	PUNCT
iajs-2767	85	29	(	(	PUNCT
iajs-2767	85	30	𝜔1	𝜔1	ADJ
iajs-2767	85	31	,	,	PUNCT
iajs-2767	85	32	0.1	0.1	NUM
iajs-2767	85	33	)	)	PUNCT
iajs-2767	85	34	,	,	PUNCT
iajs-2767	85	35	(	(	PUNCT
iajs-2767	85	36	𝜔2	𝜔2	PROPN
iajs-2767	85	37	,	,	PUNCT
iajs-2767	85	38	0.5	0.5	NUM
iajs-2767	85	39	)	)	PUNCT
iajs-2767	85	40	}	}	PUNCT
iajs-2767	85	41	,	,	PUNCT
iajs-2767	85	42	{	{	PUNCT
iajs-2767	85	43	(	(	PUNCT
iajs-2767	85	44	𝜔1	𝜔1	ADJ
iajs-2767	85	45	,	,	PUNCT
iajs-2767	85	46	0.9	0.9	NUM
iajs-2767	85	47	)	)	PUNCT
iajs-2767	85	48	,	,	PUNCT
iajs-2767	85	49	(	(	PUNCT
iajs-2767	85	50	𝜔2	𝜔2	ADV
iajs-2767	85	51	,	,	PUNCT
iajs-2767	85	52	0.5	0.5	NUM
iajs-2767	85	53	)	)	PUNCT
iajs-2767	85	54	}	}	PUNCT
iajs-2767	85	55	,	,	PUNCT
iajs-2767	85	56	{	{	PUNCT
iajs-2767	85	57	(	(	PUNCT
iajs-2767	85	58	𝜔1	𝜔1	ADJ
iajs-2767	85	59	,	,	PUNCT
iajs-2767	85	60	0.2	0.2	NUM
iajs-2767	85	61	)	)	PUNCT
iajs-2767	85	62	,	,	PUNCT
iajs-2767	85	63	(	(	PUNCT
iajs-2767	85	64	𝜔2	𝜔2	PROPN
iajs-2767	85	65	,	,	PUNCT
iajs-2767	85	66	0.4	0.4	NUM
iajs-2767	85	67	)	)	PUNCT
iajs-2767	85	68	}	}	PUNCT
iajs-2767	85	69	,	,	PUNCT
iajs-2767	85	70	{	{	PUNCT
iajs-2767	85	71	(	(	PUNCT
iajs-2767	85	72	𝜔1	𝜔1	ADJ
iajs-2767	85	73	,	,	PUNCT
iajs-2767	85	74	0.8	0.8	NUM
iajs-2767	85	75	)	)	PUNCT
iajs-2767	85	76	,	,	PUNCT
iajs-2767	85	77	(	(	PUNCT
iajs-2767	85	78	𝜔2	𝜔2	PROPN
iajs-2767	85	79	,	,	PUNCT
iajs-2767	85	80	0.6	0.6	NUM
iajs-2767	85	81	)	)	PUNCT
iajs-2767	85	82	}	}	PUNCT
iajs-2767	85	83	,	,	PUNCT
iajs-2767	85	84	𝒳∗	𝒳∗	PROPN
iajs-2767	85	85	}	}	PUNCT
iajs-2767	85	86	.	.	PUNCT
iajs-2767	86	1	put	put	NOUN
iajs-2767	86	2	,	,	PUNCT
iajs-2767	86	3	e1	e1	NOUN
iajs-2767	86	4	=	=	SYM
iajs-2767	86	5	{	{	PUNCT
iajs-2767	86	6	(	(	PUNCT
iajs-2767	86	7	𝜔1	𝜔1	ADJ
iajs-2767	86	8	,	,	PUNCT
iajs-2767	86	9	0.1	0.1	NUM
iajs-2767	86	10	)	)	PUNCT
iajs-2767	86	11	,	,	PUNCT
iajs-2767	86	12	(	(	PUNCT
iajs-2767	86	13	𝜔2	𝜔2	PROPN
iajs-2767	86	14	,	,	PUNCT
iajs-2767	86	15	0.5	0.5	NUM
iajs-2767	86	16	)	)	PUNCT
iajs-2767	86	17	}	}	PUNCT
iajs-2767	86	18	,	,	PUNCT
iajs-2767	86	19	e2	e2	PROPN
iajs-2767	86	20	=	=	SYM
iajs-2767	86	21	{	{	PUNCT
iajs-2767	86	22	(	(	PUNCT
iajs-2767	86	23	𝜔1	𝜔1	ADJ
iajs-2767	86	24	,	,	PUNCT
iajs-2767	86	25	0.9	0.9	NUM
iajs-2767	86	26	)	)	PUNCT
iajs-2767	86	27	,	,	PUNCT
iajs-2767	86	28	(	(	PUNCT
iajs-2767	86	29	𝜔2	𝜔2	ADV
iajs-2767	86	30	,	,	PUNCT
iajs-2767	86	31	0.5	0.5	NUM
iajs-2767	86	32	)	)	PUNCT
iajs-2767	86	33	}	}	PUNCT
iajs-2767	86	34	,	,	PUNCT
iajs-2767	86	35	e3	e3	NOUN
iajs-2767	86	36	=	=	SYM
iajs-2767	86	37	{	{	PUNCT
iajs-2767	86	38	(	(	PUNCT
iajs-2767	86	39	𝜔1	𝜔1	ADJ
iajs-2767	86	40	,	,	PUNCT
iajs-2767	86	41	0.2	0.2	NUM
iajs-2767	86	42	)	)	PUNCT
iajs-2767	86	43	,	,	PUNCT
iajs-2767	86	44	(	(	PUNCT
iajs-2767	86	45	𝜔2	𝜔2	PROPN
iajs-2767	86	46	,	,	PUNCT
iajs-2767	86	47	0.4	0.4	NUM
iajs-2767	86	48	)	)	PUNCT
iajs-2767	86	49	}	}	PUNCT
iajs-2767	86	50	,	,	PUNCT
iajs-2767	86	51	e4	e4	PROPN
iajs-2767	86	52	=	=	SYM
iajs-2767	86	53	{	{	PUNCT
iajs-2767	86	54	(	(	PUNCT
iajs-2767	86	55	𝜔1	𝜔1	ADJ
iajs-2767	86	56	,	,	PUNCT
iajs-2767	86	57	0.8	0.8	NUM
iajs-2767	86	58	)	)	PUNCT
iajs-2767	86	59	,	,	PUNCT
iajs-2767	86	60	(	(	PUNCT
iajs-2767	86	61	𝜔2	𝜔2	PROPN
iajs-2767	86	62	,	,	PUNCT
iajs-2767	86	63	0.6	0.6	NUM
iajs-2767	86	64	)	)	PUNCT
iajs-2767	86	65	}	}	PUNCT
iajs-2767	86	66	.then	.then	VERB
iajs-2767	86	67	ek	ek	PROPN
iajs-2767	86	68	∈	∈	PROPN
iajs-2767	86	69	ℋ1	ℋ1	PROPN
iajs-2767	86	70	∗	∗	NOUN
iajs-2767	86	71	⋃	⋃	NOUN
iajs-2767	86	72	ℋ2	ℋ2	PROPN
iajs-2767	86	73	∗	∗	NOUN
iajs-2767	86	74	for	for	ADP
iajs-2767	86	75	all	all	DET
iajs-2767	86	76	k=1,2,	k=1,2,	NOUN
iajs-2767	86	77	…	…	X
iajs-2767	86	78	,4	,4	PUNCT
iajs-2767	86	79	.	.	PUNCT
iajs-2767	87	1	so	so	ADV
iajs-2767	87	2	,	,	PUNCT
iajs-2767	87	3	we	we	PRON
iajs-2767	87	4	have	have	AUX
iajs-2767	87	5	⋃	⋃	SCONJ
iajs-2767	87	6	ek	ek	NOUN
iajs-2767	87	7	4	4	NUM
iajs-2767	87	8	k=1	k=1	NOUN
iajs-2767	87	9	=	=	PRON
iajs-2767	87	10	{	{	PUNCT
iajs-2767	87	11	(	(	PUNCT
iajs-2767	87	12	𝜔1	𝜔1	ADJ
iajs-2767	87	13	,	,	PUNCT
iajs-2767	87	14	𝑀𝑎𝑥{0.1,0.9,0.2,0.8	𝑀𝑎𝑥{0.1,0.9,0.2,0.8	PROPN
iajs-2767	87	15	}	}	PUNCT
iajs-2767	87	16	)	)	PUNCT
iajs-2767	87	17	,	,	PUNCT
iajs-2767	87	18	(	(	PUNCT
iajs-2767	87	19	𝜔2	𝜔2	PROPN
iajs-2767	87	20	,	,	PUNCT
iajs-2767	87	21	𝑀𝑎𝑥{0.5,0.5,0.4,0.6	𝑀𝑎𝑥{0.5,0.5,0.4,0.6	NOUN
iajs-2767	87	22	}	}	PUNCT
iajs-2767	87	23	)	)	PUNCT
iajs-2767	87	24	,	,	PUNCT
iajs-2767	87	25	}	}	PUNCT
iajs-2767	87	26	=	=	PRON
iajs-2767	87	27	{	{	PUNCT
iajs-2767	87	28	(	(	PUNCT
iajs-2767	87	29	𝜔1,0.9	𝜔1,0.9	NUM
iajs-2767	87	30	)	)	PUNCT
iajs-2767	87	31	,	,	PUNCT
iajs-2767	87	32	(	(	PUNCT
iajs-2767	87	33	𝜔2,0.6	𝜔2,0.6	X
iajs-2767	87	34	)	)	PUNCT
iajs-2767	87	35	}	}	PUNCT
iajs-2767	87	36	∉	∉	PROPN
iajs-2767	87	37	ℋ1	ℋ1	PROPN
iajs-2767	87	38	∗	∗	NOUN
iajs-2767	87	39	⋃	⋃	PROPN
iajs-2767	87	40	ℋ2	ℋ2	PROPN
iajs-2767	87	41	∗	∗	NOUN
iajs-2767	87	42	.	.	PUNCT
iajs-2767	88	1	thus	thus	ADV
iajs-2767	88	2	,	,	PUNCT
iajs-2767	88	3	ℋ1	ℋ1	NOUN
iajs-2767	88	4	∗	∗	NOUN
iajs-2767	88	5	⋃	⋃	NOUN
iajs-2767	88	6	ℋ2	ℋ2	PROPN
iajs-2767	88	7	∗	∗	NOUN
iajs-2767	88	8	is	be	AUX
iajs-2767	88	9	not	not	PART
iajs-2767	88	10	fuzzy	fuzzy	ADJ
iajs-2767	88	11	σ	σ	NOUN
iajs-2767	88	12	–	–	PUNCT
iajs-2767	88	13	ring	ring	NOUN
iajs-2767	88	14	over	over	ADP
iajs-2767	88	15	a	a	DET
iajs-2767	88	16	fuzzy	fuzzy	ADJ
iajs-2767	88	17	set	set	VERB
iajs-2767	88	18	𝒳∗.	𝒳∗.	PROPN
iajs-2767	88	19	definition	definition	NOUN
iajs-2767	88	20	3.3	3.3	NUM
iajs-2767	88	21	assume	assume	VERB
iajs-2767	88	22	𝒳	𝒳	NOUN
iajs-2767	88	23	≠	≠	PROPN
iajs-2767	88	24	∅	∅	NOUN
iajs-2767	88	25	and	and	CCONJ
iajs-2767	88	26	𝔗∗	𝔗∗	NUM
iajs-2767	88	27	⊆	⊆	NUM
iajs-2767	88	28	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	88	29	)	)	PUNCT
iajs-2767	88	30	,	,	PUNCT
iajs-2767	88	31	then	then	ADV
iajs-2767	88	32	the	the	DET
iajs-2767	88	33	intersection	intersection	NOUN
iajs-2767	88	34	of	of	ADP
iajs-2767	88	35	all	all	DET
iajs-2767	88	36	fuzzy	fuzzy	ADJ
iajs-2767	88	37	σ	σ	PROPN
iajs-2767	88	38	–	–	PUNCT
iajs-2767	88	39	ring	ring	NOUN
iajs-2767	88	40	over	over	ADP
iajs-2767	88	41	a	a	DET
iajs-2767	88	42	fuzzy	fuzzy	ADJ
iajs-2767	88	43	set	set	VERB
iajs-2767	88	44	𝒳∗	𝒳∗	PROPN
iajs-2767	88	45	,	,	PUNCT
iajs-2767	88	46	which	which	PRON
iajs-2767	88	47	includes	include	VERB
iajs-2767	88	48	𝔗∗	𝔗∗	NUM
iajs-2767	88	49	is	be	AUX
iajs-2767	88	50	said	say	VERB
iajs-2767	88	51	to	to	PART
iajs-2767	88	52	be	be	AUX
iajs-2767	88	53	the	the	DET
iajs-2767	88	54	fuzzy	fuzzy	ADJ
iajs-2767	88	55	σ	σ	PROPN
iajs-2767	88	56	–	–	PUNCT
iajs-2767	88	57	ring	ring	NOUN
iajs-2767	88	58	over	over	ADP
iajs-2767	88	59	a	a	DET
iajs-2767	88	60	fuzzy	fuzzy	ADJ
iajs-2767	88	61	set	set	NOUN
iajs-2767	88	62	𝒳∗	𝒳∗	PRON
iajs-2767	88	63	generated	generate	VERB
iajs-2767	88	64	by	by	ADP
iajs-2767	88	65	𝔗∗	𝔗∗	PUNCT
iajs-2767	88	66	and	and	CCONJ
iajs-2767	88	67	denoted	denote	VERB
iajs-2767	88	68	by	by	ADP
iajs-2767	88	69	σ𝑟(𝔗∗	σ𝑟(𝔗∗	NOUN
iajs-2767	88	70	)	)	PUNCT
iajs-2767	88	71	,	,	PUNCT
iajs-2767	88	72	that	that	PRON
iajs-2767	88	73	is	be	AUX
iajs-2767	88	74	:	:	PUNCT
iajs-2767	88	75	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	88	76	)	)	PUNCT
iajs-2767	89	1	=	=	SYM
iajs-2767	89	2	⋂{ℋi	⋂{ℋi	NOUN
iajs-2767	89	3	∗	∗	NOUN
iajs-2767	89	4	:	:	PUNCT
iajs-2767	90	1	ℋi	ℋi	ADJ
iajs-2767	90	2	∗	∗	NOUN
iajs-2767	90	3	is	be	AUX
iajs-2767	90	4	a	a	DET
iajs-2767	90	5	fuzzy	fuzzy	ADJ
iajs-2767	90	6	σ	σ	NOUN
iajs-2767	90	7	–	–	PUNCT
iajs-2767	90	8	ring	ring	NOUN
iajs-2767	90	9	over	over	ADP
iajs-2767	90	10	a	a	DET
iajs-2767	90	11	fuzzy	fuzzy	ADJ
iajs-2767	90	12	set	set	NOUN
iajs-2767	90	13	𝒳∗and	𝒳∗and	PROPN
iajs-2767	91	1	ℋi	ℋi	PROPN
iajs-2767	91	2	∗	∗	X
iajs-2767	91	3	⊇	⊇	NOUN
iajs-2767	91	4	𝔗∗	𝔗∗	NOUN
iajs-2767	91	5	,	,	PUNCT
iajs-2767	91	6	∀i	∀i	NOUN
iajs-2767	91	7	∈	∈	NOUN
iajs-2767	91	8	ι	ι	X
iajs-2767	91	9	}	}	PUNCT
iajs-2767	91	10	.	.	PUNCT
iajs-2767	92	1	proposition	proposition	NOUN
iajs-2767	92	2	3.1	3.1	NUM
iajs-2767	92	3	assume	assume	VERB
iajs-2767	92	4	𝒳	𝒳	PROPN
iajs-2767	92	5	≠	≠	PROPN
iajs-2767	92	6	∅	∅	NOUN
iajs-2767	92	7	and	and	CCONJ
iajs-2767	92	8	𝔗∗	𝔗∗	NUM
iajs-2767	92	9	⊆	⊆	NUM
iajs-2767	92	10	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	92	11	)	)	PUNCT
iajs-2767	92	12	.	.	PUNCT
iajs-2767	93	1	then	then	ADV
iajs-2767	93	2	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	93	3	)	)	PUNCT
iajs-2767	93	4	is	be	AUX
iajs-2767	93	5	the	the	DET
iajs-2767	93	6	smallest	small	ADJ
iajs-2767	93	7	fuzzy	fuzzy	ADJ
iajs-2767	93	8	σ	σ	NOUN
iajs-2767	93	9	–	–	PUNCT
iajs-2767	93	10	ring	ring	NOUN
iajs-2767	93	11	over	over	ADP
iajs-2767	93	12	a	a	DET
iajs-2767	93	13	fuzzy	fuzzy	ADJ
iajs-2767	93	14	set	set	NOUN
iajs-2767	93	15	𝒳∗	𝒳∗	PRON
iajs-2767	93	16	that	that	PRON
iajs-2767	93	17	includes	include	VERB
iajs-2767	93	18	𝔗∗.	𝔗∗.	PROPN
iajs-2767	93	19	proof	proof	NOUN
iajs-2767	93	20	the	the	DET
iajs-2767	93	21	result	result	NOUN
iajs-2767	93	22	is	be	AUX
iajs-2767	93	23	directed	direct	VERB
iajs-2767	93	24	by	by	ADP
iajs-2767	93	25	the	the	DET
iajs-2767	93	26	definition	definition	NOUN
iajs-2767	93	27	of	of	ADP
iajs-2767	93	28	σ𝑟(𝔗∗	σ𝑟(𝔗∗	NOUN
iajs-2767	93	29	)	)	PUNCT
iajs-2767	93	30	and	and	CCONJ
iajs-2767	93	31	lemma	lemma	PROPN
iajs-2767	93	32	3.1	3.1	NUM
iajs-2767	93	33	.	.	PUNCT
iajs-2767	94	1	ibn	ibn	PROPN
iajs-2767	94	2	al	al	PROPN
iajs-2767	94	3	-	-	PUNCT
iajs-2767	94	4	haitham	haitham	PROPN
iajs-2767	94	5	jour	jour	X
iajs-2767	94	6	.	.	PROPN
iajs-2767	94	7	for	for	ADP
iajs-2767	94	8	pure	pure	ADJ
iajs-2767	94	9	&	&	CCONJ
iajs-2767	94	10	appl	appl	PROPN
iajs-2767	94	11	.	.	PUNCT
iajs-2767	95	1	sci	sci	PROPN
iajs-2767	95	2	.	.	PROPN
iajs-2767	96	1	53	53	NUM
iajs-2767	96	2	(	(	PUNCT
iajs-2767	96	3	2)2022	2)2022	NOUN
iajs-2767	96	4	41	41	NUM
iajs-2767	96	5	example	example	NOUN
iajs-2767	96	6	3.5	3.5	NUM
iajs-2767	96	7	let	let	VERB
iajs-2767	96	8	𝒳	𝒳	PRON
iajs-2767	96	9	=	=	PRON
iajs-2767	96	10	{	{	PUNCT
iajs-2767	96	11	𝜔1	𝜔1	ADJ
iajs-2767	96	12	,	,	PUNCT
iajs-2767	96	13	𝜔2	𝜔2	PROPN
iajs-2767	96	14	}	}	PUNCT
iajs-2767	96	15	and	and	CCONJ
iajs-2767	96	16	𝔗∗	𝔗∗	X
iajs-2767	96	17	=	=	X
iajs-2767	96	18	{	{	PUNCT
iajs-2767	96	19	{	{	PUNCT
iajs-2767	96	20	(	(	PUNCT
iajs-2767	96	21	𝜔1	𝜔1	ADJ
iajs-2767	96	22	,	,	PUNCT
iajs-2767	96	23	0.1	0.1	NUM
iajs-2767	96	24	)	)	PUNCT
iajs-2767	96	25	,	,	PUNCT
iajs-2767	96	26	(	(	PUNCT
iajs-2767	96	27	𝜔2	𝜔2	PROPN
iajs-2767	96	28	,	,	PUNCT
iajs-2767	96	29	0.5	0.5	NUM
iajs-2767	96	30	)	)	PUNCT
iajs-2767	96	31	}	}	PUNCT
iajs-2767	96	32	,	,	PUNCT
iajs-2767	96	33	{	{	PUNCT
iajs-2767	96	34	(	(	PUNCT
iajs-2767	96	35	𝜔1	𝜔1	ADJ
iajs-2767	96	36	,	,	PUNCT
iajs-2767	96	37	0.9	0.9	NUM
iajs-2767	96	38	)	)	PUNCT
iajs-2767	96	39	,	,	PUNCT
iajs-2767	96	40	(	(	PUNCT
iajs-2767	96	41	𝜔2	𝜔2	ADV
iajs-2767	96	42	,	,	PUNCT
iajs-2767	96	43	0.5	0.5	NUM
iajs-2767	96	44	)	)	PUNCT
iajs-2767	96	45	}	}	PUNCT
iajs-2767	96	46	}	}	PUNCT
iajs-2767	96	47	.	.	PUNCT
iajs-2767	97	1	then	then	ADV
iajs-2767	97	2	,	,	PUNCT
iajs-2767	97	3	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	97	4	)	)	PUNCT
iajs-2767	98	1	=	=	NOUN
iajs-2767	98	2	{	{	PUNCT
iajs-2767	98	3	∅∗	∅∗	PROPN
iajs-2767	98	4	,	,	PUNCT
iajs-2767	98	5	{	{	PUNCT
iajs-2767	98	6	(	(	PUNCT
iajs-2767	98	7	𝜔1,0.1	𝜔1,0.1	NOUN
iajs-2767	98	8	)	)	PUNCT
iajs-2767	98	9	,	,	PUNCT
iajs-2767	98	10	(	(	PUNCT
iajs-2767	98	11	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	98	12	)	)	PUNCT
iajs-2767	98	13	}	}	PUNCT
iajs-2767	98	14	,	,	PUNCT
iajs-2767	98	15	{	{	PUNCT
iajs-2767	98	16	(	(	PUNCT
iajs-2767	98	17	𝜔1,0.9	𝜔1,0.9	NUM
iajs-2767	98	18	)	)	PUNCT
iajs-2767	98	19	,	,	PUNCT
iajs-2767	98	20	(	(	PUNCT
iajs-2767	98	21	𝜔2,0.5)},𝒳∗	𝜔2,0.5)},𝒳∗	PROPN
iajs-2767	98	22	}	}	PUNCT
iajs-2767	98	23	is	be	AUX
iajs-2767	98	24	the	the	DET
iajs-2767	98	25	smallest	small	ADJ
iajs-2767	98	26	fuzzy	fuzzy	ADJ
iajs-2767	98	27	σ	σ	NOUN
iajs-2767	98	28	–	–	PUNCT
iajs-2767	98	29	ring	ring	NOUN
iajs-2767	98	30	over	over	ADP
iajs-2767	98	31	a	a	DET
iajs-2767	98	32	fuzzy	fuzzy	ADJ
iajs-2767	98	33	set	set	NOUN
iajs-2767	98	34	𝒳∗	𝒳∗	PRON
iajs-2767	98	35	that	that	PRON
iajs-2767	98	36	include	include	VERB
iajs-2767	98	37	𝔗∗.	𝔗∗.	PROPN
iajs-2767	98	38	proposition	proposition	NOUN
iajs-2767	98	39	3.2	3.2	NUM
iajs-2767	98	40	assume	assume	VERB
iajs-2767	98	41	𝒳	𝒳	PROPN
iajs-2767	98	42	≠	≠	PROPN
iajs-2767	98	43	∅	∅	NOUN
iajs-2767	98	44	and	and	CCONJ
iajs-2767	98	45	𝔗∗	𝔗∗	NUM
iajs-2767	98	46	⊆	⊆	NUM
iajs-2767	98	47	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	98	48	)	)	PUNCT
iajs-2767	98	49	,	,	PUNCT
iajs-2767	98	50	then	then	ADV
iajs-2767	98	51	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	98	52	)	)	PUNCT
iajs-2767	99	1	=	=	PUNCT
iajs-2767	99	2	𝔗∗	𝔗∗	PUNCT
iajs-2767	99	3	if	if	SCONJ
iajs-2767	99	4	and	and	CCONJ
iajs-2767	99	5	only	only	ADV
iajs-2767	99	6	if	if	SCONJ
iajs-2767	99	7	𝔗∗	𝔗∗	NOUN
iajs-2767	99	8	is	be	AUX
iajs-2767	99	9	fuzzy	fuzzy	ADJ
iajs-2767	99	10	σ	σ	NOUN
iajs-2767	99	11	–	–	PUNCT
iajs-2767	99	12	ring	ring	NOUN
iajs-2767	99	13	over	over	ADP
iajs-2767	99	14	a	a	DET
iajs-2767	99	15	fuzzy	fuzzy	ADJ
iajs-2767	99	16	set	set	NOUN
iajs-2767	99	17	𝒳∗.	𝒳∗.	PROPN
iajs-2767	99	18	proof	proof	NOUN
iajs-2767	99	19	the	the	DET
iajs-2767	99	20	result	result	NOUN
iajs-2767	99	21	direct	direct	ADJ
iajs-2767	99	22	by	by	ADP
iajs-2767	99	23	definition	definition	NOUN
iajs-2767	99	24	of	of	ADP
iajs-2767	99	25	σ𝑟(𝔗∗	σ𝑟(𝔗∗	NOUN
iajs-2767	99	26	)	)	PUNCT
iajs-2767	99	27	and	and	CCONJ
iajs-2767	99	28	proposition	proposition	NOUN
iajs-2767	99	29	3.1	3.1	NUM
iajs-2767	99	30	.	.	PUNCT
iajs-2767	100	1	proposition	proposition	NOUN
iajs-2767	100	2	3.3	3.3	NUM
iajs-2767	100	3	every	every	DET
iajs-2767	100	4	fuzzy	fuzzy	ADJ
iajs-2767	100	5	σ	σ	PROPN
iajs-2767	100	6	–	–	PUNCT
iajs-2767	100	7	algebra	algebra	NOUN
iajs-2767	100	8	over	over	ADP
iajs-2767	100	9	a	a	DET
iajs-2767	100	10	fuzzy	fuzzy	ADJ
iajs-2767	100	11	set	set	NOUN
iajs-2767	100	12	𝒳∗	𝒳∗	NOUN
iajs-2767	100	13	is	be	AUX
iajs-2767	100	14	a	a	DET
iajs-2767	100	15	fuzzy	fuzzy	ADJ
iajs-2767	100	16	σ	σ	NOUN
iajs-2767	100	17	–	–	PUNCT
iajs-2767	100	18	ring	ring	NOUN
iajs-2767	100	19	over	over	ADP
iajs-2767	100	20	a	a	DET
iajs-2767	100	21	fuzzy	fuzzy	ADJ
iajs-2767	100	22	set	set	NOUN
iajs-2767	100	23	𝒳∗.	𝒳∗.	PROPN
iajs-2767	100	24	proof	proof	NOUN
iajs-2767	100	25	let	let	VERB
iajs-2767	100	26	ℋ∗	ℋ∗	NOUN
iajs-2767	100	27	be	be	AUX
iajs-2767	100	28	a	a	DET
iajs-2767	100	29	fuzzy	fuzzy	ADJ
iajs-2767	100	30	σ	σ	NOUN
iajs-2767	100	31	–	–	PUNCT
iajs-2767	100	32	algebra	algebra	NOUN
iajs-2767	100	33	over	over	ADP
iajs-2767	100	34	a	a	DET
iajs-2767	100	35	fuzzy	fuzzy	ADJ
iajs-2767	100	36	set	set	NOUN
iajs-2767	100	37	𝒳∗.	𝒳∗.	PROPN
iajs-2767	100	38	then	then	ADV
iajs-2767	100	39	from	from	ADP
iajs-2767	100	40	the	the	DET
iajs-2767	100	41	definition	definition	NOUN
iajs-2767	100	42	of	of	ADP
iajs-2767	100	43	fuzzy	fuzzy	ADJ
iajs-2767	100	44	σ	σ	PROPN
iajs-2767	100	45	–	–	PUNCT
iajs-2767	100	46	algebra	algebra	NOUN
iajs-2767	100	47	we	we	PRON
iajs-2767	100	48	get	get	VERB
iajs-2767	100	49	,	,	PUNCT
iajs-2767	100	50	∅∗	∅∗	PROPN
iajs-2767	100	51	∈	∈	PROPN
iajs-2767	100	52	ℋ∗.	ℋ∗.	PROPN
iajs-2767	100	53	let	let	VERB
iajs-2767	100	54	f	f	NOUN
iajs-2767	101	1	,	,	PUNCT
iajs-2767	101	2	e	e	PROPN
iajs-2767	101	3	∈	∈	PROPN
iajs-2767	101	4	ℋ∗.	ℋ∗.	PROPN
iajs-2767	101	5	then	then	ADV
iajs-2767	101	6	e𝑐	e𝑐	PROPN
iajs-2767	101	7	∈	∈	PROPN
iajs-2767	101	8	ℋ∗	ℋ∗	NUM
iajs-2767	101	9	,	,	PUNCT
iajs-2767	101	10	hence	hence	ADV
iajs-2767	101	11	f⋂e𝑐	f⋂e𝑐	PROPN
iajs-2767	101	12	∈	∈	PROPN
iajs-2767	101	13	ℋ∗	ℋ∗	NOUN
iajs-2767	101	14	,	,	PUNCT
iajs-2767	101	15	but	but	CCONJ
iajs-2767	101	16	f⋂e𝑐	f⋂e𝑐	NOUN
iajs-2767	101	17	=	=	NOUN
iajs-2767	101	18	f\e	f\e	NOUN
iajs-2767	101	19	implies	imply	VERB
iajs-2767	101	20	that	that	SCONJ
iajs-2767	101	21	f\e	f\e	VERB
iajs-2767	101	22	∈	∈	PROPN
iajs-2767	101	23	ℋ∗.	ℋ∗.	PROPN
iajs-2767	101	24	let	let	VERB
iajs-2767	101	25	e1	e1	NOUN
iajs-2767	101	26	,	,	PUNCT
iajs-2767	101	27	e2	e2	PROPN
iajs-2767	101	28	,	,	PUNCT
iajs-2767	101	29	…	…	PUNCT
iajs-2767	101	30	∈	∈	PROPN
iajs-2767	101	31	ℋ∗.	ℋ∗.	PROPN
iajs-2767	101	32	then	then	ADV
iajs-2767	101	33	by	by	ADP
iajs-2767	101	34	definition	definition	NOUN
iajs-2767	101	35	of	of	ADP
iajs-2767	101	36	fuzzy	fuzzy	ADJ
iajs-2767	101	37	σ	σ	PROPN
iajs-2767	101	38	–	–	PUNCT
iajs-2767	101	39	algebra	algebra	NOUN
iajs-2767	101	40	we	we	PRON
iajs-2767	101	41	have	have	VERB
iajs-2767	101	42	,	,	PUNCT
iajs-2767	101	43	ek	ek	PROPN
iajs-2767	101	44	𝑐	𝑐	PROPN
iajs-2767	101	45	(	(	PUNCT
iajs-2767	101	46	for	for	ADP
iajs-2767	101	47	all	all	PRON
iajs-2767	101	48	k	k	NOUN
iajs-2767	101	49	=	=	SYM
iajs-2767	101	50	1,2	1,2	NUM
iajs-2767	101	51	,	,	PUNCT
iajs-2767	101	52	…	…	PUNCT
iajs-2767	101	53	)	)	PUNCT
iajs-2767	101	54	and	and	CCONJ
iajs-2767	101	55	⋂	⋂	PROPN
iajs-2767	101	56	ek	ek	X
iajs-2767	101	57	𝑐∞	𝑐∞	PUNCT
iajs-2767	101	58	k=1	k=1	PROPN
iajs-2767	101	59	∈	∈	PROPN
iajs-2767	101	60	ℋ∗	ℋ∗	PROPN
iajs-2767	102	1	and	and	CCONJ
iajs-2767	102	2	(	(	PUNCT
iajs-2767	102	3	⋂	⋂	PROPN
iajs-2767	102	4	ek	ek	X
iajs-2767	102	5	𝑐∞	𝑐∞	PUNCT
iajs-2767	102	6	k=1	k=1	X
iajs-2767	102	7	)	)	PUNCT
iajs-2767	102	8	𝑐	𝑐	PROPN
iajs-2767	102	9	∈	∈	PROPN
iajs-2767	102	10	ℋ∗.	ℋ∗.	PROPN
iajs-2767	102	11	by	by	ADP
iajs-2767	102	12	de	de	PROPN
iajs-2767	102	13	-	-	PROPN
iajs-2767	102	14	morgan	morgan	PROPN
iajs-2767	102	15	law	law	NOUN
iajs-2767	102	16	,	,	PUNCT
iajs-2767	102	17	we	we	PRON
iajs-2767	102	18	get	get	VERB
iajs-2767	102	19	(	(	PUNCT
iajs-2767	102	20	⋂	⋂	PROPN
iajs-2767	102	21	ek	ek	X
iajs-2767	102	22	𝑐∞	𝑐∞	PUNCT
iajs-2767	102	23	k=1	k=1	PROPN
iajs-2767	102	24	)	)	PUNCT
iajs-2767	102	25	𝑐	𝑐	NOUN
iajs-2767	102	26	=	=	PUNCT
iajs-2767	103	1	⋃	⋃	VERB
iajs-2767	103	2	ek	ek	NOUN
iajs-2767	103	3	∞	∞	PROPN
iajs-2767	103	4	k=1	k=1	X
iajs-2767	103	5	,	,	PUNCT
iajs-2767	103	6	hence	hence	ADV
iajs-2767	103	7	⋃	⋃	PUNCT
iajs-2767	103	8	ek	ek	NOUN
iajs-2767	103	9	∞	∞	PROPN
iajs-2767	103	10	k=1	k=1	PUNCT
iajs-2767	104	1	∈	∈	PROPN
iajs-2767	104	2	ℋ∗.	ℋ∗.	PROPN
iajs-2767	104	3	therefore	therefore	ADV
iajs-2767	104	4	,	,	PUNCT
iajs-2767	104	5	ℋ∗	ℋ∗	NOUN
iajs-2767	104	6	be	be	AUX
iajs-2767	104	7	a	a	DET
iajs-2767	104	8	fuzzy	fuzzy	ADJ
iajs-2767	104	9	σ	σ	NOUN
iajs-2767	104	10	–	–	PUNCT
iajs-2767	104	11	ring	ring	NOUN
iajs-2767	104	12	over	over	ADP
iajs-2767	104	13	a	a	DET
iajs-2767	104	14	fuzzy	fuzzy	ADJ
iajs-2767	104	15	set	set	VERB
iajs-2767	104	16	𝒳∗.	𝒳∗.	PROPN
iajs-2767	104	17	while	while	SCONJ
iajs-2767	104	18	the	the	DET
iajs-2767	104	19	converse	converse	NOUN
iajs-2767	104	20	is	be	AUX
iajs-2767	104	21	not	not	PART
iajs-2767	104	22	true	true	ADJ
iajs-2767	104	23	as	as	SCONJ
iajs-2767	104	24	shown	show	VERB
iajs-2767	104	25	below	below	ADP
iajs-2767	104	26	:	:	PUNCT
iajs-2767	104	27	example	example	NOUN
iajs-2767	104	28	3.6	3.6	NUM
iajs-2767	104	29	let	let	VERB
iajs-2767	104	30	𝒳	𝒳	PRON
iajs-2767	104	31	=	=	PRON
iajs-2767	104	32	{	{	PUNCT
iajs-2767	104	33	𝜔1	𝜔1	ADJ
iajs-2767	104	34	,	,	PUNCT
iajs-2767	104	35	𝜔2	𝜔2	PROPN
iajs-2767	104	36	}	}	PUNCT
iajs-2767	104	37	and	and	CCONJ
iajs-2767	104	38	ℋ∗	ℋ∗	NUM
iajs-2767	104	39	=	=	SYM
iajs-2767	104	40	{	{	PUNCT
iajs-2767	104	41	∅∗	∅∗	PROPN
iajs-2767	104	42	,	,	PUNCT
iajs-2767	104	43	{	{	PUNCT
iajs-2767	104	44	(	(	PUNCT
iajs-2767	104	45	𝜔1	𝜔1	ADJ
iajs-2767	104	46	,	,	PUNCT
iajs-2767	104	47	0	0	NUM
iajs-2767	104	48	)	)	PUNCT
iajs-2767	104	49	,	,	PUNCT
iajs-2767	104	50	(	(	PUNCT
iajs-2767	104	51	𝜔2	𝜔2	PROPN
iajs-2767	104	52	,	,	PUNCT
iajs-2767	104	53	0.5	0.5	NUM
iajs-2767	104	54	)	)	PUNCT
iajs-2767	104	55	}	}	PUNCT
iajs-2767	104	56	,	,	PUNCT
iajs-2767	104	57	{	{	PUNCT
iajs-2767	104	58	(	(	PUNCT
iajs-2767	104	59	𝜔1	𝜔1	ADJ
iajs-2767	104	60	,	,	PUNCT
iajs-2767	104	61	0.6	0.6	NUM
iajs-2767	104	62	)	)	PUNCT
iajs-2767	104	63	,	,	PUNCT
iajs-2767	104	64	(	(	PUNCT
iajs-2767	104	65	𝜔2	𝜔2	PROPN
iajs-2767	104	66	,	,	PUNCT
iajs-2767	104	67	0.5	0.5	NUM
iajs-2767	104	68	)	)	PUNCT
iajs-2767	104	69	}	}	PUNCT
iajs-2767	104	70	,	,	PUNCT
iajs-2767	104	71	{	{	PUNCT
iajs-2767	104	72	(	(	PUNCT
iajs-2767	104	73	𝜔1	𝜔1	ADJ
iajs-2767	104	74	,	,	PUNCT
iajs-2767	104	75	0.4	0.4	NUM
iajs-2767	104	76	)	)	PUNCT
iajs-2767	104	77	,	,	PUNCT
iajs-2767	104	78	(	(	PUNCT
iajs-2767	104	79	𝜔2	𝜔2	PROPN
iajs-2767	104	80	,	,	PUNCT
iajs-2767	104	81	0.5	0.5	NUM
iajs-2767	104	82	)	)	PUNCT
iajs-2767	104	83	}	}	PUNCT
iajs-2767	104	84	}	}	PUNCT
iajs-2767	104	85	.	.	PUNCT
iajs-2767	105	1	put	put	VERB
iajs-2767	105	2	f=	f=	NOUN
iajs-2767	105	3	{	{	PUNCT
iajs-2767	105	4	(	(	PUNCT
iajs-2767	105	5	𝜔1,0	𝜔1,0	PROPN
iajs-2767	105	6	)	)	PUNCT
iajs-2767	105	7	,	,	PUNCT
iajs-2767	105	8	(	(	PUNCT
iajs-2767	105	9	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	105	10	)	)	PUNCT
iajs-2767	105	11	}	}	PUNCT
iajs-2767	105	12	,	,	PUNCT
iajs-2767	105	13	then	then	ADV
iajs-2767	105	14	f	f	PROPN
iajs-2767	105	15	\∅∗	\∅∗	PROPN
iajs-2767	105	16	=	=	SYM
iajs-2767	105	17	{	{	PUNCT
iajs-2767	105	18	(	(	PUNCT
iajs-2767	105	19	𝜔1	𝜔1	ADJ
iajs-2767	105	20	,	,	PUNCT
iajs-2767	105	21	𝑀𝑖𝑛{0,1	𝑀𝑖𝑛{0,1	ADV
iajs-2767	105	22	−	−	NOUN
iajs-2767	105	23	0	0	NUM
iajs-2767	105	24	}	}	PUNCT
iajs-2767	105	25	)	)	PUNCT
iajs-2767	105	26	,	,	PUNCT
iajs-2767	105	27	(	(	PUNCT
iajs-2767	105	28	𝜔2	𝜔2	ADV
iajs-2767	105	29	,	,	PUNCT
iajs-2767	105	30	𝑀𝑖𝑛{0.5,1	𝑀𝑖𝑛{0.5,1	ADV
iajs-2767	105	31	−	−	PROPN
iajs-2767	105	32	0	0	NUM
iajs-2767	105	33	}	}	PUNCT
iajs-2767	105	34	)	)	PUNCT
iajs-2767	105	35	}	}	PUNCT
iajs-2767	106	1	=	=	SYM
iajs-2767	106	2	{	{	PUNCT
iajs-2767	106	3	(	(	PUNCT
iajs-2767	106	4	𝜔1	𝜔1	ADJ
iajs-2767	106	5	,	,	PUNCT
iajs-2767	106	6	0	0	NUM
iajs-2767	106	7	)	)	PUNCT
iajs-2767	106	8	,	,	PUNCT
iajs-2767	106	9	(	(	PUNCT
iajs-2767	106	10	𝜔2	𝜔2	PROPN
iajs-2767	106	11	,	,	PUNCT
iajs-2767	106	12	0.5)}=	0.5)}=	NOUN
iajs-2767	106	13	f	f	PROPN
iajs-2767	106	14	in	in	ADP
iajs-2767	106	15	the	the	DET
iajs-2767	106	16	same	same	ADJ
iajs-2767	106	17	way	way	NOUN
iajs-2767	106	18	,	,	PUNCT
iajs-2767	106	19	we	we	PRON
iajs-2767	106	20	get	get	VERB
iajs-2767	106	21	∅∗\	∅∗\	NOUN
iajs-2767	106	22	f	f	PROPN
iajs-2767	107	1	=	=	X
iajs-2767	107	2	∅∗.	∅∗.	NUM
iajs-2767	107	3	since	since	SCONJ
iajs-2767	107	4	fc	fc	PROPN
iajs-2767	107	5	=	=	PUNCT
iajs-2767	107	6	{	{	PUNCT
iajs-2767	107	7	(	(	PUNCT
iajs-2767	107	8	𝜔1,1	𝜔1,1	NOUN
iajs-2767	107	9	)	)	PUNCT
iajs-2767	107	10	,	,	PUNCT
iajs-2767	107	11	(	(	PUNCT
iajs-2767	107	12	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	107	13	)	)	PUNCT
iajs-2767	107	14	}	}	PUNCT
iajs-2767	107	15	,	,	PUNCT
iajs-2767	107	16	then	then	ADV
iajs-2767	107	17	𝓋𝐹(𝜔	𝓋𝐹(𝜔	NUM
iajs-2767	107	18	)	)	PUNCT
iajs-2767	107	19	≤	≤	NUM
iajs-2767	108	1	𝓋fc(𝜔	𝓋fc(𝜔	X
iajs-2767	108	2	)	)	PUNCT
iajs-2767	108	3	,	,	PUNCT
iajs-2767	108	4	hence	hence	ADV
iajs-2767	108	5	f	f	PROPN
iajs-2767	108	6	⊆	⊆	NUM
iajs-2767	108	7	fc	fc	PROPN
iajs-2767	108	8	and	and	CCONJ
iajs-2767	108	9	f\	f\	PUNCT
iajs-2767	108	10	f	f	PROPN
iajs-2767	109	1	=	=	PUNCT
iajs-2767	109	2	f.	f.	PROPN
iajs-2767	109	3	if	if	SCONJ
iajs-2767	109	4	e	e	PROPN
iajs-2767	109	5	=	=	PRON
iajs-2767	109	6	{	{	PUNCT
iajs-2767	109	7	(	(	PUNCT
iajs-2767	109	8	𝜔1,0.6	𝜔1,0.6	NOUN
iajs-2767	109	9	)	)	PUNCT
iajs-2767	109	10	,	,	PUNCT
iajs-2767	109	11	(	(	PUNCT
iajs-2767	109	12	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	109	13	)	)	PUNCT
iajs-2767	109	14	}	}	PUNCT
iajs-2767	109	15	,	,	PUNCT
iajs-2767	109	16	then	then	ADV
iajs-2767	109	17	e\∅∗	e\∅∗	VERB
iajs-2767	109	18	=	=	SYM
iajs-2767	109	19	e	e	NOUN
iajs-2767	109	20	and	and	CCONJ
iajs-2767	109	21	∅∗\	∅∗\	NOUN
iajs-2767	109	22	e	e	PROPN
iajs-2767	109	23	=	=	PROPN
iajs-2767	109	24	∅∗	∅∗	PROPN
iajs-2767	110	1	and	and	CCONJ
iajs-2767	110	2	if	if	SCONJ
iajs-2767	110	3	f	f	PROPN
iajs-2767	110	4	=	=	PRON
iajs-2767	110	5	{	{	PUNCT
iajs-2767	110	6	(	(	PUNCT
iajs-2767	110	7	𝜔1,0	𝜔1,0	PROPN
iajs-2767	110	8	)	)	PUNCT
iajs-2767	110	9	,	,	PUNCT
iajs-2767	110	10	(	(	PUNCT
iajs-2767	110	11	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	110	12	)	)	PUNCT
iajs-2767	110	13	}	}	PUNCT
iajs-2767	110	14	and	and	CCONJ
iajs-2767	110	15	e	e	X
iajs-2767	110	16	=	=	SYM
iajs-2767	110	17	{	{	PUNCT
iajs-2767	110	18	(	(	PUNCT
iajs-2767	110	19	𝜔1,0.6	𝜔1,0.6	NOUN
iajs-2767	110	20	)	)	PUNCT
iajs-2767	110	21	,	,	PUNCT
iajs-2767	110	22	(	(	PUNCT
iajs-2767	110	23	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	110	24	)	)	PUNCT
iajs-2767	110	25	}	}	PUNCT
iajs-2767	110	26	,	,	PUNCT
iajs-2767	110	27	then	then	ADV
iajs-2767	110	28	ec	ec	PROPN
iajs-2767	110	29	=	=	PRON
iajs-2767	110	30	{	{	PUNCT
iajs-2767	110	31	(	(	PUNCT
iajs-2767	110	32	𝜔1,0.4	𝜔1,0.4	NUM
iajs-2767	110	33	)	)	PUNCT
iajs-2767	110	34	,	,	PUNCT
iajs-2767	110	35	(	(	PUNCT
iajs-2767	110	36	𝜔2,0.5	𝜔2,0.5	NOUN
iajs-2767	110	37	)	)	PUNCT
iajs-2767	110	38	}	}	PUNCT
iajs-2767	110	39	and	and	CCONJ
iajs-2767	110	40	e	e	X
iajs-2767	110	41	\e	\e	PROPN
iajs-2767	110	42	=	=	SYM
iajs-2767	110	43	{	{	PUNCT
iajs-2767	110	44	(	(	PUNCT
iajs-2767	110	45	𝜔1	𝜔1	ADJ
iajs-2767	110	46	,	,	PUNCT
iajs-2767	110	47	𝑀𝑖𝑛{0.6,0.4	𝑀𝑖𝑛{0.6,0.4	NOUN
iajs-2767	110	48	}	}	PUNCT
iajs-2767	110	49	)	)	PUNCT
iajs-2767	110	50	,	,	PUNCT
iajs-2767	110	51	(	(	PUNCT
iajs-2767	110	52	𝜔2	𝜔2	PROPN
iajs-2767	110	53	,	,	PUNCT
iajs-2767	110	54	𝑀𝑖𝑛{0.5,0.5	𝑀𝑖𝑛{0.5,0.5	ADJ
iajs-2767	110	55	}	}	PUNCT
iajs-2767	110	56	)	)	PUNCT
iajs-2767	110	57	}	}	PUNCT
iajs-2767	110	58	=	=	SYM
iajs-2767	110	59	{	{	PUNCT
iajs-2767	110	60	(	(	PUNCT
iajs-2767	110	61	𝜔1	𝜔1	ADJ
iajs-2767	110	62	,	,	PUNCT
iajs-2767	110	63	0.4	0.4	NUM
iajs-2767	110	64	)	)	PUNCT
iajs-2767	110	65	,	,	PUNCT
iajs-2767	110	66	(	(	PUNCT
iajs-2767	110	67	𝜔2	𝜔2	PROPN
iajs-2767	110	68	,	,	PUNCT
iajs-2767	110	69	0.5	0.5	NUM
iajs-2767	110	70	)	)	PUNCT
iajs-2767	110	71	}	}	PUNCT
iajs-2767	111	1	=	=	NUM
iajs-2767	111	2	ec	ec	NOUN
iajs-2767	111	3	f\	f\	PUNCT
iajs-2767	111	4	e	e	PROPN
iajs-2767	111	5	=	=	PRON
iajs-2767	111	6	{	{	PUNCT
iajs-2767	111	7	(	(	PUNCT
iajs-2767	111	8	𝜔1	𝜔1	ADJ
iajs-2767	111	9	,	,	PUNCT
iajs-2767	111	10	𝑀𝑖𝑛{0,1	𝑀𝑖𝑛{0,1	ADV
iajs-2767	111	11	−	−	NOUN
iajs-2767	111	12	0.6	0.6	NUM
iajs-2767	111	13	}	}	PUNCT
iajs-2767	111	14	)	)	PUNCT
iajs-2767	111	15	,	,	PUNCT
iajs-2767	111	16	(	(	PUNCT
iajs-2767	111	17	𝜔2	𝜔2	ADV
iajs-2767	111	18	,	,	PUNCT
iajs-2767	111	19	𝑀𝑖𝑛{0.5,1	𝑀𝑖𝑛{0.5,1	ADV
iajs-2767	111	20	−	−	PROPN
iajs-2767	111	21	0.5	0.5	NUM
iajs-2767	111	22	}	}	PUNCT
iajs-2767	111	23	)	)	PUNCT
iajs-2767	111	24	}	}	PUNCT
iajs-2767	111	25	=	=	SYM
iajs-2767	111	26	{	{	PUNCT
iajs-2767	111	27	(	(	PUNCT
iajs-2767	111	28	𝜔1	𝜔1	ADJ
iajs-2767	111	29	,	,	PUNCT
iajs-2767	111	30	0	0	NUM
iajs-2767	111	31	)	)	PUNCT
iajs-2767	111	32	,	,	PUNCT
iajs-2767	111	33	(	(	PUNCT
iajs-2767	111	34	𝜔2	𝜔2	PROPN
iajs-2767	111	35	,	,	PUNCT
iajs-2767	111	36	0.5)}=	0.5)}=	NOUN
iajs-2767	111	37	f	f	PROPN
iajs-2767	111	38	similarly	similarly	ADV
iajs-2767	111	39	,	,	PUNCT
iajs-2767	111	40	e\	e\	PROPN
iajs-2767	111	41	f	f	PROPN
iajs-2767	111	42	=	=	SYM
iajs-2767	111	43	e	e	PROPN
iajs-2767	111	44	and	and	CCONJ
iajs-2767	111	45	ec\f	ec\f	NOUN
iajs-2767	111	46	=	=	PROPN
iajs-2767	111	47	ec	ec	PROPN
iajs-2767	111	48	.	.	PUNCT
iajs-2767	111	49	.	.	PUNCT
iajs-2767	112	1	now	now	ADV
iajs-2767	112	2	,	,	PUNCT
iajs-2767	112	3	f⋃	f⋃	AUX
iajs-2767	112	4	e⋃ec	e⋃ec	X
iajs-2767	113	1	=	=	PRON
iajs-2767	113	2	{	{	PUNCT
iajs-2767	113	3	(	(	PUNCT
iajs-2767	113	4	𝜔1	𝜔1	ADJ
iajs-2767	113	5	,	,	PUNCT
iajs-2767	113	6	𝑆𝑢𝑝{0,0,0.6,0.4	𝑆𝑢𝑝{0,0,0.6,0.4	NOUN
iajs-2767	113	7	}	}	PUNCT
iajs-2767	113	8	)	)	PUNCT
iajs-2767	113	9	,	,	PUNCT
iajs-2767	113	10	(	(	PUNCT
iajs-2767	113	11	𝜔2	𝜔2	PROPN
iajs-2767	113	12	,	,	PUNCT
iajs-2767	113	13	𝑆𝑢𝑝{0,0.5,0.5,0.5	𝑆𝑢𝑝{0,0.5,0.5,0.5	PROPN
iajs-2767	113	14	}	}	PUNCT
iajs-2767	113	15	)	)	PUNCT
iajs-2767	113	16	}	}	PUNCT
iajs-2767	113	17	=	=	SYM
iajs-2767	113	18	{	{	PUNCT
iajs-2767	113	19	(	(	PUNCT
iajs-2767	113	20	𝜔1	𝜔1	ADJ
iajs-2767	113	21	,	,	PUNCT
iajs-2767	113	22	0.6	0.6	NUM
iajs-2767	113	23	)	)	PUNCT
iajs-2767	113	24	,	,	PUNCT
iajs-2767	113	25	(	(	PUNCT
iajs-2767	113	26	𝜔2	𝜔2	PROPN
iajs-2767	113	27	,	,	PUNCT
iajs-2767	113	28	0.5)}=	0.5)}=	PROPN
iajs-2767	113	29	e.	e.	PROPN
iajs-2767	113	30	therefore	therefore	ADV
iajs-2767	113	31	,	,	PUNCT
iajs-2767	113	32	ℋ∗	ℋ∗	PROPN
iajs-2767	113	33	is	be	AUX
iajs-2767	113	34	a	a	DET
iajs-2767	113	35	fuzzy	fuzzy	ADJ
iajs-2767	113	36	σ	σ	NOUN
iajs-2767	113	37	–	–	PUNCT
iajs-2767	113	38	ring	ring	NOUN
iajs-2767	113	39	over	over	ADP
iajs-2767	113	40	a	a	DET
iajs-2767	113	41	fuzzy	fuzzy	ADJ
iajs-2767	113	42	set	set	VERB
iajs-2767	113	43	𝒳∗.	𝒳∗.	PROPN
iajs-2767	113	44	in	in	ADP
iajs-2767	113	45	contrast	contrast	NOUN
iajs-2767	113	46	,	,	PUNCT
iajs-2767	113	47	ℋ∗	ℋ∗	NUM
iajs-2767	113	48	is	be	AUX
iajs-2767	113	49	not	not	PART
iajs-2767	113	50	fuzzy	fuzzy	ADJ
iajs-2767	113	51	σ	σ	NOUN
iajs-2767	113	52	–	–	PUNCT
iajs-2767	113	53	algebra	algebra	NOUN
iajs-2767	113	54	over	over	ADP
iajs-2767	113	55	a	a	DET
iajs-2767	113	56	fuzzy	fuzzy	ADJ
iajs-2767	113	57	set	set	VERB
iajs-2767	113	58	𝒳∗	𝒳∗	PROPN
iajs-2767	113	59	,	,	PUNCT
iajs-2767	113	60	because	because	SCONJ
iajs-2767	113	61	{	{	PUNCT
iajs-2767	113	62	(	(	PUNCT
iajs-2767	113	63	𝜔1,0),(𝜔2,0.5)}∈	𝜔1,0),(𝜔2,0.5)}∈	PUNCT
iajs-2767	113	64	ℋ∗	ℋ∗	NUM
iajs-2767	113	65	,	,	PUNCT
iajs-2767	113	66	but	but	CCONJ
iajs-2767	113	67	{	{	PUNCT
iajs-2767	113	68	(	(	PUNCT
iajs-2767	113	69	𝜔1	𝜔1	ADJ
iajs-2767	113	70	,	,	PUNCT
iajs-2767	113	71	0	0	NUM
iajs-2767	113	72	)	)	PUNCT
iajs-2767	113	73	,	,	PUNCT
iajs-2767	113	74	(	(	PUNCT
iajs-2767	113	75	𝜔2	𝜔2	PROPN
iajs-2767	113	76	,	,	PUNCT
iajs-2767	113	77	0.5)}c	0.5)}c	X
iajs-2767	113	78	=	=	SYM
iajs-2767	113	79	{	{	PUNCT
iajs-2767	113	80	(	(	PUNCT
iajs-2767	113	81	𝜔1	𝜔1	ADJ
iajs-2767	113	82	,	,	PUNCT
iajs-2767	113	83	1	1	NUM
iajs-2767	113	84	)	)	PUNCT
iajs-2767	113	85	,	,	PUNCT
iajs-2767	113	86	(	(	PUNCT
iajs-2767	113	87	𝜔2	𝜔2	PROPN
iajs-2767	113	88	,	,	PUNCT
iajs-2767	113	89	0.5	0.5	NUM
iajs-2767	113	90	)	)	PUNCT
iajs-2767	113	91	}	}	PUNCT
iajs-2767	113	92	∉	∉	PROPN
iajs-2767	113	93	ℋ∗.	ℋ∗.	PROPN
iajs-2767	113	94	ibn	ibn	PROPN
iajs-2767	113	95	al	al	PROPN
iajs-2767	113	96	-	-	PUNCT
iajs-2767	113	97	haitham	haitham	PROPN
iajs-2767	113	98	jour	jour	X
iajs-2767	113	99	.	.	PROPN
iajs-2767	114	1	for	for	ADP
iajs-2767	114	2	pure	pure	ADJ
iajs-2767	114	3	&	&	CCONJ
iajs-2767	114	4	appl	appl	PROPN
iajs-2767	114	5	.	.	PUNCT
iajs-2767	115	1	sci	sci	PROPN
iajs-2767	115	2	.	.	PROPN
iajs-2767	116	1	53	53	NUM
iajs-2767	116	2	(	(	PUNCT
iajs-2767	116	3	2)2022	2)2022	VERB
iajs-2767	116	4	42	42	NUM
iajs-2767	116	5	theorem	theorem	ADJ
iajs-2767	116	6	3.1	3.1	NUM
iajs-2767	116	7	assume	assume	VERB
iajs-2767	116	8	𝒳	𝒳	PROPN
iajs-2767	116	9	≠	≠	PROPN
iajs-2767	116	10	∅	∅	NOUN
iajs-2767	116	11	and	and	CCONJ
iajs-2767	116	12	ℋ∗	ℋ∗	NUM
iajs-2767	116	13	⊆	⊆	NUM
iajs-2767	116	14	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	116	15	)	)	PUNCT
iajs-2767	116	16	with	with	ADP
iajs-2767	116	17	𝒳∗	𝒳∗	PROPN
iajs-2767	116	18	∈	∈	PROPN
iajs-2767	116	19	ℋ∗.	ℋ∗.	PROPN
iajs-2767	116	20	then	then	ADV
iajs-2767	116	21	ℋ∗	ℋ∗	NUM
iajs-2767	116	22	is	be	AUX
iajs-2767	116	23	fuzzy	fuzzy	ADJ
iajs-2767	116	24	σ	σ	NOUN
iajs-2767	116	25	–	–	PUNCT
iajs-2767	116	26	algebra	algebra	NOUN
iajs-2767	116	27	over	over	ADP
iajs-2767	116	28	a	a	DET
iajs-2767	116	29	fuzzy	fuzzy	ADJ
iajs-2767	116	30	set	set	NOUN
iajs-2767	116	31	𝒳∗	𝒳∗	PUNCT
iajs-2767	117	1	if	if	SCONJ
iajs-2767	117	2	and	and	CCONJ
iajs-2767	117	3	only	only	ADV
iajs-2767	117	4	if	if	SCONJ
iajs-2767	117	5	ℋ∗	ℋ∗	NUM
iajs-2767	117	6	is	be	AUX
iajs-2767	117	7	a	a	DET
iajs-2767	117	8	fuzzy	fuzzy	ADJ
iajs-2767	117	9	σ	σ	NOUN
iajs-2767	117	10	–	–	PUNCT
iajs-2767	117	11	ring	ring	NOUN
iajs-2767	117	12	over	over	ADP
iajs-2767	117	13	a	a	DET
iajs-2767	117	14	fuzzy	fuzzy	ADJ
iajs-2767	117	15	set	set	NOUN
iajs-2767	117	16	𝒳∗.	𝒳∗.	PROPN
iajs-2767	117	17	proof	proof	NOUN
iajs-2767	117	18	assume	assume	VERB
iajs-2767	117	19	that	that	SCONJ
iajs-2767	117	20	ℋ∗	ℋ∗	NOUN
iajs-2767	117	21	is	be	AUX
iajs-2767	117	22	fuzzy	fuzzy	ADJ
iajs-2767	117	23	σ	σ	NOUN
iajs-2767	117	24	–	–	PUNCT
iajs-2767	117	25	algebra	algebra	NOUN
iajs-2767	117	26	over	over	ADP
iajs-2767	117	27	a	a	DET
iajs-2767	117	28	fuzzy	fuzzy	ADJ
iajs-2767	117	29	set	set	VERB
iajs-2767	117	30	𝒳∗	𝒳∗	PROPN
iajs-2767	117	31	,	,	PUNCT
iajs-2767	117	32	then	then	ADV
iajs-2767	117	33	by	by	ADP
iajs-2767	117	34	proposition	proposition	NOUN
iajs-2767	117	35	3.3	3.3	NUM
iajs-2767	117	36	we	we	PRON
iajs-2767	117	37	get	get	VERB
iajs-2767	117	38	ℋ∗	ℋ∗	NUM
iajs-2767	117	39	is	be	AUX
iajs-2767	117	40	fuzzy	fuzzy	ADJ
iajs-2767	117	41	σ	σ	NOUN
iajs-2767	117	42	–	–	PUNCT
iajs-2767	117	43	ring	ring	NOUN
iajs-2767	117	44	over	over	ADP
iajs-2767	117	45	a	a	DET
iajs-2767	117	46	fuzzy	fuzzy	ADJ
iajs-2767	117	47	set	set	NOUN
iajs-2767	117	48	𝒳∗.	𝒳∗.	PROPN
iajs-2767	117	49	conversely	conversely	ADV
iajs-2767	117	50	:	:	PUNCT
iajs-2767	117	51	suppose	suppose	VERB
iajs-2767	117	52	that	that	SCONJ
iajs-2767	117	53	ℋ∗	ℋ∗	PROPN
iajs-2767	117	54	is	be	AUX
iajs-2767	117	55	a	a	DET
iajs-2767	117	56	fuzzy	fuzzy	ADJ
iajs-2767	117	57	σ	σ	NOUN
iajs-2767	117	58	–	–	PUNCT
iajs-2767	117	59	ring	ring	NOUN
iajs-2767	117	60	over	over	ADP
iajs-2767	117	61	a	a	DET
iajs-2767	117	62	fuzzy	fuzzy	ADJ
iajs-2767	117	63	set	set	NOUN
iajs-2767	117	64	𝒳∗	𝒳∗	PRON
iajs-2767	117	65	such	such	ADJ
iajs-2767	117	66	that	that	SCONJ
iajs-2767	117	67	𝒳∗	𝒳∗	PROPN
iajs-2767	117	68	∈	∈	PROPN
iajs-2767	117	69	ℋ∗	ℋ∗	PROPN
iajs-2767	117	70	,	,	PUNCT
iajs-2767	117	71	then	then	ADV
iajs-2767	117	72	∅∗	∅∗	PROPN
iajs-2767	117	73	∈	∈	PROPN
iajs-2767	117	74	ℋ∗.	ℋ∗.	PROPN
iajs-2767	117	75	let	let	VERB
iajs-2767	117	76	e	e	PRON
iajs-2767	117	77	∈	∈	PROPN
iajs-2767	117	78	ℋ∗.	ℋ∗.	PROPN
iajs-2767	117	79	then	then	ADV
iajs-2767	117	80	𝒳∗\e	𝒳∗\e	PROPN
iajs-2767	117	81	∈	∈	PROPN
iajs-2767	117	82	ℋ∗	ℋ∗	PROPN
iajs-2767	117	83	,	,	PUNCT
iajs-2767	117	84	but	but	CCONJ
iajs-2767	117	85	𝒳∗\e	𝒳∗\e	NOUN
iajs-2767	118	1	=	=	NOUN
iajs-2767	119	1	𝒳∗⋂e𝑐	𝒳∗⋂e𝑐	NOUN
iajs-2767	119	2	=	=	SYM
iajs-2767	119	3	{	{	PUNCT
iajs-2767	119	4	(	(	PUNCT
iajs-2767	119	5	ω	ω	PROPN
iajs-2767	119	6	,	,	PUNCT
iajs-2767	119	7	min	min	PROPN
iajs-2767	119	8	{	{	PUNCT
iajs-2767	119	9	𝓋𝒳∗	𝓋𝒳∗	PROPN
iajs-2767	119	10	,	,	PUNCT
iajs-2767	119	11	𝓋e𝑐	𝓋e𝑐	X
iajs-2767	119	12	(	(	PUNCT
iajs-2767	119	13	ω	ω	NOUN
iajs-2767	119	14	)	)	PUNCT
iajs-2767	119	15	}	}	PUNCT
iajs-2767	119	16	)	)	PUNCT
iajs-2767	120	1	∶	∶	NOUN
iajs-2767	120	2	∀ω	∀ω	NOUN
iajs-2767	120	3	∈	∈	PROPN
iajs-2767	120	4	𝒳	𝒳	PROPN
iajs-2767	120	5	}	}	PUNCT
iajs-2767	120	6	=	=	SYM
iajs-2767	120	7	{	{	PUNCT
iajs-2767	120	8	(	(	PUNCT
iajs-2767	120	9	ω	ω	PROPN
iajs-2767	120	10	,	,	PUNCT
iajs-2767	120	11	min	min	PROPN
iajs-2767	120	12	{	{	PUNCT
iajs-2767	120	13	1,1	1,1	NUM
iajs-2767	120	14	−	−	NOUN
iajs-2767	120	15	𝓋e	𝓋e	INTJ
iajs-2767	120	16	(	(	PUNCT
iajs-2767	120	17	ω	ω	NOUN
iajs-2767	120	18	)	)	PUNCT
iajs-2767	120	19	}	}	PUNCT
iajs-2767	120	20	)	)	PUNCT
iajs-2767	121	1	∶	∶	NOUN
iajs-2767	121	2	∀ω	∀ω	NOUN
iajs-2767	121	3	∈	∈	PROPN
iajs-2767	121	4	𝒳	𝒳	PROPN
iajs-2767	121	5	}	}	PUNCT
iajs-2767	121	6	=	=	SYM
iajs-2767	121	7	{	{	PUNCT
iajs-2767	121	8	(	(	PUNCT
iajs-2767	121	9	ω	ω	NOUN
iajs-2767	121	10	,	,	PUNCT
iajs-2767	121	11	1	1	NUM
iajs-2767	121	12	−	−	NOUN
iajs-2767	121	13	𝓋e	𝓋e	INTJ
iajs-2767	121	14	(	(	PUNCT
iajs-2767	121	15	ω	ω	NOUN
iajs-2767	121	16	)	)	PUNCT
iajs-2767	121	17	)	)	PUNCT
iajs-2767	122	1	∶	∶	NOUN
iajs-2767	122	2	∀ω	∀ω	NOUN
iajs-2767	122	3	∈	∈	PROPN
iajs-2767	122	4	𝒳	𝒳	PROPN
iajs-2767	122	5	}	}	PUNCT
iajs-2767	122	6	=	=	PUNCT
iajs-2767	122	7	e𝑐	e𝑐	PRON
iajs-2767	122	8	this	this	PRON
iajs-2767	122	9	implies	imply	VERB
iajs-2767	122	10	that	that	SCONJ
iajs-2767	122	11	,	,	PUNCT
iajs-2767	122	12	𝒳∗\e	𝒳∗\e	NOUN
iajs-2767	122	13	=	=	PRON
iajs-2767	122	14	e𝑐	e𝑐	PROPN
iajs-2767	122	15	∈	∈	PROPN
iajs-2767	122	16	ℋ∗.	ℋ∗.	PROPN
iajs-2767	122	17	let	let	VERB
iajs-2767	122	18	e1	e1	NOUN
iajs-2767	122	19	,	,	PUNCT
iajs-2767	122	20	e2	e2	PROPN
iajs-2767	122	21	,	,	PUNCT
iajs-2767	122	22	…	…	PUNCT
iajs-2767	122	23	∈	∈	PROPN
iajs-2767	122	24	ℋ∗.	ℋ∗.	PROPN
iajs-2767	122	25	then	then	ADV
iajs-2767	122	26	as	as	SCONJ
iajs-2767	122	27	shown	show	VERB
iajs-2767	122	28	above	above	ADP
iajs-2767	122	29	we	we	PRON
iajs-2767	122	30	have	have	VERB
iajs-2767	122	31	,	,	PUNCT
iajs-2767	122	32	ek	ek	PROPN
iajs-2767	122	33	𝑐	𝑐	PROPN
iajs-2767	122	34	(	(	PUNCT
iajs-2767	122	35	for	for	ADP
iajs-2767	122	36	all	all	PRON
iajs-2767	122	37	k	k	NOUN
iajs-2767	122	38	=	=	SYM
iajs-2767	122	39	1,2	1,2	NUM
iajs-2767	122	40	,	,	PUNCT
iajs-2767	122	41	…	…	PUNCT
iajs-2767	122	42	)	)	PUNCT
iajs-2767	122	43	.	.	PUNCT
iajs-2767	123	1	hence	hence	ADV
iajs-2767	123	2	by	by	ADP
iajs-2767	123	3	definition	definition	NOUN
iajs-2767	123	4	of	of	ADP
iajs-2767	123	5	a	a	DET
iajs-2767	123	6	fuzzy	fuzzy	ADJ
iajs-2767	123	7	σ	σ	NOUN
iajs-2767	123	8	–	–	PUNCT
iajs-2767	123	9	ring	ring	NOUN
iajs-2767	123	10	we	we	PRON
iajs-2767	123	11	get	get	VERB
iajs-2767	123	12	⋃	⋃	ADP
iajs-2767	123	13	ek	ek	NOUN
iajs-2767	123	14	𝑐∞	𝑐∞	PUNCT
iajs-2767	123	15	k=1	k=1	PROPN
iajs-2767	123	16	∈	∈	PROPN
iajs-2767	123	17	ℋ∗	ℋ∗	PROPN
iajs-2767	123	18	,	,	PUNCT
iajs-2767	123	19	thus	thus	ADV
iajs-2767	123	20	(	(	PUNCT
iajs-2767	123	21	⋃	⋃	SCONJ
iajs-2767	123	22	ek	ek	VERB
iajs-2767	123	23	𝑐∞	𝑐∞	PUNCT
iajs-2767	123	24	k=1	k=1	X
iajs-2767	123	25	)	)	PUNCT
iajs-2767	123	26	𝑐	𝑐	PROPN
iajs-2767	123	27	∈	∈	PROPN
iajs-2767	123	28	ℋ∗.	ℋ∗.	PROPN
iajs-2767	123	29	by	by	ADP
iajs-2767	123	30	de	de	PROPN
iajs-2767	123	31	-	-	PROPN
iajs-2767	123	32	morgan	morgan	PROPN
iajs-2767	123	33	law	law	NOUN
iajs-2767	123	34	,	,	PUNCT
iajs-2767	123	35	we	we	PRON
iajs-2767	123	36	get	get	VERB
iajs-2767	123	37	(	(	PUNCT
iajs-2767	123	38	⋃	⋃	VERB
iajs-2767	123	39	ek	ek	VERB
iajs-2767	123	40	𝑐∞	𝑐∞	PUNCT
iajs-2767	123	41	k=1	k=1	X
iajs-2767	123	42	)	)	PUNCT
iajs-2767	124	1	𝑐	𝑐	PROPN
iajs-2767	124	2	=	=	SYM
iajs-2767	124	3	⋂	⋂	PROPN
iajs-2767	124	4	ek	ek	NOUN
iajs-2767	124	5	∞	∞	PROPN
iajs-2767	124	6	k=1	k=1	PROPN
iajs-2767	125	1	hence	hence	ADV
iajs-2767	125	2	⋂	⋂	PROPN
iajs-2767	125	3	ek	ek	NOUN
iajs-2767	125	4	∞	∞	PROPN
iajs-2767	125	5	k=1	k=1	PUNCT
iajs-2767	126	1	∈	∈	PROPN
iajs-2767	126	2	ℋ∗.	ℋ∗.	PROPN
iajs-2767	126	3	therefore	therefore	ADV
iajs-2767	126	4	,	,	PUNCT
iajs-2767	126	5	ℋ∗	ℋ∗	NOUN
iajs-2767	126	6	be	be	AUX
iajs-2767	126	7	a	a	DET
iajs-2767	126	8	fuzzy	fuzzy	ADJ
iajs-2767	126	9	σ	σ	NOUN
iajs-2767	126	10	–	–	PUNCT
iajs-2767	126	11	algebra	algebra	NOUN
iajs-2767	126	12	over	over	ADP
iajs-2767	126	13	a	a	DET
iajs-2767	126	14	fuzzy	fuzzy	ADJ
iajs-2767	126	15	set	set	VERB
iajs-2767	126	16	𝒳∗.	𝒳∗.	PROPN
iajs-2767	126	17	proposition	proposition	NOUN
iajs-2767	126	18	3.4	3.4	NUM
iajs-2767	126	19	let	let	VERB
iajs-2767	126	20	𝒳	𝒳	PRON
iajs-2767	126	21	be	be	AUX
iajs-2767	126	22	a	a	DET
iajs-2767	126	23	nonempty	nonempty	ADV
iajs-2767	126	24	set	set	VERB
iajs-2767	126	25	and	and	CCONJ
iajs-2767	126	26	𝔗∗	𝔗∗	NUM
iajs-2767	126	27	⊆	⊆	NUM
iajs-2767	126	28	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	126	29	)	)	PUNCT
iajs-2767	126	30	.	.	PUNCT
iajs-2767	127	1	then	then	ADV
iajs-2767	127	2	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	127	3	)	)	PUNCT
iajs-2767	128	1	⊆	⊆	NUM
iajs-2767	128	2	σ(𝔗∗	σ(𝔗∗	NOUN
iajs-2767	128	3	)	)	PUNCT
iajs-2767	128	4	,	,	PUNCT
iajs-2767	128	5	where	where	SCONJ
iajs-2767	128	6	σ(𝔗∗	σ(𝔗∗	NOUN
iajs-2767	128	7	)	)	PUNCT
iajs-2767	128	8	is	be	AUX
iajs-2767	128	9	the	the	DET
iajs-2767	128	10	smallest	small	ADJ
iajs-2767	128	11	fuzzy	fuzzy	ADJ
iajs-2767	128	12	σ	σ	NOUN
iajs-2767	128	13	–	–	PUNCT
iajs-2767	128	14	algebra	algebra	NOUN
iajs-2767	128	15	over	over	ADP
iajs-2767	128	16	a	a	DET
iajs-2767	128	17	fuzzy	fuzzy	ADJ
iajs-2767	128	18	set	set	NOUN
iajs-2767	128	19	𝒳∗	𝒳∗	PRON
iajs-2767	128	20	that	that	PRON
iajs-2767	128	21	includes	include	VERB
iajs-2767	128	22	𝔗∗.	𝔗∗.	PROPN
iajs-2767	128	23	proof	proof	NOUN
iajs-2767	128	24	clearly	clearly	ADV
iajs-2767	128	25	.	.	PUNCT
iajs-2767	129	1	proposition	proposition	NOUN
iajs-2767	129	2	3.5	3.5	NUM
iajs-2767	129	3	let	let	VERB
iajs-2767	129	4	𝒳	𝒳	PRON
iajs-2767	129	5	≠	≠	PROPN
iajs-2767	129	6	∅	∅	NOUN
iajs-2767	129	7	and	and	CCONJ
iajs-2767	129	8	𝔗∗	𝔗∗	NUM
iajs-2767	129	9	⊆	⊆	NUM
iajs-2767	129	10	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	129	11	)	)	PUNCT
iajs-2767	129	12	.	.	PUNCT
iajs-2767	130	1	if	if	SCONJ
iajs-2767	130	2	𝒳∗	𝒳∗	PRON
iajs-2767	130	3	∈	∈	PROPN
iajs-2767	130	4	σ𝑟(𝔗∗	σ𝑟(𝔗∗	NOUN
iajs-2767	130	5	)	)	PUNCT
iajs-2767	130	6	,	,	PUNCT
iajs-2767	130	7	then	then	ADV
iajs-2767	130	8	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	130	9	)	)	PUNCT
iajs-2767	131	1	=	=	SYM
iajs-2767	131	2	σ(𝔗∗	σ(𝔗∗	NOUN
iajs-2767	131	3	)	)	PUNCT
iajs-2767	131	4	.	.	PUNCT
iajs-2767	132	1	proof	proof	NOUN
iajs-2767	132	2	the	the	DET
iajs-2767	132	3	proof	proof	NOUN
iajs-2767	132	4	follows	follow	VERB
iajs-2767	132	5	from	from	ADP
iajs-2767	132	6	theorem	theorem	ADJ
iajs-2767	132	7	3.1	3.1	NUM
iajs-2767	132	8	.	.	PUNCT
iajs-2767	133	1	definition	definition	NOUN
iajs-2767	133	2	3.4	3.4	NUM
iajs-2767	133	3	let	let	AUX
iajs-2767	133	4	assume	assume	VERB
iajs-2767	133	5	𝒳	𝒳	NOUN
iajs-2767	133	6	≠	≠	PROPN
iajs-2767	133	7	∅	∅	NOUN
iajs-2767	133	8	.	.	PUNCT
iajs-2767	134	1	a	a	DET
iajs-2767	134	2	class	class	NOUN
iajs-2767	134	3	ℋ∗	ℋ∗	NOUN
iajs-2767	134	4	⊆	⊆	NUM
iajs-2767	134	5	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	134	6	)	)	PUNCT
iajs-2767	134	7	is	be	AUX
iajs-2767	134	8	said	say	VERB
iajs-2767	134	9	to	to	PART
iajs-2767	134	10	be	be	AUX
iajs-2767	134	11	a	a	DET
iajs-2767	134	12	fuzzy	fuzzy	ADJ
iajs-2767	134	13	ring	ring	NOUN
iajs-2767	134	14	over	over	ADP
iajs-2767	134	15	a	a	DET
iajs-2767	134	16	fuzzy	fuzzy	ADJ
iajs-2767	134	17	set	set	VERB
iajs-2767	134	18	𝒳∗	𝒳∗	INTJ
iajs-2767	134	19	,	,	PUNCT
iajs-2767	134	20	if	if	SCONJ
iajs-2767	134	21	1	1	NUM
iajs-2767	134	22	.	.	PUNCT
iajs-2767	135	1	∅∗	∅∗	PROPN
iajs-2767	135	2	∈	∈	PROPN
iajs-2767	135	3	ℋ∗.	ℋ∗.	PROPN
iajs-2767	135	4	2.if	2.if	NUM
iajs-2767	135	5	f	f	NOUN
iajs-2767	135	6	,	,	PUNCT
iajs-2767	135	7	e	e	PROPN
iajs-2767	135	8	∈	∈	PROPN
iajs-2767	135	9	ℋ∗	ℋ∗	NUM
iajs-2767	135	10	,	,	PUNCT
iajs-2767	135	11	then	then	ADV
iajs-2767	135	12	f	f	X
iajs-2767	135	13	∖	∖	X
iajs-2767	135	14	e	e	PROPN
iajs-2767	135	15	∈	∈	PROPN
iajs-2767	135	16	ℋ∗.	ℋ∗.	PROPN
iajs-2767	135	17	3.if	3.if	NUM
iajs-2767	135	18	e1	e1	PROPN
iajs-2767	135	19	,	,	PUNCT
iajs-2767	135	20	e2	e2	PROPN
iajs-2767	135	21	,	,	PUNCT
iajs-2767	135	22	,	,	PUNCT
iajs-2767	135	23	en	en	PROPN
iajs-2767	135	24	∈	∈	NOUN
iajs-2767	135	25	ℋ∗	ℋ∗	NUM
iajs-2767	135	26	,	,	PUNCT
iajs-2767	135	27	then	then	ADV
iajs-2767	135	28	⋃	⋃	ADP
iajs-2767	135	29	ek	ek	NOUN
iajs-2767	135	30	𝑛	𝑛	PRON
iajs-2767	135	31	𝑘=1	𝑘=1	PUNCT
iajs-2767	135	32	∈	∈	PROPN
iajs-2767	135	33	ℋ∗.	ℋ∗.	PROPN
iajs-2767	135	34	definition	definition	NOUN
iajs-2767	135	35	3.5	3.5	NUM
iajs-2767	135	36	a	a	DET
iajs-2767	135	37	fuzzy	fuzzy	ADJ
iajs-2767	135	38	measurable	measurable	ADJ
iajs-2767	135	39	space	space	NOUN
iajs-2767	135	40	relatively	relatively	ADV
iajs-2767	135	41	to	to	ADP
iajs-2767	135	42	fuzzy	fuzzy	ADJ
iajs-2767	135	43	ring	ring	NOUN
iajs-2767	135	44	is	be	AUX
iajs-2767	135	45	an	an	DET
iajs-2767	135	46	ordered	order	VERB
iajs-2767	135	47	pair	pair	NOUN
iajs-2767	135	48	(	(	PUNCT
iajs-2767	135	49	𝒳∗	𝒳∗	X
iajs-2767	135	50	,	,	PUNCT
iajs-2767	135	51	ℋ∗	ℋ∗	NUM
iajs-2767	135	52	)	)	PUNCT
iajs-2767	135	53	,	,	PUNCT
iajs-2767	135	54	where	where	SCONJ
iajs-2767	135	55	𝒳	𝒳	PROPN
iajs-2767	135	56	is	be	AUX
iajs-2767	135	57	a	a	DET
iajs-2767	135	58	nonempty	nonempty	ADV
iajs-2767	135	59	set	set	VERB
iajs-2767	135	60	and	and	CCONJ
iajs-2767	135	61	ℋ∗	ℋ∗	NUM
iajs-2767	135	62	⊆	⊆	NUM
iajs-2767	135	63	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	135	64	)	)	PUNCT
iajs-2767	135	65	be	be	VERB
iajs-2767	135	66	a	a	DET
iajs-2767	135	67	fuzzy	fuzzy	ADJ
iajs-2767	135	68	ring	ring	NOUN
iajs-2767	135	69	over	over	ADP
iajs-2767	135	70	a	a	DET
iajs-2767	135	71	fuzzy	fuzzy	ADJ
iajs-2767	135	72	set	set	VERB
iajs-2767	135	73	𝒳∗	𝒳∗	PROPN
iajs-2767	135	74	and	and	CCONJ
iajs-2767	135	75	an	an	DET
iajs-2767	135	76	element	element	NOUN
iajs-2767	135	77	of	of	ADP
iajs-2767	135	78	ℋ∗	ℋ∗	NUM
iajs-2767	135	79	is	be	AUX
iajs-2767	135	80	called	call	VERB
iajs-2767	135	81	a	a	DET
iajs-2767	135	82	measurable	measurable	ADJ
iajs-2767	135	83	set	set	VERB
iajs-2767	135	84	relatively	relatively	ADV
iajs-2767	135	85	to	to	ADP
iajs-2767	135	86	fuzzy	fuzzy	ADJ
iajs-2767	135	87	ring	ring	NOUN
iajs-2767	135	88	.	.	PUNCT
iajs-2767	136	1	example	example	NOUN
iajs-2767	136	2	3.7	3.7	NUM
iajs-2767	136	3	suppose	suppose	VERB
iajs-2767	136	4	𝒳	𝒳	PROPN
iajs-2767	136	5	≠	≠	PROPN
iajs-2767	136	6	∅.	∅.	VERB
iajs-2767	136	7	then	then	ADV
iajs-2767	136	8	each	each	PRON
iajs-2767	136	9	of	of	ADP
iajs-2767	136	10	∅∗	∅∗	PROPN
iajs-2767	136	11	and	and	CCONJ
iajs-2767	136	12	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	136	13	)	)	PUNCT
iajs-2767	136	14	is	be	AUX
iajs-2767	136	15	a	a	DET
iajs-2767	136	16	fuzzy	fuzzy	ADJ
iajs-2767	136	17	ring	ring	NOUN
iajs-2767	136	18	over	over	ADP
iajs-2767	136	19	a	a	DET
iajs-2767	136	20	fuzzy	fuzzy	ADJ
iajs-2767	136	21	set	set	NOUN
iajs-2767	136	22	𝒳∗	𝒳∗	PROPN
iajs-2767	136	23	.	.	PUNCT
iajs-2767	137	1	ibn	ibn	PROPN
iajs-2767	137	2	al	al	PROPN
iajs-2767	137	3	-	-	PUNCT
iajs-2767	137	4	haitham	haitham	PROPN
iajs-2767	137	5	jour	jour	X
iajs-2767	137	6	.	.	PROPN
iajs-2767	137	7	for	for	ADP
iajs-2767	137	8	pure	pure	ADJ
iajs-2767	137	9	&	&	CCONJ
iajs-2767	137	10	appl	appl	PROPN
iajs-2767	137	11	.	.	PUNCT
iajs-2767	138	1	sci	sci	PROPN
iajs-2767	138	2	.	.	PROPN
iajs-2767	139	1	53	53	NUM
iajs-2767	139	2	(	(	PUNCT
iajs-2767	139	3	2)2022	2)2022	NUM
iajs-2767	139	4	43	43	NUM
iajs-2767	139	5	example	example	NOUN
iajs-2767	139	6	3.8	3.8	NUM
iajs-2767	139	7	let	let	VERB
iajs-2767	139	8	𝒳	𝒳	PRON
iajs-2767	139	9	=	=	PRON
iajs-2767	139	10	{	{	PUNCT
iajs-2767	139	11	𝜔1	𝜔1	ADJ
iajs-2767	139	12	,	,	PUNCT
iajs-2767	139	13	𝜔2	𝜔2	PROPN
iajs-2767	139	14	}	}	PUNCT
iajs-2767	139	15	and	and	CCONJ
iajs-2767	139	16	ℋ∗	ℋ∗	NUM
iajs-2767	140	1	=	=	SYM
iajs-2767	140	2	{	{	PUNCT
iajs-2767	140	3	∅∗	∅∗	PROPN
iajs-2767	140	4	,	,	PUNCT
iajs-2767	140	5	{	{	PUNCT
iajs-2767	140	6	(	(	PUNCT
iajs-2767	140	7	𝜔1,0.4	𝜔1,0.4	NUM
iajs-2767	140	8	)	)	PUNCT
iajs-2767	140	9	,	,	PUNCT
iajs-2767	140	10	(	(	PUNCT
iajs-2767	140	11	𝜔2,0.3	𝜔2,0.3	NOUN
iajs-2767	140	12	)	)	PUNCT
iajs-2767	140	13	}	}	PUNCT
iajs-2767	140	14	,	,	PUNCT
iajs-2767	140	15	{	{	PUNCT
iajs-2767	140	16	(	(	PUNCT
iajs-2767	140	17	𝜔1,0.6	𝜔1,0.6	NUM
iajs-2767	140	18	)	)	PUNCT
iajs-2767	140	19	,	,	PUNCT
iajs-2767	140	20	(	(	PUNCT
iajs-2767	140	21	𝜔2,0.7	𝜔2,0.7	PROPN
iajs-2767	140	22	)	)	PUNCT
iajs-2767	140	23	}	}	PUNCT
iajs-2767	140	24	}	}	PUNCT
iajs-2767	140	25	.	.	PUNCT
iajs-2767	141	1	then	then	ADV
iajs-2767	141	2	ℋ	ℋ	VERB
iajs-2767	141	3	∗	∗	NOUN
iajs-2767	141	4	and	and	CCONJ
iajs-2767	141	5	is	be	AUX
iajs-2767	141	6	a	a	DET
iajs-2767	141	7	fuzzy	fuzzy	ADJ
iajs-2767	141	8	ring	ring	NOUN
iajs-2767	141	9	over	over	ADP
iajs-2767	141	10	a	a	DET
iajs-2767	141	11	fuzzy	fuzzy	ADJ
iajs-2767	141	12	set	set	NOUN
iajs-2767	141	13	𝒳∗.	𝒳∗.	PROPN
iajs-2767	141	14	example	example	NOUN
iajs-2767	141	15	3.9	3.9	NUM
iajs-2767	141	16	let	let	VERB
iajs-2767	141	17	𝒳	𝒳	PRON
iajs-2767	141	18	=	=	PRON
iajs-2767	141	19	{	{	PUNCT
iajs-2767	141	20	𝜔1	𝜔1	ADJ
iajs-2767	141	21	,	,	PUNCT
iajs-2767	141	22	𝜔2	𝜔2	PROPN
iajs-2767	141	23	}	}	PUNCT
iajs-2767	141	24	and	and	CCONJ
iajs-2767	141	25	ℋ∗	ℋ∗	NUM
iajs-2767	142	1	=	=	SYM
iajs-2767	142	2	{	{	PUNCT
iajs-2767	142	3	∅∗	∅∗	PROPN
iajs-2767	142	4	,	,	PUNCT
iajs-2767	142	5	{	{	PUNCT
iajs-2767	142	6	(	(	PUNCT
iajs-2767	142	7	𝜔1,0.3),(𝜔2,0.7	𝜔1,0.3),(𝜔2,0.7	ADJ
iajs-2767	142	8	)	)	PUNCT
iajs-2767	142	9	}	}	PUNCT
iajs-2767	142	10	,	,	PUNCT
iajs-2767	142	11	{	{	PUNCT
iajs-2767	142	12	(	(	PUNCT
iajs-2767	142	13	𝜔1,0.2	𝜔1,0.2	NUM
iajs-2767	142	14	)	)	PUNCT
iajs-2767	142	15	,	,	PUNCT
iajs-2767	142	16	(	(	PUNCT
iajs-2767	142	17	𝜔2,0.8	𝜔2,0.8	NOUN
iajs-2767	142	18	)	)	PUNCT
iajs-2767	142	19	}	}	PUNCT
iajs-2767	142	20	,	,	PUNCT
iajs-2767	142	21	𝒳∗	𝒳∗	PROPN
iajs-2767	142	22	}	}	PUNCT
iajs-2767	142	23	.	.	PUNCT
iajs-2767	143	1	then	then	ADV
iajs-2767	143	2	ℋ	ℋ	NOUN
iajs-2767	143	3	∗	∗	NOUN
iajs-2767	143	4	is	be	AUX
iajs-2767	143	5	not	not	PART
iajs-2767	143	6	a	a	DET
iajs-2767	143	7	fuzzy	fuzzy	ADJ
iajs-2767	143	8	ring	ring	NOUN
iajs-2767	143	9	over	over	ADP
iajs-2767	143	10	a	a	DET
iajs-2767	143	11	fuzzy	fuzzy	ADJ
iajs-2767	143	12	set	set	VERB
iajs-2767	143	13	𝒳∗	𝒳∗	PROPN
iajs-2767	143	14	,	,	PUNCT
iajs-2767	143	15	because	because	SCONJ
iajs-2767	143	16	{	{	PUNCT
iajs-2767	143	17	(	(	PUNCT
iajs-2767	143	18	𝜔1,0.3),(𝜔2,0.7	𝜔1,0.3),(𝜔2,0.7	ADJ
iajs-2767	143	19	)	)	PUNCT
iajs-2767	143	20	}	}	PUNCT
iajs-2767	143	21	and	and	CCONJ
iajs-2767	143	22	{	{	PUNCT
iajs-2767	143	23	(	(	PUNCT
iajs-2767	143	24	𝜔1,0.2	𝜔1,0.2	NUM
iajs-2767	143	25	)	)	PUNCT
iajs-2767	143	26	,	,	PUNCT
iajs-2767	143	27	(	(	PUNCT
iajs-2767	143	28	𝜔2,0.8	𝜔2,0.8	NOUN
iajs-2767	143	29	)	)	PUNCT
iajs-2767	143	30	}	}	PUNCT
iajs-2767	143	31	∈	∈	PROPN
iajs-2767	143	32	ℋ∗	ℋ∗	NUM
iajs-2767	143	33	,	,	PUNCT
iajs-2767	143	34	but	but	CCONJ
iajs-2767	143	35	{	{	PUNCT
iajs-2767	143	36	(	(	PUNCT
iajs-2767	143	37	𝜔1	𝜔1	ADJ
iajs-2767	143	38	,	,	PUNCT
iajs-2767	143	39	0.3	0.3	NUM
iajs-2767	143	40	)	)	PUNCT
iajs-2767	143	41	,	,	PUNCT
iajs-2767	143	42	(	(	PUNCT
iajs-2767	143	43	𝜔2	𝜔2	PROPN
iajs-2767	143	44	,	,	PUNCT
iajs-2767	143	45	0.7)}⋃	0.7)}⋃	NOUN
iajs-2767	143	46	{	{	PUNCT
iajs-2767	143	47	(	(	PUNCT
iajs-2767	143	48	𝜔1	𝜔1	ADJ
iajs-2767	143	49	,	,	PUNCT
iajs-2767	143	50	0.2	0.2	NUM
iajs-2767	143	51	)	)	PUNCT
iajs-2767	143	52	,	,	PUNCT
iajs-2767	143	53	(	(	PUNCT
iajs-2767	143	54	𝜔2	𝜔2	PROPN
iajs-2767	143	55	,	,	PUNCT
iajs-2767	143	56	0.8	0.8	NUM
iajs-2767	143	57	)	)	PUNCT
iajs-2767	143	58	}	}	PUNCT
iajs-2767	143	59	=	=	SYM
iajs-2767	143	60	{	{	PUNCT
iajs-2767	143	61	(	(	PUNCT
iajs-2767	143	62	𝜔1	𝜔1	ADJ
iajs-2767	143	63	,	,	PUNCT
iajs-2767	143	64	0.3	0.3	NUM
iajs-2767	143	65	)	)	PUNCT
iajs-2767	143	66	,	,	PUNCT
iajs-2767	143	67	(	(	PUNCT
iajs-2767	143	68	𝜔2	𝜔2	PROPN
iajs-2767	143	69	,	,	PUNCT
iajs-2767	143	70	0.8	0.8	NUM
iajs-2767	143	71	)	)	PUNCT
iajs-2767	143	72	}	}	PUNCT
iajs-2767	143	73	∉	∉	PROPN
iajs-2767	144	1	ℋ∗.	ℋ∗.	PROPN
iajs-2767	144	2	lemma	lemma	PROPN
iajs-2767	144	3	3.2	3.2	NUM
iajs-2767	144	4	let	let	VERB
iajs-2767	144	5	{	{	PUNCT
iajs-2767	144	6	ℋi	ℋi	PROPN
iajs-2767	144	7	∗}i∈ι	∗}i∈ι	PROPN
iajs-2767	144	8	be	be	AUX
iajs-2767	144	9	a	a	DET
iajs-2767	144	10	nonempty	nonempty	ADJ
iajs-2767	144	11	collection	collection	NOUN
iajs-2767	144	12	of	of	ADP
iajs-2767	144	13	a	a	DET
iajs-2767	144	14	fuzzy	fuzzy	ADJ
iajs-2767	144	15	ring	ring	NOUN
iajs-2767	144	16	over	over	ADP
iajs-2767	144	17	a	a	DET
iajs-2767	144	18	fuzzy	fuzzy	ADJ
iajs-2767	144	19	set	set	NOUN
iajs-2767	144	20	𝒳∗	𝒳∗	PROPN
iajs-2767	144	21	.	.	PUNCT
iajs-2767	145	1	then	then	ADV
iajs-2767	145	2	⋂	⋂	PROPN
iajs-2767	145	3	ℋi	ℋi	PROPN
iajs-2767	145	4	∗	∗	NOUN
iajs-2767	145	5	i∈ι	i∈ι	NOUN
iajs-2767	145	6	is	be	AUX
iajs-2767	145	7	a	a	DET
iajs-2767	145	8	fuzzy	fuzzy	ADJ
iajs-2767	145	9	ring	ring	NOUN
iajs-2767	145	10	over	over	ADP
iajs-2767	145	11	a	a	DET
iajs-2767	145	12	fuzzy	fuzzy	ADJ
iajs-2767	145	13	set	set	NOUN
iajs-2767	145	14	𝒳∗.	𝒳∗.	PROPN
iajs-2767	145	15	proof	proof	NOUN
iajs-2767	145	16	direct	direct	VERB
iajs-2767	145	17	.	.	PUNCT
iajs-2767	146	1	definition	definition	NOUN
iajs-2767	146	2	3.6	3.6	NUM
iajs-2767	146	3	assume	assume	VERB
iajs-2767	146	4	𝒳	𝒳	PROPN
iajs-2767	146	5	≠	≠	PROPN
iajs-2767	146	6	∅	∅	NOUN
iajs-2767	146	7	and	and	CCONJ
iajs-2767	146	8	𝔗∗	𝔗∗	NUM
iajs-2767	146	9	⊆	⊆	NUM
iajs-2767	146	10	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	146	11	)	)	PUNCT
iajs-2767	146	12	.	.	PUNCT
iajs-2767	147	1	then	then	ADV
iajs-2767	147	2	the	the	DET
iajs-2767	147	3	intersection	intersection	NOUN
iajs-2767	147	4	of	of	ADP
iajs-2767	147	5	all	all	DET
iajs-2767	147	6	fuzzy	fuzzy	ADJ
iajs-2767	147	7	rings	ring	NOUN
iajs-2767	147	8	over	over	ADP
iajs-2767	147	9	a	a	DET
iajs-2767	147	10	fuzzy	fuzzy	ADJ
iajs-2767	147	11	set	set	VERB
iajs-2767	147	12	𝒳∗	𝒳∗	PROPN
iajs-2767	147	13	,	,	PUNCT
iajs-2767	147	14	which	which	PRON
iajs-2767	147	15	includes	include	VERB
iajs-2767	147	16	𝔗∗	𝔗∗	NUM
iajs-2767	147	17	is	be	AUX
iajs-2767	147	18	said	say	VERB
iajs-2767	147	19	to	to	PART
iajs-2767	147	20	be	be	AUX
iajs-2767	147	21	the	the	DET
iajs-2767	147	22	fuzzy	fuzzy	ADJ
iajs-2767	147	23	ring	ring	NOUN
iajs-2767	147	24	over	over	ADP
iajs-2767	147	25	a	a	DET
iajs-2767	147	26	fuzzy	fuzzy	ADJ
iajs-2767	147	27	set	set	NOUN
iajs-2767	147	28	𝒳∗	𝒳∗	PRON
iajs-2767	147	29	generated	generate	VERB
iajs-2767	147	30	by	by	ADP
iajs-2767	147	31	𝔗∗	𝔗∗	PUNCT
iajs-2767	147	32	and	and	CCONJ
iajs-2767	147	33	denoted	denote	VERB
iajs-2767	147	34	by	by	ADP
iajs-2767	147	35	r(𝔗∗	r(𝔗∗	NOUN
iajs-2767	147	36	)	)	PUNCT
iajs-2767	147	37	,	,	PUNCT
iajs-2767	147	38	that	that	ADV
iajs-2767	147	39	is	is	ADV
iajs-2767	147	40	,	,	PUNCT
iajs-2767	147	41	r(𝔗∗)=	r(𝔗∗)=	PROPN
iajs-2767	147	42	⋂{ℋi	⋂{ℋi	NOUN
iajs-2767	147	43	∗	∗	NOUN
iajs-2767	147	44	:	:	PUNCT
iajs-2767	148	1	ℋi	ℋi	ADJ
iajs-2767	148	2	∗	∗	NOUN
iajs-2767	148	3	is	be	AUX
iajs-2767	148	4	a	a	DET
iajs-2767	148	5	fuzzy	fuzzy	ADJ
iajs-2767	148	6	ring	ring	NOUN
iajs-2767	148	7	over	over	ADP
iajs-2767	148	8	a	a	DET
iajs-2767	148	9	fuzzy	fuzzy	ADJ
iajs-2767	148	10	set	set	NOUN
iajs-2767	148	11	𝒳∗and	𝒳∗and	PROPN
iajs-2767	149	1	ℋi	ℋi	PROPN
iajs-2767	149	2	∗	∗	X
iajs-2767	149	3	⊇	⊇	NOUN
iajs-2767	149	4	𝔗∗	𝔗∗	NOUN
iajs-2767	149	5	,	,	PUNCT
iajs-2767	149	6	∀i	∀i	NOUN
iajs-2767	149	7	∈	∈	NOUN
iajs-2767	149	8	ι	ι	X
iajs-2767	149	9	}	}	PUNCT
iajs-2767	149	10	.	.	PUNCT
iajs-2767	150	1	proposition	proposition	NOUN
iajs-2767	150	2	3.6	3.6	NUM
iajs-2767	150	3	if	if	SCONJ
iajs-2767	150	4	𝒳	𝒳	PROPN
iajs-2767	150	5	≠	≠	PROPN
iajs-2767	150	6	∅	∅	NOUN
iajs-2767	150	7	and	and	CCONJ
iajs-2767	150	8	𝔗∗	𝔗∗	NUM
iajs-2767	150	9	⊆	⊆	NUM
iajs-2767	150	10	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	150	11	)	)	PUNCT
iajs-2767	150	12	.	.	PUNCT
iajs-2767	151	1	then	then	ADV
iajs-2767	151	2	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	151	3	)	)	PUNCT
iajs-2767	151	4	is	be	AUX
iajs-2767	151	5	the	the	DET
iajs-2767	151	6	smallest	small	ADJ
iajs-2767	151	7	fuzzy	fuzzy	ADJ
iajs-2767	151	8	ring	ring	NOUN
iajs-2767	151	9	over	over	ADP
iajs-2767	151	10	a	a	DET
iajs-2767	151	11	fuzzy	fuzzy	ADJ
iajs-2767	151	12	set	set	NOUN
iajs-2767	151	13	𝒳∗	𝒳∗	PRON
iajs-2767	151	14	that	that	PRON
iajs-2767	151	15	includes	include	VERB
iajs-2767	151	16	𝔗∗.	𝔗∗.	PROPN
iajs-2767	151	17	proof	proof	NOUN
iajs-2767	151	18	the	the	DET
iajs-2767	151	19	result	result	NOUN
iajs-2767	151	20	is	be	AUX
iajs-2767	151	21	directed	direct	VERB
iajs-2767	151	22	by	by	ADP
iajs-2767	151	23	the	the	DET
iajs-2767	151	24	definition	definition	NOUN
iajs-2767	151	25	of	of	ADP
iajs-2767	151	26	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	151	27	)	)	PUNCT
iajs-2767	151	28	and	and	CCONJ
iajs-2767	151	29	lemma	lemma	PROPN
iajs-2767	151	30	3.2	3.2	NUM
iajs-2767	151	31	.	.	PUNCT
iajs-2767	152	1	proposition	proposition	NOUN
iajs-2767	152	2	3.7	3.7	NUM
iajs-2767	152	3	every	every	DET
iajs-2767	152	4	fuzzy	fuzzy	ADJ
iajs-2767	152	5	σ	σ	PROPN
iajs-2767	152	6	–	–	PUNCT
iajs-2767	152	7	ring	ring	NOUN
iajs-2767	152	8	over	over	ADP
iajs-2767	152	9	a	a	DET
iajs-2767	152	10	fuzzy	fuzzy	ADJ
iajs-2767	152	11	set	set	NOUN
iajs-2767	152	12	𝒳∗	𝒳∗	NOUN
iajs-2767	152	13	is	be	AUX
iajs-2767	152	14	a	a	DET
iajs-2767	152	15	fuzzy	fuzzy	ADJ
iajs-2767	152	16	ring	ring	NOUN
iajs-2767	152	17	over	over	ADP
iajs-2767	152	18	a	a	DET
iajs-2767	152	19	fuzzy	fuzzy	ADJ
iajs-2767	152	20	set	set	NOUN
iajs-2767	152	21	𝒳∗.	𝒳∗.	PROPN
iajs-2767	152	22	proof	proof	NOUN
iajs-2767	152	23	let	let	VERB
iajs-2767	152	24	ℋ∗	ℋ∗	NOUN
iajs-2767	152	25	be	be	AUX
iajs-2767	152	26	a	a	DET
iajs-2767	152	27	fuzzy	fuzzy	ADJ
iajs-2767	152	28	σ	σ	NOUN
iajs-2767	152	29	–	–	PUNCT
iajs-2767	152	30	ring	ring	NOUN
iajs-2767	152	31	over	over	ADP
iajs-2767	152	32	a	a	DET
iajs-2767	152	33	fuzzy	fuzzy	ADJ
iajs-2767	152	34	set	set	NOUN
iajs-2767	152	35	𝒳∗.	𝒳∗.	PROPN
iajs-2767	152	36	then	then	ADV
iajs-2767	152	37	by	by	ADP
iajs-2767	152	38	definition	definition	NOUN
iajs-2767	152	39	of	of	ADP
iajs-2767	152	40	fuzzy	fuzzy	ADJ
iajs-2767	152	41	σ	σ	PROPN
iajs-2767	152	42	–	–	PUNCT
iajs-2767	152	43	ring	ring	NOUN
iajs-2767	152	44	we	we	PRON
iajs-2767	152	45	have	have	VERB
iajs-2767	152	46	,	,	PUNCT
iajs-2767	152	47	∅∗	∅∗	PROPN
iajs-2767	152	48	∈	∈	PROPN
iajs-2767	152	49	ℋ∗.	ℋ∗.	PROPN
iajs-2767	152	50	let	let	VERB
iajs-2767	152	51	f	f	NOUN
iajs-2767	152	52	,	,	PUNCT
iajs-2767	152	53	e	e	PROPN
iajs-2767	152	54	∈	∈	PROPN
iajs-2767	152	55	ℋ∗.	ℋ∗.	PROPN
iajs-2767	152	56	then	then	ADV
iajs-2767	152	57	f\e	f\e	VERB
iajs-2767	152	58	∈	∈	PROPN
iajs-2767	152	59	ℋ∗.	ℋ∗.	PROPN
iajs-2767	152	60	let	let	VERB
iajs-2767	152	61	e1	e1	PROPN
iajs-2767	152	62	,	,	PUNCT
iajs-2767	152	63	e2	e2	PROPN
iajs-2767	152	64	,	,	PUNCT
iajs-2767	152	65	en	en	ADP
iajs-2767	152	66	∈	∈	PROPN
iajs-2767	152	67	ℋ∗.	ℋ∗.	PROPN
iajs-2767	152	68	consider	consider	VERB
iajs-2767	152	69	,	,	PUNCT
iajs-2767	152	70	em	em	PRON
iajs-2767	152	71	=	=	PUNCT
iajs-2767	152	72	∅∗	∅∗	PROPN
iajs-2767	152	73	for	for	ADP
iajs-2767	152	74	all	all	DET
iajs-2767	152	75	m	m	PROPN
iajs-2767	152	76	>	>	X
iajs-2767	152	77	n	n	CCONJ
iajs-2767	152	78	,	,	PUNCT
iajs-2767	152	79	then	then	ADV
iajs-2767	152	80	we	we	PRON
iajs-2767	152	81	get	get	VERB
iajs-2767	152	82	e1	e1	NOUN
iajs-2767	152	83	,	,	PUNCT
iajs-2767	152	84	e2	e2	PROPN
iajs-2767	152	85	,	,	PUNCT
iajs-2767	152	86	e3	e3	NOUN
iajs-2767	152	87	,	,	PUNCT
iajs-2767	152	88	…	…	PUNCT
iajs-2767	152	89	∈	∈	NOUN
iajs-2767	152	90	ℋ∗	ℋ∗	NOUN
iajs-2767	152	91	and	and	CCONJ
iajs-2767	152	92	hence	hence	ADV
iajs-2767	152	93	from	from	ADP
iajs-2767	152	94	the	the	DET
iajs-2767	152	95	definition	definition	NOUN
iajs-2767	152	96	of	of	ADP
iajs-2767	152	97	fuzzy	fuzzy	ADJ
iajs-2767	152	98	σ	σ	PROPN
iajs-2767	152	99	–	–	PUNCT
iajs-2767	152	100	ring	ring	NOUN
iajs-2767	152	101	,	,	PUNCT
iajs-2767	152	102	we	we	PRON
iajs-2767	152	103	have	have	VERB
iajs-2767	152	104	⋃	⋃	VERB
iajs-2767	152	105	ek	ek	NOUN
iajs-2767	152	106	∞	∞	PROPN
iajs-2767	152	107	k=1	k=1	PUNCT
iajs-2767	153	1	∈	∈	PROPN
iajs-2767	153	2	ℋ∗	ℋ∗	NOUN
iajs-2767	153	3	,	,	PUNCT
iajs-2767	153	4	but	but	CCONJ
iajs-2767	153	5	⋃	⋃	ADP
iajs-2767	153	6	ek	ek	X
iajs-2767	153	7	∞	∞	PROPN
iajs-2767	153	8	k=1	k=1	PUNCT
iajs-2767	154	1	=	=	PUNCT
iajs-2767	154	2	⋃	⋃	ADP
iajs-2767	154	3	ek	ek	NOUN
iajs-2767	154	4	n	n	ADV
iajs-2767	154	5	k=1	k=1	ADJ
iajs-2767	154	6	⋃en+1⋃en+2⋃	⋃en+1⋃en+2⋃	PROPN
iajs-2767	154	7	…	…	PUNCT
iajs-2767	154	8	=	=	SYM
iajs-2767	154	9	⋃	⋃	ADP
iajs-2767	154	10	ek	ek	NOUN
iajs-2767	154	11	n	n	PROPN
iajs-2767	154	12	k=1	k=1	PROPN
iajs-2767	154	13	⋃∅∗⋃∅∗⋃	⋃∅∗⋃∅∗⋃	X
iajs-2767	154	14	…	…	PUNCT
iajs-2767	154	15	=	=	SYM
iajs-2767	154	16	⋃	⋃	ADP
iajs-2767	154	17	ek	ek	NOUN
iajs-2767	154	18	n	n	X
iajs-2767	154	19	k=1	k=1	X
iajs-2767	154	20	.	.	PUNCT
iajs-2767	155	1	thus	thus	ADV
iajs-2767	155	2	e1⋃e2	e1⋃e2	VERB
iajs-2767	155	3	∈	∈	PROPN
iajs-2767	155	4	ℋ∗.	ℋ∗.	PROPN
iajs-2767	155	5	therefore	therefore	ADV
iajs-2767	155	6	,	,	PUNCT
iajs-2767	155	7	ℋ∗	ℋ∗	NUM
iajs-2767	155	8	is	be	AUX
iajs-2767	155	9	a	a	DET
iajs-2767	155	10	fuzzy	fuzzy	ADJ
iajs-2767	155	11	ring	ring	NOUN
iajs-2767	155	12	over	over	ADP
iajs-2767	155	13	a	a	DET
iajs-2767	155	14	fuzzy	fuzzy	ADJ
iajs-2767	155	15	set	set	VERB
iajs-2767	155	16	𝒳∗.	𝒳∗.	PROPN
iajs-2767	155	17	in	in	ADP
iajs-2767	155	18	general	general	ADJ
iajs-2767	155	19	,	,	PUNCT
iajs-2767	155	20	the	the	DET
iajs-2767	155	21	converse	converse	NOUN
iajs-2767	155	22	of	of	ADP
iajs-2767	155	23	the	the	DET
iajs-2767	155	24	above	above	ADJ
iajs-2767	155	25	proposition	proposition	NOUN
iajs-2767	155	26	is	be	AUX
iajs-2767	155	27	not	not	PART
iajs-2767	155	28	true	true	ADJ
iajs-2767	155	29	as	as	SCONJ
iajs-2767	155	30	shown	show	VERB
iajs-2767	155	31	in	in	ADP
iajs-2767	155	32	the	the	DET
iajs-2767	155	33	following	follow	VERB
iajs-2767	155	34	example	example	NOUN
iajs-2767	155	35	:	:	PUNCT
iajs-2767	155	36	example	example	NOUN
iajs-2767	155	37	3.10	3.10	NUM
iajs-2767	155	38	let	let	VERB
iajs-2767	155	39	𝒳	𝒳	NOUN
iajs-2767	155	40	=	=	SYM
iajs-2767	155	41	ℝ	ℝ	PROPN
iajs-2767	155	42	and	and	CCONJ
iajs-2767	156	1	𝒥	𝒥	NOUN
iajs-2767	157	1	=	=	NOUN
iajs-2767	158	1	finite	finite	PROPN
iajs-2767	158	2	disjoint	disjoint	PROPN
iajs-2767	158	3	union	union	PROPN
iajs-2767	158	4	of	of	ADP
iajs-2767	158	5	right	right	ADJ
iajs-2767	158	6	–	–	PUNCT
iajs-2767	158	7	semi	semi	ADJ
iajs-2767	158	8	-	-	ADJ
iajs-2767	158	9	closed	closed	ADJ
iajs-2767	158	10	intervals	interval	NOUN
iajs-2767	158	11	.	.	PUNCT
iajs-2767	159	1	assume	assume	VERB
iajs-2767	159	2	that	that	SCONJ
iajs-2767	159	3	ℋ∗	ℋ∗	NOUN
iajs-2767	160	1	=	=	PUNCT
iajs-2767	160	2	{	{	PUNCT
iajs-2767	160	3	all	all	PRON
iajs-2767	160	4	(	(	PUNCT
iajs-2767	160	5	𝒥	𝒥	PROPN
iajs-2767	160	6	,	,	PUNCT
iajs-2767	160	7	𝓋𝒥	𝓋𝒥	NOUN
iajs-2767	160	8	)	)	PUNCT
iajs-2767	160	9	}	}	PUNCT
iajs-2767	160	10	.	.	PUNCT
iajs-2767	161	1	then	then	ADV
iajs-2767	161	2	ℋ∗	ℋ∗	NUM
iajs-2767	161	3	is	be	AUX
iajs-2767	161	4	a	a	DET
iajs-2767	161	5	fuzzy	fuzzy	ADJ
iajs-2767	161	6	ring	ring	NOUN
iajs-2767	161	7	over	over	ADP
iajs-2767	161	8	a	a	DET
iajs-2767	161	9	fuzzy	fuzzy	ADJ
iajs-2767	161	10	set	set	VERB
iajs-2767	161	11	ℝ∗	ℝ∗	NOUN
iajs-2767	161	12	,	,	PUNCT
iajs-2767	161	13	but	but	CCONJ
iajs-2767	161	14	ℋ∗	ℋ∗	NOUN
iajs-2767	161	15	is	be	AUX
iajs-2767	161	16	not	not	PART
iajs-2767	161	17	fuzzy	fuzzy	ADJ
iajs-2767	161	18	σ	σ	NOUN
iajs-2767	161	19	–	–	PUNCT
iajs-2767	161	20	ring	ring	NOUN
iajs-2767	161	21	over	over	ADP
iajs-2767	161	22	a	a	DET
iajs-2767	161	23	ibn	ibn	PROPN
iajs-2767	161	24	al	al	PROPN
iajs-2767	161	25	-	-	PUNCT
iajs-2767	161	26	haitham	haitham	PROPN
iajs-2767	161	27	jour	jour	X
iajs-2767	161	28	.	.	PROPN
iajs-2767	162	1	for	for	ADP
iajs-2767	162	2	pure	pure	ADJ
iajs-2767	162	3	&	&	CCONJ
iajs-2767	162	4	appl	appl	PROPN
iajs-2767	162	5	.	.	PUNCT
iajs-2767	163	1	sci	sci	PROPN
iajs-2767	163	2	.	.	PROPN
iajs-2767	164	1	53	53	NUM
iajs-2767	164	2	(	(	PUNCT
iajs-2767	164	3	2)2022	2)2022	NUM
iajs-2767	164	4	44	44	NUM
iajs-2767	164	5	fuzzy	fuzzy	ADJ
iajs-2767	164	6	set	set	VERB
iajs-2767	164	7	ℝ∗.	ℝ∗.	PROPN
iajs-2767	164	8	because	because	SCONJ
iajs-2767	164	9	if	if	SCONJ
iajs-2767	164	10	we	we	PRON
iajs-2767	164	11	take	take	VERB
iajs-2767	164	12	ek	ek	NOUN
iajs-2767	164	13	=	=	PUNCT
iajs-2767	164	14	{	{	PUNCT
iajs-2767	164	15	(	(	PUNCT
iajs-2767	164	16	(	(	PUNCT
iajs-2767	164	17	0	0	NUM
iajs-2767	164	18	,	,	PUNCT
iajs-2767	164	19	1(1∕k	1(1∕k	NUM
iajs-2767	164	20	)	)	PUNCT
iajs-2767	164	21	]	]	PUNCT
iajs-2767	164	22	,	,	PUNCT
iajs-2767	164	23	𝓋ek	𝓋ek	NOUN
iajs-2767	164	24	)	)	PUNCT
iajs-2767	164	25	}	}	PUNCT
iajs-2767	164	26	,	,	PUNCT
iajs-2767	164	27	k=1,2	k=1,2	PROPN
iajs-2767	164	28	,	,	PUNCT
iajs-2767	164	29	…	…	PUNCT
iajs-2767	164	30	,	,	PUNCT
iajs-2767	164	31	then	then	ADV
iajs-2767	164	32	ek	ek	PROPN
iajs-2767	164	33	∈	∈	PROPN
iajs-2767	164	34	ℋ∗	ℋ∗	X
iajs-2767	165	1	∀n	∀n	CCONJ
iajs-2767	165	2	,	,	PUNCT
iajs-2767	165	3	but	but	CCONJ
iajs-2767	165	4	⋃	⋃	ADP
iajs-2767	165	5	ek	ek	X
iajs-2767	165	6	∞	∞	PROPN
iajs-2767	165	7	k=1	k=1	PUNCT
iajs-2767	166	1	=	=	PUNCT
iajs-2767	166	2	{	{	PUNCT
iajs-2767	166	3	(	(	PUNCT
iajs-2767	166	4	(	(	PUNCT
iajs-2767	166	5	0,1	0,1	NUM
iajs-2767	166	6	)	)	PUNCT
iajs-2767	166	7	,	,	PUNCT
iajs-2767	166	8	𝓋ek	𝓋ek	NOUN
iajs-2767	166	9	)	)	PUNCT
iajs-2767	166	10	}	}	PUNCT
iajs-2767	166	11	∉	∉	PROPN
iajs-2767	166	12	ℋ∗.	ℋ∗.	PROPN
iajs-2767	166	13	proposition	proposition	NOUN
iajs-2767	166	14	3.8	3.8	NUM
iajs-2767	166	15	assume	assume	VERB
iajs-2767	166	16	𝒳	𝒳	PROPN
iajs-2767	166	17	≠	≠	PROPN
iajs-2767	166	18	∅	∅	NOUN
iajs-2767	166	19	and	and	CCONJ
iajs-2767	166	20	𝔗∗	𝔗∗	NUM
iajs-2767	166	21	⊆	⊆	NUM
iajs-2767	166	22	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	166	23	)	)	PUNCT
iajs-2767	166	24	.	.	PUNCT
iajs-2767	167	1	then	then	ADV
iajs-2767	167	2	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	167	3	)	)	PUNCT
iajs-2767	167	4	⊆	⊆	NUM
iajs-2767	167	5	σ𝑟(𝔗∗	σ𝑟(𝔗∗	NOUN
iajs-2767	167	6	)	)	PUNCT
iajs-2767	167	7	.	.	PUNCT
iajs-2767	168	1	proof	proof	NOUN
iajs-2767	168	2	the	the	DET
iajs-2767	168	3	proof	proof	NOUN
iajs-2767	168	4	is	be	AUX
iajs-2767	168	5	directed	direct	VERB
iajs-2767	168	6	by	by	ADP
iajs-2767	168	7	proposition	proposition	NOUN
iajs-2767	168	8	3.7	3.7	NUM
iajs-2767	168	9	and	and	CCONJ
iajs-2767	168	10	proposition	proposition	NOUN
iajs-2767	168	11	3.6	3.6	NUM
iajs-2767	168	12	.	.	PUNCT
iajs-2767	169	1	proposition	proposition	NOUN
iajs-2767	169	2	3.9	3.9	NUM
iajs-2767	169	3	every	every	DET
iajs-2767	169	4	fuzzy	fuzzy	ADJ
iajs-2767	169	5	algebra	algebra	NOUN
iajs-2767	169	6	over	over	ADP
iajs-2767	169	7	a	a	DET
iajs-2767	169	8	fuzzy	fuzzy	ADJ
iajs-2767	169	9	set	set	NOUN
iajs-2767	169	10	𝒳∗	𝒳∗	NOUN
iajs-2767	169	11	is	be	AUX
iajs-2767	169	12	a	a	DET
iajs-2767	169	13	fuzzy	fuzzy	ADJ
iajs-2767	169	14	ring	ring	NOUN
iajs-2767	169	15	over	over	ADP
iajs-2767	169	16	a	a	DET
iajs-2767	169	17	fuzzy	fuzzy	ADJ
iajs-2767	169	18	set	set	NOUN
iajs-2767	169	19	𝒳∗.	𝒳∗.	PROPN
iajs-2767	169	20	proof	proof	NOUN
iajs-2767	169	21	let	let	VERB
iajs-2767	169	22	ℋ∗	ℋ∗	NOUN
iajs-2767	169	23	be	be	AUX
iajs-2767	169	24	a	a	DET
iajs-2767	169	25	fuzzy	fuzzy	ADJ
iajs-2767	169	26	algebra	algebra	NOUN
iajs-2767	169	27	over	over	ADP
iajs-2767	169	28	a	a	DET
iajs-2767	169	29	fuzzy	fuzzy	ADJ
iajs-2767	169	30	set	set	NOUN
iajs-2767	169	31	𝒳∗.	𝒳∗.	PROPN
iajs-2767	169	32	then	then	ADV
iajs-2767	169	33	by	by	ADP
iajs-2767	169	34	the	the	DET
iajs-2767	169	35	definition	definition	NOUN
iajs-2767	169	36	of	of	ADP
iajs-2767	169	37	fuzzy	fuzzy	ADJ
iajs-2767	169	38	algebra	algebra	NOUN
iajs-2767	169	39	we	we	PRON
iajs-2767	169	40	have	have	VERB
iajs-2767	169	41	,	,	PUNCT
iajs-2767	169	42	𝒳∗	𝒳∗	PROPN
iajs-2767	169	43	∈	∈	PROPN
iajs-2767	169	44	ℋ∗	ℋ∗	PROPN
iajs-2767	169	45	,	,	PUNCT
iajs-2767	169	46	hence	hence	ADV
iajs-2767	169	47	∅∗	∅∗	PROPN
iajs-2767	169	48	=	=	SYM
iajs-2767	169	49	𝒳∗𝑐	𝒳∗𝑐	PROPN
iajs-2767	169	50	∈	∈	PROPN
iajs-2767	169	51	ℋ∗	ℋ∗	NOUN
iajs-2767	169	52	let	let	VERB
iajs-2767	169	53	f	f	NOUN
iajs-2767	169	54	,	,	PUNCT
iajs-2767	169	55	e	e	PROPN
iajs-2767	169	56	∈	∈	PROPN
iajs-2767	169	57	ℋ∗.	ℋ∗.	PROPN
iajs-2767	169	58	then	then	ADV
iajs-2767	169	59	e𝑐	e𝑐	PROPN
iajs-2767	169	60	∈	∈	PROPN
iajs-2767	169	61	ℋ∗	ℋ∗	NUM
iajs-2767	169	62	,	,	PUNCT
iajs-2767	169	63	hence	hence	ADV
iajs-2767	169	64	f⋂e𝑐	f⋂e𝑐	PROPN
iajs-2767	169	65	∈	∈	PROPN
iajs-2767	169	66	ℋ∗	ℋ∗	NOUN
iajs-2767	169	67	,	,	PUNCT
iajs-2767	169	68	but	but	CCONJ
iajs-2767	170	1	f⋂e𝑐	f⋂e𝑐	NOUN
iajs-2767	170	2	=	=	NOUN
iajs-2767	170	3	f\e	f\e	NOUN
iajs-2767	170	4	implies	imply	VERB
iajs-2767	170	5	that	that	SCONJ
iajs-2767	170	6	f\e	f\e	VERB
iajs-2767	170	7	∈	∈	PROPN
iajs-2767	170	8	ℋ∗.	ℋ∗.	PROPN
iajs-2767	170	9	let	let	VERB
iajs-2767	170	10	e1	e1	NOUN
iajs-2767	170	11	,	,	PUNCT
iajs-2767	170	12	e2	e2	PROPN
iajs-2767	170	13	∈	∈	PROPN
iajs-2767	170	14	ℋ∗.	ℋ∗.	PROPN
iajs-2767	170	15	then	then	ADV
iajs-2767	170	16	by	by	ADP
iajs-2767	170	17	definition	definition	NOUN
iajs-2767	170	18	of	of	ADP
iajs-2767	170	19	fuzzy	fuzzy	ADJ
iajs-2767	170	20	algebra	algebra	NOUN
iajs-2767	170	21	implies	imply	VERB
iajs-2767	170	22	that	that	SCONJ
iajs-2767	170	23	e1⋃e2	e1⋃e2	NOUN
iajs-2767	170	24	∈	∈	PROPN
iajs-2767	170	25	ℋ∗.	ℋ∗.	PROPN
iajs-2767	170	26	therefore	therefore	ADV
iajs-2767	170	27	,	,	PUNCT
iajs-2767	170	28	ℋ∗	ℋ∗	NOUN
iajs-2767	170	29	be	be	AUX
iajs-2767	170	30	a	a	DET
iajs-2767	170	31	fuzzy	fuzzy	ADJ
iajs-2767	170	32	ring	ring	NOUN
iajs-2767	170	33	over	over	ADP
iajs-2767	170	34	a	a	DET
iajs-2767	170	35	fuzzy	fuzzy	ADJ
iajs-2767	170	36	set	set	VERB
iajs-2767	170	37	𝒳∗.	𝒳∗.	PROPN
iajs-2767	170	38	while	while	SCONJ
iajs-2767	170	39	the	the	DET
iajs-2767	170	40	converse	converse	NOUN
iajs-2767	170	41	is	be	AUX
iajs-2767	170	42	not	not	PART
iajs-2767	170	43	true	true	ADJ
iajs-2767	170	44	as	as	SCONJ
iajs-2767	170	45	shown	show	VERB
iajs-2767	170	46	in	in	ADP
iajs-2767	170	47	the	the	DET
iajs-2767	170	48	next	next	ADJ
iajs-2767	170	49	example	example	NOUN
iajs-2767	170	50	:	:	PUNCT
iajs-2767	170	51	example	example	NOUN
iajs-2767	170	52	3.11	3.11	NUM
iajs-2767	170	53	let	let	VERB
iajs-2767	170	54	𝒳	𝒳	PRON
iajs-2767	170	55	=	=	PRON
iajs-2767	170	56	{	{	PUNCT
iajs-2767	170	57	𝑎	𝑎	PROPN
iajs-2767	170	58	,	,	PUNCT
iajs-2767	170	59	𝑏	𝑏	NOUN
iajs-2767	170	60	}	}	PUNCT
iajs-2767	170	61	and	and	CCONJ
iajs-2767	170	62	ℋ∗	ℋ∗	NUM
iajs-2767	171	1	=	=	SYM
iajs-2767	171	2	{	{	PUNCT
iajs-2767	171	3	∅∗	∅∗	PROPN
iajs-2767	171	4	,	,	PUNCT
iajs-2767	171	5	{	{	PUNCT
iajs-2767	171	6	(	(	PUNCT
iajs-2767	171	7	𝑎,0	𝑎,0	NOUN
iajs-2767	171	8	)	)	PUNCT
iajs-2767	171	9	,	,	PUNCT
iajs-2767	171	10	(	(	PUNCT
iajs-2767	171	11	𝑏,0.5	𝑏,0.5	PROPN
iajs-2767	171	12	)	)	PUNCT
iajs-2767	171	13	}	}	PUNCT
iajs-2767	171	14	,	,	PUNCT
iajs-2767	171	15	{	{	PUNCT
iajs-2767	171	16	(	(	PUNCT
iajs-2767	171	17	𝑎,0.6	𝑎,0.6	PROPN
iajs-2767	171	18	)	)	PUNCT
iajs-2767	171	19	,	,	PUNCT
iajs-2767	171	20	(	(	PUNCT
iajs-2767	171	21	𝑏,0.5	𝑏,0.5	PROPN
iajs-2767	171	22	)	)	PUNCT
iajs-2767	171	23	}	}	PUNCT
iajs-2767	171	24	,	,	PUNCT
iajs-2767	171	25	{	{	PUNCT
iajs-2767	171	26	(	(	PUNCT
iajs-2767	171	27	𝑎,0.4	𝑎,0.4	PROPN
iajs-2767	171	28	)	)	PUNCT
iajs-2767	171	29	,	,	PUNCT
iajs-2767	171	30	(	(	PUNCT
iajs-2767	171	31	𝑏,0.5	𝑏,0.5	PROPN
iajs-2767	171	32	)	)	PUNCT
iajs-2767	171	33	}	}	PUNCT
iajs-2767	171	34	}	}	PUNCT
iajs-2767	171	35	.	.	PUNCT
iajs-2767	172	1	then	then	ADV
iajs-2767	172	2	,	,	PUNCT
iajs-2767	172	3	ℋ∗	ℋ∗	NUM
iajs-2767	172	4	is	be	AUX
iajs-2767	172	5	a	a	DET
iajs-2767	172	6	fuzzy	fuzzy	ADJ
iajs-2767	172	7	ring	ring	NOUN
iajs-2767	172	8	over	over	ADP
iajs-2767	172	9	a	a	DET
iajs-2767	172	10	fuzzy	fuzzy	ADJ
iajs-2767	172	11	set	set	VERB
iajs-2767	172	12	𝒳∗.	𝒳∗.	PROPN
iajs-2767	172	13	in	in	ADP
iajs-2767	172	14	contrast	contrast	NOUN
iajs-2767	172	15	,	,	PUNCT
iajs-2767	172	16	ℋ∗	ℋ∗	NUM
iajs-2767	172	17	is	be	AUX
iajs-2767	172	18	a	a	DET
iajs-2767	172	19	fuzzy	fuzzy	ADJ
iajs-2767	172	20	algebra	algebra	NOUN
iajs-2767	172	21	over	over	ADP
iajs-2767	172	22	a	a	DET
iajs-2767	172	23	fuzzy	fuzzy	ADJ
iajs-2767	172	24	set	set	VERB
iajs-2767	172	25	𝒳∗	𝒳∗	PROPN
iajs-2767	172	26	,	,	PUNCT
iajs-2767	172	27	because	because	SCONJ
iajs-2767	172	28	{	{	PUNCT
iajs-2767	172	29	(	(	PUNCT
iajs-2767	172	30	𝑎,0	𝑎,0	NOUN
iajs-2767	172	31	)	)	PUNCT
iajs-2767	172	32	,	,	PUNCT
iajs-2767	172	33	(	(	PUNCT
iajs-2767	172	34	𝑏,0.5)}∈	𝑏,0.5)}∈	PROPN
iajs-2767	172	35	ℋ∗	ℋ∗	PROPN
iajs-2767	172	36	,	,	PUNCT
iajs-2767	172	37	but	but	CCONJ
iajs-2767	172	38	{	{	PUNCT
iajs-2767	172	39	(	(	PUNCT
iajs-2767	172	40	𝑎	𝑎	X
iajs-2767	172	41	,	,	PUNCT
iajs-2767	172	42	0	0	NUM
iajs-2767	172	43	)	)	PUNCT
iajs-2767	172	44	,	,	PUNCT
iajs-2767	172	45	(	(	PUNCT
iajs-2767	172	46	𝑏	𝑏	NOUN
iajs-2767	172	47	,	,	PUNCT
iajs-2767	172	48	0.5)}c	0.5)}c	NOUN
iajs-2767	172	49	=	=	SYM
iajs-2767	172	50	{	{	PUNCT
iajs-2767	172	51	(	(	PUNCT
iajs-2767	172	52	𝑎	𝑎	X
iajs-2767	172	53	,	,	PUNCT
iajs-2767	172	54	1	1	NUM
iajs-2767	172	55	)	)	PUNCT
iajs-2767	172	56	,	,	PUNCT
iajs-2767	172	57	(	(	PUNCT
iajs-2767	172	58	𝑏	𝑏	NOUN
iajs-2767	172	59	,	,	PUNCT
iajs-2767	172	60	0.5	0.5	NUM
iajs-2767	172	61	)	)	PUNCT
iajs-2767	172	62	}	}	PUNCT
iajs-2767	172	63	∉	∉	PROPN
iajs-2767	172	64	ℋ∗.	ℋ∗.	PROPN
iajs-2767	172	65	proposition	proposition	NOUN
iajs-2767	172	66	3.10	3.10	NUM
iajs-2767	172	67	assume	assume	VERB
iajs-2767	172	68	𝒳	𝒳	PROPN
iajs-2767	172	69	≠	≠	PROPN
iajs-2767	172	70	∅	∅	NOUN
iajs-2767	172	71	and	and	CCONJ
iajs-2767	172	72	𝔗∗	𝔗∗	NUM
iajs-2767	172	73	⊆	⊆	NUM
iajs-2767	172	74	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	172	75	)	)	PUNCT
iajs-2767	172	76	,	,	PUNCT
iajs-2767	172	77	then	then	ADV
iajs-2767	172	78	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	172	79	)	)	PUNCT
iajs-2767	172	80	⊆	⊆	NUM
iajs-2767	172	81	al(𝔗∗	al(𝔗∗	NOUN
iajs-2767	172	82	)	)	PUNCT
iajs-2767	172	83	where	where	SCONJ
iajs-2767	172	84	al(𝔗∗	al(𝔗∗	NOUN
iajs-2767	172	85	)	)	PUNCT
iajs-2767	172	86	is	be	AUX
iajs-2767	172	87	the	the	DET
iajs-2767	172	88	smallest	small	ADJ
iajs-2767	172	89	fuzzy	fuzzy	ADJ
iajs-2767	172	90	algebra	algebra	NOUN
iajs-2767	172	91	over	over	ADP
iajs-2767	172	92	a	a	DET
iajs-2767	172	93	fuzzy	fuzzy	ADJ
iajs-2767	172	94	set	set	NOUN
iajs-2767	172	95	𝒳∗	𝒳∗	PRON
iajs-2767	173	1	that	that	PRON
iajs-2767	173	2	include	include	VERB
iajs-2767	173	3	𝔗∗.	𝔗∗.	PRON
iajs-2767	173	4	proof	proof	NOUN
iajs-2767	173	5	the	the	DET
iajs-2767	173	6	proof	proof	NOUN
iajs-2767	173	7	is	be	AUX
iajs-2767	173	8	followed	follow	VERB
iajs-2767	173	9	by	by	ADP
iajs-2767	173	10	proposition	proposition	NOUN
iajs-2767	173	11	3.9	3.9	NUM
iajs-2767	173	12	with	with	ADP
iajs-2767	173	13	proposition	proposition	NOUN
iajs-2767	173	14	3.6	3.6	NUM
iajs-2767	173	15	.	.	PUNCT
iajs-2767	174	1	proposition	proposition	NOUN
iajs-2767	174	2	3.11	3.11	NUM
iajs-2767	174	3	assume	assume	VERB
iajs-2767	174	4	𝒳	𝒳	PRON
iajs-2767	174	5	to	to	PART
iajs-2767	174	6	be	be	AUX
iajs-2767	174	7	a	a	DET
iajs-2767	174	8	non	non	ADJ
iajs-2767	174	9	-	-	ADJ
iajs-2767	174	10	empty	empty	ADJ
iajs-2767	174	11	set	set	NOUN
iajs-2767	174	12	and	and	CCONJ
iajs-2767	174	13	𝔗∗	𝔗∗	NUM
iajs-2767	174	14	⊆	⊆	NUM
iajs-2767	174	15	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	174	16	)	)	PUNCT
iajs-2767	174	17	.	.	PUNCT
iajs-2767	175	1	then	then	ADV
iajs-2767	175	2	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	175	3	)	)	PUNCT
iajs-2767	175	4	⊆	⊆	NUM
iajs-2767	175	5	σ(𝔗∗	σ(𝔗∗	NOUN
iajs-2767	175	6	)	)	PUNCT
iajs-2767	175	7	.	.	PUNCT
iajs-2767	176	1	proof	proof	NOUN
iajs-2767	176	2	observe	observe	VERB
iajs-2767	176	3	.	.	PUNCT
iajs-2767	177	1	theorem	theorem	ADJ
iajs-2767	177	2	3.2	3.2	NUM
iajs-2767	177	3	assume	assume	VERB
iajs-2767	177	4	𝒳	𝒳	PROPN
iajs-2767	177	5	≠	≠	PROPN
iajs-2767	177	6	∅	∅	NOUN
iajs-2767	177	7	and	and	CCONJ
iajs-2767	177	8	ℋ∗	ℋ∗	NUM
iajs-2767	177	9	⊆	⊆	NUM
iajs-2767	177	10	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	177	11	)	)	PUNCT
iajs-2767	177	12	such	such	ADJ
iajs-2767	177	13	that	that	SCONJ
iajs-2767	177	14	𝒳∗	𝒳∗	VERB
iajs-2767	177	15	∈	∈	PROPN
iajs-2767	177	16	ℋ∗.	ℋ∗.	PROPN
iajs-2767	177	17	then	then	ADV
iajs-2767	177	18	ℋ∗	ℋ∗	NUM
iajs-2767	177	19	is	be	AUX
iajs-2767	177	20	a	a	DET
iajs-2767	177	21	fuzzy	fuzzy	ADJ
iajs-2767	177	22	algebra	algebra	NOUN
iajs-2767	177	23	over	over	ADP
iajs-2767	177	24	a	a	DET
iajs-2767	177	25	fuzzy	fuzzy	ADJ
iajs-2767	177	26	set	set	NOUN
iajs-2767	177	27	𝒳∗	𝒳∗	PUNCT
iajs-2767	178	1	if	if	SCONJ
iajs-2767	178	2	and	and	CCONJ
iajs-2767	178	3	only	only	ADV
iajs-2767	178	4	if	if	SCONJ
iajs-2767	178	5	ℋ∗	ℋ∗	NUM
iajs-2767	178	6	is	be	AUX
iajs-2767	178	7	fuzzy	fuzzy	ADJ
iajs-2767	178	8	ring	ring	NOUN
iajs-2767	178	9	over	over	ADP
iajs-2767	178	10	a	a	DET
iajs-2767	178	11	fuzzy	fuzzy	ADJ
iajs-2767	178	12	set	set	NOUN
iajs-2767	178	13	𝒳∗.	𝒳∗.	PROPN
iajs-2767	178	14	proof	proof	NOUN
iajs-2767	178	15	assume	assume	VERB
iajs-2767	178	16	that	that	SCONJ
iajs-2767	178	17	ℋ∗	ℋ∗	NOUN
iajs-2767	178	18	is	be	AUX
iajs-2767	178	19	fuzzy	fuzzy	ADJ
iajs-2767	178	20	algebra	algebra	NOUN
iajs-2767	178	21	over	over	ADP
iajs-2767	178	22	a	a	DET
iajs-2767	178	23	fuzzy	fuzzy	ADJ
iajs-2767	178	24	set	set	VERB
iajs-2767	178	25	𝒳∗	𝒳∗	PROPN
iajs-2767	178	26	,	,	PUNCT
iajs-2767	178	27	then	then	ADV
iajs-2767	178	28	by	by	ADP
iajs-2767	178	29	proposition	proposition	NOUN
iajs-2767	178	30	3.9	3.9	NUM
iajs-2767	178	31	,	,	PUNCT
iajs-2767	178	32	we	we	PRON
iajs-2767	178	33	get	get	VERB
iajs-2767	178	34	ℋ∗	ℋ∗	NUM
iajs-2767	178	35	is	be	AUX
iajs-2767	178	36	a	a	DET
iajs-2767	178	37	fuzzy	fuzzy	ADJ
iajs-2767	178	38	ring	ring	NOUN
iajs-2767	178	39	over	over	ADP
iajs-2767	178	40	a	a	DET
iajs-2767	178	41	fuzzy	fuzzy	ADJ
iajs-2767	178	42	set	set	NOUN
iajs-2767	178	43	𝒳∗.	𝒳∗.	PROPN
iajs-2767	178	44	ibn	ibn	PROPN
iajs-2767	178	45	al	al	PROPN
iajs-2767	178	46	-	-	PUNCT
iajs-2767	178	47	haitham	haitham	PROPN
iajs-2767	178	48	jour	jour	X
iajs-2767	178	49	.	.	PROPN
iajs-2767	179	1	for	for	ADP
iajs-2767	179	2	pure	pure	ADJ
iajs-2767	179	3	&	&	CCONJ
iajs-2767	179	4	appl	appl	PROPN
iajs-2767	179	5	.	.	PUNCT
iajs-2767	180	1	sci	sci	PROPN
iajs-2767	180	2	.	.	PROPN
iajs-2767	181	1	53	53	NUM
iajs-2767	181	2	(	(	PUNCT
iajs-2767	181	3	2)2022	2)2022	VERB
iajs-2767	181	4	45	45	NUM
iajs-2767	181	5	conversely	conversely	ADV
iajs-2767	181	6	:	:	PUNCT
iajs-2767	181	7	suppose	suppose	VERB
iajs-2767	181	8	that	that	SCONJ
iajs-2767	181	9	ℋ∗	ℋ∗	NOUN
iajs-2767	181	10	is	be	AUX
iajs-2767	181	11	fuzzy	fuzzy	ADJ
iajs-2767	181	12	ring	ring	NOUN
iajs-2767	181	13	over	over	ADP
iajs-2767	181	14	a	a	DET
iajs-2767	181	15	fuzzy	fuzzy	ADJ
iajs-2767	181	16	set	set	NOUN
iajs-2767	181	17	𝒳∗	𝒳∗	PRON
iajs-2767	181	18	such	such	ADJ
iajs-2767	181	19	that	that	SCONJ
iajs-2767	181	20	𝒳∗	𝒳∗	VERB
iajs-2767	182	1	∈	∈	PROPN
iajs-2767	182	2	ℋ∗.	ℋ∗.	PROPN
iajs-2767	182	3	let	let	VERB
iajs-2767	182	4	e	e	PRON
iajs-2767	182	5	∈	∈	PROPN
iajs-2767	182	6	ℋ∗.	ℋ∗.	PROPN
iajs-2767	182	7	then	then	ADV
iajs-2767	182	8	𝒳∗\e	𝒳∗\e	PROPN
iajs-2767	182	9	∈	∈	PROPN
iajs-2767	182	10	ℋ∗	ℋ∗	PROPN
iajs-2767	182	11	,	,	PUNCT
iajs-2767	182	12	but	but	CCONJ
iajs-2767	182	13	𝒳∗\e	𝒳∗\e	NOUN
iajs-2767	183	1	=	=	NOUN
iajs-2767	184	1	𝒳∗⋂e𝑐	𝒳∗⋂e𝑐	NOUN
iajs-2767	184	2	=	=	SYM
iajs-2767	184	3	{	{	PUNCT
iajs-2767	184	4	(	(	PUNCT
iajs-2767	184	5	ω	ω	PROPN
iajs-2767	184	6	,	,	PUNCT
iajs-2767	184	7	min	min	PROPN
iajs-2767	184	8	{	{	PUNCT
iajs-2767	184	9	𝓋𝒳∗	𝓋𝒳∗	PROPN
iajs-2767	184	10	,	,	PUNCT
iajs-2767	184	11	𝓋e𝑐	𝓋e𝑐	X
iajs-2767	184	12	(	(	PUNCT
iajs-2767	184	13	ω	ω	NOUN
iajs-2767	184	14	)	)	PUNCT
iajs-2767	184	15	}	}	PUNCT
iajs-2767	184	16	)	)	PUNCT
iajs-2767	185	1	∶	∶	NOUN
iajs-2767	185	2	∀ω	∀ω	NOUN
iajs-2767	185	3	∈	∈	PROPN
iajs-2767	185	4	𝒳	𝒳	PROPN
iajs-2767	185	5	}	}	PUNCT
iajs-2767	185	6	=	=	SYM
iajs-2767	185	7	{	{	PUNCT
iajs-2767	185	8	(	(	PUNCT
iajs-2767	185	9	ω	ω	PROPN
iajs-2767	185	10	,	,	PUNCT
iajs-2767	185	11	min	min	PROPN
iajs-2767	185	12	{	{	PUNCT
iajs-2767	185	13	1,1	1,1	NUM
iajs-2767	185	14	−	−	NOUN
iajs-2767	185	15	𝓋e	𝓋e	INTJ
iajs-2767	185	16	(	(	PUNCT
iajs-2767	185	17	ω	ω	NOUN
iajs-2767	185	18	)	)	PUNCT
iajs-2767	185	19	}	}	PUNCT
iajs-2767	185	20	)	)	PUNCT
iajs-2767	186	1	∶	∶	NOUN
iajs-2767	186	2	∀ω	∀ω	NOUN
iajs-2767	186	3	∈	∈	PROPN
iajs-2767	186	4	𝒳	𝒳	PROPN
iajs-2767	186	5	}	}	PUNCT
iajs-2767	186	6	=	=	SYM
iajs-2767	186	7	{	{	PUNCT
iajs-2767	186	8	(	(	PUNCT
iajs-2767	186	9	ω	ω	NOUN
iajs-2767	186	10	,	,	PUNCT
iajs-2767	186	11	1	1	NUM
iajs-2767	186	12	−	−	NOUN
iajs-2767	186	13	𝓋e	𝓋e	INTJ
iajs-2767	186	14	(	(	PUNCT
iajs-2767	186	15	ω	ω	NOUN
iajs-2767	186	16	)	)	PUNCT
iajs-2767	186	17	)	)	PUNCT
iajs-2767	187	1	∶	∶	NOUN
iajs-2767	187	2	∀ω	∀ω	NOUN
iajs-2767	187	3	∈	∈	PROPN
iajs-2767	187	4	𝒳	𝒳	PROPN
iajs-2767	187	5	}	}	PUNCT
iajs-2767	187	6	=	=	PUNCT
iajs-2767	187	7	e𝑐	e𝑐	PROPN
iajs-2767	187	8	which	which	PRON
iajs-2767	187	9	implies	imply	VERB
iajs-2767	187	10	that	that	PRON
iajs-2767	187	11	,	,	PUNCT
iajs-2767	187	12	𝒳∗\e	𝒳∗\e	NOUN
iajs-2767	187	13	=	=	PRON
iajs-2767	187	14	e𝑐	e𝑐	PROPN
iajs-2767	187	15	∈	∈	PROPN
iajs-2767	187	16	ℋ∗.	ℋ∗.	PROPN
iajs-2767	187	17	let	let	VERB
iajs-2767	187	18	e1	e1	NOUN
iajs-2767	187	19	,	,	PUNCT
iajs-2767	187	20	e2	e2	PROPN
iajs-2767	187	21	,	,	PUNCT
iajs-2767	187	22	…	…	PUNCT
iajs-2767	187	23	,	,	PUNCT
iajs-2767	187	24	en	en	X
iajs-2767	187	25	∈	∈	PROPN
iajs-2767	187	26	ℋ∗.	ℋ∗.	PROPN
iajs-2767	187	27	then	then	ADV
iajs-2767	187	28	⋃	⋃	ADP
iajs-2767	187	29	ek	ek	PROPN
iajs-2767	187	30	n	n	INTJ
iajs-2767	187	31	k=1	k=1	PUNCT
iajs-2767	187	32	∈	∈	PROPN
iajs-2767	187	33	ℋ∗.	ℋ∗.	PROPN
iajs-2767	187	34	therefore	therefore	ADV
iajs-2767	187	35	,	,	PUNCT
iajs-2767	187	36	ℋ∗	ℋ∗	NOUN
iajs-2767	187	37	be	be	AUX
iajs-2767	187	38	a	a	DET
iajs-2767	187	39	fuzzy	fuzzy	ADJ
iajs-2767	187	40	algebra	algebra	NOUN
iajs-2767	187	41	over	over	ADP
iajs-2767	187	42	a	a	DET
iajs-2767	187	43	fuzzy	fuzzy	ADJ
iajs-2767	187	44	set	set	VERB
iajs-2767	187	45	𝒳∗.	𝒳∗.	PROPN
iajs-2767	187	46	proposition	proposition	NOUN
iajs-2767	187	47	3.12	3.12	NUM
iajs-2767	187	48	suppose	suppose	VERB
iajs-2767	187	49	𝒳	𝒳	PRON
iajs-2767	187	50	be	be	VERB
iajs-2767	187	51	a	a	DET
iajs-2767	187	52	non	non	ADJ
iajs-2767	187	53	-	-	ADJ
iajs-2767	187	54	empty	empty	ADJ
iajs-2767	187	55	set	set	NOUN
iajs-2767	187	56	and	and	CCONJ
iajs-2767	187	57	𝔗∗	𝔗∗	NUM
iajs-2767	187	58	⊆	⊆	NUM
iajs-2767	187	59	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	187	60	)	)	PUNCT
iajs-2767	187	61	.	.	PUNCT
iajs-2767	188	1	if	if	SCONJ
iajs-2767	188	2	𝒳∗	𝒳∗	PROPN
iajs-2767	188	3	∈	∈	PROPN
iajs-2767	188	4	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	188	5	)	)	PUNCT
iajs-2767	188	6	,	,	PUNCT
iajs-2767	188	7	then	then	ADV
iajs-2767	188	8	r(𝔗∗	r(𝔗∗	PROPN
iajs-2767	188	9	)	)	PUNCT
iajs-2767	188	10	=	=	SYM
iajs-2767	188	11	al(𝔗∗	al(𝔗∗	PROPN
iajs-2767	188	12	)	)	PUNCT
iajs-2767	188	13	.	.	PUNCT
iajs-2767	189	1	proof	proof	NOUN
iajs-2767	189	2	the	the	DET
iajs-2767	189	3	proof	proof	NOUN
iajs-2767	189	4	is	be	AUX
iajs-2767	189	5	followed	follow	VERB
iajs-2767	189	6	by	by	ADP
iajs-2767	189	7	theorem	theorem	ADJ
iajs-2767	189	8	3.2	3.2	NUM
iajs-2767	189	9	.	.	PUNCT
iajs-2767	190	1	proposition	proposition	NOUN
iajs-2767	190	2	3.13	3.13	NUM
iajs-2767	190	3	every	every	DET
iajs-2767	190	4	fuzzy	fuzzy	ADJ
iajs-2767	190	5	σ	σ	PROPN
iajs-2767	190	6	–	–	PUNCT
iajs-2767	190	7	algebra	algebra	NOUN
iajs-2767	190	8	over	over	ADP
iajs-2767	190	9	a	a	DET
iajs-2767	190	10	fuzzy	fuzzy	ADJ
iajs-2767	190	11	set	set	NOUN
iajs-2767	190	12	𝒳∗	𝒳∗	NOUN
iajs-2767	190	13	is	be	AUX
iajs-2767	190	14	a	a	DET
iajs-2767	190	15	fuzzy	fuzzy	ADJ
iajs-2767	190	16	ring	ring	NOUN
iajs-2767	190	17	over	over	ADP
iajs-2767	190	18	a	a	DET
iajs-2767	190	19	fuzzy	fuzzy	ADJ
iajs-2767	190	20	set	set	NOUN
iajs-2767	190	21	𝒳∗.	𝒳∗.	PROPN
iajs-2767	190	22	4	4	NUM
iajs-2767	190	23	.	.	PUNCT
iajs-2767	190	24	conclusions	conclusion	NOUN
iajs-2767	190	25	we	we	PRON
iajs-2767	190	26	will	will	AUX
iajs-2767	190	27	try	try	VERB
iajs-2767	190	28	to	to	PART
iajs-2767	190	29	generalize	generalize	VERB
iajs-2767	190	30	the	the	DET
iajs-2767	190	31	concept	concept	NOUN
iajs-2767	190	32	of	of	ADP
iajs-2767	190	33	fuzzy	fuzzy	ADJ
iajs-2767	190	34	σ	σ	PROPN
iajs-2767	190	35	–	–	PUNCT
iajs-2767	190	36	ring	ring	NOUN
iajs-2767	190	37	to	to	ADP
iajs-2767	190	38	some	some	DET
iajs-2767	190	39	other	other	ADJ
iajs-2767	190	40	concepts	concept	NOUN
iajs-2767	190	41	in	in	ADP
iajs-2767	190	42	future	future	ADJ
iajs-2767	190	43	works	work	NOUN
iajs-2767	190	44	.	.	PUNCT
iajs-2767	191	1	we	we	PRON
iajs-2767	191	2	define	define	VERB
iajs-2767	191	3	the	the	DET
iajs-2767	191	4	concept	concept	NOUN
iajs-2767	191	5	of	of	ADP
iajs-2767	191	6	measure	measure	NOUN
iajs-2767	191	7	on	on	ADP
iajs-2767	191	8	fuzzy	fuzzy	ADJ
iajs-2767	191	9	σ	σ	PROPN
iajs-2767	191	10	–	–	PUNCT
iajs-2767	191	11	ring	ring	NOUN
iajs-2767	191	12	and	and	CCONJ
iajs-2767	191	13	discuss	discuss	VERB
iajs-2767	191	14	many	many	ADJ
iajs-2767	191	15	properties	property	NOUN
iajs-2767	191	16	of	of	ADP
iajs-2767	191	17	this	this	DET
iajs-2767	191	18	concept	concept	NOUN
iajs-2767	191	19	.	.	PUNCT
iajs-2767	192	1	in	in	ADP
iajs-2767	192	2	this	this	DET
iajs-2767	192	3	study	study	NOUN
iajs-2767	192	4	,	,	PUNCT
iajs-2767	192	5	the	the	DET
iajs-2767	192	6	concepts	concept	NOUN
iajs-2767	192	7	of	of	ADP
iajs-2767	192	8	fuzzy	fuzzy	ADJ
iajs-2767	192	9	σ	σ	PROPN
iajs-2767	192	10	–	–	PUNCT
iajs-2767	192	11	ring	ring	NOUN
iajs-2767	192	12	and	and	CCONJ
iajs-2767	192	13	fuzzy	fuzzy	ADJ
iajs-2767	192	14	ring	ring	NOUN
iajs-2767	192	15	over	over	ADP
iajs-2767	192	16	a	a	DET
iajs-2767	192	17	fuzzy	fuzzy	ADJ
iajs-2767	192	18	set	set	NOUN
iajs-2767	192	19	𝒳∗	𝒳∗	PRON
iajs-2767	192	20	weakly	weakly	ADJ
iajs-2767	192	21	are	be	AUX
iajs-2767	192	22	introduced	introduce	VERB
iajs-2767	192	23	as	as	ADP
iajs-2767	192	24	a	a	DET
iajs-2767	192	25	generalization	generalization	NOUN
iajs-2767	192	26	of	of	ADP
iajs-2767	192	27	fuzzy	fuzzy	ADJ
iajs-2767	192	28	σ	σ	PROPN
iajs-2767	192	29	–	–	PUNCT
iajs-2767	192	30	algebra	algebra	NOUN
iajs-2767	192	31	and	and	CCONJ
iajs-2767	192	32	fuzzy	fuzzy	ADJ
iajs-2767	192	33	algebra	algebra	NOUN
iajs-2767	192	34	over	over	ADP
iajs-2767	192	35	the	the	DET
iajs-2767	192	36	same	same	ADJ
iajs-2767	192	37	fuzzy	fuzzy	NOUN
iajs-2767	192	38	set	set	VERB
iajs-2767	192	39	𝒳∗.	𝒳∗.	PROPN
iajs-2767	192	40	furthermore	furthermore	ADV
iajs-2767	192	41	,	,	PUNCT
iajs-2767	192	42	some	some	DET
iajs-2767	192	43	properties	property	NOUN
iajs-2767	192	44	of	of	ADP
iajs-2767	192	45	these	these	DET
iajs-2767	192	46	concepts	concept	NOUN
iajs-2767	192	47	are	be	AUX
iajs-2767	192	48	investigated	investigate	VERB
iajs-2767	192	49	such	such	ADJ
iajs-2767	192	50	as	as	ADP
iajs-2767	192	51	:	:	PUNCT
iajs-2767	192	52	1	1	NUM
iajs-2767	192	53	.	.	X
iajs-2767	192	54	every	every	DET
iajs-2767	192	55	fuzzy	fuzzy	ADJ
iajs-2767	192	56	σ	σ	PROPN
iajs-2767	192	57	–	–	PUNCT
iajs-2767	192	58	algebra	algebra	NOUN
iajs-2767	192	59	over	over	ADP
iajs-2767	192	60	a	a	DET
iajs-2767	192	61	fuzzy	fuzzy	ADJ
iajs-2767	192	62	set	set	NOUN
iajs-2767	192	63	𝒳∗	𝒳∗	NOUN
iajs-2767	192	64	is	be	AUX
iajs-2767	192	65	a	a	DET
iajs-2767	192	66	fuzzy	fuzzy	ADJ
iajs-2767	192	67	σ	σ	NOUN
iajs-2767	192	68	–	–	PUNCT
iajs-2767	192	69	ring	ring	NOUN
iajs-2767	192	70	over	over	ADP
iajs-2767	192	71	fuzzy	fuzzy	ADJ
iajs-2767	192	72	set	set	VERB
iajs-2767	192	73	𝒳∗.	𝒳∗.	PROPN
iajs-2767	192	74	2.assume	2.assume	NUM
iajs-2767	192	75	𝒳	𝒳	PROPN
iajs-2767	192	76	≠	≠	PROPN
iajs-2767	192	77	∅	∅	NOUN
iajs-2767	192	78	and	and	CCONJ
iajs-2767	192	79	𝔗∗	𝔗∗	NUM
iajs-2767	192	80	⊆	⊆	NUM
iajs-2767	192	81	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	192	82	)	)	PUNCT
iajs-2767	192	83	.	.	PUNCT
iajs-2767	193	1	then	then	ADV
iajs-2767	193	2	σ𝑟(𝔗∗	σ𝑟(𝔗∗	X
iajs-2767	193	3	)	)	PUNCT
iajs-2767	193	4	is	be	AUX
iajs-2767	193	5	the	the	DET
iajs-2767	193	6	smallest	small	ADJ
iajs-2767	193	7	fuzzy	fuzzy	ADJ
iajs-2767	193	8	σ	σ	NOUN
iajs-2767	193	9	–	–	PUNCT
iajs-2767	193	10	ring	ring	NOUN
iajs-2767	193	11	over	over	ADP
iajs-2767	193	12	a	a	DET
iajs-2767	193	13	fuzzy	fuzzy	ADJ
iajs-2767	193	14	set	set	NOUN
iajs-2767	193	15	𝒳∗	𝒳∗	PRON
iajs-2767	193	16	that	that	PRON
iajs-2767	193	17	include	include	VERB
iajs-2767	193	18	𝔗∗.	𝔗∗.	PRON
iajs-2767	193	19	3.assume	3.assume	NUM
iajs-2767	193	20	𝒳	𝒳	PROPN
iajs-2767	193	21	≠	≠	PROPN
iajs-2767	193	22	∅	∅	NOUN
iajs-2767	193	23	and	and	CCONJ
iajs-2767	193	24	ℋ∗	ℋ∗	NUM
iajs-2767	193	25	⊆	⊆	NUM
iajs-2767	193	26	𝒫∗(𝒳	𝒫∗(𝒳	NOUN
iajs-2767	193	27	)	)	PUNCT
iajs-2767	193	28	such	such	ADJ
iajs-2767	193	29	that	that	SCONJ
iajs-2767	193	30	𝒳∗	𝒳∗	PROPN
iajs-2767	193	31	∈	∈	PROPN
iajs-2767	193	32	ℋ∗	ℋ∗	PROPN
iajs-2767	193	33	.	.	PUNCT
iajs-2767	194	1	then	then	ADV
iajs-2767	194	2	ℋ∗	ℋ∗	NUM
iajs-2767	194	3	is	be	AUX
iajs-2767	194	4	a	a	DET
iajs-2767	194	5	fuzzy	fuzzy	ADJ
iajs-2767	194	6	σ	σ	NOUN
iajs-2767	194	7	–	–	PUNCT
iajs-2767	194	8	algebra	algebra	NOUN
iajs-2767	194	9	over	over	ADP
iajs-2767	194	10	a	a	DET
iajs-2767	194	11	fuzzy	fuzzy	ADJ
iajs-2767	194	12	set	set	NOUN
iajs-2767	194	13	𝒳∗	𝒳∗	PUNCT
iajs-2767	195	1	if	if	SCONJ
iajs-2767	195	2	and	and	CCONJ
iajs-2767	195	3	only	only	ADV
iajs-2767	195	4	if	if	SCONJ
iajs-2767	195	5	ℋ∗	ℋ∗	NUM
iajs-2767	195	6	is	be	AUX
iajs-2767	195	7	fuzzy	fuzzy	ADJ
iajs-2767	195	8	σ	σ	NOUN
iajs-2767	195	9	–	–	PUNCT
iajs-2767	195	10	ring	ring	NOUN
iajs-2767	195	11	over	over	ADP
iajs-2767	195	12	a	a	DET
iajs-2767	195	13	fuzzy	fuzzy	ADJ
iajs-2767	195	14	set	set	VERB
iajs-2767	195	15	𝒳∗.	𝒳∗.	PROPN
iajs-2767	195	16	4.every	4.every	NUM
iajs-2767	195	17	fuzzy	fuzzy	ADJ
iajs-2767	195	18	σ	σ	PROPN
iajs-2767	195	19	–	–	PUNCT
iajs-2767	195	20	ring	ring	NOUN
iajs-2767	195	21	over	over	ADP
iajs-2767	195	22	a	a	DET
iajs-2767	195	23	fuzzy	fuzzy	ADJ
iajs-2767	195	24	set	set	NOUN
iajs-2767	195	25	𝒳∗	𝒳∗	NOUN
iajs-2767	195	26	is	be	AUX
iajs-2767	195	27	a	a	DET
iajs-2767	195	28	fuzzy	fuzzy	ADJ
iajs-2767	195	29	ring	ring	NOUN
iajs-2767	195	30	over	over	ADP
iajs-2767	195	31	a	a	DET
iajs-2767	195	32	fuzzy	fuzzy	ADJ
iajs-2767	195	33	set	set	VERB
iajs-2767	195	34	𝒳∗.	𝒳∗.	PROPN
iajs-2767	195	35	5.every	5.every	NUM
iajs-2767	195	36	fuzzy	fuzzy	ADJ
iajs-2767	195	37	algebra	algebra	NOUN
iajs-2767	195	38	over	over	ADP
iajs-2767	195	39	a	a	DET
iajs-2767	195	40	fuzzy	fuzzy	ADJ
iajs-2767	195	41	set	set	NOUN
iajs-2767	195	42	𝒳∗	𝒳∗	NOUN
iajs-2767	195	43	is	be	AUX
iajs-2767	195	44	a	a	DET
iajs-2767	195	45	fuzzy	fuzzy	ADJ
iajs-2767	195	46	ring	ring	NOUN
iajs-2767	195	47	over	over	ADP
iajs-2767	195	48	a	a	DET
iajs-2767	195	49	fuzzy	fuzzy	ADJ
iajs-2767	195	50	set	set	VERB
iajs-2767	195	51	𝒳∗.	𝒳∗.	PROPN
iajs-2767	195	52	references	reference	NOUN
iajs-2767	195	53	1.ahmed	1.ahmed	NUM
iajs-2767	195	54	,	,	PUNCT
iajs-2767	195	55	i.s	i.s	PROPN
iajs-2767	195	56	.	.	PROPN
iajs-2767	195	57	;	;	PUNCT
iajs-2767	195	58	ebrahim	ebrahim	PROPN
iajs-2767	195	59	,	,	PUNCT
iajs-2767	195	60	h.h	h.h	PROPN
iajs-2767	195	61	.	.	PROPN
iajs-2767	195	62	generalizations	generalization	NOUN
iajs-2767	195	63	of	of	ADP
iajs-2767	195	64	σ	σ	NOUN
iajs-2767	195	65	-	-	PUNCT
iajs-2767	195	66	field	field	NOUN
iajs-2767	195	67	and	and	CCONJ
iajs-2767	195	68	new	new	ADJ
iajs-2767	195	69	collections	collection	NOUN
iajs-2767	195	70	of	of	ADP
iajs-2767	195	71	sets	set	NOUN
iajs-2767	195	72	noted	note	VERB
iajs-2767	195	73	by	by	ADP
iajs-2767	195	74	δfield	δfield	PROPN
iajs-2767	195	75	,	,	PUNCT
iajs-2767	195	76	aip	aip	PROPN
iajs-2767	195	77	conf	conf	PROPN
iajs-2767	195	78	proc	proc	PROPN
iajs-2767	195	79	.	.	PROPN
iajs-2767	196	1	2019	2019	NUM
iajs-2767	196	2	,	,	PUNCT
iajs-2767	196	3	2096	2096	NUM
iajs-2767	196	4	,	,	PUNCT
iajs-2767	196	5	(	(	PUNCT
iajs-2767	196	6	020019	020019	NUM
iajs-2767	196	7	-	-	SYM
iajs-2767	196	8	1	1	NUM
iajs-2767	196	9	)	)	PUNCT
iajs-2767	196	10	-(020019	-(020019	NOUN
iajs-2767	196	11	-	-	PUNCT
iajs-2767	196	12	6	6	NUM
iajs-2767	196	13	)	)	PUNCT
iajs-2767	196	14	.	.	PUNCT
iajs-2767	197	1	2.ahmed	2.ahmed	NUM
iajs-2767	197	2	,	,	PUNCT
iajs-2767	197	3	i.s	i.s	PROPN
iajs-2767	197	4	.	.	PUNCT
iajs-2767	197	5	;	;	PUNCT
iajs-2767	197	6	asaad	asaad	PROPN
iajs-2767	197	7	,	,	PUNCT
iajs-2767	197	8	s.h	s.h	PROPN
iajs-2767	197	9	.	.	PROPN
iajs-2767	197	10	;	;	PUNCT
iajs-2767	197	11	ebrahim	ebrahim	PROPN
iajs-2767	197	12	,	,	PUNCT
iajs-2767	197	13	h.h	h.h	PROPN
iajs-2767	197	14	.	.	PROPN
iajs-2767	198	1	some	some	DET
iajs-2767	198	2	new	new	ADJ
iajs-2767	198	3	properties	property	NOUN
iajs-2767	198	4	of	of	ADP
iajs-2767	198	5	an	an	DET
iajs-2767	198	6	outer	outer	ADJ
iajs-2767	198	7	measure	measure	NOUN
iajs-2767	198	8	on	on	ADP
iajs-2767	198	9	a	a	DET
iajs-2767	198	10	σ	σ	NOUN
iajs-2767	198	11	–	–	PUNCT
iajs-2767	198	12	field	field	NOUN
iajs-2767	198	13	,	,	PUNCT
iajs-2767	198	14	journal	journal	NOUN
iajs-2767	198	15	of	of	ADP
iajs-2767	198	16	interdisciplinary	interdisciplinary	ADJ
iajs-2767	198	17	mathematics	mathematic	NOUN
iajs-2767	198	18	.	.	PUNCT
iajs-2767	198	19	2021	2021	NUM
iajs-2767	198	20	,	,	PUNCT
iajs-2767	198	21	24	24	NUM
iajs-2767	198	22	(	(	PUNCT
iajs-2767	198	23	4	4	NUM
iajs-2767	198	24	)	)	PUNCT
iajs-2767	198	25	,	,	PUNCT
iajs-2767	198	26	947–952	947–952	NUM
iajs-2767	198	27	.	.	PUNCT
iajs-2767	199	1	3.endou	3.endou	NUM
iajs-2767	199	2	,	,	PUNCT
iajs-2767	199	3	n.	n.	NOUN
iajs-2767	199	4	;	;	PUNCT
iajs-2767	199	5	nakasho	nakasho	PROPN
iajs-2767	199	6	,	,	PUNCT
iajs-2767	199	7	k.	k.	PROPN
iajs-2767	199	8	;	;	PUNCT
iajs-2767	199	9	shidama	shidama	PROPN
iajs-2767	199	10	,	,	PUNCT
iajs-2767	199	11	y.	y.	PROPN
iajs-2767	199	12	σ	σ	PROPN
iajs-2767	199	13	-	-	PUNCT
iajs-2767	199	14	ring	ring	NOUN
iajs-2767	199	15	and	and	CCONJ
iajs-2767	199	16	σ	σ	NOUN
iajs-2767	199	17	-	-	PUNCT
iajs-2767	199	18	algebra	algebra	NOUN
iajs-2767	199	19	of	of	ADP
iajs-2767	199	20	sets	set	NOUN
iajs-2767	199	21	,	,	PUNCT
iajs-2767	199	22	formaliz	formaliz	ADJ
iajs-2767	199	23	.	.	PUNCT
iajs-2767	199	24	math	math	NOUN
iajs-2767	199	25	.	.	PUNCT
iajs-2767	200	1	2015	2015	NUM
iajs-2767	200	2	,	,	PUNCT
iajs-2767	200	3	23	23	NUM
iajs-2767	200	4	(	(	PUNCT
iajs-2767	200	5	1	1	NUM
iajs-2767	200	6	)	)	PUNCT
iajs-2767	200	7	,	,	PUNCT
iajs-2767	200	8	51–57	51–57	NUM
iajs-2767	200	9	.	.	PUNCT
iajs-2767	201	1	4.ahmed	4.ahmed	NUM
iajs-2767	201	2	,	,	PUNCT
iajs-2767	201	3	i.s	i.s	PROPN
iajs-2767	201	4	.	.	PROPN
iajs-2767	201	5	;	;	PUNCT
iajs-2767	201	6	ebrahim	ebrahim	PROPN
iajs-2767	201	7	,	,	PUNCT
iajs-2767	201	8	h.h	h.h	PROPN
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iajs-2767	201	12	-	-	PUNCT
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iajs-2767	201	14	and	and	CCONJ
iajs-2767	201	15	β	β	NOUN
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iajs-2767	201	18	,	,	PUNCT
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iajs-2767	202	1	conf	conf	PROPN
iajs-2767	202	2	.	.	PUNCT
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iajs-2767	202	4	.	.	PROPN
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iajs-2767	202	6	,	,	PUNCT
iajs-2767	202	7	1294	1294	NUM
iajs-2767	202	8	,	,	PUNCT
iajs-2767	202	9	1	1	NUM
iajs-2767	202	10	-	-	SYM
iajs-2767	202	11	8	8	NUM
iajs-2767	202	12	.	.	PUNCT
iajs-2767	203	1	5.ebrahim	5.ebrahim	NUM
iajs-2767	203	2	,	,	PUNCT
iajs-2767	203	3	h.h	h.h	PROPN
iajs-2767	203	4	.	.	PROPN
iajs-2767	203	5	;	;	PUNCT
iajs-2767	203	6	ahmed	ahmed	PROPN
iajs-2767	203	7	,	,	PUNCT
iajs-2767	203	8	i.s	i.s	PROPN
iajs-2767	203	9	.	.	PROPN
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iajs-2767	204	2	a	a	DET
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iajs-2767	204	4	kind	kind	NOUN
iajs-2767	204	5	of	of	ADP
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iajs-2767	204	12	–	–	PUNCT
iajs-2767	204	13	field	field	NOUN
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iajs-2767	204	16	concepts	concept	NOUN
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iajs-2767	204	18	on	on	ADP
iajs-2767	204	19	δ	δ	PROPN
iajs-2767	204	20	–	–	PUNCT
iajs-2767	204	21	field	field	NOUN
iajs-2767	204	22	,	,	PUNCT
iajs-2767	204	23	ibn	ibn	PROPN
iajs-2767	204	24	al	al	PROPN
iajs-2767	204	25	haitham	haitham	PROPN
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iajs-2767	204	28	pure	pure	ADJ
iajs-2767	204	29	and	and	CCONJ
iajs-2767	204	30	applied	applied	ADJ
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iajs-2767	204	32	.	.	PUNCT
iajs-2767	205	1	2019	2019	NUM
iajs-2767	205	2	,	,	PUNCT
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iajs-2767	205	4	(	(	PUNCT
iajs-2767	205	5	2	2	NUM
iajs-2767	205	6	)	)	PUNCT
iajs-2767	205	7	,	,	PUNCT
iajs-2767	205	8	62	62	NUM
iajs-2767	205	9	-	-	SYM
iajs-2767	205	10	70	70	NUM
iajs-2767	205	11	.	.	PUNCT
iajs-2767	206	1	ibn	ibn	PROPN
iajs-2767	206	2	al	al	PROPN
iajs-2767	206	3	-	-	PUNCT
iajs-2767	206	4	haitham	haitham	PROPN
iajs-2767	206	5	jour	jour	X
iajs-2767	206	6	.	.	PROPN
iajs-2767	206	7	for	for	ADP
iajs-2767	206	8	pure	pure	ADJ
iajs-2767	206	9	&	&	CCONJ
iajs-2767	206	10	appl	appl	PROPN
iajs-2767	206	11	.	.	PUNCT
iajs-2767	207	1	sci	sci	PROPN
iajs-2767	207	2	.	.	PROPN
iajs-2767	208	1	53	53	NUM
iajs-2767	208	2	(	(	PUNCT
iajs-2767	208	3	2)2022	2)2022	VERB
iajs-2767	208	4	46	46	NUM
iajs-2767	208	5	6.ebrahim	6.ebrahim	NUM
iajs-2767	208	6	,	,	PUNCT
iajs-2767	208	7	h.h	h.h	PROPN
iajs-2767	208	8	.	.	PROPN
iajs-2767	208	9	;	;	PUNCT
iajs-2767	208	10	rusul	rusul	PROPN
iajs-2767	208	11	,	,	PUNCT
iajs-2767	208	12	a.a	a.a	PROPN
iajs-2767	208	13	.	.	PROPN
iajs-2767	208	14	λ	λ	PROPN
iajs-2767	208	15	–	–	PUNCT
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iajs-2767	208	17	with	with	ADP
iajs-2767	208	18	some	some	PRON
iajs-2767	208	19	of	of	ADP
iajs-2767	208	20	their	their	PRON
iajs-2767	208	21	properties	property	NOUN
iajs-2767	208	22	,	,	PUNCT
iajs-2767	208	23	ibn	ibn	PROPN
iajs-2767	208	24	al	al	PROPN
iajs-2767	208	25	haitham	haitham	PROPN
iajs-2767	208	26	journal	journal	PROPN
iajs-2767	208	27	for	for	ADP
iajs-2767	208	28	pure	pure	ADJ
iajs-2767	208	29	and	and	CCONJ
iajs-2767	208	30	applied	applied	ADJ
iajs-2767	208	31	science	science	NOUN
iajs-2767	208	32	.	.	PUNCT
iajs-2767	209	1	2020	2020	NUM
iajs-2767	209	2	,	,	PUNCT
iajs-2767	209	3	33	33	NUM
iajs-2767	209	4	(	(	PUNCT
iajs-2767	209	5	2	2	NUM
iajs-2767	209	6	)	)	PUNCT
iajs-2767	209	7	,	,	PUNCT
iajs-2767	209	8	72	72	NUM
iajs-2767	209	9	-	-	SYM
iajs-2767	209	10	80	80	NUM
iajs-2767	209	11	.	.	PUNCT
iajs-2767	210	1	7.zadeh	7.zadeh	NUM
iajs-2767	210	2	,	,	PUNCT
iajs-2767	210	3	l.	l.	PROPN
iajs-2767	210	4	fuzzy	fuzzy	PROPN
iajs-2767	210	5	sets	set	NOUN
iajs-2767	210	6	,	,	PUNCT
iajs-2767	210	7	information	information	NOUN
iajs-2767	210	8	and	and	CCONJ
iajs-2767	210	9	control	control	NOUN
iajs-2767	210	10	.	.	PUNCT
iajs-2767	211	1	1965	1965	NUM
iajs-2767	211	2	,	,	PUNCT
iajs-2767	211	3	8	8	NUM
iajs-2767	211	4	,	,	PUNCT
iajs-2767	211	5	338–353	338–353	NUM
iajs-2767	211	6	.	.	PUNCT
iajs-2767	212	1	8.brown	8.brown	NUM
iajs-2767	212	2	,	,	PUNCT
iajs-2767	212	3	j.g	j.g	PROPN
iajs-2767	212	4	.	.	PROPN
iajs-2767	213	1	a	a	DET
iajs-2767	213	2	note	note	NOUN
iajs-2767	213	3	on	on	ADP
iajs-2767	213	4	fuzzy	fuzzy	ADJ
iajs-2767	213	5	sets	set	NOUN
iajs-2767	213	6	,	,	PUNCT
iajs-2767	213	7	inf	inf	PROPN
iajs-2767	213	8	.	.	PUNCT
iajs-2767	213	9	control	control	PROPN
iajs-2767	213	10	.	.	PUNCT
iajs-2767	214	1	1971	1971	NUM
iajs-2767	214	2	,	,	PUNCT
iajs-2767	214	3	18	18	NUM
iajs-2767	214	4	(	(	PUNCT
iajs-2767	214	5	1	1	NUM
iajs-2767	214	6	)	)	PUNCT
iajs-2767	214	7	,	,	PUNCT
iajs-2767	215	1	32–39	32–39	NUM
iajs-2767	215	2	.	.	PUNCT
iajs-2767	216	1	9.wang	9.wang	NUM
iajs-2767	216	2	,	,	PUNCT
iajs-2767	216	3	z.	z.	PROPN
iajs-2767	216	4	;	;	PUNCT
iajs-2767	216	5	klir	klir	VERB
iajs-2767	216	6	,	,	PUNCT
iajs-2767	216	7	g.j	g.j	PROPN
iajs-2767	216	8	.	.	PROPN
iajs-2767	216	9	fuzzy	fuzzy	ADJ
iajs-2767	216	10	measure	measure	NOUN
iajs-2767	216	11	theory	theory	NOUN
iajs-2767	216	12	;	;	PUNCT
iajs-2767	216	13	1st	1st	ADJ
iajs-2767	216	14	ed	ed	NOUN
iajs-2767	216	15	.	.	PUNCT
iajs-2767	216	16	;	;	PUNCT
iajs-2767	216	17	springer	springer	NOUN
iajs-2767	216	18	science	science	NOUN
iajs-2767	216	19	and	and	CCONJ
iajs-2767	216	20	business	business	NOUN
iajs-2767	216	21	media	medium	NOUN
iajs-2767	216	22	,	,	PUNCT
iajs-2767	216	23	llc	llc	PROPN
iajs-2767	216	24	,	,	PUNCT
iajs-2767	216	25	new	new	PROPN
iajs-2767	216	26	york	york	PROPN
iajs-2767	216	27	,	,	PUNCT
iajs-2767	216	28	1995	1995	NUM
iajs-2767	216	29	;	;	PUNCT
iajs-2767	216	30	isbn	isbn	ADJ
iajs-2767	216	31	978	978	NUM
iajs-2767	216	32	-	-	SYM
iajs-2767	216	33	1	1	NUM
iajs-2767	216	34	-	-	PUNCT
iajs-2767	216	35	4419	4419	NUM
iajs-2767	216	36	-	-	PUNCT
iajs-2767	216	37	3225	3225	NUM
iajs-2767	216	38	-	-	SYM
iajs-2767	216	39	9	9	NUM
iajs-2767	216	40	.	.	PUNCT
iajs-2767	217	1	10.ahmed	10.ahmed	NUM
iajs-2767	217	2	,	,	PUNCT
iajs-2767	217	3	i.s	i.s	PROPN
iajs-2767	217	4	.	.	PROPN
iajs-2767	217	5	;	;	PUNCT
iajs-2767	218	1	ebrahim	ebrahim	PROPN
iajs-2767	218	2	,	,	PUNCT
iajs-2767	218	3	h.h	h.h	PROPN
iajs-2767	218	4	.	.	PROPN
iajs-2767	218	5	;	;	PUNCT
iajs-2767	218	6	al	al	PROPN
iajs-2767	218	7	-	-	PUNCT
iajs-2767	218	8	fayadh	fayadh	NOUN
iajs-2767	218	9	,	,	PUNCT
iajs-2767	218	10	a.	a.	NOUN
iajs-2767	218	11	fuzzy	fuzzy	PROPN
iajs-2767	218	12	σ	σ	PROPN
iajs-2767	218	13	–	–	PUNCT
iajs-2767	218	14	algebra	algebra	NOUN
iajs-2767	218	15	and	and	CCONJ
iajs-2767	218	16	some	some	DET
iajs-2767	218	17	related	related	ADJ
iajs-2767	218	18	concepts	concept	NOUN
iajs-2767	218	19	,	,	PUNCT
iajs-2767	218	20	journal	journal	NOUN
iajs-2767	218	21	of	of	ADP
iajs-2767	218	22	physics	physics	PROPN
iajs-2767	218	23	:	:	PUNCT
iajs-2767	218	24	conference	conference	NOUN
iajs-2767	218	25	series	series	NOUN
iajs-2767	218	26	,	,	PUNCT
iajs-2767	218	27	that	that	PRON
iajs-2767	218	28	will	will	AUX
iajs-2767	218	29	come	come	VERB
iajs-2767	218	30	out	out	ADP
iajs-2767	218	31	in	in	ADP
iajs-2767	218	32	january	january	PROPN
iajs-2767	218	33	2022	2022	NUM
iajs-2767	218	34	.	.	PUNCT
iajs-2767	219	1	https://iopscience.iop.org/journal/1742-6596	https://iopscience.iop.org/journal/1742-6596	ADP