On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems

Received 25 July 2024, accepted 6 August 2024, date of publication 12 August 2024, date of current version 20 August 2024. Digital Object Identifier 10.1109/ACCESS.2024.3441940 On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems PASCUAL JARA1,2, LUIS MERINO 1,2, GABRIEL NAVARRO 1,2,3, AND EVANGELINA SANTOS 1,2 1Department of Algebra, University of Granada, 18071 Granada, Spain 2Mathematics Institute of the University of Granada (IMAG), 18001 Granada, Spain 3Research Centre for Information and Communication Technologies (CITIC-UGR), 18014 Granada, Spain Corresponding author: Gabriel Navarro ([email protected]) This work was supported in part by the Programa Operativo FEDER 2014-2020 and Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía, Spain, under Grant A-FQM-394-UGR20; and in part by the ‘‘María de Maeztu’’ Excellence Unit IMAG, funded by MCIN/AEI/10.13039/501100011033, under Grant CEX2020-001105-M. ABSTRACT We analyze the connection between two perspectives when defining fuzzy sets: the viewpoint of mappings and the viewpoint of families of level cuts. This analysis is mathematically supported by the framework of a categorical adjunction, which serves as a dictionary between these two perspectives. We prove that hesitant fuzzy sets and gradual sets are strongly related through this connection. This allows concepts and operations to be transferred from one class to the other, and vice versa. Concretely, as an application, we provide lattice operations on gradual sets, compatible with Zadeh’s max-min operations on fuzzy sets, when considering them as families of level cuts. We discuss the well-known representation theorem for fuzzy sets within this framework, and we show that the representation of fuzzy sets as gradual sets depends on the chosen embedding of fuzzy sets as hesitant fuzzy sets. Hence, distinct embeddings yield diverse representations of fuzzy sets as collections of subsets. Furthermore, we extend this methodology to include other classes of extended fuzzy sets. As a consequence, a representation theorem for interval-valued fuzzy sets is provided. INDEX TERMS Fuzzy sets, gradual sets, hesitant fuzzy sets, interval-valued fuzzy sets, level cuts, representation theorems, set-valued fuzzy sets. I. INTRODUCTION The relevance of fuzzy sets [1] is nowadays worldwide recognized. This lies in their ability to tackle real-world problems characterized by ambiguity, vagueness, and imprecision. By providing a mathematical framework that goes beyond crisp set theory, this approach offers a more natural and realistic representation of uncertainty, contributing to the advancement of certain areas that need its management. To expand their expressiveness, some authors have generalized the original concept, yielding different types of extended fuzzy sets. For instance, interval-valued fuzzy sets [2],[3] model imprecision by specifying a range or interval. Hence, this representation allows for a more The associate editor coordinating the review of this manuscript and approving it for publication was Wai-Keung Fung . flexible description of the uncertainty, comprising both the lower and upper bounds of the possible membership values. Analogously, hesitant fuzzy sets [4] (originally called set-valued fuzzy sets in [5]) can manage multiple possibilities or sources of information, since, rather than assigning a single value, these represent the membership degree as a collection of them. For this reason, they have proved useful in modeling certain systems, such as those related to multicriteria decision-making, where different experts or criteria can contribute assigning different membership degrees to an object. Another interesting types appearing in the literature are, for instance, intuitionistic fuzzy sets [6], or type-2 fuzzy sets [3], among many others. Although these extensions understand the representation of uncertainty differently, mainly each one adapted to certain specific scenarios, they share a common functional 111158 2024 The Authors. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ VOLUME 12, 2024 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems perspective. A fuzzy set is mostly viewed as a mapping assigning, to any element on the universe set, a membership degree in the reference lattice, standardly the unit interval [0,1]. Hence, from a suitable replacement of the codomain, mainly by a partial ordered set to allow comparisons between degrees, it results the above-commented generalizations as mappings valued in these new spaces. Nevertheless, a fuzzy set may also be described by means of the collection of its level cuts, which leads to the well-known representation theorem stated by Negoita and Ralescu in [7, Theorem 1]. This offers a different point-of-view that we may call the level perspective. Following this idea, gradual sets [8] were introduced as an extension of fuzzy sets by Dubois and Prade. Instead of degrees of membership, gradual sets focus on representing gradual transitions or gradual boundaries between membership and non-membership. They capture the idea that the transition of an element from non-membership to membership can occur gradually or incrementally, rather than as an abrupt change. This generalizes the well-known fact that every fuzzy set can be described by its level cuts. In this paper, we analyze these two perspectives and describe the mathematical connection between them. This novel approach not only generalizes existing theories but also provides a comprehensive framework for understanding and extending fuzzy set concepts from the viewpoint of category theory, which has not been done before. In this sense, we follow the standard deductive methodology. We progress from general results to more specific ones. At a higher level of abstraction, this connection is modeled by a well-known categorical concept, namely, an adjunction between two functors. The adjunction reveals a one-to-one mapping, effectively serving as a dictionary between both points of view. Consequently, we may establish relationships between different extensions of fuzzy sets from both perspectives. In particular, we show that the concepts of hesitant fuzzy sets and gradual sets are strongly interconnected. In fact, according to the definitions proposed by some authors, these concepts refer to the same objects when considering the functional and level perspectives. Hence, in a lower level of abstraction, suitable properties or algebraic structures may be endowed to these objects via the above-commented bijection. For instance, lattice structures on gradual sets can be derived from known structures on hesitant fuzzy sets, enriching the level perspective. On the other hand, this mathematical framework naturally supports the renowned representation theorem of fuzzy sets as stated in [7]. Therefore, it becomes pertinent to ask about the precise specific formulation of the theorem within the scheme developed earlier. Concerning this question, we show that the representation of fuzzy sets as gradual sets depends on the chosen embedding of fuzzy sets as hesitant fuzzy sets. Consequently, we should not refer to a single fuzzy set representation theorem, but to a collection of them depending on the embedding applied. Furthermore, the theory developed here allows to state representation theorems for other classes contained on hesitant fuzzy sets, as intervalvalued fuzzy sets. The main contributions of this paper can be summarized as follows. •This work analyzes in detail the notions of hesitant fuzzy sets and gradual sets, both as extensions of classical fuzzy sets, and describes the mathematical connection that relates them. It shows that some studies in the literature refer to the same class of objects, viewed from different perspectives. This opens up the possibility of relating these studies together and improving the existing theories. •The paper introduces a lattice structure on gradual sets whose restriction to the class of fuzzy sets, viewed as chains of subsets, becomes Zadeh’s standard lattice structure. This enriches the level perspective of gradual sets, providing operations between its objects and allowing a systematic manipulation of gradual sets consistent with the operations on fuzzy sets. In particular, it provides a common framework, coherent from an operational point of view, for working with fuzzy objects whose fuzziness is measured differently. •The paper supports the well-known representation theorem of fuzzy sets, and clarifies that the representation of fuzzy sets as gradual sets depends on the chosen embedding of fuzzy sets as hesitant fuzzy sets, leading to a collection of representation theorems based on different embeddings. •The developed theory allows for the formulation of representation theorems for various classes of extended fuzzy sets. Specifically, the paper provides a representation theorem for interval-valued fuzzy sets. The paper is structured as follows. In section II we establish the needed results to develop the content of the paper. Concretely, we introduce some notation and review some basic facts, and present the main theoretical key-tool: a categorical adjunction. This adjunction defines an isomorphism which serves as a dictionary-like correspondence between the two above-commented perspectives. In Section III, by using this bijection, we derive a framework for exploring the relationships between several classes of extended fuzzy sets. We then prove a strong connection between the classes of hesitant fuzzy sets and gradual sets. As different authors have proposed distinct definitions for these concepts, we show the precise mathematical relation for several possibilities. In Section IV we apply the previous theoretical results in order to develop a lattice order on gradual sets. By means of the results given in [9], we infer a lattice order on the class of (what we have called) full gradual sets, and, as a consequence, on gradual sets as defined in [8]. It is worth noting that its restriction to fuzzy sets corresponds to the classical order proposed by Zadeh, so it allows to operate with different treatments of uncertainty simultaneously. In Section Vwe explain the role of the well-known representation theorem for fuzzy sets within the framework established in Section III. Based on this abstraction, we demonstrate the potential to VOLUME 12, 2024 111159 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems derive additional representation theorems for other classes of extended fuzzy sets. In particular, we prove a representation theorem for interval-valued fuzzy sets. Finally, Section VI presents our concluding remarks. II. A CATEGORICAL ADJUNCTION Let Ybe an arbitrary set, it is well known that the powerset P(Y) is endowed with a complete lattice structure provided by the classic set operations (union, intersection) with ∅ and Ybeing the minimum and maximum, respectively. Additionally, P(Y) is isomorphic, as a lattice, to 2 Y= Map(Y, 2 ), the class of set mappings from Yto 2 , where 2 is the Boolean algebra with two elements. Here, the lattice structure on Map(Y, 2 ) is provided by the partial order ≤M given by, for each maps f,g∈Map(Y, 2 ), f≤Mgif and only if f(y)≤g(y) for any y∈Y, (1) where ≤is the order on 2 . It used to say that ≤Mis inherited from ≤. In general, if Lis a (bounded, complete) lattice, then, for any set Y, Map(Y,L) is endowed with a (bounded, complete) lattice structure, inherited from L, defined similarly as done in (1). Let now Xand Lbe two sets, so we may consider the powersets P(X) and P(L), and the classes of set mappings Map(X,P(L)) and Map(L,P(X)). As commented above, both are lattices whose lattice structures are inherited from P(L) and P(X), respectively. Remark 1: Note that, if L=[0,1], the elements in Map(X,P(L)) represent a slight generalization of the concept of set-valued fuzzy set as defined in [10, Definition 5.1]. In that definition, the codomain is restricted to P(L)\ ∅. Nevertheless, in such a case, the set-theoretic intersection is not well-defined. In [5], a set-valued fuzzy set is defined exactly as a map in Map(X,P(L)), allowing the empty set as the nonsense membership degree, and avoiding this problem. In [4] these objects are called hesitant fuzzy sets and different operations are provided. In this paper we shall consider both terms as synonyms, and we reserve the terminology ‘‘set-valued fuzzy set’’ or ‘‘hesitant fuzzy set’’ for an element in Map(X,P(L)\ ∅). This choice is made because, as a generalization of classical fuzzy sets, settheoretic operations are not well-behaved with the max-min operations on fuzzy sets, whilst Torra’s operations do not provide a lattice structure. On the other hand, the elements in Map([0,1],P(X)) provide an alternative notion of the concept of gradual set given in [11, Definition 3.2], since, if I⊆[0,1] and G: I→P(X), the mapping G:[0,1] →P(X) defined as G(α)=(G(α),if α∈I, ∅,otherwise, verifies that the restriction G|Iequals G. This allows us an easier handling of this kind of structures, since there is no need of using different domains, and all of them are defined in [0,1]. Additionally, it extends the concept of representation level in [12, Definitions 2.1 and 2.3]. Indeed, given (3, ρ) with 3= {1=α1>· · · > αm+1=0}, simply consider the mapping G(α)=ρ(α) if αi≥α > αi+1 for each α∈(0,1] and G(0) =ρ(αm+1). These two constructions are strongly related. There exists a one-to-one mapping between Map(X,P(L)) and Map(L,P(X)). Concretely, consider the mapping 8:Map(X,P(L)) −→ Map(L,P(X)) (2) defined as 8(f)(α)= {x∈Xsuch that α∈f(x)}, for any f∈Map(X,P(L)) and any α∈L. The inverse of 8is given by the mapping 9:Map(L,P(X)) −→ Map(X,P(L)),(3) defined as 9(g)(x)= {α∈Lsuch that x∈g(α)}, for any g∈Map(L,P(X)) and any x∈X. It is not difficult to see that both mappings are inverse one to each other. Indeed, for any f∈Map(X,P(L)) and x∈X, by definition, α∈98(f)(x) if and only if x∈8(f)(α) if and only if α∈f(x), so that 98(f)(x)=f(x) for any x∈X. Then 98(f)=ffor any f∈Map(X,P(L)). Hence 98 =IdMap(X,P(L)). One may prove almost verbatim that 89 =IdMap(L,P(X)). Remark 2: The above property follows from the adjunction between the functors Map and cartesian product ×. For completeness, we recall briefly this fact, see [13] for basic definitions. Let Set be the category formed by all sets. For some set X, the functor Map(X,−):Set →Set maps a set Y to Map(X,Y), and each morphism f:Y→Y′to Map(X,f): Map(X,Y)→Map(X,Y′) given by Map(X,f)(g)=gf , the composition of mappings, for any g∈Map(X,Y). On the other hand, the functor X× − : Set →Set maps a set Yto the cartesian product X×Y, and each morphism f:Y→Y′ to X×f:X×Y→X×Y′given by (X×f)(a,b)=(a,f(b)) for any a∈Xand b∈Y. The functor X× − is left adjoint to Map(X,−), meaning that, for each sets Land Z, there exists a natural bijection φ:Map(X×L,Z)→Map(L,Map(X,Z)). That is to say, Z(X×L)∼ =(ZX)L, where we have denoted Map(A,B) by its well-known power set notation BA. Now, observe that X×L∼ = L×Xby the standard flip map, so combining the bijections (ZX)L∼ =Z(X×L)∼ =Z(L×X)∼ =(ZL)X. Whenever Z= 2 , Map(L,P(X)) =( 2 X)L∼ =( 2 L)X= Map(X,P(L)), yielding the desired bijective map. From the definition given by Goguen in [14], we may see the elements in Map(X,P(L)) as P(L)-fuzzy sets. Then, by virtue of Remark 2, each P(L)-fuzzy set can be seen in Map(X,P(L)) or in Map(L,P(X)), yielding a vertical and a horizontal representations of the corresponding P(L)-fuzzy set. Let us illustrate this with an example. 111160 VOLUME 12, 2024 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems Example 1: Consider X= {a,b,c,d}and L=[0,1], and the mapping f∈Map(X,P([0,1]) given by f(a)=[0.2,0.4], f(b)=[0.3,0.5], f(c)=[0.5,0.7], f(d)=[0,0.6]. Then fis an interval-valued fuzzy set over Xand this pointof-view corresponds to the vertical representation described in Fig. 1. On the other hand, its image under 8, the horizontal representation, 8(f):[0,1] →P(X), is given by 8(f)(x)=                        {d}if 0 ≤x<0.2, {a,d}if 0.2≤x<0.3, {a,b,d}if 0.3≤x≤0.4, {b,d}if 0.4<x<0.5, {b,c,d}if x=0.5, {c,d}if 0.5<x≤0.6, {c}if 0.6<x≤0.7, ∅if x>0.7, FIGURE 1. Vertical representation of f. FIGURE 2. Horizontal representation of f. see Fig. 2. The above-described bijections are also lattice isomorphisms when considering the standard orders on P(L) and P(X). Theorem 1: The maps 8and 9are isomorphisms of bounded lattice with respect to the order inherited by the set-theoretic operations in the powersets. Proof: We only need to prove that 8respects the partial order. Let f,g∈Map(X,P(L)) such that f≤Mg, which means f(x)⊆g(x) for all x∈X. Then 8(f)(α)= {x∈ Xsuch that α∈f(x)}⊆{x∈Xsuch that α∈g(x)} = 8(g)(α) for any α∈L, so that 8(f)≤M8(g). On the other hand, the minimum and maximum in Map(X,P(L)) are the constant maps 0and 1given by 0(x)= ∅and 1(x)=Lfor any x∈X, respectively. It is also clear that 8(0)(α)= ∅ and 8(1)(α)=Xfor any α∈L. Analogously, we may prove 9holds the same properties. Remark 3: Although the proof of Theorem 1concerns the partial order, we could have proved it by using the lattice operations, say ∧Mand ∨Mfor both lattice, which are given by (f∧Mg)(x)=f(x)∩g(x) and (f∨Mg)(x)=f(x)∪g(x) for any x∈X(or any x∈L). Therefore, since 8and 9are isomorphisms, 8(f∧Mg)=8(f)∧M8(g) and 8(f∨Mg)= 8(f)∨M8(g), for any f,g∈Map(X,P(L)), and 9(f∧M g)=9(f)∧M9(g) and 9(f∨Mg)=9(f)∨M9(g), for any f,g∈Map(L,P(X)). III. GRADUAL SETS AND HESITANT FUZZY SETS Our aim now is to show how the bijection given by the adjunction in Remark 2relates the notions of hesitant fuzzy set and gradual set. We recall from [8] that, given a bounded lattice L, an L-gradual subset of Xis a mapping from L+to P(X) (see also [15], or [16] with an additional structure of group), where L+=L\ {0}, and 0 denotes the minimum of the bounded lattice L. Let us denote by GSL(X) the class of all L-gradual sets (gradual sets, for short, when the context is clear enough) of X. We also recall from [10] that a hesitant L-fuzzy set over a universe Xis a mapping f:X→P∗(L), where by P∗(X) we mean the class of non-empty subsets of X. Let us denote by HFSL(X) the class of all hesitant L-fuzzy sets. Let us now set the class FGSL(X)=(f∈Map(L,P(X)) with [ α∈L f(α)=X), and we refer to it as the class of full L-gradual sets (full gradual sets, for short) of X. Lemma 1: The mappings 9and 8determine a bijection between the classes HFSL(X) and FGSL(X). Proof: By Remark 2we only have to show that the restrictions of 9and 8are well-defined. Indeed, let f∈ HFSL(X) and x∈X. Then f(x) is a non-empty set in L. In particular, there exists some α∈f(x), so x∈8(f)(α). Hence X⊆ ∪α∈L8(f)(α) and thus 8(f)∈FGSL(X). Conversely, if g∈FGSL(X), then, for any x∈X, there exists an α∈Lsuch that x∈g(α). Hence, α∈9(g)(x), and then 9(g)(x) is a non-empty set. Thus 9(g)∈HFSL(X). VOLUME 12, 2024 111161 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems Let us insert the well-known class of L-fuzzy sets [14] in this scheme. We may always embed a bounded lattice L into its powerset P(L) by the mapping each element αto the interval [0, α]= {β∈Lsuch that β≤α}. Observe that this is clearly a order-preserving mapping with respect to the order on Land the set-theoretical order on P(L). Also, [0, α1]∩[0, α2]=[0, α1∧α2], so that it also preserves meets. Indeed, for 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]. Nevertheless, that is not the case for the joint operator. We may only say, β∈[0, α1]∪[0, α2]⇔β∈[0, α1] or β∈[0, α2] ⇔β≤α1or β≤α2 ⇒β≤α1∨α2 ⇔β∈[0, α1∨α2]. In general, the implication (β≤α1∨α2)⇒(β≤α1or β≤α2) does not hold. For instance, consider the following toy example, Note that we may get this property if Lis a totally ordered lattice. Let us denote by FSL(X) the class of all L-fuzzy sets Map(X,L). Proposition 1: The mapping ϕ:FSL(X)→HFSL(X), defined as ϕ(f)(x)=[0,f(x)] for any f∈FSL(X) and any x∈X, is an order and meet preserving injective mapping. Moreover, if Lis totally ordered, ϕis a lattice embedding. Proof: It follows from the above discussion. We have then the following commutative diagram, see Figure 3. The third row in Figure 3corresponds to the isomorphisms described in Section II for the lattice L+. For simplicity, we have also denoted them by 8and 9. The mapping p simply maps each f∈Map(L,P(X)) to its restriction to L+, so it is surjective. Following this diagram, we find that the classes HFSL(X) and GSL(X) can be embedded into Map(L,P(X)), so we may try to compare them. Indeed, via 8, the hesitant fuzzy sets can be seen as the mappings in FGSL(X). On the other hand, since pis surjective, there exist an injection j:GSL(X)→ Map(L,P(X)) such that pj is the identity map. FIGURE 3. Relation between hesitant fuzzy sets and gradual sets. The map jhas not to be unique, and there exist as many copies of GSL(X) inside Map(L,P(X)) as possible mappings j. Observe that, given f∈GSL(X), since pj(f)=f, then pj(f)(α)=j(f)(α)=f(α) for any α∈L+. Thus, the map j is simply determined by choosing an image of 0 ∈Lfor any element in GSL(X). For our purpose, we fix the mapping jX:GSL(X)→Map(L,P(X)) given by jX(f)(α)=f(α),if α= 0, X,if α=0, for any f∈GSL(X) and α∈L. Thus jX(GSL(X)) ⊆ i(FGSL(X)), and then, GSL(X)⊂pi(FGSL(X)) = pi8(HFSL(X)). That is, in other words, the map pi8:HFSL(X)→GSL(X) is surjective. Hence GSL(X) is isomorphic to some quotient of HFSL(X). Definition 1: Let f,g∈HFSL(X) we say that f∼0gif f(x)∪ {0} = g(x)∪ {0}for all x∈X, or, equivalently, f(x)\ {0} = g(x)\ {0}for all x∈X. Clearly, ∼0is an equivalence relation on HFSL(X). Theorem 2: HFSL(X)/∼0∼ =GSL(X). Proof: From the above discussion, it is enough to see that the relation ∼0is the equivalence relation associated to the composition pi8. Let f,g∈HFSL(X), hence, f∼0g if and only if f(x)\ {0} = g(x)\ {0}for all x∈X, which is equivalent to α∈f(x) if and only if α∈g(x) for all x∈ Xand all α∈L+, and also to x∈8(f)(α) if and only if x∈ 8(g)(α) for all x∈Xand all α∈L+. Analogously, this last statement is equivalent 8(f)(α)=8(g)(α) for all α∈L+, and, then, to (pi8)(f)(α)=(pi8)(g)(α) for all α∈L+, and, finally, to (pi8)(f)=(pi8)(g). Remark 4: Note that FGSL(X)/∼0∼ =GSL(X), where f∼0gif and only if f(α)=g(α) for all α∈L+. Observe that, the image of GSL(X) via 9jXcorresponds to the class AL(X)=(f∈Map(X,P(L)) such that 0 ∈\ x∈X f(x)) ⊆Map(L,P(X)). 111162 VOLUME 12, 2024 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems Indeed, if f∈AL(X), then 8(f)(0) = {x∈Xsuch that 0 ∈f(x)} = X so 8(f) is in jX(GSL(X)). Thus f∈9jX(GSL(X)). Conversely, let g∈9jX(GSL(X)), then 8(g)∈jX(GSL(X)) and then 8(g)(0) =X. Therefore 0 ∈g(x) for all x∈Xand, consequently, g∈AL(X). Thus we may understand gradual sets as hesitant fuzzy sets in which the minimum 0 belongs to the membership degree of any element. Remark 5: To summarize this section, and for the sake of clarity, it is worth noting that different notions of hesitant fuzzy sets (or set-valued fuzzy sets) and gradual sets have been explored in the literature. Henceforth, although closely related, the explicit mathematical relationship between these extensions may differ slightly depending on how they are considered. For instance, Grattan-Guinness [5] defines a hesitant fuzzy set as an element in Map(X,P([0,1])), whilst Bustince et al. [10] define it as a member of Map(X,P∗([0,1])). Similarly, Dubois and Prade [8] define a gradual set as a mapping in Map((0,1],P(X)), whilst Wu [11] considers Map([0,1],P(X)) as the class of gradual sets. Suppose that, for a given reference set Xand lattice L, we denote by Sand Gthe class of hesitant fuzzy sets and gradual sets, respectively. Hence: •If S=Map(X,P(L)) and G=Map(L,P(X)), Remark 2ensures that both concepts describe exactly the same class of objects, viewed from two different perspectives, that we may call the horizontal and vertical representations. •If S=Map(X,P(L)) and G=Map(L+,P(X)), by the latter comments, Gcorresponds to the subclass of Scomposed by those objects verifying that the membership degree of each element in Xcontains the value 0. •If S=Map(X,P∗(L)) and G=Map(L,P(X)), by Lemma 1,Scorresponds to the subclass of G composed by the mappings that are full. •If S=Map(X,P∗(L)) and G=Map(L+,P(X)), by Theorem 2,Gis a quotient of Sgiven by the equivalent relation that joins those mappings that share the same images except for the value 0. In our opinion, the most suitable option is the third one. As we shall explain in the next section, in this case, we may find bounded lattice structures compatible with the Zadeh’s operations on fuzzy sets. IV. LATTICE STRUCTURES ON GRADUAL SETS Let us now show an application of the connections established in the diagram of Figure 3. Concretely, we shall define new lattice orders on the class of gradual sets and full gradual sets. All along this section Lis the lattice [0,1] endowed with the usual operations. As pointed out in Section II, when working with hesitant fuzzy sets, the inherited operations from the set-theoretic union and intersection on P∗([0,1]) are not well-defined. An obvious solution is to extend the definition of hesitant fuzzy set to be an element in Map(X,P([0,1])), allowing the somehow-called ‘‘nonsense’’ membership degree. Nevertheless, as a counterpart, the set inclusion seems not to be a suitable order to sort the membership degrees. Simply observe that the set {1}should be the greatest membership degree, which is contained in [0.5,1] and not related to {0}. Additionally, as commented in [10, Remark 2], see also [17], these do not extend the classical operations on fuzzy sets (when single-valued assignments are treated as singletons), and on interval-valued fuzzy sets provided by Zadeh. Torra’s paper [4] discusses the definition of new operations compatible with Zadeh’s lattice structure on fuzzy sets, under the name of hesitant fuzzy sets. Nevertheless, Torra’s proposal does not endow hesitant fuzzy sets with a lattice structure. In [9], an answer is given by defining the so-called symmetric order ≤0on the class of non-empty subsets P∗([0,1]). This yields a lattice order on HFS[0,1](X) whose restriction to fuzzy sets and interval-valued fuzzy sets equals the Zadeh’s structures. Given two subsets X,Y⊆[0,1], we say that X<Yif x<yfor any x∈Xand y∈Y. This trivially holds if one of them is the empty set. Definition 2 (Symmetric Order): [9] Given two nonempty subsets Aand Bin [0,1], we say that A≤0Bif and only if A\B<Band A<B\A. The relation ≤0is a bounded lattice order on P∗([0,1]) extending the min-max operations on [0,1], so, as done in (1), the following relation on HFS[0,1](X) provides a bounded lattice order extending the min-max operations on fuzzy sets. Definition 3: Let f,g∈HFS[0,1](X), we say that f≤0g if and only if f(x)≤0g(x) for all x∈X. Therefore the following relation ≤9, derived from the bijection 9in Figure 3, also provides a bounded lattice order on FGS[0,1](X). Indeed, for any maps f,g∈FGS[0,1](X), we say f≤9gif and only if 9(f)≤09(g). Nevertheless, in practice, it is more convenient to have an explicit description. Definition 4: Let f,g∈FGS[0,1](X) be two full gradual sets, we say that f≤9gif and only if, for any α≤βin [0,1], f(β)∩g(α)⊆f(α)∩g(β). Theorem 3: The relation ≤9is an order on FGS[0,1](X). Proof: The reflexive property holds trivially. Let now f,g∈FGS[0,1](X) such that f≤9gand g≤9f. Hence, for any α≤β,f(β)∩g(α)⊆f(α)∩g(β) and f(α)∩g(β)⊆f(β)∩g(α). That is to say, for any α, β ∈[0,1], f(β)∩g(α)=f(α)∩g(β).(4) Then, for any α∈[0,1], g(α)=X∩g(α) = [ β∈[0,1] f(β) ∩g(α) =[ β∈[0,1] (f(β)∩g(α)) VOLUME 12, 2024 111163 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems =[ β∈[0,1] (f(α)∩g(β)) =f(α)∩ [ β∈[0,1] g(β)  =f(α)∩X =f(α). Thus f=g, and the antisymmetric property is satisfied. Finally, for the transitivity, let f,g,h∈FGS[0,1](X) such that f≤9gand g≤9h. Then, for any α≤βin [0,1], f(β)∩g(α)⊆f(α)∩g(β),(5) and g(β)∩h(α)⊆g(α)∩h(β).(6) Observe that f(β)∩h(α)=f(β)∩X∩h(α) =f(β)∩ [ δ∈[0,1] g(δ) ∩h(α) =[ δ∈[0,1] (f(β)∩g(δ)∩h(α))(7) Now, if δ < α, f(β)∩g(δ)∩h(α)(5) ⊆f(δ)∩g(β)∩h(α)⊆g(β) and, trivially, f(β)∩g(δ)∩h(α)⊆f(β)∩h(α), so f(β)∩g(δ)∩h(α)⊆f(β)∩g(β)∩h(α). On the other hand, if δ > β, f(β)∩g(δ)∩h(α)(6) ⊆f(β)∩g(α)∩h(δ)⊆g(α), and f(β)∩g(δ)∩h(α)⊆f(β)∩h(α), so f(β)∩g(δ)∩h(α)⊆f(β)∩g(α)∩h(α). Thus, (7) =[ δ∈[α,β] (f(β)∩g(δ)∩h(α)) (5) ⊆[ δ∈[α,β] (f(δ)∩g(β)∩h(α)) (6) ⊆[ δ∈[α,β] (f(δ)∩g(α)∩h(β)) (5) ⊆[ δ∈[α,β] (f(α)∩g(δ)∩h(β)) =f(α)∩ [ δ∈[α,β] g(δ) ∩h(β) ⊆f(α)∩ [ δ∈[0,1] g(δ) ∩h(β) =f(α)∩X∩h(β) =f(α)∩h(β) Then f≤9h. This concludes the proof. The order ≤9is a bounded order. Observe that the minimum here is the full gradual set 0defined by 0(α)=∅if α= 0, Xif α=0, whilst the maximum is the full gradual set 1defined by 1(α)=(∅if α= 1, Xif α=1. Theorem 4: The maps 8and 9provide bounded lattice isomorphisms between FGS[0,1](X) under the order ≤9and HFS[0,1](X) under the order ≤0. Proof: It is enough to prove that 8and 9preserve the orders on FGS[0,1](X) and HFS[0,1](X). Let then Fand Gbe two hesitant fuzzy sets over Xand consider f=8(F) and g=8(G). Recall that f(α)= {x∈Xsuch that α∈F(x)} and g(α)= {x∈Xsuch that α∈G(x)}. Assume that F≤0 G, that is, F(x)≤0G(x) for any x∈X. Then i)F(x)<G(x)\F(x), and ii)F(x)\G(x)<G(x), for any x∈X. Let now α≤βand x∈f(β)∩g(α), then β∈F(x) and α∈G(x). Suppose that β /∈G(x), hence, by ii), β < α, a contradiction. Therefore, β∈G(x), and then x∈g(β). Similarly, if α /∈F(x), hence, by i), β < α, again a contradiction. Then, α∈F(x), so x∈f(α). As a consequence, x∈g(β)∩f(α). Thus f≤9g. Conversely, let fand gbe to gradual sets over X. Denote F=9(f) and G=9(g), so that F(x)= {α∈[0,1] such that x∈f(α)}and F(x)= {α∈ [0,1] such that x∈g(α)}. Suppose that f≤9g, that is, if α≤β,f(β)∩g(α)⊆g(β)∩f(α). Let α∈F(x) and β∈G(x)\F(x). Then α∈F(x), β∈G(x) and β /∈F(x). Hence x∈f(α), x∈g(β) and x/∈f(β), so f(α)∩g(β)⊈g(α)∩f(β). By hypothesis, βcannot be less or equal than α. So F(x)<G(x)\F(x). Similarly, let α∈F(x)\G(x) and β∈G(x). Hence, x∈f(α), x∈g(β) and x/∈g(α), so f(α)∩g(β)⊈g(α)∩f(β), and, by hypothesis, β≰α, so F(x)\G(x)<G(x). Thus, 9(f)≤09(g). Now, we recall from Section III that jX(GS[0,1](X)) is inside FGS[0,1](X), viewed in Map([0,1],P(X)). Therefore, we may define a lattice order on GS[0,1](X) as follows. Definition 5: Let f,g∈GS[0,1](X), we say f≤Ggif and only if jX(f)≤9jX(g). Despite gradual sets can be endowed with the order described in Definition 5, this is not a bounded lattice, since 111164 VOLUME 12, 2024 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems it has no minimum. This drawback suggests the convenience of considering gradual sets as functions f:L→P(X). V. REPRESENTATION THEOREMS In this section we deal with the notion of level cut and its relation with the representation theorem of fuzzy sets. Concretely, we examine their role within the framework established in the preceding sections in order to generalize them to some classes of extended fuzzy sets. In particular, we establish a well-founded definition of level cuts for interval-valued fuzzy sets. We also show that the classical notion of level cut depends on a given embedding, so varying this inclusion yields other options for level cuts. For simplicity, and coherency with some of the comments made above, we shall treat level cuts as maps in Map(L,P(X)) rather than gradual sets. We shall interchangeably view a family of subsets of Xindexed by Las a mapping in Map(L,P(X)), and vice versa. We recall from Figure 3that the class of L-fuzzy sets can be embebed into the class of hesitant L-fuzzy sets by means of a mapping ϕdefined as ϕ(f)(x)=[0,f(x)] for any f∈FSL(X) and x∈X. Then, for a given fuzzy set f∈FSL(X), we may associate to it the collection of subsets of X, indexed in L, described as tϕ(f)(α)= {x∈Xsuch that α∈ϕ(f)(x)} = {x∈Xsuch that α∈[0,f(x)]} = {x∈Xsuch that f(x)≥α} =f≥α, for any α∈L, getting the usual description of fuzzy sets by means of the standard level cuts. Nevertheless, depending on the embedding, we may obtain another possibilities, so, actually, we should be more precise when referring to level cuts. Definition 6: Let λbe a one-to-one mapping from FSL(X) to Map(X,P(L)) and f∈FSL(X), the class of λ-level cuts associated to fis the family {fλ α}α∈Lof subsets of X, such that, for any α∈L, fλ α={x∈Xsuch that α∈λ(f)(x)}. Example 2: a) Dually the standard case, we may also consider ϕop :FSL(X)→HFSL(X), given by ϕop(f)(x)=[f(x),1] for any f∈FSL(X) and x∈X. In this case the associated subsets fϕop α= {x∈Xsuch that f(x)≤α} = f≤α, for any α∈L, consist of the dual of the standard level cuts. Note these are simply the same objects that the standard level cuts when considering the opposite lattice Lop. b) Consider the mapping defined as φ(f)(x)= {f(x)}for any f∈FSL(X) and x∈X. The φ-level cuts of fconsists of the family {x∈Xsuch that f(x)=α}α∈L. c) Consider the right open version of the above-described mapping ϕ. Let ˜ϕ:FSL(X)→Map(X,P(L)) given by ˜ϕ(f)(x)=[0,f(x)) for any f∈FSL(X) and x∈X. Hence, the associated ˜ϕ-level cuts are defined as f˜ϕ α= {x∈Xsuch that α∈ ˜ϕ(f)(x)} = {x∈Xsuch that α∈[0,f(x))} = {x∈Xsuch that f(x)> α} =f>α, for any α∈L, i. e. the usual strict level cuts. In this context, a representation theorem for fuzzy sets as λlevel cuts is a description of those mappings in Map(L,P(X)) coming from a fuzzy set, via λ. That is to say, a description of the image of the composition For instance, the classical result given by Negoita and Ralescu [7, Theorem 1]is a representation theorem for ϕ-level cuts. We recall that, if Lis a join-complete lattice, ϕhas a left inverse mapping δgiven by δ(g)(x)=sup g(x) for any g∈Map(X,P(L)) and x∈X. Then, for a given mapping in Map(L,P(X)), we may obtain an associated L-fuzzy set using the composition Map(L,P(X)) 9 −→ Map(X,P(L)) δ −→ FSL(X), which verifies that δ98ϕ is the identity mapping. It is necessary to consider the property on L, (∗) for any family {αi}i∈I⊆L, if α < ∨i∈Iαithen there exists β∈Isuch that α≤αβ. Theorem 5: [7] Let Lbe a join-complete lattice verifying (∗). Let F∈Map(L,P(X)). There exists some f∈FSL(X) such that F=8ϕ(f) if and only if a)F(0) =X, and b)F(∨i∈Iαi)= ∩i∈IF(αi) for any family {αi}i∈I⊆L. Furthermore, if Fverifies these properties, the associated fuzzy set fis defined as f(x)=sup{α∈Lsuch that x∈F(α)}. In particular, when L=[0,1], we may rewrite Theorem 5, or [18, Theorem 4.3], as follows. Theorem 6 (Representation theorem for ϕ-level cuts): A class of subsets A= {Aα}α∈[0,1] of a universe set Xis the class of ϕ-level cuts of a fuzzy set if and only if Averifies the following conditions: i)Ais full, i. e., [ α∈[0,1] Aα=X, ii)Ais nested, i. e., if α≤βthen Aβ⊆Aα, iii)Ais upper closed, i. e., \ α<β Aα=Aβfor any β∈[0,1]. Furthermore, if Averifies these conditions, the associated fuzzy set f:X→[0,1] is defined as, for any x∈X, f(x)=sup{α∈[0,1] such that x∈Aα}. Proof: To recover the original statement, simply observe that the property b) in Theorem 5implies that the family is nested, and that Abeing nested and full is equivalent to be nested and A0=X. Similarly, by using the embedding φdescribed in Example 2b), we may prove the following result. VOLUME 12, 2024 111165 P. Jara et al.: On Gradual Sets, Hesitant Fuzzy Sets, and Representation Theorems Theorem 7 (Representation Theorem for φ-Level Cuts): Let Lbe a lattice, a class of subsets A= {Aα}α∈Lof a universe set Xis the class of φ-level cuts of a L-fuzzy set if and only if the non-empty elements in Aform a partition of X. Furthermore, if Averifies this condition, the associated L-fuzzy set f:X→Lis defined as, for any x∈X,f(x)=α, where αis the (unique) element in Lsuch that x∈Aα. Proof: By Example 2b), the φ-level cuts associated to fare given by the family {x∈Xsuch that f(x)=α}α∈Lfor any α∈L. That is, the inverse images by fof the elements in L. Since fis a function, these clearly form a partition of X. Conversely, if Aproduces a partition of X, the L-fuzzy set f described in the statement verifies that 8φ(f)=A, viewed as a map in Map(L,P(X)). In the literature, there have been studies that examine the concept of level cuts and aim to establish representation theorems for certain types of extended fuzzy sets, such as interval-valued fuzzy sets (IVFSs) or intuitionistic fuzzy sets (see [19]). These studies primarily focus on subsets that satisfy specific inequalities. However, in this work, we adopt the methodology discussed earlier to tackle this problem. Given a hesitant L-fuzzy set f∈HFSL(X), we define the family of level cuts associated to fas fα={x∈Xsuch that α∈f(x)}=8(f)(α), for any α∈L. Therefore, Lemma 1provides literally the corresponding representation theorem. Theorem 8 (Representation Theorem for Hesitant Fuzzy Sets): Let Lbe a bounded lattice. A family of subsets A= {Aα}α∈Lof a universe set Xis the class of level cuts of a hesitant L-fuzzy set if and only if Ais full. In this context, any subclass Fembedded in HFSL(X) is then likely to define the level cuts of its objects as the image throughout the composition as it has been done previously for L-fuzzy sets. Assume hereon that L=[0,1] and let us denote by Ithe set of all closed intervals in [0,1]. An Interval-Valued Fuzzy Set (IVFS) on Xis then a map A:X−→ I. Denote by IVFS[0,1](X) the class formed by all interval-valued fuzzy sets of X. The standard inclusion (say µ) of IVFS[0,1](X) into HFS[0,1](X) is just inhered of considering an interval as a non-empty subset in [0,1]. In this sense, the family of µ-level cuts associated to an IVFS fis given by fµ α= {x∈Xsuch that ax≤α≤bx}, where f(x)=[ax,bx] for any x∈X, for any α∈L. Lemma 2. Let f∈HFS[0,1](X) and F=8(f)∈ Map([0,1],P(X)). The following assertions are equivalent: a) for any x∈X,f(x) is convex, b) for any γ < β < δ ∈[0,1], F(γ)∩F(δ)⊂F(β). Proof: Let us suppose that f(x) is convex for any x∈ X, and let γ < β < δ ∈[0,1]. Given x∈F(γ)∩F(δ), hence γand δbelong to f(x). Since it is convex, β∈f(x), so x∈F(β). Conversely, let x∈X. If f(x)= {α}, then we are done. If not, take γ, δ ∈f(x) such that γ < δ and let β∈[0,1] with γ < β < δ. Since x∈F(γ)∩F(δ), by hypothesis, x∈F(β), and then β∈f(x), so f(x) is convex. Lemma 3. Let f∈HFS[0,1](X) and F=8(f)∈ Map([0,1],P(X)). The following assertions are equivalent: a) for any x∈X,f(x) is a closed subset in [0,1]. b) for any sequence {αn}n∈Nin [0,1] with {αn}n∈N→α, it holds that Tn∈NF(αn)⊆F(α). Proof: Suppose that f(x) is closed for any x∈X. Let {αn}n∈N→αa converging sequence and x∈Tn∈NF(αn). That means {αn}n∈N⊂f(x), so α∈f(x). Hence, x∈F(α). Conversely, take {αn}n∈N→αwith {αn}n∈N⊆f(x). Then x∈Tn∈NF(αn)⊆F(α), that is, α∈f(x), so f(x) is closed. Corollary 1 (Representation theorem for Interval-Valued Fuzzy Sets via µ): A class of subsets A= {Aα}α∈[0,1] of a universe set Xis the class of µ-level cuts of an interval-valued fuzzy set if and only if Averifies the following conditions: a)Ais full. b)Aγ∩Aδ⊂Aβfor any γ < β < δ ∈[0,1]. c) For any sequence {αn}n∈Nin [0,1] with {αn}n∈N→α, it holds that Tn∈NAαn⊆Aα. Furthermore, if Averifies these conditions, the associated interval-valued fuzzy set f:X→Iis defined as f(x)=[ax,bx] for any x∈X, where ax= inf{α∈[0,1] such that x∈Aα}and bx=sup{α∈ [0,1] such that x∈Aα}. Proof: It follows from Lemmas 2and 3. Beyond the subclasses of hesitant fuzzy sets, we would need to extend the working space. A possible solution is to enlarge the lattices in the adjunction described in Section II. We may exchange the Boolean algebra 2 by any other larger lattice as, for instance, the own lattice L. Hence, we find that Map(L,Map(X,L)) ∼ =Map(X,Map(L,L)), that is to say, denoting by T2FSL(X) the class of type-2 L-fuzzy sets, Map(L,FSL(X)) ∼ =T2FSL(X). So, therefore, each type-2 L-fuzzy set is represented by a family, indexed in L, of L-fuzzy sets, which can be considered its level cuts. In general, following the same principle, one may obtain that a suitable notion of level cuts for type-n L-fuzzy sets is to consider a family of type-(n−1) L-fuzzy sets. VI. CONCLUSION In this paper we have carried out an analysis of the notions of gradual set and hesitant fuzzy set. Both aiming to extend the classical theory of fuzzy sets from two different viewpoints. We have shown that these two perspectives are linked by a one-to-one mapping, coming from the categorical adjunction between the hom and the cartesian product functors. Hence we have seen that the classes of gradual sets and hesitant fuzzy sets are strongly related. However, since there are some disparities in the literature regarding their precise definitions, 111166 VOLUME 12, 2024