# Gradual Sets: An Approach to Fuzzy Sets

Authors: Josefa M. García; Pascual Jara
Journal: Advances in Fuzzy Systems
 (2023)

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## Abstract

In the fuzzy theory of sets and groups, the use ofα-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. The 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.


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## Body

## 1. Introduction

### 1.1. Gradual Elements

LetX be a set, every subset S⊆X is defined by its characteristic function χS:X⟶0,1, which is defined by χSx=1,ifx∈S,0,ifx∉S.. Thus, the concept of membership is exclusive. However, we can consider, in a wider environment, different 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 different degree of membership.The 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 difficulties.Our approach to the fuzzy concept starts from the definition of a fuzzy element: we adopt the definition 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. This 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] ⟶ PX) (see also [6, 7]). Nevertheless, we have preferred to maintain the former one as when applying to sets, groups, and other structures, it defines canonically a ground set, group, and so on which is an ambient object suitable for working.For a greater flexibility in the definition, we may assume that not all possible degrees are reached, so a gradual element is given by a partial mapping from (0, 1] toX: we will call it a partial gradual element. If ε is a partial gradual element with definition domain domε⊆0,1, for its study, we need to relate partial gradual elements to each other. The problem that arises is when two gradual elements ε1 and ε2 are equal? It is clear that we can only compare ε1 and ε2 where they are defined, that is, in domε1∩domε2.This definition of equality is too weak. In fact, we are more interested in knowing ifε1 and ε2 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 ε1β and ε2β are the same or different; we are more interested in knowing whenever ε1 and ε2 coincide for values of β close to 1.Thus, we extend the equality relation to the case, previously indicated, of values in an intervalα,1. In this way, a relationship is obtained: ε1Rαε2 if(1)ε1|α,1∩domε1∩domε2=ε2|α,1∩domε1∩domε2.However, this relation is not necessarily an equivalence relation because it depends heavily ondomε1∩domε2. So, if we want to compare partial gradual elements, we must standardize the definition domain. In other words, we must, for instance, extend domε to the whole (0, 1].There is a standard method of doing this, consisting of, givenα∈domε such that ε is not defined in β,α, defining εγ=εα for all γ∈β,α. The condition that has seemed most efficient forces to restrict the partial gradual elements to those whose definition domain verifies that for every α∈0,1, there is a minimum ζ of α,1∩domε, to, in this way, extend ε to all (0, 1], defining εα=εζ. We have called inf–compact the subsets of (0, 1] containing 1 and verifying this property. In this way, every partial gradual element ε, with inf-compact definition domain, can be extended, in a unique way, to a gradual element ε¯ with definition domain (0, 1]. We call ε¯ the extended gradual element of ε.We define a total gradual element as a mapε: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 Rα is an equivalence relation.


### 1.2. Gradual Subsets and Gradual Subgroups

The next step of complexity is to consider a binary operation∗ in the set X and extend it to gradual elements. The standard method is to define ε1∗ε2α=ε1α∗ε2α for any α∈0,1.We have that ifX,∗ 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 groupG,∗, not the set of elements of G, but the set SG of all the subgroups of G. We have an inclusion SG⊆PG, in the powerset of G, and the elements of SG are the nonempty subsets P∈PG verifying: S∗S⊆S and S−1⊆S. When considering gradual elements σ,σ1,σ2 of PG∖∅, we have new gradual elements: σ1∗σ2 and σ−1, and naturally the notions of gradual subgroup and gradual subset appear.A gradual subset of a setX is a gradual element σ of PX, and a gradual subgroup of a group G is a gradual element σ of PX∖∅ which is a subgroup, i.e, it will be a gradual element of SG. 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 define the image of any element in0,1∖domσ equals either ∅, for subsets, or e, for subgroups. Therefore, in Sections 3 and 4, we shall consider only total gradual subsets and subgroups.This 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 in-depth study that relates gradual and fuzzy sets and subsets.In the setX of the gradual subsets of X, we define a closure operator σ↦σc=∪σβ|β≥α. A gradual subset σ will be a decreasing gradual subset if σ=σc. And in the set JX of all decreasing gradual subsets of X, we define an interior operator σ↦σd=∪σβ|β>α; a decreasing gradual subset σ will be a strict decreasing gradual subset whenever σ=σd.Associated to any fuzzy subsetμ of X, we have a decreasing gradual subset σμ, defined σμα=μα, the α–level of μ, for any α∈0,1, and a strict decreasing gradual subset σ˜μ=σμd, which is the strong α-level or strong α-cut, of μ. The map μ↦σμ does not preserve unions of infinite families, and the map μ↦σ˜μ does not preserve infinite intersections; hence, after modifying the intersection, we establish an injective correspondence, preserving union and intersection, from fuzzy subsets to strict decreasing gradual subsets, and find 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 categorySet of sets which allow a functorial framework of both theories of gradual subsets and of fuzzy subsets.This theory, first 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 specific class of gradual subgroups and contravariant functors. In particular, this bijection will allow a functorial treatment of fuzzy groups.This 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 defined from binary operation in the ground setX. 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 reflects 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. This study ends in Theorem 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, first we consider binary operations inPX, 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 PX. 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. These contravariant functors, which are identified 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.The fourth section is devoted to study the more complex example of gradual groups. After studying the different concepts related to group theory, we establish the most important result, Theorem2, showing a bijection between equivalence classes of fuzzy subgroups and some specific strict gradual subgroups. This 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. The 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.


## 1.1. Gradual Elements

LetX be a set, every subset S⊆X is defined by its characteristic function χS:X⟶0,1, which is defined by χSx=1,ifx∈S,0,ifx∉S.. Thus, the concept of membership is exclusive. However, we can consider, in a wider environment, different 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 different degree of membership.The 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 difficulties.Our approach to the fuzzy concept starts from the definition of a fuzzy element: we adopt the definition 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. This 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] ⟶ PX) (see also [6, 7]). Nevertheless, we have preferred to maintain the former one as when applying to sets, groups, and other structures, it defines canonically a ground set, group, and so on which is an ambient object suitable for working.For a greater flexibility in the definition, we may assume that not all possible degrees are reached, so a gradual element is given by a partial mapping from (0, 1] toX: we will call it a partial gradual element. If ε is a partial gradual element with definition domain domε⊆0,1, for its study, we need to relate partial gradual elements to each other. The problem that arises is when two gradual elements ε1 and ε2 are equal? It is clear that we can only compare ε1 and ε2 where they are defined, that is, in domε1∩domε2.This definition of equality is too weak. In fact, we are more interested in knowing ifε1 and ε2 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 ε1β and ε2β are the same or different; we are more interested in knowing whenever ε1 and ε2 coincide for values of β close to 1.Thus, we extend the equality relation to the case, previously indicated, of values in an intervalα,1. In this way, a relationship is obtained: ε1Rαε2 if(1)ε1|α,1∩domε1∩domε2=ε2|α,1∩domε1∩domε2.However, this relation is not necessarily an equivalence relation because it depends heavily ondomε1∩domε2. So, if we want to compare partial gradual elements, we must standardize the definition domain. In other words, we must, for instance, extend domε to the whole (0, 1].There is a standard method of doing this, consisting of, givenα∈domε such that ε is not defined in β,α, defining εγ=εα for all γ∈β,α. The condition that has seemed most efficient forces to restrict the partial gradual elements to those whose definition domain verifies that for every α∈0,1, there is a minimum ζ of α,1∩domε, to, in this way, extend ε to all (0, 1], defining εα=εζ. We have called inf–compact the subsets of (0, 1] containing 1 and verifying this property. In this way, every partial gradual element ε, with inf-compact definition domain, can be extended, in a unique way, to a gradual element ε¯ with definition domain (0, 1]. We call ε¯ the extended gradual element of ε.We define a total gradual element as a mapε: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 Rα is an equivalence relation.


## 1.2. Gradual Subsets and Gradual Subgroups

The next step of complexity is to consider a binary operation∗ in the set X and extend it to gradual elements. The standard method is to define ε1∗ε2α=ε1α∗ε2α for any α∈0,1.We have that ifX,∗ 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 groupG,∗, not the set of elements of G, but the set SG of all the subgroups of G. We have an inclusion SG⊆PG, in the powerset of G, and the elements of SG are the nonempty subsets P∈PG verifying: S∗S⊆S and S−1⊆S. When considering gradual elements σ,σ1,σ2 of PG∖∅, we have new gradual elements: σ1∗σ2 and σ−1, and naturally the notions of gradual subgroup and gradual subset appear.A gradual subset of a setX is a gradual element σ of PX, and a gradual subgroup of a group G is a gradual element σ of PX∖∅ which is a subgroup, i.e, it will be a gradual element of SG. 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 define the image of any element in0,1∖domσ equals either ∅, for subsets, or e, for subgroups. Therefore, in Sections 3 and 4, we shall consider only total gradual subsets and subgroups.This 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 in-depth study that relates gradual and fuzzy sets and subsets.In the setX of the gradual subsets of X, we define a closure operator σ↦σc=∪σβ|β≥α. A gradual subset σ will be a decreasing gradual subset if σ=σc. And in the set JX of all decreasing gradual subsets of X, we define an interior operator σ↦σd=∪σβ|β>α; a decreasing gradual subset σ will be a strict decreasing gradual subset whenever σ=σd.Associated to any fuzzy subsetμ of X, we have a decreasing gradual subset σμ, defined σμα=μα, the α–level of μ, for any α∈0,1, and a strict decreasing gradual subset σ˜μ=σμd, which is the strong α-level or strong α-cut, of μ. The map μ↦σμ does not preserve unions of infinite families, and the map μ↦σ˜μ does not preserve infinite intersections; hence, after modifying the intersection, we establish an injective correspondence, preserving union and intersection, from fuzzy subsets to strict decreasing gradual subsets, and find 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 categorySet of sets which allow a functorial framework of both theories of gradual subsets and of fuzzy subsets.This theory, first 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 specific class of gradual subgroups and contravariant functors. In particular, this bijection will allow a functorial treatment of fuzzy groups.This 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 defined from binary operation in the ground setX. 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 reflects 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. This study ends in Theorem 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, first we consider binary operations inPX, 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 PX. 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. These contravariant functors, which are identified 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.The fourth section is devoted to study the more complex example of gradual groups. After studying the different concepts related to group theory, we establish the most important result, Theorem2, showing a bijection between equivalence classes of fuzzy subgroups and some specific strict gradual subgroups. This 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. The 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.


## 2. Gradual Elements

### 2.1. Definition of Gradual Elements

Definition 1.
LetX 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, defined 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 elementε:L⊆0,1⟶X, we call L the definition domain of ε and represent it by domε. We represent by X the set of all total gradual elements of X and by X¯the set of all partial gradual elements.
A gradual elementε′ is an extension of the gradual element ε if ε|domε′=ε.
There is a particularly useful method of extending a partial gradual elementε to a total gradual one; this is the case in which for any α∈0,1, there exists Minα,1∩domε; then, we define a new gradual element ε¯ as follows:(2)ε¯α=εζ,beingζ=Minα,1∩domε.
See Example2 in which examples of extensions of partial gradual elements appear. Another example is provided whenever we consider the partial gradual element ε:1⟶a,b, defined by ε1=a. In this case, an extension ε¯:0,1⟶a,b is defined by ε¯α=a for any α∈0,1; the constant map equals to a.
A subsetL⊆0,1, containing 1, such that there exists Minα,1∩L, for any α∈0,1, is named an inf-compact subset of (0, 1].
The following are examples of inf-compact subsets of (0, 1]:(1)
Any compact subsetC⊆0,1, containing 1, is inf-compact. In particular, any finite subset and any closed (in ℝ) subset of (0, 1], containing 1, are inf-compact(2)
Any ascending sequence in (0, 1], union with1, is inf-compact(3)
Any intervala,b⊆0,1, union with 1, is inf-compact(4)
Any union of finitely 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.Lemma 1.
IfCi|i∈I is a family of inf-compact subsets, containing 1, then ∩iCi is inf-compact.Proof.
For anyα∈0,1, let ξ=Infα,1∩∩iCi, and define ξi=Minξ,1∩Ci; hence ξ≤ξi, for any i∈I. On the other hand, since α,1∩∩iCi=ξ,1∩∩iCi⊆ξ,1∩Ci, then Minξ,1∩Ci=Infξ,1∩Ci≤Infξ,1∩∩iCi=ξ. In consequence, ξ=ξi for any i∈I.
For any elementa∈X, there exists a partial gradual element, which we represent by εa, with domεa=1, and defined by εa1=a. We denote also by εa the extension εa¯. Without lost of generality, we may identify the element a∈X and the gradual element εa∈X and denote them simply by a.
In this way, a gradual element is nothing more than a collection of elements ofX, each one with a degree of membership; thus, if ε is a gradual element, then εα is an element of X with membership degree α. Since ε1 is always defined, we have it as an element of X with the highest membership degree; since ε0 is not defined, then there is no any element with zero membership degree.


### 2.2. Relations between Gradual Elements

For anyα∈0,1, in the set of all partial gradual elements, we define a relation Rα as, for partial gradual elements ε1 and ε2 of X, we say ε1Rαε2 if(3)ε1|α,1∩domε1∩domε2=ε2|α,1∩domε1∩domε2.Observe that ifα∈0,1 and ε1,ε2 are total gradual elements, then ε1Rαε2, whenever ε1|α,1=ε2|α,1.Lemma 2.
For anyα,β∈0,1, we have(1)
Ifα,β∈0,1 satisfies α≤β, then Rα⊆Rβ(2)
The relationRα is an equivalence relation in the set of all total gradual elementsThe equivalence relationRα indicates us when two gradual elements are equal at a certain level. For instance,(1)
Ifα=1, then we only have an equivalence class for each element of X(2)
Ifα=0, then two gradual elements belong to the same equivalence class if, and only if, they coincide in their definition domainsIt is necessary to remark that these equivalence relationsRα are not compatible with the extension process. Indeed, if ε1Rαε2, not necessarily ε¯1Rαε¯2 as the following example shows.Example 1.
LetX=a,b, and define(4)ε1δ=b,ifδ=1,a,if12≤δ<1,ε2δ=b,ifδ=1,a,if0<δ≤12.
Then,ε1Rαε2, for any α∈0,1, but ε¯1Rαε¯2 if and only if α=1.
Letf:X⟶Y be a map between two sets.(1)
For any total gradual (resp., partial gradual) elementε of X, we have a total gradual (resp., partial gradual) element of Y defined by the composition fε:domε⟶X⟶Y; we call fε the image of ε by f.(2)
For any gradual elementε′ of Y, a gradual element ε of X is an inverse image of ε′ if ε′=fε


### 2.3. Binary Operations and Gradual Elements

There is another method to relate gradual elements of a setX; this is the case in which there exists a binary operation in X.LetG be a set together a binary operation, say ∗, and ε1,ε2 gradual elements of G, we define a new gradual element ε1∗ε2 as(5)ε1∗ε2α=ε1α∗ε2α,for anyα∈domε1∩domε2.In the case of partial gradual elementsε1,ε2, we have that domε1∗ε2=domε1∩domε2.This operation non necessarily is compatible with the extension construction.The following example shows that if we start from two partial gradual elementsε1 and ε2, then not necessarily we have the equality: ε1∗ε2¯=ε¯1∗ε¯2, i.e., then extension map is not necessarily a homomorphism with respect to the binary operation ∗.Example 2.
Letε1,ε2 be partial gradual elements defined on ℤ, defined as:(6)ε1α=2,if12≤α≤1,1,if110≤α≤13,ε2α=2,if23≤α≤1.
In this case, we have(7)ε1+ε2α=4,if23≤α≤1.and the extended gradual elements are(8)ε1¯α=2,if13<α≤1,1,ifα≤13,ε2¯α=2,ifα≤1,ε1+ε2¯α=4,ifα≤1.
On the other hand, we have(9)ε1¯+ε2¯α=4,if13<α≤1,3,ifα≤13.
On the other hand, this operation is compatible with the equivalence relationsRα.Lemma 3.
LetG be a set together a binary operation ∗, for any α∈0,1 the relations Rα 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 elements ε,ε1,ε2 of G, if ε1Rαε2, then ε∗ε1Rαε∗ε2 and ε1∗εRαε2∗ε.Proof.
Letε1,ε2,ε be gradual elements such that ε1Rαε2, for any β∈α,1∩domε1∩domε2∩domε, we have(10)ε1∗εβ=ε1β∗εβ=ε2β∗εβ=ε2∗εβ.
In some cases, in whichG has a richer structure, this structure could be inherited by the sets of gradual elements. Let us show an example.Lemma 4.
LetG be a group, with binary operation ∗ and neutral element e, the following statements hold:(1)
The setG of all total gradual elements is a group with neutral element e, i.e., the total gradual element εe(2)
For anyα∈0,1, we have that G/Rα is a groupProof.
For anyε,ε1,ε2∈G, we define, for any α∈0,1:(11)ε1∗ε2α=ε1α∗ε2α,andε−1α=εα−1.(1)
InG, the operation is associative and e is the neutral element. For any ε∈G, we have ε−1 the inverse of ε. Therefore, G is a group, which is abelian whenever G is.(2)
It is a direct consequence of beingRα a compatible equivalence relation.Remark 1.
(The particular case of partial gradual elements). LetG be a group, in the set G¯ of all partial gradual elements, we have an associative binary operation, ε1∗ε2α=ε1α∗ε2α for any α∈domε1∩domε2, but we have “many” possible neutral elements. Thus, to get a useful structure, we must define before an equivalence relation to put together all of them. For instance, given two partial gradual elements ε1,ε2, since ε1∗ε2 is defined on domε1∩domε2, there are three possible neutral elements: e|domε1,e|domε2 and finally e|domε1∩domε2, which are different two to two.
We can try to fix this problem in defining an equivalence relationR in G¯ generated by(12)ε1Rε2,if there is an inf−compact subsetC⊆0,1containing1,such thatε1|C=ε2|C.
With the relationR, the problem is that we may have domε1∩domε2=1, and this trivialize this relation.
Hence, to obtain a well-defined structure on partial gradual elements, we may consider only special types of partial gradual elements, for instance, the subset ofG¯ constituted by those partial gradual elements who have the same (inf-compact) domain containing 1.
Thus, we can extend Lemma 4 to consider gradual elements defined on an inf-compact subset containing 1.Proposition 1.
LetG be a group, and let C⊆0,1 be an inf-compact subset containing 1. If G¯ be the set of all partial gradual elements whose domain is C, the following statements hold:(1)
For anyα∈C, the relation Rα is an equivalent relation in G¯(2)
InG¯, we have an associative operation(3)
The extending map fromG¯ to G is a group monomorphism(4)
For anyα∈C, the equivalence relation Rα in G¯ is compatible(5)
The groupsG¯ and G¯/Rα are abelian whenever G isProof.
(1)
It is reflexive and symmetric, and obviously it is transitive as the domain is the whole setC.(2)
It is evident as the product is defined componentwise.(3)
Letε1,ε2∈G¯, and ξ=Minα,1∩C, then(13)ε1∗ε2¯α=ε∗ε2ξ=ε1ξ∗ε2ξ=ε¯1α∗ε¯2α.(4)
It is similar to the proof on Lemma3.(5)
It is evident as the product is defined componentwise.
It is clear that it is better to consider total gradual elements instead of partial gradual elements and therefore work inG.
If the groupG has e as neutral element and for any α in [0, 1] we consider the equivalence relation Rα, we may rewrite Lemma 3 obtaining a filtration of subgroups of G.Proposition 2.
LetG be a group with neutral element e, if for any α∈0,1, we define(14)Gα=ε∈G|εRαe.Then, we have(1)
For eachα∈0,1, the subgroup Gα⊆G is a normal.(2)
There is a filtrationGα|α∈0,1 where Gα⊆Gβ if α≤β.(3)
We have inclusions:G0⊆Gα⊆G1⊆G, and surjective group homomorphisms:(15)GG0⟶GGα⟶GG1≅G.Observe that in all these examples, it seems that the way to define an operation on gradual elements is to define it componentwise.If the base setX 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 different 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.


## 2.1. Definition of Gradual Elements

Definition 1.
LetX 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, defined 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 elementε:L⊆0,1⟶X, we call L the definition domain of ε and represent it by domε. We represent by X the set of all total gradual elements of X and by X¯the set of all partial gradual elements.
A gradual elementε′ is an extension of the gradual element ε if ε|domε′=ε.
There is a particularly useful method of extending a partial gradual elementε to a total gradual one; this is the case in which for any α∈0,1, there exists Minα,1∩domε; then, we define a new gradual element ε¯ as follows:(2)ε¯α=εζ,beingζ=Minα,1∩domε.
See Example2 in which examples of extensions of partial gradual elements appear. Another example is provided whenever we consider the partial gradual element ε:1⟶a,b, defined by ε1=a. In this case, an extension ε¯:0,1⟶a,b is defined by ε¯α=a for any α∈0,1; the constant map equals to a.
A subsetL⊆0,1, containing 1, such that there exists Minα,1∩L, for any α∈0,1, is named an inf-compact subset of (0, 1].
The following are examples of inf-compact subsets of (0, 1]:(1)
Any compact subsetC⊆0,1, containing 1, is inf-compact. In particular, any finite subset and any closed (in ℝ) subset of (0, 1], containing 1, are inf-compact(2)
Any ascending sequence in (0, 1], union with1, is inf-compact(3)
Any intervala,b⊆0,1, union with 1, is inf-compact(4)
Any union of finitely 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.Lemma 1.
IfCi|i∈I is a family of inf-compact subsets, containing 1, then ∩iCi is inf-compact.Proof.
For anyα∈0,1, let ξ=Infα,1∩∩iCi, and define ξi=Minξ,1∩Ci; hence ξ≤ξi, for any i∈I. On the other hand, since α,1∩∩iCi=ξ,1∩∩iCi⊆ξ,1∩Ci, then Minξ,1∩Ci=Infξ,1∩Ci≤Infξ,1∩∩iCi=ξ. In consequence, ξ=ξi for any i∈I.
For any elementa∈X, there exists a partial gradual element, which we represent by εa, with domεa=1, and defined by εa1=a. We denote also by εa the extension εa¯. Without lost of generality, we may identify the element a∈X and the gradual element εa∈X and denote them simply by a.
In this way, a gradual element is nothing more than a collection of elements ofX, each one with a degree of membership; thus, if ε is a gradual element, then εα is an element of X with membership degree α. Since ε1 is always defined, we have it as an element of X with the highest membership degree; since ε0 is not defined, then there is no any element with zero membership degree.


## 2.2. Relations between Gradual Elements

For anyα∈0,1, in the set of all partial gradual elements, we define a relation Rα as, for partial gradual elements ε1 and ε2 of X, we say ε1Rαε2 if(3)ε1|α,1∩domε1∩domε2=ε2|α,1∩domε1∩domε2.Observe that ifα∈0,1 and ε1,ε2 are total gradual elements, then ε1Rαε2, whenever ε1|α,1=ε2|α,1.Lemma 2.
For anyα,β∈0,1, we have(1)
Ifα,β∈0,1 satisfies α≤β, then Rα⊆Rβ(2)
The relationRα is an equivalence relation in the set of all total gradual elementsThe equivalence relationRα indicates us when two gradual elements are equal at a certain level. For instance,(1)
Ifα=1, then we only have an equivalence class for each element of X(2)
Ifα=0, then two gradual elements belong to the same equivalence class if, and only if, they coincide in their definition domainsIt is necessary to remark that these equivalence relationsRα are not compatible with the extension process. Indeed, if ε1Rαε2, not necessarily ε¯1Rαε¯2 as the following example shows.Example 1.
LetX=a,b, and define(4)ε1δ=b,ifδ=1,a,if12≤δ<1,ε2δ=b,ifδ=1,a,if0<δ≤12.
Then,ε1Rαε2, for any α∈0,1, but ε¯1Rαε¯2 if and only if α=1.
Letf:X⟶Y be a map between two sets.(1)
For any total gradual (resp., partial gradual) elementε of X, we have a total gradual (resp., partial gradual) element of Y defined by the composition fε:domε⟶X⟶Y; we call fε the image of ε by f.(2)
For any gradual elementε′ of Y, a gradual element ε of X is an inverse image of ε′ if ε′=fε


## 2.3. Binary Operations and Gradual Elements

There is another method to relate gradual elements of a setX; this is the case in which there exists a binary operation in X.LetG be a set together a binary operation, say ∗, and ε1,ε2 gradual elements of G, we define a new gradual element ε1∗ε2 as(5)ε1∗ε2α=ε1α∗ε2α,for anyα∈domε1∩domε2.In the case of partial gradual elementsε1,ε2, we have that domε1∗ε2=domε1∩domε2.This operation non necessarily is compatible with the extension construction.The following example shows that if we start from two partial gradual elementsε1 and ε2, then not necessarily we have the equality: ε1∗ε2¯=ε¯1∗ε¯2, i.e., then extension map is not necessarily a homomorphism with respect to the binary operation ∗.Example 2.
Letε1,ε2 be partial gradual elements defined on ℤ, defined as:(6)ε1α=2,if12≤α≤1,1,if110≤α≤13,ε2α=2,if23≤α≤1.
In this case, we have(7)ε1+ε2α=4,if23≤α≤1.and the extended gradual elements are(8)ε1¯α=2,if13<α≤1,1,ifα≤13,ε2¯α=2,ifα≤1,ε1+ε2¯α=4,ifα≤1.
On the other hand, we have(9)ε1¯+ε2¯α=4,if13<α≤1,3,ifα≤13.
On the other hand, this operation is compatible with the equivalence relationsRα.Lemma 3.
LetG be a set together a binary operation ∗, for any α∈0,1 the relations Rα 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 elements ε,ε1,ε2 of G, if ε1Rαε2, then ε∗ε1Rαε∗ε2 and ε1∗εRαε2∗ε.Proof.
Letε1,ε2,ε be gradual elements such that ε1Rαε2, for any β∈α,1∩domε1∩domε2∩domε, we have(10)ε1∗εβ=ε1β∗εβ=ε2β∗εβ=ε2∗εβ.
In some cases, in whichG has a richer structure, this structure could be inherited by the sets of gradual elements. Let us show an example.Lemma 4.
LetG be a group, with binary operation ∗ and neutral element e, the following statements hold:(1)
The setG of all total gradual elements is a group with neutral element e, i.e., the total gradual element εe(2)
For anyα∈0,1, we have that G/Rα is a groupProof.
For anyε,ε1,ε2∈G, we define, for any α∈0,1:(11)ε1∗ε2α=ε1α∗ε2α,andε−1α=εα−1.(1)
InG, the operation is associative and e is the neutral element. For any ε∈G, we have ε−1 the inverse of ε. Therefore, G is a group, which is abelian whenever G is.(2)
It is a direct consequence of beingRα a compatible equivalence relation.Remark 1.
(The particular case of partial gradual elements). LetG be a group, in the set G¯ of all partial gradual elements, we have an associative binary operation, ε1∗ε2α=ε1α∗ε2α for any α∈domε1∩domε2, but we have “many” possible neutral elements. Thus, to get a useful structure, we must define before an equivalence relation to put together all of them. For instance, given two partial gradual elements ε1,ε2, since ε1∗ε2 is defined on domε1∩domε2, there are three possible neutral elements: e|domε1,e|domε2 and finally e|domε1∩domε2, which are different two to two.
We can try to fix this problem in defining an equivalence relationR in G¯ generated by(12)ε1Rε2,if there is an inf−compact subsetC⊆0,1containing1,such thatε1|C=ε2|C.
With the relationR, the problem is that we may have domε1∩domε2=1, and this trivialize this relation.
Hence, to obtain a well-defined structure on partial gradual elements, we may consider only special types of partial gradual elements, for instance, the subset ofG¯ constituted by those partial gradual elements who have the same (inf-compact) domain containing 1.
Thus, we can extend Lemma 4 to consider gradual elements defined on an inf-compact subset containing 1.Proposition 1.
LetG be a group, and let C⊆0,1 be an inf-compact subset containing 1. If G¯ be the set of all partial gradual elements whose domain is C, the following statements hold:(1)
For anyα∈C, the relation Rα is an equivalent relation in G¯(2)
InG¯, we have an associative operation(3)
The extending map fromG¯ to G is a group monomorphism(4)
For anyα∈C, the equivalence relation Rα in G¯ is compatible(5)
The groupsG¯ and G¯/Rα are abelian whenever G isProof.
(1)
It is reflexive and symmetric, and obviously it is transitive as the domain is the whole setC.(2)
It is evident as the product is defined componentwise.(3)
Letε1,ε2∈G¯, and ξ=Minα,1∩C, then(13)ε1∗ε2¯α=ε∗ε2ξ=ε1ξ∗ε2ξ=ε¯1α∗ε¯2α.(4)
It is similar to the proof on Lemma3.(5)
It is evident as the product is defined componentwise.
It is clear that it is better to consider total gradual elements instead of partial gradual elements and therefore work inG.
If the groupG has e as neutral element and for any α in [0, 1] we consider the equivalence relation Rα, we may rewrite Lemma 3 obtaining a filtration of subgroups of G.Proposition 2.
LetG be a group with neutral element e, if for any α∈0,1, we define(14)Gα=ε∈G|εRαe.Then, we have(1)
For eachα∈0,1, the subgroup Gα⊆G is a normal.(2)
There is a filtrationGα|α∈0,1 where Gα⊆Gβ if α≤β.(3)
We have inclusions:G0⊆Gα⊆G1⊆G, and surjective group homomorphisms:(15)GG0⟶GGα⟶GG1≅G.Observe that in all these examples, it seems that the way to define an operation on gradual elements is to define it componentwise.If the base setX 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 different 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.


## 3. Gradual Subsets

Once we have established the notion of gradual element of a setX, we shall apply it to define new objects. If we consider a set X and the power set PX, we can study gradual elements of PX, thereby the concept of gradual subset appears.


### 3.1. Definition of Gradual Subsets

Definition 2.
LetX be a set, and let PX be the power set of X, i.e., PX=S|Sis a subset ofX. We define a gradual subset of X as a gradual element of PX. We represent by σ:0,1→PXa gradual subset of X.
Throughout this section, we follow the same assumptions used for gradual elements in the previous section. In this way, we have defined partial gradual subsets and total gradual subsets.
In some sense, gradual subsets are a generalization of gradual elements. Thus, for any gradual elementε and any gradual subset σ, we say ε belongs to σ if for any α∈domε∩domσ⊆0,1, we have εα∈σα and write ε∈σ. In the same way, given two gradual subsets σ1,σ2, we say that σ1 is a subset of σ2 if σ1α⊆σ2α for any α∈domσ1∩domσ2, we write σ1⊆σ2.
In general, for any gradual subsetσ and elements α,β∈domσ such that α≤β, we have no information about the relationship of σα and σβ. 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, first we introduce the following definitions that reflect toe order existing in (0, 1].
Letσ be a gradual subset of X, we say σ is(1)
Increasing if for anyα,β∈domσ such that α≤β, we have σα⊆σβ. For any increasing gradual subset σ de X, if ζ=Mindomσ, then σζ⊆σα⊆σ1 for any α∈domσ.(2)
Decreasing if for anyα,β∈domσ such that α≤β, we have σα⊇σβ. For any decreasing gradual subset σ de X, we have σ1⊆σα for any α∈domσ.
Let us show some examples of decreasing gradual subsets.Example 3.
Letμ be a fuzzy subset of X, i.e., a map μ:X⟶0,1 that we assume it is not constant equal to 0. For any α∈0,1, we define (1)
α-level of μ as μα=x∈X|μx≥α.    In this case, we have a decreasing gradual subset σμ, defined σμα=μα for any α∈0,1.(2)
Strictα-level (or strong α-level) of μ as μ˜α=x∈X|μx>α.     In this case, we have a decreasing gradual subset σ˜μ, defined as(16)σ˜μα=μ˜α,for anyα∈0,1,and,μ1,ifα=1.(3)
Letμ¯ the fuzzy subset defined by μ¯x=1−μx, for any x∈X; the α-levels of μ¯ define a decreasing gradual subset σμ¯α=x∈X|μx≤1−α.(4)
Inverseα-level of μ as μα=x∈X|μx≤α. In this case, we have an increasing gradual subset τμ defined τμα=μα for any α∈0,1.


### 3.2. Operators on Gradual Subsets

The following are examples of constructions that can be carried out for any gradual subset, which will be useful in their study.Letσ be a gradual subset of X, associated to σ, we define two new gradual subsets:(1)
The accumulationσc of σ is(17)σcα=∪σβ|α≤β∈domσ,for anyα∈domσ.It is clear that for any gradual subsetσ, the accumulation σc is a decreasing gradual subset, and a gradual subset σ is decreasing if and only if, σ=σc.For any gradual subsetσ, we have σ⊆σc=σcc.(2)
The strict accumulationσd of σ is(18)σdα=σ1,ifα=1,∪σβ|α<β∈domσ,ifα∈domσ∖1.It is clear that for any gradual subsetσ, the strict accumulation σd is a decreasing gradual subset and σd⊆σc. In general, σ⊈σd.Thus, we have an operator,c, on gradual subsets: σ↦σc. The behaviour of c is reflected in the following lemma.Lemma 5.
LetX be a set, for any gradual subsets σ1,σ2,σ of X, the following statements hold:(1)
σ⊆σc(2)
σc=σcc(3)
Ifσ1⊆σ2, then σ1c⊆σ2c(4)
σc is the smallest decreasing gradual subset containing σProof. (1), (2), and (3) are easy(5)
Letτ be a decreasing gradual subset such that σ⊆τ, then σc⊆τc=τ.This means that the operatorc 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≤cp for any p∈P(2)
For anyp1,p2∈P such that p1≤p2 we have cp1≤cp2(3)
cp=ccp for any p∈PThe elementsp∈P such that cp=p are named the c–closed elements. Thus, the gradual subsets which are closed for the operator c are the decreasing gradual subsets. Let us denote by JX the set of all decreasing gradual subsets of X.In the same way, we may consider the operatord, defined as σ↦σd; its behaviour is reflected in the following lemma.Lemma 6.
LetX be a set, for any gradual subsets σ1,σ2,σ of X, the following statements hold:(1)
σd⊆σc(2)
Ifσ1⊆σ2, then σ1d⊆σ2d(3)
σd=σdd=σcd=σdcProof.
(1) and (2) are easy. (3) Indeed, for anyα∈0,1, we have(19)σddα=∪σdβ|β>α=∪∪σγ|γ>β|β>α=∪σβ|β>α=σdα.
In the same way, for anyα∈0,1, we have(20)σcdα=∪σcβ|β>α=∪∪σγ|γ≤β|β>α,=∪σβ|β>α,=σdα.
A gradual subsetσ is an strict decreasing gradual subset if σ=σd. We haveLemma 7.
For any gradual subsetσ, the following statements hold:(1)
σd is the smallest strict decreasing gradual subset contained in σc(2)
σ is a decreasing gradual subset non strict decreasing if, and only if, σd⊊σcProof.
Letτ be a strict decreasing gradual subset such that τ⊆σc, then τ=τd⊆σcd=σd.
This means that the operatord is an interior operator in the set of all decreasing gradual subsets of X.
Remember that an interior operator in a posetP is a map d:P⟶P satisfying(1)
dp≤p for any p∈P(2)
For anyp1,p2∈P such that p1≤p2, we have dp1≤dp2(3)
dp=ddp for any p∈P
The elementsp∈P such that dp=p is named the d–open elements. Thus, the decreasing gradual subsets open for the operator d are the strict decreasing gradual subsets. Let us denote byJdX the set of all d-open (strict) decreasing gradual subsets.Remark 2.
Inspired in these constructions, we consider a new construction of a gradual subset from a partial gradual subset that allows us to avoid the initial restriction of inf-compact in the domain of definition of partial gradual elements.
Letσ:0,1→PX be a partial map defined at 1, i.e., 1∈domσ⊆0,1, and such that domσ is not necessarily inf-compact, we may extend σ to all (0, 1] simply defining σβ=∅ if β∉domσ. The decreasing gradual subset associated to σis σc:0,1→PX defined as(21)σcα=∪σβ|β≥α,β∈domσ,for anyα∈0,1.
The 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 subsetσ on X to σ¯ on the whole interval [0, 1], in defining σ¯α=∪σβ|α≤β∈0,1,ifα∈0,1,∪σβ|β∈0,1,ifα=0, for any α∈0,1. In consequence, we may consider also decreasing gradual subsets as maps from [0, 1] to X.


### 3.3. The Algebra of Gradual Subsets

There is a natural relationship between gradual elements and gradual subsets of a given setX. Thus, for any partial gradual element ε, we may define a unitary partial gradual subset σε as σεα=εα, for any α∈domε. As we point out before, we have ε∈σ if, and only if, σε⊆σ, for any gradual subset σ.In the setPX, there are two operations: the intersection and the union; thus, we can translate these two operations to gradual subsets as did in the first section. Following this line, we define, for any gradual subsets, σ1 and σ2:(1)
The intersection,σ1∩σ2, as σ1∩σ2α=σ1α∩σ2α, for any α∈domσ1∩domσ2(2)
The union,σ1∪σ2, as σ1∪σ2α=σ1α∪σ2α, for any α∈domσ1∩domσ2In this way, we may consider the algebra of gradual subsets of a given setX with respect to intersection and union.The definition of intersection and union can also be extended to arbitrary families of gradual subsets. Letσi|i∈I be a family of gradual subsets.(1)
The intersection∩iσi is defined as ∩iσiα=∩iσiα, for any α∈0,1(2)
The union∪iσi defined as ∪iσiα=∪iσiα, for any α∈0,1Letμi|i∈I be a family of fuzzy subsets of a set X, the union, ∨iμi, and the intersection, ∧iμi, are the fuzzy subsets defined by(22)∨iμia=∨iμia=Supμia|i∈I,for anya∈X,∧iμia=∧iμia=Infμia|i∈I,for anya∈X.Example 4.
LetX=a,b be a set, for any n∈ℕ∖0,1, we define μn:a,b⟶0,1 by μna=1 and μnb=1/2−1/2n. We have(1)
∨nμna=1 and ∨nμnb=1/2(2)
σ∨nμnδ=X,ifδ>1/2,a,ifδ≤1/2,(3)
∪nσμnδ=X,ifδ≥1/2,a,ifδ<1/2,
This shows that the inclusionσ∨nμn⊇∪nσμn is proper.
In the same line, we have a similar situation forσ˜ and the intersection.Example 5.
LetX=a,b be a set, for any n∈ℕ∖0,1, we define μn:a,b⟶0,1 by μna=1 and μnb=1/2+1/2n. We have(1)
∧nμna=1 and ∨nμnb=1/2(2)
σ˜∧nμnδ=X,ifδ≥1/2,a,ifδ<1/2,(3)
∩nσ˜μnδ=X,ifδ>1/2,a,ifδ≤1/2,
This shows that the inclusionσ˜∧nμn⊆∩nσ˜μn is proper.
In the setJdX of all strict decreasing gradual subsets of X, we define two new operations: intersection: ∧¯iσi=∩iσid, and maintain the old union: ∨¯iσi=∩iσi, for every family σi|i∈I of strict decreasing gradual subsets of X. With these definitions, we have the following.Proposition 3.
The union and intersection of strict decreasing gradual subsets are compatible with the union and intersection of fuzzy subsets via the gradual subsetσ˜μ, i.e., for any family of fuzzy subsets μi|i∈I, we have(1)
∨¯iσ˜μi=σ˜∨iμi(2)
∧¯iσ˜μi=σ˜∧iμiProof.
For anyα∈0,1, we have(23)σ˜∨iμiα=a∈X|∨iμia>α=a∈X|∨iμia>α,=∪a∈X|there existsisuch thatμia>α=∪iσ˜μiα=∨¯iσ˜μiα.
In the same way, we can prove the case ofσ˜∧iμiα.
From this point of view, strictα-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 defining a map from one to the other. In this context, first 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).
Letf:X⟶Y be a map and denote by f the induced map from PX to PY, the following statements hold:(1)
For every gradual elementε of X, we have that fε:domε⟶Y is a gradual element of Y.(2)
Letσ be a gradual subset of X, then fσ:domσ⟶PY is a gradual subset of Y. And we have a map f∗:X⟶Y defined f∗σ=fσ for any σ∈X.(3)
Letσ be a gradual subset of X, then f∗σc=f∗σc.
In addition, ifε∈σ, then fε∈fσ.Lemma 9 (Inverse image).
Letf:X⟶Y be a map and denote by f−1:PY⟶PX the induced map. For any gradual subset τ of Y, we have that f∗τ:domτ⟶PX, defined as f∗τα=f−1τα, for any α∈domτ, is a partial gradual subset of X. Thus, we have a map f∗:Y⟶X, defined f∗τ=f∗τ, for any τ∈Y.In particular, for any gradual subsetτ of Y, we have f∗τc=f∗τc.Since every element ofX 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 haveLemma 10.
Letf:X⟶Y be a map, then(1)
f is injective if, and only if, f∗°f∗=idX(2)
f is surjective if, and only if, f∗°f∗=idY
Our aim will be to establish maps between gradual sets instead of between gradual subsets, i.e., leave out the ground setX and use only the subsets σα|α∈0,1. However, we postpone it until the point in which we change the paradigm introducing these gradual sets.Remark 4.
A gradual subsetσ of a set X is just a family σα|α∈0,1 of subsets, indexed in (0, 1]. There are particular types of gradual subsets, as decreasing gradual subsets, in which, for any α,β∈0,1, α≤β, there exists a map jα,β:σβ⟶σα: the inclusion, satisfying jα,βjβ,γ=jα,γ whenever α≤β≤γ. In some sense, decreasing gradual subsets are gradual subsets enriched with a family of maps jα,β|α,β∈0,1,α≤β satisfying the above conditions and compatible with the inclusions in X. Thus, we may work with these enriched gradual subsets of X.
An enriched gradual subset ofX is a gradual subset σ together with a family of maps fα,β|α,β∈0,1,α≤β satisfying(1)
fα,β:σβ⟶σα(2)
fα,βfβ,γ=fα,γ whenever α≤β≤γ(3)
ifjα:σα⟶X is the inclusion, for any α∈0,1, then jαfα,β=jβ, whenever α≤β
Observe that, as a consequence of (3), eachfα,β is an injective map. In particular, enriched gradual subsets are just the decreasing gradual subsets. See also Remark 7.


### 3.5. Gradual Quotient Sets

The same technique we used to introduce gradual subsets can be applied to define quotient gradual sets of a given setX.Remember that ifX is a set, a subset S⊆X is an equivalence class in the class of all injective maps i,Y|i:Y⟶Xinjective, whenever we consider the equivalence relation: i1,Y1∼i2,Y2 if there exists a bijective map b:Y1⟶Y2 such that i1=i2b.Dually, a quotient set ofX is an equivalence class in the class of all surjective maps Z,p|p:X⟶Zsurjective when we consider the equivalence relation: Z1,p1∼Z2,p2 if there exists a bijective map b:Z1⟶Z2 such that p2=bp1.The set of all subsets ofX is represented by PX, and there exists a bijective correspondence between PX and 2X. The set of all quotient set of X will be represented by QX, and for any element Z∈QX, we have(1)
A surjective mapp:X⟶Z(2)
An equivalence relationRp in X defined as xRpy if px=py(3)
A partition ofX into the equivalence classes defined by a relation R.Each equivalence relationR in X is a subset of X×X satisfying the properties reflexive, symmetric and transitive. Hence, QX is in bijection with a subset of PX×X. If we call QX×X this subset, it is constituted by all the equivalence relations in X.A gradual quotient set ofX is a gradual element of QX or equivalently of QX×X. We represent by ρ a gradual quotient set of X.


### 3.6. 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 Proposition3; 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.In addition, the gradual subsetsσμ and σ˜μ are related in a strong way: σ˜μ=σμd⊆σμc=σμ, using the interior and closure operator. Also, these gradual subsets satisfy the following properties:(1)
μx=Maxα|x∈σμα, for any x∈X(2)
μx=Infα|x∉σ˜μα, for any x∈XWe say(1)
A decreasing gradual subsetσ satisfies property (F) if there exists Maxα|x∈σα for every x∈∪α|α∈0,1(2)
A strict decreasing gradual subsetσ satisfies property (inf-F) if Infα|x∉σdα=β satisfies x∈σβ, for every x∈∪α|α∈0,1As a consequence, we have the following result.Lemma 11.
Letσ be a decreasing gradual subset, not strict decreasing gradual subset, the following statements are equivalent:(i)
σ satisfies property (F)(ii)
σd satisfies property (inf-F)(iii)
∪α∈0,1σα=∪•α∈0,1σcα∖σdα (the disjoint union)Proof.
By the hypothesis, we haveσd⊊σc=σ.(i)
⇒ (ii) Let β=Maxα|x∈σα and γ=Infα|x∉σdα, then x∉σδ for every δ>γ. Since σ is decreasing, β≤γ. If β<γ, for any ω such that β<ω<γ, we have x∉σω, hence γ≠Infα|x∉σdα, which is a contradiction.(ii)
⇒ (i) Let β=Supα|x∈σα and γ=Infα|x∉σdα. Since σ is decreasing, β≤γ. If β<γ, for any ω such that β<ω<γ, we have x∈σω and β≠Supα|x∈σα, which is a contradiction.(i)
⇒ (iii) One inclusion is obvious. Otherwise, if x∈∪α∈0,1σα, let β=Maxα|x∈σα, then x∉σγ for any γ>β; hence, x∉σdβ and x∈∪•α∈0,1σcα∖σdα.(iii)
⇒ (i) Let x∈X, if x∉∪α∈0,1σα, then either x∈σ1, and there exists Maxα∣x∈σα=1 or x∉∪α∈0,1σα and Maxα|x∈σα=0. Otherwise, if x∈∪α∈0,1σα=∪•α∈0,1σcα∖σdα, there exists α such that x∈σα∖σdα, hence Maxα|x∈σα=α.Remark 5.
(1)
As a consequence of this result, for any decreasing nonstrict decreasing gradual subset, satisfying property (F), we haveMaxα|x∈σα=Infα|x∉σdα for any x∈X(2)
In the case of a strict decreasing gradual subset satisfying property (inf-F), we have that it also satisfies property (F); hence the following equalities hold:Maxα|x∈σα=Infα|x∉σdα for any x∈X
Now, we are going to establish correspondences between fuzzy subsets and strict decreasing gradual subsets, which preserves union and intersection. The following result, for finite unions, is well known.Theorem 1.
LetX be a set, the following statements hold:(1)
The mapν:μ↦σμ associates to any fuzzy subset μ of X a decreasing gradual subset satisfying property (F), σμ of X, defined σμα=x|μx≥α, and it preserves intersections and finite unions.(2)
The mapυ:σ↦μσ associates to any decreasing gradual subset σ=σc, satisfying property (F); a fuzzy subset μσ is defined as follows:(24)μσx=Maxα|x∈σα.In addition, we haveν°υ=id and υ°ν=id, and they preserve finite unions and intersections.The behaviour with respect to infinite unions can be solved using only strict decreasing gradual subsets instead of decreasing gradual subsets.Theorem 2.
LetX be a set, the following statements hold:(1)
The mapν:μ↦σ˜μ associates to any fuzzy subset μ of X a strict decreasing gradual subset σ˜μ of X, defined σ˜μα=x|μx>α, and it preserves arbitrary unions and intersections.(2)
The mapυ:σ↦μ˜σ associates to any strict decreasing gradual subset σ=σd, satisfying property (inf-F); a fuzzy subset μ˜σ is defined as follows:(25)μ˜σx=Infα|x∉σα.In addition, we haveνυ=id and υν=id.Proof.
We had already studied the mapν in Proposition 3.
The mapυ is well defined due to Theorem 2. Now, we check that the compositions are the identity.
Letμ be a fuzzy subset of X, for any x∈X, we have(26)υ°νμx=μ˜σ˜μx=Infα|x∉σ˜μα=Infα|μx≤α=μx.
On the other hand, letσ be a strict decreasing gradual subset, and α∈0,1, we have(27)ν°υσα=σ˜μ˜σα=x|μ˜σx>α=x|Infβ|x∉σβ>α.
IfInfβ|x∉σβ=δx, then δx>α; since σ satisfies property (inf-F), then x∈σδx⊆σα. Otherwise, if x∈σα=σdα=∪σβ|β>α, there exists γ>α such that x∈σγ, hence Infβ|x∉σβ>γ>α, and we have the other inclusion.
As a consequence, we have the final result that establishes an isomorphism between the two lattices. See Proposition3.Corollary 1.
LetX 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.


### 3.7. A Functorial Interpretation

Let us consider (0, 1] as a category whose objects are the elements of (0, 1] and homomorphisms: only one,fα,β, from α to β whenever α≤β and the obvious composition.LetF:0,1⟶Set a contravariant functor from the category (0, 1] to the category of sets. Indeed, Fα|α∈0,1 is a directed system with maps Ffα,β:Fβ⟶Fα, if α≤β. Let D=lim⟶F be the direct limit of this system.Let us remember the definition of the direct limit,D=lim⟶F. First, we consider the disjoint union, ∪•Fα, of the family of sets Fα|α∈0,1 and in it; the equivalence relation R generated by a∈Fα is related with b∈Fβ if either α≤β and Ffα,βb=a or β≤α and Ffβ,αa=b. Let p:∪•Fα⟶∪•Fα/R be the canonical projection, iβ:Fβ⟶∪•Fα be the inclusion, for any β∈0,1, and qβ=piβ the composition.The pairD,pα|α∈0,1 satisfies the corresponding universal property of the direct limit.In the particular case in which every mapFfα,β is injective, then every qα is also injective; this means that we can consider every Fα as a subset of D. Therefore, we have that F defines 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.This allows to give an interpretation of decreasing gradual subsets in terms of contravariant functors. If we start from a decreasing gradual subset of a setX, then σα|α∈0,1, together with the family of inclusions, is a directed system, and if jα:σα⟶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 hD=Imh the union of the family of the subsets σα|α∈0,1.Taking into account this construction, we may show that a decreasing gradual subsetσ 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 define a gradual set as a contravariant functorF:0,1⟶Set and a decreasing gradual set as a gradual set such that each map Ffα,β, whenever α≤β, is injective.Given two gradual setsF1 and F2, a map from F1 to F2 is just a natural transformation θ:F1⟶F2, i.e., a set of maps θα|α∈0,1 such that each diagram commutes, whenever α≤β.The contravariant functors from (0, 1] toSet, i.e., the gradual sets, constitute a category that we shall denote by Set0,1. The class of all decreasing gradual sets defines a full subcategory of Set0,1, and it is closed under (finite and infinite) unions and intersections. Let us call J this subcategory.In the subcategoryJ, we define an interior operator, d:F↦Fd, as follows:(28)Fdα=∪Fγ|γ>α=lim⟶α,1Fγ,and if α≤β, then there is an inclusion functor β,1→α,1 and a natural map from Fdβ=lim⟶β,1Fγ to Fdα=lim⟶α,1Fγ.Proposition 4.
LetF be a decreasing gradual set and θ:F1⟶F2 be a decreasing gradual set map. The following statements hold:(1)
Fd is a contravariant functor from (0, 1] to Set; hence, it is a gradual set(2)
Ifα≤β, the natural map Fdβ⟶Fdα is injective; hence, Fd is a decreasing gradual set(3)
There exists a natural mapθd:F1d⟶F2d, defined, for every β∈0,1, as the only map making commutative the following diagram.(4)
d defines an endofunctor of J, which is an interior operator in J.A decreasing gradual setF is an strict decreasing gradual set whenever F=Fd and satisfies property (F) if D=lim⟶F=∪•Fα∖Fdα, where the union is taken in D.By the relationship between (inf-F) and (F) properties, we may define 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 ofJ.As a consequence, we have the following result.Theorem 3.
LetF be a gradual set.(1)
The following statements are equivalent:(a)
F is a strict decreasing gradual set(b)
The pairF,lim⟶F is a strict decreasing gradual subset of D=lim⟶F(2)
The following statements are equivalent:(a)
F is a strict decreasing gradual set satisfying property (inf-F), i.e., F is a fuzzy set(b)
The pairF,lim⟶F is a strict decreasing gradual subset of D=lim⟶F satisfying property (inf-F)Remark 6.
The 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 categorySet0,1. 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 setX, our first candidate was the functor category Set0,1. 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 Xdefined by σδ=X,ifδ=1/2,∅,ifδ≠1/2.. 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). This have been overcome when we consider enriched gradual sets. This model works perfectly and meets all our expectations whenever we consider decreasing gradual subsets.Remark 8.
Observe that in our construction, we have fixed the categoriesSet 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 final object; the other is that there are objects A and B such that HomSetA,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).


## 3.1. Definition of Gradual Subsets

Definition 2.
LetX be a set, and let PX be the power set of X, i.e., PX=S|Sis a subset ofX. We define a gradual subset of X as a gradual element of PX. We represent by σ:0,1→PXa gradual subset of X.
Throughout this section, we follow the same assumptions used for gradual elements in the previous section. In this way, we have defined partial gradual subsets and total gradual subsets.
In some sense, gradual subsets are a generalization of gradual elements. Thus, for any gradual elementε and any gradual subset σ, we say ε belongs to σ if for any α∈domε∩domσ⊆0,1, we have εα∈σα and write ε∈σ. In the same way, given two gradual subsets σ1,σ2, we say that σ1 is a subset of σ2 if σ1α⊆σ2α for any α∈domσ1∩domσ2, we write σ1⊆σ2.
In general, for any gradual subsetσ and elements α,β∈domσ such that α≤β, we have no information about the relationship of σα and σβ. 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, first we introduce the following definitions that reflect toe order existing in (0, 1].
Letσ be a gradual subset of X, we say σ is(1)
Increasing if for anyα,β∈domσ such that α≤β, we have σα⊆σβ. For any increasing gradual subset σ de X, if ζ=Mindomσ, then σζ⊆σα⊆σ1 for any α∈domσ.(2)
Decreasing if for anyα,β∈domσ such that α≤β, we have σα⊇σβ. For any decreasing gradual subset σ de X, we have σ1⊆σα for any α∈domσ.
Let us show some examples of decreasing gradual subsets.Example 3.
Letμ be a fuzzy subset of X, i.e., a map μ:X⟶0,1 that we assume it is not constant equal to 0. For any α∈0,1, we define (1)
α-level of μ as μα=x∈X|μx≥α.    In this case, we have a decreasing gradual subset σμ, defined σμα=μα for any α∈0,1.(2)
Strictα-level (or strong α-level) of μ as μ˜α=x∈X|μx>α.     In this case, we have a decreasing gradual subset σ˜μ, defined as(16)σ˜μα=μ˜α,for anyα∈0,1,and,μ1,ifα=1.(3)
Letμ¯ the fuzzy subset defined by μ¯x=1−μx, for any x∈X; the α-levels of μ¯ define a decreasing gradual subset σμ¯α=x∈X|μx≤1−α.(4)
Inverseα-level of μ as μα=x∈X|μx≤α. In this case, we have an increasing gradual subset τμ defined τμα=μα for any α∈0,1.


## 3.2. Operators on Gradual Subsets

The following are examples of constructions that can be carried out for any gradual subset, which will be useful in their study.Letσ be a gradual subset of X, associated to σ, we define two new gradual subsets:(1)
The accumulationσc of σ is(17)σcα=∪σβ|α≤β∈domσ,for anyα∈domσ.It is clear that for any gradual subsetσ, the accumulation σc is a decreasing gradual subset, and a gradual subset σ is decreasing if and only if, σ=σc.For any gradual subsetσ, we have σ⊆σc=σcc.(2)
The strict accumulationσd of σ is(18)σdα=σ1,ifα=1,∪σβ|α<β∈domσ,ifα∈domσ∖1.It is clear that for any gradual subsetσ, the strict accumulation σd is a decreasing gradual subset and σd⊆σc. In general, σ⊈σd.Thus, we have an operator,c, on gradual subsets: σ↦σc. The behaviour of c is reflected in the following lemma.Lemma 5.
LetX be a set, for any gradual subsets σ1,σ2,σ of X, the following statements hold:(1)
σ⊆σc(2)
σc=σcc(3)
Ifσ1⊆σ2, then σ1c⊆σ2c(4)
σc is the smallest decreasing gradual subset containing σProof. (1), (2), and (3) are easy(5)
Letτ be a decreasing gradual subset such that σ⊆τ, then σc⊆τc=τ.This means that the operatorc 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≤cp for any p∈P(2)
For anyp1,p2∈P such that p1≤p2 we have cp1≤cp2(3)
cp=ccp for any p∈PThe elementsp∈P such that cp=p are named the c–closed elements. Thus, the gradual subsets which are closed for the operator c are the decreasing gradual subsets. Let us denote by JX the set of all decreasing gradual subsets of X.In the same way, we may consider the operatord, defined as σ↦σd; its behaviour is reflected in the following lemma.Lemma 6.
LetX be a set, for any gradual subsets σ1,σ2,σ of X, the following statements hold:(1)
σd⊆σc(2)
Ifσ1⊆σ2, then σ1d⊆σ2d(3)
σd=σdd=σcd=σdcProof.
(1) and (2) are easy. (3) Indeed, for anyα∈0,1, we have(19)σddα=∪σdβ|β>α=∪∪σγ|γ>β|β>α=∪σβ|β>α=σdα.
In the same way, for anyα∈0,1, we have(20)σcdα=∪σcβ|β>α=∪∪σγ|γ≤β|β>α,=∪σβ|β>α,=σdα.
A gradual subsetσ is an strict decreasing gradual subset if σ=σd. We haveLemma 7.
For any gradual subsetσ, the following statements hold:(1)
σd is the smallest strict decreasing gradual subset contained in σc(2)
σ is a decreasing gradual subset non strict decreasing if, and only if, σd⊊σcProof.
Letτ be a strict decreasing gradual subset such that τ⊆σc, then τ=τd⊆σcd=σd.
This means that the operatord is an interior operator in the set of all decreasing gradual subsets of X.
Remember that an interior operator in a posetP is a map d:P⟶P satisfying(1)
dp≤p for any p∈P(2)
For anyp1,p2∈P such that p1≤p2, we have dp1≤dp2(3)
dp=ddp for any p∈P
The elementsp∈P such that dp=p is named the d–open elements. Thus, the decreasing gradual subsets open for the operator d are the strict decreasing gradual subsets. Let us denote byJdX the set of all d-open (strict) decreasing gradual subsets.Remark 2.
Inspired in these constructions, we consider a new construction of a gradual subset from a partial gradual subset that allows us to avoid the initial restriction of inf-compact in the domain of definition of partial gradual elements.
Letσ:0,1→PX be a partial map defined at 1, i.e., 1∈domσ⊆0,1, and such that domσ is not necessarily inf-compact, we may extend σ to all (0, 1] simply defining σβ=∅ if β∉domσ. The decreasing gradual subset associated to σis σc:0,1→PX defined as(21)σcα=∪σβ|β≥α,β∈domσ,for anyα∈0,1.
The 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 subsetσ on X to σ¯ on the whole interval [0, 1], in defining σ¯α=∪σβ|α≤β∈0,1,ifα∈0,1,∪σβ|β∈0,1,ifα=0, for any α∈0,1. In consequence, we may consider also decreasing gradual subsets as maps from [0, 1] to X.


## 3.3. The Algebra of Gradual Subsets

There is a natural relationship between gradual elements and gradual subsets of a given setX. Thus, for any partial gradual element ε, we may define a unitary partial gradual subset σε as σεα=εα, for any α∈domε. As we point out before, we have ε∈σ if, and only if, σε⊆σ, for any gradual subset σ.In the setPX, there are two operations: the intersection and the union; thus, we can translate these two operations to gradual subsets as did in the first section. Following this line, we define, for any gradual subsets, σ1 and σ2:(1)
The intersection,σ1∩σ2, as σ1∩σ2α=σ1α∩σ2α, for any α∈domσ1∩domσ2(2)
The union,σ1∪σ2, as σ1∪σ2α=σ1α∪σ2α, for any α∈domσ1∩domσ2In this way, we may consider the algebra of gradual subsets of a given setX with respect to intersection and union.The definition of intersection and union can also be extended to arbitrary families of gradual subsets. Letσi|i∈I be a family of gradual subsets.(1)
The intersection∩iσi is defined as ∩iσiα=∩iσiα, for any α∈0,1(2)
The union∪iσi defined as ∪iσiα=∪iσiα, for any α∈0,1Letμi|i∈I be a family of fuzzy subsets of a set X, the union, ∨iμi, and the intersection, ∧iμi, are the fuzzy subsets defined by(22)∨iμia=∨iμia=Supμia|i∈I,for anya∈X,∧iμia=∧iμia=Infμia|i∈I,for anya∈X.Example 4.
LetX=a,b be a set, for any n∈ℕ∖0,1, we define μn:a,b⟶0,1 by μna=1 and μnb=1/2−1/2n. We have(1)
∨nμna=1 and ∨nμnb=1/2(2)
σ∨nμnδ=X,ifδ>1/2,a,ifδ≤1/2,(3)
∪nσμnδ=X,ifδ≥1/2,a,ifδ<1/2,
This shows that the inclusionσ∨nμn⊇∪nσμn is proper.
In the same line, we have a similar situation forσ˜ and the intersection.Example 5.
LetX=a,b be a set, for any n∈ℕ∖0,1, we define μn:a,b⟶0,1 by μna=1 and μnb=1/2+1/2n. We have(1)
∧nμna=1 and ∨nμnb=1/2(2)
σ˜∧nμnδ=X,ifδ≥1/2,a,ifδ<1/2,(3)
∩nσ˜μnδ=X,ifδ>1/2,a,ifδ≤1/2,
This shows that the inclusionσ˜∧nμn⊆∩nσ˜μn is proper.
In the setJdX of all strict decreasing gradual subsets of X, we define two new operations: intersection: ∧¯iσi=∩iσid, and maintain the old union: ∨¯iσi=∩iσi, for every family σi|i∈I of strict decreasing gradual subsets of X. With these definitions, we have the following.Proposition 3.
The union and intersection of strict decreasing gradual subsets are compatible with the union and intersection of fuzzy subsets via the gradual subsetσ˜μ, i.e., for any family of fuzzy subsets μi|i∈I, we have(1)
∨¯iσ˜μi=σ˜∨iμi(2)
∧¯iσ˜μi=σ˜∧iμiProof.
For anyα∈0,1, we have(23)σ˜∨iμiα=a∈X|∨iμia>α=a∈X|∨iμia>α,=∪a∈X|there existsisuch thatμia>α=∪iσ˜μiα=∨¯iσ˜μiα.
In the same way, we can prove the case ofσ˜∧iμiα.
From this point of view, strictα-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 defining a map from one to the other. In this context, first 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).
Letf:X⟶Y be a map and denote by f the induced map from PX to PY, the following statements hold:(1)
For every gradual elementε of X, we have that fε:domε⟶Y is a gradual element of Y.(2)
Letσ be a gradual subset of X, then fσ:domσ⟶PY is a gradual subset of Y. And we have a map f∗:X⟶Y defined f∗σ=fσ for any σ∈X.(3)
Letσ be a gradual subset of X, then f∗σc=f∗σc.
In addition, ifε∈σ, then fε∈fσ.Lemma 9 (Inverse image).
Letf:X⟶Y be a map and denote by f−1:PY⟶PX the induced map. For any gradual subset τ of Y, we have that f∗τ:domτ⟶PX, defined as f∗τα=f−1τα, for any α∈domτ, is a partial gradual subset of X. Thus, we have a map f∗:Y⟶X, defined f∗τ=f∗τ, for any τ∈Y.In particular, for any gradual subsetτ of Y, we have f∗τc=f∗τc.Since every element ofX 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 haveLemma 10.
Letf:X⟶Y be a map, then(1)
f is injective if, and only if, f∗°f∗=idX(2)
f is surjective if, and only if, f∗°f∗=idY
Our aim will be to establish maps between gradual sets instead of between gradual subsets, i.e., leave out the ground setX and use only the subsets σα|α∈0,1. However, we postpone it until the point in which we change the paradigm introducing these gradual sets.Remark 4.
A gradual subsetσ of a set X is just a family σα|α∈0,1 of subsets, indexed in (0, 1]. There are particular types of gradual subsets, as decreasing gradual subsets, in which, for any α,β∈0,1, α≤β, there exists a map jα,β:σβ⟶σα: the inclusion, satisfying jα,βjβ,γ=jα,γ whenever α≤β≤γ. In some sense, decreasing gradual subsets are gradual subsets enriched with a family of maps jα,β|α,β∈0,1,α≤β satisfying the above conditions and compatible with the inclusions in X. Thus, we may work with these enriched gradual subsets of X.
An enriched gradual subset ofX is a gradual subset σ together with a family of maps fα,β|α,β∈0,1,α≤β satisfying(1)
fα,β:σβ⟶σα(2)
fα,βfβ,γ=fα,γ whenever α≤β≤γ(3)
ifjα:σα⟶X is the inclusion, for any α∈0,1, then jαfα,β=jβ, whenever α≤β
Observe that, as a consequence of (3), eachfα,β is an injective map. In particular, enriched gradual subsets are just the decreasing gradual subsets. See also Remark 7.


## 3.5. Gradual Quotient Sets

The same technique we used to introduce gradual subsets can be applied to define quotient gradual sets of a given setX.Remember that ifX is a set, a subset S⊆X is an equivalence class in the class of all injective maps i,Y|i:Y⟶Xinjective, whenever we consider the equivalence relation: i1,Y1∼i2,Y2 if there exists a bijective map b:Y1⟶Y2 such that i1=i2b.Dually, a quotient set ofX is an equivalence class in the class of all surjective maps Z,p|p:X⟶Zsurjective when we consider the equivalence relation: Z1,p1∼Z2,p2 if there exists a bijective map b:Z1⟶Z2 such that p2=bp1.The set of all subsets ofX is represented by PX, and there exists a bijective correspondence between PX and 2X. The set of all quotient set of X will be represented by QX, and for any element Z∈QX, we have(1)
A surjective mapp:X⟶Z(2)
An equivalence relationRp in X defined as xRpy if px=py(3)
A partition ofX into the equivalence classes defined by a relation R.Each equivalence relationR in X is a subset of X×X satisfying the properties reflexive, symmetric and transitive. Hence, QX is in bijection with a subset of PX×X. If we call QX×X this subset, it is constituted by all the equivalence relations in X.A gradual quotient set ofX is a gradual element of QX or equivalently of QX×X. We represent by ρ a gradual quotient set of X.


## 3.6. 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 Proposition3; 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.In addition, the gradual subsetsσμ and σ˜μ are related in a strong way: σ˜μ=σμd⊆σμc=σμ, using the interior and closure operator. Also, these gradual subsets satisfy the following properties:(1)
μx=Maxα|x∈σμα, for any x∈X(2)
μx=Infα|x∉σ˜μα, for any x∈XWe say(1)
A decreasing gradual subsetσ satisfies property (F) if there exists Maxα|x∈σα for every x∈∪α|α∈0,1(2)
A strict decreasing gradual subsetσ satisfies property (inf-F) if Infα|x∉σdα=β satisfies x∈σβ, for every x∈∪α|α∈0,1As a consequence, we have the following result.Lemma 11.
Letσ be a decreasing gradual subset, not strict decreasing gradual subset, the following statements are equivalent:(i)
σ satisfies property (F)(ii)
σd satisfies property (inf-F)(iii)
∪α∈0,1σα=∪•α∈0,1σcα∖σdα (the disjoint union)Proof.
By the hypothesis, we haveσd⊊σc=σ.(i)
⇒ (ii) Let β=Maxα|x∈σα and γ=Infα|x∉σdα, then x∉σδ for every δ>γ. Since σ is decreasing, β≤γ. If β<γ, for any ω such that β<ω<γ, we have x∉σω, hence γ≠Infα|x∉σdα, which is a contradiction.(ii)
⇒ (i) Let β=Supα|x∈σα and γ=Infα|x∉σdα. Since σ is decreasing, β≤γ. If β<γ, for any ω such that β<ω<γ, we have x∈σω and β≠Supα|x∈σα, which is a contradiction.(i)
⇒ (iii) One inclusion is obvious. Otherwise, if x∈∪α∈0,1σα, let β=Maxα|x∈σα, then x∉σγ for any γ>β; hence, x∉σdβ and x∈∪•α∈0,1σcα∖σdα.(iii)
⇒ (i) Let x∈X, if x∉∪α∈0,1σα, then either x∈σ1, and there exists Maxα∣x∈σα=1 or x∉∪α∈0,1σα and Maxα|x∈σα=0. Otherwise, if x∈∪α∈0,1σα=∪•α∈0,1σcα∖σdα, there exists α such that x∈σα∖σdα, hence Maxα|x∈σα=α.Remark 5.
(1)
As a consequence of this result, for any decreasing nonstrict decreasing gradual subset, satisfying property (F), we haveMaxα|x∈σα=Infα|x∉σdα for any x∈X(2)
In the case of a strict decreasing gradual subset satisfying property (inf-F), we have that it also satisfies property (F); hence the following equalities hold:Maxα|x∈σα=Infα|x∉σdα for any x∈X
Now, we are going to establish correspondences between fuzzy subsets and strict decreasing gradual subsets, which preserves union and intersection. The following result, for finite unions, is well known.Theorem 1.
LetX be a set, the following statements hold:(1)
The mapν:μ↦σμ associates to any fuzzy subset μ of X a decreasing gradual subset satisfying property (F), σμ of X, defined σμα=x|μx≥α, and it preserves intersections and finite unions.(2)
The mapυ:σ↦μσ associates to any decreasing gradual subset σ=σc, satisfying property (F); a fuzzy subset μσ is defined as follows:(24)μσx=Maxα|x∈σα.In addition, we haveν°υ=id and υ°ν=id, and they preserve finite unions and intersections.The behaviour with respect to infinite unions can be solved using only strict decreasing gradual subsets instead of decreasing gradual subsets.Theorem 2.
LetX be a set, the following statements hold:(1)
The mapν:μ↦σ˜μ associates to any fuzzy subset μ of X a strict decreasing gradual subset σ˜μ of X, defined σ˜μα=x|μx>α, and it preserves arbitrary unions and intersections.(2)
The mapυ:σ↦μ˜σ associates to any strict decreasing gradual subset σ=σd, satisfying property (inf-F); a fuzzy subset μ˜σ is defined as follows:(25)μ˜σx=Infα|x∉σα.In addition, we haveνυ=id and υν=id.Proof.
We had already studied the mapν in Proposition 3.
The mapυ is well defined due to Theorem 2. Now, we check that the compositions are the identity.
Letμ be a fuzzy subset of X, for any x∈X, we have(26)υ°νμx=μ˜σ˜μx=Infα|x∉σ˜μα=Infα|μx≤α=μx.
On the other hand, letσ be a strict decreasing gradual subset, and α∈0,1, we have(27)ν°υσα=σ˜μ˜σα=x|μ˜σx>α=x|Infβ|x∉σβ>α.
IfInfβ|x∉σβ=δx, then δx>α; since σ satisfies property (inf-F), then x∈σδx⊆σα. Otherwise, if x∈σα=σdα=∪σβ|β>α, there exists γ>α such that x∈σγ, hence Infβ|x∉σβ>γ>α, and we have the other inclusion.
As a consequence, we have the final result that establishes an isomorphism between the two lattices. See Proposition3.Corollary 1.
LetX 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.


## 3.7. A Functorial Interpretation

Let us consider (0, 1] as a category whose objects are the elements of (0, 1] and homomorphisms: only one,fα,β, from α to β whenever α≤β and the obvious composition.LetF:0,1⟶Set a contravariant functor from the category (0, 1] to the category of sets. Indeed, Fα|α∈0,1 is a directed system with maps Ffα,β:Fβ⟶Fα, if α≤β. Let D=lim⟶F be the direct limit of this system.Let us remember the definition of the direct limit,D=lim⟶F. First, we consider the disjoint union, ∪•Fα, of the family of sets Fα|α∈0,1 and in it; the equivalence relation R generated by a∈Fα is related with b∈Fβ if either α≤β and Ffα,βb=a or β≤α and Ffβ,αa=b. Let p:∪•Fα⟶∪•Fα/R be the canonical projection, iβ:Fβ⟶∪•Fα be the inclusion, for any β∈0,1, and qβ=piβ the composition.The pairD,pα|α∈0,1 satisfies the corresponding universal property of the direct limit.In the particular case in which every mapFfα,β is injective, then every qα is also injective; this means that we can consider every Fα as a subset of D. Therefore, we have that F defines 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.This allows to give an interpretation of decreasing gradual subsets in terms of contravariant functors. If we start from a decreasing gradual subset of a setX, then σα|α∈0,1, together with the family of inclusions, is a directed system, and if jα:σα⟶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 hD=Imh the union of the family of the subsets σα|α∈0,1.Taking into account this construction, we may show that a decreasing gradual subsetσ 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 define a gradual set as a contravariant functorF:0,1⟶Set and a decreasing gradual set as a gradual set such that each map Ffα,β, whenever α≤β, is injective.Given two gradual setsF1 and F2, a map from F1 to F2 is just a natural transformation θ:F1⟶F2, i.e., a set of maps θα|α∈0,1 such that each diagram commutes, whenever α≤β.The contravariant functors from (0, 1] toSet, i.e., the gradual sets, constitute a category that we shall denote by Set0,1. The class of all decreasing gradual sets defines a full subcategory of Set0,1, and it is closed under (finite and infinite) unions and intersections. Let us call J this subcategory.In the subcategoryJ, we define an interior operator, d:F↦Fd, as follows:(28)Fdα=∪Fγ|γ>α=lim⟶α,1Fγ,and if α≤β, then there is an inclusion functor β,1→α,1 and a natural map from Fdβ=lim⟶β,1Fγ to Fdα=lim⟶α,1Fγ.Proposition 4.
LetF be a decreasing gradual set and θ:F1⟶F2 be a decreasing gradual set map. The following statements hold:(1)
Fd is a contravariant functor from (0, 1] to Set; hence, it is a gradual set(2)
Ifα≤β, the natural map Fdβ⟶Fdα is injective; hence, Fd is a decreasing gradual set(3)
There exists a natural mapθd:F1d⟶F2d, defined, for every β∈0,1, as the only map making commutative the following diagram.(4)
d defines an endofunctor of J, which is an interior operator in J.A decreasing gradual setF is an strict decreasing gradual set whenever F=Fd and satisfies property (F) if D=lim⟶F=∪•Fα∖Fdα, where the union is taken in D.By the relationship between (inf-F) and (F) properties, we may define 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 ofJ.As a consequence, we have the following result.Theorem 3.
LetF be a gradual set.(1)
The following statements are equivalent:(a)
F is a strict decreasing gradual set(b)
The pairF,lim⟶F is a strict decreasing gradual subset of D=lim⟶F(2)
The following statements are equivalent:(a)
F is a strict decreasing gradual set satisfying property (inf-F), i.e., F is a fuzzy set(b)
The pairF,lim⟶F is a strict decreasing gradual subset of D=lim⟶F satisfying property (inf-F)Remark 6.
The 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 categorySet0,1. 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 setX, our first candidate was the functor category Set0,1. 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 Xdefined by σδ=X,ifδ=1/2,∅,ifδ≠1/2.. 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). This have been overcome when we consider enriched gradual sets. This model works perfectly and meets all our expectations whenever we consider decreasing gradual subsets.Remark 8.
Observe that in our construction, we have fixed the categoriesSet 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 final object; the other is that there are objects A and B such that HomSetA,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).


## 4. Gradual Subgroups

In Section 2.3, we have studied gradual subsets ofPX for any set X and considered in PX the operations: intersections and union. We can repeat the same procedure whenever we have a binary operation in X and translate it into PX, or a subset of PX, in the natural way. Thus, 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.


### 4.1. Gradual Subgroups

LetX be a nonempty set with a binary operation ∗, we define in PX∖∅ new binary and unary operations by(29)S1∗S2=s1∗s2|si∈Si,for everyS1,S2∈PX∖∅and,S−1=s−1|s∈S,for everyS,∈PX∖∅.Thus, we may define an operation on gradual subsets ofX (for simplicity, in this section, for a set X, a gradual subset of X is a gradual element of PX∖∅) by(30)σ1∗σ2α=σ1α∗σ2α,for everyσ1,σ2andα∈0,1,σ−1α=σα−1,for everyσandα∈0,1.Definition 3.
LetG 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(1)
σ∗σ⊆σ(2)
σ−1⊆σProposition 5.
Lete be the neutral element of a group G and σ a gradual subgroup, the following statements hold:(1)
e∈σα for any α∈0,1(2)
σα is a subgroup of G for any α∈0,1
Therefore, ifSG is the set of all subgroups of G, a gradual subgroup of G is just a gradual element of SG.Proof.
(1)
Leta∈σα, then a−1∈σα; hence, e=a∗a−1∈σα(2)
Leta,b∈σα, then b−1∈σα; hence, a∗b−1∈σα
Ifε is a gradual element of a group G, for any α∈0,1, we define εα=εα, 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.
We may also define finitely generated gradual subgroups: a gradual subgroupσ is finitely generated if there are gradual elements ε1,…,εt such that for any α∈L we have σα=ε1α,…,εtα. We represent this σ simply as ε1,…,εt.Proposition 6.
Letσ be a gradual subgroup of a group G. The following statements are equivalent:(i)
σ is finitely generated(ii)
There exist gradual elementsε1,…,εt such that σ=ε1,…,εt(iii)
There exists a positive integert such that each subgroup σα can be generated by t elements
Observe that due to Proposition5, gradual subgroups of G can be identified with subgroups of the direct product, indexed in (0, 1], of copies of G.


### 4.2. Normal Gradual Subgroups

By the aforementioned identification, the study of gradual subgroups is very simple. Thus, a gradual subgroup of a groupG is normal if for any α∈0,1, we have that σα⊆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/σα.Letσ be a normal gradual subgroup of G, for every α∈0,1, we have a gradual quotient group G/σα; hence, a gradual quotient set ρ of G is defined as ρα=G/σα, for every α∈0,1. We may represent it also by G/σ. For every α∈0,1, there is a group homomorphism G⟶G/σα=G/σα.We define a gradual quotient group ofG a gradual quotient set η of G such that for every α∈0,1, the projection pα:G⟶ηα is a group homomorphism.Proposition 7.
LetG be a group, then(1)
For every normal gradual subgroupσ of G, we have G/σ a gradual quotient group of G(2)
For every gradual quotient groupη of G, there is a normal gradual subgroup κ of G, defined κα=KerG⟶ηα, for any α∈0,1Lemma 12.
Letf:G⟶G′ be a group homomorphism.(1)
For any gradual subgroupσ of G, we have that f∗σ:0,1⟶G′, defined f∗σα=fσα as a gradual subgroup of G′(2)
For any gradual subgroupτ of G′, we have that f∗τ:0,1⟶G, defined f∗τα=f∗τα as a gradual subgroup of G(3)
Ifτ is normal, then f∗τ is normalLetσ1,σ2 be gradual subgroups of G, we define σ1⊆σ2 and say σ1 is a subgroup of σ2, if σ1α⊆σ2α for any α∈0,1.Lemma 13.
Letσ1,σ2 be normal gradual subgroups of G, the following statements are equivalent:(i)
σ1⊆σ2(ii)
For anyα∈0,1, there exist group homomorphisms hα such that the following diagrams commuteRemark 9.
In consequence, to includeG/σ2 inside this theory, we could introduce the notion of gradual quotient group of a gradual quotient group G/σ1; hence, study gradual objects which are not related to an ambient group. The same can be done in considering gradual subgroups. Thus, we could introduce the notion of gradual group or enriched gradual group, in a similar way as we did for gradual subsets and sets.
Ifσ1 and σ2 are gradual subgroups of G, we have a gradual subset σ1σ2 of G and not necessarily a gradual subgroup; we get a gradual subgroup whenever one of them is normal and, in this case, we have the following.Lemma 14.
Letσ1,σ2 be gradual subgroups of G such that σ1 is normal, then(1)
σ1σ2 is a gradual subgroup of G(2)
σ1σ2/σ1 defined σ1σ2/σ1α=σ1ασ2α/σ1α is a gradual subgroup of G/σ1
This theory can be enriched whenever we consider maps between the differentσα’s, i.e., enriched gradual subgroups. For instance, when there is an inclusion, σβ⊆σα whenever α≤β.


### 4.3. Decreasing Gradual Subgroups

A gradual subgroupσ of G is decreasing if for any α,β∈0,1 such that α≤β, we have σβ⊆σα. For any decreasing gradual subgroup σ of G, for every α∈0,1, we have that σ1⊆σα.Letσ be a gradual subgroup, we define the accumulation σc of σ as(31)σcα=∪σβ|β≥α,for anyα∈0,1.It is clear thatσc is a decreasing gradual subgroup, and a gradual subgroup σ is decreasing if, and only if, σ=σc. In particular, we have the following properties of the operator σ↦σc.Lemma 15.
LetG be a group, for every gradual subgroups σ,σ1,σ2, the following statements hold:(1)
σ⊆σc(2)
σc=σcc(3)
Ifσ1⊆σ2, then σ1c⊆σ2c(4)
σc is the smallest decreasing gradual subgroup containing σ(5)
Ifσ is a normal gradual subgroup, then σc is a normal subgroup and σcα is the set of all products of elements in ∪σβ|β≥α for any α∈0,1Proof.
Each element of∪σβ|β≥α is a product a1∗⋯∗at, for some ai∈σβi, and βi≥α. For any b∈G, we have g∗a1∗⋯∗at∗g−1=g∗a1∗g−1∗⋯∗g∗at∗g−1∈∪σβ|β≥α.
This means that the mapσ↦σc is a closure operator in the set G of all gradual subgroups of G which is compatible with the product in G. The set of all c-closed gradual subgroups of G is denoted by JG.Proposition 8.
Letσ1,σ2 be gradual subgroups of G, then σ1σ2c=σ1cσ2c. In addition, if either σ1 or σ2 is normal, then σ1σ2c=σ1cσ2c.Proof.
In fact, we have that bothσ1σ2cα and σ1cσ2cα are the subgroup generated by the subset ∪σ1β∪σ2β|β≥α.
Sinceσ1 is normal, each element of σ1∗σ2cα is a product a∗b; for a∈σ1cβ and b∈σ2cγ, some β,γ≥α is formed, then a∗b∈σ1cα∗σ2cγ. The converse is similar.
In the same way, we may define the strict accumulationσd of σas(32)σdα=σ1,ifα=1,∪σβ|β>α,for anyα∈0,1.
We have thatσd is a decreasing gradual subgroup and σd is normal whenever σ is. Some properties of the operator σ↦σd are the following, whose proof is similar to the proof of Lemma 15 and Proposition 8.Lemma 16.
LetG be a group, for any gradual subgroups σ,σ1,σ2, the following statements hold:(1)
σd⊆σc.(2)
Ifσ1⊆σ2, then σ1d⊆σ2d.(3)
σd=σdd=σcd=σcd.(4)
Ifσ is a normal gradual subgroup, then σd is a normal subgroup, and σcα is the set of all products of elements in ∪σβ|β>α for any α∈0,1.(5)
σ1σ2d=σ1dσ2d. In addition, if either σ1 or σ2 is normal, then σ1σd=σ1dσ2d.A gradual subgroupσ is an strict decreasing gradual subgroup whenever σ=σd, and we have the following.Lemma 17.
For any gradual subgroupσ, we have that σd is the biggest strict decreasing gradual subgroup contained in σc.These results mean that the mapσ↦σd is an interior operator in the set JdX of all decreasing gradual subgroups of G.


### 4.4. Gradual Subgroups and Fuzzy Subgroups

We shall show that there exists a strong relationship between gradual subgroups of a groupG and fuzzy subgroups of G. Remember that if G is a group, a fuzzy subgroup μ of G is a nonconstant, equal to 0, map μ:G⟶0,1 satisfying μxy−1≥μx∧μy, for any x,y∈G. In particular, if e is the neutral element of G, then μe≥μx and μx=μx−1 for any x∈G. Our aim is to identify fuzzy subgroups with some particular decreasing gradual subgroups.First, we need to realize some modifications to have well-defined gradual groups starting from a fuzzy group.In the set of all fuzzy subgroupsμ of G, we define a equivalence relation: μ1∼μ2 if μ1x=μ2x for any x≠e. In order to choose a canonical element in each equivalence class, following an idea in [8], for any fuzzy subgroup μ, we define μ1 as follows: μ1x=μx,ifx≠e,1,ifx=e.. Observe that each equivalence class μ has a unique element of the shape μ1, whenever μ1 is a fuzzy subgroup.Lemma 18.
Letμ be a fuzzy subgroup of a group G, then μ1 is a fuzzy subgroup and μ1 is normal whenever μ is.Proof.
Letx,y∈G, then μ1xy−1=μxy−1≥μx∧μy=μ1x∧μ1y whenever x,y,xy−1≠e. On the other hand, if xy−1=e, then μ1xy−1=1≥μ1x∧μ1y; if x≠e, y=e, then μ1y=1≥μ1x, and we have μ1xy−1=μ1x=μ1x∧μ1y.
A decreasing gradual subgroupσ satisfies property (F) if there exists Maxα|x∈σα for every x∈∪ασα. An strict decreasing gradual subgroup σ satisfies property (inf-F) if γ=Infβ|x∉σdβ satisfies x∈σγ for any x∈G.
Letσ be a decreasing gradual subgroup; let us denote σ∗α=σcα∖σdα, the difference set, for every α∈0,1.Lemma 19.
Letσ be a decreasing gradual subgroup such that σc≠σd; the following statements are equivalent:(a)
σc satisfies property (F)(b)
σd satisfies property (inf-F)(c)
∪•α∈0,1σ∗α=∪σα|α∈0,1
Letμ be the equivalence class of the fuzzy subgroup μ; we define a gradual subset σμ of G by(33)σμα=x∈G|μ1x≥α.Lemma 20.
The mapν:μ↦σμ is well defined, and σμ is a decreasing gradual subgroup satisfying property (F).Proof.
Observe that for any classμ, there exists only one fuzzy subgroup μ such that μ=μ1.
Letσ be a decreasing gradual subgroup satisfying property (F); we define a fuzzy subset μσ by(34)μσx=Maxγ|x∈σγ,ifx≠e,1,ifx=e.Lemma 21.
With the above notation,μσ is a fuzzy subgroup, and we have a map υ:σ↦μσ from the set of all decreasing gradual subsets satisfying property (F) to the set of all classes of fuzzy subgroups.Now, we have the announced relationship of gradual subgroups and fuzzy subgroups.Theorem 4.
LetG be a group, the maps ν:μ↦σμ and υ:σ↦μσ define a bijective correspondence between the following:(1)
Equivalence classes of fuzzy subgroupsμ of G(2)
Descending gradual subgroupsσ of G satisfying property (F)Proof.
Letμ=μ1 be a fuzzy group, for any x∈G, we have(35)μσμx=Maxα|x∈σμα,=Maxα|α≤μx,=μx.
On the other hand, letσ be a decreasing gradual subgroup satisfying property (F), for any α∈0,1, we have(36)σμσα=x|μσx≥α,=x|Maxβ|x∈σβ≥α,=σα.Lemma 22.
Letμ1,μ2 fuzzy subgroups; let us define μ1μ2 as μ1μ2.Proof.
The productμ1μ2 is well defined. Let μ=μ′, for any μ2, we have(37)μμ2x=Supμy∧μ2z|yz=x,=Supμ′y∧μ2z|yz=x,=μ′μ2x.Remark 10.
Unfortunately, in Theorem4, the map ν is not a homomorphism with respect to the product of classes of fuzzy subgroups. Indeed, for any μ1,μ2, we have(38)σμ1σμ2α=σμ1xσμ2α,=x|μ1x≥αx|μ2x≥α,⊆x|Supμ1y∧μ2z|yz=x≥α=x|μ1μ2x≥α,=σμ1μ2x.
This inclusion could be strict as the following example shows.Example 6.
We define fuzzy subgroupsμ1 and μ2 of ℤ as follows:(39)μ1x=0,ifx∈ℤ∖2ℤ,1−2t3t,ifx∈2tℤ∖2t+1ℤ,1,ifx=0.μ2x=0,ifx∈ℤ∖3ℤ,12−13t,ifx∈3tℤ∖3t+1ℤ,1,ifx=0.
We claimμ1+μ22=Supμ1y∧μ22−y|y∈ℤ≤1/2. Indeed, we have two possibilities:(1)
μ1y>1/2, then y∈4ℤ, i.e., there exists k∈ℤ such that y=4k. Hence, μ22−y=μ22−4k=μ221−2k<1/2 as 2−y≠0.(2)
μ1y<1/2.
In both cases, we haveμ1y∧μ22−y<1/2. In addition, we can choose y such that μ1y∧μ22−y is as closed to 1/2 as we desire. For any 2≤t,s∈ℕ, there exist k,h∈ℤ such that 2t−1k−3sh=1; hence, 2−2tk=21−2t−1k=3sh; now, if we take y=2tk, then μ1y≥1−1/3t and μ22−y≥1/2−1/3t. In consequence, 1/2>μ1y∧μ22−y≥1/2−1/3t, which implies that μ1+μ22=1/2 and 2∈μ1+μ21/2. On the other hand, we have μ11/2+μ21/2=4ℤ and 2∉μ11/2+μ21/2.
We shall change the assignation defined byν to consider ν˜:μ↦σ˜μ, in which σ˜μ is a strict decreasing gradual subgroup satisfying property (inf-F) and is defined by(40)σ˜μα=x∈G|μ1x>α,ifα≠1,x∈G|μ1x=1,ifα=1,whose inverse is υ˜:σ↦μ˜σ, defined as(41)μ˜σx=Infγ|x∉σγ,ifx≠e,1,ifx=e.
Thus, we have the following theorem.Theorem 5.
With the above notation, we have(1)
σ˜μ is a strict decreasing gradual subgroup satisfying property (inf-F), and ν˜ is well defined.(2)
μ˜σ is a fuzzy subgroup.(3)
The mapsν˜ and υ˜ define a bijective correspondence between equivalence classes of fuzzy subgroups μ of G and strict descending gradual subgroups σ of G satisfying property (inf-F).(4)
ν˜ is a homomorphism with respect to the product of classes of fuzzy subgroups.Proof.
(1)
First, we observe thatσ˜μ=σμd; hence, it is a strict gradual subgroup, and by Lemma 19, it satisfies property (inf-F). It is well defined as σμ is uniquely defined; hence, it is σ˜μ.(2)
It is a direct consequence of Lemma16.(3)
We can mimic the proof of Theorem4.(4)
For anyμ1,μ2, we have(42)σμ1σμ2α=σμ1xσμ2α,=x|μ1x>αx|μ2x>α,=x|Supμ1y∧μ2z|yz=x>α,=x|μ1μ2x>α,=σμ1μ2x.


### 4.5. Normal Fuzzy Subgroups

A fuzzy subgroupμ of a group G is normal if μη=ημ for any fuzzy subset η or equivalently if μxy=μyx for any x,y∈G (see [9]). We are interested in relating normal fuzzy subgroups and normal gradual subgroups. We have defined a gradual subgroup σ to be normal if σα⊆G is a normal subgroup for any α∈0,1.Lemma 23.
Letμ1,μ2 be a fuzzy subgroup such that μ1∼μ2 and μ1 is normal, then μ2 is normal.Proof.
By hypothesisμ1xy=μ1yx for every x,y∈G, if xy,yx≠e, then μ2xy=μ2yx. If xy=e, then x=y−1, hence yx=e, and we have μ2xy=μ2yx.
As a consequence, ifμ is a normal fuzzy subgroup, then every fuzzy subgroup in μ is normal; in particular, μ1 is normal.Theorem 6.
Letμ be a fuzzy subgroup, the following statements are equivalent:(i)
μ is normal(ii)
σ˜μ is normalProof.
We may assume, without loss of generality, thatμ=μ1. Let g∈G and let us consider the fuzzy subset ηg defined as the characteristic function of g, then(43)ηgμx=Supηgx1∧μx2|x=x1x2=ηgg∧μg−1x=μg−1x,and in the same way μηgx=μxg−1. Then,(44)σ˜μ=σ˜ηg−1μηg=σ˜ηg−1σ˜μσ˜ηg=g−1σ˜μg.
Therefore,σ˜μ is normal. Conversely, if σ˜μ is normal, for any element g∈G, we have g−1σ˜μg=σ˜μ; hence, ηg−1μηg=μ, and μ is a normal fuzzy subgroup.


### 4.6. Gradual Groups

From any decreasing gradual subgroupσ of a group G, we have two families: one σα|α∈0,1 is a family of groups and the other iα,β:σβ⟶σα|α≤β is the family of the inclusions. To include these objects inside a more general theory, we shall consider contravariant functors from (0, 1] to Gr, the category of groups.For any contravariant functorF:0,1→Gr and every α≤β, we have now a group homomorphism from Fβ to Fα, and the pair Fα|α∈0,1,Ffα,β|α≤β is a direct systems of groups and group homomorphisms; hence, there exists its direct limit, say lim⟶F.We define a gradual group as a contravariant functorF:0,1→Gr, and a gradual group homomorphisms from F1 to F2 is just a natural transformation from F1 to F2. Therefore, 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 subgroupσ of a group G. In this case, the direct limit lim⟶σ is isomorphic to a subgroup of G; indeed, it is the union ∪σα|α∈0,1.Following this example, for any arbitrary gradual groupF, we say F is a decreasing gradual group whenever each Ffα,β, for α≤β. The 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, defined on objects as follows: for any F∈J, we define Fdα=lim⟶α,1Fγ, 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.
LetF be a decreasing gradual group and θ:F1⟶F2 be a decreasing gradual map. The following statements hold:(1)
Fd 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 groupF such that F=Fd.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 defines. This 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 identified with adequate subcategories; see the forthcoming paper [10], in which we study gradual and fuzzy modules over a ring.


## 4.1. Gradual Subgroups

LetX be a nonempty set with a binary operation ∗, we define in PX∖∅ new binary and unary operations by(29)S1∗S2=s1∗s2|si∈Si,for everyS1,S2∈PX∖∅and,S−1=s−1|s∈S,for everyS,∈PX∖∅.Thus, we may define an operation on gradual subsets ofX (for simplicity, in this section, for a set X, a gradual subset of X is a gradual element of PX∖∅) by(30)σ1∗σ2α=σ1α∗σ2α,for everyσ1,σ2andα∈0,1,σ−1α=σα−1,for everyσandα∈0,1.Definition 3.
LetG 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(1)
σ∗σ⊆σ(2)
σ−1⊆σProposition 5.
Lete be the neutral element of a group G and σ a gradual subgroup, the following statements hold:(1)
e∈σα for any α∈0,1(2)
σα is a subgroup of G for any α∈0,1
Therefore, ifSG is the set of all subgroups of G, a gradual subgroup of G is just a gradual element of SG.Proof.
(1)
Leta∈σα, then a−1∈σα; hence, e=a∗a−1∈σα(2)
Leta,b∈σα, then b−1∈σα; hence, a∗b−1∈σα
Ifε is a gradual element of a group G, for any α∈0,1, we define εα=εα, 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.
We may also define finitely generated gradual subgroups: a gradual subgroupσ is finitely generated if there are gradual elements ε1,…,εt such that for any α∈L we have σα=ε1α,…,εtα. We represent this σ simply as ε1,…,εt.Proposition 6.
Letσ be a gradual subgroup of a group G. The following statements are equivalent:(i)
σ is finitely generated(ii)
There exist gradual elementsε1,…,εt such that σ=ε1,…,εt(iii)
There exists a positive integert such that each subgroup σα can be generated by t elements
Observe that due to Proposition5, gradual subgroups of G can be identified with subgroups of the direct product, indexed in (0, 1], of copies of G.


## 4.2. Normal Gradual Subgroups

By the aforementioned identification, the study of gradual subgroups is very simple. Thus, a gradual subgroup of a groupG is normal if for any α∈0,1, we have that σα⊆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/σα.Letσ be a normal gradual subgroup of G, for every α∈0,1, we have a gradual quotient group G/σα; hence, a gradual quotient set ρ of G is defined as ρα=G/σα, for every α∈0,1. We may represent it also by G/σ. For every α∈0,1, there is a group homomorphism G⟶G/σα=G/σα.We define a gradual quotient group ofG a gradual quotient set η of G such that for every α∈0,1, the projection pα:G⟶ηα is a group homomorphism.Proposition 7.
LetG be a group, then(1)
For every normal gradual subgroupσ of G, we have G/σ a gradual quotient group of G(2)
For every gradual quotient groupη of G, there is a normal gradual subgroup κ of G, defined κα=KerG⟶ηα, for any α∈0,1Lemma 12.
Letf:G⟶G′ be a group homomorphism.(1)
For any gradual subgroupσ of G, we have that f∗σ:0,1⟶G′, defined f∗σα=fσα as a gradual subgroup of G′(2)
For any gradual subgroupτ of G′, we have that f∗τ:0,1⟶G, defined f∗τα=f∗τα as a gradual subgroup of G(3)
Ifτ is normal, then f∗τ is normalLetσ1,σ2 be gradual subgroups of G, we define σ1⊆σ2 and say σ1 is a subgroup of σ2, if σ1α⊆σ2α for any α∈0,1.Lemma 13.
Letσ1,σ2 be normal gradual subgroups of G, the following statements are equivalent:(i)
σ1⊆σ2(ii)
For anyα∈0,1, there exist group homomorphisms hα such that the following diagrams commuteRemark 9.
In consequence, to includeG/σ2 inside this theory, we could introduce the notion of gradual quotient group of a gradual quotient group G/σ1; hence, study gradual objects which are not related to an ambient group. The same can be done in considering gradual subgroups. Thus, we could introduce the notion of gradual group or enriched gradual group, in a similar way as we did for gradual subsets and sets.
Ifσ1 and σ2 are gradual subgroups of G, we have a gradual subset σ1σ2 of G and not necessarily a gradual subgroup; we get a gradual subgroup whenever one of them is normal and, in this case, we have the following.Lemma 14.
Letσ1,σ2 be gradual subgroups of G such that σ1 is normal, then(1)
σ1σ2 is a gradual subgroup of G(2)
σ1σ2/σ1 defined σ1σ2/σ1α=σ1ασ2α/σ1α is a gradual subgroup of G/σ1
This theory can be enriched whenever we consider maps between the differentσα’s, i.e., enriched gradual subgroups. For instance, when there is an inclusion, σβ⊆σα whenever α≤β.


## 4.3. Decreasing Gradual Subgroups

A gradual subgroupσ of G is decreasing if for any α,β∈0,1 such that α≤β, we have σβ⊆σα. For any decreasing gradual subgroup σ of G, for every α∈0,1, we have that σ1⊆σα.Letσ be a gradual subgroup, we define the accumulation σc of σ as(31)σcα=∪σβ|β≥α,for anyα∈0,1.It is clear thatσc is a decreasing gradual subgroup, and a gradual subgroup σ is decreasing if, and only if, σ=σc. In particular, we have the following properties of the operator σ↦σc.Lemma 15.
LetG be a group, for every gradual subgroups σ,σ1,σ2, the following statements hold:(1)
σ⊆σc(2)
σc=σcc(3)
Ifσ1⊆σ2, then σ1c⊆σ2c(4)
σc is the smallest decreasing gradual subgroup containing σ(5)
Ifσ is a normal gradual subgroup, then σc is a normal subgroup and σcα is the set of all products of elements in ∪σβ|β≥α for any α∈0,1Proof.
Each element of∪σβ|β≥α is a product a1∗⋯∗at, for some ai∈σβi, and βi≥α. For any b∈G, we have g∗a1∗⋯∗at∗g−1=g∗a1∗g−1∗⋯∗g∗at∗g−1∈∪σβ|β≥α.
This means that the mapσ↦σc is a closure operator in the set G of all gradual subgroups of G which is compatible with the product in G. The set of all c-closed gradual subgroups of G is denoted by JG.Proposition 8.
Letσ1,σ2 be gradual subgroups of G, then σ1σ2c=σ1cσ2c. In addition, if either σ1 or σ2 is normal, then σ1σ2c=σ1cσ2c.Proof.
In fact, we have that bothσ1σ2cα and σ1cσ2cα are the subgroup generated by the subset ∪σ1β∪σ2β|β≥α.
Sinceσ1 is normal, each element of σ1∗σ2cα is a product a∗b; for a∈σ1cβ and b∈σ2cγ, some β,γ≥α is formed, then a∗b∈σ1cα∗σ2cγ. The converse is similar.
In the same way, we may define the strict accumulationσd of σas(32)σdα=σ1,ifα=1,∪σβ|β>α,for anyα∈0,1.
We have thatσd is a decreasing gradual subgroup and σd is normal whenever σ is. Some properties of the operator σ↦σd are the following, whose proof is similar to the proof of Lemma 15 and Proposition 8.Lemma 16.
LetG be a group, for any gradual subgroups σ,σ1,σ2, the following statements hold:(1)
σd⊆σc.(2)
Ifσ1⊆σ2, then σ1d⊆σ2d.(3)
σd=σdd=σcd=σcd.(4)
Ifσ is a normal gradual subgroup, then σd is a normal subgroup, and σcα is the set of all products of elements in ∪σβ|β>α for any α∈0,1.(5)
σ1σ2d=σ1dσ2d. In addition, if either σ1 or σ2 is normal, then σ1σd=σ1dσ2d.A gradual subgroupσ is an strict decreasing gradual subgroup whenever σ=σd, and we have the following.Lemma 17.
For any gradual subgroupσ, we have that σd is the biggest strict decreasing gradual subgroup contained in σc.These results mean that the mapσ↦σd is an interior operator in the set JdX of all decreasing gradual subgroups of G.


## 4.4. Gradual Subgroups and Fuzzy Subgroups

We shall show that there exists a strong relationship between gradual subgroups of a groupG and fuzzy subgroups of G. Remember that if G is a group, a fuzzy subgroup μ of G is a nonconstant, equal to 0, map μ:G⟶0,1 satisfying μxy−1≥μx∧μy, for any x,y∈G. In particular, if e is the neutral element of G, then μe≥μx and μx=μx−1 for any x∈G. Our aim is to identify fuzzy subgroups with some particular decreasing gradual subgroups.First, we need to realize some modifications to have well-defined gradual groups starting from a fuzzy group.In the set of all fuzzy subgroupsμ of G, we define a equivalence relation: μ1∼μ2 if μ1x=μ2x for any x≠e. In order to choose a canonical element in each equivalence class, following an idea in [8], for any fuzzy subgroup μ, we define μ1 as follows: μ1x=μx,ifx≠e,1,ifx=e.. Observe that each equivalence class μ has a unique element of the shape μ1, whenever μ1 is a fuzzy subgroup.Lemma 18.
Letμ be a fuzzy subgroup of a group G, then μ1 is a fuzzy subgroup and μ1 is normal whenever μ is.Proof.
Letx,y∈G, then μ1xy−1=μxy−1≥μx∧μy=μ1x∧μ1y whenever x,y,xy−1≠e. On the other hand, if xy−1=e, then μ1xy−1=1≥μ1x∧μ1y; if x≠e, y=e, then μ1y=1≥μ1x, and we have μ1xy−1=μ1x=μ1x∧μ1y.
A decreasing gradual subgroupσ satisfies property (F) if there exists Maxα|x∈σα for every x∈∪ασα. An strict decreasing gradual subgroup σ satisfies property (inf-F) if γ=Infβ|x∉σdβ satisfies x∈σγ for any x∈G.
Letσ be a decreasing gradual subgroup; let us denote σ∗α=σcα∖σdα, the difference set, for every α∈0,1.Lemma 19.
Letσ be a decreasing gradual subgroup such that σc≠σd; the following statements are equivalent:(a)
σc satisfies property (F)(b)
σd satisfies property (inf-F)(c)
∪•α∈0,1σ∗α=∪σα|α∈0,1
Letμ be the equivalence class of the fuzzy subgroup μ; we define a gradual subset σμ of G by(33)σμα=x∈G|μ1x≥α.Lemma 20.
The mapν:μ↦σμ is well defined, and σμ is a decreasing gradual subgroup satisfying property (F).Proof.
Observe that for any classμ, there exists only one fuzzy subgroup μ such that μ=μ1.
Letσ be a decreasing gradual subgroup satisfying property (F); we define a fuzzy subset μσ by(34)μσx=Maxγ|x∈σγ,ifx≠e,1,ifx=e.Lemma 21.
With the above notation,μσ is a fuzzy subgroup, and we have a map υ:σ↦μσ from the set of all decreasing gradual subsets satisfying property (F) to the set of all classes of fuzzy subgroups.Now, we have the announced relationship of gradual subgroups and fuzzy subgroups.Theorem 4.
LetG be a group, the maps ν:μ↦σμ and υ:σ↦μσ define a bijective correspondence between the following:(1)
Equivalence classes of fuzzy subgroupsμ of G(2)
Descending gradual subgroupsσ of G satisfying property (F)Proof.
Letμ=μ1 be a fuzzy group, for any x∈G, we have(35)μσμx=Maxα|x∈σμα,=Maxα|α≤μx,=μx.
On the other hand, letσ be a decreasing gradual subgroup satisfying property (F), for any α∈0,1, we have(36)σμσα=x|μσx≥α,=x|Maxβ|x∈σβ≥α,=σα.Lemma 22.
Letμ1,μ2 fuzzy subgroups; let us define μ1μ2 as μ1μ2.Proof.
The productμ1μ2 is well defined. Let μ=μ′, for any μ2, we have(37)μμ2x=Supμy∧μ2z|yz=x,=Supμ′y∧μ2z|yz=x,=μ′μ2x.Remark 10.
Unfortunately, in Theorem4, the map ν is not a homomorphism with respect to the product of classes of fuzzy subgroups. Indeed, for any μ1,μ2, we have(38)σμ1σμ2α=σμ1xσμ2α,=x|μ1x≥αx|μ2x≥α,⊆x|Supμ1y∧μ2z|yz=x≥α=x|μ1μ2x≥α,=σμ1μ2x.
This inclusion could be strict as the following example shows.Example 6.
We define fuzzy subgroupsμ1 and μ2 of ℤ as follows:(39)μ1x=0,ifx∈ℤ∖2ℤ,1−2t3t,ifx∈2tℤ∖2t+1ℤ,1,ifx=0.μ2x=0,ifx∈ℤ∖3ℤ,12−13t,ifx∈3tℤ∖3t+1ℤ,1,ifx=0.
We claimμ1+μ22=Supμ1y∧μ22−y|y∈ℤ≤1/2. Indeed, we have two possibilities:(1)
μ1y>1/2, then y∈4ℤ, i.e., there exists k∈ℤ such that y=4k. Hence, μ22−y=μ22−4k=μ221−2k<1/2 as 2−y≠0.(2)
μ1y<1/2.
In both cases, we haveμ1y∧μ22−y<1/2. In addition, we can choose y such that μ1y∧μ22−y is as closed to 1/2 as we desire. For any 2≤t,s∈ℕ, there exist k,h∈ℤ such that 2t−1k−3sh=1; hence, 2−2tk=21−2t−1k=3sh; now, if we take y=2tk, then μ1y≥1−1/3t and μ22−y≥1/2−1/3t. In consequence, 1/2>μ1y∧μ22−y≥1/2−1/3t, which implies that μ1+μ22=1/2 and 2∈μ1+μ21/2. On the other hand, we have μ11/2+μ21/2=4ℤ and 2∉μ11/2+μ21/2.
We shall change the assignation defined byν to consider ν˜:μ↦σ˜μ, in which σ˜μ is a strict decreasing gradual subgroup satisfying property (inf-F) and is defined by(40)σ˜μα=x∈G|μ1x>α,ifα≠1,x∈G|μ1x=1,ifα=1,whose inverse is υ˜:σ↦μ˜σ, defined as(41)μ˜σx=Infγ|x∉σγ,ifx≠e,1,ifx=e.
Thus, we have the following theorem.Theorem 5.
With the above notation, we have(1)
σ˜μ is a strict decreasing gradual subgroup satisfying property (inf-F), and ν˜ is well defined.(2)
μ˜σ is a fuzzy subgroup.(3)
The mapsν˜ and υ˜ define a bijective correspondence between equivalence classes of fuzzy subgroups μ of G and strict descending gradual subgroups σ of G satisfying property (inf-F).(4)
ν˜ is a homomorphism with respect to the product of classes of fuzzy subgroups.Proof.
(1)
First, we observe thatσ˜μ=σμd; hence, it is a strict gradual subgroup, and by Lemma 19, it satisfies property (inf-F). It is well defined as σμ is uniquely defined; hence, it is σ˜μ.(2)
It is a direct consequence of Lemma16.(3)
We can mimic the proof of Theorem4.(4)
For anyμ1,μ2, we have(42)σμ1σμ2α=σμ1xσμ2α,=x|μ1x>αx|μ2x>α,=x|Supμ1y∧μ2z|yz=x>α,=x|μ1μ2x>α,=σμ1μ2x.


## 4.5. Normal Fuzzy Subgroups

A fuzzy subgroupμ of a group G is normal if μη=ημ for any fuzzy subset η or equivalently if μxy=μyx for any x,y∈G (see [9]). We are interested in relating normal fuzzy subgroups and normal gradual subgroups. We have defined a gradual subgroup σ to be normal if σα⊆G is a normal subgroup for any α∈0,1.Lemma 23.
Letμ1,μ2 be a fuzzy subgroup such that μ1∼μ2 and μ1 is normal, then μ2 is normal.Proof.
By hypothesisμ1xy=μ1yx for every x,y∈G, if xy,yx≠e, then μ2xy=μ2yx. If xy=e, then x=y−1, hence yx=e, and we have μ2xy=μ2yx.
As a consequence, ifμ is a normal fuzzy subgroup, then every fuzzy subgroup in μ is normal; in particular, μ1 is normal.Theorem 6.
Letμ be a fuzzy subgroup, the following statements are equivalent:(i)
μ is normal(ii)
σ˜μ is normalProof.
We may assume, without loss of generality, thatμ=μ1. Let g∈G and let us consider the fuzzy subset ηg defined as the characteristic function of g, then(43)ηgμx=Supηgx1∧μx2|x=x1x2=ηgg∧μg−1x=μg−1x,and in the same way μηgx=μxg−1. Then,(44)σ˜μ=σ˜ηg−1μηg=σ˜ηg−1σ˜μσ˜ηg=g−1σ˜μg.
Therefore,σ˜μ is normal. Conversely, if σ˜μ is normal, for any element g∈G, we have g−1σ˜μg=σ˜μ; hence, ηg−1μηg=μ, and μ is a normal fuzzy subgroup.


## 4.6. Gradual Groups

From any decreasing gradual subgroupσ of a group G, we have two families: one σα|α∈0,1 is a family of groups and the other iα,β:σβ⟶σα|α≤β is the family of the inclusions. To include these objects inside a more general theory, we shall consider contravariant functors from (0, 1] to Gr, the category of groups.For any contravariant functorF:0,1→Gr and every α≤β, we have now a group homomorphism from Fβ to Fα, and the pair Fα|α∈0,1,Ffα,β|α≤β is a direct systems of groups and group homomorphisms; hence, there exists its direct limit, say lim⟶F.We define a gradual group as a contravariant functorF:0,1→Gr, and a gradual group homomorphisms from F1 to F2 is just a natural transformation from F1 to F2. Therefore, 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 subgroupσ of a group G. In this case, the direct limit lim⟶σ is isomorphic to a subgroup of G; indeed, it is the union ∪σα|α∈0,1.Following this example, for any arbitrary gradual groupF, we say F is a decreasing gradual group whenever each Ffα,β, for α≤β. The 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, defined on objects as follows: for any F∈J, we define Fdα=lim⟶α,1Fγ, 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.
LetF be a decreasing gradual group and θ:F1⟶F2 be a decreasing gradual map. The following statements hold:(1)
Fd 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 groupF such that F=Fd.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 defines. This 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 identified with adequate subcategories; see the forthcoming paper [10], in which we study gradual and fuzzy modules over a ring.


## 5. Conclusion

Our goal in this article has been to introduce more general notions than the fuzzy subset in order to find 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 satisfies property (inf-F). The 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 define the notion of fuzzy and gradual sets without the use of an ambient set. Thus, we have three degrees of abstraction, the first 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 different but compatible ways (1) defining contravariant functors to the category of groups;Gr0,1, and (2) defining groups in the functor category Set0,1.One of the goals of this paper was to find 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.



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