Embeddings between lattices of fuzzy sets: An application of closed-valued fuzzy sets

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Fuzzy Sets and Systems ••• (••••) •••–•••

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Embeddings between lattices of fuzzy sets: An application of closed-valued fuzzy sets ✩

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F.J. Lobillo , Luis Merino , Gabriel Navarro

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, Evangelina Santos

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a CITIC and Department of Algebra, University of Granada, Spain b Department of Algebra, University of Granada, Spain

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c Department of Computer Science and Artificial Intelligence and CITIC, University of Granada, Spain

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Received 5 October 2017; received in revised form 12 February 2018; accepted 18 April 2018

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Abstract

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In this paper we deal with the problem of extending Zadeh’s operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton viewpoint is required, we reformulate the intersection on hesitant fuzzy sets and introduce what we have called closed-valued fuzzy sets. This new type of fuzzy sets extends standard union and intersection on FSs. In addition, it allows handling together membership degrees of different nature as, for instance, closed intervals and finite sets. Finally, all these constructions are viewed as T2FSs forming a chain of lattices. © 2018 Elsevier B.V. All rights reserved.

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Keywords: Fuzzy sets; Interval-valued fuzzy sets; Set-valued fuzzy sets; Type-2 fuzzy sets; Closed-valued fuzzy sets

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1. Introduction

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The relevance of the different types of fuzzy sets, as models to handle uncertainty, is worldwide recognized. This is yet another example of the continuing need for mathematical formalisms in computer science-related fields. Coming from Zadeh’s notion of Fuzzy Set (FS) [25], some of these types have been also defined in order to solve the problem of the attribution of membership degree: whenever the objects of the universe of discourse are connected to imprecise concepts, the design the membership functions might be hard to achieve. Nevertheless, as solutions to specific

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* Corresponding author at: Department of Computer Science and Artificial Intelligence, University of Granada, c/Periodista Daniel Saucedo

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The authors are listed in alphabetical order. All the authors contributed equally to this work.

Aranda s/n, E18071 Granada, Spain. E-mail addresses: [email protected] (F.J. Lobillo), [email protected] (L. Merino), [email protected] (G. Navarro), [email protected] (E. Santos). https://doi.org/10.1016/j.fss.2018.04.014 0165-0114/© 2018 Elsevier B.V. All rights reserved.

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problems of different nature, their algebraic structures seem to be disconnected or, at least, their relationships are not well-understood. In the literature there exist some attempts aiming to establish a common framework where these types of fuzzy sets could be inserted gradually into a chain of lattice structures, see the recently published survey [8] and the references therein. Nevertheless, for some cases, the compatibility between lattice structures and embedding homomorphisms is a tricky problem. The origin of our work is motivated by the problem stated in [8, Remark 2]: given a fixed universe set X, is there a lattice structure on the class of Set-Valued Fuzzy Sets (SVFSs) over X [11] extending Zadeh’s union and intersection on FSs over X? Actually, this question can also be stated in the realm of Type-2 Fuzzy Sets (T2FSs) [27]. The key-point here is how we may see FSs as SVFSs, or as T2FSs. In general, FSs are viewed as SVFSs for which membership degrees are given by a singleton on [0, 1]. This is not consistent with the standard lattice operators, and this is why it is proposed to change the algebraic structure. Nevertheless, from an algebraic point of view, what is important are the relations (lattice morphisms) between the objects (lattices) of the category. That is, a different approach to a practical solution could then be to change the embedding and keep the lattice structures. In this paper we study the embeddings between certain lattice structures on several types of fuzzy sets over a common universe. Concretely, we focus on FSs and the generalizations given by Interval-Valued Fuzzy Sets (IVFSs) [21], SVFSs and T2FSs. The embeddings and lattice operators under consideration are derived pointwise from their corresponding set-theoretical versions defined on the spaces where uncertainty is measured. Our main goal is to present two possible solutions to the above question. One keeps the standard lattice structures whilst the other keeps the usual embedding from FSs to SVFSs. In particular, the aims of our work are the following:

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1. To show, from a mathematical perspective, why Zadeh’s lattice operators on FSs and set theoretical-derived operators on SVFSs are not compatible by way of the embedding from [0, 1] into its power set as singletons. 2. To provide a collection of embeddings from FSs to SVFSs making these structures compatible. 3. To reformulate the meet operator on Hesitant Fuzzy Sets (HFSs) [22] in order to obtain a lattice structure for most of their practical applications. 4. To introduce Closed-Valued Fuzzy Sets (CVFSs) in order to extend the lattice of FSs via the embedding from [0, 1] into its power set as singletons. 5. To embed all these classes into T2FSs forming a chain of lattice embeddings. The paper is organized in increasing order of generality of the different types of fuzzy sets. Namely, in Section 2 we recall the notion of FS and explain the recurrent and basic categorical constructions of the paper. Readers non-familiar with category theory may consult, for instance, the reference [16]. Section 3 concerns IVFSs. We then provide a family of embeddings from FSs to IVFSs respecting the usual meet and join operators. Sections 4 and 5 deal with the problem stated in [8, Remark 2]. In the former we describe a family of lattice embeddings from FSs to SVFSs, where SVFSs are endowed with the lattice operators derived pointwise from the set union and intersection. In the latter we show an alternative solution. We define the class of CVFSs with a lattice structure inspired by the celebrated notion of HFS. Then CVFSs become a lattice which extends the lattice of FSs through the “singleton embedding”. Finally, in Section 6, we embed the achieved chains of lattices inside the class of T2FSs endowed with the structure defined in [9]. Due to the number of lattice structures we shall manage, we condense the notation used for the partial orders and the lattice operators in Table 1. There, the left column contains the operators over the classes where the uncertainty is measured, whilst the right column includes their corresponding pointwise-induced operators on the types of fuzzy sets.

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In 1965, inspired by the idea of multivalued logic, Zadeh [25] proposes the theory of Fuzzy Sets, an extension of the classical set theory, for some classes of objects whose criteria of membership are not precisely defined. In what follows, X denotes a nonempty universe set and [0, 1] the closed unit interval.

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2. Preliminaries

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Definition 1. A Fuzzy Set (FS) A on X is a set mapping A : X → [0, 1]. This is also called a Type-1 Fuzzy Set (T1FS).

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Table 1 Partial orders and lattice structures in this paper.

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[0, 1] I([0, 1]) C([0, 1]) 2[0,1] 2[0,1] \∅b [0, 1][0,1]

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a Empty set, as membership degree, allowed. b It is only a partially ordered set, since ∩ is not a meet operator. S

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That is, the fuzzy sets on X is the class of set maps from X to [0, 1]. The standard order ≤ on the unit interval [0, 1] can be lifted pointwise to a partial order on FS(X). For any A, B ∈ FS(X) A ≤F B ⇐⇒ A(x) ≤ B(x) for all x ∈ X.

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This inherited partial order is a particular case of a categorical construction. We recall that a category C (see for instance [16, Definition 1.1.1]) is a collection of objects together with a collection of homomorphisms, or arrows, between these objects verifying:

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• There exists an operator composition ◦ that assigns, for any objects A, B, C of C and homomorphisms f : A → B and g : B → C, an homomorphism g ◦ f from A to C. • The operator ◦ is associative, i.e. (f ◦ g) ◦ h = f ◦ (g ◦ h) for any composable homomorphisms f, g, h. • For any object A, there exists an identity homomorphism idA verifying that, for any homomorphism f : A → B, idB ◦ f = f and f ◦ idA = f .

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For instance, we may consider the category Set [16, Example 1.1.3] formed by the class of all sets and set maps between them. In this paper, it will be relevant for us the category Poset [16, Example 1.1.5] of partially ordered sets (posets) and homomorphisms between them (i.e. set maps preserving the partial order) and Lat formed by lattice and lattice homomorphisms (see below). Two categories C and D can be related by a (covariant) functor [16, Definition 2.1.1] F : C → D. This is a correspondence which maps an object C in C to an object F (C) in D, and an homomorphism f : A → B in C to an homomorphism F (f ) : F (A) → F (B) in D. F verifies the following properties: F (idA ) = idF (A) for any object A, and F (f ◦ g) = F (f ) ◦ F (g) for any composable homomorphisms f and g. In this paper we mostly make use of the Hom-functor [16, Example 2.1.11]. Formally, given a category C and an object A in C, we may consider HomC (A, −) : C → Set which maps any B in C to HomC (A, B) the set of homomorphisms from A to B, and any homomorphism f : B → C to HomC (A, f ) the map which assigns any homomorphism g : A → B to f ◦ g : A → C. In this paper, the objects of any considered category are sets (possibly with an additional algebraic structure) and then the homomorphisms are also maps between sets. Hence, there is always an inclusion of sets HomC (A, B) ⊆ HomS et (A, B) for any objects A and B in C. For brevity, given two objects A and B, we simply write Hom(A, B) or B A instead of HomS et (A, B). We use the first notation when we want to highlight the categorical viewpoint. A key point in this paper is the fact that, in the above discussion, if B is a poset, Hom(A, B) becomes also a poset. More concretely, for an arbitrary nonempty set S, we may consider Hom(S, −) : Poset → Poset

(2)

which maps any poset (P , ≤) to (Hom(S, P ), ), where is the partial order defined by, for each f, g ∈ Hom(S, P ), f g ⇐⇒ f (s) ≤ g(s) for all s ∈ S.

(3)

We say then that the partial order in Hom(S, P ) is pointwise derived (or induced, or lifted) from ≤ in P . This construction generalizes the one in (1).

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Remark 2. Of course, there exist partial orders on Hom(S, P ) which do not come from a partial order on P . Nevertheless, in such a case, the poset structure of P is forgotten and P is treated merely as a set. For this reason, all along this paper, Hom posets are endowed with the pointwise partial order lifted from the codomain.

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We recall (see for instance [12]) that a lattice (L, ≤, ∨, ∧) is a poset in which every pair of elements a, b ∈ L has a supremum, or joint, a ∨ b and an infimum, or meet, a ∧ b. The joint and meet operators satisfy the following laws:

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• Commutativity, a ∨ b = b ∨ a and a ∧ b = b ∧ a for any a, b ∈ L. • Associativity, (a ∨ b) ∨ c = a ∨ (b ∨ c) and (a ∧ b) ∧ c = a ∧ (b ∧ c) for any a, b ∈ L. • Absorption laws, a ∨ (a ∧ b) = a and a ∧ (a ∨ b) = a for any a, b ∈ L.

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Since a ≤ b if and only if a ∨ b = b if and only if a ∧ b = a, the partial order is determined by the joint and meet operators. Hence, we simply write (L, ∨, ∧) for denoting a lattice. We denote by 2 = {0, 1} the two-element lattice, where 0 ≤ 1. The power set of A is therefore identified with 2A via the characteristic map. Given two lattices (L1 , ∨1 , ∧1 ) and (L2 , ∨2 , ∧2 ), a set map f : L1 → L2 is called a lattice homomorphism if it is closed for the joint and meet operators, that is, f (a ∨1 b) = f (a) ∨2 f (b) and f (a ∧1 b) = f (a) ∧2 f (b) for any a, b ∈ L1 . The class of all lattices and their homomorphisms form a category which we denote by Lat. We denote by HomLat (L1 , L2 ) the set of lattice homomorphisms from L1 to L2 . Finally, we say that f is a lattice embedding if it is an injective lattice homomorphism. The standard meet and join operators, minimum ∧ and maximum ∨, associated to the order on [0, 1] provide a lattice structure on FS(X) given by (4)

for each A, B ∈ FS(X) and any x ∈ X. Obviously, this is coherent with the partial order defined in (1). In general, for any lattice (L, ∧, ∨) and any nonempty set S, Hom(S, L) becomes a lattice with meet and join operators defined pointwise, i.e. (f g)(s) = f (s) ∨ g(s) and (f  g)(s) = f (s) ∧ g(s),

(5)

for any f, g ∈ Hom(S, L) and any s ∈ S, which is coherent with (3). This simply means that (2) can be restricted to a functor

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Hom(S, −) : Lat → Lat

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on the category of lattices. In addition, if S is a chain (or totally ordered set), HomLat (S, L) becomes a lattice by restricting the operators and . In such a case we may also consider the functor HomLat (S, −) : Lat → Lat.

(7)

The practical utility of FSs depends heavily on a suitable estimation of the membership degrees and, in some cases, it becomes a hard problem. Zadeh himself realizes this point in [27] and suggests to make use of more complex objects as membership degrees rather than a solely real value. Several extensions (types) of FSs have arisen from this idea. Mainly, the generalizations invoke implicitly the Hom functor of (6), or the HomLat functor of (7), for a lattice L constructed from [0, 1]. In the forthcoming sections we describe and relate those concerning this work.

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3. Interval-Valued Fuzzy Sets

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(A ∪F B)(x) = A(x) ∨ B(x) and (A ∩F B)(x) = A(x) ∧ B(x),

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A closed interval in [0, 1] is given by a couple of elements α, β ∈ [0, 1] such that α ≤ β. So the set of closed intervals is in one-to-one correspondence with the set of lattice homomorphisms HomLat (2, [0, 1]). Here, each f ∈ HomLat (2, [0, 1]) is identified with the interval [f (0), f (1)]. We shall denote by I([0, 1]) the class of all closed intervals in [0, 1]. The idea of assigning a closed interval as membership degree can be found in Zadeh’s paper [27], although it is also introduced independently by Grattan-Guinness [11], Jahn [13] and Sambuc [21]. Many practical applications related to this concept have been developed, see e.g. [8, Section VI]. Formally:

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Definition 3. An Interval-Valued Fuzzy Set (IVFS)  A on X is a mapping A : X → I([0, 1]), i.e. the set of all IVFSs  on X is IVFS(X) = Hom X, HomLat (2, [0, 1]) .

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A I B ⇐⇒ A(x) ≤I B(x) for any x ∈ X.

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The lattice structure on IVFS(X) comes from the lattice structure on HomLat (2, [0, 1]), which is lifted from [0, 1]. Namely, for each closed intervals [a, b] and [c, d],

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[a, b] ∪I [c, d] = [a ∨ c, b ∨ d] and [a, b] ∩I [c, d] = [a ∧ c, b ∧ d],

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and then, for each A, B ∈ IVFS(X),

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(A I B)(x) = A(x) ∪I B(x) and (A I B)(x) = A(x) ∩I B(x),

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for any x ∈ X. These lattice structures are considered, for example, in [1], [6], [9] (by using the extension principle applied to ∨ and ∧) or [10]. From a categorical perspective, any one-to-one lattice homomorphism

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h : ([0, 1], ∨, ∧) → (HomLat (2, [0, 1]), ∪I , ∩I )

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yields a lattice embedding

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Hom(X, h) : (FS(X), ∪F , ∩F ) → (IVFS(X), I , I )

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by applying the functor (6) to h. Concretely, Hom(X, h)(A) = h ◦ A for each A ∈ FS(X). This is an embedding because (6) preserves monomorphisms, that is, in our context, assigns injective maps to injective maps, see [16, Proposition 1.3.2]. Here, it is interesting to see HomLat (2, [0, 1]) as a sublattice of Hom(2, [0, 1]) ≡ [0, 1]2 , the unit square in R2 , with the product order. Therefore, HomLat (2, [0, 1]) ≡ {(x, y) ∈ [0, 1] such that x ≤ y} = T . 2

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h1 , h2 : [0, 1] → [0, 1]

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such that h1 (t) ≤ h2 (t) for any t ∈ [0, 1], and one of them is strictly increasing. The embedding h is then defined as h(t) = (h1 (t), h2 (t)) for any t ∈ [0, 1]. Moreover, for any A ∈ FS(X) and any x ∈ X, the associated closed interval is [h1 (A(x)), h2 (A(x))]. For instance, set h1 , h2 , h3 : [0, 1] → [0, 1] defined by h1 (t) = t , h2 (t) = 0 and h3 (t) = 1 for any t ∈ [0, 1]. Hence, we obtain the lattice embeddings (14)

given by φ(t) = [h1 (t), h1 (t)] = [t, t], ω(t) = [h1 (t), h3 (t)] = [t, 1] and γ (t) = [h2 (t), h1 (t)] = [0, t] for any t ∈ [0, 1], see Fig. 1(a)–(c). The first two examples are instances of the same construction. Namely, we may always consider the graph of an increasing set map f : [0, 1] → [0, 1] such that f (t) ≥ t . In this case, the embedding is defined by h(t) = (t, f (t)) for any t ∈ [0, 1], see Fig. 1(d). These maps induce the corresponding lattice embeddings , , : (FS(X), ∪F , ∩F ) −→ (IVFS(X), I , I )

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So, in order to find a lattice embedding from FSs to IVFSs with respect to the above-mentioned lattice structures, it is enough to fix a lattice embedding from [0, 1] to the upper triangle T . A simple way of finding one of such an embedding is fixing two increasing maps

φ, ω, γ : ([0, 1], ∨, ∧) → (I([0, 1]), ∪I , ∩I )

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defined pointwise as (A)(x) = [A(x), A(x)], (A)(x) = [A(x), 1] and (A)(x) = [0, A(x)] for any A ∈ FS(X) and any x ∈ X. Observe that  is the inclusion of FSs into IVFSs pointwise-derived from seeing an element in [0, 1] as a singleton interval.

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The idea of representing the plausible range of the membership degrees by means of closed intervals has achieved a relative success, see [7, Section 4] and the references therein. An obvious generalization consists in allowing to choose an arbitrary nonempty subset of [0, 1], yielding the notion of set-valued fuzzy set. The following definition is given in [11].

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The insertion of SVFSs, as another piece of the chain of lattices of types of fuzzy sets, is somehow problematic. Actually, in [8, Remark 2] it is formulated the open problem of finding a lattice structure on the class of SVFSs such that the restriction to FSs preserves Zadeh’s intersection and union. Let us explain the partial solutions that we propose in this work. In this section we consider SVFSs endowed with the lattice structure derived from set intersection and union, whilst, in the next one, we shall analyze another meet and join operators.

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4.1. Some lattice embeddings from ([0, 1], ∨, ∧) to (2[0,1] , ∪, ∩)

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Definition 4. A Set-Valued Fuzzy Set (SVFS) A on X is a set mapping A : X → 2[0,1] \{∅}. Hence SVFS(X) =   [0,1] Hom X, 2 \{∅} .

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The power set of [0, 1] is endowed with the usual lattice structure defined by the set union ∪ and intersection ∩. Hence, by virtue of (6), Hom(X, 2[0,1] ) becomes a lattice with the structure defined by the operators (A B)(x) = A(x) ∪ B(x) and (A  B)(x) = A(x) ∩ B(x),

(16)

for each A, B ∈ Hom(X, 2[0,1] ) and any x ∈ X. Nevertheless, in general, SVFS(X) is not closed under these operators. Indeed, given A, B ∈ SVFS(X), there could exist an x ∈ X such that A(x) ∩ B(x) = ∅, the reader may find easily examples verifying this, so that A  B, according to Definition 4, is not in SVFS(X). This is a slight mistake committed in [8, Proposition 5.1], where it is asserted that (SVFS(X), , ) is a complete lattice. An elementary solution consists in extending the notion of SVFS by allowing the empty set as a possible membership degree. A possible interpretation of this degree would be a nonsense [23]. Hence, under this condition, (SVFS(X), , ) is a complete lattice. Via the characteristic maps, this is to say we see A and B belonging to Hom(X, Hom([0, 1], 2)), we may write (A B)(x)(t) = A(x)(t) ∨ B(x)(t) and (A  B)(x)(t) = A(x)(t) ∧ B(x)(t),

(17)

for all x ∈ X and t ∈ [0, 1]. So, this lattice structure on SVFS(X) comes from the Boolean algebra 2. This conclusion also follows from the bijection  Hom X, 2Y ) ∼ (18) = Hom(X × Y, 2),  for any sets X and Y , which sends any f ∈ Hom X, 2Y ) to the map f given by f (x, y) = f (x)(y). As observed in [8], FSs can be viewed as SVFSs via the injective map ι : FS(X) → SVFS(X) given by ι(A)(x) = {A(x)}, for any A ∈ FS(x) and x ∈ X. This map is simply the composition

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FS(X)

IVFS(X)

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Hom(X, [0, 1]) ⊆ Hom(X, 2[0,1] )

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(20)

and the map ι is a description of (20). Nevertheless, it is not a lattice injection. The isomorphism (18) explains why the lattice structures are not compatible: The structure on (FS(X), ∪F , ∩F ) is lifted from [0, 1] applying (6), and the structure on (SVFS(X), , ) is inherited from 2. Actually, this is also the reason of why i is not a lattice homomorphism. So, in order to make compatible the lattice structures, we have to look for a new way to embed FS(X) inside SVFS(X). Let A ∈ FS(X), for any t ∈ [0, 1], the level t-cut of A is defined by At = {x ∈ X | A(x) ≥ t}. The class of all level t -cuts of A is a chain, since t ≤ s implies As ⊆ At . Then A can also be viewed as a lattice homomorphism A : ([0, 1]op , ∧, ∨) → (2X , ∪, ∩),

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7 8 9 10 11 12 13 14

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(A ∪∗ B)t = At ∪ Bt and (A ∩∗ B)t = At ∩ Bt ,

(22)

for any t ∈ [0, 1]. So, from this viewpoint, the structure on (FS(X), ∪F , ∩F ) is inherited from the lattice structure of the power set 2X and, whence, from 2. Since the map c is a bijective map, the composition (18)

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(18)

FS(X) ∼ = HomLat ([0, 1]op , 2X ) ⊆ Hom([0, 1], 2X ) ∼ = Hom([0, 1] × X, 2) ∼ = Hom(X × [0, 1], 2) ∼ = SVFS(X),

27 28

provides an embedding

29

 : (FS(X), ∪F , ∩F ) → (SVFS(X), , )

(23)

with (A)(x) = [0, A(x)] for each A ∈ FS(X) and x ∈ X. Actually,  is induced pointwise from the extension of γ in (14) to subsets of [0, 1]. That is, IVFS(X)

i

SVFS(X).

(24)



Observe that the lattice structures of all intermediate sets come from 2. Therefore, as a composition of lattice homomorphisms,  is also a lattice homomorphism, and so, a lattice embedding. Nevertheless, for the convenience, here we show a direct proof.

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(21)

Zadeh’s lattice structure on FS(X) is reflected, via c, from the following one: for each A, B ∈ HomLat ([0, 1]op , 2X ),

FS(X)

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17

A −→ [t → At ] .



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c : Hom(X, [0, 1]) −→ HomLat ([0, 1]op , 2X )

c

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where by [0, 1]op we mean the set [0, 1] with the opposite order. Concretely, there is a bijective map

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1 2

where  is the lattice embedding provided in (15) and i is defined by lifting the identification of a closed interval as a subset of [0, 1]. By (6), the insertion of [0, 1] inside 2[0,1] as singleton sets implies the inclusion

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(19)

SVFS(X),

ι

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i

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Theorem 5. The map  defined in (23) is a lattice embedding. Proof. Assume first (A) = (B) for some A, B ∈ FS(X). Then, for all x ∈ X, [0, A(x)] = [0, B(x)], so A(x) = B(x). Thus A = B and  is an injective map. Now, (A ∪F B)(x) = [0, (A ∪F B)(x)]

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= [0, A(x) ∨ B(x)]

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= [0, A(x)] ∪ [0, B(x)]

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= (A)(x) ∪ (B)(x)

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= ((A) (B))(x), for any x ∈ X. Similarly, (A ∩F B) = (A)  (B). So  is a lattice homomorphism. 2

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The above discussion gives an answer to the open problem stated in [8, Remark 2]. Nevertheless, this solution does not include (IVFS(X), I , I ) as a lattice between (FS(X), ∪F , ∩F ) and (SVFS(X), , ). Although a closed interval is a subset of [0, 1], and then i : IVFS(X) → SVFS(X) is a one-to-one map, the operators and  are not closed when they are restricted to IVFS(X). For instance, the union of intervals is not always an interval. So the question is if there exists an embedding from IVFS(X) to SVFS(X) coherent with these lattice structures. That is, can we provide a commutative lattice diagram (FS(X), ∪F , ∩F )

9



12 13 14 15

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Let us denote by Q and I the set of rational and irrational numbers, respectively. Consider the map ξ : I([0, 1]) −→ 2

[0,1]

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(25)

defined by ξ([a, b]) = ([0, a] ∩ Q) ∪ ([0, b] ∩ I) for any closed interval [a, b]. All along this section we shall need the following well-known properties concerning the real numbers, see for instance [5, Theorem 2.6.13]. Density of irrationals. Given a, b ∈ R with a < b, there exists an irrational p ∈ I such that a < p < b.

(27)

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Lemma 6. Let I, J ∈ I([0, 1]) such that I is not a singleton, the following are equivalent:

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i) I ⊆ J . ii) I ∩ Q ⊆ J ∩ Q. iii) I ∩ I ⊆ J ∩ I.

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Proof. i) ⇒ ii) and i) ⇒ iii) are trivial. We prove ii) ⇒ i). Let I = [a, b] and J = [c, d], with a = b, and I ∩ Q ⊆ J . Let us first prove that a < d and c < b. Since a < b, by (26), there exists q ∈ Q such that a < q < b. Therefore q ∈ I ∩ Q ⊆ J = [c, d], so c ≤ q < b and a < q ≤ d. Now, we prove c ≤ a and b ≤ d. Suppose that a < c < b. By (26), there exists q ∈ Q such that a < q < c. Then q is in I ∩ Q, but not in J . A contradiction, so c ≤ a. Similarly, if d < b, there exists q ∈ Q such that d < q < b. Since a < d, q ∈ I ∩ Q ⊆ J , and we get a contradiction. Then b ≤ d. Therefore [a, b] ⊆ [c, d]. iii) ⇒ i) can be proved similarly by applying (27). 2

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Theorem 7. The map ξ : (I([0, 1]), ∪I , ∩I

26

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) → (2[0,1] , ∪, ∩)

is a lattice embedding

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(26)

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4.2. A lattice embedding from (I([0, 1]), ∪I , ∩I ) to (2[0,1] , ∪, ∩)

Density of rationals. Given a, b ∈ R with a < b, there exists a rational q ∈ Q such that a < q < b.

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Next, we give a theoretical positive answer. This is based on a specific property of the reals: they can be split into two disjoint and dense subsets.

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(SVFS(X), , )?

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(IVFS(X), I , I )



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Proof. Let us fix I = [a, b] and J = [c, d] two closed intervals. Suppose first that ξ([a, b]) = ξ([c, d]). Then, since Q and I are disjoint sets, [0, a] ∩ Q = [0, c] ∩ Q and [0, b] ∩ I = [0, d] ∩ I. If a and c are non-zero, by Lemma 6, [0, a] = [0, c], so a = c. Suppose one of them is zero. For instance, suppose that a = 0. Hence [0, c] ∩ Q = {0}. By the density of Q, it follows that c = 0. The equality b = d may be proved similarly by applying Lemma 6 and (27). Thus ξ is an injective map. Now, ξ(I ) ∩ ξ(J ) = (([0, a] ∩ Q) ∪ ([0, b] ∩ I)) ∩ (([0, c] ∩ Q) ∪ ([0, d] ∩ I))

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= ([0, a ∧ c] ∩ Q) ∪ ([0, b ∧ d] ∩ I)

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= ξ([a ∧ c, b ∧ d])

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= ξ(I ∩I J ) and, by a similar reasoning, ξ(I ) ∪ ξ(J ) = ξ(I ∪I J ). So ξ is a lattice homomorphism. 2

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By applying the functor (6) to ξ , we get the induced lattice embedding

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: (IVFS(X), I , I ) → (SVFS(X), , )

(28)

defined by (A) = ξ ◦ A for any A ∈ IVFS(X). Observe that the restriction to FS(X) maps an A ∈ FS(X) to (A) ∈ SVFS(X) defined by x → [0, A(x)]. Then, the following diagram of lattice homomorphisms is commutative: (FS(X), ∪F , ∩F )



(29)

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Corollary 8. The composition of with any of the maps constructed in Section 3 provides a family of lattice embeddings from (FS(X), ∪F , ∩F ) to (SVFS(X), , ).

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5. Closed-valued fuzzy sets

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(SVFS(X), , ).

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(IVFS(X), I , I )

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Let us now develop an alternative solution to the problem in [8, Remark 2]. In some practical situations it could be interesting to insert FSs inside SVFSs via the embedding ι defined in Section 4. Therefore, in order to preserve Zadeh’s operators on FS(X), we need to change the lattice structure on SVFS(X). The most promising attempt in this direction is the notion of Hesitant Fuzzy Set (HFS) defined by Torra in [22]. The class HFS(X) is constructed by replacing the lattice structure on SVFS(X) by the following operators:

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(A H B)(x) = {t ∈ A(x) ∪ B(x) | t ≥ A(x) ∨ B(x)} and (A H B)(x) = {t ∈ A(x) ∪ B(x) | t ≤ A(x) ∧ B(x)},

22

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for each A, B ∈ HFS(X) and any x ∈ X, where, for any nonempty subset S of [0, 1], we denote by S the infimum of S (in the natural order) and, by S, its supremum. In particular, S ⊆ [S, S]. Actually, these operators are derived pointwise from

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S ∪H T = {t ∈ S ∪ T | t ≥ S ∨ T } and S ∩H T = {t ∈ S ∪ T | t ≤ S ∧ T },

(31)

29

for any S and T nonempty subsets of [0, 1]. HFSs have been successfully applied, for instance, to group decision making, decision support systems, computing with words or cluster analysis, see e.g. [17,14,18–20]. From a mathematical point of view, the operators on HFS(X) preserve Zadeh’s lattice operators on FS(X) and I and I on IVFS(X). Nevertheless, (HFS(X), ∪H , ∩H ) is not a lattice, see [8, Section 5.C] for a simple example. The reason is that ∩H is not a meet operator. On the contrary, ∪H does provide a join-semilattice, see Theorem 15 below. Our aim now is to give a reformulation of ∩H . Let us first define a new partial order on SVFS(X). We shall manage the original definition of Grattan-Guinness in [11], that is, a SVFS on X is a map X → 2[0,1] \∅.

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Definition 9. Let S and T be nonempty subsets of [0, 1], we say that S ≤S T if and only if:

38

i) S ≤ T . ii) S ≤ T . iii) S ∩ [T , S] ⊆ T .

40

31 32 33 34 35 36 37

39

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2[0,1] \∅.

44

Proof. The relation is clearly reflexive. Let S and T be nonempty subsets of [0, 1] such that S ≤S T and T ≤S S. Hence, i) and ii) in Definition 9 imply that S = T and S = T . From iii), S ∩ [T , S] = S ∩ [S, S] = S ⊆ T and T ∩ [S, T ] = T ∩ [T , T ] = T ⊆ S, so S = T and ≤S is antisymmetric. For the transitive property, let S, T and U be nonempty subsets of [0, 1] such that S ≤S T and T ≤S U . By i) and ii) in Definition 9, S ≤ T ≤ U and S ≤ T ≤ U . Now,

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Proposition 10. The relation ≤S is a partial order on

S ∩ [U , S] ⊆ S ∩ [T , S] ⊆ T ,

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Fig. 2. Operators ∩H and ∩S .

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and hence,

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S ∩ [U , S] ⊆ T ∩ [U , S] ⊆ T ∩ [U , T ] ⊆ U.

12 13 14 15 16 17 18 19 20 21

Thus S ≤S U , and the result follows.

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From the partial order ≤S , the class SVFS(X) inherits the following pointwise partial order.

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Definition 11. For each A, B ∈ SVFS(X), we say A S B ⇐⇒ A(x) ≤S B(x) for any x ∈ X.

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It is easy to check that the partial order ≤S restricts to ≤I when working on closed intervals in [0, 1]. Therefore, via the Hom(X, −) functor, we obtain the chain of poset homomorphisms

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(FS(X), ≤F )

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(IVFS(X), I )

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(SVFS(X), S ),

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Definition 12. Let S and T be two nonempty subsets of [0, 1], we define:  (S ∩ [T , S]) ∪ T if S ≤ T , S ∪S T = (T ∩ [S, T ]) ∪ S if T ≤ S

25 26

(33)



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S ∩S T =

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(34)

Observe that ∪S is the same operator as ∪H . This is not the case for ∩S and ∩H .

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Example 14. Let us consider the subsets     1 1 S= | n ≥ 1 and T = |n≥1 . 2n 2n + 1

41

Then S ∩S T = ∅.

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The operator ∩S is not closed in the class 2[0,1] \∅ as the following example shows.

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Example 13. Let S = [0.3, 0.7] and T = {0.4, 0.5, 0.6}, hence S ∩H T = [0.3, 0.6] and S ∩S T = [0.3, 0.4] ∪{0.5, 0.6}, see Fig. 2.

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S ∩ ([S, T ] ∪ T ) if S ≤ T , T ∩ ([T , S] ∪ S) if T ≤ S.

35 36

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and

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which embeds FS(X) into SVFS(X) as those whose membership degrees are singletons.

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The reader might wonder if adding the empty set as a lower bound of all nonempty sets, the class 2[0,1] becomes a lattice with ∪S and ∩S . Notwithstanding, we may see that this also fails by an easy reformulation of the former example. Indeed, simply set S = {0} and T as in Example 14. Hence, S ∩S T = S, but S is not a lower bound of T . The underlying problem is of topological nature: there is a sequence inside T whose limit does not belong to T . This suggests that the suitable class should be formed by closed sets under the standard topology on [0, 1]. Let us denote by C([0, 1]) the class of all nonempty closed subsets of [0, 1].

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Theorem 15. The operators ∪S and ∩S endow the class C([0, 1]) with the lattice structure given by the order ≤S .

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Proof. Let us fix S and T in C([0, 1]). Firstly, the operators are closed for C([0, 1]). By definition, S ∪S T and S ∩S T are constructed as a finite intersection and union of closed sets, so they are also closed. Additionally, S ∪S T = ∅, since S or T are contained in S ∪S T , and S ∩S T = ∅, since S or T are in S ∩S T . For brevity, denote I = S ∪S T . We suppose that S ≤ T , since the other case can be proved symmetrically. Therefore I = (S ∩ [T , S]) ∪ T . The minimum I = T since T is lower or equal than the minimum of S ∩ [T , S], and T ∈ T . The maximum I is the greatest among the maxima of the sets S ∩ [T , S] and T , which are S and T , respectively. So I = S ∨ T . Therefore I ≥ S, T and I ≥ S, T . Now,

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S ∩ [I , S] = S ∩ [T , S] ⊆ (S ∩ [T , S]) ∪ T = I

12 13

and

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thus S ≤S I and T ≤S I , so I is an upper bound of both. Let J ∈ C([0, 1]) such that S ≤S J and T ≤S J . Then J ≥ T = I and J ≥ S ∨ T = I . Therefore,

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I ∩ [J , I ] ⊆ I ∩ [T , J ]

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T ∩ [I , T ] = T ∩ [T , T ] = T ⊆ (S ∩ [T , S]) ∪ T = I,

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= ((S ∩ [T , S]) ∪ T ) ∩ [T , J ]

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= (S ∩ [T , S] ∩ [T , J ]) ∪ (T ∩ [T , J ])

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⊆ (S ∩ [T , S] ∩ [S, J ]) ∪ J

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⊆ ([T , S] ∩ J ) ∪ J

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⊆ J.

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Thus I ≤S J , i.e. I is the lowest upper bound. Let us now denote I = S ∩S T . Again, we suppose that S ≤ T , so I = S ∩ ([S, T ] ∪ T ). The minimum of I is S, since S is the minimum of [S, T ] ∪ T and S ∈ S. Since I ⊆ S and I ⊆ [S, T ] ∪ T , hence I ≤ S and I ≤ T . Now,

30

I ∩ [S, I ] = I ∩ [I , I ] = I = S ∩ ([S, T ] ∪ T ) ⊆ S

31 32 33 34 35 36 37 38 39 40 41 42

and

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I ∩ [T , I ] = S ∩ ([S, T ] ∪ T ) ∩ [T , I ] = (S ∩ [S, T ] ∩ [T , I ]) ∪ (S ∩ T ∩ [T , I ]) ⊆ {T } ∪ T = T, so I is a lower bound of S and T . Let J ∈ C([0, 1]) such that J ≤S S and J ≤S T . Therefore, J ≤ S = I . Before proving the condition on the maxima, let us show that J ∩ [I , J ] ⊆ I , with I = S. Since J ≤S S, J ∩ [S, J ] ⊆ S. We claim that J ∩ [S, J ] ⊆ [S, T ] ∪ T , and consequently, J ∩ [S, T ] ⊆ S ∩ ([S, T ] ∪ T ) = I . We distinguish three cases:

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Finally, let j ∈ J . If j ≤ I then j ≤ I . Otherwise, j ∈ J ∩ [I , J ] ⊆ I , so j ≤ I . Thus J ≤ I . This proves that I is the greatest lower bound, and finishes the proof. 2

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• If J < S, then J ∩ [S, J ] = ∅ ⊆ [S, T ] ∪ T . • If S ≤ J ≤ T , then J ∩ [S, J ] ⊆ [S, T ] ⊆ [S, T ] ∪ T . • If T ≤ J , then J ∩ [S, J ] = (J ∩ [S, T ]) ∪ (J ∩ [T , J ]) ⊆ [S, T ] ∪ T , since J ≤S T .

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Definition 16. A Closed-Valued Fuzzy Set (CVFS) A on X is a mapping A : X → C([0, 1]). Then the set of all closed-valued fuzzy sets over X is CVFS(X) = Hom(X, C([0, 1])).

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By (2), CVFS(X) becomes a partially ordered set with the pointwise partial order inherited from ≤S . Also, by (6), it is a lattice by means of the operators derived from ∪S and ∩S . For consistency with the notation established in the former sections, let us denote by ≤C the partial order ≤S when restricting to the class of closed sets. Then we mean by C the restriction of S to the class CVFS(X). Analogously, we shall use the symbols ∪C and ∩C instead of ∪S and ∩S , respectively. Then, for each A, B ∈ CVFS(X), we define the operators C and C as: (A C B)(x) = A(x) ∪C B(x) and (A C B)(x) = A(x) ∩C B(x),

11

for any x ∈ X.

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(FS(X), ∪F , ∩F )

(36)

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(IVFS(X), I , I )

i

(CVFS(X), C , C )

is a chain of lattice embeddings, where i is lifted from the identification of a closed interval as a nonempty closed set, and  is the one defined in (15).

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Proof. The operators C and C equals I and I , respectively, when restricting to IVFSs. Clearly, this is a consequence of the same property for the corresponding operators on nonempty closed sets and closed intervals in [0, 1]. Indeed, let [a, b] and [c, d] be closed intervals with, for instance, a ≤ c. Then [a, b] ∪C [c, d] = ([a, b] ∩ [c, b]) ∪ [c, d] = [c, b] ∪ [c, d] = [c, b ∨ d] = [a, b] ∪I [c, d]. Similarly, one may prove that [a, b] ∩C [c, d] = [a, b] ∩I [c, d]. Thus, the statement is a consequence of the results of this section. 2

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Remark 19. We may also substitute the map  by any of the lattice embeddings defined in Section 3. Their compositions with i yield a family of lattice extensions of Zadeh’s union and intersection.

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Although this solution to [8, Remark 2] is not complete, it covers most of the practical applications of HFSs. As commented in [20], these only make use of nonempty finite sets, i.e. the so-called Typical Hesitant Fuzzy Sets (THFSs) [3,4]. Clearly, THFSs are CVFSs. Additionally, the approach by CVFSs allows to work with different philosophies of understanding the membership degrees. For instance, we may include in the same framework THFSs and IVFSs. One could handle a CVFS whose membership degrees are given by closed intervals for some elements, whilst, for others, they are given by finite sets. An interesting problem is to find a negation on CVFSs that extends any of the ones considered in [2] for IVFSs. A different approach to this problem may be inspired from [24]. There, a negation is proposed for type-2 fuzzy sets, although the lattice structure is lost.

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6. Type-2 Fuzzy Sets

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Theorem 18. The sequence of maps

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Proof. It follows directly from Theorem 15.

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Corollary 17. The operators C and C endow the class CVFS(X) with the lattice structure associated to the partial order C .

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We finish the paper analyzing how to place the former notions into the class of Type-2 Fuzzy Sets (T2FSs). This type of fuzzy sets is defined by Zadeh in [27]. Actually, there it is defined something more general, the class of type-n

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fuzzy sets. Nevertheless, here we focus on n = 2, since, as far as our knowledge goes, no practical application has been developed for n > 2. A T2FS is a fuzzy set in which the membership degrees are given by FSs on [0, 1]. Formally speaking,

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by the isomorphism (18). Observe that, looking at the right part of the isomorphism, a T2FS can be interpreted as a time-varying membership degree assignation, i.e. for any element x ∈ X and time t ∈ [0, 1], we assign a membership degree A(x, t) ∈ [0, 1].

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A way of viewing SVFSs inside T2FSs may be obtained by making use of the obvious lattice inclusion 2 → [0, 1], which maps 0 → 0 and 1 → 1. This can be lifted to

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μ : Hom([0, 1], 2) → Hom([0, 1], [0, 1]),

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30

by means of the functor Hom([0, 1], −), which maps a set S ⊆ [0, 1] to its characteristic function. Hence, applying the functor Hom(X, −), we get μ : SVFS(X) → T2FS(X) defined by  1 if t ∈ A(x), μ(A)(x)(t) = (38) 0 otherwise,

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for each A ∈ SVFS(X) and any x ∈ X. Now, any injection from FS(X) into SVFS(X) provides, by composition with μ, an embedding from FS(X) into T2FS(X). For instance, the composition

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Definition 20. A type-2 fuzzy set A on X is a mapping A : X → [0, 1][0,1] , that is, the class of all T2FSs on X is   T2FS(X) = Hom X, [0, 1][0,1] ∼ = Hom(X × [0, 1], [0, 1]),

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where  is the map defined in (15), yields the embedding that sees FSs as T2FSs whose membership degrees are singleton maps. That is,  1 if t = A(x), (A)(x)(t) = (40) 0 otherwise, for each A ∈ FS(X) and any x ∈ X, see Fig. 3(a). We may also see FS(X) inside T2FS(X) through any lattice injection FS(X) → IVFS(X). For instance, consider : FS(X) → T2FS(X) defined by  1 if t ≤ A(x), (A)(x)(t) = (41) 0 otherwise,

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for any A ∈ FS(X) and x ∈ X. Observe that here we have used the map : FS(X) → IVFS(X) in (15), which is derived from the map γ in (14). So that we have the commutative diagram, derived from (24),

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Albeit any of those described in Section 3 could be taken, see Fig. 3(b) and (c). Observe that we may also substitute i : IVFS(X) → SVFS(X) by the map defined in (28). A different methodology can be followed by taking into account the philosophy of seeing a T2FS as a timevarying FS. The construction is dual to the ones developed in this paper. Fix a set map f : A → B. Then, any map h ∈ Hom(B, [0, 1]) yields a map h ◦ f ∈ Hom(A, [0, 1]). In this particular case, consider the projection to the first coordinate p1 : X × [0, 1] → X defined by (x, t) → x for any x ∈ X and t ∈ [0, 1]. Therefore, there is a map  : FS(X) → T2FS(X) defined as (A)(x)(t) = A(x)

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for each A ∈ FS(X) and any x ∈ X and t ∈ [0, 1], see Fig. 3(d).

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Consider now the lattice structures. As for SVFSs, some authors have treated the problem of endowing the class T2FS(X) with a lattice structure whose restriction to FS(X), via  in (40), becomes Zadeh’s union and intersection. In [9] and [15] it is proposed the lattice structure given by (A T 2 B)(x) = A(x) ∪F B(x) and (A T 2 B)(x) = A(x) ∩F B(x),

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(44)

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for each A, B ∈ T2FS(X) and x ∈ X. Nevertheless, although (T2FS(X), T 2 , T 2 ) is a complete lattice, as proved in [8, Proposition 4.1], Zadeh’s operators on FSs are not recovered, see [9], or [8, Section IV.B] for a simple counterexample. Since it can be proved that μ, defined in (38), is a lattice embedding, the underlying problem is that ι in (19) is not a lattice homomorphism. Hence, replacing ι by any other lattice embedding from (FS(X), ∪F , ∩F ) to (SVFS(X), , ), we may produce a family of lattice embeddings which preserve Zadeh’s operators. We have then proved the following result.

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(T2FS(x), T 2 , T 2 )

(45)

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(IVFS(X), I , I )

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Theorem 21. The composition of maps (FS(X), ∪F , ∩F )

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provides a family of lattice embeddings from (FS(X), ∪F , ∩F ) to (T2FS(X), T 2 , T 2 ), where  is any of the embeddings defined in Section 3, μ is defined in (38) and is defined in (28). Additionally, we may prove the following theorem. Theorem 22. The map  : (FS(X), ∪F , ∩F ) → (T2FS(X), T 2 , T 2 ), defined in (43), is a lattice embedding. Proof. It is clear that  is one-to-one. Now, ((A) ∪T 2 (B))(x)(t) = (A)(x)(t) ∨ (B)(x)(t) = A(x) ∨ B(x) = (A ∪F B)(x) = (A ∪F B)(x)(t),

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for any A, B ∈ FS(X), x ∈ X and t ∈ [0, 1]. Similarly, (A) ∩T 2 (B) = (A ∩F B). So  is a lattice homomorphism. 2

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for any t ∈ [0, 1]. The map δ is injective. Indeed, given two closed sets C and D, if δ(C) = δ(D), then t ∈ C if and only if t = δ(C)(t) = δ(D)(t) if and only if t ∈ D.

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Remark 23. The map δ is well-defined when applying to arbitrary nonempty subsets. However, in this case, it is no longer injective. For instance, the sets S = {1/n with n ≥ 1} and T = S ∪ {0} share the same image under δ.

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Proposition 24. The map δ : (C([0, 1]), ∪C , ∩C ) → ([0, 1][0,1] , ∪F , ∩F ) is a lattice embedding.

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We may provide a different solution. In Section 5 we treat the problem of seeing FSs as SVFSs by means of the so-called CVFSs. Here we give an embedding from CVFS(X) into T2FS(X). Consequently, its composition with any of the maps described in Remark 19 yields a lattice extension of FSs as T2FSs preserving Zadeh’s union and intersection. Let us define a map δ : C([0, 1]) → [0, 1][0,1] as follows. For any nonempty closed set C, ⎧ ⎪ ⎨ C if t ≤ C, δ(C)(t) = t if t > C and t ∈ C, (46) ⎪ ⎩ 0 if t > C and t ∈ / C,

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Proof. Let C and D be nonempty closed sets in [0, 1]. We prove first that δ(C ∪C D) = δ(C) ∪F δ(D). Without loss of generality, we may suppose that C ≤ D. Then C ∪C D = (C ∩ [D, C]) ∪ D and C ∪C D = D. Observe that

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= C ∪C D.

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= δ(C)(t) ∨ δ(D)(t)

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= (δ(C) ∪F δ(D))(t).

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Then, whenever D ≤ t ≤ C ∨ D, t ∈ C ∪ D if and only if t ∈ C ∪C D. Hence, for any t ∈ [0, 1], ⎧ ⎪ ⎨ D if t ≤ D, if D < t and t ∈ C ∪C D, δ(C ∪C D)(t) = t ⎪ ⎩ 0 if D < t and t ∈ / C ∪C D ⎧ ⎪ ⎨ D if t ≤ D, if D < t and t ∈ C ∪ D, = t ⎪ ⎩ 0 if D < t and t ∈ / C ∪D

The equality δ(C ∩C D) = δ(C) ∩F δ(D) can be proved similarly.

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Therefore, by applying (6), we get the injective lattice homomorphism  : (CVFS(X), C , C ) → (T2FS(X), T 2 , T 2 )

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defined pointwise from δ. Therefore, from Proposition 24, we obtain the following.

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Corollary 25. The composition of lattice homomorphisms (FS(X), ∪F , ∩F )

(T2FS(x), T 2 , T 2 )



(IVFS(X), I , I )

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(48)

 i

(CVFS(X), C , C ),

where  is any of the embeddings defined in Section 3, provides a family of lattice embeddings from (FS(X), ∪F , ∩F ) to (T2FS(X), T 2 , T 2 ).

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Fig. 4. Membership degrees of a fuzzy set A viewed as a type-2 fuzzy set through some of the compositions of Corollary 25.

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Remark 26. Let f : [0, 1] → [0, 1] be a strictly increasing map with f (0) = 0 and f (1) = 1. We may modify δ and define δf : C([0, 1]) → [0, 1][0,1] as, for any C ∈ C([0, 1]), ⎧ ⎪ ⎨ f (C) if t ≤ C, δf (C)(t) = f (t) if t > C and t ∈ C, (49) ⎪ ⎩0 if t > C and t ∈ / C, ([0, 1][0,1] , ∪

for any t ∈ [0, 1]. This is also an injective lattice homomorphism from (C([0, 1]), ∪C , ∩C ) to F , ∩F ). Therefore it also defines a lattice embedding f from (CVFS(X), C , C ) to (T2FS(X), T 2 , T 2 ). Thus, Corollary 25 also remains valid if we change  by f . In Fig. 4 we show examples of how a FS can be viewed as a T2FS by means of some of these embeddings.

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7. Conclusion In this paper we have analyzed the relationships, as lattices, of the following types of fuzzy sets: FSs, IVFSs, SVFSs and T2FSs. The use of the language of category theory has allowed us to understand the underlying mathematical problem of inserting FSs inside SVFSs and T2FSs. The solutions that we have shown are then in accord with this viewpoint. It seems more convenient, mathematically speaking, to exchange the way of seeing FSs as SVFS or T2FS rather than substituting the standard lattice structures. We think that this way of reasoning may be extrapolated to other models of handling uncertainty and imperfect information. On the other hand, the success of HFSs suggests that, in some particular situations, it may be convenient to modify the usual lattice structures. In this sense, we have adjusted the meet operator defined by Torra for HFSs and defined CVFSs. This generalization covers most of the applications of HFSs, since it may handle THFSs and IVFSs. Additionally, CVFSs over a fixed universe set become a lattice.

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Acknowledgements Gabriel Navarro has been partially supported by grant TIN2013-41990-R from the Ministerio de Economía y Competitividad of the Spanish Government and from Fondo Europeo de Desarrollo Regional FEDER.

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