id	sid	tid	token	lemma	pos
ejpam-4561	1	1	european	european	PROPN
ejpam-4561	1	2	journal	journal	PROPN
ejpam-4561	1	3	of	of	ADP
ejpam-4561	1	4	pure	pure	ADJ
ejpam-4561	1	5	and	and	CCONJ
ejpam-4561	1	6	applied	apply	VERB
ejpam-4561	1	7	mathematics	mathematic	NOUN
ejpam-4561	1	8	vol	vol	NOUN
ejpam-4561	1	9	.	.	PROPN
ejpam-4561	2	1	15	15	NUM
ejpam-4561	2	2	,	,	PUNCT
ejpam-4561	2	3	no	no	INTJ
ejpam-4561	2	4	.	.	NOUN
ejpam-4561	2	5	4	4	NUM
ejpam-4561	2	6	,	,	PUNCT
ejpam-4561	2	7	2022	2022	NUM
ejpam-4561	2	8	,	,	PUNCT
ejpam-4561	2	9	1957	1957	NUM
ejpam-4561	2	10	-	-	SYM
ejpam-4561	2	11	1965	1965	NUM
ejpam-4561	2	12	issn	issn	PROPN
ejpam-4561	2	13	1307	1307	NUM
ejpam-4561	2	14	-	-	SYM
ejpam-4561	2	15	5543	5543	NUM
ejpam-4561	2	16	–	–	PUNCT
ejpam-4561	2	17	ejpam.com	ejpam.com	X
ejpam-4561	2	18	published	publish	VERB
ejpam-4561	2	19	by	by	ADP
ejpam-4561	2	20	new	new	PROPN
ejpam-4561	2	21	york	york	PROPN
ejpam-4561	2	22	business	business	PROPN
ejpam-4561	2	23	global	global	ADJ
ejpam-4561	2	24	fuzzification	fuzzification	NOUN
ejpam-4561	2	25	of	of	ADP
ejpam-4561	2	26	the	the	DET
ejpam-4561	2	27	dual	dual	ADJ
ejpam-4561	2	28	b	b	NOUN
ejpam-4561	2	29	-	-	PUNCT
ejpam-4561	2	30	algebra	algebra	NOUN
ejpam-4561	2	31	ma	ma	PROPN
ejpam-4561	2	32	.	.	PROPN
ejpam-4561	2	33	rizal	rizal	PROPN
ejpam-4561	2	34	v.	v.	ADP
ejpam-4561	2	35	dicen1,∗	dicen1,∗	PROPN
ejpam-4561	2	36	,	,	PUNCT
ejpam-4561	2	37	katrina	katrina	PROPN
ejpam-4561	2	38	e.	e.	PROPN
ejpam-4561	2	39	belleza	belleza	PROPN
ejpam-4561	2	40	-	-	PUNCT
ejpam-4561	2	41	fuentes1	fuentes1	PROPN
ejpam-4561	2	42	1	1	NUM
ejpam-4561	2	43	department	department	NOUN
ejpam-4561	2	44	of	of	ADP
ejpam-4561	2	45	computer	computer	NOUN
ejpam-4561	2	46	,	,	PUNCT
ejpam-4561	2	47	information	information	NOUN
ejpam-4561	2	48	sciences	science	NOUN
ejpam-4561	2	49	and	and	CCONJ
ejpam-4561	2	50	mathematics	mathematic	NOUN
ejpam-4561	2	51	,	,	PUNCT
ejpam-4561	2	52	school	school	NOUN
ejpam-4561	2	53	of	of	ADP
ejpam-4561	2	54	arts	art	NOUN
ejpam-4561	2	55	and	and	CCONJ
ejpam-4561	2	56	sciences	science	NOUN
ejpam-4561	2	57	,	,	PUNCT
ejpam-4561	2	58	university	university	NOUN
ejpam-4561	2	59	of	of	ADP
ejpam-4561	2	60	san	san	PROPN
ejpam-4561	2	61	carlos	carlos	PROPN
ejpam-4561	2	62	,	,	PUNCT
ejpam-4561	2	63	6000	6000	NUM
ejpam-4561	2	64	cebu	cebu	NOUN
ejpam-4561	2	65	city	city	NOUN
ejpam-4561	2	66	,	,	PUNCT
ejpam-4561	2	67	philippines	philippine	NOUN
ejpam-4561	2	68	abstract	abstract	ADJ
ejpam-4561	2	69	.	.	PUNCT
ejpam-4561	3	1	this	this	DET
ejpam-4561	3	2	paper	paper	NOUN
ejpam-4561	3	3	introduces	introduce	VERB
ejpam-4561	3	4	the	the	DET
ejpam-4561	3	5	fuzzification	fuzzification	NOUN
ejpam-4561	3	6	of	of	ADP
ejpam-4561	3	7	the	the	DET
ejpam-4561	3	8	dual	dual	ADJ
ejpam-4561	3	9	b	b	NOUN
ejpam-4561	3	10	-	-	PUNCT
ejpam-4561	3	11	algebra	algebra	NOUN
ejpam-4561	3	12	and	and	CCONJ
ejpam-4561	3	13	presented	present	VERB
ejpam-4561	3	14	some	some	PRON
ejpam-4561	3	15	of	of	ADP
ejpam-4561	3	16	its	its	PRON
ejpam-4561	3	17	related	related	ADJ
ejpam-4561	3	18	properties	property	NOUN
ejpam-4561	3	19	.	.	PUNCT
ejpam-4561	4	1	a	a	DET
ejpam-4561	4	2	characterization	characterization	NOUN
ejpam-4561	4	3	of	of	ADP
ejpam-4561	4	4	a	a	PRON
ejpam-4561	4	5	fuzzy	fuzzy	ADJ
ejpam-4561	4	6	(	(	PUNCT
ejpam-4561	4	7	normal	normal	ADJ
ejpam-4561	4	8	)	)	PUNCT
ejpam-4561	4	9	dual	dual	ADJ
ejpam-4561	4	10	b	b	X
ejpam-4561	4	11	-	-	PUNCT
ejpam-4561	4	12	algebra	algebra	NOUN
ejpam-4561	4	13	is	be	AUX
ejpam-4561	4	14	also	also	ADV
ejpam-4561	4	15	investigated	investigate	VERB
ejpam-4561	4	16	.	.	PUNCT
ejpam-4561	5	1	2020	2020	NUM
ejpam-4561	5	2	mathematics	mathematic	NOUN
ejpam-4561	5	3	subject	subject	NOUN
ejpam-4561	5	4	classifications	classification	NOUN
ejpam-4561	5	5	:	:	PUNCT
ejpam-4561	5	6	47l45	47l45	NUM
ejpam-4561	5	7	,	,	PUNCT
ejpam-4561	5	8	08c05	08c05	NOUN
ejpam-4561	5	9	,	,	PUNCT
ejpam-4561	5	10	08a72	08a72	NOUN
ejpam-4561	5	11	key	key	ADJ
ejpam-4561	5	12	words	word	NOUN
ejpam-4561	5	13	and	and	CCONJ
ejpam-4561	5	14	phrases	phrase	NOUN
ejpam-4561	5	15	:	:	PUNCT
ejpam-4561	5	16	dual	dual	ADJ
ejpam-4561	5	17	algebras	algebra	NOUN
ejpam-4561	5	18	,	,	PUNCT
ejpam-4561	5	19	categories	category	NOUN
ejpam-4561	5	20	of	of	ADP
ejpam-4561	5	21	algebras	algebra	NOUN
ejpam-4561	5	22	,	,	PUNCT
ejpam-4561	5	23	fuzzy	fuzzy	ADJ
ejpam-4561	5	24	algebraic	algebraic	ADJ
ejpam-4561	5	25	structures	structure	NOUN
ejpam-4561	5	26	1	1	NUM
ejpam-4561	5	27	.	.	PUNCT
ejpam-4561	5	28	introduction	introduction	NOUN
ejpam-4561	5	29	in	in	ADP
ejpam-4561	5	30	1965	1965	NUM
ejpam-4561	5	31	,	,	PUNCT
ejpam-4561	5	32	after	after	SCONJ
ejpam-4561	5	33	the	the	DET
ejpam-4561	5	34	publication	publication	NOUN
ejpam-4561	5	35	of	of	ADP
ejpam-4561	5	36	lotfi	lotfi	PROPN
ejpam-4561	5	37	zadeh	zadeh	PROPN
ejpam-4561	5	38	’s	’s	PART
ejpam-4561	5	39	article	article	NOUN
ejpam-4561	5	40	entitled	entitle	VERB
ejpam-4561	5	41	”	"	PUNCT
ejpam-4561	5	42	fuzzy	fuzzy	ADJ
ejpam-4561	5	43	sets	set	NOUN
ejpam-4561	5	44	”	"	PUNCT
ejpam-4561	5	45	,	,	PUNCT
ejpam-4561	5	46	the	the	DET
ejpam-4561	5	47	era	era	NOUN
ejpam-4561	5	48	of	of	ADP
ejpam-4561	5	49	fuzzy	fuzzy	ADJ
ejpam-4561	5	50	mathematics	mathematic	NOUN
ejpam-4561	5	51	had	have	AUX
ejpam-4561	5	52	started	start	VERB
ejpam-4561	5	53	.	.	PUNCT
ejpam-4561	6	1	fuzzy	fuzzy	ADJ
ejpam-4561	6	2	mathematics	mathematics	PROPN
ejpam-4561	6	3	is	be	AUX
ejpam-4561	6	4	the	the	DET
ejpam-4561	6	5	branch	branch	NOUN
ejpam-4561	6	6	of	of	ADP
ejpam-4561	6	7	mathematics	mathematic	NOUN
ejpam-4561	6	8	which	which	PRON
ejpam-4561	6	9	includes	include	VERB
ejpam-4561	6	10	fuzzy	fuzzy	ADJ
ejpam-4561	6	11	set	set	NOUN
ejpam-4561	6	12	theory	theory	NOUN
ejpam-4561	6	13	and	and	CCONJ
ejpam-4561	6	14	fuzzy	fuzzy	ADJ
ejpam-4561	6	15	logic	logic	NOUN
ejpam-4561	6	16	that	that	PRON
ejpam-4561	6	17	deals	deal	VERB
ejpam-4561	6	18	with	with	ADP
ejpam-4561	6	19	partial	partial	ADJ
ejpam-4561	6	20	inclusion	inclusion	NOUN
ejpam-4561	6	21	of	of	ADP
ejpam-4561	6	22	elements	element	NOUN
ejpam-4561	6	23	in	in	ADP
ejpam-4561	6	24	a	a	DET
ejpam-4561	6	25	set	set	NOUN
ejpam-4561	6	26	on	on	ADP
ejpam-4561	6	27	a	a	DET
ejpam-4561	6	28	spectrum	spectrum	NOUN
ejpam-4561	6	29	,	,	PUNCT
ejpam-4561	6	30	as	as	SCONJ
ejpam-4561	6	31	opposed	oppose	VERB
ejpam-4561	6	32	to	to	ADP
ejpam-4561	6	33	simple	simple	ADJ
ejpam-4561	6	34	binary	binary	NOUN
ejpam-4561	6	35	”	"	PUNCT
ejpam-4561	6	36	yes	yes	NOUN
ejpam-4561	6	37	”	"	PUNCT
ejpam-4561	6	38	or	or	CCONJ
ejpam-4561	6	39	”	"	PUNCT
ejpam-4561	6	40	no	no	NOUN
ejpam-4561	6	41	”	"	PUNCT
ejpam-4561	6	42	(	(	PUNCT
ejpam-4561	6	43	0	0	NUM
ejpam-4561	6	44	or	or	CCONJ
ejpam-4561	6	45	1	1	NUM
ejpam-4561	6	46	)	)	PUNCT
ejpam-4561	6	47	inclusion	inclusion	NOUN
ejpam-4561	6	48	[	[	X
ejpam-4561	6	49	8	8	NUM
ejpam-4561	6	50	]	]	PUNCT
ejpam-4561	6	51	.	.	PUNCT
ejpam-4561	7	1	a	a	DET
ejpam-4561	7	2	fuzzy	fuzzy	ADJ
ejpam-4561	7	3	set	set	NOUN
ejpam-4561	7	4	is	be	AUX
ejpam-4561	7	5	a	a	DET
ejpam-4561	7	6	class	class	NOUN
ejpam-4561	7	7	of	of	ADP
ejpam-4561	7	8	objects	object	NOUN
ejpam-4561	7	9	with	with	ADP
ejpam-4561	7	10	a	a	DET
ejpam-4561	7	11	continuum	continuum	NOUN
ejpam-4561	7	12	of	of	ADP
ejpam-4561	7	13	grades	grade	NOUN
ejpam-4561	7	14	of	of	ADP
ejpam-4561	7	15	membership	membership	NOUN
ejpam-4561	7	16	[	[	X
ejpam-4561	7	17	8	8	NUM
ejpam-4561	7	18	]	]	PUNCT
ejpam-4561	7	19	.	.	PUNCT
ejpam-4561	8	1	the	the	DET
ejpam-4561	8	2	term	term	NOUN
ejpam-4561	8	3	fuzzification	fuzzification	NOUN
ejpam-4561	8	4	refers	refer	VERB
ejpam-4561	8	5	to	to	ADP
ejpam-4561	8	6	the	the	DET
ejpam-4561	8	7	process	process	NOUN
ejpam-4561	8	8	of	of	ADP
ejpam-4561	8	9	mapping	map	VERB
ejpam-4561	8	10	the	the	DET
ejpam-4561	8	11	numerical	numerical	ADJ
ejpam-4561	8	12	input	input	NOUN
ejpam-4561	8	13	of	of	ADP
ejpam-4561	8	14	a	a	DET
ejpam-4561	8	15	system	system	NOUN
ejpam-4561	8	16	to	to	ADP
ejpam-4561	8	17	fuzzy	fuzzy	ADJ
ejpam-4561	8	18	sets	set	NOUN
ejpam-4561	8	19	with	with	ADP
ejpam-4561	8	20	some	some	DET
ejpam-4561	8	21	degree	degree	NOUN
ejpam-4561	8	22	of	of	ADP
ejpam-4561	8	23	membership	membership	NOUN
ejpam-4561	8	24	.	.	PUNCT
ejpam-4561	9	1	this	this	DET
ejpam-4561	9	2	degree	degree	NOUN
ejpam-4561	9	3	of	of	ADP
ejpam-4561	9	4	membership	membership	NOUN
ejpam-4561	9	5	can	can	AUX
ejpam-4561	9	6	be	be	AUX
ejpam-4561	9	7	anywhere	anywhere	ADV
ejpam-4561	9	8	within	within	ADP
ejpam-4561	9	9	the	the	DET
ejpam-4561	9	10	closed	closed	ADJ
ejpam-4561	9	11	interval	interval	NOUN
ejpam-4561	9	12	from	from	ADP
ejpam-4561	9	13	0	0	NUM
ejpam-4561	9	14	to	to	ADP
ejpam-4561	9	15	1	1	NUM
ejpam-4561	9	16	[	[	X
ejpam-4561	9	17	4	4	NUM
ejpam-4561	9	18	]	]	PUNCT
ejpam-4561	9	19	.	.	PUNCT
ejpam-4561	10	1	fuzzy	fuzzy	ADJ
ejpam-4561	10	2	technology	technology	NOUN
ejpam-4561	10	3	is	be	AUX
ejpam-4561	10	4	very	very	ADV
ejpam-4561	10	5	productive	productive	ADJ
ejpam-4561	10	6	in	in	ADP
ejpam-4561	10	7	many	many	ADJ
ejpam-4561	10	8	applications	application	NOUN
ejpam-4561	10	9	especially	especially	ADV
ejpam-4561	10	10	in	in	ADP
ejpam-4561	10	11	the	the	DET
ejpam-4561	10	12	field	field	NOUN
ejpam-4561	10	13	of	of	ADP
ejpam-4561	10	14	computer	computer	NOUN
ejpam-4561	10	15	science	science	NOUN
ejpam-4561	10	16	and	and	CCONJ
ejpam-4561	10	17	control	control	PROPN
ejpam-4561	10	18	engineering	engineering	NOUN
ejpam-4561	10	19	,	,	PUNCT
ejpam-4561	10	20	expert	expert	NOUN
ejpam-4561	10	21	system	system	NOUN
ejpam-4561	10	22	,	,	PUNCT
ejpam-4561	10	23	management	management	NOUN
ejpam-4561	10	24	science	science	NOUN
ejpam-4561	10	25	,	,	PUNCT
ejpam-4561	10	26	robotics	robotic	NOUN
ejpam-4561	10	27	,	,	PUNCT
ejpam-4561	10	28	to	to	PART
ejpam-4561	10	29	name	name	VERB
ejpam-4561	10	30	a	a	DET
ejpam-4561	10	31	few	few	ADJ
ejpam-4561	10	32	[	[	X
ejpam-4561	10	33	7	7	NUM
ejpam-4561	10	34	]	]	PUNCT
ejpam-4561	10	35	.	.	PUNCT
ejpam-4561	11	1	in	in	ADP
ejpam-4561	11	2	1999	1999	NUM
ejpam-4561	11	3	,	,	PUNCT
ejpam-4561	11	4	y.b	y.b	PROPN
ejpam-4561	11	5	.	.	PROPN
ejpam-4561	11	6	jun	jun	PROPN
ejpam-4561	11	7	,	,	PUNCT
ejpam-4561	11	8	et	et	PROPN
ejpam-4561	11	9	.	.	PUNCT
ejpam-4561	12	1	al	al	PROPN
ejpam-4561	12	2	introduced	introduce	VERB
ejpam-4561	12	3	the	the	DET
ejpam-4561	12	4	fuzzification	fuzzification	NOUN
ejpam-4561	12	5	of	of	ADP
ejpam-4561	12	6	b	b	NOUN
ejpam-4561	12	7	-	-	PUNCT
ejpam-4561	12	8	algebras	algebras	PROPN
ejpam-4561	12	9	and	and	CCONJ
ejpam-4561	12	10	at	at	ADP
ejpam-4561	12	11	the	the	DET
ejpam-4561	12	12	same	same	ADJ
ejpam-4561	12	13	time	time	NOUN
ejpam-4561	12	14	investigated	investigate	VERB
ejpam-4561	12	15	the	the	DET
ejpam-4561	12	16	characterization	characterization	NOUN
ejpam-4561	12	17	of	of	ADP
ejpam-4561	12	18	fuzzy	fuzzy	ADJ
ejpam-4561	12	19	normal	normal	ADJ
ejpam-4561	12	20	b	b	NOUN
ejpam-4561	12	21	-	-	PUNCT
ejpam-4561	12	22	algebras	algebras	X
ejpam-4561	13	1	[	[	X
ejpam-4561	13	2	5	5	NUM
ejpam-4561	13	3	]	]	PUNCT
ejpam-4561	13	4	.	.	PUNCT
ejpam-4561	14	1	by	by	ADP
ejpam-4561	14	2	2020	2020	NUM
ejpam-4561	14	3	,	,	PUNCT
ejpam-4561	14	4	m.	m.	NOUN
ejpam-4561	14	5	ahmed	ahmed	PROPN
ejpam-4561	14	6	and	and	CCONJ
ejpam-4561	14	7	e.	e.	PROPN
ejpam-4561	14	8	ahmed	ahmed	PROPN
ejpam-4561	14	9	investigated	investigate	VERB
ejpam-4561	14	10	the	the	DET
ejpam-4561	14	11	properties	property	NOUN
ejpam-4561	14	12	of	of	ADP
ejpam-4561	14	13	fuzzy	fuzzy	ADJ
ejpam-4561	14	14	bck	bck	NOUN
ejpam-4561	14	15	-	-	PUNCT
ejpam-4561	14	16	algebras	algebras	PROPN
ejpam-4561	14	17	.	.	PUNCT
ejpam-4561	15	1	in	in	ADP
ejpam-4561	15	2	their	their	PRON
ejpam-4561	15	3	research	research	NOUN
ejpam-4561	15	4	,	,	PUNCT
ejpam-4561	15	5	they	they	PRON
ejpam-4561	15	6	applied	apply	VERB
ejpam-4561	15	7	the	the	DET
ejpam-4561	15	8	concept	concept	NOUN
ejpam-4561	15	9	of	of	ADP
ejpam-4561	15	10	fuzzy	fuzzy	ADJ
ejpam-4561	15	11	sets	set	NOUN
ejpam-4561	15	12	and	and	CCONJ
ejpam-4561	15	13	gave	give	VERB
ejpam-4561	15	14	properties	property	NOUN
ejpam-4561	15	15	to	to	ADP
ejpam-4561	15	16	some	some	DET
ejpam-4561	15	17	algebraic	algebraic	ADJ
ejpam-4561	15	18	structures	structure	NOUN
ejpam-4561	15	19	such	such	ADJ
ejpam-4561	15	20	as	as	ADP
ejpam-4561	15	21	an	an	DET
ejpam-4561	15	22	ideal	ideal	ADJ
ejpam-4561	15	23	,	,	PUNCT
ejpam-4561	15	24	semilattice	semilattice	NOUN
ejpam-4561	15	25	,	,	PUNCT
ejpam-4561	15	26	lower	low	ADJ
ejpam-4561	15	27	semilattice	semilattice	NOUN
ejpam-4561	15	28	,	,	PUNCT
ejpam-4561	15	29	lattice	lattice	NOUN
ejpam-4561	15	30	and	and	CCONJ
ejpam-4561	15	31	sub	sub	NOUN
ejpam-4561	15	32	-	-	NOUN
ejpam-4561	15	33	algebra	algebra	ADJ
ejpam-4561	15	34	[	[	X
ejpam-4561	15	35	1	1	NUM
ejpam-4561	15	36	]	]	PUNCT
ejpam-4561	15	37	.	.	PUNCT
ejpam-4561	16	1	in	in	ADP
ejpam-4561	16	2	2019	2019	NUM
ejpam-4561	16	3	,	,	PUNCT
ejpam-4561	16	4	k.	k.	PROPN
ejpam-4561	16	5	belleza	belleza	PROPN
ejpam-4561	16	6	and	and	CCONJ
ejpam-4561	16	7	j.	j.	PROPN
ejpam-4561	16	8	vilela	vilela	PROPN
ejpam-4561	16	9	,	,	PUNCT
ejpam-4561	16	10	introduced	introduce	VERB
ejpam-4561	16	11	and	and	CCONJ
ejpam-4561	16	12	gave	give	VERB
ejpam-4561	16	13	the	the	DET
ejpam-4561	16	14	characterization	characterization	NOUN
ejpam-4561	16	15	of	of	ADP
ejpam-4561	16	16	the	the	DET
ejpam-4561	16	17	notion	notion	NOUN
ejpam-4561	16	18	of	of	ADP
ejpam-4561	16	19	dual	dual	ADJ
ejpam-4561	16	20	b	b	NOUN
ejpam-4561	16	21	-	-	PUNCT
ejpam-4561	16	22	algebra	algebra	NOUN
ejpam-4561	16	23	and	and	CCONJ
ejpam-4561	16	24	its	its	PRON
ejpam-4561	16	25	initial	initial	ADJ
ejpam-4561	16	26	properties	property	NOUN
ejpam-4561	16	27	[	[	X
ejpam-4561	16	28	2	2	NUM
ejpam-4561	16	29	]	]	PUNCT
ejpam-4561	16	30	.	.	PUNCT
ejpam-4561	17	1	in	in	ADP
ejpam-4561	17	2	their	their	PRON
ejpam-4561	17	3	research	research	NOUN
ejpam-4561	17	4	,	,	PUNCT
ejpam-4561	17	5	they	they	PRON
ejpam-4561	17	6	investigated	investigate	VERB
ejpam-4561	17	7	the	the	DET
ejpam-4561	17	8	relationships	relationship	NOUN
ejpam-4561	17	9	between	between	ADP
ejpam-4561	17	10	the	the	DET
ejpam-4561	17	11	dual	dual	ADJ
ejpam-4561	17	12	b	b	NOUN
ejpam-4561	17	13	-	-	PUNCT
ejpam-4561	17	14	algebra	algebra	NOUN
ejpam-4561	17	15	and	and	CCONJ
ejpam-4561	17	16	the	the	DET
ejpam-4561	17	17	bck	bck	NOUN
ejpam-4561	17	18	-	-	PUNCT
ejpam-4561	17	19	algebra	algebra	NOUN
ejpam-4561	17	20	.	.	PUNCT
ejpam-4561	18	1	this	this	DET
ejpam-4561	18	2	paper	paper	NOUN
ejpam-4561	18	3	considers	consider	VERB
ejpam-4561	18	4	the	the	DET
ejpam-4561	18	5	fuzzification	fuzzification	NOUN
ejpam-4561	18	6	of	of	ADP
ejpam-4561	18	7	the	the	DET
ejpam-4561	18	8	dual	dual	ADJ
ejpam-4561	18	9	b	b	NOUN
ejpam-4561	18	10	-	-	PUNCT
ejpam-4561	18	11	algebra	algebra	NOUN
ejpam-4561	18	12	and	and	CCONJ
ejpam-4561	18	13	its	its	PRON
ejpam-4561	18	14	related	related	ADJ
ejpam-4561	18	15	properties	property	NOUN
ejpam-4561	18	16	.	.	PUNCT
ejpam-4561	19	1	this	this	PRON
ejpam-4561	19	2	also	also	ADV
ejpam-4561	19	3	aims	aim	VERB
ejpam-4561	19	4	to	to	PART
ejpam-4561	19	5	give	give	VERB
ejpam-4561	19	6	a	a	DET
ejpam-4561	19	7	characterization	characterization	NOUN
ejpam-4561	19	8	of	of	ADP
ejpam-4561	19	9	a	a	DET
ejpam-4561	19	10	fuzzy	fuzzy	ADJ
ejpam-4561	19	11	dual	dual	ADJ
ejpam-4561	19	12	b	b	NOUN
ejpam-4561	19	13	-	-	PUNCT
ejpam-4561	19	14	algebra	algebra	NOUN
ejpam-4561	19	15	.	.	PUNCT
ejpam-4561	20	1	∗corresponding	∗corresponde	VERB
ejpam-4561	20	2	author	author	NOUN
ejpam-4561	20	3	.	.	PUNCT
ejpam-4561	21	1	doi	doi	NOUN
ejpam-4561	21	2	:	:	PUNCT
ejpam-4561	21	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4561	https://doi.org/10.29020/nybg.ejpam.v15i4.4561	NOUN
ejpam-4561	21	4	email	email	NOUN
ejpam-4561	21	5	addresses	address	VERB
ejpam-4561	21	6	:	:	PUNCT
ejpam-4561	22	1	marizverdadicen@gmail.com	marizverdadicen@gmail.com	X
ejpam-4561	22	2	(	(	PUNCT
ejpam-4561	22	3	m.	m.	NOUN
ejpam-4561	22	4	dicen	dicen	PROPN
ejpam-4561	22	5	)	)	PUNCT
ejpam-4561	22	6	,	,	PUNCT
ejpam-4561	22	7	kebelleza@usc.edu.ph	kebelleza@usc.edu.ph	PROPN
ejpam-4561	22	8	(	(	PUNCT
ejpam-4561	22	9	k.	k.	PROPN
ejpam-4561	22	10	belleza	belleza	PROPN
ejpam-4561	22	11	-	-	PUNCT
ejpam-4561	22	12	fuentes	fuentes	PROPN
ejpam-4561	22	13	)	)	PUNCT
ejpam-4561	22	14	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4561	22	15	1957	1957	NUM
ejpam-4561	23	1	©	©	PROPN
ejpam-4561	23	2	2022	2022	NUM
ejpam-4561	23	3	ejpam	ejpam	VERB
ejpam-4561	23	4	all	all	DET
ejpam-4561	23	5	rights	right	NOUN
ejpam-4561	23	6	reserved	reserve	VERB
ejpam-4561	23	7	.	.	PUNCT
ejpam-4561	24	1	k.	k.	PROPN
ejpam-4561	24	2	belleza	belleza	PROPN
ejpam-4561	24	3	,	,	PUNCT
ejpam-4561	24	4	m.	m.	PROPN
ejpam-4561	24	5	r.	r.	PROPN
ejpam-4561	24	6	v.	v.	PROPN
ejpam-4561	24	7	dicen	dicen	PROPN
ejpam-4561	24	8	/	/	SYM
ejpam-4561	24	9	eur	eur	PROPN
ejpam-4561	24	10	.	.	PUNCT
ejpam-4561	25	1	j.	j.	PROPN
ejpam-4561	25	2	pure	pure	PROPN
ejpam-4561	25	3	appl	appl	PROPN
ejpam-4561	25	4	.	.	PROPN
ejpam-4561	25	5	math	math	PROPN
ejpam-4561	25	6	,	,	PUNCT
ejpam-4561	25	7	15	15	NUM
ejpam-4561	25	8	(	(	PUNCT
ejpam-4561	25	9	4	4	NUM
ejpam-4561	25	10	)	)	PUNCT
ejpam-4561	25	11	(	(	PUNCT
ejpam-4561	25	12	2022	2022	NUM
ejpam-4561	25	13	)	)	PUNCT
ejpam-4561	25	14	,	,	PUNCT
ejpam-4561	25	15	1957	1957	NUM
ejpam-4561	25	16	-	-	SYM
ejpam-4561	25	17	1965	1965	NUM
ejpam-4561	25	18	1958	1958	NUM
ejpam-4561	25	19	2	2	NUM
ejpam-4561	25	20	.	.	PUNCT
ejpam-4561	25	21	preliminaries	preliminary	NOUN
ejpam-4561	25	22	an	an	DET
ejpam-4561	25	23	algebra	algebra	NOUN
ejpam-4561	25	24	of	of	ADP
ejpam-4561	25	25	type	type	NOUN
ejpam-4561	25	26	(	(	PUNCT
ejpam-4561	25	27	2	2	NUM
ejpam-4561	25	28	,	,	PUNCT
ejpam-4561	25	29	0	0	NUM
ejpam-4561	25	30	)	)	PUNCT
ejpam-4561	25	31	is	be	AUX
ejpam-4561	25	32	an	an	DET
ejpam-4561	25	33	algebra	algebra	NOUN
ejpam-4561	25	34	with	with	ADP
ejpam-4561	25	35	a	a	DET
ejpam-4561	25	36	binary	binary	ADJ
ejpam-4561	25	37	operation	operation	NOUN
ejpam-4561	25	38	and	and	CCONJ
ejpam-4561	25	39	a	a	DET
ejpam-4561	25	40	constant	constant	ADJ
ejpam-4561	25	41	element	element	NOUN
ejpam-4561	25	42	.	.	PUNCT
ejpam-4561	26	1	definition	definition	NOUN
ejpam-4561	26	2	1	1	NUM
ejpam-4561	26	3	.	.	PUNCT
ejpam-4561	27	1	[	[	X
ejpam-4561	27	2	3	3	X
ejpam-4561	27	3	]	]	PUNCT
ejpam-4561	27	4	a	a	DET
ejpam-4561	27	5	binary	binary	ADJ
ejpam-4561	27	6	operation	operation	NOUN
ejpam-4561	27	7	“	"	PUNCT
ejpam-4561	27	8	∗	∗	NOUN
ejpam-4561	27	9	”	"	PUNCT
ejpam-4561	27	10	on	on	ADP
ejpam-4561	27	11	a	a	DET
ejpam-4561	27	12	set	set	NOUN
ejpam-4561	27	13	s	s	PART
ejpam-4561	27	14	is	be	AUX
ejpam-4561	27	15	a	a	DET
ejpam-4561	27	16	function	function	NOUN
ejpam-4561	27	17	mapping	map	VERB
ejpam-4561	27	18	s×s→s	s×s→s	NOUN
ejpam-4561	27	19	.	.	PROPN
ejpam-4561	27	20	for	for	ADP
ejpam-4561	27	21	each	each	DET
ejpam-4561	27	22	element	element	NOUN
ejpam-4561	27	23	(	(	PUNCT
ejpam-4561	27	24	a	a	PRON
ejpam-4561	27	25	,	,	PUNCT
ejpam-4561	27	26	b	b	NOUN
ejpam-4561	27	27	)	)	PUNCT
ejpam-4561	27	28	∈	∈	PROPN
ejpam-4561	27	29	s	s	PART
ejpam-4561	27	30	×	×	NOUN
ejpam-4561	27	31	s	s	PART
ejpam-4561	27	32	,	,	PUNCT
ejpam-4561	27	33	we	we	PRON
ejpam-4561	27	34	will	will	AUX
ejpam-4561	27	35	denote	denote	VERB
ejpam-4561	27	36	∗((a	∗((a	PROPN
ejpam-4561	27	37	,	,	PUNCT
ejpam-4561	27	38	b	b	NOUN
ejpam-4561	27	39	)	)	PUNCT
ejpam-4561	27	40	)	)	PUNCT
ejpam-4561	28	1	∈	∈	PROPN
ejpam-4561	28	2	s	s	PART
ejpam-4561	28	3	by	by	ADP
ejpam-4561	28	4	a∗b	a∗b	PROPN
ejpam-4561	28	5	.	.	PUNCT
ejpam-4561	29	1	that	that	PRON
ejpam-4561	29	2	is	be	AUX
ejpam-4561	29	3	,	,	PUNCT
ejpam-4561	29	4	∗	∗	PROPN
ejpam-4561	29	5	is	be	AUX
ejpam-4561	29	6	a	a	DET
ejpam-4561	29	7	binary	binary	ADJ
ejpam-4561	29	8	operation	operation	NOUN
ejpam-4561	29	9	on	on	ADP
ejpam-4561	29	10	s	s	PRON
ejpam-4561	29	11	if	if	SCONJ
ejpam-4561	29	12	and	and	CCONJ
ejpam-4561	29	13	only	only	ADV
ejpam-4561	29	14	if	if	SCONJ
ejpam-4561	29	15	for	for	ADP
ejpam-4561	29	16	all	all	DET
ejpam-4561	29	17	a	a	PRON
ejpam-4561	29	18	,	,	PUNCT
ejpam-4561	29	19	b	b	X
ejpam-4561	29	20	∈	∈	PROPN
ejpam-4561	29	21	s	s	PROPN
ejpam-4561	29	22	,	,	PUNCT
ejpam-4561	29	23	a	a	DET
ejpam-4561	29	24	∗	∗	NOUN
ejpam-4561	29	25	b	b	NOUN
ejpam-4561	29	26	∈	∈	PROPN
ejpam-4561	29	27	s.	s.	PROPN
ejpam-4561	29	28	definition	definition	NOUN
ejpam-4561	29	29	2	2	NUM
ejpam-4561	29	30	.	.	PUNCT
ejpam-4561	30	1	[	[	X
ejpam-4561	30	2	2	2	X
ejpam-4561	30	3	]	]	PUNCT
ejpam-4561	30	4	a	a	DET
ejpam-4561	30	5	dual	dual	ADJ
ejpam-4561	30	6	b	b	NOUN
ejpam-4561	30	7	-	-	PUNCT
ejpam-4561	30	8	algebra	algebra	NOUN
ejpam-4561	30	9	x	x	PUNCT
ejpam-4561	30	10	is	be	AUX
ejpam-4561	30	11	a	a	DET
ejpam-4561	30	12	triple	triple	ADJ
ejpam-4561	30	13	(	(	PUNCT
ejpam-4561	30	14	x	x	NOUN
ejpam-4561	30	15	,	,	PUNCT
ejpam-4561	30	16	◦	◦	NOUN
ejpam-4561	30	17	,	,	PUNCT
ejpam-4561	30	18	1	1	NUM
ejpam-4561	30	19	)	)	PUNCT
ejpam-4561	30	20	where	where	SCONJ
ejpam-4561	30	21	x	x	PRON
ejpam-4561	30	22	is	be	AUX
ejpam-4561	30	23	a	a	DET
ejpam-4561	30	24	non	non	ADJ
ejpam-4561	30	25	-	-	ADJ
ejpam-4561	30	26	empty	empty	ADJ
ejpam-4561	30	27	set	set	NOUN
ejpam-4561	30	28	with	with	ADP
ejpam-4561	30	29	a	a	DET
ejpam-4561	30	30	binary	binary	ADJ
ejpam-4561	30	31	operation	operation	NOUN
ejpam-4561	30	32	“	"	PUNCT
ejpam-4561	30	33	◦	◦	NOUN
ejpam-4561	30	34	”	"	PUNCT
ejpam-4561	30	35	and	and	CCONJ
ejpam-4561	30	36	a	a	DET
ejpam-4561	30	37	constant	constant	ADJ
ejpam-4561	30	38	“	"	PUNCT
ejpam-4561	30	39	1	1	NUM
ejpam-4561	30	40	”	"	PUNCT
ejpam-4561	30	41	satisfying	satisfy	VERB
ejpam-4561	30	42	the	the	DET
ejpam-4561	30	43	following	following	ADJ
ejpam-4561	30	44	axioms	axiom	NOUN
ejpam-4561	30	45	for	for	ADP
ejpam-4561	30	46	all	all	DET
ejpam-4561	30	47	x	x	NOUN
ejpam-4561	30	48	,	,	PUNCT
ejpam-4561	30	49	y	y	PROPN
ejpam-4561	30	50	,	,	PUNCT
ejpam-4561	30	51	z	z	VERB
ejpam-4561	30	52	in	in	ADP
ejpam-4561	30	53	x	x	NOUN
ejpam-4561	30	54	:	:	PUNCT
ejpam-4561	30	55	(	(	PUNCT
ejpam-4561	30	56	db1	db1	NOUN
ejpam-4561	30	57	)	)	PUNCT
ejpam-4561	30	58	x	x	SYM
ejpam-4561	31	1	◦	◦	NOUN
ejpam-4561	31	2	x	x	SYM
ejpam-4561	31	3	=	=	NOUN
ejpam-4561	31	4	1	1	NUM
ejpam-4561	31	5	;	;	PUNCT
ejpam-4561	31	6	(	(	PUNCT
ejpam-4561	31	7	db2	db2	NOUN
ejpam-4561	31	8	)	)	PUNCT
ejpam-4561	31	9	1	1	NUM
ejpam-4561	31	10	◦	◦	NOUN
ejpam-4561	31	11	x	x	SYM
ejpam-4561	31	12	=	=	SYM
ejpam-4561	31	13	x	x	X
ejpam-4561	31	14	;	;	PUNCT
ejpam-4561	31	15	(	(	PUNCT
ejpam-4561	31	16	db3	db3	PROPN
ejpam-4561	31	17	)	)	PUNCT
ejpam-4561	31	18	x	x	SYM
ejpam-4561	32	1	◦	◦	NOUN
ejpam-4561	32	2	(	(	PUNCT
ejpam-4561	32	3	y	y	PROPN
ejpam-4561	32	4	◦	◦	PROPN
ejpam-4561	32	5	z	z	PROPN
ejpam-4561	32	6	)	)	PUNCT
ejpam-4561	32	7	=	=	SYM
ejpam-4561	32	8	(	(	PUNCT
ejpam-4561	32	9	(	(	PUNCT
ejpam-4561	32	10	y	y	NOUN
ejpam-4561	32	11	◦	◦	NOUN
ejpam-4561	32	12	1	1	NUM
ejpam-4561	32	13	)	)	PUNCT
ejpam-4561	32	14	◦	◦	NOUN
ejpam-4561	32	15	x	x	SYM
ejpam-4561	32	16	)	)	PUNCT
ejpam-4561	32	17	◦	◦	NOUN
ejpam-4561	32	18	z.	z.	PROPN
ejpam-4561	33	1	a	a	DET
ejpam-4561	33	2	non	non	ADJ
ejpam-4561	33	3	-	-	ADJ
ejpam-4561	33	4	empty	empty	ADJ
ejpam-4561	33	5	subset	subset	NOUN
ejpam-4561	33	6	n	n	PROPN
ejpam-4561	33	7	of	of	ADP
ejpam-4561	33	8	a	a	DET
ejpam-4561	33	9	dual	dual	ADJ
ejpam-4561	33	10	b	b	NOUN
ejpam-4561	33	11	-	-	PUNCT
ejpam-4561	33	12	algebra	algebra	NOUN
ejpam-4561	33	13	x	x	PUNCT
ejpam-4561	33	14	is	be	AUX
ejpam-4561	33	15	called	call	VERB
ejpam-4561	33	16	a	a	DET
ejpam-4561	33	17	dual	dual	ADJ
ejpam-4561	33	18	b	b	NOUN
ejpam-4561	33	19	-	-	PUNCT
ejpam-4561	33	20	subalgebra	subalgebra	NOUN
ejpam-4561	33	21	of	of	ADP
ejpam-4561	33	22	x	x	PART
ejpam-4561	33	23	if	if	SCONJ
ejpam-4561	33	24	x	x	PART
ejpam-4561	33	25	◦	◦	VERB
ejpam-4561	33	26	y	y	PROPN
ejpam-4561	33	27	∈	∈	PROPN
ejpam-4561	33	28	n	n	PROPN
ejpam-4561	33	29	for	for	ADP
ejpam-4561	33	30	any	any	DET
ejpam-4561	33	31	x	x	NOUN
ejpam-4561	33	32	,	,	PUNCT
ejpam-4561	33	33	y	y	PROPN
ejpam-4561	33	34	∈	∈	PROPN
ejpam-4561	33	35	n	n	ADV
ejpam-4561	33	36	.	.	PUNCT
ejpam-4561	34	1	a	a	DET
ejpam-4561	34	2	non	non	ADJ
ejpam-4561	34	3	-	-	ADJ
ejpam-4561	34	4	empty	empty	ADJ
ejpam-4561	34	5	subset	subset	NOUN
ejpam-4561	34	6	n	n	PROPN
ejpam-4561	34	7	of	of	ADP
ejpam-4561	34	8	a	a	DET
ejpam-4561	34	9	dual	dual	ADJ
ejpam-4561	34	10	b	b	NOUN
ejpam-4561	34	11	-	-	PUNCT
ejpam-4561	34	12	algebra	algebra	NOUN
ejpam-4561	34	13	x	x	PUNCT
ejpam-4561	34	14	is	be	AUX
ejpam-4561	34	15	said	say	VERB
ejpam-4561	34	16	to	to	PART
ejpam-4561	34	17	be	be	AUX
ejpam-4561	34	18	normal	normal	ADJ
ejpam-4561	34	19	if	if	SCONJ
ejpam-4561	34	20	(	(	PUNCT
ejpam-4561	34	21	x	x	PUNCT
ejpam-4561	34	22	◦	◦	NOUN
ejpam-4561	34	23	f	f	NOUN
ejpam-4561	34	24	)	)	PUNCT
ejpam-4561	34	25	◦	◦	NOUN
ejpam-4561	34	26	(	(	PUNCT
ejpam-4561	34	27	y	y	NOUN
ejpam-4561	34	28	◦	◦	VERB
ejpam-4561	34	29	g	g	NOUN
ejpam-4561	34	30	)	)	PUNCT
ejpam-4561	34	31	∈	∈	PROPN
ejpam-4561	35	1	n	n	CCONJ
ejpam-4561	35	2	whenever	whenever	SCONJ
ejpam-4561	35	3	x	x	VERB
ejpam-4561	35	4	◦	◦	VERB
ejpam-4561	35	5	y	y	PROPN
ejpam-4561	35	6	∈	∈	PROPN
ejpam-4561	35	7	n	n	NOUN
ejpam-4561	35	8	and	and	CCONJ
ejpam-4561	35	9	f	f	PROPN
ejpam-4561	35	10	◦	◦	NOUN
ejpam-4561	35	11	g	g	PROPN
ejpam-4561	35	12	∈	∈	PROPN
ejpam-4561	35	13	n	n	X
ejpam-4561	35	14	.	.	PUNCT
ejpam-4561	36	1	note	note	VERB
ejpam-4561	36	2	that	that	SCONJ
ejpam-4561	36	3	any	any	DET
ejpam-4561	36	4	normal	normal	ADJ
ejpam-4561	36	5	subset	subset	NOUN
ejpam-4561	36	6	n	n	PROPN
ejpam-4561	36	7	of	of	ADP
ejpam-4561	36	8	a	a	DET
ejpam-4561	36	9	dual	dual	ADJ
ejpam-4561	36	10	b	b	NOUN
ejpam-4561	36	11	algebra	algebra	NOUN
ejpam-4561	36	12	x	x	PUNCT
ejpam-4561	36	13	is	be	AUX
ejpam-4561	36	14	a	a	DET
ejpam-4561	36	15	b	b	NOUN
ejpam-4561	36	16	-	-	PUNCT
ejpam-4561	36	17	subalgebra	subalgebra	NOUN
ejpam-4561	36	18	of	of	ADP
ejpam-4561	36	19	x	x	PRON
ejpam-4561	36	20	,	,	PUNCT
ejpam-4561	36	21	the	the	DET
ejpam-4561	36	22	converse	converse	NOUN
ejpam-4561	36	23	need	need	AUX
ejpam-4561	36	24	not	not	PART
ejpam-4561	36	25	be	be	AUX
ejpam-4561	36	26	true	true	ADJ
ejpam-4561	36	27	[	[	X
ejpam-4561	36	28	6	6	NUM
ejpam-4561	36	29	]	]	PUNCT
ejpam-4561	36	30	.	.	PUNCT
ejpam-4561	37	1	a	a	DET
ejpam-4561	37	2	non	non	ADJ
ejpam-4561	37	3	-	-	ADJ
ejpam-4561	37	4	empty	empty	ADJ
ejpam-4561	37	5	subset	subset	NOUN
ejpam-4561	37	6	n	n	PROPN
ejpam-4561	37	7	of	of	ADP
ejpam-4561	37	8	a	a	DET
ejpam-4561	37	9	dual	dual	ADJ
ejpam-4561	37	10	b	b	NOUN
ejpam-4561	37	11	-	-	PUNCT
ejpam-4561	37	12	algebra	algebra	NOUN
ejpam-4561	37	13	x	x	PUNCT
ejpam-4561	37	14	is	be	AUX
ejpam-4561	37	15	called	call	VERB
ejpam-4561	37	16	a	a	DET
ejpam-4561	37	17	normal	normal	ADJ
ejpam-4561	37	18	b	b	NOUN
ejpam-4561	37	19	-	-	PUNCT
ejpam-4561	37	20	subalgebra	subalgebra	NOUN
ejpam-4561	37	21	of	of	ADP
ejpam-4561	37	22	x	x	PRON
ejpam-4561	37	23	if	if	SCONJ
ejpam-4561	37	24	it	it	PRON
ejpam-4561	37	25	is	be	AUX
ejpam-4561	37	26	both	both	CCONJ
ejpam-4561	37	27	a	a	DET
ejpam-4561	37	28	dual	dual	ADJ
ejpam-4561	37	29	b	b	NOUN
ejpam-4561	37	30	-	-	PUNCT
ejpam-4561	37	31	subalgebra	subalgebra	NOUN
ejpam-4561	37	32	and	and	CCONJ
ejpam-4561	37	33	normal	normal	ADJ
ejpam-4561	37	34	.	.	PUNCT
ejpam-4561	38	1	lemma	lemma	PROPN
ejpam-4561	38	2	1	1	NUM
ejpam-4561	38	3	.	.	PUNCT
ejpam-4561	39	1	[	[	X
ejpam-4561	39	2	2	2	X
ejpam-4561	39	3	]	]	PUNCT
ejpam-4561	39	4	let	let	VERB
ejpam-4561	39	5	x	x	PRON
ejpam-4561	39	6	be	be	AUX
ejpam-4561	39	7	a	a	DET
ejpam-4561	39	8	dual	dual	ADJ
ejpam-4561	39	9	b	b	NOUN
ejpam-4561	39	10	-	-	PUNCT
ejpam-4561	39	11	algebra	algebra	NOUN
ejpam-4561	39	12	.	.	PUNCT
ejpam-4561	40	1	then	then	ADV
ejpam-4561	40	2	for	for	ADP
ejpam-4561	40	3	any	any	DET
ejpam-4561	40	4	x	x	NOUN
ejpam-4561	40	5	,	,	PUNCT
ejpam-4561	40	6	y	y	PROPN
ejpam-4561	40	7	∈	∈	PROPN
ejpam-4561	40	8	x	x	X
ejpam-4561	40	9	,	,	PUNCT
ejpam-4561	40	10	we	we	PRON
ejpam-4561	40	11	have	have	VERB
ejpam-4561	40	12	:	:	PUNCT
ejpam-4561	40	13	(	(	PUNCT
ejpam-4561	40	14	i	i	NOUN
ejpam-4561	40	15	)	)	PUNCT
ejpam-4561	40	16	x	x	VERB
ejpam-4561	41	1	◦	◦	NOUN
ejpam-4561	41	2	y	y	NOUN
ejpam-4561	41	3	=	=	SYM
ejpam-4561	41	4	1	1	NUM
ejpam-4561	41	5	;	;	PUNCT
ejpam-4561	41	6	(	(	PUNCT
ejpam-4561	41	7	ii	ii	NOUN
ejpam-4561	41	8	)	)	PUNCT
ejpam-4561	41	9	(	(	PUNCT
ejpam-4561	41	10	x	x	X
ejpam-4561	41	11	◦	◦	NOUN
ejpam-4561	41	12	1	1	NUM
ejpam-4561	41	13	)	)	PUNCT
ejpam-4561	41	14	◦	◦	NOUN
ejpam-4561	41	15	1	1	NUM
ejpam-4561	41	16	=	=	SYM
ejpam-4561	41	17	x.	x.	NOUN
ejpam-4561	41	18	3	3	X
ejpam-4561	41	19	.	.	PUNCT
ejpam-4561	41	20	fuzzy	fuzzy	ADJ
ejpam-4561	41	21	dual	dual	ADJ
ejpam-4561	41	22	b	b	NOUN
ejpam-4561	41	23	-	-	PUNCT
ejpam-4561	41	24	algebra	algebra	NOUN
ejpam-4561	41	25	in	in	ADP
ejpam-4561	41	26	what	what	PRON
ejpam-4561	41	27	follows	follow	VERB
ejpam-4561	41	28	,	,	PUNCT
ejpam-4561	41	29	let	let	VERB
ejpam-4561	41	30	x	x	PRON
ejpam-4561	41	31	denotes	denote	VERB
ejpam-4561	41	32	a	a	DET
ejpam-4561	41	33	dual	dual	ADJ
ejpam-4561	41	34	b	b	NOUN
ejpam-4561	41	35	-	-	PUNCT
ejpam-4561	41	36	algebra	algebra	NOUN
ejpam-4561	41	37	unless	unless	SCONJ
ejpam-4561	41	38	otherwise	otherwise	ADV
ejpam-4561	41	39	specified	specify	VERB
ejpam-4561	41	40	.	.	PUNCT
ejpam-4561	42	1	definition	definition	NOUN
ejpam-4561	42	2	3	3	NUM
ejpam-4561	42	3	.	.	PUNCT
ejpam-4561	43	1	a	a	DET
ejpam-4561	43	2	fuzzy	fuzzy	ADJ
ejpam-4561	43	3	set	set	NOUN
ejpam-4561	43	4	µ	µ	X
ejpam-4561	43	5	inx	inx	PROPN
ejpam-4561	43	6	is	be	AUX
ejpam-4561	43	7	called	call	VERB
ejpam-4561	43	8	a	a	DET
ejpam-4561	43	9	fuzzy	fuzzy	ADJ
ejpam-4561	43	10	dual	dual	ADJ
ejpam-4561	43	11	b	b	NOUN
ejpam-4561	43	12	-	-	PUNCT
ejpam-4561	43	13	algebra	algebra	NOUN
ejpam-4561	43	14	if	if	SCONJ
ejpam-4561	43	15	it	it	PRON
ejpam-4561	43	16	satisfies	satisfy	VERB
ejpam-4561	43	17	the	the	DET
ejpam-4561	43	18	following	follow	VERB
ejpam-4561	43	19	inequality	inequality	NOUN
ejpam-4561	43	20	for	for	ADP
ejpam-4561	43	21	all	all	DET
ejpam-4561	43	22	x	x	NOUN
ejpam-4561	43	23	,	,	PUNCT
ejpam-4561	43	24	y	y	PROPN
ejpam-4561	43	25	∈	∈	PROPN
ejpam-4561	44	1	x	x	X
ejpam-4561	44	2	:	:	PUNCT
ejpam-4561	44	3	µ(x	µ(x	PROPN
ejpam-4561	44	4	◦	◦	NOUN
ejpam-4561	44	5	y	y	NOUN
ejpam-4561	44	6	)	)	PUNCT
ejpam-4561	44	7	≥	≥	NOUN
ejpam-4561	44	8	min{µ(x	min{µ(x	NOUN
ejpam-4561	44	9	)	)	PUNCT
ejpam-4561	44	10	,	,	PUNCT
ejpam-4561	44	11	µ(y	µ(y	PROPN
ejpam-4561	44	12	)	)	PUNCT
ejpam-4561	44	13	}	}	PUNCT
ejpam-4561	44	14	.	.	PUNCT
ejpam-4561	45	1	example	example	NOUN
ejpam-4561	46	1	1	1	NUM
ejpam-4561	46	2	.	.	PUNCT
ejpam-4561	46	3	let	let	VERB
ejpam-4561	46	4	x	x	PUNCT
ejpam-4561	46	5	=	=	PRON
ejpam-4561	46	6	{	{	PUNCT
ejpam-4561	46	7	1	1	NUM
ejpam-4561	46	8	,	,	PUNCT
ejpam-4561	46	9	a	a	DET
ejpam-4561	46	10	,	,	PUNCT
ejpam-4561	46	11	b	b	NOUN
ejpam-4561	46	12	,	,	PUNCT
ejpam-4561	46	13	c	c	NOUN
ejpam-4561	46	14	,	,	PUNCT
ejpam-4561	46	15	d	d	NOUN
ejpam-4561	46	16	,	,	PUNCT
ejpam-4561	46	17	e	e	AUX
ejpam-4561	46	18	}	}	PUNCT
ejpam-4561	46	19	be	be	AUX
ejpam-4561	46	20	the	the	DET
ejpam-4561	46	21	set	set	NOUN
ejpam-4561	46	22	with	with	ADP
ejpam-4561	46	23	the	the	DET
ejpam-4561	46	24	following	follow	VERB
ejpam-4561	46	25	table	table	NOUN
ejpam-4561	46	26	:	:	PUNCT
ejpam-4561	46	27	◦	◦	NOUN
ejpam-4561	46	28	1	1	NUM
ejpam-4561	46	29	a	a	DET
ejpam-4561	46	30	b	b	NOUN
ejpam-4561	46	31	c	c	NOUN
ejpam-4561	46	32	d	d	X
ejpam-4561	46	33	e	e	PROPN
ejpam-4561	46	34	1	1	NUM
ejpam-4561	46	35	1	1	NUM
ejpam-4561	46	36	a	a	DET
ejpam-4561	46	37	b	b	NOUN
ejpam-4561	46	38	c	c	NOUN
ejpam-4561	46	39	d	d	PROPN
ejpam-4561	46	40	e	e	PROPN
ejpam-4561	46	41	a	a	PRON
ejpam-4561	46	42	b	b	PROPN
ejpam-4561	46	43	1	1	NUM
ejpam-4561	46	44	a	a	PRON
ejpam-4561	46	45	d	d	X
ejpam-4561	46	46	e	e	NOUN
ejpam-4561	46	47	c	c	NOUN
ejpam-4561	46	48	b	b	PROPN
ejpam-4561	46	49	a	a	DET
ejpam-4561	46	50	b	b	NOUN
ejpam-4561	46	51	1	1	NUM
ejpam-4561	46	52	e	e	NOUN
ejpam-4561	46	53	c	c	NOUN
ejpam-4561	46	54	d	d	NOUN
ejpam-4561	46	55	c	c	NOUN
ejpam-4561	46	56	c	c	NOUN
ejpam-4561	46	57	d	d	X
ejpam-4561	46	58	e	e	PROPN
ejpam-4561	46	59	1	1	NUM
ejpam-4561	46	60	a	a	DET
ejpam-4561	46	61	b	b	NOUN
ejpam-4561	46	62	d	d	X
ejpam-4561	46	63	d	d	PROPN
ejpam-4561	46	64	e	e	PROPN
ejpam-4561	46	65	c	c	NOUN
ejpam-4561	46	66	b	b	PROPN
ejpam-4561	46	67	1	1	NUM
ejpam-4561	46	68	a	a	DET
ejpam-4561	46	69	e	e	NOUN
ejpam-4561	46	70	e	e	NOUN
ejpam-4561	46	71	c	c	NOUN
ejpam-4561	46	72	d	d	X
ejpam-4561	46	73	a	a	PRON
ejpam-4561	46	74	b	b	NOUN
ejpam-4561	46	75	1	1	NUM
ejpam-4561	46	76	then	then	ADV
ejpam-4561	46	77	(	(	PUNCT
ejpam-4561	46	78	x	x	NOUN
ejpam-4561	46	79	,	,	PUNCT
ejpam-4561	46	80	◦	◦	NOUN
ejpam-4561	46	81	,	,	PUNCT
ejpam-4561	46	82	1	1	NUM
ejpam-4561	46	83	)	)	PUNCT
ejpam-4561	46	84	is	be	AUX
ejpam-4561	46	85	a	a	DET
ejpam-4561	46	86	dual	dual	ADJ
ejpam-4561	46	87	b	b	NOUN
ejpam-4561	46	88	-	-	PUNCT
ejpam-4561	46	89	algebra	algebra	NOUN
ejpam-4561	46	90	by	by	ADP
ejpam-4561	46	91	routine	routine	ADJ
ejpam-4561	46	92	calculations	calculation	NOUN
ejpam-4561	46	93	.	.	PUNCT
ejpam-4561	47	1	define	define	VERB
ejpam-4561	47	2	a	a	DET
ejpam-4561	47	3	fuzzy	fuzzy	ADJ
ejpam-4561	47	4	set	set	VERB
ejpam-4561	47	5	µ	µ	NOUN
ejpam-4561	47	6	:	:	PUNCT
ejpam-4561	47	7	x	x	SYM
ejpam-4561	47	8	→	→	SYM
ejpam-4561	47	9	[	[	X
ejpam-4561	47	10	0	0	NUM
ejpam-4561	47	11	,	,	PUNCT
ejpam-4561	47	12	1	1	NUM
ejpam-4561	47	13	]	]	PUNCT
ejpam-4561	47	14	by	by	ADP
ejpam-4561	47	15	µ(1)=µ(c)=	µ(1)=µ(c)=	PROPN
ejpam-4561	47	16	0.8	0.8	NUM
ejpam-4561	47	17	>	>	SYM
ejpam-4561	47	18	0.3	0.3	NUM
ejpam-4561	47	19	=	=	SYM
ejpam-4561	47	20	µ(x	µ(x	X
ejpam-4561	47	21	)	)	PUNCT
ejpam-4561	47	22	for	for	ADP
ejpam-4561	47	23	all	all	DET
ejpam-4561	47	24	x	x	SYM
ejpam-4561	47	25	∈	∈	PROPN
ejpam-4561	47	26	x\{1	x\{1	PROPN
ejpam-4561	47	27	,	,	PUNCT
ejpam-4561	47	28	c	c	NOUN
ejpam-4561	47	29	}	}	PUNCT
ejpam-4561	47	30	.	.	PUNCT
ejpam-4561	48	1	then	then	ADV
ejpam-4561	48	2	,	,	PUNCT
ejpam-4561	48	3	µ	µ	X
ejpam-4561	48	4	is	be	AUX
ejpam-4561	48	5	a	a	DET
ejpam-4561	48	6	fuzzy	fuzzy	ADJ
ejpam-4561	48	7	dual	dual	ADJ
ejpam-4561	48	8	b	b	NOUN
ejpam-4561	48	9	-	-	PUNCT
ejpam-4561	48	10	algebra	algebra	NOUN
ejpam-4561	48	11	.	.	PUNCT
ejpam-4561	49	1	k.	k.	PROPN
ejpam-4561	49	2	belleza	belleza	PROPN
ejpam-4561	49	3	,	,	PUNCT
ejpam-4561	49	4	m.	m.	PROPN
ejpam-4561	49	5	r.	r.	PROPN
ejpam-4561	49	6	v.	v.	PROPN
ejpam-4561	49	7	dicen	dicen	PROPN
ejpam-4561	49	8	/	/	SYM
ejpam-4561	49	9	eur	eur	PROPN
ejpam-4561	49	10	.	.	PUNCT
ejpam-4561	50	1	j.	j.	PROPN
ejpam-4561	50	2	pure	pure	PROPN
ejpam-4561	50	3	appl	appl	PROPN
ejpam-4561	50	4	.	.	PROPN
ejpam-4561	50	5	math	math	PROPN
ejpam-4561	50	6	,	,	PUNCT
ejpam-4561	50	7	15	15	NUM
ejpam-4561	50	8	(	(	PUNCT
ejpam-4561	50	9	4	4	NUM
ejpam-4561	50	10	)	)	PUNCT
ejpam-4561	50	11	(	(	PUNCT
ejpam-4561	50	12	2022	2022	NUM
ejpam-4561	50	13	)	)	PUNCT
ejpam-4561	50	14	,	,	PUNCT
ejpam-4561	50	15	1957	1957	NUM
ejpam-4561	50	16	-	-	SYM
ejpam-4561	50	17	1965	1965	NUM
ejpam-4561	50	18	1959	1959	NUM
ejpam-4561	50	19	proposition	proposition	NOUN
ejpam-4561	50	20	1	1	NUM
ejpam-4561	50	21	.	.	PUNCT
ejpam-4561	51	1	every	every	DET
ejpam-4561	51	2	fuzzy	fuzzy	ADJ
ejpam-4561	51	3	dual	dual	ADJ
ejpam-4561	51	4	balgebra	balgebra	NOUN
ejpam-4561	51	5	µ	µ	PRON
ejpam-4561	51	6	satisfies	satisfie	NOUN
ejpam-4561	51	7	the	the	DET
ejpam-4561	51	8	inequality	inequality	NOUN
ejpam-4561	51	9	µ(1	µ(1	PROPN
ejpam-4561	51	10	)	)	PUNCT
ejpam-4561	51	11	≥	≥	NOUN
ejpam-4561	51	12	µ(x	µ(x	VERB
ejpam-4561	51	13	)	)	PUNCT
ejpam-4561	51	14	for	for	ADP
ejpam-4561	51	15	all	all	DET
ejpam-4561	51	16	x	x	SYM
ejpam-4561	51	17	∈	∈	ADJ
ejpam-4561	51	18	x.	x.	NOUN
ejpam-4561	51	19	proof	proof	NOUN
ejpam-4561	51	20	.	.	PUNCT
ejpam-4561	52	1	note	note	VERB
ejpam-4561	52	2	that	that	SCONJ
ejpam-4561	52	3	by	by	ADP
ejpam-4561	52	4	(	(	PUNCT
ejpam-4561	52	5	db1	db1	NOUN
ejpam-4561	52	6	)	)	PUNCT
ejpam-4561	52	7	x	x	SYM
ejpam-4561	52	8	◦	◦	NOUN
ejpam-4561	52	9	x	x	SYM
ejpam-4561	52	10	=	=	SYM
ejpam-4561	52	11	1	1	NUM
ejpam-4561	52	12	for	for	ADP
ejpam-4561	52	13	all	all	DET
ejpam-4561	52	14	x	x	SYM
ejpam-4561	52	15	∈	∈	PROPN
ejpam-4561	52	16	x	x	X
ejpam-4561	52	17	,	,	PUNCT
ejpam-4561	52	18	which	which	PRON
ejpam-4561	52	19	implies	imply	VERB
ejpam-4561	52	20	that	that	SCONJ
ejpam-4561	52	21	,	,	PUNCT
ejpam-4561	52	22	µ(1	µ(1	PROPN
ejpam-4561	52	23	)	)	PUNCT
ejpam-4561	52	24	=	=	NOUN
ejpam-4561	52	25	µ(x	µ(x	PUNCT
ejpam-4561	52	26	◦	◦	NOUN
ejpam-4561	52	27	x	x	SYM
ejpam-4561	52	28	)	)	PUNCT
ejpam-4561	52	29	≥	≥	PROPN
ejpam-4561	52	30	min	min	NOUN
ejpam-4561	52	31	{	{	PUNCT
ejpam-4561	52	32	µ(x	µ(x	PROPN
ejpam-4561	52	33	)	)	PUNCT
ejpam-4561	52	34	,	,	PUNCT
ejpam-4561	52	35	µ(x	µ(x	NOUN
ejpam-4561	52	36	)	)	PUNCT
ejpam-4561	52	37	}	}	PUNCT
ejpam-4561	52	38	=	=	SYM
ejpam-4561	52	39	µ(x	µ(x	NUM
ejpam-4561	52	40	)	)	PUNCT
ejpam-4561	52	41	.	.	PUNCT
ejpam-4561	53	1	this	this	PRON
ejpam-4561	53	2	proves	prove	VERB
ejpam-4561	53	3	that	that	SCONJ
ejpam-4561	53	4	µ(1	µ(1	PROPN
ejpam-4561	53	5	)	)	PUNCT
ejpam-4561	53	6	≥	≥	NOUN
ejpam-4561	53	7	µ(x	µ(x	VERB
ejpam-4561	53	8	)	)	PUNCT
ejpam-4561	53	9	for	for	ADP
ejpam-4561	53	10	all	all	PRON
ejpam-4561	53	11	x	x	SYM
ejpam-4561	53	12	∈	∈	NOUN
ejpam-4561	53	13	x.	x.	NOUN
ejpam-4561	54	1	the	the	DET
ejpam-4561	54	2	following	follow	VERB
ejpam-4561	54	3	corollary	corollary	NOUN
ejpam-4561	54	4	follows	follow	VERB
ejpam-4561	54	5	immediately	immediately	ADV
ejpam-4561	54	6	from	from	ADP
ejpam-4561	54	7	proposition	proposition	NOUN
ejpam-4561	54	8	1	1	NUM
ejpam-4561	54	9	.	.	PUNCT
ejpam-4561	54	10	corollary	corollary	ADJ
ejpam-4561	54	11	1	1	NUM
ejpam-4561	54	12	.	.	PUNCT
ejpam-4561	55	1	let	let	VERB
ejpam-4561	55	2	x	x	PRON
ejpam-4561	55	3	be	be	AUX
ejpam-4561	55	4	a	a	DET
ejpam-4561	55	5	fuzzy	fuzzy	ADJ
ejpam-4561	55	6	dual	dual	ADJ
ejpam-4561	55	7	b	b	NOUN
ejpam-4561	55	8	-	-	PUNCT
ejpam-4561	55	9	algebra	algebra	NOUN
ejpam-4561	55	10	.	.	PUNCT
ejpam-4561	56	1	then	then	ADV
ejpam-4561	56	2	for	for	ADP
ejpam-4561	56	3	all	all	DET
ejpam-4561	56	4	x	x	SYM
ejpam-4561	56	5	∈	∈	PROPN
ejpam-4561	56	6	x	x	NOUN
ejpam-4561	56	7	,	,	PUNCT
ejpam-4561	56	8	µ(x	µ(x	ADJ
ejpam-4561	56	9	)	)	PUNCT
ejpam-4561	56	10	=	=	NOUN
ejpam-4561	56	11	µ(x	µ(x	PUNCT
ejpam-4561	56	12	◦	◦	NOUN
ejpam-4561	56	13	1	1	NUM
ejpam-4561	56	14	)	)	PUNCT
ejpam-4561	56	15	.	.	PUNCT
ejpam-4561	57	1	proof	proof	NOUN
ejpam-4561	57	2	.	.	PUNCT
ejpam-4561	58	1	note	note	VERB
ejpam-4561	58	2	that	that	SCONJ
ejpam-4561	58	3	x	x	X
ejpam-4561	59	1	=	=	PRON
ejpam-4561	59	2	(	(	PUNCT
ejpam-4561	59	3	x	x	SYM
ejpam-4561	59	4	◦	◦	NOUN
ejpam-4561	59	5	1	1	NUM
ejpam-4561	59	6	)	)	PUNCT
ejpam-4561	59	7	◦	◦	NOUN
ejpam-4561	59	8	1	1	NUM
ejpam-4561	59	9	[	[	X
ejpam-4561	59	10	by	by	ADP
ejpam-4561	59	11	lemma	lemma	PROPN
ejpam-4561	59	12	1(ii	1(ii	NUM
ejpam-4561	59	13	)	)	PUNCT
ejpam-4561	59	14	]	]	PUNCT
ejpam-4561	59	15	.	.	PUNCT
ejpam-4561	60	1	if	if	SCONJ
ejpam-4561	60	2	µ	µ	NOUN
ejpam-4561	60	3	is	be	AUX
ejpam-4561	60	4	a	a	DET
ejpam-4561	60	5	fuzzy	fuzzy	ADJ
ejpam-4561	60	6	dual	dual	ADJ
ejpam-4561	60	7	b	b	NOUN
ejpam-4561	60	8	-	-	PUNCT
ejpam-4561	60	9	algebra	algebra	NOUN
ejpam-4561	60	10	,	,	PUNCT
ejpam-4561	60	11	then	then	ADV
ejpam-4561	60	12	,	,	PUNCT
ejpam-4561	60	13	µ(x	µ(x	X
ejpam-4561	60	14	)	)	PUNCT
ejpam-4561	60	15	=	=	SYM
ejpam-4561	60	16	µ	µ	X
ejpam-4561	60	17	(	(	PUNCT
ejpam-4561	60	18	(	(	PUNCT
ejpam-4561	60	19	x	x	SYM
ejpam-4561	60	20	◦	◦	NOUN
ejpam-4561	60	21	1	1	NUM
ejpam-4561	60	22	)	)	PUNCT
ejpam-4561	60	23	◦	◦	NOUN
ejpam-4561	60	24	1	1	NUM
ejpam-4561	60	25	)	)	PUNCT
ejpam-4561	61	1	[	[	X
ejpam-4561	61	2	by	by	ADP
ejpam-4561	61	3	lemma	lemma	PROPN
ejpam-4561	61	4	1(ii	1(ii	NUM
ejpam-4561	61	5	)	)	PUNCT
ejpam-4561	61	6	]	]	PUNCT
ejpam-4561	61	7	≥	≥	X
ejpam-4561	61	8	min{µ(x	min{µ(x	NOUN
ejpam-4561	61	9	◦	◦	NOUN
ejpam-4561	61	10	1	1	NUM
ejpam-4561	61	11	)	)	PUNCT
ejpam-4561	61	12	,	,	PUNCT
ejpam-4561	61	13	µ(1	µ(1	PROPN
ejpam-4561	61	14	)	)	PUNCT
ejpam-4561	61	15	}	}	PUNCT
ejpam-4561	62	1	[	[	X
ejpam-4561	62	2	by	by	ADP
ejpam-4561	62	3	definition	definition	NOUN
ejpam-4561	62	4	3	3	NUM
ejpam-4561	62	5	]	]	PUNCT
ejpam-4561	62	6	≥	≥	NOUN
ejpam-4561	62	7	min{min{µ(x	min{min{µ(x	NOUN
ejpam-4561	62	8	)	)	PUNCT
ejpam-4561	62	9	,	,	PUNCT
ejpam-4561	62	10	µ(1	µ(1	PROPN
ejpam-4561	62	11	)	)	PUNCT
ejpam-4561	62	12	}	}	PUNCT
ejpam-4561	62	13	,	,	PUNCT
ejpam-4561	62	14	µ(1	µ(1	PROPN
ejpam-4561	62	15	)	)	PUNCT
ejpam-4561	62	16	}	}	PUNCT
ejpam-4561	62	17	≥	≥	NOUN
ejpam-4561	62	18	min{µ(x	min{µ(x	NOUN
ejpam-4561	62	19	)	)	PUNCT
ejpam-4561	62	20	,	,	PUNCT
ejpam-4561	62	21	µ(1	µ(1	PROPN
ejpam-4561	62	22	)	)	PUNCT
ejpam-4561	62	23	}	}	PUNCT
ejpam-4561	63	1	[	[	X
ejpam-4561	63	2	by	by	ADP
ejpam-4561	63	3	proposition	proposition	NOUN
ejpam-4561	63	4	1	1	NUM
ejpam-4561	63	5	]	]	X
ejpam-4561	63	6	=	=	SYM
ejpam-4561	63	7	µ(x	µ(x	PUNCT
ejpam-4561	63	8	◦	◦	NOUN
ejpam-4561	63	9	1	1	NUM
ejpam-4561	63	10	)	)	PUNCT
ejpam-4561	63	11	.	.	PUNCT
ejpam-4561	64	1	that	that	PRON
ejpam-4561	64	2	is	be	AUX
ejpam-4561	64	3	,	,	PUNCT
ejpam-4561	64	4	µ(x	µ(x	ADJ
ejpam-4561	64	5	)	)	PUNCT
ejpam-4561	64	6	=	=	NOUN
ejpam-4561	64	7	µ(x	µ(x	PUNCT
ejpam-4561	64	8	◦	◦	NOUN
ejpam-4561	64	9	1	1	NUM
ejpam-4561	64	10	)	)	PUNCT
ejpam-4561	64	11	for	for	ADP
ejpam-4561	64	12	all	all	PRON
ejpam-4561	64	13	x	x	SYM
ejpam-4561	64	14	∈	∈	ADJ
ejpam-4561	64	15	x.	x.	NOUN
ejpam-4561	64	16	for	for	ADP
ejpam-4561	64	17	any	any	DET
ejpam-4561	64	18	elements	element	NOUN
ejpam-4561	64	19	x	x	PUNCT
ejpam-4561	64	20	and	and	CCONJ
ejpam-4561	64	21	y	y	PROPN
ejpam-4561	64	22	of	of	ADP
ejpam-4561	64	23	x	x	PRON
ejpam-4561	64	24	,	,	PUNCT
ejpam-4561	64	25	let	let	VERB
ejpam-4561	64	26	us	we	PRON
ejpam-4561	64	27	write	write	VERB
ejpam-4561	64	28	n∏	n∏	PROPN
ejpam-4561	64	29	x	x	PUNCT
ejpam-4561	64	30	◦	◦	NOUN
ejpam-4561	64	31	y	y	PRON
ejpam-4561	64	32	for	for	ADP
ejpam-4561	64	33	(	(	PUNCT
ejpam-4561	64	34	(	(	PUNCT
ejpam-4561	64	35	(	(	PUNCT
ejpam-4561	64	36	(	(	PUNCT
ejpam-4561	64	37	x	x	SYM
ejpam-4561	64	38	◦	◦	VERB
ejpam-4561	64	39	y	y	NOUN
ejpam-4561	64	40	)	)	PUNCT
ejpam-4561	64	41	◦	◦	NOUN
ejpam-4561	64	42	y	y	NOUN
ejpam-4561	64	43	)	)	PUNCT
ejpam-4561	64	44	◦	◦	NOUN
ejpam-4561	64	45	y	y	PROPN
ejpam-4561	64	46	)	)	PUNCT
ejpam-4561	64	47	...	...	PUNCT
ejpam-4561	64	48	)	)	PUNCT
ejpam-4561	64	49	where	where	SCONJ
ejpam-4561	64	50	y	y	PROPN
ejpam-4561	64	51	occurs	occur	VERB
ejpam-4561	64	52	n	n	PRON
ejpam-4561	64	53	times	time	NOUN
ejpam-4561	64	54	.	.	PUNCT
ejpam-4561	65	1	proposition	proposition	NOUN
ejpam-4561	65	2	2	2	NUM
ejpam-4561	65	3	.	.	PUNCT
ejpam-4561	66	1	let	let	VERB
ejpam-4561	66	2	a	a	DET
ejpam-4561	66	3	fuzzy	fuzzy	ADJ
ejpam-4561	66	4	set	set	VERB
ejpam-4561	66	5	µ	µ	NOUN
ejpam-4561	66	6	in	in	ADP
ejpam-4561	66	7	x	x	VERB
ejpam-4561	66	8	be	be	AUX
ejpam-4561	66	9	a	a	DET
ejpam-4561	66	10	fuzzy	fuzzy	ADJ
ejpam-4561	66	11	dual	dual	ADJ
ejpam-4561	66	12	b	b	NOUN
ejpam-4561	66	13	-	-	PUNCT
ejpam-4561	66	14	algebra	algebra	NOUN
ejpam-4561	66	15	and	and	CCONJ
ejpam-4561	66	16	let	let	VERB
ejpam-4561	66	17	n	n	PRON
ejpam-4561	66	18	∈	∈	PROPN
ejpam-4561	66	19	n.	n.	NOUN
ejpam-4561	66	20	then	then	ADV
ejpam-4561	66	21	for	for	ADP
ejpam-4561	66	22	all	all	DET
ejpam-4561	66	23	x	x	SYM
ejpam-4561	66	24	∈	∈	PROPN
ejpam-4561	66	25	x	x	X
ejpam-4561	66	26	,	,	PUNCT
ejpam-4561	66	27	(	(	PUNCT
ejpam-4561	66	28	i	i	NOUN
ejpam-4561	66	29	)	)	PUNCT
ejpam-4561	66	30	µ	µ	PROPN
ejpam-4561	66	31	(	(	PUNCT
ejpam-4561	66	32	n∏	n∏	PROPN
ejpam-4561	66	33	x	x	SYM
ejpam-4561	66	34	◦	◦	NOUN
ejpam-4561	66	35	x	x	SYM
ejpam-4561	66	36	)	)	PUNCT
ejpam-4561	66	37	≥	≥	NOUN
ejpam-4561	66	38	µ(x	µ(x	NOUN
ejpam-4561	66	39	)	)	PUNCT
ejpam-4561	66	40	whenever	whenever	SCONJ
ejpam-4561	66	41	n	n	X
ejpam-4561	66	42	is	be	AUX
ejpam-4561	66	43	odd	odd	ADJ
ejpam-4561	66	44	,	,	PUNCT
ejpam-4561	66	45	and	and	CCONJ
ejpam-4561	66	46	(	(	PUNCT
ejpam-4561	66	47	ii	ii	NOUN
ejpam-4561	66	48	)	)	PUNCT
ejpam-4561	66	49	µ	µ	X
ejpam-4561	66	50	(	(	PUNCT
ejpam-4561	66	51	n∏	n∏	PROPN
ejpam-4561	66	52	x	x	SYM
ejpam-4561	66	53	◦	◦	NOUN
ejpam-4561	66	54	x	x	SYM
ejpam-4561	66	55	)	)	PUNCT
ejpam-4561	66	56	=	=	SYM
ejpam-4561	67	1	µ(x	µ(x	X
ejpam-4561	67	2	)	)	PUNCT
ejpam-4561	67	3	whenever	whenever	SCONJ
ejpam-4561	67	4	n	n	PRON
ejpam-4561	67	5	is	be	AUX
ejpam-4561	67	6	even	even	ADV
ejpam-4561	67	7	.	.	PUNCT
ejpam-4561	68	1	k.	k.	PROPN
ejpam-4561	68	2	belleza	belleza	PROPN
ejpam-4561	68	3	,	,	PUNCT
ejpam-4561	68	4	m.	m.	PROPN
ejpam-4561	68	5	r.	r.	PROPN
ejpam-4561	68	6	v.	v.	PROPN
ejpam-4561	68	7	dicen	dicen	PROPN
ejpam-4561	68	8	/	/	SYM
ejpam-4561	68	9	eur	eur	PROPN
ejpam-4561	68	10	.	.	PUNCT
ejpam-4561	69	1	j.	j.	PROPN
ejpam-4561	69	2	pure	pure	PROPN
ejpam-4561	69	3	appl	appl	PROPN
ejpam-4561	69	4	.	.	PROPN
ejpam-4561	69	5	math	math	PROPN
ejpam-4561	69	6	,	,	PUNCT
ejpam-4561	69	7	15	15	NUM
ejpam-4561	69	8	(	(	PUNCT
ejpam-4561	69	9	4	4	NUM
ejpam-4561	69	10	)	)	PUNCT
ejpam-4561	69	11	(	(	PUNCT
ejpam-4561	69	12	2022	2022	NUM
ejpam-4561	69	13	)	)	PUNCT
ejpam-4561	69	14	,	,	PUNCT
ejpam-4561	69	15	1957	1957	NUM
ejpam-4561	69	16	-	-	SYM
ejpam-4561	69	17	1965	1965	NUM
ejpam-4561	69	18	1960	1960	NUM
ejpam-4561	69	19	proof	proof	NOUN
ejpam-4561	69	20	.	.	PUNCT
ejpam-4561	70	1	let	let	VERB
ejpam-4561	70	2	x	x	SYM
ejpam-4561	70	3	∈	∈	PROPN
ejpam-4561	70	4	x.	x.	NOUN
ejpam-4561	70	5	(	(	PUNCT
ejpam-4561	70	6	i	i	NOUN
ejpam-4561	70	7	)	)	PUNCT
ejpam-4561	70	8	suppose	suppose	VERB
ejpam-4561	70	9	that	that	SCONJ
ejpam-4561	70	10	n	n	PRON
ejpam-4561	70	11	is	be	AUX
ejpam-4561	70	12	odd	odd	ADJ
ejpam-4561	70	13	.	.	PUNCT
ejpam-4561	71	1	then	then	ADV
ejpam-4561	71	2	n	n	PROPN
ejpam-4561	71	3	=	=	SYM
ejpam-4561	71	4	2k−1	2k−1	NUM
ejpam-4561	71	5	for	for	ADP
ejpam-4561	71	6	some	some	DET
ejpam-4561	71	7	positive	positive	ADJ
ejpam-4561	71	8	integer	integer	NOUN
ejpam-4561	71	9	k.	k.	PROPN
ejpam-4561	71	10	note	note	VERB
ejpam-4561	71	11	that	that	SCONJ
ejpam-4561	71	12	by	by	ADP
ejpam-4561	71	13	db1	db1	NOUN
ejpam-4561	71	14	and	and	CCONJ
ejpam-4561	71	15	proposition	proposition	NOUN
ejpam-4561	71	16	1	1	NUM
ejpam-4561	71	17	,	,	PUNCT
ejpam-4561	71	18	µ(x	µ(x	ADJ
ejpam-4561	71	19	◦	◦	NOUN
ejpam-4561	71	20	x	x	SYM
ejpam-4561	71	21	)	)	PUNCT
ejpam-4561	71	22	=	=	SYM
ejpam-4561	71	23	µ(1	µ(1	PROPN
ejpam-4561	71	24	)	)	PUNCT
ejpam-4561	71	25	≥	≥	NOUN
ejpam-4561	71	26	µ(x	µ(x	NOUN
ejpam-4561	71	27	)	)	PUNCT
ejpam-4561	71	28	when	when	SCONJ
ejpam-4561	71	29	k	k	PROPN
ejpam-4561	71	30	=	=	SYM
ejpam-4561	71	31	1	1	X
ejpam-4561	71	32	.	.	X
ejpam-4561	71	33	assume	assume	VERB
ejpam-4561	71	34	that	that	SCONJ
ejpam-4561	71	35	µ	µ	X
ejpam-4561	71	36	(	(	PUNCT
ejpam-4561	71	37	2k−1∏	2k−1∏	NUM
ejpam-4561	71	38	x	x	SYM
ejpam-4561	71	39	◦	◦	NOUN
ejpam-4561	71	40	x	x	SYM
ejpam-4561	71	41	)	)	PUNCT
ejpam-4561	71	42	≥	≥	NOUN
ejpam-4561	71	43	µ(x	µ(x	VERB
ejpam-4561	71	44	)	)	PUNCT
ejpam-4561	71	45	for	for	ADP
ejpam-4561	71	46	a	a	DET
ejpam-4561	71	47	positive	positive	ADJ
ejpam-4561	71	48	integer	integer	NOUN
ejpam-4561	72	1	k.	k.	PROPN
ejpam-4561	73	1	then	then	ADV
ejpam-4561	73	2	,	,	PUNCT
ejpam-4561	73	3	µ	µ	X
ejpam-4561	73	4	(	(	PUNCT
ejpam-4561	73	5	2(k+1)−1∏	2(k+1)−1∏	NOUN
ejpam-4561	73	6	x	x	PUNCT
ejpam-4561	73	7	◦	◦	NOUN
ejpam-4561	73	8	x	x	SYM
ejpam-4561	73	9	)	)	PUNCT
ejpam-4561	74	1	=	=	SYM
ejpam-4561	74	2	µ	µ	X
ejpam-4561	74	3	(	(	PUNCT
ejpam-4561	74	4	2k+1∏	2k+1∏	NUM
ejpam-4561	74	5	x	x	SYM
ejpam-4561	74	6	◦	◦	NOUN
ejpam-4561	74	7	x	x	SYM
ejpam-4561	74	8	)	)	PUNCT
ejpam-4561	74	9	=	=	SYM
ejpam-4561	74	10	µ	µ	X
ejpam-4561	74	11	(	(	PUNCT
ejpam-4561	74	12	2k−1∏	2k−1∏	NUM
ejpam-4561	74	13	(	(	PUNCT
ejpam-4561	74	14	(	(	PUNCT
ejpam-4561	74	15	x	x	PART
ejpam-4561	74	16	◦	◦	NOUN
ejpam-4561	74	17	x	x	NOUN
ejpam-4561	74	18	)	)	PUNCT
ejpam-4561	74	19	◦	◦	NOUN
ejpam-4561	74	20	x	x	SYM
ejpam-4561	74	21	)	)	PUNCT
ejpam-4561	74	22	◦	◦	NOUN
ejpam-4561	74	23	x	x	SYM
ejpam-4561	74	24	)	)	PUNCT
ejpam-4561	75	1	=	=	SYM
ejpam-4561	75	2	µ	µ	X
ejpam-4561	75	3	(	(	PUNCT
ejpam-4561	75	4	2k−1∏	2k−1∏	NUM
ejpam-4561	75	5	(	(	PUNCT
ejpam-4561	75	6	1	1	NUM
ejpam-4561	75	7	◦	◦	NOUN
ejpam-4561	75	8	x	x	NOUN
ejpam-4561	75	9	)	)	PUNCT
ejpam-4561	75	10	◦	◦	NOUN
ejpam-4561	75	11	x	x	SYM
ejpam-4561	75	12	)	)	PUNCT
ejpam-4561	76	1	[	[	X
ejpam-4561	76	2	by	by	ADP
ejpam-4561	76	3	(	(	PUNCT
ejpam-4561	76	4	db1	db1	NOUN
ejpam-4561	76	5	)	)	PUNCT
ejpam-4561	76	6	]	]	PUNCT
ejpam-4561	76	7	=	=	SYM
ejpam-4561	76	8	µ	µ	X
ejpam-4561	76	9	(	(	PUNCT
ejpam-4561	76	10	2k−1∏	2k−1∏	NUM
ejpam-4561	76	11	x	x	SYM
ejpam-4561	76	12	◦	◦	NOUN
ejpam-4561	76	13	x	x	SYM
ejpam-4561	76	14	)	)	PUNCT
ejpam-4561	77	1	[	[	X
ejpam-4561	77	2	by	by	ADP
ejpam-4561	77	3	(	(	PUNCT
ejpam-4561	77	4	db2	db2	PROPN
ejpam-4561	77	5	)	)	PUNCT
ejpam-4561	77	6	]	]	PUNCT
ejpam-4561	77	7	=	=	PUNCT
ejpam-4561	77	8	µ(1	µ(1	PROPN
ejpam-4561	77	9	)	)	PUNCT
ejpam-4561	77	10	≥	≥	NOUN
ejpam-4561	77	11	µ(x	µ(x	VERB
ejpam-4561	77	12	)	)	PUNCT
ejpam-4561	77	13	.	.	PUNCT
ejpam-4561	78	1	[	[	X
ejpam-4561	78	2	by	by	ADP
ejpam-4561	78	3	proposition	proposition	NOUN
ejpam-4561	78	4	1	1	NUM
ejpam-4561	78	5	]	]	PUNCT
ejpam-4561	78	6	hence	hence	ADV
ejpam-4561	78	7	,	,	PUNCT
ejpam-4561	78	8	the	the	DET
ejpam-4561	78	9	above	above	ADJ
ejpam-4561	78	10	implies	imply	VERB
ejpam-4561	78	11	that	that	SCONJ
ejpam-4561	78	12	(	(	PUNCT
ejpam-4561	78	13	i	i	NOUN
ejpam-4561	78	14	)	)	PUNCT
ejpam-4561	78	15	holds	hold	VERB
ejpam-4561	78	16	.	.	PUNCT
ejpam-4561	79	1	(	(	PUNCT
ejpam-4561	79	2	ii	ii	NOUN
ejpam-4561	79	3	)	)	PUNCT
ejpam-4561	79	4	suppose	suppose	VERB
ejpam-4561	79	5	that	that	SCONJ
ejpam-4561	79	6	n	n	PRON
ejpam-4561	79	7	is	be	AUX
ejpam-4561	79	8	even	even	ADV
ejpam-4561	79	9	.	.	PUNCT
ejpam-4561	80	1	then	then	ADV
ejpam-4561	80	2	n	n	PROPN
ejpam-4561	80	3	=	=	SYM
ejpam-4561	80	4	2k	2k	PROPN
ejpam-4561	80	5	for	for	ADP
ejpam-4561	80	6	some	some	DET
ejpam-4561	80	7	positive	positive	ADJ
ejpam-4561	80	8	integer	integer	NOUN
ejpam-4561	80	9	k.	k.	PROPN
ejpam-4561	80	10	for	for	ADP
ejpam-4561	80	11	k	k	PROPN
ejpam-4561	80	12	=	=	SYM
ejpam-4561	80	13	1	1	NUM
ejpam-4561	80	14	,	,	PUNCT
ejpam-4561	80	15	n	n	NOUN
ejpam-4561	80	16	=	=	SYM
ejpam-4561	80	17	2	2	NUM
ejpam-4561	80	18	,	,	PUNCT
ejpam-4561	80	19	we	we	PRON
ejpam-4561	80	20	have	have	AUX
ejpam-4561	80	21	,	,	PUNCT
ejpam-4561	80	22	µ((x	µ((x	VERB
ejpam-4561	80	23	◦	◦	NOUN
ejpam-4561	80	24	x	x	SYM
ejpam-4561	80	25	)	)	PUNCT
ejpam-4561	80	26	◦	◦	NOUN
ejpam-4561	80	27	x	x	SYM
ejpam-4561	80	28	)	)	PUNCT
ejpam-4561	80	29	=	=	PUNCT
ejpam-4561	80	30	µ(1	µ(1	PROPN
ejpam-4561	80	31	◦	◦	NOUN
ejpam-4561	80	32	x	x	NOUN
ejpam-4561	80	33	)	)	PUNCT
ejpam-4561	80	34	=	=	SYM
ejpam-4561	80	35	µ(x	µ(x	X
ejpam-4561	80	36	)	)	PUNCT
ejpam-4561	80	37	by	by	ADP
ejpam-4561	80	38	(	(	PUNCT
ejpam-4561	80	39	db1	db1	NOUN
ejpam-4561	80	40	)	)	PUNCT
ejpam-4561	80	41	and	and	CCONJ
ejpam-4561	80	42	(	(	PUNCT
ejpam-4561	80	43	db2	db2	PROPN
ejpam-4561	80	44	)	)	PUNCT
ejpam-4561	80	45	.	.	PUNCT
ejpam-4561	80	46	suppose	suppose	VERB
ejpam-4561	80	47	that	that	SCONJ
ejpam-4561	80	48	µ	µ	X
ejpam-4561	80	49	(	(	PUNCT
ejpam-4561	80	50	2k∏	2k∏	NUM
ejpam-4561	80	51	x	x	SYM
ejpam-4561	80	52	◦	◦	NOUN
ejpam-4561	80	53	x	x	SYM
ejpam-4561	80	54	)	)	PUNCT
ejpam-4561	80	55	=	=	SYM
ejpam-4561	80	56	µ(x	µ(x	X
ejpam-4561	80	57	)	)	PUNCT
ejpam-4561	80	58	for	for	ADP
ejpam-4561	80	59	a	a	DET
ejpam-4561	80	60	positive	positive	ADJ
ejpam-4561	80	61	integer	integer	NOUN
ejpam-4561	80	62	k.	k.	PROPN
ejpam-4561	80	63	then	then	ADV
ejpam-4561	80	64	,	,	PUNCT
ejpam-4561	80	65	µ	µ	X
ejpam-4561	80	66	(	(	PUNCT
ejpam-4561	80	67	2(k+1)∏	2(k+1)∏	NUM
ejpam-4561	80	68	x	x	SYM
ejpam-4561	80	69	◦	◦	NOUN
ejpam-4561	80	70	x	x	SYM
ejpam-4561	80	71	)	)	PUNCT
ejpam-4561	80	72	=	=	SYM
ejpam-4561	80	73	µ	µ	X
ejpam-4561	80	74	(	(	PUNCT
ejpam-4561	80	75	2k+2∏	2k+2∏	NUM
ejpam-4561	80	76	x	x	SYM
ejpam-4561	80	77	◦	◦	NOUN
ejpam-4561	80	78	x	x	SYM
ejpam-4561	80	79	)	)	PUNCT
ejpam-4561	80	80	=	=	SYM
ejpam-4561	80	81	µ	µ	X
ejpam-4561	80	82	(	(	PUNCT
ejpam-4561	80	83	2k∏	2k∏	NUM
ejpam-4561	80	84	(	(	PUNCT
ejpam-4561	80	85	(	(	PUNCT
ejpam-4561	80	86	x	x	PART
ejpam-4561	80	87	◦	◦	NOUN
ejpam-4561	80	88	x	x	NOUN
ejpam-4561	80	89	)	)	PUNCT
ejpam-4561	80	90	◦	◦	NOUN
ejpam-4561	80	91	x	x	SYM
ejpam-4561	80	92	)	)	PUNCT
ejpam-4561	80	93	◦	◦	NOUN
ejpam-4561	80	94	x	x	SYM
ejpam-4561	80	95	)	)	PUNCT
ejpam-4561	80	96	=	=	SYM
ejpam-4561	80	97	µ	µ	X
ejpam-4561	80	98	(	(	PUNCT
ejpam-4561	80	99	2k∏	2k∏	NUM
ejpam-4561	80	100	(	(	PUNCT
ejpam-4561	80	101	1	1	NUM
ejpam-4561	80	102	◦	◦	NOUN
ejpam-4561	80	103	x	x	NOUN
ejpam-4561	80	104	)	)	PUNCT
ejpam-4561	80	105	◦	◦	NOUN
ejpam-4561	80	106	x	x	SYM
ejpam-4561	80	107	)	)	PUNCT
ejpam-4561	81	1	[	[	X
ejpam-4561	81	2	by	by	ADP
ejpam-4561	81	3	(	(	PUNCT
ejpam-4561	81	4	db1	db1	NOUN
ejpam-4561	81	5	)	)	PUNCT
ejpam-4561	81	6	]	]	PUNCT
ejpam-4561	81	7	=	=	SYM
ejpam-4561	81	8	µ	µ	X
ejpam-4561	81	9	(	(	PUNCT
ejpam-4561	81	10	2k∏	2k∏	NUM
ejpam-4561	81	11	(	(	PUNCT
ejpam-4561	81	12	x	x	PART
ejpam-4561	81	13	◦	◦	NOUN
ejpam-4561	81	14	x	x	NUM
ejpam-4561	81	15	)	)	PUNCT
ejpam-4561	81	16	)	)	PUNCT
ejpam-4561	82	1	[	[	X
ejpam-4561	82	2	by	by	ADP
ejpam-4561	82	3	(	(	PUNCT
ejpam-4561	82	4	db2	db2	PROPN
ejpam-4561	82	5	)	)	PUNCT
ejpam-4561	82	6	]	]	PUNCT
ejpam-4561	82	7	=	=	SYM
ejpam-4561	82	8	µ(x	µ(x	X
ejpam-4561	82	9	)	)	PUNCT
ejpam-4561	82	10	.	.	PUNCT
ejpam-4561	83	1	hence	hence	ADV
ejpam-4561	83	2	,	,	PUNCT
ejpam-4561	83	3	the	the	DET
ejpam-4561	83	4	above	above	ADJ
ejpam-4561	83	5	implies	imply	VERB
ejpam-4561	83	6	that	that	SCONJ
ejpam-4561	83	7	(	(	PUNCT
ejpam-4561	83	8	ii	ii	NOUN
ejpam-4561	83	9	)	)	PUNCT
ejpam-4561	83	10	holds	hold	VERB
ejpam-4561	83	11	.	.	PUNCT
ejpam-4561	84	1	k.	k.	PROPN
ejpam-4561	84	2	belleza	belleza	PROPN
ejpam-4561	84	3	,	,	PUNCT
ejpam-4561	84	4	m.	m.	PROPN
ejpam-4561	84	5	r.	r.	PROPN
ejpam-4561	84	6	v.	v.	PROPN
ejpam-4561	84	7	dicen	dicen	PROPN
ejpam-4561	84	8	/	/	SYM
ejpam-4561	84	9	eur	eur	PROPN
ejpam-4561	84	10	.	.	PUNCT
ejpam-4561	85	1	j.	j.	PROPN
ejpam-4561	85	2	pure	pure	PROPN
ejpam-4561	85	3	appl	appl	PROPN
ejpam-4561	85	4	.	.	PROPN
ejpam-4561	85	5	math	math	PROPN
ejpam-4561	85	6	,	,	PUNCT
ejpam-4561	85	7	15	15	NUM
ejpam-4561	85	8	(	(	PUNCT
ejpam-4561	85	9	4	4	NUM
ejpam-4561	85	10	)	)	PUNCT
ejpam-4561	85	11	(	(	PUNCT
ejpam-4561	85	12	2022	2022	NUM
ejpam-4561	85	13	)	)	PUNCT
ejpam-4561	85	14	,	,	PUNCT
ejpam-4561	85	15	1957	1957	NUM
ejpam-4561	85	16	-	-	SYM
ejpam-4561	85	17	1965	1965	NUM
ejpam-4561	85	18	1961	1961	NUM
ejpam-4561	85	19	proposition	proposition	NOUN
ejpam-4561	85	20	3	3	X
ejpam-4561	85	21	.	.	PUNCT
ejpam-4561	86	1	if	if	SCONJ
ejpam-4561	86	2	a	a	DET
ejpam-4561	86	3	fuzzy	fuzzy	ADJ
ejpam-4561	86	4	set	set	VERB
ejpam-4561	86	5	µ	µ	NOUN
ejpam-4561	86	6	in	in	ADP
ejpam-4561	86	7	x	x	VERB
ejpam-4561	86	8	is	be	AUX
ejpam-4561	86	9	a	a	DET
ejpam-4561	86	10	fuzzy	fuzzy	ADJ
ejpam-4561	86	11	dual	dual	ADJ
ejpam-4561	86	12	b	b	NOUN
ejpam-4561	86	13	-	-	PUNCT
ejpam-4561	86	14	algebra	algebra	NOUN
ejpam-4561	86	15	,	,	PUNCT
ejpam-4561	86	16	then	then	ADV
ejpam-4561	86	17	for	for	ADP
ejpam-4561	86	18	all	all	DET
ejpam-4561	86	19	x	x	NOUN
ejpam-4561	86	20	,	,	PUNCT
ejpam-4561	86	21	y	y	PROPN
ejpam-4561	86	22	∈	∈	PROPN
ejpam-4561	86	23	x	x	X
ejpam-4561	86	24	,	,	PUNCT
ejpam-4561	86	25	(	(	PUNCT
ejpam-4561	86	26	fdb1	fdb1	PROPN
ejpam-4561	86	27	)	)	PUNCT
ejpam-4561	86	28	µ(x	µ(x	PUNCT
ejpam-4561	86	29	◦	◦	NOUN
ejpam-4561	86	30	1	1	NUM
ejpam-4561	86	31	)	)	PUNCT
ejpam-4561	86	32	≥	≥	NOUN
ejpam-4561	86	33	µ(x	µ(x	VERB
ejpam-4561	86	34	)	)	PUNCT
ejpam-4561	86	35	;	;	PUNCT
ejpam-4561	86	36	(	(	PUNCT
ejpam-4561	86	37	fdb2	fdb2	NOUN
ejpam-4561	86	38	)	)	PUNCT
ejpam-4561	86	39	µ((y	µ((y	VERB
ejpam-4561	87	1	◦	◦	NOUN
ejpam-4561	87	2	1	1	NUM
ejpam-4561	87	3	)	)	PUNCT
ejpam-4561	87	4	◦	◦	NOUN
ejpam-4561	87	5	x	x	SYM
ejpam-4561	87	6	)	)	PUNCT
ejpam-4561	87	7	≥	≥	NOUN
ejpam-4561	87	8	min{µ(x	min{µ(x	NOUN
ejpam-4561	87	9	)	)	PUNCT
ejpam-4561	87	10	,	,	PUNCT
ejpam-4561	87	11	µ(y	µ(y	PROPN
ejpam-4561	87	12	)	)	PUNCT
ejpam-4561	87	13	}	}	PUNCT
ejpam-4561	87	14	.	.	PUNCT
ejpam-4561	88	1	proof	proof	NOUN
ejpam-4561	88	2	.	.	PUNCT
ejpam-4561	89	1	suppose	suppose	VERB
ejpam-4561	89	2	x	x	PRON
ejpam-4561	89	3	is	be	AUX
ejpam-4561	89	4	a	a	DET
ejpam-4561	89	5	fuzzy	fuzzy	ADJ
ejpam-4561	89	6	dual	dual	ADJ
ejpam-4561	89	7	b	b	NOUN
ejpam-4561	89	8	-	-	PUNCT
ejpam-4561	89	9	algebra	algebra	NOUN
ejpam-4561	89	10	and	and	CCONJ
ejpam-4561	89	11	µ	µ	NOUN
ejpam-4561	89	12	is	be	AUX
ejpam-4561	89	13	a	a	DET
ejpam-4561	89	14	fuzzy	fuzzy	ADJ
ejpam-4561	89	15	set	set	NOUN
ejpam-4561	89	16	for	for	ADP
ejpam-4561	89	17	all	all	DET
ejpam-4561	89	18	x	x	NOUN
ejpam-4561	89	19	,	,	PUNCT
ejpam-4561	89	20	y	y	PROPN
ejpam-4561	89	21	∈	∈	PROPN
ejpam-4561	89	22	x.	x.	NOUN
ejpam-4561	89	23	(	(	PUNCT
ejpam-4561	89	24	fdb1	fdb1	PROPN
ejpam-4561	89	25	):	):	PUNCT
ejpam-4561	89	26	by	by	ADP
ejpam-4561	89	27	definition	definition	NOUN
ejpam-4561	89	28	3	3	NUM
ejpam-4561	89	29	,	,	PUNCT
ejpam-4561	89	30	µ(x	µ(x	ADJ
ejpam-4561	89	31	◦	◦	NOUN
ejpam-4561	89	32	1	1	NUM
ejpam-4561	89	33	)	)	PUNCT
ejpam-4561	89	34	≥	≥	NOUN
ejpam-4561	89	35	min{µ(x	min{µ(x	NOUN
ejpam-4561	89	36	)	)	PUNCT
ejpam-4561	89	37	,	,	PUNCT
ejpam-4561	89	38	µ(1	µ(1	PROPN
ejpam-4561	89	39	)	)	PUNCT
ejpam-4561	89	40	}	}	PUNCT
ejpam-4561	89	41	≥	≥	NOUN
ejpam-4561	89	42	µ(x	µ(x	VERB
ejpam-4561	89	43	)	)	PUNCT
ejpam-4561	89	44	,	,	PUNCT
ejpam-4561	89	45	by	by	ADP
ejpam-4561	89	46	proposition	proposition	NOUN
ejpam-4561	89	47	1	1	NUM
ejpam-4561	89	48	.	.	PUNCT
ejpam-4561	90	1	hence	hence	ADV
ejpam-4561	90	2	,	,	PUNCT
ejpam-4561	90	3	(	(	PUNCT
ejpam-4561	90	4	fdb1	fdb1	PROPN
ejpam-4561	90	5	)	)	PUNCT
ejpam-4561	90	6	holds	hold	VERB
ejpam-4561	90	7	.	.	PUNCT
ejpam-4561	91	1	(	(	PUNCT
ejpam-4561	91	2	fdb2	fdb2	NOUN
ejpam-4561	91	3	):	):	PUNCT
ejpam-4561	91	4	by	by	ADP
ejpam-4561	91	5	definition	definition	NOUN
ejpam-4561	91	6	3	3	NUM
ejpam-4561	91	7	,	,	PUNCT
ejpam-4561	91	8	µ((y	µ((y	VERB
ejpam-4561	91	9	◦	◦	NOUN
ejpam-4561	91	10	1	1	NUM
ejpam-4561	91	11	)	)	PUNCT
ejpam-4561	91	12	◦	◦	NOUN
ejpam-4561	91	13	x	x	SYM
ejpam-4561	91	14	)	)	PUNCT
ejpam-4561	91	15	≥	≥	NOUN
ejpam-4561	91	16	min{µ(y	min{µ(y	PUNCT
ejpam-4561	91	17	◦	◦	NOUN
ejpam-4561	91	18	1	1	NUM
ejpam-4561	91	19	)	)	PUNCT
ejpam-4561	91	20	,	,	PUNCT
ejpam-4561	91	21	µ(x	µ(x	NOUN
ejpam-4561	91	22	)	)	PUNCT
ejpam-4561	91	23	}	}	PUNCT
ejpam-4561	91	24	≥	≥	PROPN
ejpam-4561	91	25	min{min{µ(y	min{min{µ(y	PROPN
ejpam-4561	91	26	)	)	PUNCT
ejpam-4561	91	27	,	,	PUNCT
ejpam-4561	91	28	µ(1	µ(1	PROPN
ejpam-4561	91	29	)	)	PUNCT
ejpam-4561	91	30	}	}	PUNCT
ejpam-4561	91	31	,	,	PUNCT
ejpam-4561	91	32	µ(x	µ(x	NOUN
ejpam-4561	91	33	)	)	PUNCT
ejpam-4561	91	34	}	}	PUNCT
ejpam-4561	91	35	≥	≥	NOUN
ejpam-4561	91	36	min{µ(y	min{µ(y	PROPN
ejpam-4561	91	37	)	)	PUNCT
ejpam-4561	91	38	,	,	PUNCT
ejpam-4561	91	39	µ(x	µ(x	NOUN
ejpam-4561	91	40	)	)	PUNCT
ejpam-4561	91	41	}	}	PUNCT
ejpam-4561	92	1	[	[	X
ejpam-4561	92	2	by	by	ADP
ejpam-4561	92	3	proposition	proposition	NOUN
ejpam-4561	92	4	1	1	NUM
ejpam-4561	92	5	]	]	PUNCT
ejpam-4561	92	6	≥	≥	NOUN
ejpam-4561	92	7	min{µ(x	min{µ(x	NOUN
ejpam-4561	92	8	)	)	PUNCT
ejpam-4561	92	9	,	,	PUNCT
ejpam-4561	92	10	µ(y	µ(y	PROPN
ejpam-4561	92	11	)	)	PUNCT
ejpam-4561	92	12	}	}	PUNCT
ejpam-4561	92	13	.	.	PUNCT
ejpam-4561	93	1	hence	hence	ADV
ejpam-4561	93	2	,	,	PUNCT
ejpam-4561	93	3	(	(	PUNCT
ejpam-4561	93	4	fdb2	fdb2	NOUN
ejpam-4561	93	5	)	)	PUNCT
ejpam-4561	93	6	holds	hold	VERB
ejpam-4561	93	7	.	.	PUNCT
ejpam-4561	94	1	the	the	DET
ejpam-4561	94	2	next	next	ADJ
ejpam-4561	94	3	theorem	theorem	NOUN
ejpam-4561	94	4	is	be	AUX
ejpam-4561	94	5	a	a	DET
ejpam-4561	94	6	characterization	characterization	NOUN
ejpam-4561	94	7	of	of	ADP
ejpam-4561	94	8	a	a	DET
ejpam-4561	94	9	fuzzy	fuzzy	ADJ
ejpam-4561	94	10	dual	dual	ADJ
ejpam-4561	94	11	b	b	NOUN
ejpam-4561	94	12	-	-	PUNCT
ejpam-4561	94	13	algebra	algebra	NOUN
ejpam-4561	94	14	which	which	PRON
ejpam-4561	94	15	is	be	AUX
ejpam-4561	94	16	the	the	DET
ejpam-4561	94	17	converse	converse	NOUN
ejpam-4561	94	18	of	of	ADP
ejpam-4561	94	19	proposition	proposition	NOUN
ejpam-4561	94	20	3	3	NUM
ejpam-4561	94	21	.	.	PUNCT
ejpam-4561	94	22	theorem	theorem	NOUN
ejpam-4561	94	23	1	1	NUM
ejpam-4561	94	24	.	.	PUNCT
ejpam-4561	95	1	if	if	SCONJ
ejpam-4561	95	2	a	a	DET
ejpam-4561	95	3	fuzzy	fuzzy	ADJ
ejpam-4561	95	4	set	set	VERB
ejpam-4561	95	5	µ	µ	NOUN
ejpam-4561	95	6	in	in	ADP
ejpam-4561	95	7	x	x	X
ejpam-4561	95	8	satisfies	satisfie	NOUN
ejpam-4561	95	9	(	(	PUNCT
ejpam-4561	95	10	fdb1	fdb1	PROPN
ejpam-4561	95	11	)	)	PUNCT
ejpam-4561	95	12	and	and	CCONJ
ejpam-4561	95	13	(	(	PUNCT
ejpam-4561	95	14	fdb2	fdb2	PROPN
ejpam-4561	95	15	)	)	PUNCT
ejpam-4561	95	16	,	,	PUNCT
ejpam-4561	95	17	then	then	ADV
ejpam-4561	95	18	µ	µ	X
ejpam-4561	95	19	is	be	AUX
ejpam-4561	95	20	a	a	DET
ejpam-4561	95	21	fuzzy	fuzzy	ADJ
ejpam-4561	95	22	dual	dual	ADJ
ejpam-4561	95	23	b	b	NOUN
ejpam-4561	95	24	-	-	PUNCT
ejpam-4561	95	25	algebra	algebra	NOUN
ejpam-4561	95	26	.	.	PUNCT
ejpam-4561	96	1	proof	proof	NOUN
ejpam-4561	96	2	.	.	PUNCT
ejpam-4561	97	1	suppose	suppose	VERB
ejpam-4561	97	2	that	that	SCONJ
ejpam-4561	97	3	a	a	DET
ejpam-4561	97	4	fuzzy	fuzzy	ADJ
ejpam-4561	97	5	set	set	VERB
ejpam-4561	97	6	µ	µ	NOUN
ejpam-4561	97	7	in	in	ADV
ejpam-4561	97	8	x	x	PART
ejpam-4561	97	9	satisfies	satisfie	NOUN
ejpam-4561	97	10	the	the	DET
ejpam-4561	97	11	conditions	condition	NOUN
ejpam-4561	97	12	in	in	ADP
ejpam-4561	97	13	(	(	PUNCT
ejpam-4561	97	14	fdb1	fdb1	PROPN
ejpam-4561	97	15	)	)	PUNCT
ejpam-4561	97	16	and	and	CCONJ
ejpam-4561	97	17	(	(	PUNCT
ejpam-4561	97	18	fdb2	fdb2	NOUN
ejpam-4561	97	19	)	)	PUNCT
ejpam-4561	97	20	.	.	PUNCT
ejpam-4561	98	1	let	let	VERB
ejpam-4561	98	2	x	x	PRON
ejpam-4561	98	3	,	,	PUNCT
ejpam-4561	98	4	y	y	PROPN
ejpam-4561	98	5	∈	∈	PROPN
ejpam-4561	98	6	x.	x.	NOUN
ejpam-4561	99	1	then	then	ADV
ejpam-4561	99	2	we	we	PRON
ejpam-4561	99	3	have	have	VERB
ejpam-4561	99	4	,	,	PUNCT
ejpam-4561	99	5	µ(x	µ(x	VERB
ejpam-4561	99	6	◦	◦	NOUN
ejpam-4561	99	7	y	y	NOUN
ejpam-4561	99	8	)	)	PUNCT
ejpam-4561	99	9	=	=	SYM
ejpam-4561	100	1	µ(((x	µ(((x	PRON
ejpam-4561	100	2	◦	◦	NOUN
ejpam-4561	100	3	1	1	NUM
ejpam-4561	100	4	)	)	PUNCT
ejpam-4561	100	5	◦	◦	NOUN
ejpam-4561	100	6	1	1	NUM
ejpam-4561	100	7	)	)	PUNCT
ejpam-4561	100	8	◦	◦	NOUN
ejpam-4561	100	9	y	y	NOUN
ejpam-4561	100	10	)	)	PUNCT
ejpam-4561	101	1	[	[	X
ejpam-4561	101	2	by	by	ADP
ejpam-4561	101	3	lemma	lemma	PROPN
ejpam-4561	101	4	1(i	1(i	NUM
ejpam-4561	101	5	)	)	PUNCT
ejpam-4561	101	6	]	]	PUNCT
ejpam-4561	101	7	≥	≥	X
ejpam-4561	101	8	min{µ((x	min{µ((x	DET
ejpam-4561	101	9	◦	◦	NOUN
ejpam-4561	101	10	1	1	NUM
ejpam-4561	101	11	)	)	PUNCT
ejpam-4561	101	12	◦	◦	NOUN
ejpam-4561	101	13	1	1	NUM
ejpam-4561	101	14	)	)	PUNCT
ejpam-4561	101	15	,	,	PUNCT
ejpam-4561	101	16	µ(y	µ(y	PROPN
ejpam-4561	101	17	)	)	PUNCT
ejpam-4561	101	18	}	}	PUNCT
ejpam-4561	101	19	≥	≥	NOUN
ejpam-4561	101	20	min{min{µ(x	min{min{µ(x	ADJ
ejpam-4561	101	21	◦	◦	NOUN
ejpam-4561	101	22	1	1	NUM
ejpam-4561	101	23	)	)	PUNCT
ejpam-4561	101	24	,	,	PUNCT
ejpam-4561	101	25	µ(1	µ(1	PROPN
ejpam-4561	101	26	)	)	PUNCT
ejpam-4561	101	27	}	}	PUNCT
ejpam-4561	101	28	,	,	PUNCT
ejpam-4561	101	29	µ(y	µ(y	PROPN
ejpam-4561	101	30	)	)	PUNCT
ejpam-4561	101	31	}	}	PUNCT
ejpam-4561	101	32	≥	≥	NOUN
ejpam-4561	101	33	min{µ(x	min{µ(x	NOUN
ejpam-4561	101	34	◦	◦	NOUN
ejpam-4561	101	35	1	1	NUM
ejpam-4561	101	36	)	)	PUNCT
ejpam-4561	101	37	,	,	PUNCT
ejpam-4561	101	38	µ(y	µ(y	PROPN
ejpam-4561	101	39	)	)	PUNCT
ejpam-4561	101	40	}	}	PUNCT
ejpam-4561	102	1	[	[	X
ejpam-4561	102	2	by	by	ADP
ejpam-4561	102	3	(	(	PUNCT
ejpam-4561	102	4	fdb2	fdb2	NOUN
ejpam-4561	102	5	)	)	PUNCT
ejpam-4561	102	6	]	]	PUNCT
ejpam-4561	102	7	=	=	PUNCT
ejpam-4561	102	8	min{µ(x	min{µ(x	NOUN
ejpam-4561	102	9	)	)	PUNCT
ejpam-4561	102	10	,	,	PUNCT
ejpam-4561	102	11	µ(y	µ(y	PROPN
ejpam-4561	102	12	)	)	PUNCT
ejpam-4561	102	13	}	}	PUNCT
ejpam-4561	102	14	.	.	PUNCT
ejpam-4561	103	1	[	[	X
ejpam-4561	103	2	by	by	ADP
ejpam-4561	103	3	(	(	PUNCT
ejpam-4561	103	4	fdb1	fdb1	PROPN
ejpam-4561	103	5	)	)	PUNCT
ejpam-4561	103	6	]	]	PUNCT
ejpam-4561	103	7	hence	hence	ADV
ejpam-4561	103	8	,	,	PUNCT
ejpam-4561	103	9	µ	µ	X
ejpam-4561	103	10	is	be	AUX
ejpam-4561	103	11	a	a	DET
ejpam-4561	103	12	fuzzy	fuzzy	ADJ
ejpam-4561	103	13	dual	dual	ADJ
ejpam-4561	103	14	b	b	NOUN
ejpam-4561	103	15	-	-	PUNCT
ejpam-4561	103	16	algebra	algebra	NOUN
ejpam-4561	103	17	.	.	PUNCT
ejpam-4561	104	1	in	in	ADP
ejpam-4561	104	2	what	what	PRON
ejpam-4561	104	3	follows	follow	VERB
ejpam-4561	104	4	is	be	AUX
ejpam-4561	104	5	an	an	DET
ejpam-4561	104	6	example	example	NOUN
ejpam-4561	104	7	of	of	ADP
ejpam-4561	104	8	the	the	DET
ejpam-4561	104	9	above	above	ADJ
ejpam-4561	104	10	theorem	theorem	PROPN
ejpam-4561	104	11	.	.	PROPN
ejpam-4561	104	12	example	example	NOUN
ejpam-4561	105	1	2	2	NUM
ejpam-4561	105	2	.	.	PUNCT
ejpam-4561	105	3	the	the	DET
ejpam-4561	105	4	fuzzy	fuzzy	ADJ
ejpam-4561	105	5	set	set	VERB
ejpam-4561	105	6	µ	µ	ADP
ejpam-4561	105	7	discussed	discuss	VERB
ejpam-4561	105	8	in	in	ADP
ejpam-4561	105	9	example	example	NOUN
ejpam-4561	105	10	1	1	NUM
ejpam-4561	105	11	satisfies	satisfie	NOUN
ejpam-4561	105	12	(	(	PUNCT
ejpam-4561	105	13	fdb1	fdb1	PROPN
ejpam-4561	105	14	)	)	PUNCT
ejpam-4561	105	15	and	and	CCONJ
ejpam-4561	105	16	(	(	PUNCT
ejpam-4561	105	17	fdb2	fdb2	NOUN
ejpam-4561	105	18	)	)	PUNCT
ejpam-4561	105	19	.	.	PUNCT
ejpam-4561	106	1	considering	consider	VERB
ejpam-4561	106	2	the	the	DET
ejpam-4561	106	3	example	example	NOUN
ejpam-4561	106	4	below	below	ADV
ejpam-4561	106	5	:	:	PUNCT
ejpam-4561	106	6	let	let	VERB
ejpam-4561	106	7	x	x	PUNCT
ejpam-4561	106	8	=	=	PUNCT
ejpam-4561	106	9	a	a	PROPN
ejpam-4561	106	10	and	and	CCONJ
ejpam-4561	106	11	y	y	PROPN
ejpam-4561	106	12	=	=	PROPN
ejpam-4561	106	13	b.	b.	PROPN
ejpam-4561	106	14	note	note	VERB
ejpam-4561	106	15	that	that	SCONJ
ejpam-4561	106	16	µ(a	µ(a	PROPN
ejpam-4561	106	17	◦	◦	NOUN
ejpam-4561	106	18	b	b	NOUN
ejpam-4561	106	19	)	)	PUNCT
ejpam-4561	106	20	=	=	SYM
ejpam-4561	107	1	µ(((a	µ(((a	PROPN
ejpam-4561	107	2	◦	◦	NOUN
ejpam-4561	107	3	1	1	NUM
ejpam-4561	107	4	)	)	PUNCT
ejpam-4561	107	5	◦	◦	NOUN
ejpam-4561	107	6	1	1	NUM
ejpam-4561	107	7	)	)	PUNCT
ejpam-4561	107	8	◦	◦	NOUN
ejpam-4561	107	9	b	b	NOUN
ejpam-4561	107	10	)	)	PUNCT
ejpam-4561	107	11	=	=	SYM
ejpam-4561	107	12	µ(a	µ(a	PROPN
ejpam-4561	107	13	)	)	PUNCT
ejpam-4561	107	14	=	=	PUNCT
ejpam-4561	108	1	0.3	0.3	NUM
ejpam-4561	108	2	.	.	PUNCT
ejpam-4561	109	1	then	then	ADV
ejpam-4561	109	2	we	we	PRON
ejpam-4561	109	3	have	have	VERB
ejpam-4561	109	4	µ(a	µ(a	PROPN
ejpam-4561	109	5	◦	◦	NOUN
ejpam-4561	109	6	b	b	NOUN
ejpam-4561	109	7	)	)	PUNCT
ejpam-4561	109	8	=	=	SYM
ejpam-4561	109	9	µ(((a	µ(((a	PROPN
ejpam-4561	109	10	◦	◦	NOUN
ejpam-4561	109	11	1	1	NUM
ejpam-4561	109	12	)	)	PUNCT
ejpam-4561	109	13	◦	◦	NOUN
ejpam-4561	109	14	1	1	NUM
ejpam-4561	109	15	)	)	PUNCT
ejpam-4561	109	16	◦	◦	NOUN
ejpam-4561	109	17	b	b	NOUN
ejpam-4561	109	18	)	)	PUNCT
ejpam-4561	109	19	k.	k.	PROPN
ejpam-4561	109	20	belleza	belleza	PROPN
ejpam-4561	109	21	,	,	PUNCT
ejpam-4561	109	22	m.	m.	PROPN
ejpam-4561	109	23	r.	r.	PROPN
ejpam-4561	109	24	v.	v.	PROPN
ejpam-4561	109	25	dicen	dicen	PROPN
ejpam-4561	109	26	/	/	SYM
ejpam-4561	109	27	eur	eur	PROPN
ejpam-4561	109	28	.	.	PUNCT
ejpam-4561	110	1	j.	j.	PROPN
ejpam-4561	110	2	pure	pure	PROPN
ejpam-4561	110	3	appl	appl	PROPN
ejpam-4561	110	4	.	.	PROPN
ejpam-4561	110	5	math	math	PROPN
ejpam-4561	110	6	,	,	PUNCT
ejpam-4561	110	7	15	15	NUM
ejpam-4561	110	8	(	(	PUNCT
ejpam-4561	110	9	4	4	NUM
ejpam-4561	110	10	)	)	PUNCT
ejpam-4561	110	11	(	(	PUNCT
ejpam-4561	110	12	2022	2022	NUM
ejpam-4561	110	13	)	)	PUNCT
ejpam-4561	110	14	,	,	PUNCT
ejpam-4561	110	15	1957	1957	NUM
ejpam-4561	110	16	-	-	SYM
ejpam-4561	110	17	1965	1965	NUM
ejpam-4561	110	18	1962	1962	NUM
ejpam-4561	110	19	≥	≥	NOUN
ejpam-4561	110	20	min{µ((a	min{µ((a	NOUN
ejpam-4561	110	21	◦	◦	NOUN
ejpam-4561	110	22	1	1	NUM
ejpam-4561	110	23	)	)	PUNCT
ejpam-4561	110	24	◦	◦	NOUN
ejpam-4561	110	25	1	1	NUM
ejpam-4561	110	26	)	)	PUNCT
ejpam-4561	110	27	,	,	PUNCT
ejpam-4561	110	28	µ(b	µ(b	PROPN
ejpam-4561	110	29	)	)	PUNCT
ejpam-4561	110	30	}	}	PUNCT
ejpam-4561	110	31	≥	≥	NUM
ejpam-4561	110	32	min{µ(b	min{µ(b	PROPN
ejpam-4561	110	33	)	)	PUNCT
ejpam-4561	110	34	,	,	PUNCT
ejpam-4561	110	35	µ(b	µ(b	PROPN
ejpam-4561	110	36	)	)	PUNCT
ejpam-4561	110	37	}	}	PUNCT
ejpam-4561	110	38	=	=	PUNCT
ejpam-4561	110	39	0.3	0.3	NUM
ejpam-4561	110	40	.	.	PUNCT
ejpam-4561	111	1	hence	hence	ADV
ejpam-4561	111	2	,	,	PUNCT
ejpam-4561	111	3	µ	µ	PRON
ejpam-4561	111	4	is	be	AUX
ejpam-4561	111	5	indeed	indeed	ADV
ejpam-4561	111	6	a	a	DET
ejpam-4561	111	7	fuzzy	fuzzy	ADJ
ejpam-4561	111	8	dual	dual	ADJ
ejpam-4561	111	9	b	b	NOUN
ejpam-4561	111	10	-	-	PUNCT
ejpam-4561	111	11	algebra	algebra	NOUN
ejpam-4561	111	12	.	.	PUNCT
ejpam-4561	112	1	definition	definition	NOUN
ejpam-4561	112	2	4	4	NUM
ejpam-4561	112	3	.	.	PUNCT
ejpam-4561	113	1	a	a	DET
ejpam-4561	113	2	fuzzy	fuzzy	ADJ
ejpam-4561	113	3	set	set	VERB
ejpam-4561	113	4	µ	µ	NOUN
ejpam-4561	113	5	in	in	ADP
ejpam-4561	113	6	x	x	AUX
ejpam-4561	113	7	is	be	AUX
ejpam-4561	113	8	said	say	VERB
ejpam-4561	113	9	to	to	PART
ejpam-4561	113	10	be	be	AUX
ejpam-4561	113	11	fuzzy	fuzzy	ADJ
ejpam-4561	113	12	normal	normal	ADJ
ejpam-4561	113	13	if	if	SCONJ
ejpam-4561	113	14	it	it	PRON
ejpam-4561	113	15	satisfies	satisfy	VERB
ejpam-4561	113	16	the	the	DET
ejpam-4561	113	17	following	follow	VERB
ejpam-4561	113	18	inequality	inequality	NOUN
ejpam-4561	113	19	for	for	ADP
ejpam-4561	113	20	all	all	DET
ejpam-4561	113	21	f	f	PROPN
ejpam-4561	113	22	,	,	PUNCT
ejpam-4561	113	23	g	g	PROPN
ejpam-4561	113	24	,	,	PUNCT
ejpam-4561	113	25	x	x	PRON
ejpam-4561	113	26	,	,	PUNCT
ejpam-4561	113	27	y	y	PROPN
ejpam-4561	113	28	∈	∈	PROPN
ejpam-4561	114	1	x	x	NOUN
ejpam-4561	114	2	:	:	PUNCT
ejpam-4561	114	3	µ((f	µ((f	VERB
ejpam-4561	114	4	◦	◦	NOUN
ejpam-4561	114	5	x	x	NOUN
ejpam-4561	114	6	)	)	PUNCT
ejpam-4561	114	7	◦	◦	NOUN
ejpam-4561	114	8	(	(	PUNCT
ejpam-4561	114	9	g	g	PROPN
ejpam-4561	114	10	◦	◦	NOUN
ejpam-4561	114	11	y	y	PROPN
ejpam-4561	114	12	)	)	PUNCT
ejpam-4561	114	13	)	)	PUNCT
ejpam-4561	115	1	≥	≥	NOUN
ejpam-4561	115	2	min{µ(f	min{µ(f	PROPN
ejpam-4561	115	3	◦	◦	VERB
ejpam-4561	115	4	g	g	NOUN
ejpam-4561	115	5	)	)	PUNCT
ejpam-4561	115	6	,	,	PUNCT
ejpam-4561	115	7	µ(x	µ(x	VERB
ejpam-4561	115	8	◦	◦	NOUN
ejpam-4561	115	9	y	y	NOUN
ejpam-4561	115	10	)	)	PUNCT
ejpam-4561	115	11	}	}	PUNCT
ejpam-4561	115	12	.	.	PUNCT
ejpam-4561	116	1	example	example	NOUN
ejpam-4561	117	1	3	3	NUM
ejpam-4561	117	2	.	.	X
ejpam-4561	117	3	refer	refer	VERB
ejpam-4561	117	4	to	to	ADP
ejpam-4561	117	5	the	the	DET
ejpam-4561	117	6	dual	dual	ADJ
ejpam-4561	117	7	b	b	NOUN
ejpam-4561	117	8	-	-	PUNCT
ejpam-4561	117	9	algebra	algebra	NOUN
ejpam-4561	117	10	from	from	ADP
ejpam-4561	117	11	example	example	NOUN
ejpam-4561	117	12	1	1	NUM
ejpam-4561	117	13	.	.	X
ejpam-4561	118	1	define	define	VERB
ejpam-4561	118	2	a	a	DET
ejpam-4561	118	3	fuzzy	fuzzy	ADJ
ejpam-4561	118	4	set	set	VERB
ejpam-4561	118	5	δ	δ	NOUN
ejpam-4561	118	6	:	:	PUNCT
ejpam-4561	118	7	x	x	X
ejpam-4561	118	8	→	→	PUNCT
ejpam-4561	119	1	[	[	X
ejpam-4561	119	2	0	0	NUM
ejpam-4561	119	3	,	,	PUNCT
ejpam-4561	119	4	1	1	NUM
ejpam-4561	119	5	]	]	PUNCT
ejpam-4561	119	6	by	by	ADP
ejpam-4561	119	7	δ(1	δ(1	NOUN
ejpam-4561	119	8	)	)	PUNCT
ejpam-4561	119	9	=	=	SYM
ejpam-4561	119	10	δ(a	δ(a	PROPN
ejpam-4561	119	11	)	)	PUNCT
ejpam-4561	119	12	=	=	SYM
ejpam-4561	119	13	δ(b	δ(b	PROPN
ejpam-4561	119	14	)	)	PUNCT
ejpam-4561	119	15	=	=	SYM
ejpam-4561	119	16	0.6	0.6	NUM
ejpam-4561	119	17	and	and	CCONJ
ejpam-4561	119	18	δ(c	δ(c	PROPN
ejpam-4561	119	19	)	)	PUNCT
ejpam-4561	119	20	=	=	SYM
ejpam-4561	119	21	δ(d	δ(d	PROPN
ejpam-4561	119	22	)	)	PUNCT
ejpam-4561	119	23	=	=	PUNCT
ejpam-4561	120	1	δ(e	δ(e	X
ejpam-4561	120	2	)	)	PUNCT
ejpam-4561	120	3	=	=	SYM
ejpam-4561	120	4	0.1	0.1	NUM
ejpam-4561	120	5	,	,	PUNCT
ejpam-4561	120	6	then	then	ADV
ejpam-4561	120	7	δ	δ	PROPN
ejpam-4561	120	8	is	be	AUX
ejpam-4561	120	9	a	a	DET
ejpam-4561	120	10	fuzzy	fuzzy	ADJ
ejpam-4561	120	11	normal	normal	ADJ
ejpam-4561	120	12	set	set	NOUN
ejpam-4561	120	13	in	in	ADP
ejpam-4561	120	14	x	x	PUNCT
ejpam-4561	120	15	by	by	ADP
ejpam-4561	120	16	routine	routine	ADJ
ejpam-4561	120	17	calculations	calculation	NOUN
ejpam-4561	120	18	.	.	PUNCT
ejpam-4561	121	1	example	example	NOUN
ejpam-4561	122	1	4	4	X
ejpam-4561	122	2	.	.	PUNCT
ejpam-4561	122	3	let	let	VERB
ejpam-4561	122	4	x	x	PUNCT
ejpam-4561	122	5	=	=	PRON
ejpam-4561	122	6	{	{	PUNCT
ejpam-4561	122	7	1	1	NUM
ejpam-4561	122	8	,	,	PUNCT
ejpam-4561	122	9	a	a	DET
ejpam-4561	122	10	,	,	PUNCT
ejpam-4561	122	11	b	b	NOUN
ejpam-4561	122	12	,	,	PUNCT
ejpam-4561	122	13	c	c	AUX
ejpam-4561	122	14	}	}	PUNCT
ejpam-4561	122	15	be	be	AUX
ejpam-4561	122	16	a	a	DET
ejpam-4561	122	17	set	set	NOUN
ejpam-4561	122	18	with	with	ADP
ejpam-4561	122	19	the	the	DET
ejpam-4561	122	20	following	follow	VERB
ejpam-4561	122	21	table	table	NOUN
ejpam-4561	122	22	:	:	PUNCT
ejpam-4561	122	23	◦	◦	NOUN
ejpam-4561	122	24	1	1	NUM
ejpam-4561	122	25	a	a	DET
ejpam-4561	122	26	b	b	NOUN
ejpam-4561	122	27	c	c	NOUN
ejpam-4561	122	28	1	1	NUM
ejpam-4561	122	29	1	1	NUM
ejpam-4561	122	30	a	a	DET
ejpam-4561	122	31	b	b	NOUN
ejpam-4561	122	32	c	c	NOUN
ejpam-4561	122	33	a	a	DET
ejpam-4561	122	34	c	c	NOUN
ejpam-4561	122	35	1	1	NUM
ejpam-4561	122	36	a	a	DET
ejpam-4561	122	37	b	b	PROPN
ejpam-4561	122	38	b	b	PROPN
ejpam-4561	122	39	b	b	PROPN
ejpam-4561	122	40	c	c	PROPN
ejpam-4561	122	41	1	1	NUM
ejpam-4561	122	42	a	a	DET
ejpam-4561	122	43	c	c	NOUN
ejpam-4561	122	44	a	a	DET
ejpam-4561	122	45	b	b	NOUN
ejpam-4561	122	46	c	c	NOUN
ejpam-4561	122	47	1	1	NUM
ejpam-4561	122	48	then	then	ADV
ejpam-4561	122	49	(	(	PUNCT
ejpam-4561	122	50	x	x	NOUN
ejpam-4561	122	51	,	,	PUNCT
ejpam-4561	122	52	◦	◦	NOUN
ejpam-4561	122	53	,	,	PUNCT
ejpam-4561	122	54	1	1	NUM
ejpam-4561	122	55	)	)	PUNCT
ejpam-4561	122	56	is	be	AUX
ejpam-4561	122	57	a	a	DET
ejpam-4561	122	58	dual	dual	ADJ
ejpam-4561	122	59	b	b	NOUN
ejpam-4561	122	60	-	-	PUNCT
ejpam-4561	122	61	algebra	algebra	NOUN
ejpam-4561	122	62	by	by	ADP
ejpam-4561	122	63	routine	routine	ADJ
ejpam-4561	122	64	calculations	calculation	NOUN
ejpam-4561	122	65	.	.	PUNCT
ejpam-4561	123	1	suppose	suppose	VERB
ejpam-4561	123	2	we	we	PRON
ejpam-4561	123	3	define	define	VERB
ejpam-4561	123	4	a	a	DET
ejpam-4561	123	5	map	map	NOUN
ejpam-4561	123	6	µ	µ	X
ejpam-4561	123	7	:	:	PUNCT
ejpam-4561	123	8	x	x	SYM
ejpam-4561	123	9	→	→	SYM
ejpam-4561	124	1	[	[	X
ejpam-4561	124	2	0	0	NUM
ejpam-4561	124	3	,	,	PUNCT
ejpam-4561	124	4	1	1	NUM
ejpam-4561	124	5	]	]	PUNCT
ejpam-4561	124	6	by	by	ADP
ejpam-4561	124	7	µ(1	µ(1	PROPN
ejpam-4561	124	8	)	)	PUNCT
ejpam-4561	124	9	>	>	X
ejpam-4561	124	10	µ(b	µ(b	PROPN
ejpam-4561	124	11	)	)	PUNCT
ejpam-4561	124	12	>	>	PUNCT
ejpam-4561	124	13	µ(a	µ(a	PROPN
ejpam-4561	124	14	)	)	PUNCT
ejpam-4561	124	15	=	=	SYM
ejpam-4561	124	16	µ(c	µ(c	PROPN
ejpam-4561	124	17	)	)	PUNCT
ejpam-4561	124	18	,	,	PUNCT
ejpam-4561	124	19	then	then	ADV
ejpam-4561	124	20	µ	µ	X
ejpam-4561	124	21	is	be	AUX
ejpam-4561	124	22	a	a	DET
ejpam-4561	124	23	fuzzy	fuzzy	ADJ
ejpam-4561	124	24	normal	normal	ADJ
ejpam-4561	124	25	set	set	NOUN
ejpam-4561	124	26	in	in	ADP
ejpam-4561	124	27	x.	x.	NOUN
ejpam-4561	124	28	and	and	CCONJ
ejpam-4561	124	29	also	also	ADV
ejpam-4561	124	30	,	,	PUNCT
ejpam-4561	124	31	if	if	SCONJ
ejpam-4561	124	32	we	we	PRON
ejpam-4561	124	33	define	define	VERB
ejpam-4561	124	34	a	a	DET
ejpam-4561	124	35	map	map	NOUN
ejpam-4561	124	36	φ	φ	X
ejpam-4561	124	37	:	:	PUNCT
ejpam-4561	124	38	x	x	X
ejpam-4561	124	39	→	→	SYM
ejpam-4561	125	1	[	[	X
ejpam-4561	125	2	0	0	NUM
ejpam-4561	125	3	,	,	PUNCT
ejpam-4561	125	4	1	1	NUM
ejpam-4561	125	5	]	]	PUNCT
ejpam-4561	125	6	by	by	ADP
ejpam-4561	125	7	φ(1	φ(1	PROPN
ejpam-4561	125	8	)	)	PUNCT
ejpam-4561	125	9	=	=	PUNCT
ejpam-4561	126	1	φ(b	φ(b	PROPN
ejpam-4561	126	2	)	)	PUNCT
ejpam-4561	126	3	>	>	X
ejpam-4561	126	4	φ(a	φ(a	PROPN
ejpam-4561	126	5	)	)	PUNCT
ejpam-4561	126	6	=	=	SYM
ejpam-4561	126	7	φ(c	φ(c	NOUN
ejpam-4561	126	8	)	)	PUNCT
ejpam-4561	126	9	,	,	PUNCT
ejpam-4561	126	10	then	then	ADV
ejpam-4561	126	11	φ	φ	PROPN
ejpam-4561	126	12	is	be	AUX
ejpam-4561	126	13	also	also	ADV
ejpam-4561	126	14	a	a	DET
ejpam-4561	126	15	fuzzy	fuzzy	ADJ
ejpam-4561	126	16	normal	normal	ADJ
ejpam-4561	126	17	set	set	NOUN
ejpam-4561	126	18	in	in	ADP
ejpam-4561	126	19	x.	x.	NOUN
ejpam-4561	126	20	theorem	theorem	VERB
ejpam-4561	126	21	2	2	NUM
ejpam-4561	126	22	.	.	PUNCT
ejpam-4561	127	1	every	every	DET
ejpam-4561	127	2	fuzzy	fuzzy	ADJ
ejpam-4561	127	3	normal	normal	ADJ
ejpam-4561	127	4	set	set	VERB
ejpam-4561	127	5	µ	µ	NOUN
ejpam-4561	127	6	in	in	ADP
ejpam-4561	127	7	x	x	VERB
ejpam-4561	127	8	is	be	AUX
ejpam-4561	127	9	a	a	DET
ejpam-4561	127	10	fuzzy	fuzzy	ADJ
ejpam-4561	127	11	dual	dual	ADJ
ejpam-4561	127	12	b	b	NOUN
ejpam-4561	127	13	-	-	PUNCT
ejpam-4561	127	14	algebra	algebra	NOUN
ejpam-4561	127	15	.	.	PUNCT
ejpam-4561	128	1	proof	proof	NOUN
ejpam-4561	128	2	.	.	PUNCT
ejpam-4561	129	1	since	since	SCONJ
ejpam-4561	129	2	µ	µ	NOUN
ejpam-4561	129	3	is	be	AUX
ejpam-4561	129	4	fuzzy	fuzzy	ADJ
ejpam-4561	129	5	normal	normal	ADJ
ejpam-4561	129	6	,	,	PUNCT
ejpam-4561	129	7	recall	recall	VERB
ejpam-4561	129	8	definition	definition	NOUN
ejpam-4561	129	9	4	4	NUM
ejpam-4561	129	10	.	.	PUNCT
ejpam-4561	130	1	for	for	ADP
ejpam-4561	130	2	any	any	DET
ejpam-4561	130	3	x	x	NOUN
ejpam-4561	130	4	,	,	PUNCT
ejpam-4561	130	5	y	y	PROPN
ejpam-4561	130	6	∈	∈	PROPN
ejpam-4561	130	7	x	x	X
ejpam-4561	130	8	,	,	PUNCT
ejpam-4561	130	9	we	we	PRON
ejpam-4561	130	10	have	have	VERB
ejpam-4561	130	11	,	,	PUNCT
ejpam-4561	130	12	µ((x	µ((x	VERB
ejpam-4561	130	13	◦	◦	NOUN
ejpam-4561	130	14	y	y	NOUN
ejpam-4561	130	15	)	)	PUNCT
ejpam-4561	130	16	=	=	VERB
ejpam-4561	130	17	µ((1	µ((1	NOUN
ejpam-4561	130	18	◦	◦	VERB
ejpam-4561	130	19	1	1	NUM
ejpam-4561	130	20	)	)	PUNCT
ejpam-4561	130	21	◦	◦	NOUN
ejpam-4561	130	22	(	(	PUNCT
ejpam-4561	130	23	x	x	PART
ejpam-4561	130	24	◦	◦	VERB
ejpam-4561	130	25	y	y	PROPN
ejpam-4561	130	26	)	)	PUNCT
ejpam-4561	130	27	)	)	PUNCT
ejpam-4561	130	28	≥	≥	NOUN
ejpam-4561	130	29	min{µ(1	min{µ(1	PROPN
ejpam-4561	130	30	◦	◦	NOUN
ejpam-4561	130	31	x	x	X
ejpam-4561	130	32	)	)	PUNCT
ejpam-4561	130	33	,	,	PUNCT
ejpam-4561	130	34	µ(1	µ(1	PROPN
ejpam-4561	130	35	◦	◦	NOUN
ejpam-4561	130	36	y	y	PROPN
ejpam-4561	130	37	)	)	PUNCT
ejpam-4561	130	38	}	}	PUNCT
ejpam-4561	131	1	=	=	SYM
ejpam-4561	131	2	min{µ(x	min{µ(x	NOUN
ejpam-4561	131	3	)	)	PUNCT
ejpam-4561	131	4	,	,	PUNCT
ejpam-4561	131	5	µ(y	µ(y	PROPN
ejpam-4561	131	6	)	)	PUNCT
ejpam-4561	131	7	}	}	PUNCT
ejpam-4561	131	8	.	.	PUNCT
ejpam-4561	132	1	[	[	X
ejpam-4561	132	2	by	by	ADP
ejpam-4561	132	3	(	(	PUNCT
ejpam-4561	132	4	db2	db2	PROPN
ejpam-4561	132	5	)	)	PUNCT
ejpam-4561	132	6	]	]	PUNCT
ejpam-4561	132	7	hence	hence	ADV
ejpam-4561	132	8	,	,	PUNCT
ejpam-4561	132	9	µ	µ	X
ejpam-4561	132	10	is	be	AUX
ejpam-4561	132	11	a	a	DET
ejpam-4561	132	12	fuzzy	fuzzy	ADJ
ejpam-4561	132	13	dual	dual	ADJ
ejpam-4561	132	14	b	b	NOUN
ejpam-4561	132	15	-	-	PUNCT
ejpam-4561	132	16	algebra	algebra	NOUN
ejpam-4561	132	17	.	.	PUNCT
ejpam-4561	133	1	remark	remark	NOUN
ejpam-4561	133	2	1	1	NUM
ejpam-4561	133	3	.	.	PUNCT
ejpam-4561	134	1	the	the	DET
ejpam-4561	134	2	converse	converse	NOUN
ejpam-4561	134	3	of	of	ADP
ejpam-4561	134	4	theorem	theorem	ADJ
ejpam-4561	134	5	2	2	NUM
ejpam-4561	134	6	is	be	AUX
ejpam-4561	134	7	not	not	PART
ejpam-4561	134	8	always	always	ADV
ejpam-4561	134	9	true	true	ADJ
ejpam-4561	134	10	.	.	PUNCT
ejpam-4561	135	1	example	example	NOUN
ejpam-4561	135	2	1	1	NUM
ejpam-4561	135	3	above	above	ADV
ejpam-4561	135	4	is	be	AUX
ejpam-4561	135	5	the	the	DET
ejpam-4561	135	6	correct	correct	ADJ
ejpam-4561	135	7	representation	representation	NOUN
ejpam-4561	135	8	of	of	ADP
ejpam-4561	135	9	this	this	DET
ejpam-4561	135	10	remark	remark	NOUN
ejpam-4561	135	11	.	.	PUNCT
ejpam-4561	136	1	now	now	ADV
ejpam-4561	136	2	,	,	PUNCT
ejpam-4561	136	3	note	note	VERB
ejpam-4561	136	4	that	that	SCONJ
ejpam-4561	136	5	µ((c	µ((c	PROPN
ejpam-4561	136	6	◦	◦	NOUN
ejpam-4561	136	7	d	d	NOUN
ejpam-4561	136	8	)	)	PUNCT
ejpam-4561	136	9	◦	◦	NOUN
ejpam-4561	136	10	(	(	PUNCT
ejpam-4561	136	11	1	1	NUM
ejpam-4561	136	12	◦	◦	NOUN
ejpam-4561	136	13	b	b	NOUN
ejpam-4561	136	14	)	)	PUNCT
ejpam-4561	136	15	)	)	PUNCT
ejpam-4561	137	1	=	=	PUNCT
ejpam-4561	137	2	µ(b	µ(b	NOUN
ejpam-4561	137	3	◦	◦	VERB
ejpam-4561	137	4	a	a	X
ejpam-4561	137	5	)	)	PUNCT
ejpam-4561	137	6	=	=	SYM
ejpam-4561	137	7	µ(a	µ(a	PROPN
ejpam-4561	137	8	)	)	PUNCT
ejpam-4561	137	9	=	=	PUNCT
ejpam-4561	137	10	0.3	0.3	NUM
ejpam-4561	137	11	.	.	PUNCT
ejpam-4561	138	1	then	then	ADV
ejpam-4561	138	2	we	we	PRON
ejpam-4561	138	3	have	have	VERB
ejpam-4561	138	4	,	,	PUNCT
ejpam-4561	138	5	µ((c	µ((c	NOUN
ejpam-4561	138	6	◦	◦	NOUN
ejpam-4561	138	7	d	d	NOUN
ejpam-4561	138	8	)	)	PUNCT
ejpam-4561	138	9	◦	◦	NOUN
ejpam-4561	138	10	(	(	PUNCT
ejpam-4561	138	11	1	1	NUM
ejpam-4561	138	12	◦	◦	NOUN
ejpam-4561	138	13	b	b	NOUN
ejpam-4561	138	14	)	)	PUNCT
ejpam-4561	138	15	)	)	PUNCT
ejpam-4561	139	1	=	=	SYM
ejpam-4561	139	2	min{µ(c	min{µ(c	PROPN
ejpam-4561	139	3	◦	◦	NOUN
ejpam-4561	139	4	1	1	NUM
ejpam-4561	139	5	)	)	PUNCT
ejpam-4561	139	6	,	,	PUNCT
ejpam-4561	139	7	µ(d	µ(d	PROPN
ejpam-4561	139	8	◦	◦	NOUN
ejpam-4561	139	9	b	b	NOUN
ejpam-4561	139	10	)	)	PUNCT
ejpam-4561	139	11	}	}	PUNCT
ejpam-4561	139	12	=	=	SYM
ejpam-4561	139	13	min{µ(c	min{µ(c	PROPN
ejpam-4561	139	14	)	)	PUNCT
ejpam-4561	139	15	,	,	PUNCT
ejpam-4561	139	16	µ(c	µ(c	PROPN
ejpam-4561	139	17	)	)	PUNCT
ejpam-4561	139	18	}	}	PUNCT
ejpam-4561	139	19	<	<	X
ejpam-4561	139	20	0.8	0.8	NUM
ejpam-4561	139	21	.	.	PUNCT
ejpam-4561	140	1	k.	k.	PROPN
ejpam-4561	140	2	belleza	belleza	PROPN
ejpam-4561	140	3	,	,	PUNCT
ejpam-4561	140	4	m.	m.	PROPN
ejpam-4561	140	5	r.	r.	PROPN
ejpam-4561	140	6	v.	v.	PROPN
ejpam-4561	140	7	dicen	dicen	PROPN
ejpam-4561	140	8	/	/	SYM
ejpam-4561	140	9	eur	eur	PROPN
ejpam-4561	140	10	.	.	PUNCT
ejpam-4561	141	1	j.	j.	PROPN
ejpam-4561	141	2	pure	pure	PROPN
ejpam-4561	141	3	appl	appl	PROPN
ejpam-4561	141	4	.	.	PROPN
ejpam-4561	141	5	math	math	PROPN
ejpam-4561	141	6	,	,	PUNCT
ejpam-4561	141	7	15	15	NUM
ejpam-4561	141	8	(	(	PUNCT
ejpam-4561	141	9	4	4	NUM
ejpam-4561	141	10	)	)	PUNCT
ejpam-4561	141	11	(	(	PUNCT
ejpam-4561	141	12	2022	2022	NUM
ejpam-4561	141	13	)	)	PUNCT
ejpam-4561	141	14	,	,	PUNCT
ejpam-4561	141	15	1957	1957	NUM
ejpam-4561	141	16	-	-	SYM
ejpam-4561	141	17	1965	1965	NUM
ejpam-4561	141	18	1963	1963	NUM
ejpam-4561	141	19	manipulation	manipulation	NOUN
ejpam-4561	141	20	above	above	ADV
ejpam-4561	141	21	shows	show	VERB
ejpam-4561	141	22	that	that	SCONJ
ejpam-4561	141	23	the	the	DET
ejpam-4561	141	24	fuzzy	fuzzy	ADJ
ejpam-4561	141	25	dual	dual	ADJ
ejpam-4561	141	26	b	b	NOUN
ejpam-4561	141	27	-	-	PUNCT
ejpam-4561	141	28	algebra	algebra	NOUN
ejpam-4561	141	29	is	be	AUX
ejpam-4561	141	30	not	not	PART
ejpam-4561	141	31	fuzzy	fuzzy	ADJ
ejpam-4561	141	32	normal	normal	ADJ
ejpam-4561	141	33	.	.	PUNCT
ejpam-4561	142	1	thus	thus	ADV
ejpam-4561	142	2	,	,	PUNCT
ejpam-4561	142	3	the	the	DET
ejpam-4561	142	4	remark	remark	NOUN
ejpam-4561	142	5	holds	hold	VERB
ejpam-4561	142	6	true	true	ADJ
ejpam-4561	142	7	.	.	PUNCT
ejpam-4561	143	1	definition	definition	NOUN
ejpam-4561	143	2	5	5	NUM
ejpam-4561	143	3	.	.	PUNCT
ejpam-4561	144	1	a	a	DET
ejpam-4561	144	2	fuzzy	fuzzy	ADJ
ejpam-4561	144	3	set	set	VERB
ejpam-4561	144	4	µ	µ	NOUN
ejpam-4561	144	5	in	in	ADP
ejpam-4561	144	6	x	x	AUX
ejpam-4561	144	7	is	be	AUX
ejpam-4561	144	8	called	call	VERB
ejpam-4561	144	9	fuzzy	fuzzy	ADJ
ejpam-4561	144	10	normal	normal	ADJ
ejpam-4561	144	11	dual	dual	ADJ
ejpam-4561	144	12	b	b	NOUN
ejpam-4561	144	13	-	-	PUNCT
ejpam-4561	144	14	algebra	algebra	NOUN
ejpam-4561	144	15	if	if	SCONJ
ejpam-4561	144	16	it	it	PRON
ejpam-4561	144	17	is	be	AUX
ejpam-4561	144	18	a	a	DET
ejpam-4561	144	19	fuzzy	fuzzy	ADJ
ejpam-4561	144	20	dual	dual	ADJ
ejpam-4561	144	21	b	b	NOUN
ejpam-4561	144	22	-	-	PUNCT
ejpam-4561	144	23	algebra	algebra	NOUN
ejpam-4561	144	24	which	which	PRON
ejpam-4561	144	25	is	be	AUX
ejpam-4561	144	26	fuzzy	fuzzy	ADJ
ejpam-4561	144	27	normal	normal	ADJ
ejpam-4561	144	28	.	.	PUNCT
ejpam-4561	144	29	example	example	NOUN
ejpam-4561	145	1	5	5	NUM
ejpam-4561	145	2	.	.	PUNCT
ejpam-4561	146	1	the	the	DET
ejpam-4561	146	2	fuzzy	fuzzy	ADJ
ejpam-4561	146	3	sets	set	NOUN
ejpam-4561	146	4	discussed	discuss	VERB
ejpam-4561	146	5	in	in	ADP
ejpam-4561	146	6	example	example	NOUN
ejpam-4561	146	7	3	3	NUM
ejpam-4561	146	8	and	and	CCONJ
ejpam-4561	146	9	example	example	NOUN
ejpam-4561	146	10	4	4	NUM
ejpam-4561	146	11	are	be	AUX
ejpam-4561	146	12	indeed	indeed	ADV
ejpam-4561	146	13	fuzzy	fuzzy	ADJ
ejpam-4561	146	14	normal	normal	ADJ
ejpam-4561	146	15	dual	dual	ADJ
ejpam-4561	146	16	b	b	NOUN
ejpam-4561	146	17	-	-	PUNCT
ejpam-4561	146	18	algebras	algebra	VERB
ejpam-4561	146	19	by	by	ADP
ejpam-4561	146	20	routine	routine	ADJ
ejpam-4561	146	21	calculations	calculation	NOUN
ejpam-4561	146	22	.	.	PUNCT
ejpam-4561	147	1	proposition	proposition	NOUN
ejpam-4561	147	2	4	4	NUM
ejpam-4561	147	3	.	.	PUNCT
ejpam-4561	148	1	if	if	SCONJ
ejpam-4561	148	2	a	a	DET
ejpam-4561	148	3	fuzzy	fuzzy	ADJ
ejpam-4561	148	4	set	set	VERB
ejpam-4561	148	5	µ	µ	NOUN
ejpam-4561	148	6	in	in	ADP
ejpam-4561	148	7	x	x	VERB
ejpam-4561	148	8	is	be	AUX
ejpam-4561	148	9	a	a	DET
ejpam-4561	148	10	fuzzy	fuzzy	ADJ
ejpam-4561	148	11	normal	normal	ADJ
ejpam-4561	148	12	dual	dual	ADJ
ejpam-4561	148	13	b	b	NOUN
ejpam-4561	148	14	-	-	PUNCT
ejpam-4561	148	15	algebra	algebra	NOUN
ejpam-4561	148	16	,	,	PUNCT
ejpam-4561	148	17	then	then	ADV
ejpam-4561	148	18	µ(x	µ(x	VERB
ejpam-4561	148	19	◦	◦	NOUN
ejpam-4561	148	20	y	y	NOUN
ejpam-4561	148	21	)	)	PUNCT
ejpam-4561	148	22	=	=	SYM
ejpam-4561	148	23	µ(y	µ(y	PROPN
ejpam-4561	148	24	◦	◦	NOUN
ejpam-4561	148	25	x	x	X
ejpam-4561	148	26	)	)	PUNCT
ejpam-4561	148	27	for	for	ADP
ejpam-4561	148	28	all	all	DET
ejpam-4561	148	29	x	x	NOUN
ejpam-4561	148	30	,	,	PUNCT
ejpam-4561	148	31	y	y	PROPN
ejpam-4561	148	32	∈	∈	PROPN
ejpam-4561	148	33	x.	x.	NOUN
ejpam-4561	148	34	proof	proof	NOUN
ejpam-4561	148	35	.	.	PUNCT
ejpam-4561	149	1	let	let	VERB
ejpam-4561	149	2	x	x	PRON
ejpam-4561	149	3	,	,	PUNCT
ejpam-4561	149	4	y	y	PROPN
ejpam-4561	149	5	∈	∈	PROPN
ejpam-4561	149	6	x.	x.	NOUN
ejpam-4561	149	7	then	then	ADV
ejpam-4561	149	8	µ(x	µ(x	X
ejpam-4561	149	9	◦	◦	NOUN
ejpam-4561	149	10	y	y	NOUN
ejpam-4561	149	11	)	)	PUNCT
ejpam-4561	150	1	=	=	PUNCT
ejpam-4561	150	2	µ(1	µ(1	PROPN
ejpam-4561	150	3	◦	◦	NOUN
ejpam-4561	150	4	(	(	PUNCT
ejpam-4561	150	5	x	x	SYM
ejpam-4561	150	6	◦	◦	VERB
ejpam-4561	150	7	y	y	PROPN
ejpam-4561	150	8	)	)	PUNCT
ejpam-4561	150	9	)	)	PUNCT
ejpam-4561	151	1	[	[	X
ejpam-4561	151	2	by	by	ADP
ejpam-4561	151	3	db2	db2	PROPN
ejpam-4561	151	4	]	]	PUNCT
ejpam-4561	151	5	=	=	PUNCT
ejpam-4561	151	6	µ((y	µ((y	VERB
ejpam-4561	151	7	◦	◦	VERB
ejpam-4561	151	8	y	y	NOUN
ejpam-4561	151	9	)	)	PUNCT
ejpam-4561	151	10	◦	◦	NOUN
ejpam-4561	151	11	(	(	PUNCT
ejpam-4561	151	12	x	x	PART
ejpam-4561	151	13	◦	◦	VERB
ejpam-4561	151	14	y	y	PROPN
ejpam-4561	151	15	)	)	PUNCT
ejpam-4561	151	16	)	)	PUNCT
ejpam-4561	152	1	[	[	X
ejpam-4561	152	2	by	by	ADP
ejpam-4561	152	3	db1	db1	PROPN
ejpam-4561	152	4	]	]	X
ejpam-4561	152	5	≥	≥	NOUN
ejpam-4561	152	6	min{µ(y	min{µ(y	PUNCT
ejpam-4561	152	7	◦	◦	NOUN
ejpam-4561	152	8	x	x	X
ejpam-4561	152	9	)	)	PUNCT
ejpam-4561	152	10	,	,	PUNCT
ejpam-4561	152	11	µ(y	µ(y	PROPN
ejpam-4561	152	12	◦	◦	NOUN
ejpam-4561	152	13	y	y	NOUN
ejpam-4561	152	14	)	)	PUNCT
ejpam-4561	152	15	}	}	PUNCT
ejpam-4561	153	1	[	[	X
ejpam-4561	153	2	since	since	SCONJ
ejpam-4561	153	3	µ	µ	NOUN
ejpam-4561	153	4	is	be	AUX
ejpam-4561	153	5	fuzzy	fuzzy	ADJ
ejpam-4561	153	6	normal	normal	ADJ
ejpam-4561	153	7	]	]	X
ejpam-4561	154	1	=	=	PUNCT
ejpam-4561	154	2	min{µ(y	min{µ(y	PROPN
ejpam-4561	154	3	◦	◦	NOUN
ejpam-4561	154	4	x	x	X
ejpam-4561	154	5	)	)	PUNCT
ejpam-4561	154	6	,	,	PUNCT
ejpam-4561	154	7	µ(1	µ(1	PROPN
ejpam-4561	154	8	)	)	PUNCT
ejpam-4561	154	9	}	}	PUNCT
ejpam-4561	155	1	[	[	X
ejpam-4561	155	2	by	by	ADP
ejpam-4561	155	3	db1	db1	NOUN
ejpam-4561	155	4	]	]	PUNCT
ejpam-4561	155	5	≥	≥	NOUN
ejpam-4561	155	6	µ(y	µ(y	PROPN
ejpam-4561	155	7	◦	◦	NOUN
ejpam-4561	155	8	x	x	X
ejpam-4561	155	9	)	)	PUNCT
ejpam-4561	155	10	.	.	PUNCT
ejpam-4561	156	1	[	[	X
ejpam-4561	156	2	by	by	ADP
ejpam-4561	156	3	proposition	proposition	NOUN
ejpam-4561	156	4	1	1	NUM
ejpam-4561	156	5	]	]	PUNCT
ejpam-4561	156	6	interchanging	interchange	VERB
ejpam-4561	156	7	x	x	PUNCT
ejpam-4561	156	8	with	with	ADP
ejpam-4561	156	9	y	y	PROPN
ejpam-4561	156	10	,	,	PUNCT
ejpam-4561	156	11	we	we	PRON
ejpam-4561	156	12	obtain	obtain	VERB
ejpam-4561	156	13	µ(y	µ(y	PROPN
ejpam-4561	156	14	◦	◦	NOUN
ejpam-4561	156	15	x	x	X
ejpam-4561	156	16	)	)	PUNCT
ejpam-4561	156	17	=	=	PUNCT
ejpam-4561	156	18	µ(1	µ(1	PROPN
ejpam-4561	156	19	◦	◦	NOUN
ejpam-4561	156	20	(	(	PUNCT
ejpam-4561	156	21	y	y	PROPN
ejpam-4561	156	22	◦	◦	NOUN
ejpam-4561	156	23	x	x	NOUN
ejpam-4561	156	24	)	)	PUNCT
ejpam-4561	156	25	)	)	PUNCT
ejpam-4561	157	1	[	[	X
ejpam-4561	157	2	by	by	ADP
ejpam-4561	157	3	db2	db2	PROPN
ejpam-4561	157	4	]	]	X
ejpam-4561	157	5	=	=	PUNCT
ejpam-4561	157	6	µ((x	µ((x	SYM
ejpam-4561	157	7	◦	◦	NOUN
ejpam-4561	157	8	x	x	SYM
ejpam-4561	157	9	)	)	PUNCT
ejpam-4561	157	10	◦	◦	NOUN
ejpam-4561	157	11	(	(	PUNCT
ejpam-4561	157	12	y	y	PROPN
ejpam-4561	157	13	◦	◦	NOUN
ejpam-4561	157	14	x	x	NOUN
ejpam-4561	157	15	)	)	PUNCT
ejpam-4561	157	16	)	)	PUNCT
ejpam-4561	158	1	[	[	X
ejpam-4561	158	2	by	by	ADP
ejpam-4561	158	3	db1	db1	NOUN
ejpam-4561	158	4	]	]	PUNCT
ejpam-4561	158	5	≥	≥	NOUN
ejpam-4561	158	6	min{µ(x	min{µ(x	PROPN
ejpam-4561	158	7	◦	◦	NOUN
ejpam-4561	158	8	y	y	PROPN
ejpam-4561	158	9	)	)	PUNCT
ejpam-4561	158	10	,	,	PUNCT
ejpam-4561	158	11	µ(x	µ(x	VERB
ejpam-4561	158	12	◦	◦	NOUN
ejpam-4561	158	13	x	x	NOUN
ejpam-4561	158	14	)	)	PUNCT
ejpam-4561	158	15	}	}	PUNCT
ejpam-4561	159	1	[	[	X
ejpam-4561	159	2	since	since	SCONJ
ejpam-4561	159	3	µ	µ	NOUN
ejpam-4561	159	4	is	be	AUX
ejpam-4561	159	5	fuzzy	fuzzy	ADJ
ejpam-4561	159	6	normal	normal	ADJ
ejpam-4561	159	7	]	]	X
ejpam-4561	159	8	=	=	SYM
ejpam-4561	159	9	min{µ(x	min{µ(x	PROPN
ejpam-4561	159	10	◦	◦	NOUN
ejpam-4561	159	11	y	y	PROPN
ejpam-4561	159	12	)	)	PUNCT
ejpam-4561	159	13	,	,	PUNCT
ejpam-4561	159	14	µ(1	µ(1	PROPN
ejpam-4561	159	15	)	)	PUNCT
ejpam-4561	159	16	}	}	PUNCT
ejpam-4561	160	1	[	[	X
ejpam-4561	160	2	db1	db1	NOUN
ejpam-4561	160	3	]	]	PUNCT
ejpam-4561	160	4	≥	≥	NOUN
ejpam-4561	160	5	µ(x	µ(x	PUNCT
ejpam-4561	160	6	◦	◦	NOUN
ejpam-4561	160	7	y	y	NOUN
ejpam-4561	160	8	)	)	PUNCT
ejpam-4561	160	9	.	.	PUNCT
ejpam-4561	161	1	[	[	X
ejpam-4561	161	2	by	by	ADP
ejpam-4561	161	3	proposition	proposition	NOUN
ejpam-4561	161	4	1	1	NUM
ejpam-4561	161	5	]	]	PUNCT
ejpam-4561	161	6	by	by	ADP
ejpam-4561	161	7	the	the	DET
ejpam-4561	161	8	manipulation	manipulation	NOUN
ejpam-4561	161	9	above	above	ADV
ejpam-4561	161	10	,	,	PUNCT
ejpam-4561	161	11	we	we	PRON
ejpam-4561	161	12	conclude	conclude	VERB
ejpam-4561	161	13	that	that	SCONJ
ejpam-4561	161	14	µ(x	µ(x	VERB
ejpam-4561	161	15	◦	◦	NOUN
ejpam-4561	161	16	y	y	NOUN
ejpam-4561	161	17	)	)	PUNCT
ejpam-4561	161	18	=	=	SYM
ejpam-4561	161	19	µ(y	µ(y	PROPN
ejpam-4561	161	20	◦	◦	NOUN
ejpam-4561	161	21	x	x	X
ejpam-4561	161	22	)	)	PUNCT
ejpam-4561	161	23	for	for	ADP
ejpam-4561	161	24	all	all	DET
ejpam-4561	161	25	x	x	NOUN
ejpam-4561	161	26	,	,	PUNCT
ejpam-4561	161	27	y	y	PROPN
ejpam-4561	161	28	∈	∈	PROPN
ejpam-4561	161	29	x.	x.	NOUN
ejpam-4561	162	1	the	the	DET
ejpam-4561	162	2	following	following	ADJ
ejpam-4561	162	3	result	result	NOUN
ejpam-4561	162	4	will	will	AUX
ejpam-4561	162	5	be	be	AUX
ejpam-4561	162	6	used	use	VERB
ejpam-4561	162	7	in	in	ADP
ejpam-4561	162	8	the	the	DET
ejpam-4561	162	9	characterization	characterization	NOUN
ejpam-4561	162	10	of	of	ADP
ejpam-4561	162	11	fuzzy	fuzzy	ADJ
ejpam-4561	162	12	normal	normal	ADJ
ejpam-4561	162	13	dual	dual	ADJ
ejpam-4561	162	14	b	b	NOUN
ejpam-4561	162	15	-	-	PUNCT
ejpam-4561	162	16	algebra	algebra	NOUN
ejpam-4561	162	17	.	.	PUNCT
ejpam-4561	163	1	theorem	theorem	NOUN
ejpam-4561	163	2	3	3	X
ejpam-4561	163	3	.	.	PUNCT
ejpam-4561	164	1	let	let	VERB
ejpam-4561	164	2	µ	µ	X
ejpam-4561	164	3	be	be	AUX
ejpam-4561	164	4	a	a	DET
ejpam-4561	164	5	fuzzy	fuzzy	ADJ
ejpam-4561	164	6	normal	normal	ADJ
ejpam-4561	164	7	dual	dual	ADJ
ejpam-4561	164	8	b	b	NOUN
ejpam-4561	164	9	-	-	PUNCT
ejpam-4561	164	10	algebra	algebra	NOUN
ejpam-4561	164	11	.	.	PUNCT
ejpam-4561	165	1	then	then	ADV
ejpam-4561	165	2	the	the	DET
ejpam-4561	165	3	following	follow	VERB
ejpam-4561	165	4	set	set	NOUN
ejpam-4561	165	5	is	be	AUX
ejpam-4561	165	6	a	a	DET
ejpam-4561	165	7	normal	normal	ADJ
ejpam-4561	165	8	dual	dual	ADJ
ejpam-4561	165	9	b	b	NOUN
ejpam-4561	165	10	-	-	PUNCT
ejpam-4561	165	11	subalgebra	subalgebra	NOUN
ejpam-4561	165	12	of	of	ADP
ejpam-4561	165	13	x	x	PRON
ejpam-4561	165	14	,	,	PUNCT
ejpam-4561	165	15	xµ	xµ	X
ejpam-4561	165	16	=	=	SYM
ejpam-4561	165	17	{	{	PUNCT
ejpam-4561	165	18	x	x	PUNCT
ejpam-4561	165	19	∈	∈	PROPN
ejpam-4561	165	20	x	x	X
ejpam-4561	165	21	|	|	ADV
ejpam-4561	165	22	µ(x	µ(x	VERB
ejpam-4561	165	23	)	)	PUNCT
ejpam-4561	165	24	=	=	SYM
ejpam-4561	165	25	µ(1	µ(1	PROPN
ejpam-4561	165	26	)	)	PUNCT
ejpam-4561	165	27	}	}	PUNCT
ejpam-4561	165	28	.	.	PUNCT
ejpam-4561	166	1	proof	proof	NOUN
ejpam-4561	166	2	.	.	PUNCT
ejpam-4561	167	1	it	it	PRON
ejpam-4561	167	2	is	be	AUX
ejpam-4561	167	3	sufficient	sufficient	ADJ
ejpam-4561	167	4	to	to	PART
ejpam-4561	167	5	show	show	VERB
ejpam-4561	167	6	that	that	SCONJ
ejpam-4561	167	7	xµ	xµ	PROPN
ejpam-4561	167	8	is	be	AUX
ejpam-4561	167	9	normal	normal	ADJ
ejpam-4561	167	10	since	since	SCONJ
ejpam-4561	167	11	µ	µ	NOUN
ejpam-4561	167	12	is	be	AUX
ejpam-4561	167	13	a	a	DET
ejpam-4561	167	14	fuzzy	fuzzy	ADJ
ejpam-4561	167	15	normal	normal	ADJ
ejpam-4561	167	16	dual	dual	ADJ
ejpam-4561	167	17	balgebra	balgebra	NOUN
ejpam-4561	167	18	.	.	PUNCT
ejpam-4561	168	1	let	let	VERB
ejpam-4561	168	2	f	f	X
ejpam-4561	168	3	,	,	PUNCT
ejpam-4561	168	4	g	g	PROPN
ejpam-4561	168	5	,	,	PUNCT
ejpam-4561	168	6	x	x	PRON
ejpam-4561	168	7	,	,	PUNCT
ejpam-4561	168	8	y	y	PROPN
ejpam-4561	168	9	∈	∈	PROPN
ejpam-4561	168	10	x	x	AUX
ejpam-4561	168	11	be	be	AUX
ejpam-4561	168	12	such	such	ADJ
ejpam-4561	168	13	that	that	SCONJ
ejpam-4561	168	14	x	x	SYM
ejpam-4561	168	15	◦	◦	VERB
ejpam-4561	168	16	y	y	PROPN
ejpam-4561	168	17	∈	∈	PROPN
ejpam-4561	168	18	xµ	xµ	X
ejpam-4561	168	19	and	and	CCONJ
ejpam-4561	168	20	f	f	PROPN
ejpam-4561	168	21	◦	◦	NOUN
ejpam-4561	168	22	g	g	NOUN
ejpam-4561	168	23	∈	∈	PROPN
ejpam-4561	168	24	xµ.	xµ.	PROPN
ejpam-4561	168	25	then	then	ADV
ejpam-4561	168	26	µ(x	µ(x	VERB
ejpam-4561	168	27	◦	◦	NOUN
ejpam-4561	168	28	y	y	NOUN
ejpam-4561	168	29	)	)	PUNCT
ejpam-4561	169	1	=	=	PUNCT
ejpam-4561	169	2	µ(1	µ(1	PROPN
ejpam-4561	169	3	)	)	PUNCT
ejpam-4561	169	4	=	=	PUNCT
ejpam-4561	170	1	µ(f	µ(f	PROPN
ejpam-4561	170	2	◦	◦	NOUN
ejpam-4561	170	3	g	g	NOUN
ejpam-4561	170	4	)	)	PUNCT
ejpam-4561	170	5	.	.	PUNCT
ejpam-4561	171	1	considering	consider	VERB
ejpam-4561	171	2	µ	µ	X
ejpam-4561	171	3	is	be	AUX
ejpam-4561	171	4	fuzzy	fuzzy	ADJ
ejpam-4561	171	5	normal	normal	ADJ
ejpam-4561	171	6	,	,	PUNCT
ejpam-4561	171	7	it	it	PRON
ejpam-4561	171	8	follows	follow	VERB
ejpam-4561	171	9	that	that	SCONJ
ejpam-4561	171	10	µ((x	µ((x	NOUN
ejpam-4561	171	11	◦	◦	NOUN
ejpam-4561	171	12	f	f	X
ejpam-4561	171	13	)	)	PUNCT
ejpam-4561	171	14	◦	◦	NOUN
ejpam-4561	171	15	(	(	PUNCT
ejpam-4561	171	16	y	y	NOUN
ejpam-4561	171	17	◦	◦	VERB
ejpam-4561	171	18	g	g	NOUN
ejpam-4561	171	19	)	)	PUNCT
ejpam-4561	171	20	)	)	PUNCT
ejpam-4561	172	1	≥	≥	X
ejpam-4561	173	1	min{µ(x	min{µ(x	NOUN
ejpam-4561	173	2	◦	◦	NOUN
ejpam-4561	173	3	y	y	PROPN
ejpam-4561	173	4	)	)	PUNCT
ejpam-4561	173	5	,	,	PUNCT
ejpam-4561	173	6	µ(f	µ(f	PROPN
ejpam-4561	173	7	◦	◦	VERB
ejpam-4561	173	8	g	g	NOUN
ejpam-4561	173	9	)	)	PUNCT
ejpam-4561	173	10	}	}	PUNCT
ejpam-4561	173	11	≥	≥	NOUN
ejpam-4561	173	12	min{µ(1	min{µ(1	NOUN
ejpam-4561	173	13	)	)	PUNCT
ejpam-4561	173	14	,	,	PUNCT
ejpam-4561	173	15	µ(1	µ(1	PROPN
ejpam-4561	173	16	)	)	PUNCT
ejpam-4561	173	17	}	}	PUNCT
ejpam-4561	174	1	=	=	SYM
ejpam-4561	174	2	µ(1	µ(1	PROPN
ejpam-4561	174	3	)	)	PUNCT
ejpam-4561	174	4	.	.	PUNCT
ejpam-4561	175	1	now	now	ADV
ejpam-4561	175	2	,	,	PUNCT
ejpam-4561	175	3	by	by	ADP
ejpam-4561	175	4	proposition	proposition	NOUN
ejpam-4561	175	5	1	1	NUM
ejpam-4561	175	6	,	,	PUNCT
ejpam-4561	175	7	µ(1	µ(1	PROPN
ejpam-4561	175	8	)	)	PUNCT
ejpam-4561	175	9	≥	≥	NOUN
ejpam-4561	176	1	min{µ(x	min{µ(x	PROPN
ejpam-4561	176	2	◦	◦	NOUN
ejpam-4561	176	3	y	y	PROPN
ejpam-4561	176	4	)	)	PUNCT
ejpam-4561	176	5	,	,	PUNCT
ejpam-4561	176	6	µ(f	µ(f	PROPN
ejpam-4561	176	7	◦	◦	VERB
ejpam-4561	176	8	g	g	NOUN
ejpam-4561	176	9	)	)	PUNCT
ejpam-4561	176	10	}	}	PUNCT
ejpam-4561	176	11	≥	≥	X
ejpam-4561	176	12	µ((x	µ((x	NOUN
ejpam-4561	176	13	◦	◦	NOUN
ejpam-4561	176	14	f	f	X
ejpam-4561	176	15	)	)	PUNCT
ejpam-4561	176	16	◦	◦	NOUN
ejpam-4561	176	17	(	(	PUNCT
ejpam-4561	176	18	y	y	NOUN
ejpam-4561	176	19	◦	◦	VERB
ejpam-4561	176	20	g	g	NOUN
ejpam-4561	176	21	)	)	PUNCT
ejpam-4561	176	22	)	)	PUNCT
ejpam-4561	176	23	.	.	PUNCT
ejpam-4561	177	1	k.	k.	PROPN
ejpam-4561	177	2	belleza	belleza	PROPN
ejpam-4561	177	3	,	,	PUNCT
ejpam-4561	177	4	m.	m.	PROPN
ejpam-4561	177	5	r.	r.	PROPN
ejpam-4561	177	6	v.	v.	PROPN
ejpam-4561	177	7	dicen	dicen	PROPN
ejpam-4561	177	8	/	/	SYM
ejpam-4561	177	9	eur	eur	PROPN
ejpam-4561	177	10	.	.	PUNCT
ejpam-4561	178	1	j.	j.	PROPN
ejpam-4561	178	2	pure	pure	PROPN
ejpam-4561	178	3	appl	appl	PROPN
ejpam-4561	178	4	.	.	PROPN
ejpam-4561	178	5	math	math	PROPN
ejpam-4561	178	6	,	,	PUNCT
ejpam-4561	178	7	15	15	NUM
ejpam-4561	178	8	(	(	PUNCT
ejpam-4561	178	9	4	4	NUM
ejpam-4561	178	10	)	)	PUNCT
ejpam-4561	178	11	(	(	PUNCT
ejpam-4561	178	12	2022	2022	NUM
ejpam-4561	178	13	)	)	PUNCT
ejpam-4561	178	14	,	,	PUNCT
ejpam-4561	178	15	1957	1957	NUM
ejpam-4561	178	16	-	-	SYM
ejpam-4561	178	17	1965	1965	NUM
ejpam-4561	178	18	1964	1964	NUM
ejpam-4561	178	19	therefore	therefore	ADV
ejpam-4561	178	20	,	,	PUNCT
ejpam-4561	178	21	by	by	ADP
ejpam-4561	178	22	the	the	DET
ejpam-4561	178	23	manipulation	manipulation	NOUN
ejpam-4561	178	24	above	above	ADV
ejpam-4561	178	25	,	,	PUNCT
ejpam-4561	178	26	µ((x	µ((x	NOUN
ejpam-4561	178	27	◦	◦	NOUN
ejpam-4561	178	28	f	f	X
ejpam-4561	178	29	)	)	PUNCT
ejpam-4561	178	30	◦	◦	NOUN
ejpam-4561	178	31	(	(	PUNCT
ejpam-4561	178	32	y	y	NOUN
ejpam-4561	178	33	◦	◦	VERB
ejpam-4561	178	34	g	g	NOUN
ejpam-4561	178	35	)	)	PUNCT
ejpam-4561	178	36	)	)	PUNCT
ejpam-4561	179	1	=	=	PUNCT
ejpam-4561	179	2	µ(1	µ(1	PROPN
ejpam-4561	179	3	)	)	PUNCT
ejpam-4561	179	4	,	,	PUNCT
ejpam-4561	179	5	which	which	PRON
ejpam-4561	179	6	implies	imply	VERB
ejpam-4561	179	7	that	that	SCONJ
ejpam-4561	179	8	(	(	PUNCT
ejpam-4561	179	9	x	x	PUNCT
ejpam-4561	179	10	◦	◦	NOUN
ejpam-4561	179	11	f	f	NOUN
ejpam-4561	179	12	)	)	PUNCT
ejpam-4561	179	13	◦	◦	NOUN
ejpam-4561	179	14	(	(	PUNCT
ejpam-4561	179	15	y	y	NOUN
ejpam-4561	179	16	◦	◦	VERB
ejpam-4561	179	17	g	g	NOUN
ejpam-4561	179	18	)	)	PUNCT
ejpam-4561	179	19	∈	∈	PROPN
ejpam-4561	179	20	xµ.	xµ.	PROPN
ejpam-4561	179	21	theorem	theorem	VERB
ejpam-4561	179	22	4	4	NUM
ejpam-4561	179	23	.	.	PUNCT
ejpam-4561	180	1	the	the	DET
ejpam-4561	180	2	intersection	intersection	NOUN
ejpam-4561	180	3	of	of	ADP
ejpam-4561	180	4	any	any	DET
ejpam-4561	180	5	set	set	NOUN
ejpam-4561	180	6	of	of	ADP
ejpam-4561	180	7	a	a	DET
ejpam-4561	180	8	fuzzy	fuzzy	ADJ
ejpam-4561	180	9	normal	normal	ADJ
ejpam-4561	180	10	dual	dual	ADJ
ejpam-4561	180	11	b	b	NOUN
ejpam-4561	180	12	-	-	PUNCT
ejpam-4561	180	13	algebras	algebras	PROPN
ejpam-4561	180	14	is	be	AUX
ejpam-4561	180	15	also	also	ADV
ejpam-4561	180	16	fuzzy	fuzzy	ADJ
ejpam-4561	180	17	normal	normal	ADJ
ejpam-4561	180	18	dual	dual	ADJ
ejpam-4561	180	19	b	b	NOUN
ejpam-4561	180	20	-	-	PUNCT
ejpam-4561	180	21	algebra	algebra	NOUN
ejpam-4561	180	22	.	.	PUNCT
ejpam-4561	181	1	proof	proof	NOUN
ejpam-4561	181	2	.	.	PUNCT
ejpam-4561	182	1	let	let	VERB
ejpam-4561	182	2	{	{	PUNCT
ejpam-4561	182	3	µω	µω	VERB
ejpam-4561	182	4	|	|	PROPN
ejpam-4561	182	5	ω	ω	PROPN
ejpam-4561	182	6	∈	∈	PROPN
ejpam-4561	183	1	i	i	PRON
ejpam-4561	183	2	}	}	PUNCT
ejpam-4561	183	3	be	be	VERB
ejpam-4561	183	4	a	a	DET
ejpam-4561	183	5	family	family	NOUN
ejpam-4561	183	6	of	of	ADP
ejpam-4561	183	7	fuzzy	fuzzy	ADJ
ejpam-4561	183	8	normal	normal	ADJ
ejpam-4561	183	9	dual	dual	ADJ
ejpam-4561	183	10	b	b	NOUN
ejpam-4561	183	11	-	-	PUNCT
ejpam-4561	183	12	algebras	algebras	PROPN
ejpam-4561	183	13	and	and	CCONJ
ejpam-4561	183	14	let	let	VERB
ejpam-4561	183	15	f	f	X
ejpam-4561	183	16	,	,	PUNCT
ejpam-4561	183	17	g	g	PROPN
ejpam-4561	183	18	,	,	PUNCT
ejpam-4561	183	19	x	x	PRON
ejpam-4561	183	20	,	,	PUNCT
ejpam-4561	183	21	y	y	PROPN
ejpam-4561	183	22	∈	∈	PROPN
ejpam-4561	183	23	x.	x.	NOUN
ejpam-4561	183	24	then	then	ADV
ejpam-4561	183	25	by	by	ADP
ejpam-4561	183	26	the	the	DET
ejpam-4561	183	27	greatest	greatest	ADV
ejpam-4561	183	28	lower	low	ADJ
ejpam-4561	183	29	bound	bind	VERB
ejpam-4561	183	30	of	of	ADP
ejpam-4561	183	31	fuzzy	fuzzy	ADJ
ejpam-4561	183	32	set	set	NOUN
ejpam-4561	183	33	,	,	PUNCT
ejpam-4561	183	34	definition	definition	NOUN
ejpam-4561	183	35	of	of	ADP
ejpam-4561	183	36	fuzzy	fuzzy	ADJ
ejpam-4561	183	37	normal	normal	ADJ
ejpam-4561	183	38	,	,	PUNCT
ejpam-4561	183	39	and	and	CCONJ
ejpam-4561	183	40	by	by	ADP
ejpam-4561	183	41	the	the	DET
ejpam-4561	183	42	minimum	minimum	ADJ
ejpam-4561	183	43	or	or	CCONJ
ejpam-4561	183	44	infimum	infimum	ADJ
ejpam-4561	183	45	property	property	NOUN
ejpam-4561	183	46	,	,	PUNCT
ejpam-4561	183	47	we	we	PRON
ejpam-4561	183	48	have	have	VERB
ejpam-4561	183	49	:(	:(	PUNCT
ejpam-4561	183	50	⋂	⋂	PROPN
ejpam-4561	183	51	ω∈i	ω∈i	NOUN
ejpam-4561	183	52	µω	µω	PROPN
ejpam-4561	183	53	)	)	PUNCT
ejpam-4561	183	54	(	(	PUNCT
ejpam-4561	183	55	(	(	PUNCT
ejpam-4561	183	56	x	x	SYM
ejpam-4561	183	57	◦	◦	NOUN
ejpam-4561	183	58	f	f	NOUN
ejpam-4561	183	59	)	)	PUNCT
ejpam-4561	183	60	◦	◦	NOUN
ejpam-4561	183	61	(	(	PUNCT
ejpam-4561	183	62	y	y	NOUN
ejpam-4561	183	63	◦	◦	VERB
ejpam-4561	183	64	g	g	NOUN
ejpam-4561	183	65	)	)	PUNCT
ejpam-4561	183	66	)	)	PUNCT
ejpam-4561	184	1	=	=	SYM
ejpam-4561	184	2	inf	inf	PROPN
ejpam-4561	184	3	ω∈i	ω∈i	NOUN
ejpam-4561	184	4	µω((x	µω((x	VERB
ejpam-4561	185	1	◦	◦	NOUN
ejpam-4561	185	2	f	f	X
ejpam-4561	185	3	)	)	PUNCT
ejpam-4561	185	4	◦	◦	NOUN
ejpam-4561	185	5	(	(	PUNCT
ejpam-4561	185	6	y	y	NOUN
ejpam-4561	185	7	◦	◦	VERB
ejpam-4561	185	8	g	g	PROPN
ejpam-4561	185	9	)	)	PUNCT
ejpam-4561	185	10	)	)	PUNCT
ejpam-4561	185	11	≥	≥	PROPN
ejpam-4561	185	12	inf	inf	PROPN
ejpam-4561	185	13	ω∈i	ω∈i	NOUN
ejpam-4561	185	14	{	{	PUNCT
ejpam-4561	185	15	min{µω(x	min{µω(x	X
ejpam-4561	185	16	◦	◦	VERB
ejpam-4561	185	17	y	y	PROPN
ejpam-4561	185	18	)	)	PUNCT
ejpam-4561	185	19	,	,	PUNCT
ejpam-4561	185	20	µω(f	µω(f	ADJ
ejpam-4561	185	21	◦	◦	NOUN
ejpam-4561	185	22	g	g	NOUN
ejpam-4561	185	23	)	)	PUNCT
ejpam-4561	185	24	}	}	PUNCT
ejpam-4561	185	25	}	}	PUNCT
ejpam-4561	185	26	=	=	SYM
ejpam-4561	185	27	min	min	PROPN
ejpam-4561	185	28	{	{	PUNCT
ejpam-4561	185	29	inf	inf	NOUN
ejpam-4561	185	30	ω∈i	ω∈i	NOUN
ejpam-4561	185	31	(	(	PUNCT
ejpam-4561	185	32	x	x	SYM
ejpam-4561	185	33	◦	◦	VERB
ejpam-4561	185	34	y	y	PROPN
ejpam-4561	185	35	)	)	PUNCT
ejpam-4561	185	36	,	,	PUNCT
ejpam-4561	185	37	inf	inf	NOUN
ejpam-4561	185	38	ω∈i	ω∈i	NUM
ejpam-4561	185	39	(	(	PUNCT
ejpam-4561	185	40	f	f	X
ejpam-4561	185	41	◦	◦	VERB
ejpam-4561	185	42	g	g	NOUN
ejpam-4561	185	43	)	)	PUNCT
ejpam-4561	185	44	}	}	PUNCT
ejpam-4561	185	45	=	=	SYM
ejpam-4561	185	46	min	min	NOUN
ejpam-4561	185	47	{	{	PUNCT
ejpam-4561	185	48	(	(	PUNCT
ejpam-4561	185	49	⋂	⋂	PROPN
ejpam-4561	185	50	ω∈i	ω∈i	VERB
ejpam-4561	185	51	µω	µω	PROPN
ejpam-4561	185	52	)	)	PUNCT
ejpam-4561	185	53	(	(	PUNCT
ejpam-4561	185	54	x	x	X
ejpam-4561	185	55	◦	◦	VERB
ejpam-4561	185	56	y	y	PROPN
ejpam-4561	185	57	)	)	PUNCT
ejpam-4561	185	58	,	,	PUNCT
ejpam-4561	185	59	(	(	PUNCT
ejpam-4561	185	60	⋂	⋂	PROPN
ejpam-4561	185	61	ω∈i	ω∈i	VERB
ejpam-4561	185	62	µω	µω	PROPN
ejpam-4561	185	63	)	)	PUNCT
ejpam-4561	185	64	(	(	PUNCT
ejpam-4561	185	65	f	f	X
ejpam-4561	185	66	◦	◦	VERB
ejpam-4561	185	67	g	g	NOUN
ejpam-4561	185	68	)	)	PUNCT
ejpam-4561	185	69	}	}	PUNCT
ejpam-4561	185	70	,	,	PUNCT
ejpam-4561	185	71	which	which	PRON
ejpam-4561	185	72	implies	imply	VERB
ejpam-4561	185	73	that	that	SCONJ
ejpam-4561	185	74	⋂	⋂	PROPN
ejpam-4561	185	75	ω∈i	ω∈i	VERB
ejpam-4561	185	76	µω	µω	PROPN
ejpam-4561	185	77	is	be	AUX
ejpam-4561	185	78	a	a	DET
ejpam-4561	185	79	fuzzy	fuzzy	ADJ
ejpam-4561	185	80	normal	normal	ADJ
ejpam-4561	185	81	set	set	NOUN
ejpam-4561	185	82	in	in	ADP
ejpam-4561	185	83	x.	x.	NOUN
ejpam-4561	185	84	by	by	ADP
ejpam-4561	185	85	theorem	theorem	NOUN
ejpam-4561	185	86	2	2	NUM
ejpam-4561	185	87	,	,	PUNCT
ejpam-4561	185	88	therefore	therefore	ADV
ejpam-4561	185	89	⋂	⋂	PROPN
ejpam-4561	185	90	ω∈i	ω∈i	VERB
ejpam-4561	185	91	µω	µω	PROPN
ejpam-4561	185	92	is	be	AUX
ejpam-4561	185	93	a	a	DET
ejpam-4561	185	94	fuzzy	fuzzy	ADJ
ejpam-4561	185	95	normal	normal	ADJ
ejpam-4561	185	96	dual	dual	ADJ
ejpam-4561	185	97	b	b	NOUN
ejpam-4561	185	98	-	-	PUNCT
ejpam-4561	185	99	algebra	algebra	NOUN
ejpam-4561	185	100	.	.	PUNCT
ejpam-4561	186	1	remark	remark	NOUN
ejpam-4561	186	2	2	2	NUM
ejpam-4561	186	3	.	.	PUNCT
ejpam-4561	187	1	the	the	DET
ejpam-4561	187	2	union	union	NOUN
ejpam-4561	187	3	of	of	ADP
ejpam-4561	187	4	any	any	DET
ejpam-4561	187	5	set	set	NOUN
ejpam-4561	187	6	of	of	ADP
ejpam-4561	187	7	fuzzy	fuzzy	ADJ
ejpam-4561	187	8	dual	dual	ADJ
ejpam-4561	187	9	b	b	NOUN
ejpam-4561	187	10	-	-	PUNCT
ejpam-4561	187	11	algebras	algebras	PROPN
ejpam-4561	187	12	may	may	AUX
ejpam-4561	187	13	not	not	PART
ejpam-4561	187	14	be	be	AUX
ejpam-4561	187	15	a	a	DET
ejpam-4561	187	16	fuzzy	fuzzy	ADJ
ejpam-4561	187	17	dual	dual	ADJ
ejpam-4561	187	18	b	b	NOUN
ejpam-4561	187	19	-	-	PUNCT
ejpam-4561	187	20	algebra	algebra	NOUN
ejpam-4561	187	21	as	as	SCONJ
ejpam-4561	187	22	shown	show	VERB
ejpam-4561	187	23	in	in	ADP
ejpam-4561	187	24	the	the	DET
ejpam-4561	187	25	next	next	ADJ
ejpam-4561	187	26	example	example	NOUN
ejpam-4561	187	27	.	.	PUNCT
ejpam-4561	188	1	example	example	NOUN
ejpam-4561	188	2	6	6	NUM
ejpam-4561	188	3	.	.	PUNCT
ejpam-4561	188	4	referring	refer	VERB
ejpam-4561	188	5	to	to	ADP
ejpam-4561	188	6	example	example	NOUN
ejpam-4561	188	7	1	1	NUM
ejpam-4561	188	8	.	.	PUNCT
ejpam-4561	188	9	define	define	VERB
ejpam-4561	188	10	a	a	DET
ejpam-4561	188	11	fuzzy	fuzzy	ADJ
ejpam-4561	188	12	set	set	VERB
ejpam-4561	188	13	δ	δ	NOUN
ejpam-4561	188	14	:	:	PUNCT
ejpam-4561	189	1	x	x	X
ejpam-4561	189	2	→	→	PUNCT
ejpam-4561	189	3	[	[	X
ejpam-4561	189	4	0	0	NUM
ejpam-4561	189	5	,	,	PUNCT
ejpam-4561	189	6	1	1	NUM
ejpam-4561	189	7	]	]	PUNCT
ejpam-4561	189	8	by	by	ADP
ejpam-4561	189	9	δ(1	δ(1	NOUN
ejpam-4561	189	10	)	)	PUNCT
ejpam-4561	189	11	=	=	SYM
ejpam-4561	189	12	δ(d	δ(d	PROPN
ejpam-4561	189	13	)	)	PUNCT
ejpam-4561	189	14	=	=	PUNCT
ejpam-4561	190	1	0.6	0.6	NUM
ejpam-4561	190	2	>	>	SYM
ejpam-4561	190	3	0.1	0.1	NUM
ejpam-4561	190	4	=	=	SYM
ejpam-4561	190	5	δ(a	δ(a	PROPN
ejpam-4561	190	6	)	)	PUNCT
ejpam-4561	190	7	=	=	SYM
ejpam-4561	190	8	δ(b	δ(b	PROPN
ejpam-4561	190	9	)	)	PUNCT
ejpam-4561	190	10	=	=	SYM
ejpam-4561	190	11	δ(c	δ(c	PROPN
ejpam-4561	190	12	)	)	PUNCT
ejpam-4561	190	13	=	=	PUNCT
ejpam-4561	191	1	δ(e	δ(e	X
ejpam-4561	191	2	)	)	PUNCT
ejpam-4561	191	3	,	,	PUNCT
ejpam-4561	191	4	then	then	ADV
ejpam-4561	191	5	it	it	PRON
ejpam-4561	191	6	is	be	AUX
ejpam-4561	191	7	also	also	ADV
ejpam-4561	191	8	a	a	DET
ejpam-4561	191	9	fuzzy	fuzzy	ADJ
ejpam-4561	191	10	dual	dual	ADJ
ejpam-4561	191	11	b	b	NOUN
ejpam-4561	191	12	-	-	PUNCT
ejpam-4561	191	13	algebra	algebra	NOUN
ejpam-4561	191	14	.	.	PUNCT
ejpam-4561	192	1	now	now	ADV
ejpam-4561	192	2	,	,	PUNCT
ejpam-4561	192	3	(	(	PUNCT
ejpam-4561	192	4	µ	µ	X
ejpam-4561	192	5	∪	∪	X
ejpam-4561	192	6	δ)(c	δ)(c	X
ejpam-4561	192	7	◦	◦	NOUN
ejpam-4561	192	8	d	d	NOUN
ejpam-4561	192	9	)	)	PUNCT
ejpam-4561	192	10	=	=	SYM
ejpam-4561	192	11	(	(	PUNCT
ejpam-4561	192	12	µ	µ	X
ejpam-4561	192	13	∪	∪	X
ejpam-4561	192	14	δ)(a	δ)(a	NOUN
ejpam-4561	192	15	)	)	PUNCT
ejpam-4561	192	16	=	=	SYM
ejpam-4561	193	1	0.3	0.3	NUM
ejpam-4561	194	1	[	[	PUNCT
ejpam-4561	194	2	by	by	ADP
ejpam-4561	194	3	union	union	NOUN
ejpam-4561	194	4	property	property	NOUN
ejpam-4561	194	5	of	of	ADP
ejpam-4561	194	6	fuzzy	fuzzy	ADJ
ejpam-4561	194	7	set	set	NOUN
ejpam-4561	194	8	]	]	PUNCT
ejpam-4561	194	9	and	and	CCONJ
ejpam-4561	194	10	min{(µ	min{(µ	PROPN
ejpam-4561	194	11	∪	∪	ADP
ejpam-4561	194	12	σ)(c	σ)(c	NOUN
ejpam-4561	194	13	)	)	PUNCT
ejpam-4561	194	14	,	,	PUNCT
ejpam-4561	194	15	(	(	PUNCT
ejpam-4561	194	16	µ	µ	X
ejpam-4561	194	17	∪	∪	X
ejpam-4561	194	18	σ)(d	σ)(d	NOUN
ejpam-4561	194	19	)	)	PUNCT
ejpam-4561	194	20	}	}	PUNCT
ejpam-4561	194	21	=	=	SYM
ejpam-4561	194	22	min{0.8	min{0.8	PROPN
ejpam-4561	194	23	,	,	PUNCT
ejpam-4561	194	24	0.6	0.6	NUM
ejpam-4561	194	25	}	}	PUNCT
ejpam-4561	194	26	=	=	SYM
ejpam-4561	194	27	0.6	0.6	X
ejpam-4561	194	28	.	.	PUNCT
ejpam-4561	194	29	notice	notice	VERB
ejpam-4561	194	30	that	that	SCONJ
ejpam-4561	194	31	(	(	PUNCT
ejpam-4561	194	32	µ	µ	X
ejpam-4561	194	33	∪	∪	X
ejpam-4561	194	34	δ)(c	δ)(c	X
ejpam-4561	194	35	◦	◦	NOUN
ejpam-4561	194	36	d	d	NOUN
ejpam-4561	194	37	)	)	PUNCT
ejpam-4561	194	38	<	<	X
ejpam-4561	194	39	min{(µ	min{(µ	X
ejpam-4561	194	40	∪	∪	ADP
ejpam-4561	194	41	σ)(c	σ)(c	NOUN
ejpam-4561	194	42	)	)	PUNCT
ejpam-4561	194	43	,	,	PUNCT
ejpam-4561	194	44	(	(	PUNCT
ejpam-4561	194	45	µ	µ	X
ejpam-4561	194	46	∪	∪	X
ejpam-4561	194	47	σ)(d	σ)(d	NOUN
ejpam-4561	194	48	)	)	PUNCT
ejpam-4561	194	49	}	}	PUNCT
ejpam-4561	194	50	which	which	PRON
ejpam-4561	194	51	does	do	AUX
ejpam-4561	194	52	not	not	PART
ejpam-4561	194	53	satisfy	satisfy	VERB
ejpam-4561	194	54	the	the	DET
ejpam-4561	194	55	definition	definition	NOUN
ejpam-4561	194	56	.	.	PUNCT
ejpam-4561	195	1	thus	thus	ADV
ejpam-4561	195	2	,	,	PUNCT
ejpam-4561	195	3	µ	µ	X
ejpam-4561	195	4	∪	∪	X
ejpam-4561	195	5	δ	δ	PROPN
ejpam-4561	195	6	is	be	AUX
ejpam-4561	195	7	not	not	PART
ejpam-4561	195	8	a	a	DET
ejpam-4561	195	9	fuzzy	fuzzy	ADJ
ejpam-4561	195	10	dual	dual	ADJ
ejpam-4561	195	11	b	b	NOUN
ejpam-4561	195	12	-	-	PUNCT
ejpam-4561	195	13	algebra	algebra	NOUN
ejpam-4561	195	14	.	.	PUNCT
ejpam-4561	196	1	recall	recall	NOUN
ejpam-4561	196	2	from	from	ADP
ejpam-4561	196	3	theorem	theorem	ADJ
ejpam-4561	196	4	2	2	NUM
ejpam-4561	196	5	that	that	SCONJ
ejpam-4561	196	6	every	every	DET
ejpam-4561	196	7	fuzzy	fuzzy	ADJ
ejpam-4561	196	8	normal	normal	ADJ
ejpam-4561	196	9	dual	dual	ADJ
ejpam-4561	196	10	b	b	NOUN
ejpam-4561	196	11	-	-	PUNCT
ejpam-4561	196	12	algebra	algebra	NOUN
ejpam-4561	196	13	is	be	AUX
ejpam-4561	196	14	a	a	DET
ejpam-4561	196	15	fuzzy	fuzzy	ADJ
ejpam-4561	196	16	dual	dual	ADJ
ejpam-4561	196	17	b	b	NOUN
ejpam-4561	196	18	-	-	PUNCT
ejpam-4561	196	19	algebra	algebra	NOUN
ejpam-4561	196	20	.	.	PUNCT
ejpam-4561	197	1	however	however	ADV
ejpam-4561	197	2	,	,	PUNCT
ejpam-4561	197	3	in	in	ADP
ejpam-4561	197	4	view	view	NOUN
ejpam-4561	197	5	of	of	ADP
ejpam-4561	197	6	remark	remark	NOUN
ejpam-4561	197	7	2	2	NUM
ejpam-4561	197	8	,	,	PUNCT
ejpam-4561	197	9	the	the	DET
ejpam-4561	197	10	union	union	NOUN
ejpam-4561	197	11	of	of	ADP
ejpam-4561	197	12	fuzzy	fuzzy	ADJ
ejpam-4561	197	13	normal	normal	ADJ
ejpam-4561	197	14	dual	dual	ADJ
ejpam-4561	197	15	b	b	NOUN
ejpam-4561	197	16	-	-	PUNCT
ejpam-4561	197	17	algebras	algebras	PROPN
ejpam-4561	197	18	may	may	AUX
ejpam-4561	197	19	not	not	PART
ejpam-4561	197	20	be	be	AUX
ejpam-4561	197	21	a	a	DET
ejpam-4561	197	22	fuzzy	fuzzy	ADJ
ejpam-4561	197	23	normal	normal	ADJ
ejpam-4561	197	24	b	b	NOUN
ejpam-4561	197	25	-	-	PUNCT
ejpam-4561	197	26	algebra	algebra	NOUN
ejpam-4561	197	27	.	.	PUNCT
ejpam-4561	198	1	4	4	X
ejpam-4561	198	2	.	.	X
ejpam-4561	198	3	conclusion	conclusion	NOUN
ejpam-4561	198	4	in	in	ADP
ejpam-4561	198	5	this	this	DET
ejpam-4561	198	6	paper	paper	NOUN
ejpam-4561	198	7	,	,	PUNCT
ejpam-4561	198	8	the	the	DET
ejpam-4561	198	9	notion	notion	NOUN
ejpam-4561	198	10	of	of	ADP
ejpam-4561	198	11	a	a	PRON
ejpam-4561	198	12	fuzzy	fuzzy	ADJ
ejpam-4561	198	13	dual	dual	ADJ
ejpam-4561	198	14	b	b	NOUN
ejpam-4561	198	15	-	-	PUNCT
ejpam-4561	198	16	algebra	algebra	NOUN
ejpam-4561	198	17	is	be	AUX
ejpam-4561	198	18	presented	present	VERB
ejpam-4561	198	19	together	together	ADV
ejpam-4561	198	20	with	with	ADP
ejpam-4561	198	21	some	some	PRON
ejpam-4561	198	22	of	of	ADP
ejpam-4561	198	23	its	its	PRON
ejpam-4561	198	24	properties	property	NOUN
ejpam-4561	198	25	and	and	CCONJ
ejpam-4561	198	26	characterizations	characterization	NOUN
ejpam-4561	198	27	.	.	PUNCT
ejpam-4561	199	1	here	here	ADV
ejpam-4561	199	2	it	it	PRON
ejpam-4561	199	3	is	be	AUX
ejpam-4561	199	4	considered	consider	VERB
ejpam-4561	199	5	that	that	SCONJ
ejpam-4561	199	6	every	every	DET
ejpam-4561	199	7	fuzzy	fuzzy	ADJ
ejpam-4561	199	8	normal	normal	ADJ
ejpam-4561	199	9	set	set	NOUN
ejpam-4561	199	10	is	be	AUX
ejpam-4561	199	11	a	a	DET
ejpam-4561	199	12	fuzzy	fuzzy	ADJ
ejpam-4561	199	13	dual	dual	ADJ
ejpam-4561	199	14	b	b	NOUN
ejpam-4561	199	15	algebra	algebra	NOUN
ejpam-4561	199	16	but	but	CCONJ
ejpam-4561	199	17	the	the	DET
ejpam-4561	199	18	converse	converse	NOUN
ejpam-4561	199	19	is	be	AUX
ejpam-4561	199	20	not	not	PART
ejpam-4561	199	21	always	always	ADV
ejpam-4561	199	22	true	true	ADJ
ejpam-4561	199	23	.	.	PUNCT
ejpam-4561	200	1	while	while	SCONJ
ejpam-4561	200	2	undergoing	undergo	VERB
ejpam-4561	200	3	an	an	DET
ejpam-4561	200	4	operation	operation	NOUN
ejpam-4561	200	5	in	in	ADP
ejpam-4561	200	6	fuzzy	fuzzy	ADJ
ejpam-4561	200	7	sets	set	NOUN
ejpam-4561	200	8	,	,	PUNCT
ejpam-4561	200	9	the	the	DET
ejpam-4561	200	10	union	union	NOUN
ejpam-4561	200	11	of	of	ADP
ejpam-4561	200	12	any	any	DET
ejpam-4561	200	13	set	set	NOUN
ejpam-4561	200	14	of	of	ADP
ejpam-4561	200	15	fuzzy	fuzzy	ADJ
ejpam-4561	200	16	dual	dual	ADJ
ejpam-4561	200	17	b	b	NOUN
ejpam-4561	200	18	-	-	PUNCT
ejpam-4561	200	19	algebra	algebra	NOUN
ejpam-4561	200	20	may	may	AUX
ejpam-4561	200	21	not	not	PART
ejpam-4561	200	22	be	be	AUX
ejpam-4561	200	23	a	a	DET
ejpam-4561	200	24	fuzzy	fuzzy	ADJ
ejpam-4561	200	25	dual	dual	ADJ
ejpam-4561	200	26	balgebra	balgebra	NOUN
ejpam-4561	200	27	.	.	PUNCT
ejpam-4561	201	1	however	however	ADV
ejpam-4561	201	2	,	,	PUNCT
ejpam-4561	201	3	the	the	DET
ejpam-4561	201	4	intersection	intersection	NOUN
ejpam-4561	201	5	of	of	ADP
ejpam-4561	201	6	any	any	DET
ejpam-4561	201	7	set	set	NOUN
ejpam-4561	201	8	of	of	ADP
ejpam-4561	201	9	a	a	DET
ejpam-4561	201	10	fuzzy	fuzzy	ADJ
ejpam-4561	201	11	normal	normal	ADJ
ejpam-4561	201	12	dual	dual	ADJ
ejpam-4561	201	13	b	b	NOUN
ejpam-4561	201	14	-	-	PUNCT
ejpam-4561	201	15	algebra	algebra	NOUN
ejpam-4561	201	16	is	be	AUX
ejpam-4561	201	17	also	also	ADV
ejpam-4561	201	18	a	a	DET
ejpam-4561	201	19	fuzzy	fuzzy	ADJ
ejpam-4561	201	20	normal	normal	ADJ
ejpam-4561	201	21	dual	dual	ADJ
ejpam-4561	201	22	b	b	NOUN
ejpam-4561	201	23	-	-	PUNCT
ejpam-4561	201	24	algebra	algebra	NOUN
ejpam-4561	201	25	.	.	PUNCT
ejpam-4561	202	1	the	the	DET
ejpam-4561	202	2	difference	difference	NOUN
ejpam-4561	202	3	between	between	ADP
ejpam-4561	202	4	a	a	DET
ejpam-4561	202	5	fuzzy	fuzzy	ADJ
ejpam-4561	202	6	dual	dual	ADJ
ejpam-4561	202	7	b	b	NOUN
ejpam-4561	202	8	-	-	PUNCT
ejpam-4561	202	9	algebra	algebra	NOUN
ejpam-4561	202	10	and	and	CCONJ
ejpam-4561	202	11	fuzzy	fuzzy	ADJ
ejpam-4561	202	12	normal	normal	ADJ
ejpam-4561	202	13	dual	dual	ADJ
ejpam-4561	202	14	b	b	NOUN
ejpam-4561	202	15	-	-	PUNCT
ejpam-4561	202	16	algebra	algebra	NOUN
ejpam-4561	202	17	is	be	AUX
ejpam-4561	202	18	also	also	ADV
ejpam-4561	202	19	presented	present	VERB
ejpam-4561	202	20	.	.	PUNCT
ejpam-4561	203	1	references	reference	NOUN
ejpam-4561	203	2	1965	1965	NUM
ejpam-4561	203	3	acknowledgements	acknowledgement	NOUN
ejpam-4561	203	4	the	the	DET
ejpam-4561	203	5	authors	author	NOUN
ejpam-4561	203	6	are	be	AUX
ejpam-4561	203	7	grateful	grateful	ADJ
ejpam-4561	203	8	to	to	ADP
ejpam-4561	203	9	the	the	DET
ejpam-4561	203	10	department	department	NOUN
ejpam-4561	203	11	of	of	ADP
ejpam-4561	203	12	science	science	NOUN
ejpam-4561	203	13	and	and	CCONJ
ejpam-4561	203	14	technology	technology	NOUN
ejpam-4561	203	15	(	(	PUNCT
ejpam-4561	203	16	dost	dost	NOUN
ejpam-4561	203	17	)	)	PUNCT
ejpam-4561	203	18	through	through	ADP
ejpam-4561	203	19	the	the	DET
ejpam-4561	203	20	accelerated	accelerated	ADJ
ejpam-4561	203	21	science	science	NOUN
ejpam-4561	203	22	and	and	CCONJ
ejpam-4561	203	23	technology	technology	NOUN
ejpam-4561	203	24	human	human	ADJ
ejpam-4561	203	25	resource	resource	NOUN
ejpam-4561	203	26	development	development	NOUN
ejpam-4561	203	27	program	program	NOUN
ejpam-4561	203	28	(	(	PUNCT
ejpam-4561	203	29	asthrdp	asthrdp	PROPN
ejpam-4561	203	30	)	)	PUNCT
ejpam-4561	203	31	and	and	CCONJ
ejpam-4561	203	32	its	its	PRON
ejpam-4561	203	33	partner	partner	NOUN
ejpam-4561	203	34	university	university	NOUN
ejpam-4561	203	35	,	,	PUNCT
ejpam-4561	203	36	university	university	PROPN
ejpam-4561	203	37	of	of	ADP
ejpam-4561	203	38	san	san	PROPN
ejpam-4561	203	39	carlos	carlos	PROPN
ejpam-4561	203	40	who	who	PRON
ejpam-4561	203	41	funded	fund	VERB
ejpam-4561	203	42	and	and	CCONJ
ejpam-4561	203	43	made	make	VERB
ejpam-4561	203	44	this	this	DET
ejpam-4561	203	45	research	research	NOUN
ejpam-4561	203	46	possible	possible	ADJ
ejpam-4561	203	47	.	.	PUNCT
ejpam-4561	204	1	references	reference	NOUN
ejpam-4561	204	2	[	[	X
ejpam-4561	204	3	1	1	NUM
ejpam-4561	204	4	]	]	PUNCT
ejpam-4561	204	5	e	e	X
ejpam-4561	204	6	ahmed	ahmed	PROPN
ejpam-4561	204	7	and	and	CCONJ
ejpam-4561	204	8	m	m	PROPN
ejpam-4561	204	9	ahmed	ahme	VERB
ejpam-4561	204	10	.	.	PUNCT
ejpam-4561	205	1	fuzzy	fuzzy	ADJ
ejpam-4561	205	2	bck	bck	PROPN
ejpam-4561	205	3	-	-	PUNCT
ejpam-4561	205	4	algebras	algebras	PROPN
ejpam-4561	205	5	.	.	PUNCT
ejpam-4561	205	6	journal	journal	PROPN
ejpam-4561	205	7	of	of	ADP
ejpam-4561	205	8	applied	apply	VERB
ejpam-4561	205	9	mathematics	mathematic	NOUN
ejpam-4561	205	10	and	and	CCONJ
ejpam-4561	205	11	physics	physics	NOUN
ejpam-4561	205	12	,	,	PUNCT
ejpam-4561	205	13	08(05):927–932	08(05):927–932	NUM
ejpam-4561	205	14	,	,	PUNCT
ejpam-4561	205	15	2020	2020	NUM
ejpam-4561	205	16	.	.	PUNCT
ejpam-4561	206	1	[	[	X
ejpam-4561	206	2	2	2	NUM
ejpam-4561	206	3	]	]	X
ejpam-4561	206	4	k	k	PROPN
ejpam-4561	206	5	belleza	belleza	PROPN
ejpam-4561	206	6	and	and	CCONJ
ejpam-4561	206	7	j	j	PROPN
ejpam-4561	206	8	vilela	vilela	NOUN
ejpam-4561	206	9	.	.	PUNCT
ejpam-4561	207	1	the	the	DET
ejpam-4561	207	2	dual	dual	ADJ
ejpam-4561	207	3	b	b	NOUN
ejpam-4561	207	4	-	-	PUNCT
ejpam-4561	207	5	algebra	algebra	NOUN
ejpam-4561	207	6	.	.	PUNCT
ejpam-4561	208	1	european	european	ADJ
ejpam-4561	208	2	journal	journal	PROPN
ejpam-4561	208	3	of	of	ADP
ejpam-4561	208	4	pure	pure	ADJ
ejpam-4561	208	5	and	and	CCONJ
ejpam-4561	208	6	applied	applied	ADJ
ejpam-4561	208	7	mathematics	mathematic	NOUN
ejpam-4561	208	8	,	,	PUNCT
ejpam-4561	208	9	12(04):1497–1507	12(04):1497–1507	NUM
ejpam-4561	208	10	,	,	PUNCT
ejpam-4561	208	11	2019	2019	NUM
ejpam-4561	208	12	.	.	PUNCT
ejpam-4561	209	1	[	[	X
ejpam-4561	209	2	3	3	X
ejpam-4561	209	3	]	]	X
ejpam-4561	209	4	j	j	PROPN
ejpam-4561	209	5	fraleigh	fraleigh	PROPN
ejpam-4561	209	6	.	.	PUNCT
ejpam-4561	210	1	a	a	DET
ejpam-4561	210	2	first	first	ADJ
ejpam-4561	210	3	course	course	NOUN
ejpam-4561	210	4	in	in	ADP
ejpam-4561	210	5	abstract	abstract	ADJ
ejpam-4561	210	6	algebra	algebra	NOUN
ejpam-4561	210	7	.	.	PUNCT
ejpam-4561	211	1	pearson	pearson	PROPN
ejpam-4561	211	2	education	education	PROPN
ejpam-4561	211	3	limited	limit	VERB
ejpam-4561	211	4	,	,	PUNCT
ejpam-4561	211	5	london	london	PROPN
ejpam-4561	211	6	,	,	PUNCT
ejpam-4561	211	7	united	united	ADJ
ejpam-4561	211	8	kingdom	kingdom	PROPN
ejpam-4561	211	9	,	,	PUNCT
ejpam-4561	211	10	2014	2014	NUM
ejpam-4561	211	11	.	.	PUNCT
ejpam-4561	212	1	[	[	X
ejpam-4561	212	2	4	4	NUM
ejpam-4561	212	3	]	]	X
ejpam-4561	212	4	z	z	X
ejpam-4561	212	5	guo	guo	PROPN
ejpam-4561	212	6	,	,	PUNCT
ejpam-4561	212	7	s	s	PROPN
ejpam-4561	212	8	leung	leung	PROPN
ejpam-4561	212	9	,	,	PUNCT
ejpam-4561	212	10	and	and	CCONJ
ejpam-4561	212	11	w	w	PROPN
ejpam-4561	212	12	wong	wong	PROPN
ejpam-4561	212	13	.	.	PUNCT
ejpam-4561	213	1	optimizing	optimize	VERB
ejpam-4561	213	2	decision	decision	NOUN
ejpam-4561	213	3	making	making	NOUN
ejpam-4561	213	4	in	in	ADP
ejpam-4561	213	5	the	the	DET
ejpam-4561	213	6	apparel	apparel	NOUN
ejpam-4561	213	7	supply	supply	NOUN
ejpam-4561	213	8	chain	chain	NOUN
ejpam-4561	213	9	using	use	VERB
ejpam-4561	213	10	artificial	artificial	ADJ
ejpam-4561	213	11	intelligence	intelligence	NOUN
ejpam-4561	213	12	.	.	PUNCT
ejpam-4561	214	1	woodhead	woodhead	PROPN
ejpam-4561	214	2	publishing	publish	VERB
ejpam-4561	214	3	limited	limit	VERB
ejpam-4561	214	4	,	,	PUNCT
ejpam-4561	214	5	sawston	sawston	PROPN
ejpam-4561	214	6	,	,	PUNCT
ejpam-4561	214	7	united	united	ADJ
ejpam-4561	214	8	kingdom	kingdom	PROPN
ejpam-4561	214	9	,	,	PUNCT
ejpam-4561	214	10	2013	2013	NUM
ejpam-4561	214	11	.	.	PUNCT
ejpam-4561	215	1	[	[	X
ejpam-4561	215	2	5	5	NUM
ejpam-4561	215	3	]	]	X
ejpam-4561	215	4	y	y	PROPN
ejpam-4561	215	5	jun	jun	PROPN
ejpam-4561	215	6	,	,	PUNCT
ejpam-4561	215	7	e	e	PROPN
ejpam-4561	215	8	roh	roh	PROPN
ejpam-4561	215	9	,	,	PUNCT
ejpam-4561	215	10	and	and	CCONJ
ejpam-4561	215	11	h	h	PROPN
ejpam-4561	215	12	kim	kim	PROPN
ejpam-4561	215	13	.	.	PUNCT
ejpam-4561	216	1	on	on	ADP
ejpam-4561	216	2	fuzzy	fuzzy	ADJ
ejpam-4561	216	3	b	b	NOUN
ejpam-4561	216	4	-	-	PUNCT
ejpam-4561	216	5	algebra	algebra	NOUN
ejpam-4561	216	6	.	.	PUNCT
ejpam-4561	217	1	czechoslovak	czechoslovak	ADJ
ejpam-4561	217	2	mathematical	mathematical	PROPN
ejpam-4561	217	3	journal	journal	PROPN
ejpam-4561	217	4	,	,	PUNCT
ejpam-4561	217	5	52(2):375–384	52(2):375–384	NOUN
ejpam-4561	217	6	,	,	PUNCT
ejpam-4561	217	7	2002	2002	NUM
ejpam-4561	217	8	.	.	PUNCT
ejpam-4561	218	1	[	[	X
ejpam-4561	218	2	6	6	NUM
ejpam-4561	218	3	]	]	PUNCT
ejpam-4561	218	4	j	j	PROPN
ejpam-4561	218	5	neggers	negger	NOUN
ejpam-4561	218	6	.	.	PUNCT
ejpam-4561	219	1	a	a	DET
ejpam-4561	219	2	fundamental	fundamental	ADJ
ejpam-4561	219	3	theorem	theorem	NOUN
ejpam-4561	219	4	of	of	ADP
ejpam-4561	219	5	b	b	NOUN
ejpam-4561	219	6	-	-	PUNCT
ejpam-4561	219	7	homomorphism	homomorphism	NOUN
ejpam-4561	219	8	for	for	ADP
ejpam-4561	219	9	b	b	NOUN
ejpam-4561	219	10	-	-	PUNCT
ejpam-4561	219	11	algebras	algebras	PROPN
ejpam-4561	219	12	.	.	PUNCT
ejpam-4561	220	1	onternation	onternation	PROPN
ejpam-4561	220	2	journal	journal	PROPN
ejpam-4561	220	3	of	of	ADP
ejpam-4561	220	4	mathematicsl	mathematicsl	PROPN
ejpam-4561	220	5	,	,	PUNCT
ejpam-4561	220	6	2:215–219	2:215–219	NUM
ejpam-4561	220	7	,	,	PUNCT
ejpam-4561	220	8	2002	2002	NUM
ejpam-4561	220	9	.	.	PUNCT
ejpam-4561	221	1	[	[	X
ejpam-4561	221	2	7	7	X
ejpam-4561	221	3	]	]	X
ejpam-4561	221	4	a	a	DET
ejpam-4561	221	5	sangalli	sangalli	NOUN
ejpam-4561	221	6	.	.	PUNCT
ejpam-4561	222	1	fuzzy	fuzzy	ADJ
ejpam-4561	222	2	logic	logic	NOUN
ejpam-4561	222	3	.	.	PUNCT
ejpam-4561	223	1	in	in	ADP
ejpam-4561	223	2	encyclopedia	encyclopedia	PROPN
ejpam-4561	223	3	britannica	britannica	PROPN
ejpam-4561	223	4	.	.	PROPN
ejpam-4561	224	1	2019	2019	NUM
ejpam-4561	224	2	.	.	PUNCT
ejpam-4561	225	1	[	[	X
ejpam-4561	225	2	8	8	NUM
ejpam-4561	225	3	]	]	X
ejpam-4561	225	4	l	l	PROPN
ejpam-4561	225	5	zadeh	zadeh	PROPN
ejpam-4561	225	6	.	.	PUNCT
ejpam-4561	225	7	fuzzy	fuzzy	ADJ
ejpam-4561	225	8	sets	set	NOUN
ejpam-4561	225	9	.	.	PUNCT
ejpam-4561	226	1	information	information	NOUN
ejpam-4561	226	2	and	and	CCONJ
ejpam-4561	226	3	control	control	NOUN
ejpam-4561	226	4	,	,	PUNCT
ejpam-4561	226	5	08:338–365	08:338–365	PROPN
ejpam-4561	226	6	,	,	PUNCT
ejpam-4561	226	7	1965	1965	NUM
ejpam-4561	226	8	.	.	PUNCT