Several methods have been developed for problems related to incomplete or inaccurate information. Examples of such methods are, for example, the theory of probability, fuzzy set theory, interval mathematics, and rough set theory. These can be considered mathematical methods for handling imperfect knowledge, and each have their own theoretical foundation and range of potential applications. However, it should be noted that even though the last three theories are quite different, they share several common features, and under certain conditions, we can transform theoretical results of one theory into results of the other. This process then leads to the emergence of new theories at the borders of these basic theories. This enables the use of diverse theoretical tools, thus increasing their application potential.
An example of such a boundary theory that does not fully fit in these basic areas is soft set theory. This is a rather recent approach for modeling uncertainty and was introduced by Molodtsov [1] in 1999, who established the fundamental results and proposed possible applications. Further extensions of the theoretical part and applications of soft sets subsequently appeared, for example, Maji et al. [2, 3], Majumdar and Samanta [4], Aktas and Cagman [5], Feng et al. [6], Chen et al. [7], and Mushrif et al. [8]. Both fuzzy set theory and soft set theory are concerned with problems that involve vagueness and uncertainties from different areas. Hence, the combination of these two approaches naturally led to the concept of fuzzy soft sets as a fuzzy generalization of soft sets. This was introduced by Maji et al. [9], where the basic properties of fuzzy soft sets were provided. Several studies have since appeared, extending the theoretical features of fuzzy soft sets (e.g., [10–13]).
Recently, the properties and applications of soft and fuzzy soft sets have attracted considerable attention. Xiao et al. [13] studied a synthetic evaluation method for business competitive capacity. Zou and Xiao [14] exploited the link between soft sets and data analysis in incomplete information systems. Pei and Miao [15] demonstrated that soft sets are a class of special information systems. Mushrif et al. [8] presented a new algorithm based on soft set theory for classification of natural textures. Kovkov et al. [16] considered optimization problems in the framework of soft set theory. Zou and Xiao [14] presented data analysis approaches for soft sets under incomplete information. Finally, Majumdar and Samanta [4] studied similarity measures for soft sets.
From this review, it follows that fuzzy soft set theory is a highly lively research area with great application potential. However, fuzzy soft set theory itself and its position in uncertainty theories have not been sufficiently investigated. Only a limited number of studies are concerned with the theoretical properties of fuzzy soft sets and their relation to, for example, classical fuzzy sets, or general categorical properties of fuzzy soft sets. Fuzzy set theory already has a large system of theoretical techniques that allow combining fuzzy sets with, for example, standard algebraic and topological methods; however, in the case of fuzzy soft sets, such techniques have not been systematically developed.
In fuzzy set theory, one of the key theoretical techniques that allows the application of standard algebraic and topological methods to fuzzy sets is powerset theory. Powerset theories are widely used in algebra, logic, topology, and computer science. The standard example of a powerset theory is based on the powerset object P(X) = {A : A ⊆ X} of a set X, and the corresponding (Zadeh’s) extension of a mapping f : X → Y to the map f_{P⃗} : P(X) → P(Y) is widely used in almost all branches of mathematics and their applications, including computer science. As classical set theory can be considered a special case of fuzzy set theory, powerset objects associated with fuzzy sets were soon investigated as generalizations of classical powerset objects. This was first carried out by Zadeh [17], who defined [0, 1]^{X} (or L^{X}, where L is an appropriate ordered structure) as a new powerset object Z(X) instead of P(X), and introduced the new powerset operator f_{Z⃗} : Z(X) → Z(Y). Zadeh’s extension was extensively studied by Rodabaugh [18] for lattice-valued fuzzy sets. in [18], the author provided a serious basis for further research on powerset objects and operators. This new approach to powerset theories was based on the application of the theory of monads in clone form, introduced by Manes [19]. A special example of monads in clone form was introduced by Rodabaugh [20] as a special structure describing powerset objects.
Instead of a monad in clone form, a more explicit powerset theory was introduced by Rodabaugh [20] as a structure describing powerset objects. A slight modification of this structure defined in a category
Unfortunately, such a powerset theory is not available for fuzzy soft sets. The existence of such a theory would allow not only the application of standard algebraic and topological techniques to this area, but also the precise definition of the relationships between classical fuzzy sets and fuzzy soft sets.
Accordingly, this study focuses on obtaining a fuzzy soft set powerset theory. Our objective is not only to provide an analog of classical powerset theory for fuzzy soft sets, as in the case of classical fuzzy sets, but also to handle the case of L-valued fuzzy soft sets for different lattices L. Hence, instead of the so-called fixed-basis fuzzy soft set powerset theory, where the value lattice L is fixed, we consider the so-called variable-basis fuzzy soft set powerset theory, where the value lattices can be different. To this end, we first prove that, as in the case of fixed-basis fuzzy set powerset theory, a variable-basis fuzzy set powerset theory exists and is defined by a suitable monad. Subsequently, we show that there exists a variable-basis fuzzy soft set powerset theory, which is a generalization of fuzzy set powerset theory, and we prove that it is also defined by a suitable monad. This allows us to use a series of constructs from fuzzy set theory that, by their very nature, require the existence of a powerset theory defined by a monad. To illustrate this, we use fuzzy soft set powerset theory and monads to define the theory of fuzzy soft relations and the composition of these relations.
All these results prove that fuzzy set powerset theory is a special case of fuzzy soft set powerset theory. However, we also prove that fuzzy soft set powerset theory can be understood as a special case of fuzzy set powerset theory, as instead of fixed-basis theories, we consider a variable-basis theory. Thereby, we obtain a detailed description of the relationships between the two theories.
Let U be an initial universe, and X be the set of all possible parameters, which are properties of objects in U. An L-valued fuzzy set in U is a mapping s : U → L, where L is a complete lattice, the largest and smallest elements of which are 1_{L} and 0_{L}, respectively. The category of complete lattices with complete lattice homomorphisms is denoted by . For , the set of all L-valued fuzzy sets in U is denoted by L^{X}. Instead of a lattice L, we occasionally consider a lattice L with a t-norm (denoted by ⊗) such that a ⊗ ⋁_{i}_{∈}_{I} a_{i} = ⋁_{i}_{∈}_{I} (a ⊗ a_{i}). An example of a complete lattice with a t-norm is a complete residuated lattice (see, e.g., [21]).
We also use the category CSLAT of complete⋁-semilatices with complete⋁-preserving homomorphisms. As S ∈ CSLAT is a complete semilattice, it is also a complete lattice. The only difference between the category of complete lattices and the category of complete ⋁-semilattices CSLAT is that the morphisms in the former preserve both complete suprema and infima, whereas in the latter, we require that the morphisms preserve only complete suprema.
We use the following definition of a fuzzy soft set, introduced in [9].
A pair (E, s) is called a fuzzy soft set in a space (X, U, L) if E ⊆ X and s : E → L^{U}.
The interpretation of individual sets in this definition implies that a fuzzy set s(e) ∈ L^{U}, where e ∈ E, represents a fuzzy evaluation of how individual objects in U correspond to the parameter e in E, that is, s(e)(u) ∈ L is an L-valued measure of how the object u ∈ U corresponds to parameter e ∈ E. We now present an example of an application of fuzzy soft sets in image processing.
(Color image segmentation as a fuzzy soft set)
Let (U, d) represent a finite metric space of pixels of a color image with metric d. For each pixel u ∈ U, we define a value S(u) representing the color of u. This value can be represented in different ways. For instance, we can use the so-called HSV representation, that is, a vector S(u) = [h, s, v], where h represents the hue of the color, s represents the saturation dimension, corresponding to various shades of brightly colored paint, and v represents the value dimension, corresponding to a mixture with varying amounts of black or white paint. Accordingly, the color image can be represented as a mapping S : U → E = {
However, we can use a different interpretation of S, particularly when we are interested in segmenting a given image because of its color. Let E_{S} = S(U) = {
On the set E of colors, we can define a fuzzy similarity relation σ : E × E → [0, 1] such that σ(
which defines how true it is that in the vicinity of pixel u, there are sufficiently many pixels with a color similar to
Therefore, from this perspective, the color segmentation of an image S can be identified with the fuzzy soft set (E_{S}, s) in a set E, or with the α-cut (E_{S}, s)(
As mentioned in Introduction, powerset theories are widely used in algebra, logic, topology, and computer science. Numerous studies have been concerned with Zadeh’s extension and its generalizations, which could be considered the first example of a powerset operator in fuzzy set theory. The theoretical justification of Zadeh’s extension principle was first presented by Rodabaugh [18].
As general powerset theories and monads are not commonly used in the community of readers interested in fuzzy set structures, we repeat the basic definitions of these notions. For more details on category theory, readers are referred to, for example, [23]. The following definition by Rodabaugh [18] introduces a CSLAT-powerset theory. If f : X → Y and g : Y → Z are morphisms in a category, g.f : X → Z denotes their composition. Morphisms in a category
Let
1) T :
2) For each morphism f : A → B in
3) V :
4) For each object X ∈
5) For each f : X → Y in
For simplicity, a CSLAT-powerset theory will be referred to as a powerset theory. A set T(A) is called a powerset object of X ∈
The powerset theory
1) The object function P :
2) For each mapping f : X → Y in
3) For each X ∈
The powerset theory
1) The object function Z :
2) For each mapping f : X → Y in
3) For each X ∈
These basic powerset theories
The following definition of a monad in clone form was introduced by Manes [19].
1) T̃ :
2) η is a system of
3) For each pair of
4) For every
5) ◊ is compatible with composition of morphisms of
It should be noted that if (T̃, ◊, η) is a monad in a category
Moreover, η represents identities on both sides for ◊, that is, for each A → T̃(B),
In this case, η : 1
Let us recall some simple properties of Kleisli composition.
Let
We now present well-known examples of monads in the category
The structure
1) For each object X ∈
2) For each object X ∈
3) For each f : X → P̃(Y), g : Y → P̃(Z), g◊f : X → P̃(Z) is defined by
Let L be a complete lattice with a t-norm ⊗. The structure
1) For each object X ∈
2) For each X ∈
3) For each f : X → Z̃(Y) and g : Y → Z̃(Q) in
As mentioned previously, some powerset theories have additional properties related to special properties of fuzzy sets. Examples of such theories are powerset theories in a category defined by monads. We present a general definition of such a powerset theory.
Let
where for an arbitrary
Let us consider the following simple example of a powerset theory defined by a monad.
The powerset theory
and the diagram from Definition 4 commutes. Hence,
It is also clear that the powerset theory
Herein, we show that, as classical fuzzy sets in a set X define Zadeh’s powerset theory with objects Z(X) in the category
Let the object function W : and the functor V : be defined by
where ≤ is the pointwise ordering, and for an arbitrary morphism (f, ϕ) : (X, L) → (Y, M), the powerset extension (f, ϕ)_{W⃗} : W(X, L) → W(Y, M) is defined by
Moreover, for (X, L), the map χ_{(}_{X}_{,}_{L}_{)} : X → W(X, L) is defined by
The following theorem ensuring the existence of variable-base fuzzy set powerset theory was proven by Rodabaugh [24].
Let be the category of complete lattices with complete lattice-preserving homomorphisms. Then,
As we showed in Example 6, the classical fixed-basis Zadeh’s power set theory
Let be the category of complete residuated lattices. Then, there exists a monad
Let the object function W̃ : be defined by
For (X, L), we set ν_{(}_{X}_{,}_{L}_{)} = (χ_{X}, id_{L}) : (X, L) → W̃(X, L), where χ_{X} is the characteristic function of subsets in X. Finally, for two morphisms (f, ϕ) : (X, L) → W̃(Y, M) and (g, ψ) : (Y, M) → W̃(Z, N), we set
By a simple computation, it can be proven that the operation Δ is associative, and that the following equalities hold:
by the identity g ◇ (χ_{Y}.f) = (g ◇ χ_{Y} ).f = g.f. Hence,
We now show that the powerset theory
where for an arbitrary morphism (f, ϕ) : (X, L) → (Y, M),
1) W(f, ϕ) := (f, ϕ)_{W→ },
2) W̃(f, ϕ) = (ν_{(}_{Y}_{,}_{M}_{)}.(f, ϕ) Δ 1_{W̃}_{(}_{X}_{,}_{L}_{)}.
We have VW̃(f, ϕ) = V(χ_{Y}.f ◇ 1_{W}_{(}_{X}_{,}_{L}_{)}, id_{L}) = χ_{Y}.f ◇ 1_{W}_{(}_{X}_{,}_{L}_{)}, and for arbitrary s ∈ W(X, L), y ∈ Y, we obtain
Therefore,
to study the properties of powerset objects of fuzzy soft sets, Definition 1 implies that we cannot define them for objects of the category , but we must use a “richer” category. Instead of the category , we consider some subcategories of . This reduction of the category involves two steps. In the first, we construct a fuzzy soft set powerset theory analogous to the classical fuzzy set powerset theory. In this step, instead of the category , we use the subcategory , where
This restriction of the category
Analogously, the relationships between fuzzy soft sets are derived from the morphisms between the corresponding objects of the basic spaces of the fuzzy soft sets, that is, from morphisms (f, g, ϕ) : (X, U, L) → (Y, V, M). A typical example of such a relationship between fuzzy soft sets may be a change in the value structures by which we evaluate individual objects, that is, instead of a lattice L we use another lattice M, but the evaluated objects remain unchanged. In this case, we use a morphism (1_{X}, 1_{U}, ϕ) from . We can proceed analogously if we change the criteria from the set X and replace them with other criteria. The morphism that we use in this case is (f, 1_{U}, ϕ). In all these cases, we use morphisms from the category .
Under these assumptions, we will show that, as in variable-basis fuzzy set powerset theory, fuzzy soft sets define a variable-base powerset theory, and the corresponding powerset theory is also defined by a monad.
In [11], the set of all L-valued fuzzy soft sets in a space (X, U) was introduced, and an extension of maps between two spaces (X, U) and (Y, V) to maps between the corresponding sets of fuzzy soft sets was defined. Additional properties of extended maps between powerset objects of fuzzy soft sets were also investigated in [10]. To compare the relationships between fuzzy sets and fuzzy soft sets in the variable-basis case, we should define sets of variable-basis powerset objects of objects from the category sets and extensions of the corresponding morphisms. The sets of variable-basis powerset objects can be defined by the object function T : , where
According to Definition 1, the elements (E, s) of the set T(X, U, L) are called fuzzy soft sets in a space (X, U, L). In the next theorem, we show that this object function is an object function of a CSLAT-powerset theory in the category .
Let be the category of complete lattices with complete lattice homomorphisms. The object function T can be extended to the CSLAT-powerset theory
Let (X, U, L) be an object in the category . In the set T(X, U, L), we define an order relation by
It is clear that T(X, U, L) is a complete ⋁-semilattice with respect to this ordering, where for a system {(E_{i}, s_{i}) : i ∈ I} ⊆ T(X, U, L), we have
Let R : be a forgetful functor defined by R(X, U, L) = X, R(f, g, ϕ) = f.
For an arbitrary object (X, U, L) and x, z ∈ X, we set
where
for all (E, s) ∈ T(X, U, L), y ∈ f(E), v ∈ V. As y ∈ f(E), we have f^{−1}(y) ∩ E ≠ ∅︀, and therefore (f, g, ϕ)_{T⃗} is well defined. It should be mentioned that the map (f, g,ϕ)_{T⃗} represents a variable-basis extension of a map introduced in [11]. As ϕ from a morphism (f, g, ϕ) is a complete lattice homomorphism, by a simple computation, we can prove that (f, g,ϕ)_{T⃗} is a morphism in the category CSLAT. For a morphism (f, g, ϕ) : (X, U, L) → (Y, V, M), let us consider the following diagram:
We prove that this diagram commutes. In fact, as g is a surjective map, for x ∈ X, v ∈ V, we have
Hence, ({f(x)}, η_{→}) = ({f(x)}, η_{{}_{f}_{(}_{x}_{)}}), and the diagram commutes. Therefore,
We prove that, as in a variable-basis fuzzy set powerset theory, the variable-basis fuzzy soft set powerset theory is defined by a monad. However, to prove this, we should consider the restriction of the powerset theory
Unlike the proof of this statement in the case of fuzzy sets, the proof for fuzzy soft sets is more complicated.
Let be the category of complete residuated lattices. There exists a monad
Let the object function T̃ : be defined by
where T(X, U, L) is identified with the underlying set of a complete lattice from Theorem 2. Let the morphism ξ_{(}_{X}_{,}_{U}_{,}_{L}_{)} be defined by
where η_{(}_{X}_{,}_{U}_{,}_{L}_{)} is from Theorem 2. For morphisms
we define the composition
where ◊ is a composition map that is defined as follows. If f : X → T(Y, V, M) is a map, then for x ∈ X, we use the notation
where
where
◊ is well defined because j is a bijection. We now prove that
We prove that the operation ⊙ is associative. We have
and we need only prove that h◊(g◊f) = (h◊g)◊f : X → T(R, Q, S). Let x ∈ X. Then, using relations
By a simple calculation, it can be proven that
We prove that ((h◊g)◊f)_{x} = (h◊(g◊f))_{x}. Using the notation
For
where
Analogously, we obtain
where
As j and m are bijective maps, we obtain A=B, (h◊(g◊f))_{x} = ((h◊g)◊f)_{x}, and the operation ◊ is associative. Thus, the operation ⊙ is associative.
We show that for arbitrary morphism (f, k, ϕ) : (X, U, L) → T̃(Y, V, M), the identity
holds. In fact, from the definition of η and relations
We show that for morphisms (f, k, ϕ):(X, U, L)→(Y, V, M) and (g, j, ψ) : (Y, V, M) → T̃(Z, W, M), the following identity holds:
In fact, we have
Hence, we should prove that g◊(η_{(}_{Y}_{,}_{V}_{,}_{M}_{)}.f) = g.f. For simplicity, instead of η_{(}_{Y}_{,}_{V}_{,}_{M}_{)}, we write only η. For x ∈ X, according to relations
Let
Furthermore, we have
Hence, identity
Finally, we prove that the monad
where for a morphism (f, k, ϕ) : (X, U, L) → (Y, V, M) in the category ,
It is clear that the diagram commutes for objects of the category. According to the definition of the powerset theory in Theorem 2, for (E, s) ∈ T(X, U, L), we have
According to relations
Therefore, for a ∈ f(E), v ∈ V, we obtain
Therefore, RT̃(f, k, ϕ)(E, s) = (f(E), s^{→}) = T(f, k, ϕ)(E, s), and the functor diagram commutes. Hence,
As expected, any L-valued fuzzy set (i.e., an element of W(X, L)) can also be considered a fuzzy soft set in the space (X, {*}, L), that is, an element of T(X, {*}, L). In fact, if s ∈ W(X, L), then s can be identified with (E_{s}, [s]) ∈ T(X, {*}, L), where E_{s} = {x ∈ X : s(x) ≠ 0_{L}} and [s] : E_{s} → L^{{*}} is defined by [s](x)(*) = s(x) ∈ L. Moreover, the embedding W(X, L) ↪ T(X, {*}, L) can be extended to a more precisely defined relationship between W and T, using the notion of a morphism between the powerset theories. The existence of a morphism between these powerset theories allows for a more precise understanding of the relationship between classical fuzzy sets and fuzzy soft sets.
The notion of a morphism of powerset theories is introduced in the following definition. We recall that the commutativity of the diagram
implies that g.u ≥ v.f, where D is an object with ordering ≤.
Let
1) G :
2) Φ_{X} : T(X) → R(G(X)) is a morphism in the category CSLAT.
3) Ψ_{X} : V (X) → U(G(X)) is a map in the category
4) For each morphism f : X → Y in
that is, G(f)^{⇒}.Φ_{X} ≥ Φ_{Y}.f_{T⃗}, where ≤ is the ordering in the CSLAT object R(G(Y)).
A morphism (G, Φ, Ψ) is called an embedding if R :
We now prove that there is an embedding morphism between the powerset theories
Let be the category of complete lattices. Then, there exists an embedding morphism of CSLAT-powerset pairs
Therefore, the variable-basis fuzzy set powerset theory is a special case of the variable-basis fuzzy soft set powerset theory.
We define the functor G : by
where 1_{*} : {*} → {*}. It is clear that G is an embedding functor. For , we set
It is clear that Φ_{(}_{X}_{,}_{L}_{)} is a complete ⋁-semilattice homomorphism. In fact, we have
where [s_{i}] is considered an L^{{*}}-valued fuzzy set in E_{⋁i}_{s}_{i}. We show that the following diagram commutes:
Let x ∈ X. We obtain
Hence, the diagram commutes. We show that for an arbitrary morphism (f,ϕ) : (X, L) → (Y, M), inequality holds in the diagram
Let s ∈ W(X, L). From the proof of Theorem 2, we have
Moreover, we have
As F ⊆ f(E_{s}), for f(x) ∈ F, we obtain
and (F, t^{→}) ⪯ (f(E_{s}), [s]^{→}). Therefore,
is a morphism of powerset pairs.
In Proposition 2, we showed that any fuzzy set can be considered a special fuzzy soft set. In the next proposition, we prove the converse: Any fuzzy soft set can be extended to a fuzzy set, and this extension process is a morphism from the powerset theory
To this end, we recall that for an arbitrary lattice L and a set X, L^{X} is also a lattice with the ordering defined pointwise and with properties analogous to those of the lattice L.
Let be the category of complete ⋁-semilattices with complete semilattice homomorphisms. Then, there exists a morphism of powerset pairs
called a transformation of a fuzzy soft sets to fuzzy sets.
For arbitrary morphism (f, g, ϕ) : (X, U, L) → (Y, V, M), let the functor H : be defined by
where (g, ϕ)_{W⃗} is a morphism from the definition of the variable-basis powerset theory
It can be easily proven that Λ_{(}_{X}_{,}_{U}_{,}_{L}_{)} is a complete ⋁-semilattice homomorphism, that is, Λ_{(}_{X}_{,}_{U}_{,}_{L}_{)}(⋁_{i}(E_{i}, s_{i})) = ⋁_{i} Λ_{(}_{X}_{,}_{U}_{,}_{L}_{)} (E_{i}, s_{i}) holds. We prove that the following diagram commutes:
For x, z ∈ X, u ∈ U,
Moreover,
and the diagram commutes. Finally, we prove that for any morphism (f, g, ϕ) : (X, U, L) → (Y, V, M), the following diagram commutes:
We recall that g is a bijection map. For (E, s) ∈ T(X, U, L) and y ∈ Y, v ∈ V, we have
Moreover, we have
Therefore, the diagram commutes, and (H, Λ, Θ) is a morphism of powerset pairs.
By a simple calculation we obtain the following corollary.
The composition
is the identity morphism of powerset pairs.
In set and fuzzy set theory, there are a number of concepts and methods that are, in fact, defined using powerset objects. To illustrate this, let us mention at least two typical powerset applications: relations and topology. If
1) a ∈ T(X), c(a) ≥ a,
2) a, b ∈ T(X), c(a ∨ b) = c(a) ∨ c(b),
3) a ∈ T(X), c(c(a)) = c(a).
The set {a ∈ T(X) : c(a) = a} represents the set of all closed elements in T(X). A morphism between co-topological spaces (X, c) and (Y, d) is a morphism f : X → Y in a category
If, for example, instead of a category
Herein, we focus on the relationship between powerset theory and relations, particularly between the fuzzy soft set powerset theory
To clarify this relationship, we first consider the general situation of the powerset objects in a category that is defined by a monad in this category.
Hence, let
Let
1) Objects of
2) For arbitrary objects A, B ∈
3) A composition of morphisms f : A ⇝ B, g : B ⇝ C is defined by g◊f.
If we understand the objects T̃(X) as an analog of the cloud of fuzzy objects over the object X, then the Klesli category in fact represents a category analogous to that of fuzzy morphisms between objects. Thus, this category is also used in areas other than the classical theory of fuzzy sets, for example, in computer science. This category also allows us to introduce the notion of a relation between objects of this category. Let us consider the following general definition.
Let
1) Let X be an object of
2) Let R and S be T-relations on X and Y, respectively. A morphism f : R → S between relations is a morphism f : X → Y in
we have
where ≤ is the pointwise ordering inherited from the semilattice T(X).
This definition directly implies the possibility of composing two T-relations. In fact, let R and S be T-relations on X. Then, there exist morphisms R̃ : X → T̃(X) and S̃ : X → T̃(X) such that V (R̃) = R, V (S̃) = S. The composition S ∘ R : V (X) → T(X) can be defined by the composition of morphisms R̃ and S̃ in the Kleisli category
The following examples illustrate the importance of T-relations in classical set and fuzzy set theories.
Let
that is, (x, y) ∈ R ⇒ (f(x), f(y)) ∈ S holds for arbitrary x, y ∈ X. As
Therefore, the category of P-relations is isomorphic to the classical category of sets with relations with the corresponding morphisms. This category is isomorphic to the Kleisli category
Let
It can be easily proven that f is a morphism if and only if
for each x, x′ ∈ X. As
Both Examples 7 and 8 indicate that the classical relations in sets or fuzzy relations can be fully replaced by morphisms in Kleisli categories. This now provides the opportunity to consider fuzzy soft relations defined on the basic objects of fuzzy soft sets from the perspective of Kleisli categories. Although it is common practice in fuzzy set theory to explicitly define new terms without justifying motivation, it is always advisable to attempt to interpret such an established concept in the context of a broader theory. An example of this procedure can be the aforementioned notion of relation in fuzzy structures. Using the standard procedure, we can independently define a fuzzy soft relation in a space (X, U, L), even in various ways. However, if we realize the previous relationship between relations in sets and the monad
If we use the common procedure in the theory of fuzzy sets, we can introduce, for example, the following explicit definition of a fuzzy soft relation in a space (X, U, L), a composition of such fuzzy soft relations and morphisms between fuzzy soft relations in different spaces.
Let (X, U, L) be an object in .
1) A fuzzy soft relation in (X, U, L) is a fuzzy soft set in the space (X × X, U, L).
2) If (E, R) and (F, S) are fuzzy soft relations in (X, U, L), their composition is a fuzzy soft relation defined by
3) Let (E, R) and (F, S) be fuzzy soft relations in (X, U, L) and (Y, V, M), respectively. Then, (f, g, ϕ) : (E, R) → (F, S) is a morphism of fuzzy soft relations if
(a) (f, g, ϕ) : (X, U, L) → (Y, V, M) is a morphism in the category
(b) For arbitrary (a, b)∈E and u∈U, we have (f(a), f(b)) ∈ F and
However, to maintain the analogy of the relationships between sets, fuzzy sets, and fuzzy soft sets, we should show that, as in the case of relations in sets or fuzzy relations in fuzzy sets, fuzzy soft relations can be derived from morphisms in the Kleisli category of the monad
Let
1) There exists a bijective map
2) If , then
where the clone composition ◊ is defined by relations
1) Let (E, S) be a fuzzy soft relation in defined in Definition 8. To define the map α, for arbitrary x ∈ X, we set
Conversely, let s : be a map, where, using the notation
By a simple calculation, we obtain α^{−1}.α(E, S) = (E, S) and α.α^{−1}(s) = s.
2) Let (E, Q), (F, S) be fuzzy soft relations in (X, U, L), and let
According to identities
It follows that
In this study, we focused on constructing a fuzzy soft set powerset theory, including its variable-basis variant, that enables the use of L-valued fuzzy soft sets for various lattices L. We proved that there exists a variable-basis fuzzy soft set powerset theory in the category , which is a generalization of both the classical and fuzzy set powerset theory in the category
This research was partially supported by the ERDF/ESF project CZ.02.1.01/0.0/0.0/17-049/0008414.
No potential conflict of interest relevant to this article was reported.
E-mail: mockor@osu.cz