Keywords: Fuzzy convexity; L-fuzzy sets; Hesitant fuzzy sets; Symmetric order; Scalar observables

1 Introduction↩︎

Convexity is one of the basic structural notions in several branches of mathematics, including optimization, analysis, geometry, and decision theory. In fuzzy set theory, convexity emerged naturally from the seminal work of Zadeh [1]. Several extensions of the concept have been proposed since then, in order to adapt it to richer models of uncertainty, such as interval-valued fuzzy sets (IVFSs) [2] or hesitant fuzzy sets (HFSs) [3]. However, many of these proposals have been formulated in broad terms, without explicitly accounting for the specific structural features of the fuzzy-set model under consideration. Thus, a natural question arises: from a structural perspective, are the existing notions of convexity equally appropriate for all fuzzy-set models?

A substantial part of the literature on fuzzy convexity has traditionally focused on scalar descriptions of it. For instance, in the classical setting of [1], min-based fuzzy convexity was introduced for comparing membership degrees directly in the unit interval through the minimum operator. [4] generalized this idea by introducing an operator-based viewpoint through the notion of \(F\)-convexity, which replaces the minimum with a general operator \(F:[0,1]^2\to[0,1]\). In turn, in the hesitant fuzzy setting, [5] introduced a scalar-based notion of convexity for typical hesitant fuzzy sets (THFSs), namely fuzzy sets whose membership degrees are finite nonempty subsets of \([0,1]\), commonly called typical hesitant fuzzy elements (THFEs) [6]. Their proposal relied on the arithmetic mean score function introduced by [7]. However, [8] showed that this proposal was not stable under intersections in general, and [9] addressed this limitation by proposing a general aggregation-based approach.

A different line of research has avoided the former preliminary scalarization by relying directly on order comparisons. In this approach, endpoint and intermediate values are compared through a suitable order on the codomain, rather than through a score or an aggregation function. Order-based convexities have been studied in particular for IVFSs [10] and for THFSs [11].

Thus, the existing literature reveals two complementary lines of development: a scalar or observable approach, based on numerical summaries, and an order-theoretic approach, based on direct comparison. One of the aims of this article is to place both viewpoints within a common framework that may help determine when scalar observables can faithfully preserve the intrinsic order-theoretic structure. This clarification is especially relevant in set-valued contexts, where membership degrees are nonempty subsets of \([0,1]\) rather than single numbers. In such settings, the difficulty of working with suitable lattice structures has often encouraged the use of scores and aggregation functions, even though these scalar summaries may conceal structurally relevant information.

To address this issue, we study convexity on lattice-valued mappings over point-convex segment-generated abstract convexity spaces. This setting allows us to separate the geometric structure of the domain from the order-theoretic structure of the codomain, while recovering as particular cases ordinary segment convexity in real vector spaces, interval convexity on linearly ordered universes, and more general non-linearly ordered betweenness structures. On the codomain side, taken throughout to be a lattice \(L\), we distinguish intrinsic convexity, defined through the lattice meet, from relational and observable convexities induced by preorders or scalar maps. We then determine when scalar observables preserve or reconstruct intrinsic convexity, and which codomain operators \(T:L^m\to L\) preserve it through their pointwise extensions on \(L^X\). These results show that preservation by scalar observables is governed by compatibility with finite meets as minima, exact scalar reconstruction requires order reflection, and preservation under pointwise induced operators depends on compatibility with the meet operation of the codomain lattice. The latter result follows the general framework of induced operators on lattices developed by [12] and [13].

The main structural development of the article is obtained by specializing the previous framework to the symmetric lattice of HFSs. The relevance of this choice is reinforced by [14], who identified as a foundational challenge the construction of HFS operations that recover the classical behaviour on standard fuzzy extensions while providing a lattice structure. In our setting, this structural perspective allows us to recover classical fuzzy and interval-valued convexities as natural restrictions of a common hesitant formulation. More importantly, by combining a reconstruction theorem for families of observables with an order-dimension obstruction, we show that the symmetric order contains finite subposets of arbitrarily large order dimension already within the class of two-point THFEs. Consequently, symmetric hesitant convexity cannot, in general, be globally reconstructed by any finite family of scalar observables. More concretely, every such family admits a three-point THFS profile that is scalar-convex for all selected descriptions but not symmetrically convex. This obstruction shows that the symmetric order cannot be globally recovered from finitely many scalar observables, in sharp contrast with chain-valued settings.

The article makes four main contributions. First, we formulate a lattice-theoretic framework for fuzzy convexity over point-convex segment-generated abstract convexity spaces, distinguishing an operational perspective based on codomain operators from a relational perspective based on preorders and scalar observables. This formulation provides a common perspective on several existing approaches to fuzzy convexity. Second, we establish the conditions under which scalar observables preserve or reconstruct intrinsic lattice convexity, and clarify its relation with lattice-order convexity. Third, we characterize the codomain operators whose pointwise extensions preserve intrinsic convexity. Finally, we specialize the framework to HFSs endowed with the symmetric lattice of [15] and derive the finite scalar non-representability results that constitute the central structural contribution of the article.

The structure of this article is as follows. Section 2 provides the necessary background and reviews the main convexity viewpoints introduced in the literature. Section 3 introduces the operational and relational convexity perspectives on \(L\)-fuzzy sets. Section 4 studies the bridges between intrinsic and scalarized convexity. Section 5 characterizes the pointwise operators that preserve intrinsic convexity. Section 6 develops the hesitant specialization through the symmetric lattice and establishes the finite scalar non-representability results. Finally, Section 7 concludes the article and outlines future research directions.

2 Preliminaries↩︎

Throughout this article, we use the following mathematical background. A partially ordered set, or poset, is a pair \((P,\leq)\), where \(\leq\) is reflexive, antisymmetric, and transitive. If antisymmetry is omitted, one obtains a preordered set. A preorder or order is called total if, for any \(a,b\in P\), either \(a\leq b\) or \(b\leq a\). Given a poset \((P,\leq)\), a subset \(U\subseteq P\) is called an up-set if \(a\in U\) and \(a\leq b\) imply \(b\in U\).

A lattice is a poset \((L,\leq)\) such that every pair of elements \(a,b\in L\) has an infimum \(a\wedge b\) and a supremum \(a\vee b\). A lattice is bounded if it has a least element \(0\) and a greatest element \(1\). It is complete if every subset \(S\subseteq L\) has both an infimum \(\bigwedge S\) and a supremum \(\bigvee S\). In particular, every complete lattice is bounded, since \(\bigvee\varnothing\) and \(\bigwedge\varnothing\) are, respectively, the least and greatest elements of \(L\).

In this article, we work inside the general framework of lattice-valued fuzzy sets. If \((L,\leq)\) is a lattice and \(X\) is a universe, we denote by \(L^X\) the class of all mappings from \(X\) to \(L\). A map \(A\in L^X\) is called an \(L\)-fuzzy set [16]. The pointwise order on \(L^X\) is defined by \[A\sqsubseteq B \iff A(x)\leq B(x)\;\text{for all }x\in X,\] and the pointwise lattice operations are given by \[(A\sqcap B)(x):=A(x)\wedge B(x), \qquad (A\sqcup B)(x):=A(x)\vee B(x), \qquad x\in X.\] For instance, taking \(L=[0,1]\), endowed with the usual order, one recovers classical fuzzy sets on \(X\) [1], i.e., a fuzzy set is a mapping \(A:X\to[0,1]\). Likewise, by choosing \(L=I([0,1])\), the family of closed intervals of \([0,1]\) endowed with the usual interval order, one obtains standard interval-valued fuzzy sets [2], [17]–[20].

2.1 Related work on fuzzy convexity↩︎

The development of fuzzy convexity can be traced back to the classical notion of convexity for crisp subsets of a linear space [21]. A subset \(C\) of a linear space \(X\) is convex if, for every \(x,y\in C\) and every \(\lambda\in[0,1]\), one has \(\lambda x+(1-\lambda)y\in C\). Given a fuzzy set \(A:X\to[0,1]\), one possible arithmetic way to transfer this idea is to impose the Jensen-type condition considered, for instance, by [22], \[\label{eq:jensen-fuzzy-convexity} A(\lambda x+(1-\lambda)y) \geq \lambda A(x)+(1-\lambda)A(y).\tag{1}\] Although natural, this formulation depends on the arithmetic structure of the codomain and is therefore not directly available in general lattice-valued settings.

The standard cut-compatible formulation in fuzzy convexity is the min-based condition introduced by [1] in the classical fuzzy setting, \[\label{eq:min-based-fuzzy-convexity} A(\lambda x+(1-\lambda)y) \geq \min\{A(x),A(y)\}.\tag{2}\] This definition is stable under intersections and is equivalent to the convexity of all \(\alpha\)-cuts, \[\label{eq:alpha-cuts-classical} [A]_\alpha=\{x\in X:A(x)\geq \alpha\}, \qquad \alpha\in(0,1].\tag{3}\] This cut-compatible viewpoint became one of the basic foundations of convex fuzzy-set theory; see, for instance, the systematic treatment of [23].

A direct lattice-valued counterpart of this min-based condition was considered by [24] in a Euclidean context. For a completely distributive lattice \(L\), an \(L\)-fuzzy set \(A:\mathbb{R}^n\to L\) is said to be \(L\)-fuzzy convex when \[\label{eq:maruyama-l-fuzzy-convexity} A(\lambda x+(1-\lambda)y) \geq A(x)\wedge A(y)\tag{4}\] for all \(x,y\in\mathbb{R}^n\) and every \(\lambda\in[0,1]\). Moreover, [24] showed that this condition is equivalent to the convexity of all lattice-valued level cuts \(\{x\in\mathbb{R}^n:a\leq A(x)\}, ~ a\in L.\) This places the classical cut-compatible viewpoint within a meet-based lattice-valued setting and anticipates the intrinsic convexity that we consider below.

An operator-oriented generalization of min-based convexity was introduced by [4] through the notion of \(F\)-convexity. Given a binary operator \(F:[0,1]^2\to[0,1]\), a fuzzy set \(A:X\to[0,1]\) is said to be \(F\)-convex when \[\label{eq:F-convexity} A(y)\geq F(A(x),A(z))\tag{5}\] for all \(x\leq y\leq z\), \(x,y,z\in X\). The usual min-based fuzzy convexity is recovered when \(F=\min\). This approach shifts the focus from a fixed convexity inequality to the choice of a codomain operator.

In hesitant fuzzy settings, the study of convexity has mainly focused on scalar summaries of typical hesitant information. [5], for instance, proposed a score-based approach for typical hesitant fuzzy sets (THFSs). We denote by \(F^*([0,1])\) the family of all nonempty finite subsets of \([0,1]\). The membership degrees of a THFS are elements of \(F^*([0,1])\), commonly called typical hesitant fuzzy elements (THFEs) [6]. Given a THFS \(A:X\to F^*([0,1])\), they used the arithmetic mean score function [7], [25] \[\label{eq:arithmetic-mean-score} s_A(x)=\frac{1}{|A(x)|}\sum_{\gamma\in A(x)}\gamma,\tag{6}\] and defined convexity by \[\label{eq:score-based-thfs-convexity} s_A(\lambda x+(1-\lambda)y) \geq \min\{s_A(x),s_A(y)\}.\tag{7}\]

This definition preserves a cut-based interpretation when the cuts are given by \([A]^s_\alpha=\{x\in X:s_A(x)\geq \alpha\}\). However, [8] showed that it is not stable under the usual intersection of THFSs. This loss of stability is relevant, since closure under intersections is a basic structural property of the classical fuzzy approach.

To address this drawback, [8] introduced a max-based notion of convexity for THFSs. Given a THFS \(A\), and \(A_{\max}(x):=\max A(x)\), the corresponding condition is given by \[\label{eq:max-based-thfs-convexity} A_{\max}(\lambda x+(1-\lambda)y) \geq \min\{A_{\max}(x),A_{\max}(y)\}.\tag{8}\]

The aggregation-based approach of [9] subsequently generalized this idea. Given an aggregation function \(\mathcal{S}\), a THFS A over a totally ordered universe is said to be \(\mathcal{S}\)-convex if \[\label{eq:aggregation-based-thfs-convexity} \mathcal{S}(A(y)) \geq \min\{\mathcal{S}(A(x)),\mathcal{S}(A(z))\}\tag{9}\] for every \(x\leq y\leq z\),  \(x,y,z\in X\). This notion recovers the arithmetic mean score approach of Rashid and Beg and the max-based approach as particular cases. However, all these formulations still reduce each typical hesitant value to a scalar quantity.

To avoid these preliminary reductions, some order-based approaches have also been proposed. These attempts have focused, in particular, on IVFSs [10] and on THFSs [11]. In these, convexity is imposed directly through a codomain order \(\preccurlyeq\). Thus, given a totally ordered universe \(X\), for an IVFS or a THFS \(A\), the condition is of the form \[\label{eq:order-based-convexity-literature} A(x)\preccurlyeq A(y) \quad\text{or}\quad A(z)\preccurlyeq A(y), \qquad x<y<z, \qquad x,y,z\in X.\tag{10}\] In other words, the intermediate value must dominate at least one endpoint value with respect to the chosen order. In this way, the comparison is carried out directly at the level of the codomain values, rather than after a scalar aggregation.

Most of the previous notions are formulated either on real vector spaces or on linearly ordered universes. Although these domains may be natural in certain concrete contexts, they become restrictive whenever the relevant notion of intermediate point is not encoded by linear combinations or by a total order, as occurs, for instance, in graph-like, geodesic, or more general betweenness structures. This is the reason why we work inside point-convex segment-generated abstract convexity spaces instead. This setting allows us to treat the classical segment \(\lambda x+(1-\lambda)y\), the order interval \([x,z]\), and other betweenness structures through a common segment mechanism.

A complementary structural perspective on lattice-valued convexity has been developed through the theory of \(L\)-convex spaces. In particular, [26] studied \(L\)-convex structures generated from convexity of level cuts. This viewpoint is related to the cut-based characterization of intrinsic convexity considered later in this article. Our focus, however, is different. Rather than constructing or classifying convex structures on \(L^X\), we fix a segment-generated abstract convexity on the domain and study concrete convexity notions for individual lattice-valued mappings from \(X\) to \(L\), together with their comparison, preservation, and reconstruction properties.

3 Two convexity perspectives on \(L\)-fuzzy sets↩︎

In this section, we introduce the two new convexity schemes. The purpose is to separate two ingredients that are often mixed in fuzzy convexity: the segment structure of the domain and the order-theoretic structure of the codomain. We begin by fixing the abstract notion of segment used on the domain.

Following [27], an abstract convexity space is a pair \((X,\Gamma)\), where \(X\) is a nonempty set and \(\Gamma\subseteq\mathcal{P}(X)\) is a family of subsets of \(X\), called convex sets, such that \(\varnothing,X\in\Gamma\), \(\Gamma\) is closed under arbitrary intersections, and \(\Gamma\) is closed under nested unions. For every \(S\subseteq X\), its convex hull is defined by \[\operatorname{co}_{\Gamma}(S):=\bigcap\{C\in\Gamma:S\subseteq C\}.\]

In this article, we restrict our attention to point-convex segment-generated abstract convexity spaces. More precisely, we assume that \(\operatorname{co}_{\Gamma}(\{x\})=\{x\}\) for every \(x\in X\), and that a subset \(C\subseteq X\) belongs to \(\Gamma\) if and only if \[x,z\in C \Longrightarrow \operatorname{co}_{\Gamma}(\{x,z\})\subseteq C.\] We write \[[x,z]_{\Gamma}:=\operatorname{co}_{\Gamma}(\{x,z\}).\] Thus, in a segment-generated setting, a subset \(C\subseteq X\) is convex if and only if \(x,z\in C\) imply \([x,z]_{\Gamma}\subseteq C\). This agrees with the interval-convexity viewpoint in betweenness structures [28], where convexity is determined by requiring that every interval between two points of a set remains inside the set. For instance, this setting includes the linearly ordered universes usually considered in the literature, by taking \([x,z]_{\Gamma}\) as the order interval between \(x\) and \(z\), and real vector spaces, by taking \([x,z]_{\Gamma}\) as the usual segment joining \(x\) and \(z\). It also allows non-linearly ordered domains, as illustrated below. From now on, \((X,\Gamma)\) will denote a point-convex segment-generated abstract convexity space.

3.1 Operational perspective: \((L,\otimes)\)-convexity↩︎

The first perspective of convexity is operational. This vision abstracts the role played by the codomain operator in min-based fuzzy convexity and in its operator-based generalizations.

In a lattice-valued context, the canonical operational choice is \(\otimes=\wedge\). This gives the intrinsic notion used throughout the article. Unlike scalar or aggregation-based approaches, this definition does not reduce the codomain value to a numerical observable.

3.1.1 Intrinsic lattice convexity↩︎

The following example shows that this definition is not restricted to linearly ordered domains.

We next record the cut characterization of intrinsic convexity. In the Euclidean setting, [24] showed that meet-based \(L\)-fuzzy convexity can be characterized through the convexity of all lattice-valued level cuts. Related cut-based analyses for lattice- and poset-valued convex mappings were later developed by [30]. The result below provides the corresponding characterization in our framework of point-convex segment-generated abstract convexity spaces.

3.2 Relational perspective: preorder-based convexity↩︎

The second perspective on convexity is relational. This viewpoint uses a comparison relation on the codomain, instead of a binary operator. As we show, this is the natural setting for order-based and scalar approaches to convexity.

This definition provides an order-theoretic analogue of the classical cut-based interpretation of fuzzy convexity in 3 and of the lattice-valued level-cut viewpoint in 4 . Moreover, when the domain convexity is induced by a linear order, it recovers the order-based condition in 10 . The following elementary characterization gives its local form.

If \((L,\preccurlyeq)\) is totally preordered, then \(\preccurlyeq\)-convexity can be written as \[\min\nolimits_{\preccurlyeq}\{A(x),A(z)\}\preccurlyeq A(y), \qquad y\in[x,z]_{\Gamma},\] where \(\min_{\preccurlyeq}\) denotes a minimal element, with respect to \(\preccurlyeq\), among \(A(x)\) and \(A(z)\).

A particular case is obtained by taking the preorder to be the lattice order.

A different specialization of Definition 4 arises when the comparison among codomain values is induced not by the lattice order itself, but by a scalar map into \([0,1]\).

3.2.1 Observable convexity↩︎

In the following paragraphs, we turn to the scalar case. Several notions of fuzzy convexity rely on numerical summaries of the codomain values. In the present framework, we regard all these maps as general observables. We adopt this term to make a clear distinction from particular constructions such as score or aggregation functions. Indeed, direct inspection shows, for instance, that every score function is an observable, whereas an observable need not satisfy the specific axioms usually required of scores [32].

The following characterization is the scalar analogue of Theorem 2.

Observable convexity is a particular case of preorder-based convexity. Indeed, every observable \(\varphi:L\to[0,1]\) induces a total preorder on \(L\), defined by \[a\preccurlyeq_{\varphi} b \Longleftrightarrow \varphi(a)\le \varphi(b).\] Since \([0,1]\) is totally ordered, \(\preccurlyeq_{\varphi}\) is always a total preorder.

The next example visualizes why observable convexity should not be confused with intrinsic lattice convexity.

In the next section, we study when such scalar descriptions preserve or reconstruct the intrinsic one.

Figure 5 summarizes the relationships between the two schemes introduced in this section.

4 Bridges between intrinsic and scalarized convexity↩︎

In the previous section we introduced two general convexity models on \(L\)-fuzzy sets. The operational perspective contains intrinsic lattice convexity as the distinguished case, while the relational perspective contains preorder-based and observable convexities. The purpose of this section is to compare these notions and to identify when a scalar or relational description is faithful to the intrinsic lattice structure.

In the converse statements below, we shall use the following non-degeneracy condition. We say that \((X,\Gamma)\) has a nontrivial segment if there exist \(p,r\in X\) and \(q\in[p,r]_{\Gamma}\setminus\{p,r\}\). This assumption only excludes degenerate domains with no genuine intermediate points. This condition is used to obtain necessity in the characterization results.

4.1 Intrinsic convexity versus observable convexity↩︎

We first study when intrinsically convex mappings remain convex after applying a scalar observable. The following result shows that this is possible only when the observable transforms lattice meets into minima.

Thus, scalarization preserves intrinsic convexity only under a meet-preservation condition. This result is particularly important because it already excludes many numerical summaries from being faithful to intrinsic convexity.

4.1.1 Reconstruction by one observable↩︎

We must note that preservation is not the same as reconstruction. We now characterize when a single observable not only preserves intrinsic convexity, but completely recovers it.

Theorem 6 gives an obstruction to scalar reconstruction. A single observable must preserve meets as minima and must also reflect the underlying lattice order. Since \([0,1]\) is totally ordered, the existence of an order-reflecting map \(\varphi:L\to[0,1]\) forces \(L\) itself to be totally ordered. Hence this reconstruction mechanism is unavailable in non-chain lattices.

The implication from the second statement in Theorem 6 to the first one remains valid without assuming the existence of a nontrivial segment.

Note that, by contrast, the converse implication in Theorem 6 requires the existence of a nontrivial segment. This assumption is necessary to derive the meet-minimum identity and the order-reflection property from exact scalar reconstruction.

4.1.2 Reconstruction by families of observables↩︎

When a single observable is not enough, one may consider a family of observables. In this case, we demonstrate that the relevant condition for the family is that it separates the order.

Observe that Theorem 7 provides the bridge between observable convexity and order representation. It shows that reconstructing intrinsic convexity by a family of scalar observables necessarily requires the same family to separate the codomain order. Hence, any obstruction to a finite scalar representation of the codomain order becomes an obstruction to a finite scalar reconstruction of intrinsic convexity.

As shown in Section 6, Corollary 2 provides the abstract criterion behind the main non-representability result of the article. Once the relevant codomain lattice contains finite subposets of arbitrarily large order dimension, no finite family of scalar observables can reconstruct the associated intrinsic convexity.

Despite the previous limitation, for finite domains, the convexity of a fixed \(L\)-fuzzy set admits a finite certificate. However, this does not represent the whole codomain order; it only provides a finite test for one profile.

4.2 Intrinsic convexity versus lattice-order convexity↩︎

We now compare intrinsic convexity with the particular relational notion obtained from the lattice order itself. This comparison clarifies why the intrinsic and relational viewpoints coincide in chains, but not in general lattices.

Thus, intrinsic convexity and lattice-order convexity coincide for every profile whenever the codomain lattice is totally ordered. Conversely, on domains with a nontrivial segment, this coincidence for all \(L\)-fuzzy sets characterizes totality of the codomain order. This explains why scalar or order-based formulations behave well in chain-like settings, but not necessarily in genuinely lattice-valued ones.

Combining this with Corollary 1, we obtain the scalar case in totally ordered codomains.

The following example illustrates the scalar reconstruction obtained in Corollary 4 for a totally ordered codomain.

5 Pointwise operators and preservation of intrinsic convexity↩︎

The previous section identified the order-theoretic conditions under which scalar observables preserve or reconstruct intrinsic convexity. We now address the complementary preservation problem of whether intrinsic convexity is stable under pointwise operators acting directly on the lattice-valued codomain. This question is relevant since, in several fuzzy contexts, profiles are often combined through aggregation, conjunction, disjunction, or other codomain operations. Related preservation problems for aggregation operators and fuzzy quasiconvexity were previously studied by [33]. Following the approach to induced operators on lattices developed in [12], [13], we characterize when pointwise combinations preserve intrinsic convexity in the present lattice-valued framework. The result below shows that preservation is governed by compatibility with the meet operation of the codomain lattice. In fact, this compatibility already forces monotonicity of the operator, while monotonicity alone is not sufficient. In any case, no completeness or distributivity assumption on \(L\) is required.

Thus, the preservation problem is controlled by the meet-compatibility of the codomain operator. The domain only provides the segments on which convexity is tested. In the classical fuzzy case \(L=[0,1]\), Theorem 10 says that an operator \(T:[0,1]^m\to[0,1]\) preserves min-based fuzzy convexity if and only if \[T(\min\{a_1,b_1\},\dots,\min\{a_m,b_m\}) = \min\{T(a_1,\dots,a_m),T(b_1,\dots,b_m)\}.\]

The condition is restrictive, but it is not vacuous. The following example shows that it is satisfied by operators other than the meet itself.

Nevertheless, it should be noted that monotonicity alone does not guarantee this condition, even for standard aggregation operators.

The example shows that pointwise aggregation may destroy intrinsic convexity even in the classical fuzzy case. Thus, operations that are natural from a numerical or aggregative viewpoint need not be faithful to the order-theoretic structure that determines intrinsic convexity.

6 Symmetric convexity on hesitant fuzzy sets↩︎

In this section, we specialize the previous framework to hesitant fuzzy sets. This specialization is the point at which the distinction between intrinsic and scalarized convexity becomes most relevant: the symmetric order on hesitant fuzzy elements carries order-theoretic information that cannot be represented globally by finitely many scalar observables.

For consistency, we use the following notation, which is standard in the literature [32], [34], [35]:

• \(P^*([0,1])\) denotes the family of all nonempty subsets of \([0,1]\);

• \(F^*([0,1])\) denotes the family of all nonempty finite subsets of \([0,1]\).

Elements of \(P^*([0,1])\) will be called hesitant fuzzy elements (HFEs), while elements of \(F^*([0,1])\) are typical hesitant fuzzy elements (THFEs) [6]. In particular, \(F^*([0,1])\subseteq P^*([0,1])\).

We work with the symmetric lattice \[\bigl(P^*([0,1]),\leq^0,\wedge_0,\vee_0\bigr)\] introduced in [15]. This lattice contains, as natural substructures, the usual fuzzy values and the interval-valued fuzzy values. Hence it allows us to treat classical fuzzy sets, interval-valued fuzzy sets, and HFSs within the same order-theoretic setting.

We recall that, for \(H,K\in P^*([0,1])\), one has \(H\leq^0 K\) if and only if \[H<K\setminus H \quad\text{and}\quad H\setminus K<K,\] where, for \(U,V\subseteq[0,1]\), \(U<V\) means that \(u<v\) for every \(u\in U\) and every \(v\in V\). As usual, this condition is interpreted vacuously when \(U=\varnothing\) or \(V=\varnothing\). Thus, if \(K\setminus H=\varnothing\), the first condition is automatically satisfied, and if \(H\setminus K=\varnothing\), the second one is automatically satisfied.

6.1 Symmetric convexity↩︎

We now specialize intrinsic lattice convexity to HFSs endowed with the symmetric lattice. Since this is the central convexity notion in the hesitant setting considered here, we refer to it as symmetric convexity.

In the next example, we illustrate this notion.

By Theorem 1, it is clear that symmetric convexity is equivalent to the convexity of all symmetric principal cuts. This gives a direct cut-based interpretation, in which the levels are hesitant values ordered by \(\leq^0\).

The next result records how the symmetric setting restricts to two familiar cases.

The next subsection shows that this finite scalar reconstruction does not extend to the genuinely hesitant regime.

6.2 Finite scalar non-representability of the symmetric order↩︎

The reconstruction results in Section 4 showed that scalar observables reconstruct intrinsic convexity only under strong order-separation and meet-compatibility requirements. We now show that, in the symmetric hesitant setting, such a reconstruction cannot be achieved globally by any finite family of scalar observables.

For \(n\ge 2\), let \(\mathcal{I}_n\) denote the set of all integer intervals \([i,j]\), with \(1\le i\le j\le n\), endowed with the order \[[i,j]\preceq_{\mathcal{I}}[k,\ell] \Longleftrightarrow [i,j]=[k,\ell]\;\text{or}\;j<k.\] This poset contains, as an induced subposet, the canonical interval order on the non-degenerate integer intervals \(1\le i<j\le n\).

Recall that an order-embedding of a poset \((P,\le_P)\) into a poset \((Q,\le_Q)\) is an injective map \(\Phi:P\to Q\) such that, for every \(x,y\in P\), \[x\le_P y \Longleftrightarrow \Phi(x)\le_Q\Phi(y).\]

In view of Corollary 2, it is enough to show that the symmetric lattice contains finite subposets of arbitrarily large order dimension. We do this by embedding canonical interval orders into \((F^*([0,1]),\leq^0)\). In fact, the construction below takes values in the fixed-cardinality subclass of two-point THFEs.

Recall that the order dimension \(\dim(P)\) of a poset \(P\) is the least cardinal number \(d\) such that \(P\) can be order-embedded into a product of \(d\) chains.

The previous impossibility result shows that symmetric convexity cannot be reconstructed by a finite family of scalar observables, even without assuming monotonicity. When the prescribed observables are monotone with respect to the symmetric order, this obstruction can be exhibited through an explicit three-point THFS profile.

Corollary 8 shows that the failure of finite scalar reconstruction can already be detected on a three-point domain and within the fixed-cardinality class of two-point THFEs. Hence the obstruction is not an artefact of infinite HFEs, large profiles, or varying hesitant cardinalities. This global obstruction should not be confused with the finite verifiability of a fixed profile. Indeed, if \(X=\{x_1,\ldots,x_n\}\) is finite, \((X,\Gamma)\) is a point-convex segment-generated abstract convexity space, and \(A:X\to P^*([0,1])\) is an HFS, then Theorem 8 shows that \(A\) is symmetrically convex if and only if, for every \[C\in\{A(x_i)\wedge_0 A(x_j):1\le i\le j\le n\},\] the symmetric principal cut \[[A]_C=\{x\in X:C\leq^0 A(x)\}\] is convex. Therefore, symmetric convexity of a finite HFS profile can always be verified through at most \(n(n+1)/2\) symmetric principal cuts, although no finite family of scalar observables represents the symmetric order globally.

7 Conclusions and future research↩︎

In this article we have developed a lattice-theoretic framework for convexity on \(L\)-fuzzy sets over point-convex segment-generated abstract convexity spaces. The framework separates the segment structure of the domain from the order-theoretic structure of the codomain. Thus, linearly ordered universes are recovered by taking segments as order intervals, while ordinary segment convexity in vector spaces and non-linearly ordered betweenness structures can be treated within the same formal setting. Within this framework, two complementary perspectives have been considered: an operational perspective, formulated through \((L,\otimes)\)-convexity, and a relational perspective, formulated through preorder-based convexity.

The operational perspective leads naturally to intrinsic lattice convexity, obtained when the codomain operator is the meet. This notion recovers classical min-based fuzzy convexity in 2 , the lattice-valued meet-based convexity considered by [24] in 4 , and the meet-based instance of \(F\)-convexity in 5 as particular cases. Inside this layer, we have characterized the codomain operators whose induced pointwise operators preserve intrinsic convexity. The resulting condition shows that preservation is governed by compatibility with the lattice meet; in particular, this compatibility already forces monotonicity, while monotonicity alone is not sufficient.

On the other hand, the relational perspective makes explicit the role of order comparisons and scalar observables. Preorder-based convexity was characterized locally, while observable convexity was shown to be a particular case induced by the total preorder associated with a numerical observable. The comparison between intrinsic and observable convexity gives the conditions under which scalar descriptions are faithful. In particular, we have shown that a single observable reconstructs intrinsic convexity only when it preserves finite meets as minima and reflects the underlying order, a requirement that is unavailable in non-chain lattices. More generally, we have demonstrated that the reconstruction by a family of observables requires both meet-compatibility and separation of the codomain order. However, for finite profiles, intrinsic convexity can still be checked through a finite family of principal cuts.

In the final part of the article, we have specialized the framework to HFSs. In this setting, symmetric convexity has been shown to recover the classical min-based fuzzy convexity on singleton-valued profiles and interval-valued convexity on interval-valued profiles. These two lower layers admit finite scalar reconstructions. Indeed, we have demonstrated that one observable suffices in the singleton-valued case, while the two endpoint observables reconstruct the interval-valued case. However, the genuinely hesitant regime behaves differently. By combining a reconstruction theorem for families of observables with an order-dimension obstruction, we have shown that the symmetric order cannot be globally represented by any finite family of scalar observables. This obstruction already appears within the class of two-point THFEs. Consequently, symmetric hesitant convexity cannot, in general, be reconstructed by a finite family of scalar observables. More concretely, every finite family of monotone scalar observables admits a three-point THFS profile that it classifies as convex although symmetric convexity fails. This obstruction may be relevant for the broader development of the theory of HFSs [14]. Indeed, it indicates that future research on this topic should take into account that finite scalar descriptions must be understood only as partial summaries, rather than as globally faithful substitutes for the underlying hesitant structure. This suggests that future work on HFSs should treat finite scalar summaries as partial descriptions rather than substitutes for the underlying order structure.

As an overview, in Table 1 we summarize how the main convexity viewpoints discussed throughout the article fit into the proposed framework and clarify the role played by each of them in the present study.

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The results raise several questions for further investigation. First, it would be interesting to further investigate codomain operators satisfying the meet-compatibility condition characterized here, especially for the concrete lattice structures that appear in fuzzy and hesitant fuzzy settings. This may clarify which natural aggregation, conjunction, or fusion mechanisms preserve intrinsic convexity and which ones necessarily destroy it. Another direction could be to relate the present framework with graded convexity structures, such as the general \((L,M)\)-fuzzy convex structures of [38] and the \(M\)-fuzzifying convexities induced by \(M\)-orders studied by [39]. It would also be useful to identify additional geometric assumptions on \((X,\Gamma)\) that lead to stronger preservation or reconstruction results. Finally, the symmetric hesitant setting suggests further questions, including more explicit procedures for handling symmetric principal cuts and the development of applications in decision-making models where preserving hesitant information is preferable to scalar aggregation.

Code availability↩︎

Not applicable.

Declaration of competing interest↩︎

The authors have no competing interests to declare that are relevant to the content of this article.

Acknowledgments↩︎

Raquel Fernandez-Peralta was funded by the EU NextGenerationEU through the Recovery and Resilience Plan for Slovakia under the project No. 09I03-03-V04-00557. Pedro Huidobro was funded by the Spanish Ministry of Science and Innovation through project MCINN-23-PID2022-139886NB-I00, by the Asturian Agency for Science, Business Competitiveness and Innovation (SEKUENS) under Grant Agreement No. SEK-25-GRU-GIC-24-018, and by project PID2024-155289NB-I00 funded by MICIU/AEI/10.13039/501100011033 and ERDF/EU.

References↩︎

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1. All authors contributed equally to this work. Corresponding author: carlosss@ugr.es.↩︎