Recei ed 25 July 2024, accep ed 6 Augus 2024, da e o publica ion 12 Augus 2024, da e o cu en e sion 20 Augus 2024.
Digi al Objec Iden i ie 10.1109/ACCESS.2024.3441940
On G adual Se s, Hesi an Fuzzy Se s,
and Rep esen a ion Theo ems
PASCUAL JARA1,2, LUIS MERINO 1,2, GABRIEL NAVARRO 1,2,3,
AND EVANGELINA SANTOS 1,2
1Depa men o Algeb a, Uni e si y o G anada, 18071 G anada, Spain
2Ma hema ics Ins i u e o he Uni e si y o G anada (IMAG), 18001 G anada, Spain
3Resea ch Cen e o In o ma ion and Communica ion Technologies (CITIC-UGR), 18014 G anada, Spain
Co esponding au ho : Gab iel Na a o (gna a o@ug .es)
This wo k was suppo ed in pa by he P og ama Ope a i o FEDER 2014-2020 and Conseje ía de Economía, Conocimien o, Emp esas y
Uni e sidad de la Jun a de Andalucía, Spain, unde G an A-FQM-394-UGR20; and in pa by he ‘‘Ma ía de Maez u’’ Excellence Uni
IMAG, unded by MCIN/AEI/10.13039/501100011033, unde G an CEX2020-001105-M.
ABSTRACT We analyze he connec ion be ween wo pe spec i es when de ining uzzy se s: he iewpoin
o mappings and he iewpoin o amilies o le el cu s. This analysis is ma hema ically suppo ed by
he amewo k o a ca ego ical adjunc ion, which se es as a dic iona y be ween hese wo pe spec i es.
We p o e ha hesi an uzzy se s and g adual se s a e s ongly ela ed h ough his connec ion. This allows
concep s and ope a ions o be ans e ed om one class o he o he , and ice e sa. Conc e ely, as an
applica ion, we p o ide la ice ope a ions on g adual se s, compa ible wi h Zadeh’s max-min ope a ions
on uzzy se s, when conside ing hem as amilies o le el cu s. We discuss he well-known ep esen a ion
heo em o uzzy se s wi hin his amewo k, and we show ha he ep esen a ion o uzzy se s as g adual
se s depends on he chosen embedding o uzzy se s as hesi an uzzy se s. Hence, dis inc embeddings yield
di e se ep esen a ions o uzzy se s as collec ions o subse s. Fu he mo e, we ex end his me hodology o
include o he classes o ex ended uzzy se s. As a consequence, a ep esen a ion heo em o in e al- alued
uzzy se s is p o ided.
INDEX TERMS Fuzzy se s, g adual se s, hesi an uzzy se s, in e al- alued uzzy se s, le el cu s,
ep esen a ion heo ems, se - alued uzzy se s.
I. INTRODUCTION
The ele ance o uzzy se s [1] is nowadays wo ldwide
ecognized. This lies in hei abili y o ackle eal-wo ld
p oblems cha ac e ized by ambigui y, agueness, and imp e-
cision. By p o iding a ma hema ical amewo k ha goes
beyond c isp se heo y, his app oach o e s a mo e na u al
and ealis ic ep esen a ion o unce ain y, con ibu ing o he
ad ancemen o ce ain a eas ha need i s managemen .
To expand hei exp essi eness, some au ho s ha e gen-
e alized he o iginal concep , yielding di e en ypes o
ex ended uzzy se s. Fo ins ance, in e al- alued uzzy
se s [2],[3] model imp ecision by speci ying a ange
o in e al. Hence, his ep esen a ion allows o a mo e
The associa e edi o coo dina ing he e iew o his manusc ip and
app o ing i o publica ion was Wai-Keung Fung .
lexible desc ip ion o he unce ain y, comp ising bo h
he lowe and uppe bounds o he possible membe ship
alues. Analogously, hesi an uzzy se s [4] (o iginally called
se - alued uzzy se s in [5]) can manage mul iple possibili ies
o sou ces o in o ma ion, since, a he han assigning a
single alue, hese ep esen he membe ship deg ee as a
collec ion o hem. Fo his eason, hey ha e p o ed use ul
in modeling ce ain sys ems, such as hose ela ed o mul i-
c i e ia decision-making, whe e di e en expe s o c i e ia
can con ibu e assigning di e en membe ship deg ees o an
objec . Ano he in e es ing ypes appea ing in he li e a u e
a e, o ins ance, in ui ionis ic uzzy se s [6], o ype-2 uzzy
se s [3], among many o he s.
Al hough hese ex ensions unde s and he ep esen a ion
o unce ain y di e en ly, mainly each one adap ed o
ce ain speci ic scena ios, hey sha e a common unc ional
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P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
pe spec i e. A uzzy se is mos ly iewed as a mapping
assigning, o any elemen on he uni e se se , a membe ship
deg ee in he e e ence la ice, s anda dly he uni in e al
[0,1]. Hence, om a sui able eplacemen o he codomain,
mainly by a pa ial o de ed se o allow compa isons be ween
deg ees, i esul s he abo e-commen ed gene aliza ions as
mappings alued in hese new spaces. Ne e heless, a uzzy
se may also be desc ibed by means o he collec ion o
i s le el cu s, which leads o he well-known ep esen a ion
heo em s a ed by Negoi a and Ralescu in [7, Theo em 1].
This o e s a di e en poin -o - iew ha we may call he
le el pe spec i e. Following his idea, g adual se s [8] we e
in oduced as an ex ension o uzzy se s by Dubois and
P ade. Ins ead o deg ees o membe ship, g adual se s ocus
on ep esen ing g adual ansi ions o g adual bounda ies
be ween membe ship and non-membe ship. They cap u e he
idea ha he ansi ion o an elemen om non-membe ship
o membe ship can occu g adually o inc emen ally, a he
han as an ab up change. This gene alizes he well-known
ac ha e e y uzzy se can be desc ibed by i s le el
cu s.
In his pape , we analyze hese wo pe spec i es and
desc ibe he ma hema ical connec ion be ween hem. This
no el app oach no only gene alizes exis ing heo ies bu also
p o ides a comp ehensi e amewo k o unde s anding and
ex ending uzzy se concep s om he iewpoin o ca ego y
heo y, which has no been done be o e. In his sense,
we ollow he s anda d deduc i e me hodology. We p og ess
om gene al esul s o mo e speci ic ones. A a highe le el
o abs ac ion, his connec ion is modeled by a well-known
ca ego ical concep , namely, an adjunc ion be ween wo
unc o s. The adjunc ion e eals a one- o-one mapping,
e ec i ely se ing as a dic iona y be ween bo h poin s o
iew. Consequen ly, we may es ablish ela ionships be ween
di e en ex ensions o uzzy se s om bo h pe spec i es.
In pa icula , we show ha he concep s o hesi an uzzy
se s and g adual se s a e s ongly in e connec ed. In ac ,
acco ding o he de ini ions p oposed by some au ho s, hese
concep s e e o he same objec s when conside ing he
unc ional and le el pe spec i es. Hence, in a lowe le el o
abs ac ion, sui able p ope ies o algeb aic s uc u es may be
endowed o hese objec s ia he abo e-commen ed bijec ion.
Fo ins ance, la ice s uc u es on g adual se s can be de i ed
om known s uc u es on hesi an uzzy se s, en iching
he le el pe spec i e. On he o he hand, his ma hema ical
amewo k na u ally suppo s he enowned ep esen a ion
heo em o uzzy se s as s a ed in [7]. The e o e, i becomes
pe inen o ask abou he p ecise speci ic o mula ion o he
heo em wi hin he scheme de eloped ea lie . Conce ning his
ques ion, we show ha he ep esen a ion o uzzy se s as
g adual se s depends on he chosen embedding o uzzy se s
as hesi an uzzy se s. Consequen ly, we should no e e o a
single uzzy se ep esen a ion heo em, bu o a collec ion o
hem depending on he embedding applied. Fu he mo e, he
heo y de eloped he e allows o s a e ep esen a ion heo ems
o o he classes con ained on hesi an uzzy se s, as in e al-
alued uzzy se s. The main con ibu ions o his pape can
be summa ized as ollows.
•This wo k analyzes in de ail he no ions o hesi an uzzy
se s and g adual se s, bo h as ex ensions o classical
uzzy se s, and desc ibes he ma hema ical connec ion
ha ela es hem. I shows ha some s udies in he
li e a u e e e o he same class o objec s, iewed
om di e en pe spec i es. This opens up he possibili y
o ela ing hese s udies oge he and imp o ing he
exis ing heo ies.
•The pape in oduces a la ice s uc u e on g adual
se s whose es ic ion o he class o uzzy se s,
iewed as chains o subse s, becomes Zadeh’s s anda d
la ice s uc u e. This en iches he le el pe spec i e o
g adual se s, p o iding ope a ions be ween i s objec s
and allowing a sys ema ic manipula ion o g adual
se s consis en wi h he ope a ions on uzzy se s.
In pa icula , i p o ides a common amewo k, cohe en
om an ope a ional poin o iew, o wo king wi h
uzzy objec s whose uzziness is measu ed di e en ly.
•The pape suppo s he well-known ep esen a ion heo-
em o uzzy se s, and cla i ies ha he ep esen a ion
o uzzy se s as g adual se s depends on he chosen
embedding o uzzy se s as hesi an uzzy se s, leading
o a collec ion o ep esen a ion heo ems based on
di e en embeddings.
•The de eloped heo y allows o he o mula ion o
ep esen a ion heo ems o a ious classes o ex ended
uzzy se s. Speci ically, he pape p o ides a ep esen a-
ion heo em o in e al- alued uzzy se s.
The pape is s uc u ed as ollows. In sec ion II we es ablish
he needed esul s o de elop he con en o he pape .
Conc e ely, we in oduce some no a ion and e iew some
basic ac s, and p esen he main heo e ical key- ool: a ca -
ego ical adjunc ion. This adjunc ion de ines an isomo phism
which se es as a dic iona y-like co espondence be ween he
wo abo e-commen ed pe spec i es. In Sec ion III, by using
his bijec ion, we de i e a amewo k o explo ing he
ela ionships be ween se e al classes o ex ended uzzy se s.
We hen p o e a s ong connec ion be ween he classes o
hesi an uzzy se s and g adual se s. As di e en au ho s ha e
p oposed dis inc de ini ions o hese concep s, we show
he p ecise ma hema ical ela ion o se e al possibili ies.
In Sec ion IV we apply he p e ious heo e ical esul s in
o de o de elop a la ice o de on g adual se s. By means o
he esul s gi en in [9], we in e a la ice o de on he class o
(wha we ha e called) ull g adual se s, and, as a consequence,
on g adual se s as de ined in [8]. I is wo h no ing ha i s
es ic ion o uzzy se s co esponds o he classical o de
p oposed by Zadeh, so i allows o ope a e wi h di e en
ea men s o unce ain y simul aneously. In Sec ion Vwe
explain he ole o he well-known ep esen a ion heo em o
uzzy se s wi hin he amewo k es ablished in Sec ion III.
Based on his abs ac ion, we demons a e he po en ial o
VOLUME 12, 2024 111159
P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
de i e addi ional ep esen a ion heo ems o o he classes o
ex ended uzzy se s. In pa icula , we p o e a ep esen a ion
heo em o in e al- alued uzzy se s. Finally, Sec ion VI
p esen s ou concluding ema ks.
II. A CATEGORICAL ADJUNCTION
Le Ybe an a bi a y se , i is well known ha he powe se
P(Y) is endowed wi h a comple e la ice s uc u e p o ided
by he classic se ope a ions (union, in e sec ion) wi h ∅
and Ybeing he minimum and maximum, espec i ely.
Addi ionally, P(Y) is isomo phic, as a la ice, o
2
Y=
Map(Y,
2
), he class o se mappings om Y o
2
, whe e
2
is he Boolean algeb a wi h wo elemen s. He e, he la ice
s uc u e on Map(Y,
2
) is p o ided by he pa ial o de ≤M
gi en by, o each maps ,g∈Map(Y,
2
),
≤Mgi and only i (y)≤g(y) o any y∈Y, (1)
whe e ≤is he o de on
2
. I used o say ha ≤Mis
inhe i ed om ≤. In gene al, i Lis a (bounded, comple e)
la ice, hen, o any se Y, Map(Y,L) is endowed wi h
a (bounded, comple e) la ice s uc u e, inhe i ed om L,
de ined simila ly as done in (1).
Le now Xand Lbe wo se s, so we may conside he
powe se s P(X) and P(L), and he classes o se mappings
Map(X,P(L)) and Map(L,P(X)). As commen ed abo e,
bo h a e la ices whose la ice s uc u es a e inhe i ed om
P(L) and P(X), espec i ely.
Rema k 1: No e ha , i L=[0,1], he elemen s in
Map(X,P(L)) ep esen a sligh gene aliza ion o he concep
o se - alued uzzy se as de ined in [10, De ini ion 5.1].
In ha de ini ion, he codomain is es ic ed o P(L) ∅.
Ne e heless, in such a case, he se - heo e ic in e sec ion
is no well-de ined. In [5], a se - alued uzzy se is de ined
exac ly as a map in Map(X,P(L)), allowing he emp y
se as he nonsense membe ship deg ee, and a oiding his
p oblem. In [4] hese objec s a e called hesi an uzzy se s
and di e en ope a ions a e p o ided. In his pape we
shall conside bo h e ms as synonyms, and we ese e he
e minology ‘‘se - alued uzzy se ’’ o ‘‘hesi an uzzy se ’’
o an elemen in Map(X,P(L) ∅). This choice is made
because, as a gene aliza ion o classical uzzy se s, se -
heo e ic ope a ions a e no well-beha ed wi h he max-min
ope a ions on uzzy se s, whils To a’s ope a ions do no
p o ide a la ice s uc u e.
On he o he hand, he elemen s in Map([0,1],P(X))
p o ide an al e na i e no ion o he concep o g adual se
gi en in [11, De ini ion 3.2], since, i I⊆[0,1] and G:
I→P(X), he mapping G:[0,1] →P(X) de ined as
G(α)=(G(α),i α∈I,
∅,o he wise,
e i ies ha he es ic ion G|Iequals G. This allows us an
easie handling o his kind o s uc u es, since he e is no
need o using di e en domains, and all o hem a e de ined in
[0,1]. Addi ionally, i ex ends he concep o ep esen a ion
le el in [12, De ini ions 2.1 and 2.3]. Indeed, gi en (3, ρ)
wi h 3= {1=α1>· · · > αm+1=0}, simply conside he
mapping
G(α)=ρ(α) i αi≥α > αi+1
o each α∈(0,1] and G(0) =ρ(αm+1).
These wo cons uc ions a e s ongly ela ed. The e
exis s a one- o-one mapping be ween Map(X,P(L)) and
Map(L,P(X)). Conc e ely, conside he mapping
8:Map(X,P(L)) −→ Map(L,P(X)) (2)
de ined as 8( )(α)= {x∈Xsuch ha α∈ (x)}, o any
∈Map(X,P(L)) and any α∈L. The in e se o 8is gi en
by he mapping
9:Map(L,P(X)) −→ Map(X,P(L)),(3)
de ined as 9(g)(x)= {α∈Lsuch ha x∈g(α)}, o any
g∈Map(L,P(X)) and any x∈X. I is no di icul o see ha
bo h mappings a e in e se one o each o he . Indeed, o any
∈Map(X,P(L)) and x∈X, by de ini ion, α∈98( )(x)
i and only i x∈8( )(α) i and only i α∈ (x), so ha
98( )(x)= (x) o any x∈X. Then 98( )= o any
∈Map(X,P(L)). Hence 98 =IdMap(X,P(L)). One may
p o e almos e ba im ha 89 =IdMap(L,P(X)).
Rema k 2: The abo e p ope y ollows om he adjunc-
ion be ween he unc o s Map and ca esian p oduc ×. Fo
comple eness, we ecall b ie ly his ac , see [13] o basic
de ini ions. Le Se be he ca ego y o med by all se s. Fo
some se X, he unc o Map(X,−):Se →Se maps a se Y
o Map(X,Y), and each mo phism :Y→Y′ o Map(X, ):
Map(X,Y)→Map(X,Y′) gi en by Map(X, )(g)=g , he
composi ion o mappings, o any g∈Map(X,Y). On he
o he hand, he unc o X× − : Se →Se maps a se Y o
he ca esian p oduc X×Y, and each mo phism :Y→Y′
o X× :X×Y→X×Y′gi en by (X× )(a,b)=(a, (b))
o any a∈Xand b∈Y.
The unc o X× − is le adjoin o Map(X,−), meaning
ha , o each se s Land Z, he e exis s a na u al bijec ion
φ:Map(X×L,Z)→Map(L,Map(X,Z)). Tha is o say,
Z(X×L)∼
=(ZX)L, whe e we ha e deno ed Map(A,B) by i s
well-known powe se no a ion BA. Now, obse e ha X×L∼
=
L×Xby he s anda d lip map, so combining he bijec ions
(ZX)L∼
=Z(X×L)∼
=Z(L×X)∼
=(ZL)X.
Whene e Z=
2
, Map(L,P(X)) =(
2
X)L∼
=(
2
L)X=
Map(X,P(L)), yielding he desi ed bijec i e map.
F om he de ini ion gi en by Goguen in [14], we may
see he elemen s in Map(X,P(L)) as P(L)- uzzy se s. Then,
by i ue o Rema k 2, each P(L)- uzzy se can be seen in
Map(X,P(L)) o in Map(L,P(X)), yielding a e ical and a
ho izon al ep esen a ions o he co esponding P(L)- uzzy
se . Le us illus a e his wi h an example.
111160 VOLUME 12, 2024
P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
Example 1: Conside X= {a,b,c,d}and L=[0,1], and
he mapping ∈Map(X,P([0,1]) gi en by
(a)=[0.2,0.4],
(b)=[0.3,0.5],
(c)=[0.5,0.7],
(d)=[0,0.6].
Then is an in e al- alued uzzy se o e Xand his poin -
o - iew co esponds o he e ical ep esen a ion desc ibed
in Fig. 1. On he o he hand, i s image unde 8, he ho izon al
ep esen a ion, 8( ):[0,1] →P(X), is gi en by
8( )(x)=























{d}i 0 ≤x<0.2,
{a,d}i 0.2≤x<0.3,
{a,b,d}i 0.3≤x≤0.4,
{b,d}i 0.4<x<0.5,
{b,c,d}i x=0.5,
{c,d}i 0.5<x≤0.6,
{c}i 0.6<x≤0.7,
∅i x>0.7,
FIGURE 1. Ve ical ep esen a ion o .
FIGURE 2. Ho izon al ep esen a ion o .
see Fig. 2.
The abo e-desc ibed bijec ions a e also la ice isomo phisms
when conside ing he s anda d o de s on P(L) and P(X).
Theo em 1: The maps 8and 9a e isomo phisms o
bounded la ice wi h espec o he o de inhe i ed by he
se - heo e ic ope a ions in he powe se s.
P oo : We only need o p o e ha 8 espec s he pa ial
o de . Le ,g∈Map(X,P(L)) such ha ≤Mg, which
means (x)⊆g(x) o all x∈X. Then 8( )(α)= {x∈
Xsuch ha α∈ (x)}⊆{x∈Xsuch ha α∈g(x)} =
8(g)(α) o any α∈L, so ha 8( )≤M8(g).
On he o he hand, he minimum and maximum in
Map(X,P(L)) a e he cons an maps 0and 1gi en by 0(x)=
∅and 1(x)=L o any x∈X, espec i ely. I is also clea
ha 8(0)(α)= ∅ and 8(1)(α)=X o any α∈L.
Analogously, we may p o e 9holds he same
p ope ies.
Rema k 3: Al hough he p oo o Theo em 1conce ns he
pa ial o de , we could ha e p o ed i by using he la ice
ope a ions, say ∧Mand ∨M o bo h la ice, which a e gi en
by ( ∧Mg)(x)= (x)∩g(x) and ( ∨Mg)(x)= (x)∪g(x)
o any x∈X(o any x∈L). The e o e, since 8and 9a e
isomo phisms, 8( ∧Mg)=8( )∧M8(g) and 8( ∨Mg)=
8( )∨M8(g), o any ,g∈Map(X,P(L)), and 9( ∧M
g)=9( )∧M9(g) and 9( ∨Mg)=9( )∨M9(g),
o any ,g∈Map(L,P(X)).
III. GRADUAL SETS AND HESITANT FUZZY SETS
Ou aim now is o show how he bijec ion gi en by he adjunc-
ion in Rema k 2 ela es he no ions o hesi an uzzy se and
g adual se . We ecall om [8] ha , gi en a bounded la ice
L, an L-g adual subse o Xis a mapping om L+ o P(X)
(see also [15], o [16] wi h an addi ional s uc u e o g oup),
whe e L+=L {0}, and 0 deno es he minimum o he
bounded la ice L. Le us deno e by GSL(X) he class o all
L-g adual se s (g adual se s, o sho , when he con ex is
clea enough) o X. We also ecall om [10] ha a hesi an
L- uzzy se o e a uni e se Xis a mapping :X→P∗(L),
whe e by P∗(X) we mean he class o non-emp y subse s
o X. Le us deno e by HFSL(X) he class o all hesi an
L- uzzy se s.
Le us now se he class
FGSL(X)=( ∈Map(L,P(X)) wi h [
α∈L
(α)=X),
and we e e o i as he class o ull L-g adual se s ( ull
g adual se s, o sho ) o X.
Lemma 1: The mappings 9and 8de e mine a bijec ion
be ween he classes HFSL(X) and FGSL(X).
P oo : By Rema k 2we only ha e o show ha he
es ic ions o 9and 8a e well-de ined. Indeed, le ∈
HFSL(X) and x∈X. Then (x) is a non-emp y se in L.
In pa icula , he e exis s some α∈ (x), so x∈8( )(α).
Hence X⊆ ∪α∈L8( )(α) and hus 8( )∈FGSL(X).
Con e sely, i g∈FGSL(X), hen, o any x∈X, he e exis s
an α∈Lsuch ha x∈g(α). Hence, α∈9(g)(x), and hen
9(g)(x) is a non-emp y se . Thus 9(g)∈HFSL(X).
VOLUME 12, 2024 111161
P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
Le us inse he well-known class o L- uzzy se s [14]
in his scheme. We may always embed a bounded la ice L
in o i s powe se P(L) by he mapping each elemen α o
he in e al [0, α]= {β∈Lsuch ha β≤α}. Obse e
ha his is clea ly a o de -p ese ing mapping wi h espec
o he o de on Land he se - heo e ical o de on P(L). Also,
[0, α1]∩[0, α2]=[0, α1∧α2], so ha i also p ese es mee s.
Indeed, o any α1, α2∈L,
β∈[0, α1]∩[0, α2]⇔β∈[0, α1] and β∈[0, α2]
⇔β≤α1and β≤α2
⇔β≤α1∧α2
⇔β∈[0, α1∧α2].
Ne e heless, ha is no he case o he join ope a o .
We may only say,
β∈[0, α1]∪[0, α2]⇔β∈[0, α1] o β∈[0, α2]
⇔β≤α1o β≤α2
⇒β≤α1∨α2
⇔β∈[0, α1∨α2].
In gene al, he implica ion
(β≤α1∨α2)⇒(β≤α1o β≤α2)
does no hold. Fo ins ance, conside he ollowing oy
example,
No e ha we may ge his p ope y i Lis a o ally o de ed
la ice. Le us deno e by FSL(X) he class o all L- uzzy se s
Map(X,L).
P oposi ion 1: The mapping ϕ:FSL(X)→HFSL(X),
de ined as ϕ( )(x)=[0, (x)] o any ∈FSL(X) and any
x∈X, is an o de and mee p ese ing injec i e mapping.
Mo eo e , i Lis o ally o de ed, ϕis a la ice embedding.
P oo : I ollows om he abo e discussion.
We ha e hen he ollowing commu a i e diag am, see
Figu e 3.
The hi d ow in Figu e 3co esponds o he isomo phisms
desc ibed in Sec ion II o he la ice L+. Fo simplici y,
we ha e also deno ed hem by 8and 9. The mapping p
simply maps each ∈Map(L,P(X)) o i s es ic ion o L+,
so i is su jec i e.
Following his diag am, we ind ha he classes HFSL(X)
and GSL(X) can be embedded in o Map(L,P(X)), so we may
y o compa e hem. Indeed, ia 8, he hesi an uzzy se s
can be seen as he mappings in FGSL(X). On he o he hand,
since pis su jec i e, he e exis an injec ion j:GSL(X)→
Map(L,P(X)) such ha pj is he iden i y map.
FIGURE 3. Rela ion be ween hesi an uzzy se s and g adual se s.
The map jhas no o be unique, and he e exis as many copies
o GSL(X) inside Map(L,P(X)) as possible mappings j.
Obse e ha , gi en ∈GSL(X), since pj( )= , hen
pj( )(α)=j( )(α)= (α) o any α∈L+. Thus, he map j
is simply de e mined by choosing an image o 0 ∈L o
any elemen in GSL(X). Fo ou pu pose, we ix he mapping
jX:GSL(X)→Map(L,P(X)) gi en by
jX( )(α)= (α),i α= 0,
X,i α=0,
o any ∈GSL(X) and α∈L. Thus jX(GSL(X)) ⊆
i(FGSL(X)), and hen, GSL(X)⊂pi(FGSL(X)) =
pi8(HFSL(X)). Tha is, in o he wo ds, he map
pi8:HFSL(X)→GSL(X)
is su jec i e. Hence GSL(X) is isomo phic o some quo ien
o HFSL(X).
De ini ion 1: Le ,g∈HFSL(X) we say ha ∼0gi
(x)∪ {0} = g(x)∪ {0} o all x∈X, o , equi alen ly, (x)
{0} = g(x) {0} o all x∈X. Clea ly, ∼0is an equi alence
ela ion on HFSL(X).
Theo em 2: HFSL(X)/∼0∼
=GSL(X).
P oo : F om he abo e discussion, i is enough o see
ha he ela ion ∼0is he equi alence ela ion associa ed o
he composi ion pi8. Le ,g∈HFSL(X), hence, ∼0g
i and only i (x) {0} = g(x) {0} o all x∈X, which
is equi alen o α∈ (x) i and only i α∈g(x) o all x∈
Xand all α∈L+, and also o x∈8( )(α) i and only i x∈
8(g)(α) o all x∈Xand all α∈L+. Analogously, his las
s a emen is equi alen 8( )(α)=8(g)(α) o all α∈L+,
and, hen, o (pi8)( )(α)=(pi8)(g)(α) o all α∈L+, and,
inally, o (pi8)( )=(pi8)(g).
Rema k 4: No e ha FGSL(X)/∼0∼
=GSL(X), whe e
∼0gi and only i (α)=g(α) o all α∈L+.
Obse e ha , he image o GSL(X) ia 9jXco esponds o
he class
AL(X)=( ∈Map(X,P(L)) such ha 0 ∈
x∈X
(x))
⊆Map(L,P(X)).
111162 VOLUME 12, 2024

P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
Indeed, i ∈AL(X), hen
8( )(0) = {x∈Xsuch ha 0 ∈ (x)} = X
so 8( ) is in jX(GSL(X)). Thus ∈9jX(GSL(X)).
Con e sely, le g∈9jX(GSL(X)), hen 8(g)∈jX(GSL(X))
and hen 8(g)(0) =X. The e o e 0 ∈g(x) o all x∈Xand,
consequen ly, g∈AL(X). Thus we may unde s and g adual
se s as hesi an uzzy se s in which he minimum 0 belongs o
he membe ship deg ee o any elemen .
Rema k 5: To summa ize his sec ion, and o he sake o
cla i y, i is wo h no ing ha di e en no ions o hesi an
uzzy se s (o se - alued uzzy se s) and g adual se s ha e
been explo ed in he li e a u e. Hence o h, al hough closely
ela ed, he explici ma hema ical ela ionship be ween hese
ex ensions may di e sligh ly depending on how hey a e
conside ed. Fo ins ance, G a an-Guinness [5] de ines a
hesi an uzzy se as an elemen in Map(X,P([0,1])),
whils Bus ince e al. [10] de ine i as a membe o
Map(X,P∗([0,1])). Simila ly, Dubois and P ade [8] de ine
a g adual se as a mapping in Map((0,1],P(X)), whils
Wu [11] conside s Map([0,1],P(X)) as he class o g adual
se s. Suppose ha , o a gi en e e ence se Xand la ice L,
we deno e by Sand G he class o hesi an uzzy se s and
g adual se s, espec i ely. Hence:
•I S=Map(X,P(L)) and G=Map(L,P(X)),
Rema k 2ensu es ha bo h concep s desc ibe exac ly
he same class o objec s, iewed om wo di e en
pe spec i es, ha we may call he ho izon al and e ical
ep esen a ions.
•I S=Map(X,P(L)) and G=Map(L+,P(X)),
by he la e commen s, Gco esponds o he subclass
o Scomposed by hose objec s e i ying ha he
membe ship deg ee o each elemen in Xcon ains he
alue 0.
•I S=Map(X,P∗(L)) and G=Map(L,P(X)),
by Lemma 1,Sco esponds o he subclass o G
composed by he mappings ha a e ull.
•I S=Map(X,P∗(L)) and G=Map(L+,P(X)),
by Theo em 2,Gis a quo ien o Sgi en by he
equi alen ela ion ha joins hose mappings ha sha e
he same images excep o he alue 0.
In ou opinion, he mos sui able op ion is he hi d one.
As we shall explain in he nex sec ion, in his case, we may
ind bounded la ice s uc u es compa ible wi h he Zadeh’s
ope a ions on uzzy se s.
IV. LATTICE STRUCTURES ON GRADUAL SETS
Le us now show an applica ion o he connec ions es ablished
in he diag am o Figu e 3. Conc e ely, we shall de ine new
la ice o de s on he class o g adual se s and ull g adual
se s. All along his sec ion Lis he la ice [0,1] endowed
wi h he usual ope a ions. As poin ed ou in Sec ion II, when
wo king wi h hesi an uzzy se s, he inhe i ed ope a ions
om he se - heo e ic union and in e sec ion on P∗([0,1])
a e no well-de ined. An ob ious solu ion is o ex end
he de ini ion o hesi an uzzy se o be an elemen
in Map(X,P([0,1])), allowing he somehow-called ‘‘non-
sense’’ membe ship deg ee. Ne e heless, as a coun e pa ,
he se inclusion seems no o be a sui able o de o so he
membe ship deg ees. Simply obse e ha he se {1}should
be he g ea es membe ship deg ee, which is con ained in
[0.5,1] and no ela ed o {0}. Addi ionally, as commen ed
in [10, Rema k 2], see also [17], hese do no ex end
he classical ope a ions on uzzy se s (when single- alued
assignmen s a e ea ed as single ons), and on in e al- alued
uzzy se s p o ided by Zadeh. To a’s pape [4] discusses
he de ini ion o new ope a ions compa ible wi h Zadeh’s
la ice s uc u e on uzzy se s, unde he name o hesi an
uzzy se s. Ne e heless, To a’s p oposal does no endow
hesi an uzzy se s wi h a la ice s uc u e. In [9], an answe
is gi en by de ining he so-called symme ic o de ≤0on he
class o non-emp y subse s P∗([0,1]). This yields a la ice
o de on HFS[0,1](X) whose es ic ion o uzzy se s and
in e al- alued uzzy se s equals he Zadeh’s s uc u es.
Gi en wo subse s X,Y⊆[0,1], we say ha X<Yi
x<y o any x∈Xand y∈Y. This i ially holds i one o
hem is he emp y se .
De ini ion 2 (Symme ic O de ): [9] Gi en wo non-
emp y subse s Aand Bin [0,1], we say ha A≤0Bi and
only i A B<Band A<B A.
The ela ion ≤0is a bounded la ice o de on P∗([0,1])
ex ending he min-max ope a ions on [0,1], so, as done in (1),
he ollowing ela ion on HFS[0,1](X) p o ides a bounded
la ice o de ex ending he min-max ope a ions on uzzy se s.
De ini ion 3: Le ,g∈HFS[0,1](X), we say ha ≤0g
i and only i (x)≤0g(x) o all x∈X.
The e o e he ollowing ela ion ≤9, de i ed om he
bijec ion 9in Figu e 3, also p o ides a bounded la ice o de
on FGS[0,1](X). Indeed, o any maps ,g∈FGS[0,1](X),
we say ≤9gi and only i 9( )≤09(g). Ne e heless,
in p ac ice, i is mo e con enien o ha e an explici
desc ip ion.
De ini ion 4: Le ,g∈FGS[0,1](X) be wo ull g adual
se s, we say ha ≤9gi and only i , o any α≤βin
[0,1], (β)∩g(α)⊆ (α)∩g(β).
Theo em 3: The ela ion ≤9is an o de on FGS[0,1](X).
P oo : The e lexi e p ope y holds i ially. Le now
,g∈FGS[0,1](X) such ha ≤9gand g≤9 .
Hence, o any α≤β, (β)∩g(α)⊆ (α)∩g(β) and
(α)∩g(β)⊆ (β)∩g(α). Tha is o say, o any α, β ∈[0,1],
(β)∩g(α)= (α)∩g(β).(4)
Then, o any α∈[0,1],
g(α)=X∩g(α)
=
[
β∈[0,1]
(β)
∩g(α)
=[
β∈[0,1]
( (β)∩g(α))
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P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
=[
β∈[0,1]
( (α)∩g(β))
= (α)∩
[
β∈[0,1]
g(β)

= (α)∩X
= (α).
Thus =g, and he an isymme ic p ope y is sa is ied.
Finally, o he ansi i i y, le ,g,h∈FGS[0,1](X) such ha
≤9gand g≤9h. Then, o any α≤βin [0,1],
(β)∩g(α)⊆ (α)∩g(β),(5)
and
g(β)∩h(α)⊆g(α)∩h(β).(6)
Obse e ha
(β)∩h(α)= (β)∩X∩h(α)
= (β)∩
[
δ∈[0,1]
g(δ)
∩h(α)
=[
δ∈[0,1]
( (β)∩g(δ)∩h(α))(7)
Now, i δ < α,
(β)∩g(δ)∩h(α)(5)
⊆ (δ)∩g(β)∩h(α)⊆g(β)
and, i ially,
(β)∩g(δ)∩h(α)⊆ (β)∩h(α),
so
(β)∩g(δ)∩h(α)⊆ (β)∩g(β)∩h(α).
On he o he hand, i δ > β,
(β)∩g(δ)∩h(α)(6)
⊆ (β)∩g(α)∩h(δ)⊆g(α),
and
(β)∩g(δ)∩h(α)⊆ (β)∩h(α),
so
(β)∩g(δ)∩h(α)⊆ (β)∩g(α)∩h(α).
Thus,
(7) =[
δ∈[α,β]
( (β)∩g(δ)∩h(α))
(5)
⊆[
δ∈[α,β]
( (δ)∩g(β)∩h(α))
(6)
⊆[
δ∈[α,β]
( (δ)∩g(α)∩h(β))
(5)
⊆[
δ∈[α,β]
( (α)∩g(δ)∩h(β))
= (α)∩
[
δ∈[α,β]
g(δ)
∩h(β)
⊆ (α)∩
[
δ∈[0,1]
g(δ)
∩h(β)
= (α)∩X∩h(β)
= (α)∩h(β)
Then ≤9h. This concludes he p oo .
The o de ≤9is a bounded o de . Obse e ha he
minimum he e is he ull g adual se 0de ined by
0(α)=∅i α= 0,
Xi α=0,
whils he maximum is he ull g adual se 1de ined by
1(α)=(∅i α= 1,
Xi α=1.
Theo em 4: The maps 8and 9p o ide bounded la ice
isomo phisms be ween FGS[0,1](X) unde he o de ≤9and
HFS[0,1](X) unde he o de ≤0.
P oo : I is enough o p o e ha 8and 9p ese e he
o de s on FGS[0,1](X) and HFS[0,1](X). Le hen Fand Gbe
wo hesi an uzzy se s o e Xand conside =8(F) and
g=8(G). Recall ha (α)= {x∈Xsuch ha α∈F(x)}
and g(α)= {x∈Xsuch ha α∈G(x)}. Assume ha F≤0
G, ha is, F(x)≤0G(x) o any x∈X. Then
i)F(x)<G(x) F(x), and
ii)F(x) G(x)<G(x),
o any x∈X. Le now α≤βand x∈ (β)∩g(α), hen
β∈F(x) and α∈G(x). Suppose ha β /∈G(x), hence,
by ii), β < α, a con adic ion. The e o e, β∈G(x), and
hen x∈g(β). Simila ly, i α /∈F(x), hence, by i), β < α,
again a con adic ion. Then, α∈F(x), so x∈ (α). As a
consequence, x∈g(β)∩ (α). Thus ≤9g.
Con e sely, le and gbe o g adual se s o e X.
Deno e F=9( ) and G=9(g), so ha F(x)=
{α∈[0,1] such ha x∈ (α)}and F(x)= {α∈
[0,1] such ha x∈g(α)}. Suppose ha ≤9g, ha is,
i α≤β, (β)∩g(α)⊆g(β)∩ (α). Le α∈F(x)
and β∈G(x) F(x). Then α∈F(x), β∈G(x) and
β /∈F(x). Hence x∈ (α), x∈g(β) and x/∈ (β),
so (α)∩g(β)⊈g(α)∩ (β). By hypo hesis, βcanno be
less o equal han α. So F(x)<G(x) F(x). Simila ly, le
α∈F(x) G(x) and β∈G(x). Hence, x∈ (α), x∈g(β) and
x/∈g(α), so (α)∩g(β)⊈g(α)∩ (β), and, by hypo hesis,
β≰α, so F(x) G(x)<G(x). Thus, 9( )≤09(g).
Now, we ecall om Sec ion III ha jX(GS[0,1](X)) is
inside FGS[0,1](X), iewed in Map([0,1],P(X)). The e o e,
we may de ine a la ice o de on GS[0,1](X) as ollows.
De ini ion 5: Le ,g∈GS[0,1](X), we say ≤Ggi and
only i jX( )≤9jX(g).
Despi e g adual se s can be endowed wi h he o de
desc ibed in De ini ion 5, his is no a bounded la ice, since
111164 VOLUME 12, 2024
P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems
i has no minimum. This d awback sugges s he con enience
o conside ing g adual se s as unc ions :L→P(X).
V. REPRESENTATION THEOREMS
In his sec ion we deal wi h he no ion o le el cu and
i s ela ion wi h he ep esen a ion heo em o uzzy se s.
Conc e ely, we examine hei ole wi hin he amewo k
es ablished in he p eceding sec ions in o de o gene alize
hem o some classes o ex ended uzzy se s. In pa icula ,
we es ablish a well- ounded de ini ion o le el cu s o
in e al- alued uzzy se s. We also show ha he classical
no ion o le el cu depends on a gi en embedding, so a ying
his inclusion yields o he op ions o le el cu s. Fo
simplici y, and cohe ency wi h some o he commen s made
abo e, we shall ea le el cu s as maps in Map(L,P(X))
a he han g adual se s. We shall in e changeably iew a
amily o subse s o Xindexed by Las a mapping in
Map(L,P(X)), and ice e sa.
We ecall om Figu e 3 ha he class o L- uzzy se s can be
embebed in o he class o hesi an L- uzzy se s by means o a
mapping ϕde ined as ϕ( )(x)=[0, (x)] o any ∈FSL(X)
and x∈X. Then, o a gi en uzzy se ∈FSL(X), we may
associa e o i he collec ion o subse s o X, indexed in L,
desc ibed as
ϕ( )(α)= {x∈Xsuch ha α∈ϕ( )(x)}
= {x∈Xsuch ha α∈[0, (x)]}
= {x∈Xsuch ha (x)≥α}
= ≥α,
o any α∈L, ge ing he usual desc ip ion o uzzy se s
by means o he s anda d le el cu s. Ne e heless, depending
on he embedding, we may ob ain ano he possibili ies, so,
ac ually, we should be mo e p ecise when e e ing o le el
cu s.
De ini ion 6: Le λbe a one- o-one mapping om FSL(X)
o Map(X,P(L)) and ∈FSL(X), he class o λ-le el cu s
associa ed o is he amily { λ
α}α∈Lo subse s o X, such
ha , o any α∈L,
λ
α={x∈Xsuch ha α∈λ( )(x)}.
Example 2: a) Dually he s anda d case, we may also
conside ϕop :FSL(X)→HFSL(X), gi en by
ϕop( )(x)=[ (x),1] o any ∈FSL(X) and x∈X.
In his case he associa ed subse s
ϕop
α= {x∈Xsuch ha (x)≤α} = ≤α,
o any α∈L, consis o he dual o he s anda d le el cu s.
No e hese a e simply he same objec s ha he s anda d
le el cu s when conside ing he opposi e la ice Lop.
b) Conside he mapping de ined as φ( )(x)= { (x)} o any
∈FSL(X) and x∈X. The φ-le el cu s o consis s o
he amily {x∈Xsuch ha (x)=α}α∈L.
c) Conside he igh open e sion o he abo e-desc ibed
mapping ϕ. Le ˜ϕ:FSL(X)→Map(X,P(L)) gi en by
˜ϕ( )(x)=[0, (x)) o any ∈FSL(X) and x∈X. Hence,
he associa ed ˜ϕ-le el cu s a e de ined as
˜ϕ
α= {x∈Xsuch ha α∈ ˜ϕ( )(x)}
= {x∈Xsuch ha α∈[0, (x))}
= {x∈Xsuch ha (x)> α}
= >α,
o any α∈L, i. e. he usual s ic le el cu s.
In his con ex , a ep esen a ion heo em o uzzy se s as λ-
le el cu s is a desc ip ion o hose mappings in Map(L,P(X))
coming om a uzzy se , ia λ. Tha is o say, a desc ip ion
o he image o he composi ion
Fo ins ance, he classical esul gi en by Negoi a and Ralescu
[7, Theo em 1]is a ep esen a ion heo em o ϕ-le el cu s.
We ecall ha , i Lis a join-comple e la ice, ϕhas a le
in e se mapping δgi en by δ(g)(x)=sup g(x) o any
g∈Map(X,P(L)) and x∈X. Then, o a gi en mapping
in Map(L,P(X)), we may ob ain an associa ed L- uzzy se
using he composi ion
Map(L,P(X)) 9
−→ Map(X,P(L)) δ
−→ FSL(X),
which e i ies ha δ98ϕ is he iden i y mapping. I is
necessa y o conside he p ope y on L,
(∗) o any amily {αi}i∈I⊆L, i α < ∨i∈Iαi hen he e
exis s β∈Isuch ha α≤αβ.
Theo em 5: [7] Le Lbe a join-comple e la ice e i ying
(∗). Le F∈Map(L,P(X)). The e exis s some ∈FSL(X)
such ha F=8ϕ( ) i and only i
a)F(0) =X, and
b)F(∨i∈Iαi)= ∩i∈IF(αi) o any amily {αi}i∈I⊆L.
Fu he mo e, i F e i ies hese p ope ies, he associa ed
uzzy se is de ined as (x)=sup{α∈Lsuch ha
x∈F(α)}.
In pa icula , when L=[0,1], we may ew i e Theo em 5,
o [18, Theo em 4.3], as ollows.
Theo em 6 (Rep esen a ion heo em o ϕ-le el cu s): A
class o subse s A= {Aα}α∈[0,1] o a uni e se se Xis he
class o ϕ-le el cu s o a uzzy se i and only i A e i ies
he ollowing condi ions:
i)Ais ull, i. e., [
α∈[0,1]
Aα=X,
ii)Ais nes ed, i. e., i α≤β hen Aβ⊆Aα,
iii)Ais uppe closed, i. e.,
α<β
Aα=Aβ o any β∈[0,1].
Fu he mo e, i A e i ies hese condi ions, he associa ed
uzzy se :X→[0,1] is de ined as, o any x∈X,
(x)=sup{α∈[0,1] such ha x∈Aα}.
P oo : To eco e he o iginal s a emen , simply obse e
ha he p ope y b) in Theo em 5implies ha he amily is
nes ed, and ha Abeing nes ed and ull is equi alen o be
nes ed and A0=X.
Simila ly, by using he embedding φdesc ibed in Exam-
ple 2b), we may p o e he ollowing esul .
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Theo em 7 (Rep esen a ion Theo em o φ-Le el Cu s):
Le Lbe a la ice, a class o subse s A= {Aα}α∈Lo a
uni e se se Xis he class o φ-le el cu s o a L- uzzy se
i and only i he non-emp y elemen s in A o m a pa i ion
o X. Fu he mo e, i A e i ies his condi ion, he associa ed
L- uzzy se :X→Lis de ined as, o any x∈X, (x)=α,
whe e αis he (unique) elemen in Lsuch ha x∈Aα.
P oo : By Example 2b), he φ-le el cu s associa ed o
a e gi en by he amily {x∈Xsuch ha (x)=α}α∈L o
any α∈L. Tha is, he in e se images by o he elemen s
in L. Since is a unc ion, hese clea ly o m a pa i ion o X.
Con e sely, i Ap oduces a pa i ion o X, he L- uzzy se
desc ibed in he s a emen e i ies ha 8φ( )=A, iewed
as a map in Map(L,P(X)).
In he li e a u e, he e ha e been s udies ha examine he
concep o le el cu s and aim o es ablish ep esen a ion
heo ems o ce ain ypes o ex ended uzzy se s, such as
in e al- alued uzzy se s (IVFSs) o in ui ionis ic uzzy se s
(see [19]). These s udies p ima ily ocus on subse s ha
sa is y speci ic inequali ies. Howe e , in his wo k, we adop
he me hodology discussed ea lie o ackle his p oblem.
Gi en a hesi an L- uzzy se ∈HFSL(X), we de ine he
amily o le el cu s associa ed o as
α={x∈Xsuch ha α∈ (x)}=8( )(α),
o any α∈L. The e o e, Lemma 1p o ides li e ally he
co esponding ep esen a ion heo em.
Theo em 8 (Rep esen a ion Theo em o Hesi an Fuzzy
Se s): Le Lbe a bounded la ice. A amily o subse s A=
{Aα}α∈Lo a uni e se se Xis he class o le el cu s o a
hesi an L- uzzy se i and only i Ais ull.
In his con ex , any subclass Fembedded in HFSL(X) is
hen likely o de ine he le el cu s o i s objec s as he image
h oughou he composi ion
as i has been done p e iously o L- uzzy se s.
Assume he eon ha L=[0,1] and le us deno e by I he
se o all closed in e als in [0,1]. An In e al-Valued Fuzzy
Se (IVFS) on Xis hen a map A:X−→ I. Deno e by
IVFS[0,1](X) he class o med by all in e al- alued uzzy
se s o X. The s anda d inclusion (say µ) o IVFS[0,1](X)
in o HFS[0,1](X) is jus inhe ed o conside ing an in e al as a
non-emp y subse in [0,1]. In his sense, he amily o µ-le el
cu s associa ed o an IVFS is gi en by
µ
α= {x∈Xsuch ha ax≤α≤bx},
whe e (x)=[ax,bx] o any x∈X, o any α∈L.
Lemma 2. Le ∈HFS[0,1](X) and F=8( )∈
Map([0,1],P(X)). The ollowing asse ions a e equi alen :
a) o any x∈X, (x) is con ex,
b) o any γ < β < δ ∈[0,1], F(γ)∩F(δ)⊂F(β).
P oo : Le us suppose ha (x) is con ex o any x∈
X, and le γ < β < δ ∈[0,1]. Gi en x∈F(γ)∩F(δ),
hence γand δbelong o (x). Since i is con ex, β∈ (x),
so x∈F(β).
Con e sely, le x∈X. I (x)= {α}, hen we a e done.
I no , ake γ, δ ∈ (x) such ha γ < δ and le β∈[0,1]
wi h γ < β < δ. Since x∈F(γ)∩F(δ), by hypo hesis,
x∈F(β), and hen β∈ (x), so (x) is con ex.
Lemma 3. Le ∈HFS[0,1](X) and F=8( )∈
Map([0,1],P(X)). The ollowing asse ions a e equi alen :
a) o any x∈X, (x) is a closed subse in [0,1].
b) o any sequence {αn}n∈Nin [0,1] wi h {αn}n∈N→α,
i holds ha Tn∈NF(αn)⊆F(α).
P oo : Suppose ha (x) is closed o any x∈X. Le
{αn}n∈N→αa con e ging sequence and x∈Tn∈NF(αn).
Tha means {αn}n∈N⊂ (x), so α∈ (x). Hence, x∈F(α).
Con e sely, ake {αn}n∈N→αwi h {αn}n∈N⊆ (x). Then
x∈Tn∈NF(αn)⊆F(α), ha is, α∈ (x), so (x) is
closed.
Co olla y 1 (Rep esen a ion heo em o In e al-Valued
Fuzzy Se s ia µ): A class o subse s A= {Aα}α∈[0,1] o a
uni e se se Xis he class o µ-le el cu s o an in e al- alued
uzzy se i and only i A e i ies he ollowing condi ions:
a)Ais ull.
b)Aγ∩Aδ⊂Aβ o any γ < β < δ ∈[0,1].
c) Fo any sequence {αn}n∈Nin [0,1] wi h {αn}n∈N→α,
i holds ha Tn∈NAαn⊆Aα.
Fu he mo e, i A e i ies hese condi ions, he associa ed
in e al- alued uzzy se :X→Iis de ined
as (x)=[ax,bx] o any x∈X, whe e ax=
in {α∈[0,1] such ha x∈Aα}and bx=sup{α∈
[0,1] such ha x∈Aα}.
P oo : I ollows om Lemmas 2and 3.
Beyond he subclasses o hesi an uzzy se s, we would
need o ex end he wo king space. A possible solu ion is o
enla ge he la ices in he adjunc ion desc ibed in Sec ion II.
We may exchange he Boolean algeb a
2
by any o he la ge
la ice as, o ins ance, he own la ice L. Hence, we ind ha
Map(L,Map(X,L)) ∼
=Map(X,Map(L,L)), ha is o say,
deno ing by T2FSL(X) he class o ype-2 L- uzzy se s,
Map(L,FSL(X)) ∼
=T2FSL(X).
So, he e o e, each ype-2 L- uzzy se is ep esen ed by a
amily, indexed in L, o L- uzzy se s, which can be conside ed
i s le el cu s. In gene al, ollowing he same p inciple, one
may ob ain ha a sui able no ion o le el cu s o ype-n
L- uzzy se s is o conside a amily o ype-(n−1) L- uzzy
se s.
VI. CONCLUSION
In his pape we ha e ca ied ou an analysis o he no ions o
g adual se and hesi an uzzy se . Bo h aiming o ex end he
classical heo y o uzzy se s om wo di e en iewpoin s.
We ha e shown ha hese wo pe spec i es a e linked by a
one- o-one mapping, coming om he ca ego ical adjunc ion
be ween he hom and he ca esian p oduc unc o s. Hence
we ha e seen ha he classes o g adual se s and hesi an
uzzy se s a e s ongly ela ed. Howe e , since he e a e some
dispa i ies in he li e a u e ega ding hei p ecise de ini ions,
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