Research Article Gradual Sets: An Approach to Fuzzy Sets

In the fuzzy theory of sets and groups, the use of α -levels is a standard to translate problems from the fuzzy to the crisp framework. Using strong α -levels, it is possible to establish a one to one correspondence which makes possible doubly, a gradual and a functorial treatment of the fuzzy theory. Te main result of this paper is to identify the class of fuzzy sets, respectively, fuzzy groups, with subcategories of the functorial categories Set (0, 1] , resp., Gr (0, 1] . In this line, the algebraic potential of this theory will be reached, in forthcoming papers.


Gradual Elements.
Let X be a set, every subset S ⊆ X is defned by its characteristic function χ S : . Tus, the concept of membership is exclusive. However, we can consider, in a wider environment, diferent degrees of membership: 1 means that the element belongs to the subset; 0 means that it does not belong, and any other real number 0 < α < 1 would mean a diferent degree of membership. Te theory, in these terms, is due to Zadeh (see [1]) who introduces a fuzzy subset of a given set X as a map μ: X ⟶ [0, 1]. From this primitive concept, we can develop a whole theory of sets, relations, maps, numbers, and so on.
In this approach to the fuzzy theory, we begin by relating various mathematical theories; this relationship is evident in the crisp framework, but which in the fuzzy theory presents, so far, some difculties.
Our approach to the fuzzy concept starts from the defnition of a fuzzy element: we adopt the defnition given by Dubois and Prade in [2] of a gradual element (if X is set, a gradual element in X is a partial map ϵ: (0, 1] ⟶ X); see also [3] in which gradual elements are applied to gradual numbers or [4] in which they are applied to build gradual intervals so that a gradual element of a set X is given by a collection of elements of X, each with a degree of membership, ranging in (0, 1]: there is always an element of the set X that has a degree of membership 1, and, possibly, other elements with other membership values, but never 0; that is, we do not determine any element of X that has zero degree of membership. Tis notion of gradual element has been extended to study several problems (see [5]). Let X be a set. A representation level (RL) of a fuzzy concept on X is a partial map ε: [0, 1] ⟶ P(X)) (see also [6,7]). Nevertheless, we have preferred to maintain the former one as when applying to sets, groups, and other structures, it defnes canonically a ground set, group, and so on which is an ambient object suitable for working.
For a greater fexibility in the defnition, we may assume that not all possible degrees are reached, so a gradual element is given by a partial mapping from (0, 1] to X: we will call it a partial gradual element. If ε is a partial gradual element with defnition domain dom(ε) ⊆ (0, 1], for its study, we need to relate partial gradual elements to each other. Te problem that arises is when two gradual elements ε 1 and ε 2 are equal? It is clear that we can only compare ε 1 and ε 2 where they are defned, that is, in dom(ε 1 ) ∩ dom(ε 2 ).
Tis defnition of equality is too weak. In fact, we are more interested in knowing if ε 1 and ε 2 take equal values in a range [α, 1], for some α ∈ (0, 1]. Taking into account that whenever β ∈ (0, 1] is very small, it is not relevant at all if that ε 1 (β) and ε 2 (β) are the same or diferent; we are more interested in knowing whenever ε 1 and ε 2 coincide for values of β close to 1.
Tus, we extend the equality relation to the case, previously indicated, of values in an interval [α, 1]. In this way, a relationship is obtained: However, this relation is not necessarily an equivalence relation because it depends heavily on dom(ε 1 ) ∩ dom(ε 2 ). So, if we want to compare partial gradual elements, we must standardize the defnition domain. In other words, we must, for instance, extend dom(ε) to the whole (0, 1].
We defne a total gradual element as a map ε: (0, 1] ⟶ X, among which we have the extended of the partial gradual elements; and denote by X the set of all total gradual elements of X. Observe that, when working with total gradual elements, the relation R α is an equivalence relation.

Gradual Subsets and Gradual
Subgroups. Te next step of complexity is to consider a binary operation * in the set X and extend it to gradual elements. Te standard method is to defne (ε 1 * ε 2 )(α) � ε 1 (α) * ε 2 (α) for any α ∈ (0, 1]. We have that if (X, * ) has a more complex algebraic structure, for example, if it is a group, a semilattice, the set X of all gradual elements can have the same property. However, this has not been the line we followed for the study of fuzzy structures in a set X, the reason is that when considering, for example, a ring structure in X, although X has a ring structure, this is of little interest, since it has too many zero-divisor elements.
We have chosen therefore to consider a greater degree of abstraction and consider, given a group (G, * ), not the set of elements of G, but the set S(G) of all the subgroups of G. We have an inclusion S(G) ⊆ P(G), in the powerset of G, and the elements of S(G) are the nonempty subsets P ∈ P(G) verifying: S * S ⊆ S and S − 1 ⊆ S. When considering gradual elements σ, σ 1 , σ 2 of P(G)\ ∅ { }, we have new gradual elements: σ 1 * σ 2 and σ − 1 , and naturally the notions of gradual subgroup and gradual subset appear.
A gradual subset of a set X is a gradual element σ of P(X), and a gradual subgroup of a group G is a gradual element σ of P(X)\ ∅ { } which is a subgroup, i.e, it will be a gradual element of S(G). Compare with the notion of RL (representation by levels) introduced in [5].
Observe that in these situations, we have solved the problem of extending partial gradual subsets or subgroups because we can defne the image of any element in (0, 1]\dom(σ) equals either ∅, for subsets, or e { }, for subgroups. Terefore, in Sections 3 and 4, we shall consider only total gradual subsets and subgroups.
Tis study will lead us to relate gradual subgroups with fuzzy subgroups and gradual groups with fuzzy groups, and the same process will allow to relate other structures: rings, modules, and so on.
In our exposition, we will try to establish a framework, for future developments, based on a categorical structure that allows to consider, not only gradual sets but also gradual algebraic and geometrical structures. Before carrying out this work, we have considered necessary to implement an indepth study that relates gradual and fuzzy sets and subsets.
In the set X of the gradual subsets of X, we defne a closure operator σ ↦ σ c � ∪ σ(β)|β ≥ α . A gradual subset σ will be a decreasing gradual subset if σ � σ c . And in the set J(X) of all decreasing gradual subsets of X, we defne an interior operator σ ↦ σ d � ∪ σ(β)|β > α ; a decreasing gradual subset σ will be a strict decreasing gradual subset whenever σ � σ d .
Associated to any fuzzy subset μ of X, we have a decreasing gradual subset σ(μ), defned σ(μ)(α) � μ α , the α -level of μ, for any α ∈ (0, 1], and a strict decreasing gradual subset σ(μ) � σ(μ) d , which is the strong α-level or strong α -cut, of μ. Te map μ ↦ σ(μ) does not preserve unions of infnite families, and the map μ ↦ σ(μ) does not preserve infnite intersections; hence, after modifying the intersection, we establish an injective correspondence, preserving union and intersection, from fuzzy subsets to strict decreasing gradual subsets, and fnd conditions on strict decreasing gradual subsets to be in the image of this map; that condition is property (inf-F). Which is important, in this situation, is that we have an isomorphism, for intersections and unions, between fuzzy subsets and strict decreasing gradual subsets satisfying property (inf-F). As a consequence, properties on fuzzy subsets can be studied via strict decreasing gradual subsets.
In addition, we consider a generalization of the theory of gradual subsets through the use of contravariant functors from the category (0, 1] to the category Set of sets which allow a functorial framework of both theories of gradual subsets and of fuzzy subsets.
Tis theory, frst developed in a context of sets, can be carried out to the more algebraic framework of groups, in which we may establish also a bijection between fuzzy subgroups and a specifc class of gradual subgroups and contravariant functors. In particular, this bijection will allow a functorial treatment of fuzzy groups.
Tis paper is organized in four sections. In the second one, we study and establish the general theory of gradual elements and introduce binary operations in the set of all gradual elements defned from binary operation in the ground set X. In particular, if we start from a group G, we get a structure of group in the set of gradual elements. Not in all cases, this structure refects the properties of G and its elements.
For this reason, to make an algebraic development later, in the third section, we study gradual subsets and operators in the set of gradual subsets that will allow to establish a close relationship, an isomorphism, between the set of fuzzy subsets and a set of gradual subsets. Tis study ends in Teorem 2 in which an isomorphism is established; observe that to obtain the isomorphism, we had to make use of the strict decreasing gradual subsets. To do that, frst we consider binary operations in P(X), the power set of X: the standard ones are the meet (intersection) and the join (union) and translate them to gradual subsets, which are noting more than gradual elements of P(X). In this section, we also identify a new type of objects through the use of contravariant functors from the category (0, 1] to the category of sets. Tese contravariant functors, which are identifed with directed systems, generalize gradual subsets and fuzzy subsets and allow a functorial framework of these two examples, which will provide a tool capable of dealing with other types of gradual and fuzzy objects such as groups, rings, and so on, and that will allow to work, by using direct limits, with gradual and fuzzy sets, instead of with gradual and fuzzy subsets.
Te fourth section is devoted to study the more complex example of gradual groups. After studying the diferent concepts related to group theory, we establish the most important result, Teorem 2, showing a bijection between equivalence classes of fuzzy subgroups and some specifc strict gradual subgroups. Tis gradual subgroups appear in a natural way after studying two operator on gradual subgroups, one a closure operator and another one an interior operator in the class of all decreasing gradual groups. Te formulation of the theory in terms of operators allows to develop a more abstract framework, in this case a functorial one, and hence obtain new properties and relationships between known objects.

Defnition of Gradual Elements
Defnition 1. Let X be a set, a total gradual element of X is a map ε: (0, 1] ⟶ X, and a partial gradual element of X is a map ε: L ⊆ (0, 1] ⟶ X, defned on a subset L ⊆ (0, 1] such that 1 ∈ L. For simplicity, depending of the context, we use gradual element to refer either to a total gradual element or to a partial gradual element. For any partial gradual element ε: L ⊆ (0, 1] ⟶ X, we call L the defnition domain of ε and represent it by dom(ε). We represent by X the set of all total gradual elements of X and by Xthe set of all partial gradual elements.
A gradual element ε ′ is an extension of the gradual el- Tere is a particularly useful method of extending a partial gradual element ε to a total gradual one; this is the case in which for any α ∈ (0, 1], there exists Min([α, 1] ∩ dom(ε)); then, we defne a new gradual element ε as follows: See Example 2 in which examples of extensions of partial gradual elements appear. Another example is provided whenever we consider the partial gradual element ε: In this case, an extension ε: { } is defned by ε(α) � a for any α ∈ (0, 1]; the constant map equals to a.

) Any union of fnitely many inf-compact subsets is inf-compact
In the following, the domain of any partial gradual element will be an inf-compact subset of (0, 1], containing 1; hence, any partial gradual element can be extended to a total gradual element. For any element a ∈ X, there exists a partial gradual element, which we represent by ε a , with dom(ε a ) � 1 { }, and defned by ε a (1) � a. We denote also by ε a the extension ε a . Without lost of generality, we may identify the element a ∈ X and the gradual element ε a ∈ X and denote them simply by a.
In this way, a gradual element is nothing more than a collection of elements of X, each one with a degree of membership; thus, if ε is a gradual element, then ε(α) is an element of X with membership degree α. Since ε(1) is always defned, we have it as an element of X with the highest membership degree; since ε(0) is not defned, then there is no any element with zero membership degree.

Relations between Gradual Elements.
For any α ∈ [0, 1], in the set of all partial gradual elements, we defne a relation R α as, for partial gradual elements ε 1 and ε 2 of X, we say Observe that if α ∈ (0, 1] and ε 1 , ε 2 are total gradual elements, then ε 1 R α ε 2  It is necessary to remark that these equivalence relations R α are not compatible with the extension process. Indeed, if ε 1 R α ε 2 , not necessarily ε 1 R α ε 2 as the following example shows.
Let f: X ⟶ Y be a map between two sets.
(1) For any total gradual (resp., partial gradual) element ε of X, we have a total gradual (resp., partial gradual) element of Y defned by the composition

Binary Operations and Gradual Elements.
Tere is another method to relate gradual elements of a set X; this is the case in which there exists a binary operation in X.
Let G be a set together a binary operation, say * , and ε 1 , ε 2 gradual elements of G, we defne a new gradual element ε 1 * ε 2 as In the case of partial gradual elements ε 1 , ε 2 , we have that Tis operation non necessarily is compatible with the extension construction.
Te following example shows that if we start from two partial gradual elements ε 1 and ε 2 , then not necessarily we have the equality: ε 1 * ε 2 � ε 1 * ε 2 , i.e., then extension map is not necessarily a homomorphism with respect to the binary operation * .
Example 2. Let ε 1 , ε 2 be partial gradual elements defned on Z, defned as: In this case, we have and the extended gradual elements are On the other hand, we have On the other hand, this operation is compatible with the equivalence relations R α .

Lemma 3.
Let G be a set together a binary operation * , for any α ∈ [0, 1] the relations R α in the set of all total gradual elements (resp., in the set of all partial gradual elements) are compatible with the binary operation, i.e., for gradual ele- In some cases, in which G has a richer structure, this structure could be inherited by the sets of gradual elements. Let us show an example. □ Lemma 4. Let G be a group, with binary operation * and neutral element e, the following statements hold: (1) Te set G of all total gradual elements is a group with neutral element e, i.e., the total gradual element ε e (2) For any α ∈ [0, 1], we have that G/R α is a group Proof. For any ε, ε 1 , ε 2 ∈ G, we defne, for any α ∈ (0, 1]: (1) In G, the operation is associative and e is the neutral element. For any ε ∈ G, we have ε − 1 the inverse of ε. Terefore, G is a group, which is abelian whenever G is. (2) It is a direct consequence of being R α a compatible equivalence relation.
□ Remark 1 (Te particular case of partial gradual elements). Let G be a group, in the set G of all partial gradual elements, we have an associative binary operation, , but we have "many" possible neutral elements. Tus, to get a useful structure, we must defne before an equivalence relation to put together all of them. For instance, given two partial gradual elements ε 1 , ε 2 , since ε 1 * ε 2 is defned on dom(ε 1 ) ∩ dom(ε 2 ), there are three possible neutral elements: e |dom(ε 1 ) , e |dom(ε 2 ) and fnally e |dom(ε 1 ) ∩ dom(ε 2 ) , which are diferent two to two. We can try to fx this problem in defning an equivalence relation R in G generated by With the relation R, the problem is that we may have dom(ε 1 ) ∩ dom(ε 2 ) � 1 { }, and this trivialize this relation. Hence, to obtain a well-defned structure on partial gradual elements, we may consider only special types of partial gradual elements, for instance, the subset of G constituted by those partial gradual elements who have the same (inf-compact) domain containing 1.
Tus, we can extend Lemma 4 to consider gradual elements defned on an inf-compact subset containing 1.

Proposition 1.
Let G be a group, and let C ⊆ (0, 1] be an infcompact subset containing 1. If G be the set of all partial gradual elements whose domain is C, the following statements hold: (1) For any α ∈ C, the relation R α is an equivalent relation in G (2) In G, we have an associative operation (3) Te extending map from G to G is a group monomorphism (4) For any α ∈ C, the equivalence relation R α in G is compatible (5) Te groups G and G /R α are abelian whenever G is Proof (1) It is refexive and symmetric, and obviously it is transitive as the domain is the whole set C.
It is clear that it is better to consider total gradual elements instead of partial gradual elements and therefore work in G.
If the group G has e as neutral element and for any α in [0, 1] we consider the equivalence relation R α , we may rewrite Lemma 3 obtaining a fltration of subgroups of G.
Ten, we have and surjective group homomorphisms: Observe that in all these examples, it seems that the way to defne an operation on gradual elements is to defne it componentwise.
If the base set X has a more richer structure, for instance, if it is a ring R, then the corresponding sets R and R are rings, but there are in these rings many elements which are zero-divisor. So, in this case, the use of gradual elements is not a good option. For that, in this and forthcoming papers, we shall develop a diferent approach to study algebraic structures. Before doing that, let us study the simplest notion of gradual subset, and after doing this we shall return to consider a set endowed with one or several binary operations, for instance, a group.

Gradual Subsets
Once we have established the notion of gradual element of a set X, we shall apply it to defne new objects. If we consider a set X and the power set P(X), we can study gradual elements of P(X), thereby the concept of gradual subset appears.

Defnition of Gradual Subsets
Defnition 2. Let X be a set, and let P(X) be the power set of X, i.e., P(X) � S|S is a subset of X { }. We defne a gradual subset of X as a gradual element of P(X). We represent by σ: (0, 1] → P(X)a gradual subset of X.
Troughout this section, we follow the same assumptions used for gradual elements in the previous section. In this way, we have defned partial gradual subsets and total gradual subsets.
In some sense, gradual subsets are a generalization of gradual elements. Tus, for any gradual element ε and any gradual subset σ, we say ε belongs to σ if for any In the same way, given two gradual subsets σ 1 , σ 2 , we say that σ 1 is a subset of In general, for any gradual subset σ and elements α, β ∈ dom(σ) such that α ≤ β, we have no information about the relationship of σ(α) and σ(β). In some cases, as in the classical one of α-levels in fuzzy set theory, there is an evident relationship, as we shall see later. To work with them, frst we introduce the following defnitions that refect toe order existing in (0, 1].
Let us show some examples of decreasing gradual subsets.

Operators on Gradual Subsets.
Te following are examples of constructions that can be carried out for any gradual subset, which will be useful in their study. Let σ be a gradual subset of X, associated to σ, we defne two new gradual subsets: It is clear that for any gradual subset σ, the accumulation σ c is a decreasing gradual subset, and a gradual subset σ is decreasing if and only if, σ � σ c . For any gradual subset σ, It is clear that for any gradual subset σ, the strict accumulation σ d is a decreasing gradual subset and σ d ⊆ σ c . In general, σ ⊈ σ d .
Tus, we have an operator, c, on gradual subsets: σ ↦ σ c . Te behaviour of c is refected in the following lemma.
Tis means that the operator c is a closure operator in the set X of all gradual subsets of X.
Remember that a closure operator in a poset (partial ordered set) P is a map c: P ⟶ P satisfying (1) p ≤ c(p) for any p ∈ P (2) For any p 1 , p 2 ∈ P such that p 1 ≤ p 2 we have c(p 1 ) ≤ c(p 2 ) (3) c(p) � cc(p) for any p ∈ P Te elements p ∈ P such that c(p) � p are named the c -closed elements. Tus, the gradual subsets which are closed for the operator c are the decreasing gradual subsets. Let us denote by J(X) the set of all decreasing gradual subsets of X.
In the same way, we may consider the operator d, defned as σ ↦ σ d ; its behaviour is refected in the following lemma. Lemma 6. Let X be a set, for any gradual subsets σ 1 , σ 2 , σ of X, the following statements hold: Proof. (1) and (2) are easy. (3) Indeed, for any α ∈ (0, 1], we have In the same way, for any α ∈ (0, 1], we have A gradual subset σ is an strict decreasing gradual subset if σ � σ d . We have □ Lemma 7. For any gradual subset σ, the following statements hold: (1) σ d is the smallest strict decreasing gradual subset contained in σ c (2) σ is a decreasing gradual subset non strict decreasing if, and only if, σ d ⊊σ c Proof. Let τ be a strict decreasing gradual subset such that Tis means that the operator d is an interior operator in the set of all decreasing gradual subsets of X.
Remember that an interior operator in a poset P is a map d: P ⟶ P satisfying (1) d(p) ≤ p for any p ∈ P (2) For any p 1 , p 2 ∈ P such that p 1 ≤ p 2 , we have □ Remark 2. Inspired in these constructions, we consider a new construction of a gradual subset from a partial gradual Advances in Fuzzy Systems subset that allows us to avoid the initial restriction of infcompact in the domain of defnition of partial gradual elements.
Let σ: (0, 1] → P(X) be a partial map defned at 1, i.e., 1 ∈ dom(σ) ⊆ (0, 1], and such that dom(σ) is not necessarily inf-compact, we may extend σ to all (0, 1] simply defning σ(β) � ∅ if β ∉ dom(σ). Te decreasing gradual subset associated to σis σ c : (0, 1] → P(X) defned as Te use of decreasing gradual subsets is due to the fact that gradual subsets are wild structures that one can not be managed, in which there is no relationship between its components. On the other hand, when studying subsets of a given set, it seems natural to impose some inclusion relationships and that these inclusions should be parameterized by the order relation in (0, 1].

Remark 3.
Observe that we may extend any gradual subset σ on X to σ on the whole interval In consequence, we may consider also decreasing gradual subsets as maps from [0, 1] to X.
In the set P(X), there are two operations: the intersection and the union; thus, we can translate these two operations to gradual subsets as did in the frst section. Following this line, we defne, for any gradual subsets, σ 1 and σ 2 : In this way, we may consider the algebra of gradual subsets of a given set X with respect to intersection and union.
Te defnition of intersection and union can also be extended to arbitrary families of gradual subsets. Let σ i |i ∈ I be a family of gradual subsets.
Let μ i |i ∈ I be a family of fuzzy subsets of a set X, the union, ∨ i μ i , and the intersection, ∧ i μ i , are the fuzzy subsets defned by { } be a set, for any n ∈ N\ 0, 1 In the same line, we have a similar situation for σ and the intersection.
In the setJ d (X) of all strict decreasing gradual subsets of X, we defne two new operations: intersection: ∧ i σ i � ( ∩ i σ i ) d , and maintain the old union: ∨ i σ i � ∩ i σ i , for every family σ i |i ∈ I of strict decreasing gradual subsets of X. With these defnitions, we have the following.

Advances in Fuzzy Systems
Proof. For any α ∈ (0, 1], we have In the same way, we can prove the case of σ( ∧ i μ i )(α). From this point of view, strict α-levels should be a suitable tool for studying the algebra of fuzzy subsets via decreasing gradual subsets. □ 3.4. Maps. In order to relate two gradual subsets, a standard method consists in defning a map from one to the other. In this context, frst we consider a map between the underlying sets containing each gradual subset; the following result show how to associate gradual subsets to gradual subsets via a map.

Lemma 8 (Direct image). Let f: X ⟶ Y be a map and denote by f the induced map from P(X) to P(Y), the following statements hold:
(1) For every gradual element ε of X, we have that fε: dom(ε) ⟶ Y is a gradual element of Y. In addition, if ε ∈ σ, then fε ∈ fσ.
Since every element of X and every element of Y are gradual elements and the same for gradual subsets, the notions of injective map and surjective map applied either to gradual elements or gradual subsets are equivalent. In the case of gradual subsets, we have Our aim will be to establish maps between gradual sets instead of between gradual subsets, i.e., leave out the ground set X and use only the subsets σ(α)|α ∈ (0, 1] { }. However, we postpone it until the point in which we change the paradigm introducing these gradual sets.
An enriched gradual subset of X is a gradual subset σ together with a family of maps f α,β |α, β ∈ (0, 1], α ≤ β satisfying Observe that, as a consequence of (3), each f α,β is an injective map. In particular, enriched gradual subsets are just the decreasing gradual subsets. See also Remark 7.

Gradual Quotient Sets.
Te same technique we used to introduce gradual subsets can be applied to defne quotient gradual sets of a given set X.
Remember that if X is a set, a subset S ⊆ X is an equivalence class in the class of all injective maps (i, Y)|i: Y { ⟶ X injective}, whenever we consider the equivalence relation: Dually, a quotient set of X is an equivalence class in the class of all surjective maps (Z, p)|p: X ⟶ Z surjective when we consider the equivalence relation: Te set of all subsets of X is represented by P(X), and there exists a bijective correspondence between P(X) and 2 X . Te set of all quotient set of X will be represented by Q(X), and for any element Z ∈ Q(X), we have

Advances in Fuzzy Systems
(2) An equivalence relation R p in X defned as xR p y if p(x) � p(y) (3) A partition of X into the equivalence classes defned by a relation R.
Each equivalence relation R in X is a subset of X × X satisfying the properties refexive, symmetric and transitive. Hence, Q(X) is in bijection with a subset of P(X × X). If we call Q(X × X) this subset, it is constituted by all the equivalence relations in X.
A gradual quotient set of X is a gradual element of Q(X) or equivalently of Q(X × X). We represent by ρ a gradual quotient set of X.

Gradual Subsets and Fuzzy Subsets.
As an example of application of the gradual subset theory, let us establish a correspondence between fuzzy subsets and enriched gradual subsets. As we had shown before, see Proposition 3; if we consider the strict decreasing gradual subset σ(μ), the correspondence μ ↦ σ(μ) is a homomorphism with respect to arbitrary union and intersection.
Lemma 11. Let σ be a decreasing gradual subset, not strict decreasing gradual subset, the following statements are equivalent: Proof. By the hypothesis, we have σ d ⊊σ c � σ.
(1) As a consequence of this result, for any decreasing nonstrict decreasing gradual subset, satisfying property (F), we have Max α|x ∈ σ(α) for any x ∈ X (2) In the case of a strict decreasing gradual subset satisfying property (inf-F), we have that it also satisfes property (F); hence the following equalities hold: for any x ∈ X Now, we are going to establish correspondences between fuzzy subsets and strict decreasing gradual subsets, which preserves union and intersection. Te following result, for fnite unions, is well known.
Te behaviour with respect to infnite unions can be solved using only strict decreasing gradual subsets instead of decreasing gradual subsets.

Advances in Fuzzy Systems
Proof. We had already studied the map ] in Proposition 3. Te map υ is well defned due to Teorem 2. Now, we check that the compositions are the identity.
Let μ be a fuzzy subset of X, for any x ∈ X, we have On the other hand, let σ be a strict decreasing gradual subset, and α ∈ (0, 1], we have and we have the other inclusion. As a consequence, we have the fnal result that establishes an isomorphism between the two lattices. See Proposition 3. □ Corollary 1. Let X be a set, there is an isomorphism between the lattice of all fuzzy subset of X and the lattice of all strict decreasing gradual subset of X, satisfying property (F), and they preserve arbitrary unions and intersections.

A Functorial Interpretation.
Let us consider (0, 1] as a category whose objects are the elements of (0, 1] and homomorphisms: only one, f α,β , from α to β whenever α ≤ β and the obvious composition.
Let F: (0, 1] ⟶ Set a contravariant functor from the category (0, 1] to the category of sets. Indeed, Let us remember the defnition of the direct limit, D � lim ⟶ F. First, we consider the disjoint union, ∪ be the inclusion, for any β ∈ (0, 1], and q β � pi β the composition. Te pair (D, p α |α ∈ (0, 1] ) satisfes the corresponding universal property of the direct limit.
In the particular case in which every map F(f α,β ) is injective, then every q α is also injective; this means that we can consider every F(α) as a subset of D. Terefore, we have that F defnes a decreasing gradual subset of D � lim ⟶ F, or more generally, a decreasing gradual subset of any overset of D and represent it by (F, lim ⟶ F). Tis allows to give an interpretation of decreasing gradual subsets in terms of contravariant functors. If we start from a decreasing gradual subset of a set X, then σ(α)|α ∈ (0, 1] { }, together with the family of inclusions, is a directed system, and if j α : σ(α) ⟶ X, for every α ∈ (0, 1], is the inclusion, then we have a commutative diagram and an inclusion h: D ⟶ X, from the direct limit D to X, being h(D) � Im(h) the union of the family of the subsets σ(α)|α ∈ (0, 1] { }. Taking into account this construction, we may show that a decreasing gradual subset σ of a set X is nothing more than a contravariant functor F: (0, 1] ⟶ Set of this shape, together with an injective map D � lim ⟶ F ⟶ X. As a consequence, we may consider contravariant functors as the central element in the study of decreasing gradual subsets. Hence, we may defne a gradual set as a contravariant functor F: (0, 1] ⟶ Set and a decreasing gradual set as a Advances in Fuzzy Systems gradual set such that each map F(f α,β ), whenever α ≤ β, is injective.
Given two gradual sets F 1 and F 2 , a map from F 1 to F 2 is just a natural transformation θ: F 1 ⟶ F 2 , i.e., a set of maps θ α |α ∈ (0, 1] such that each diagram commutes, whenever α ≤ β. Te contravariant functors from (0, 1] to Set, i.e., the gradual sets, constitute a category that we shall denote by Set (0,1] . Te class of all decreasing gradual sets defnes a full subcategory of Set (0,1] , and it is closed under (fnite and infnite) unions and intersections. Let us call J this subcategory.
In the subcategory J, we defne an interior operator, d: F ↦ F d , as follows: and if α ≤ β, then there is an inclusion functor A decreasing gradual set F is an strict decreasing gradual set whenever F � F d and satisfes property ( where the union is taken in D. By the relationship between (inf-F) and (F) properties, we may defne a fuzzy set as a strict decreasing gradual set satisfying property (inf-F). In particular, strict decreasing gradual sets satisfying property (inf-F) constitute a full subcategory of J.
As a consequence, we have the following result.

Theorem 3. Let F be a gradual set.
(1) Te following statements are equivalent: Remark 6. Te use of decreasing gradual sets allows to avoid the use of decreasing gradual subsets. Indeed, a decreasing gradual subset is, in some sense, more natural: we can build the category of decreasing gradual sets as a subcategory of the functor category Set (0,1] . Otherwise, decreasing gradual subsets are referenced to a set, the same does not happen with decreasing gradual sets; although, as we have the direct limit, the direct system itself acts as a real set. With fuzzy subsets, we have the same situation. Observe that in the fuzzy situation, when we consider the directed systems and the direct limit, we are considering only those elements with a positive, nonzero, membership degree, i.e., we do not consider those with zero membership degree. See also Remark 4.

Remark 7.
In looking for an abstract model for gradual subsets of a set X, our frst candidate was the functor category Set (0,1] . However, unfortunately, with this category, we do not obtain faithful representation of all gradual subsets. One may consider the gradual subset σ of a nonempty set . Obviously, we can not obtain σ using contravariant functors from the category (0, 1]: the reason is that there are no maps from X to ∅ (it is not an enriched gradual subset). Tis have been overcome when we consider enriched gradual sets. Tis model works perfectly and meets all our expectations whenever we consider decreasing gradual subsets.

Remark 8.
Observe that in our construction, we have fxed the categories Set and (0, 1] and considered contravariant functor. If we change contravariant for covariant, we get increasing gradual sets. On the other hand, the category Set has some peculiarities: one is that there ∅ which is an initial and not a fnal object; the other is that there are objects A and B such that Hom Set (A, B) � ∅; these force the use of increasing or decreasing gradual sets to assure writing the theory in a functor language using the usual order relation in (0, 1]. Some of these restrictions will be removed once we change the category Set for another category as Gr (the category of groups) or Mod − A (the category of right A-modules).

Gradual Subgroups
In Section 2.3, we have studied gradual subsets of P(X) for any set X and considered in P(X) the operations: intersections and union. We can repeat the same procedure whenever we have a binary operation in X and translate it into P(X), or a subset of P(X), in the natural way. Tus, our aim in this section is to study gradual subsets of a given set X, together with an additional algebraic structure in X; to do that we shall consider the simplest example of groups.

Gradual Subgroups.
Let X be a nonempty set with a binary operation * , we defne in P(X)\ ∅ { } new binary and unary operations by Tus, we may defne an operation on gradual subsets of X (for simplicity, in this section, for a set X, a gradual subset of X is a gradual element of P(X)\ ∅ { } ) by , for every σ 1 , σ 2 and α ∈ (0, 1], Defnition 3. Let G be a group (we eliminate the symbol * and represent the product just as juxtaposition); a gradual subgroup of G is a gradual subset σ of G, satisfying Let e be the neutral element of a group G and σ a gradual subgroup, the following statements hold: (1) e ∈ σ(α) for any α ∈ (0, 1] (2) σ(α) is a subgroup of G for any α ∈ (0, 1]

Terefore, if S(G) is the set of all subgroups of G, a gradual subgroup of G is just a gradual element of S(G).
Proof If ε is a gradual element of a group G, for any α ∈ (0, 1], we defne 〈ε〉(α) � 〈ε(α)〉, the gradual subgroup of G generated by ε. A gradual subgroup σ of G is cyclic if there exists a gradual element ε such that σ � 〈ε〉.

Normal Gradual Subgroups.
By the aforementioned identifcation, the study of gradual subgroups is very simple. Tus, a gradual subgroup of a group G is normal if for any α ∈ (0, 1], we have that σ(α) ⊆ G is a normal subgroup of G. If σ is a normal gradual subgroup of G, for every α ∈ (0, 1], we have a quotient group G/σ(α).
We defne a gradual quotient group of G a gradual quotient set η of G such that for every α ∈ (0, 1], the projection p(α): G ⟶ η(α) is a group homomorphism.
We shall change the assignation defned by ] to consider ]: [μ] ↦ σ(μ), in which σ(μ) is a strict decreasing gradual subgroup satisfying property (inf-F) and is defned by Tus, we have the following theorem.
(42) □ 4.5. Normal Fuzzy Subgroups. A fuzzy subgroup μ of a group G is normal if μη � ημ for any fuzzy subset η or equivalently if μ(xy) � μ(yx) for any x, y ∈ G (see [9]). We are interested in relating normal fuzzy subgroups and normal gradual subgroups. We have defned a gradual subgroup σ to be normal if σ(α) ⊆ G is a normal subgroup for any α ∈ (0, 1].
For any contravariant functor F: (0, 1] → Gr and every α ≤ β, we have now a group homomorphism from F(β) to F(α), and the pair ( F(α)|α ∈ (0, 1] { }, F(f α,β )| α ≤ β ) is a direct systems of groups and group homomorphisms; hence, there exists its direct limit, say lim ⟶ F. We defne a gradual group as a contravariant functor F: (0, 1] → Gr, and a gradual group homomorphisms from F 1 to F 2 is just a natural transformation from F 1 to F 2 . Terefore, we can consider the category of gradual groups and gradual group homomorphisms, which we denote by G.
An example of such a gradual group is provided by any decreasing gradual subgroup σ of a group G. In this case, the direct limit lim ⟶ σ is isomorphic to a subgroup of G; indeed, it is the union ∪ σ(α)|α ∈ (0, 1] { }. Following this example, for any arbitrary gradual group F, we say F is a decreasing gradual group whenever each F(f α,β ), for α ≤ β. Te class of all decreasing gradual groups is denoted by J. To well understand the structure of decreasing gradual groups, we build an operator (an endofunctor) d in J, defned on objects as follows: for any F ∈ J, we defne F d (α) � lim ⟶ (α,1] F(c), for every α ∈ (0, 1]. We collect these results in the following proposition, whose proof, after the theory developed in Section 3, is straightforward. Proposition 9. Let F be a decreasing gradual group and θ: F 1 ⟶ F 2 be a decreasing gradual map. Te following statements hold: (1) F d is a decreasing gradual group (2) d is an endofunctor of the full subcategory J of G (3) d is an interior operator in J.
A strict decreasing gradual group is a decreasing gradual group F such that F � F d .
At this point, it is convenient to remark that we have gradual groups and gradual subgroups. Contrary to decreasing gradual subgroups, that need of an ambient or a ground group, decreasing gradual groups have it included: it is the direct limit of the direct system that the gradual group defnes. Tis situation allows us to formulate a more attractive category theory of gradual objects which includes the usual constructions of the category of groups. In this context, decreasing gradual groups, strict decreasing gradual groups and fuzzy groups can be identifed with adequate subcategories; see the forthcoming paper [10], in which we study gradual and fuzzy modules over a ring.

Conclusion
Our goal in this article has been to introduce more general notions than the fuzzy subset in order to fnd a framework in which to develop a simpler theory that allows testing new techniques and establishing new results in fuzzy theory. In this sense, we start from the concept of gradual element with the goal of introducing gradual subsets. At this point, we establish a bijective correspondence between fuzzy subsets and a particular kind of gradual subsets (strictly decreasing gradual subsets), that satisfes property (inf-F). Te more interesting property of this correspondence is that it preserves arbitrary unions and intersections of fuzzy subsets.
In a second degree of abstraction, we consider a gradual subset as a contravariant functor from the category (0, 1] to the category of sets, which allows us to defne the notion of fuzzy and gradual sets without the use of an ambient set. Tus, we have three degrees of abstraction, the frst one corresponds to fuzzy subset; the second one to gradual subsets, identifying fuzzy subsets as some particular gradual subsets; and the third one to contravariant functors from (0, 1] to the category of sets, or directed systems of sets, identifying decreasing gradual subsets as those systems with injective maps. Observe that in each abstraction level, we have the objects studied in the previous one. We also establish the corresponding theory for groups in two diferent but compatible ways (1) defning contravariant functors to the category of groups; Gr (0,1] , and (2) defning groups in the functor category Set (0,1] .
One of the goals of this paper was to fnd a framework to study together the two crisp sets associated with each fuzzy set, and we have proven that groups and gradual groups allow it to do so. On the other hand, the use of category theory tools will allow to extend this working method to other structures, of which the sets and groups studied are only an example.

Data Availability
In this article, we include all the data that support this research. Te references included at the end of the article can be used to complement partial aspects of these data.

Disclosure
An older version has been submitted as arxiv in Cornell University according to the following link: https://export.arxiv.org/abs/1812.07521. Tis is an updated version of that manuscript.

Conflicts of Interest
Te authors declare that they have no conficts of interest.