On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems

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 111158 2024 The Au ho s. This wo k is licensed unde a C ea i e Commons A ibu ion-NonComme cial-NoDe i a i es 4.0 License. Fo mo e in o ma ion, see h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/ 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 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)) VOLUME 12, 2024 111163 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 . VOLUME 12, 2024 111165 P. Ja a e al.: On G adual Se s, Hesi an Fuzzy Se s, and Rep esen a ion Theo ems 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, 111166 VOLUME 12, 2024