Extending L-Topologies to Bipolar $\mathcal{L}$-Fuzzy Topologies

Abstract

We develop a model that allows for the extension of an L-topology $\tau$ on a set X (i.e., $\tau \subseteq L^X$) to a bipolar $\mathcal{L}$-fuzzy topology $\mathcal{T}$ on this set (i.e., $\mathcal{T}:L^X\to \mathcal{L}$). This model is based on the use of an additional algebraic structure on a complete infinitely distributive lattice L, and the derived lattice $\mathcal{L}$ obtained by “bipolarizing” the original lattice L. The properties of the obtained bipolar $\mathcal{L}$-fuzzy topology are studied. A number of examples show how the choice of algebraic structure on L affects the resulting bipolar $\mathcal{L}$-fuzzy topology. In particular, we consider the case when the original lattice L is enriched with a structure of a Girard monoid. In this case our construction becomes most transparent. In addition, the relationship between the extended bipolar $\mathcal{L}$-fuzzy topology and the corresponding extended bipolar $\mathcal{L}$-fuzzy co-topology in this case becomes dual. In the last section we examine the proposed model from the categorical point of view.

1. Introduction

In “classic” (non-fuzzy) mathematics it usually seems pointless to ask to what extent a given object has a certain property: a topological space is either compact or not, a metric space is either complete or not, a group is either commutative or not, etc. On the other hand, within the framework of fuzzy mathematical structures, the tools of “fuzzy logic” make it possible to give some meaning to this issue. In particular, a considerable amount of work has been conducted to estimate the degree to which a fuzzy topological space or its fuzzy subset is compact, Hausdorff, connected, etc., and to what degree a function of (fuzzy) topological spaces is continuous, among other properties; see, for example [1,2,3,4,5,6]. Such an assessment of the presence of a property is usually evaluated by a value in a complete lattice L. However, in certain situations it may be appropriate to combine the assessment of the presence of a property with the assessment of the absence of this property or with the assessment of the presence of the opposite property in a given object (see the general discussion on these issues, e.g., in [7]). As a tool for implementing this approach, more general assessment scales can be used, in particular, the scales based on the so-called “intuitionistic” fuzzy sets [8] or on bipolar-valued fuzzy sets [9,10]. The purpose of this work is to initiate the use of bipolar fuzzy estimation of specific topological properties within the framework of fuzzy mathematical structures. Specifically, the goal is to present a bipolar estimation of the properties of openness and closedness for fuzzy sets in (fuzzy) topological spaces. Realizing this goal, we develop a model allowing to extend an L-topology $\tau$ (Definition 8) on a set X to an $(L,\mathcal{L})$-fuzzy topology $\mathcal{T}:L^X\to \mathcal{L}$ (Definition 10), where $\mathcal{L}$ is the bipolar lattice generated by the lattice L enriched with an additional algebraic structure (Section 2.2). For the first time, this scheme of fuzzification was presented at the International Conference on Topology and its Applications (Nafpaktos, Greece, 3–7 July 2023) [11]. Along with the purely theoretical interest in this model, we assume that it can serve as a tool for a deeper analysis (compared to previously undertaken studies) of (fuzzy) topologies and their properties. In particular, we assume that our model can be efficiently used for the qualitative analysis of such topological properties as compactness, connectedness versus disconnectedness and fulfilment of the separation axiom.

The structure of the paper is as follows. In the following preliminary sections, which are subdivided into four subsections, we present the preparatory material necessary to develop our theory. In the first subsection we specify the terminology related to lattices and introduce additional algebraic operations on lattices. The second subsection presents the construction of the bipolar lattice $\mathcal{L}$ from the original lattice L. This lattice $\mathcal{L}$ will play the major role in our work as the scale for the estimation of the degree of the presence of topological properties. The aim of the third subsection is to introduce the measures of inclusion and non-inclusion between L-fuzzy sets and study the properties of these measures. These measures will serve as the basis for the bipolar extension of L-topology realized further in this work. The purpose of the last subsection of the preliminary section is to provide an introduction to the topic of topology in the context of fuzzy sets. Although the concepts considered here will be familiar to people working in the field of fuzzy topology, we believe it is important to provide such a brief overview, firstly, to enable topologists not working in the fuzzy field to follow the ideas of this work, and secondly, to clarify the terminology used here and to make some historical remarks.

The extension of a L-topology $\tau \subseteq L^X$ to a bipolar $\mathcal{L}$-fuzzy topology $\mathcal{T}:L^X\to \mathcal{L}$ is implemented in Section 3. This extension is realized in three stages: first, extending $\tau \subseteq L^X$ to the positive component $L^{+}$ of the bipolar lattice Ł, then extending to its negative component $L^{-}$ and finally, by combining both components, we extend $\tau \subseteq L^X$ in the bipolar lattice $\mathcal{L}$-fuzzy topology $\mathcal{T}:L^X\to \mathcal{L}$. In the last subsection of Section 3, we illustrate the obtained results by a series of examples based on different original L-topological spaces and enriching the original lattice L with different algebraic structures.

Next, in Section 4, we develop a similar extension of an L-co-topology $\sigma$ on a set X to a bipolar $\mathcal{L}$-fuzzy co-topology $\mathcal{S}:L^X\to \mathcal{L}$ and study the properties of this extension. As will be explained in Section 2.4, in the context of this work it is important to develop co-topological questions of the theory independently, and not reduce them to topological ones.

Section 5 is devoted to the special case when the original lattice L is enriched with the structure of a Girard monoid. In this case our design becomes especially transparent and allows us to obtain a number of additional results. In particular, under the assumption that L is a Girard monoid, we have established relations between the positive and the negative parts of a extended bipolar $\mathcal{L}$-fuzzy topology. Moreover, in this case, the extended bipolar $\mathcal{L}$-fuzzy co-topology can be directly reduced from the corresponding extended bipolar $\mathcal{L}$-fuzzy topology.

In Section 6, we consider the extension of L-topologies to bipolar $\mathcal{L}$-topologies within the framework of the category theory. Some preliminary results about the relations between the category of L-topological spaces and the category of their bipolar $(L,\mathcal{L})$-fuzzy extension are established.

In the final Section 7, Conclusion, we return again to the motivation for using bipolar lattices in the study of fuzzy topologies and, after a brief overview of the main results obtained, discuss some prospects for the use of bipolar lattices in the study of L-fuzzy topologies, and more generally, in the context of fuzzification of other mathematical structures.

Concluding this introductory part, let us discuss the place of our work in the spectrum of research centred in bipolar fuzzy sets and in particular, in bipolar fuzzy topological spaces.

The concept of a bipolar fuzzy set was introduced and thoroughly studied in the papers by Zhang [9,10,12]. Lee [7,13] made important contributions to the study of bipolar fuzzy sets and their methodological justification. Note, however, that in these papers bipolar lattices are generated (in the sense of our paper) by the interval $[0, 1]$ and not by an arbitrary complete, completely distributive lattice L, as is the case in our approach.

As is usually the case when new valuable concepts emerge, bipolar fuzzy sets and bipolar fuzzy lattices have aroused great interest among both theoretically oriented researchers and specialists working in applied areas of “fuzzy mathematics”. In particular, interest in the use of bipolar structures to study questions of topology can be observed for at least the last ten years. The first definition of a bipolar fuzzy topological space can be found in [14]. A full-fledged theory of bipolar fuzzy topologies (i.e., bipolar $[0,1]$-topologies according to the terminology accepted here, c.f. Definition 8) is presented in the article by Kim et al. [15]. In particular, the authors introduce the concepts of a bipolar fuzzy topology, bipolar fuzzy base and subbase and study relations between them. Based on the concept of a bipolar fuzzy neighbourhood they define continuity for mappings between bipolar fuzzy topological spaces thus effectively coming to the category of fuzzy bipolar fuzzy topological spaces. The authors prove the existence of the initial bipolar fuzzy topology in this category. Some properties, in particular compactness in bipolar fuzzy topological spaces are studied. Bipolar fuzzy supra-topologies and their neighbourhood structure is the subject of a paper by Pazar-Varol et. al [16]. This paper also contains an application of bipolar fuzzy topologies in data mining. Roy at al. [17] introduce the notion of graded bipolar fuzzy topologies (i.e., bipolar $[0,1]$-fuzzy topologies according to our terminology, cf. Definition 9) and start the study of their properties. Furthermore, the notion of a bipolar gradation preserving map is given here. The concept of a bipolar fuzzy closure operator is also introduced and its characteristic properties are studied. A decomposition theorem involving bipolar gradation of openness and Chang type bipolar fuzzy topology is established. Some categorical results concerning bipolar fuzzy topology are proved.

Regarding the fundamental difference between previously completed work in the field of bipolar fuzzy topology and our research, we would like to note the following. If in the works of other authors known to us the object of study is bipolar fuzzy topological spaces, then in our approach bipolar fuzzy topological spaces arise as a result of analysis of given fuzzy topological spaces. Consequently, the bipolar lattice $\mathcal{L}$ in our case is not given a priori, but is constructed by bipolarizing the lattice L, which is included in the definition of a bipolarizable L-topological space. In other words, referring to Definition 10, we are working with induced $(L,\mathcal{L})$-fuzzy topological spaces while the objects of study in, e.g., Ref. [17] are apriori given $(\mathfrak{I},\mathfrak{I})$-fuzzy topological spaces, where $\mathfrak{I}$ is essentially the bipolarization of the unit interval $[0,1]$.

2. Preliminary Information: The Framework of Our Studies

2.1. Lattices Enriched with Additional Algebraic Structures

The fundamental role in our studies will play an algebraic structure $\mathfrak{L}=(L,\le ,$ $\wedge ,\vee ,\ast ,\mapsto {,}^c)$. Here, $(L,\le ,$ $\wedge ,\vee )$ is a frame (known also as a complete Heyting algebra) that is a complete lattice that satisfies the following infinite distributive law:

$$\left({\bigvee }_{i\in I}{a}_i\right)\wedge b={\bigvee }_{i\in I}(a_i\wedge b)\forall b\in L\forall \{a_i\mid i\in I\}\subset L.$$

See, e.g., Refs. [18,19]. In particular, let $\mathbf{0}$ and $\mathbf{1}$ be the bottom and the top elements of the lattice L, respectively. In the structure $\mathfrak{L}$, the lattice L is enriched with two binary operators $\ast :L\times L\to L$ (conjunction) and $\mapsto :L\times L\to L$ (implication) and one unary operation ^c $:L\to L$ (complementation). These operators are assumed to satisfy the axioms clarified in the following definitions.

Definition 1. 

A binary relation $\ast :L\times L\to L$ will be called a conjunction if it satisfies the following four conditions:

• $\left({\ast }_1\right)$ it is commutative: $a\ast b=b\ast a$ for all $a,b\in L$;
• $\left({\ast }_2\right)$ it is associative: $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in L$;
• $\left({\ast }_3\right)$ it is monotone: if $a\le a_1,b\le b_1$ then $a\ast b\le a_1\ast b_1$ for all $a,b,a_1,b_1\in L$;
• $\left({\ast }_4\right)$ the top element $\mathbf{1}$ acts as the unit in the monoid $(L,\ast ),i.e.,a\ast \mathbf{1}=a$ for every $a\in L.$
• A conjunction $\ast :L\times L\to L$ is called lower semi-continuous, i.e., satisfies $\left({\ast }_5\right):$
• $\left({\ast }_5\right)$ $({\bigvee }_{i\in I}{a}_i)\ast b={\bigvee }_{i\in I}(a_i\ast b)$ for every $\{a_i\mid i\in I\}\subseteq L,b\in L$.

Note that from the definition, it follows that $a\ast \mathbf{0}=\mathbf{0}$ for every $a\in L$.

For the reader’s convenience, we recall the interpretation of conjunction ∗ in the framework of fuzzy logic. If $a,b\in L$ are truth degrees for statements $P_a$ and $P_b$, then $a\ast b$ is the truth degree for the statement $P_a\&P_b.$

Various authors require different more or less general axioms when defining implication operator. The properties of an implication required for our goals are specified in the next definition:

Definition 2. 

A binary relation $\mapsto :L\times L\to L$ is called an implication if it satisfies the following conditions:

• $\left({\mapsto }_1\right)$ $\mathbf{0}\mapsto \mathbf{0}=\mathbf{0}\mapsto \mathbf{1}=\mathbf{1}\mapsto \mathbf{1}=\mathbf{1}$ and $\mathbf{1}\mapsto \mathbf{0}=\mathbf{0}$ (boundary conditions);
• $\left({\mapsto }_2\right)$ $b\le b_1⟹a\mapsto b\ge a\mapsto b_1\forall a,b,b_1\in L$, i.e., ↦ is non-increasing on the second argument;
• $\left({\mapsto }_3\right)$ $a\le a_1⟹a\mapsto b\ge a_1\mapsto b\forall a,a_1,b\in L,$ i.e., ↦ is non-decreasing on the second argument.
• An implication ↦ is called lower semi-continuous if it satisfies conditions $\left({\mapsto }_4\right)$, and $\left({\mapsto }_5\right)$:
• $\left({\mapsto }_4\right)$ $a\mapsto \left({\bigvee }_{i\in I}{b}_i\right)={\bigwedge }_{i\in I}(a\mapsto b_i)$ for any $a\in L$ and for any $\{b_i\mid i\in I\}\subseteq L$;
• $\left({\mapsto }_5\right)$ $\left({\bigvee }_{i\in I}{a}_i\right)\mapsto b={\bigwedge }_{i\in I}(a_i\mapsto b)$ for any $b\in L$ and for any $\{a_i\mid i\in I\}\subseteq L$.
• An implication is called idempotent if it satisfies condition $\left({\mapsto }_6\right)$:
• $\left({\mapsto }_6\right)$ $a\mapsto a=\mathbf{1}$ for any $a\in L.$

Note that $\left({\mapsto }_4\right)$ implies $\left({\mapsto }_2\right)$ and $\left({\mapsto }_5\right)$ implies $\left({\mapsto }_3\right)$. In turn, $\left({\mapsto }_6\right)$ implies the first two equalities in condition $\left({\mapsto }_1\right)$.

The properties $\left({\mapsto }_1\right)-\left({\mapsto }_3\right)$ are required in all definitions of an implication known to us, while the properties $\left({\mapsto }_4\right)-\left({\mapsto }_6\right)$ are more specific and are necessary for conducting our studies.

For the reader’s convenience, we recall the interpretation of implication ↦ in the framework of fuzzy logic. Namely, if $a,b\in L$ are the truth degrees for some statements $P_a$ and $P_b$, then $a\mapsto b$ is the truth degree for the statement, “if the statement $P_a$ holds then the statement $P_b$ also holds”.

Although, generally, we do not assume any relation between conjunction and implication in the lattice $(L,\le ,\wedge ,\vee ,\ast ,\mapsto )$, it is convenient to have in mind an important case when all the above listed properties of conjunction and implication are valid. It is the case of a residuated lattice. We recall this important concept in a way slightly adapted for the goals of this work.

Definition 3 

([20]). An algebraic structure $(L,\le ,\wedge ,\vee ,\ast ,\mapsto )$ will be called a residuated lattice if $(L,\le ,\wedge ,\vee )$ is a complete frame, $\ast :L\times L\to L$ is a lower semi-continuous conjunction, and ↦ is related with ∗ via Galois connections:

$$a\ast b\le c⟺a\le b\mapsto c.$$

In this case, ↦ is efficiently defined by $a\mapsto b=\bigvee \{\lambda \in L\mid a\ast \lambda \le b\}$ and is called the residuum induced by ∗ and is a lower semi-continuous implication.

Definition 4. 

A unary relation ^c $:L\to L$ is called a complementation if it is an order reversing involution that is

• $\left({\neg }_1\right)$ $a\le b⟹b^c\le a^c$ for all $a,b\in L$;
• $\left({\neg }_2\right)$ ${\left(a^c\right)}^c=a$ for every $a\in L$.

A typical example of complementation is the subtraction operation in the unit interval; that is, $a^c=1-a$ for every $a\in [0,1]$. In the context of fuzzy logic complementation, ^c $:L\to L$ means the following: “If a statement $P_a$ is true to the degree a, then the statement not $P_a$ is true to the degree $a^c$ ”.

The following property of complementation is well known. Since it will play an important role in our work, we provide it with a proof for completeness.

Proposition 1. 

A complementation satisfies the generalized de Morgan laws:

$${\left({\bigvee }_{i\in I}{a}_i\right)}^c={\bigwedge }_{i\in I}{a}_i^c\mathrm{and}{\left({\bigwedge }_{i\in I}{a}_i\right)}^c={\bigvee }_{i\in I}{a}_i^c\mathrm{for}\mathrm{all}\{a_i\mid i\in I\}\subseteq L.$$

Proof. 

Given a family $\{a_i\mid i\in I\}\subseteq L$, from the definitions, we obviously have that ${\left({\bigvee }_{i\in I}{a}_i\right)}^c\le a_{i_0}^c$ for every $a_{i_0}$; and hence,

$${\left({\bigvee }_{i\in I}{a}_i\right)}^c\le {\bigwedge }_{i\in I}{a}_i^c.$$

In a similar way, we show that

$${\left({\bigwedge }_{i\in I}{a}_i\right)}^c\ge {\bigvee }_{i\in I}{a}_i^c.$$

Now, replacing $a_i$ with $a_i^c$ for every $i\in I$ in the second inequality and taking the complementation once more, we have

$${\bigwedge }_{i\in I}{a}_i^c\le {\left({\bigvee }_{i\in I}{a}_i\right)}^c,$$

and this completes the proof of the first of the De Morgan laws. The second law can be proved similarly. □

2.2. Construction of the Bipolar Lattice $\mathcal{L}$

In our work, we use two copies of the lattice L denoted respectively by $L^{+}$ and $L^{-}$. The elements of $L^{+}$ are denoted as $\mathbf{0},a,\mathbf{1}$, while the corresponding elements of $L^{-}$ are denoted as ${\mathbf{0}}^{-},a^{-},$ and ${\mathbf{1}}^{-}$, respectively. We consider the correspondence

$${}^{\sim }:L^{+}⟷L^{-}{\mathrm{defined} \mathrm{by} }^{\sim }a=a^{-}{\mathrm{and} }^{\sim }{a}^{-}=a.$$

and introduce the partial order on $L^{-}$ by transferring the order ≤ on $L^{+}$:

$${}^{\sim }b{\le }^{\sim }{b}^{\prime }⟺b\ge b^{\prime }.$$

Obviously, $L^{-}$ can be interpreted as the lattice $L^{op}$ and, since $L^{+}=L$ is a complete lattice, then the lattice $L^{-}$ is also complete, with the bottom element ${\mathbf{1}}^{-}$ and the top element ${\mathbf{0}}^{-}$.

Further, let $\mathcal{L}=L^{+}\times L^{-}$. We introduce a partial order ⪯ on $\mathcal{L}$ by setting

$$(a,b)⪯(a^{\prime },b^{\prime })⟺a\le a^{\prime },b\le b^{\prime }⟺a\le a^{\prime }{,}^{\sim }b{\ge }^{\sim }{b}^{\prime };$$

Given a family $\mathcal{F}=\{(a_i,b_i)\mid \in I\}\subseteq \mathcal{L}$, we define

$${\bigvee }^{\mathcal{L}}\mathcal{F}=\left({\bigvee }_{i\in I}^L{a}_i,{\bigvee }_{i\in I}^{L^{-}}{b}_i\right);{\bigwedge }^{\mathcal{L}}\mathcal{F}=\left({\bigwedge }_{i\in I}^L{a}_i,{\bigwedge }_{i\in I}^{L^{-}}{b}_i\right).$$

From the construction, one can get easily the following theorem:

Theorem 1. 

$\left(\mathcal{L},⪯,{\wedge }^{\mathcal{L}},{\vee }^{\mathcal{L}}\right)$ is a complete lattice. Its top element ${\top }_{\mathcal{L}}$ is $(\mathbf{1},{\mathbf{0}}^{-})\in \mathcal{L}$; its bottom element ${\perp }_{\mathcal{L}}$ is $(\mathbf{0},{\mathbf{1}}^{-})\in \mathcal{L}$.

2.3. Measures of Inclusion and Non-Inclusion on L-Powersets

In order to extend L-topology on a set X to a bipolar $\mathcal{L}$-fuzzy topology on X, we need two measures defined on the L-powerset $L^X$ of the set X. For the first one, the measure of inclusion is used to define the degree of openness of L-fuzzy sets. In order to define the degree of non-openness of L-fuzzy sets, we introduce the measure of non-inclusion. The parallel use of these measures is laid in the definition of the bipolar $\mathcal{L}$-fuzzy topology, extending a given L-topology.

2.3.1. Measure of Inclusion between L-Fuzzy Sets

To find a reasonable definition of the measure of inclusion of one fuzzy set into another, we analyse the inclusion relation between two crisp sets $A,B\subseteq X$:

$A\subseteq B$ if and only if for every $x\in X$ if $x\in A$ then $x\in B$.

Imitating this statement in the context of fuzzy sets, one comes to the formula evaluating the measure of inclusion of an L-fuzzy set A into an L-fuzzy set B:

Definition 5 

(cf, e.g., Refs. [21,22]). Let X be a set and $\mapsto :L\times L\to L$ a continuous idempotent implication on a complete lattice L. The measure of inclusion of an L-fuzzy set $A\in L^X$ into an L-fuzzy set $B\in L^X$ is defined by

$$A\hookrightarrow B={\bigwedge }_{x\in X}A\left(x\right)\mapsto B\left(x\right).$$

In the following proposition, we collect basic well-known properties of the operator $\hookrightarrow :L^X\to L^X$; see, e.g., Ref. [23].

Proposition 2. 

Let $\hookrightarrow :L^X\times L^X\to L$ be induced by a lower semi-continuous implication $\mapsto :L\times L\to L$. Then $\hookrightarrow :L^X\times L^X\to L$ satisfies the following properties:

(1)

$\left({\bigvee }_i{A}_i\right)\hookrightarrow B={\bigwedge }_i\left(A_i\hookrightarrow B\right)$ for all $\{A_i\mid i\in I\}\subseteq L^X$, for all $B\in L^X;$

(2)

$A\hookrightarrow \left({\bigwedge }_i{B}_i\right)={\bigwedge }_i(A\hookrightarrow B_i)$ for all $A\in L^X,$ for all $\{B_i\mid i\in I\}\subseteq L^X;$

(3)

$A\hookrightarrow B=\mathbf{1}$ whenever $A\le B$;

(4)

$A_1\le A_2⟹A_1\hookrightarrow B\ge A_2\hookrightarrow B$;

(5)

$B_1\le B_2⟹A\hookrightarrow B_1\le A\hookrightarrow B_2$;

(6)

$\left({\bigwedge }_i{A}_i\right)\hookrightarrow \left({\bigwedge }_i{B}_i\right)\ge {\bigwedge }_i(A_i\hookrightarrow B_i)$ for all $\{A_i:i\in I\}$, $\{B_i:i\in I\}\subseteq L^X;$

(7)

$\left({\bigvee }_i{A}_i\right)\hookrightarrow \left({\bigvee }_i{B}_i\right)\ge {\bigwedge }_i(A_i\hookrightarrow B_i)$ for all $\{A_i:i\in I\}$, $\{B_i:i\in I\}\subseteq L^X.$

2.3.2. Measure of Non-Inclusion between L-Fuzzy Sets

We define a measure of non-inclusion on the powerset $L^X$ based on the operation ↛ on the lattice L, which can be interpreted as a certain co-implication.

Definition 6. 

Let $L=(L,\ast ,\mapsto {,}^c)$. We define the co-implication $\nrightarrow :L\to L$ by setting

$$a\nrightarrow b=a\ast b^c\forall a,b\in L.$$

Intuitively, the relation $a\nrightarrow b$ means that

“a does not imply b equals the degree to which a holds and b does not hold”.

In the following proposition we collect the basic properties of co-implication:

Proposition 3. 

Let $a_1,a_2,a,b_1,b_2,b\in L$; $\{a_i\mid i\in I\},\{b_i\mid i\in I\}\subseteq L.$ Then

(1)

$a_1\le a_2⟹a_1\nrightarrow b\le a_2\nrightarrow b$

(2)

$b_1\le b_2⟹a\nrightarrow b_1\ge a\nrightarrow b_2$

(3)

$\left({\bigvee }_{i\in I}{a}_i\right)\nrightarrow \left({\bigvee }_{i\in I}{b}_i\right)\le {\bigvee }_{i\in I}(a_i\nrightarrow b_i)$

(4)

$\left({\bigwedge }_{i\in I}{a}_i\right)\nrightarrow \left({\bigwedge }_{i\in I}{b}_i\right)\le {\bigvee }_{i\in I}(a_i\nrightarrow b_i)$

Proof. 

Referring to the properties of conjunction $\ast :L\times L\to L$ (Definition 1) and the properties of complementation ^c $:L\to L$ (Definition 4) we have

1.

$a_1\nrightarrow b=a\ast b^c\le a_2\ast b^c=a_2\nrightarrow b^c$.

2.

$a\nrightarrow b_1=a\ast b_1^c\le a\ast b_2^c=a\nrightarrow b_2^c.$

3.

$\left({\bigvee }_{i\in I}{a}_i\right)\nrightarrow \left({\bigvee }_{i\in I}{b}_i\right)={\bigvee }_{i\in I}{a}_i\ast {\left({\bigvee }_{i\in I}{b}_i\right)}^c=$${\bigvee }_{i\in I}{a}_i\ast \left({\bigwedge }_{i\in I}{b}_i^c\right)\le \left({\bigvee }_{i\in I}{a}_i\right)\ast b_{i_0}^c={\bigvee }_{i\in I}(a_i\ast b_{i_0}^c)\le$${\bigvee }_{i\in I}(a_i\ast b_i^c)={\bigvee }_{i\in I}(a_i\nrightarrow b_i).$

4.

$\left({\bigwedge }_{i\in I}{a}_i\right)\nrightarrow \left({\bigwedge }_{i\in I}{b}_i\right)={\bigwedge }_{i\in I}{a}_i\ast {\left({\bigwedge }_{i\in I}{b}_i\right)}^c=$ ${\bigwedge }_{i\in I}{a}_i\ast \left({\bigvee }_{i\in I}{b}_i^c\right)\le a_{i_0}\ast \left({\bigvee }_{i\in I}{b}_i^c\right)={\bigvee }_{i\in I}\left(a_{i_0}\ast b_i^c\right)\le {\bigvee }_{i\in I}\left(a_i\ast b_i^c\right)\le {\bigvee }_{i\in I}(a_i\nrightarrow b_i)$.

□

Based on the operator $\nrightarrow :L\times L\to L$, we introduce the measure of non-inclusion on the powerset $L^X$:

Definition 7. 

Let $A,B\in L^X$. The measure of non-inclusion of an L-fuzzy set A into an L-fuzzy set B is defined by

$$A\hookrightarrow ̸B={\bigvee }_{x\in X}(A\left(x\right)\nrightarrow B\left(x\right)).$$

Given L-fuzzy sets A and B, the relation $A\hookrightarrow ̸B$ intuitively means the largest degree achieved at a point $x\in X$ belonging to the L-fuzzy set A and not belonging to the L-fuzzy set B.

From Proposition 3, we get the following properties for the measure of non-inclusion:

Proposition 4. 

Let $A_1,A_2,A,B_1,B_2,B\in L^X$; $\{A_i\mid i\in I\},\{B_i\mid i\in I\}\subseteq L^{.}$ Then

(1)

$A_1\le A_2⟹A_1\hookrightarrow ̸B\le A_1\hookrightarrow ̸B$

(2)

$B_1\le B_2⟹A\hookrightarrow ̸B_1\ge A\hookrightarrow ̸B_2$

(3)

$\left({\bigvee }_{i\in I}{A}_i\right)\hookrightarrow ̸\left({\bigvee }_{i\in I}{B}_i\right)\le {\bigvee }_{i\in I}\left(A_i\hookrightarrow ̸B_i\right)$

(4)

$\left({\bigwedge }_{i\in I}{A}_i\right)\hookrightarrow ̸\left({\bigwedge }_{i\in I}{B}_i\right)\le {\bigvee }_{i\in I}\left(A_i\hookrightarrow ̸B_i\right)$

Proof. 

(1) If $A_1\le A_2$, then $A_1\left(x\right)\nrightarrow B\left(x\right)\ge A_2\left(x\right)\nrightarrow B\left(x\right)$ for each $x\in X$. Taking the supremum on the both sides of this inequality, we get

$A_1\hookrightarrow ̸B={\bigvee }_{x\in X}(A_1\left(x\right)\nrightarrow B\left(x\right))\ge {\bigvee }_{x\in X}(A_2\left(x\right)\nrightarrow B\left(x\right))=A_2\hookrightarrow ̸B.$

(2)

If $B_1\le B_2$, then we have $A\left(x\right)\nrightarrow B_1\left(x\right)\le A\left(x\right)\nrightarrow B_2\left(x\right)$ for each $x\in X$. Taking the supremum on the both sides of this inequality, we get

$$A\hookrightarrow ̸B_1={\bigvee }_{x\in X}(A\left(x\right)\nrightarrow B_1\left(x\right))\le {\bigvee }_{x\in X}(A\left(x\right)\nrightarrow B_2\left(x\right))=A\hookrightarrow ̸B_2.$$

(3)

From Proposition 3(3), we have for every $x\in X$

$$\left({\bigvee }_{i\in I}{A}_i\left(x\right)\right)\nrightarrow \left({\bigvee }_{i\in I}{B}_i\left(x\right)\right)\le {\bigvee }_{i\in I}(A_i\left(x\right)\nrightarrow B_i\left(x\right)).$$

Taking the supremum on the both sides of the inequality, we get

$$\left({\bigvee }_{i\in I}{A}_i\right)\hookrightarrow ̸\left({\bigvee }_{i\in I}{B}_i\right)={\bigvee }_{x\in X}\left(\left({\bigvee }_{i\in I}{A}_i\left(x\right)\right)\nrightarrow \left({\bigvee }_{i\in I}{B}_i\left(x\right)\right)\right)\le$$

$${\bigvee }_{x\in X}\left({\bigvee }_{i\in I}(A_i\left(x\right)\nrightarrow B_i\left(x\right))\right)={\bigvee }_{i\in I}\left({\bigvee }_{x\in X}(A_i\left(x\right)\nrightarrow B_i\left(x\right))\right)=$$

(4)

From Proposition 3(4), we have for every $x\in X$

$$\left({\bigwedge }_{i\in I}{A}_i\left(x\right)\right)\nrightarrow \left({\bigwedge }_{i\in I}{B}_i\left(x\right)\right)\le {\bigvee }_{i\in I}(A_i\left(x\right)\nrightarrow B_i\left(x\right)).$$

Taking the supremum on the both sides of the inequality, we get

$$\begin{gathered} \left({\bigwedge }_{i\in I}{A}_i\right)\hookrightarrow ̸\left({\bigwedge }_{i\in I}{B}_i\right)={\bigvee }_{x\in X}\left({\bigwedge }_{i\in I}{A}_i\left(x\right)\right)\nrightarrow \left({\bigwedge }_{i\in I}{B}_i\left(x\right)\right)\le {\bigvee }_{x\in X}\left({\bigvee }_{i\in I}(A_i\left(x\right)\nrightarrow B_i\left(x\right))\right)= \\ {\bigvee }_{i\in I}\left({\bigvee }_{x\in X}(A_i\left(x\right)\nrightarrow B_i\left(x\right))\right)={\bigvee }_{i\in I}\left(A_i\hookrightarrow ̸B_i\right). \end{gathered}$$

□

2.4. Topological Structures in the Context of Fuzzy Sets

The first definition of a fuzzy topology was introduced by Zadeh’s student C.L. Chang [24] just three years after the celebrated article by Zadeh [25]. Soon after, this definition was generalized by J.A. Goguen [26] for the case of L-fuzzy sets where L is an arbitrary complete frame. Following, for example, works [27,28], such “fuzzy topologies” are usually called L-topologies now.

Definition 8 

([24,26]). Let X be a set. A family of L-fuzzy subsets $\tau \subseteq L^X$ is called an L-topology on a set X if the following conditions are satisfied:

(1)

${\mathbf{0}}_X,{\mathbf{1}}_X\in \tau ;$

(2)

$A,B\in \tau ⟹A\wedge B\in \tau ;$

(3)

$\{A_i\mid i\in I\}\subseteq L⟹{\bigvee }_{i\in I}{A}_i\in \tau .$

The corresponding pair $(X,\tau )$ is called an L-topological space. Given two L-topological spaces $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$, a mapping $f:X\to Y$ is called continuous if $V\in {\tau }_Y⟹f^{-1}\left(V\right)\in {\tau }_X.$

Note that in this definition, only sets are fuzzy while the topology itself is just a crisp subset of the powerset $L^X$. Considering this as a known inconsistency, the authors of [4,29] (independently), suggested a definition admitting both sets and the topology to be fuzzy:

Definition 9 

([4,6,29]). Let X be a set and L be a complete frame. A mapping $\mathcal{T}:L^X\to L$ is called an L-fuzzy topology on a set X if the following conditions are satisfied:

(1)

$\mathcal{T}\left({\mathbf{0}}_X\right)=\mathcal{T}\left({\mathbf{1}}_X\right)=\mathbf{1};$

(2)

$\mathcal{T}(A\wedge B)\ge \mathcal{T}\left(A\right)\wedge \mathcal{T}\left(B\right)\forall A,B\in L^X;$

(3)

$\mathcal{T}\left({\bigvee }_{i\in I}{A}_i\right)\ge {\bigwedge }_{i\in I}\mathcal{T}\left(A_i\right)\forall \{A_i\mid i\in I\}\subseteq L^X.$

The corresponding pair $(X,\mathcal{T})$ is called an L-fuzzy topological space.

(Later, some other authors came (apparently independently) to a similar definition, see [30,31].)

Finally, noticing that the role of the lattice L as the domain for L-fuzzy sets and its role as the range of fuzzy topology are unrelated, the following general notion of an $(L,M)$-fuzzy topology was introduced:

Definition 10 

([32,33,34]). Let X be a set and let $L,M$ be complete frames with ${\mathbf{0}}^L,{\mathbf{0}}^M,{\mathbf{1}}^L,$ and ${\mathbf{1}}^M$ denoting their bottom and top elements, respectively. A mapping $\mathcal{T}:L^X\to M$ is called an $(L,M)$-fuzzy topology on a set X if the following conditions are satisfied:

(1)

$\mathcal{T}\left({\mathbf{0}}_X^L\right)=\mathcal{T}\left({\mathbf{1}}_X^L\right)={\mathbf{1}}^M;$

(2)

$\mathcal{T}(A\wedge B)\ge \mathcal{T}\left(A\right)\wedge \mathcal{T}\left(B\right)\forall A,B\in L^X;$

(3)

$\mathcal{T}\left({\bigvee }_{i\in I}{A}_i\right)\ge {\bigwedge }_{i\in I}\mathcal{T}\left(A_i\right)\forall \{A_i\mid i\in I\}\subseteq L^X.$

The corresponding pair $(X,\mathcal{T})$ is called an $(L,M)$-fuzzy topological space. Given two $(L,M)$-fuzzy topological spaces $(X,{\mathcal{T}}_X)$ and $(Y,{\mathcal{T}}_Y)$, a mapping $f:X\to Y$ is called continuous if

$${\mathcal{T}}_Y\left(V\right)\le {\mathcal{T}}_X\left(f^{-1}\left(V\right)\right)\forall V\in L^Y.$$

In particular, in the case where $L=\{0,1\}=\mathbf{2}$ is a two point lattice, $(\mathbf{2},M)$-fuzzy topological spaces correspond to the case considered by U. Höhle in [35] and further rediscovered by M.S. Ying [36] by analysing axioms of topology by means of fuzzy logic rules. Such spaces are now usually called fuzzifying topological spaces.

In an ordinary, crisp topological space $(X,T)$, the co-topology S, i.e., the family of closed sets, is derived from the topology T as the family of complements of open sets. Thus, the topology T and the corresponding co-topology S mutually determine each other. On the other hand, in the fuzzy framework it is often reasonable to consider separately topological and co-topological structures on a given set X. The main reason for this is that, typically, a complete lattice L, the range for fuzzy sets, is not assumed to have an order reversing involution that is crucial for the interconnection between properties of closedness and openness. Moreover, even in the case when L is endowed with an order-reversing involution, in some cases it is important to define closedness not referring to the openness of objects. In particular, this has led to the concept of a ditopological texture space, first introduced in [37], see also [38,39] et al., where ditopological (texture) spaces are studied. This situation is also applicable to our theory. There is not a priory connection between the operators $\mapsto :L\times L\to L$, $\ast :L\times L\to L$ and ^c $:L\to L$; hence, it makes no sense to reduce problems related to co-topology to topology issues. Therefore, we consider L-co-topology without referring to L-topology defined above.

Definition 11. 

see, for example [40,41]. A family of L-fuzzy subsets $\sigma \subseteq L^X$ is called an L-co-topology on a set X if the following conditions are satisfied:

(1)

${\mathbf{0}}_X,{\mathbf{1}}_X\in \sigma ;$

(2)

$A,B\in \sigma ⟹A\vee B\in \sigma ;$

(3)

$\{A_i\mid i\in I\}\subseteq \sigma ⟹{\bigwedge }_{i\in I}{A}_i\in \sigma .$

• The corresponding pair $(X,\sigma )$ is called an L-co-topological space.

Definition 12 

([32,33,34]). A mapping $\mathcal{S}:L^X\to M$ is an $(L,M)$-fuzzy co-topology on a set X if the following conditions are satisfied:

(1)

$\mathcal{S}\left({\mathbf{0}}_X^L\right)=\mathcal{S}\left({\mathbf{1}}_X^L\right)={\mathbf{1}}^M;$

(2)

$\mathcal{S}(A\vee B)\ge \mathcal{S}\left(A\right)\wedge \mathcal{S}\left(B\right)\forall A,B\in L^X;$

(3)

$\mathcal{S}\left({\bigwedge }_{i\in I}{A}_i\right)\ge {\bigwedge }_{i\in I}\mathcal{S}\left(A_i\right)\forall \{A_i\mid i\in I\}\subseteq L^X.$

The corresponding pair $(X,\mathcal{S})$ is called an $(L,M)$-fuzzy co-topological space.

3. Bipolar Extension of an L-Topology

In this section, we extend an L-topology $\tau \subseteq L^X$ to an $(L,\mathcal{L})$-fuzzy topology ${\mathcal{T}}_{\tau }:L^X\to \mathcal{L}$. We realize this extension by separately evaluating the degrees of openness and non-openness for L-fuzzy subsets of the space $(X,\tau )$. This evaluation is considered in the following three subsections. In the first one, we consider the openness degree of L-fuzzy sets and come to $(L,L^{+})$-fuzzy topology ${\mathcal{T}}_{\tau }^{+}:L^X\to L^{+}$, the second one deals with non-openness degree, and in the result gives rise to the $(L,L^{-})$-fuzzy topology ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$. Finally, in the third and final subsection, we unite these fuzzy topologies into a combined bipolar fuzzy topology ${\mathcal{T}}_{\tau }:L^X\to \mathcal{L}.$

3.1. Openness Degree of Fuzzy Subsets in L-Topological Spaces

Let $(X,\tau )$ be an L-topological space. Given an L-fuzzy set $A\in L^X$, we define an L-fuzzy set $A^o$, called the interior of A in the space $(X,\tau )$, by setting $A^o=\bigvee \{U\mid U\in \tau ,U\le A\}$. It is well known (see, e.g., Refs. [40,42,43]) and easy to see that $A^o$ is the largest open L-fuzzy set contained in A. It is also well known that ${(A\wedge B)}^o=A^o\wedge B^o$ and ${\left(A^o\right)}^o=A^o$ for any $A,B\in L^X$ and ${\mathbf{1}}_X^o=1{\mathbf{1}}_X;{\mathbf{0}}_X^o={\mathbf{0}}_X$.

We use the interior $A^o$ of A and operator of inclusion ↪ defined by a lower semi-continuous idempotent implication ↦ (Definition 2) in order to measure the degree of openness of an L-fuzzy set A in an L-topological space $(X,\tau )$ as follows: ${\mathcal{T}}_{\tau }^{+}\left(A\right)=A\hookrightarrow A^o.$ By varying fuzzy sets A over $L^X$, we get a mapping ${\mathcal{T}}_{\tau }^{+}:L^X\to L.$

The following theorem describes the positive component of the bipolar extension of L-topology $\tau$:

Theorem 2. 

The mapping ${\mathcal{T}}_{\tau }^{+}:L^X\to L^{+}$ is an L-fuzzy topology on the set $X.$

Proof. 

(1) Referring to the Definition 2 of implication and the Definition 8 of an L-topology, we easily get that ${\mathcal{T}}_{\tau }^{+}\left({\mathbf{0}}_X\right)=\mathbf{1}$ and ${\mathcal{T}}_{\tau }^{+}\left({\mathbf{1}}_X\right)=\mathbf{1}$.

(2)

Recalling that ${(A\wedge B)}^o=A^o\wedge B^o$ for any $A,B\in L^X$, we have the following series of (in)equalities:

$$\begin{gathered} {\mathcal{T}}_{\tau }^{+}(A\wedge B)=A\wedge B\hookrightarrow {(A\wedge B)}^o=A\wedge B\hookrightarrow A^o\wedge B^o{\ge }_{Prop.2\left(7\right)}= \\ (A\hookrightarrow A^o)\wedge (B\hookrightarrow B^o)={\mathcal{T}}_{\tau }^{+}\left(A\right)\wedge {\mathcal{T}}_{\tau }^{+}\left(B\right). \end{gathered}$$

(3)

To prove the last condition, notice first that ${\left({\bigvee }_{i\in I}{A}_i\right)}^o\ge {\bigvee }_{i\in I}{A}_i^o$ for each family $\{A_i\mid i\in I\}\subseteq L^X$. Now, we have the following sequence of (in)equalities:

$$\begin{gathered} {\mathcal{T}}_{\tau }^{+}\left({\bigvee }_{i\in I}{A}_i\right)=\left({\bigvee }_{i\in I}{A}_i\right)\hookrightarrow {\left({\bigvee }_{i\in I}{A}_i\right)}^o{\ge }_{Prop.2\left(2\right)}\left({\bigvee }_{i\in I}{A}_i\right)\hookrightarrow \left({\bigvee }_{i\in I}{A}_i^o\right){=}_{Prop.2\left(2\right)} \\ {\bigwedge }_{i\in I}(A_i\hookrightarrow A_i^o)={\bigwedge }_{i\in I}{\mathcal{T}}_{\tau }^{+}\left(A_i\right). \end{gathered}$$

□

As the following important theorem shows, the continuity of a mapping between L-topological spaces implies the continuity of the mapping between the extended positive components of these spaces. This fact will play the crucial role when studying categorical properties of the bipolar extension in Section 6.

Theorem 3. 

If $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ is continuous, then $f:(X,{{\mathcal{T}}_{{\tau }_X}}^{+})\to (Y,{{\mathcal{T}}_{{\tau }_Y}}^{+})$ is continuous.

Proof. 

Recall first that if a mapping $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ is continuous, then ${\left(f^{-1}\left(V\right)\right)}^o\ge f^{-1}\left(V^o\right)$ for every $V\in L^Y$. Now, the proof follows from the following series of (in)equalities:

$$\begin{gathered} {\mathcal{T}}_X^{+}\left(f^{-1}\left(V\right)\right)=f^{-1}\left(V\right)\hookrightarrow {\left(f^{-1}\left(V\right)\right)}^o{\ge }_{Prop2.\left(2\right)} f^{-1}\left(V\right)\hookrightarrow f^{-1}\left(V^o\right))= \\ {\bigwedge }_{x\in X}\left(f^{-1}\left(V\right)\left(x\right)\mapsto f^{-1}\left(V^o\right)\left(x\right)\right)={\bigwedge }_{x\in X}\left(V\left(f\left(x\right)\right)\mapsto V^o\left(f\left(x\right)\right)\right)\ge \\ {\bigwedge }_{\in X}\left(V\left(f\left(x\right)\right)\mapsto V^o\left(f\left(x\right)\right)\right)=V\hookrightarrow V^o={\mathcal{T}}_Y^{+}\left(V\right). \end{gathered}$$

□

3.2. Non-Openness Degree of L-Fuzzy Subsets in L-Topological Spaces

Let $(X,\tau )$ be an L-topological space, let A be its L-fuzzy subset, and let $A^o=\bigvee \{U\mid U\in \tau ,U\le A\}$ be its interior.

Definition 13. 

The degree of non-openness of a fuzzy set in an L-topological space is defined by

$${\mathcal{T}}_{\tau }^{-}\left(A\right){=}^{\sim }\left(A\hookrightarrow ̸A^o\right).$$

By varying A over $L^X$, we get an operator ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ of non-openness of fuzzy subsets in the L-topological space $(X,\tau )$.

The following theorem describes the negative component of the bipolar extension of L-topology $\tau$:

Theorem 4. 

The mapping ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ is an $(L,L^{-})$-fuzzy topology on the set X.

Proof. 

(1) ${}^{\sim }\mathcal{T}_{\tau }^{-}\left({\mathbf{0}}_X\right)={\bigvee }_{x\in X}\left({\mathbf{0}}_X\left(x\right)\ast {\left({\mathbf{0}}_X^o\right)}^c\left(x\right)\right)={\bigvee }_{x\in X}\left({\mathbf{0}}_X\left(x\right)\ast {\mathbf{1}}_X\left(x\right)\right)=\mathbf{0}\in L$. Hence, ${\mathcal{T}}_{\tau }\left({\mathbf{0}}_X\right)={\mathbf{0}}^{-}$, which is the top element of the lattice $L^{-}$.

• ${}^{\sim }\mathcal{T}_{\tau }^{-}\left({\mathbf{1}}_X\right)={\bigvee }_{x\in X}\left({\mathbf{1}}_X\left(x\right)\ast {\left({\mathbf{1}}_X^o\right)}^c\left(x\right)\right)={\bigvee }_{x\in X}\left({\mathbf{1}}_X\left(x\right)\ast {\mathbf{0}}_X\left(x\right)\right)=\mathbf{0}$. Hence, ${\mathcal{T}}_{\tau }^{(}{\mathbf{1}}_X)={\mathbf{0}}^{-}$, which is the top element of $L^{-}$. Thus, the first axiom of $(L,L^{-})$-fuzzy topology for ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ is valid.
• (2) Let $A,B\in L^X$. Then referring to Proposition 4(4), we have

  $$\begin{gathered} {}^{\sim }\mathcal{T}_{\tau }^{-}(A\wedge B)=A\wedge B\hookrightarrow ̸{(A\wedge B)}^o=A\wedge B\hookrightarrow ̸A^o\wedge B^o\le \left(A\hookrightarrow ̸A^o\right)\wedge \left(B\hookrightarrow ̸B^o\right)= \\ {}^{\sim }\mathcal{T}_{\tau }^{-}\left(A\right){\vee }^{\sim }{\mathcal{T}}_{\tau }^{-}\left(B\right). \end{gathered}$$

Hence, ${\mathcal{T}}_{\tau }^{-}(A\wedge B)\ge {\mathcal{T}}_{\tau }^{-}\left(A\right)\wedge {\mathcal{T}}_{\tau }^{-}\left(B\right)$, i.e., the second axiom of the $(L,L^{-})$-fuzzy topology for ${\mathcal{T}}_{\tau }^{-}$ is valid.

• (3) Let $\{A_i\mid i\in I\}\subseteq L^X$. Then, referring to Proposition 4(3), we have

  $${}^{\sim }\mathcal{T}^{-}\left({\bigvee }_{i\in I}\left(A_i\right)\right)={\bigvee }_{i\in I}\left(A_i\right)\hookrightarrow ̸{\left({\bigvee }_{i\in I}\left(A_i\right)\right)}^o\le {\bigvee }_{i\in I}\left(A_i\hookrightarrow ̸A_i^o\right)={}^{\sim }({\bigvee }_{i\in I}{\mathcal{T}}^{-}{A}_i).$$

Thus, ${\mathcal{T}}_{\tau }^{-}\left({\bigvee }_{i\in I}{A}_i\right)\ge {\bigwedge }_{i\in I}{\mathcal{T}}^{-}\left(A_i\right)$; hence, ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ satisfies also the third axiom of an $(L,L^{-})$-fuzzy topology. □

3.3. $(L,\mathcal{L})$-Fuzzy Topology on an L-Topological Space

Above, we have defined two fuzzy topologies on a L-topological space $(X,\tau )$: an $(L,L^{+})$-fuzzy topology ${\mathcal{T}}_{\tau }^{+}:L^X\to L^{+}$ determining the openness degree of a fuzzy set $A\in L^X$ and an $(L,L^{-})$-fuzzy topology ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ determining the non-openness degree of a fuzzy set $A\in L^X$. Based on these fuzzy topologies, we define a mapping ${\mathcal{T}}_{\tau }=({\mathcal{T}}_{\tau }^{+},{\mathcal{T}}_{\tau }^{-}):L^X\to \mathcal{L}$ by setting ${\mathcal{T}}_{\tau }\left(A\right)=({\mathcal{T}}_{\tau }^{+}\left(A\right),{\mathcal{T}}_{\tau }^{-}\left(A\right))\in \mathcal{L}$ for every $A\in L^X.$ The following theorem is the main result of this section.

Theorem 5. 

The mapping ${\mathcal{T}}_{\tau }:L^X\to \mathcal{L}$ is an $(L,\mathcal{L})$-fuzzy topology on the set X.

Proof. 

(1) ${\mathcal{T}}_{\tau }\left({\mathbf{0}}_X\right)=({\mathcal{T}}_{\tau }^{+}\left({\mathbf{0}}_X\right),{\mathcal{T}}_{\tau }^{-}\left({\mathbf{0}}_X\right))=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}}$;

${\mathcal{T}}_{\tau }\left({\mathbf{1}}_X\right)=({\mathcal{T}}_{\tau }^{+}\left({\mathbf{1}}_X\right),{\mathcal{T}}_{\tau }^{-}\left({\mathbf{1}}_X\right))=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}}$.

(2) Given $A,B\in L^X$ we have:

$$\begin{gathered} {\mathcal{T}}_{\tau }(A\wedge B)=\left({\mathcal{T}}_{\tau }^{+}(A\wedge B),{\mathcal{T}}_{\tau }^{-}(A\wedge B)\right)⪰ \\ \left(({\mathcal{T}}_{\tau }^{+}\left(A\right)\wedge {\mathcal{T}}_{\tau }^{+}\left(B\right),({\mathcal{T}}_{\tau }^{-}\left(A\right)\wedge {\mathcal{T}}_{\tau }^{-}\left(B\right)\right)= \\ \left(({\mathcal{T}}_{\tau }^{+}\left(A\right),({\mathcal{T}}_{\tau }^{-}\left(A\right)\right)\wedge \left({\mathcal{T}}_{\tau }^{+}\left(B\right),{\mathcal{T}}_{\tau }^{-}\left(B\right)\right)={\mathcal{T}}_{\tau }\left(A\right)\wedge {\mathcal{T}}_{\tau }\left(B\right). \end{gathered}$$

(3) Given a family $\{A_i\mid i\in I\}$ we have:

$$\begin{gathered} {\mathcal{T}}_{\tau }\left({\bigvee }_i{A}_i\right)=({\mathcal{T}}_{\tau }^{+}\left({\bigvee }_i{A}_i\right),{\mathcal{T}}_{\tau }^{-}\left({\bigvee }_i{A}_i\right))⪰({\bigwedge }_i({\mathcal{T}}_{\tau }^{+}\left(A_i\right),{\bigwedge }_i{\mathcal{T}}_{\tau }^{-}\left(A_i\right))= \\ {\bigwedge }_i({\mathcal{T}}_{\tau }^{+}\left(A_i\right),{\mathcal{T}}_{\tau }^{-}\left(A_i\right))={\bigwedge }_i{\mathcal{T}}_{\tau }\left(A_i\right). \end{gathered}$$

□

Taking into account that the lattice $\mathcal{L}$ is actually a bipolarization of the original lattice L and wishing to emphasize the role of bipolarity in our constructions, we sometimes refer to $(L,\mathcal{L})$-fuzzy topology $\mathcal{T}:L^X\to \mathcal{L}$ as the bipolar $\mathcal{L}$-fuzzy topology that extends the L-topology $\tau \subseteq L^X$. Compare also our bipolar $\mathcal{L}$-fuzzy topology with the definition of a bipolar openness gradation introduced in [17], see Definition 10.

In [44], see also [45], Lowen introduced a stronger version of a (Chang–Goguen) L-topological space. Namely, he proposed replacing axiom (1) with a stronger axiom

$$\left(1^{\prime }\right)a_X\in \tau \mathrm{for} \mathrm{all}a\in L,where a_X\mathrm{is} \mathrm{the} \mathrm{constant} \mathrm{fuzzy} \mathrm{set} \mathrm{with} \mathrm{value} a.$$

Following articles [42,43], such L-topological spaces are called stratified. Reviewing the above reasoning, we easily get the following stratified version of Theorem 5:

Theorem 6. 

If τ is a stratified L-topology on a set X, then ${\mathcal{T}}_{\tau }:L^X\to \mathcal{L}$ is a stratified $(L,\mathcal{L})$-fuzzy topology on the set X, i.e., a bipolar $\mathcal{L}$-fuzzy topology that satisfies (1^s) ${\mathcal{T}}_{\tau }\left(a_X\right)=({\mathcal{T}}^{+}\left(a_X\right),{\mathcal{T}}^{-}\left(a_X\right))=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}} for every a\in L.$

3.4. Examples

In order to illustrate the manifestation of the choice of different operators ∗, ↦ and ^c in the process of extension of an L-topology to a bipolar $\mathcal{L}$-fuzzy topology, we present here several toy examples.

Let X be an infinite set decomposed into a family of disjoint non-empty sets $X={\bigcup }_{n=1}^{\infty }{X}_n$ and let ${\chi }_n:X\to [0,1]$ be the characteristic function of the set $X_n$. Further, let the fuzzy set $A:X\to [0,1]$ be defined as $A={\bigvee }_{n=1}^{\infty }{A}_n$ where $A_n\left(x\right)={\displaystyle \frac{n}{n+1}}·{\chi }_n\left(x\right)$ and let the fuzzy set $B:X\to [0,1]$ be defined as $B={\bigvee }_{n=1}^{\infty }{B}_n$ where $B_n\left(x\right)={\displaystyle \frac{1}{n+1}}·{\chi }_n\left(x\right)$.

1.

Let $(X,{\tau }_{ind})$ be the indiscrete L-topological space, i.e., ${\tau }_{ind}=\{0_X,1_X\}$.

• Let the $\ast =\wedge$ and let the implication be defined as the residuum ${\mapsto }_{\wedge }$; that is, $a\mapsto b=\left\{\begin{array}{cc}\hfill b& \mathrm{if}a>b\hfill \\ \hfill 1& \mathrm{otherwise}.\hfill \end{array}\right.$ Then $\mathcal{T}\left(A\right)=(\mathbf{0},{\mathbf{1}}^{-})={\perp }_{\mathcal{L}}$ and $\mathcal{T}\left(B\right)=\left(\mathbf{0},{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right).$
• Let the conjunction be defined as the product t-norm $a\ast b=a·b$ and let the implication ↦ be defined as the corresponding residuum, i.e., $a\mapsto b=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{b}{a}}& \mathrm{if}a>b\hfill \\ \hfill 1& \mathrm{otherwise}.\hfill \end{array}\right.$ Then $\mathcal{T}\left(A\right)=(\mathbf{0},{\mathbf{1}}^{-})={\perp }_{\mathcal{L}}.$ and $\mathcal{T}\left(B\right)=\left(\mathbf{0},{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right)$
• Let the conjunction be defined as the Łukasiewicz t-norm, i.e, $a{\ast }_{L}b=max\{a+b-1,0\}$ and let the implication be the corresponding residuum, i.e., $a{\mapsto }_{L}b=min\{1-a+b,0\}$. Then $\mathcal{T}\left(A\right)=(\mathbf{0},{\mathbf{1}}^{-})={\perp }_{\mathcal{L}}.$ and $\mathcal{T}\left(B\right)=\left({\displaystyle \frac{\mathbf{1}}{\mathbf{2}}},{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right).$

2.

Let $(X,{\tau }_s)$ be the indiscrete stratified L-topological space, i.e., ${\tau }_s=\{a_X:a\in L\}\subset L^X$, where $a_X$ is the constant function with the value $a\in L$. Note first that in this case $A^o={\left({\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\right)}_X$ and $B^o={\mathbf{0}}_X$. In particular, this means that the value $\mathcal{T}\left(B\right)$ will be the same as in the case of the indiscrete L-topology. Concerning the value $\mathcal{T}\left(A\right)$, we proceed as follows.

• Let the conjunction be defined by $\ast =\wedge$ and let the implication be defined as the residuum ${\mapsto }_{\wedge }$. Then $\mathcal{T}\left(A\right)=\left({\displaystyle \frac{\mathbf{1}}{\mathbf{2}}},{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right).$
• Take the product t-norm $a\ast b=a·b$ as conjunction and let the implication ↦ be the corresponding residuum. Then $\mathcal{T}\left(A\right)=\left({{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}},{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right).$
• Take the Łukasiewicz t-norm and the implication ↦ as the corresponding residuum. Then $\mathcal{T}\left(A\right)=\left({{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}},{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right).$

3.

Let $(X,{\tau }_d)$ be the discrete L-topological space, i.e., ${\tau }_d={[0,1]}^X.$

• Let the conjunction be defined by $\ast =\wedge$ and let the implication be defined as the corresponding residuum ${\mapsto }_{\wedge }$. Then $\mathcal{T}\left(A\right)=\left(\mathbf{1},{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right)$ and $\mathcal{T}\left(B\right)=\left(\mathbf{1},{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}}^{-}\right).$
• Let the conjunction be defined as the product t-norm $a\ast b=a·b$ and let implication ↦ be defined as the corresponding residuum. Then $\mathcal{T}\left(A\right)=\left(\mathbf{1},{{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}}^{-}\right)$ and $\mathcal{T}\left(B\right)=\left(\mathbf{1},{{\displaystyle \frac{\mathbf{1}}{\mathbf{4}}}}^{-}\right).$
• Let the conjunction be defined as the Łukasiewicz t-norm and implication as the corresponding residuum. Then $\mathcal{T}\left(A\right)=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}}.$ and $\mathcal{T}\left(B\right)=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}}.$

4. Bipolar Fuzzy Extension of L-Co-Topologies

As explained in the introduction, in the context of our research, which allows us to rely on arbitrary L frames enriched with a very general algebraic structure, there is no reason to derive co-topological questions from topological ones. Therefore, in this section we develop the theory of bipolar extension of L-co-topologies independently, but on the model of its topological analogue implemented in the previous section. Due to the significant similarity of reasoning, we will omit some details here.

4.1. Closedness Degrees of Fuzzy Subsets in L-Co-Topological Spaces

Let $(X,\sigma )$ be an L-co-topological space. Given an L-fuzzy set $A\in L^X$, we define the closure $\overline{A}$ of A in the space $(X,\sigma )$, by setting $\overline{A}=\bigwedge \{F\mid F\in \sigma ,F\ge A\}$. It is well-known (see, for example [40,42]) that $\overline{A}$ is the smallest closed fuzzy set containing A. Besides, $\overline{A\vee B}=\overline{A}\vee \overline{B}$ for all $A,B\in L^X$ and ${\overline{\mathbf{1}}}_X={\mathbf{1}}_X$, ${\overline{\mathbf{0}}}_X={\mathbf{0}}_X.$

We use the closure $\overline{A}$ of A to measure the degree of closedness of A in the space $(X,\sigma )$ by setting ${\mathcal{S}}_{\sigma }^{+}\left(A\right)=\overline{A}\hookrightarrow A.$ By varying fuzzy sets A over $L^X$, we get a mapping ${\mathcal{S}}_{\sigma }^{+}:L^X\to L.$

Theorem 7. 

The mapping ${\mathcal{S}}_{\sigma }^{+}:L^X\to L$ is an L-fuzzy co-topology on the set $X.$

Proof. 

(1) Referring to Definition 2 of the measure of inclusion and Definition 11 of an L-cotopology, we easily get that ${\mathcal{S}}_{\sigma }^{+}\left({\mathbf{0}}_X\right)=\mathbf{1}$ and ${\mathcal{S}}_{\sigma }^{+}\left({\mathbf{1}}_X\right)=\mathbf{1}$.

(2)

The validity of the second condition follows from the next sequence of (in)equalities ensured by the lower semi-continuity of the implication ↦:

$$\begin{gathered} {\mathcal{S}}_{\sigma }^{+}(A\vee B)=\overline{A\vee B}\hookrightarrow A\vee B=\overline{A}\vee \overline{B}\hookrightarrow A\vee B{\ge }_{Prop.2\left(7\right)} \\ (\overline{A}\hookrightarrow A)\wedge (\overline{B}\hookrightarrow B)={\mathcal{S}}_{\sigma }^{+}\left(A\right)\wedge {\mathcal{S}}_{\sigma }^{+}\left(B\right). \end{gathered}$$

(3)

To prove the third condition, notice first that $\overline{{\bigwedge }_{i\in I}{A}_i}\le {\bigwedge }_{i\in I}{\overline{A}}_i$ for each family $\{A_i\mid i\in I\}\subseteq L^X$. Now, we have the following sequence of (in)equalities:

$$\begin{gathered} {\mathcal{S}}_{\sigma }^{+}\left({\bigwedge }_{i\in I}{A}_i\right)=\left(\overline{{\bigwedge }_{i\in I}{A}_i}\right)\hookrightarrow \left({\bigwedge }_{i\in I}{A}_i\right){\ge }_{Prop.2\left(4\right)}\left({\bigwedge }_{i\in I}{\overline{A}}_i\right)\hookrightarrow \left({\bigwedge }_{i\in I}{A}_i\right){\ge }_{Prop.2\left(6\right)} \\ {\bigwedge }_{i\in I}\left({\overline{A}}_i\hookrightarrow A_i\right)={\bigwedge }_{i\in I}{\mathcal{S}}_{\sigma }^{+}\left(A_i\right). \end{gathered}$$

□

Theorem 8. 

If $f:(X,{\sigma }_X)\to (Y,{\sigma }_Y)$ is a continuous mapping of L-co-topological spaces, then $f:(X,{\mathcal{S}}_{{\sigma }_X}^{+})\to (Y,{\mathcal{S}}_{{\sigma }_Y}^{+})$ is a continuous mapping of L-fuzzy co-topological spaces.

Proof. 

Recall first that the continuity of a mapping $f:(X,{\sigma }_X)\to (Y,{\sigma }_Y)$ means that $\overline{f^{-1}\left(B\right)}\le f^{-1}\left(\overline{B}\right)$ for every $B\in L^Y$; see, for example [40]. While the continuity of a mapping $f:(X,{\mathcal{S}}_X)\to (Y,{\mathcal{S}}_Y)$ means that ${\mathcal{S}}_X\left(f^{-1}\left(B\right)\right)\ge {\mathcal{S}}_Y\left(B\right)$; see, for example [4,29,32]. Now, the statement of the theorem follows from the next series of easy provable (in)equalities:

$$\begin{gathered} {\mathcal{S}}_{{\sigma }_X}^{+}\left(f^{-1}\left(B\right)\right)=\overline{f^{-1}\left(B\right)}\hookrightarrow f^{-1}\left(B\right)={\bigwedge }_{x\in X}\left(\overline{f^{-1}\left(B\right)}\mapsto f^{-1}\left(B\right)\right)\left(x\right){\ge }_{Prop.\left(4\right)2} \\ {\bigwedge }_{x\in X}\left(f^{-1}\left(\overline{B}\right)\mapsto f^{-1}\left(B\right)\right)\left(x\right)={\bigwedge }_{x\in X}\left(\overline{B}\left(f\left(x\right)\right)\mapsto B(f\left(x\right)\right)\ge \\ {\bigwedge }_{y\in Y}\left(\overline{B}\left(y\right)\mapsto B\left(y\right)\right)=\overline{B}\hookrightarrow B={\mathcal{S}}_{{\sigma }_Y}^{+}\left(B\right). \end{gathered}$$

□

4.2. Non-Closedness Degree of Fuzzy Subsets in L-Co-Topological Spaces

Patterned after the definition of non-openness for L-fuzzy subsets of an L-topological space, we introduce here the degree of non-closedness for L-fuzzy subsets in L-co-topological spaces.

Definition 14. 

The degree of non-closedness of a fuzzy set in an L-co-topological space is defined by ${\mathcal{S}}_{\sigma }^{-}\left(A\right){=}^{\sim }\overline{A}\hookrightarrow ̸A$. By varying A over $L^X$, we get an operator ${\mathcal{S}}^{-}:L^X\to L^{-}$ of non-closedness of subsets of the space $(X,\sigma )$.

Theorem 9. 

The mapping ${\mathcal{S}}_{\sigma }^{-}:L^X\to L^{-}$ is an $(L,L^{-})$-fuzzy co-topology on the set X.

Proof. 

(1) ${}^{\sim }\mathcal{S}_{\sigma }^{-}\left({\mathbf{0}}_X\right)=\overline{\mathbf{0}}\hookrightarrow ̸\mathbf{0}={\bigvee }_{x\in X}\left({\overline{\mathbf{0}}}_X\left(x\right)\ast {{\mathbf{0}}_X}^c\left(x\right)\right)={\bigvee }_{x\in X}\left({\overline{\mathbf{0}}}_X\left(x\right)\ast {\mathbf{1}}_X\left(x\right)\right)=\mathbf{0}\ast \mathbf{1}=\mathbf{0}$. Hence, ${\mathcal{S}}_{\sigma }^{-}\left({\mathbf{0}}_X\right)={\mathbf{0}}^{-}$, and this is the top element of $L^{-}$;

• ${}^{\sim }\mathcal{S}_{\sigma }^{-}\left({\mathbf{1}}_X\right)=\overline{\mathbf{1}}\hookrightarrow ̸\mathbf{1}={\bigvee }_{x\in X}\left({\overline{\mathbf{1}}}_X\left(x\right)\ast 0_X\left(x\right)\right)={\bigvee }_{x\in X}\left({\overline{\mathbf{1}}}_X\left(x\right)\ast {{\mathbf{1}}_X}^c\left(x\right)\right)=\mathbf{1}\ast \mathbf{0}=\mathbf{0}$. Hence, ${\mathcal{S}}_{\sigma }^{-}\left({\mathbf{1}}_X\right)={\mathbf{0}}^{-}$, and this the top element of $L^{-}$. Hence, the first axiom of the $(L,L^{-})$-fuzzy co-topology for ${\mathcal{S}}_{\sigma }^{-}:L^X\to L^{-}$ is valid.

(2) Let $A,B\in L^X$. Then, noticing that $\overline{A\vee B}=\overline{A}\vee \overline{B}$, we have the following series of (in) equalities:

$$\begin{gathered} {}^{\sim }\mathcal{S}_{\sigma }^{-}(A\vee B)=\overline{A\vee B}\hookrightarrow ̸A\vee B=\overline{A}\vee \overline{B}\hookrightarrow ̸A\vee B{\le }_{Prop.4\left(3\right)}\overline{A}\hookrightarrow ̸A\vee \overline{B}\hookrightarrow ̸B= \\ {}^{\sim }\mathcal{S}_{\sigma }^{-}\left(A\right){\vee }^{\sim }{\mathcal{S}}_{\sigma }^{-}\left(B\right). \end{gathered}$$

Hence, ${\mathcal{S}}_{\sigma }^{-}(A\vee B)\ge {\mathcal{S}}_{\sigma }^{-}\left(A\right)\wedge {\mathcal{S}}_{\sigma }^{-}\left(B\right)$, i.e., the second axiom of the L-fuzzy cotopology ${\mathcal{S}}_{\sigma }^{-}:L^X\to L^{-}$ is valid.

(3) Let $\{A_i\mid i\in I\}\subseteq L^X$. Noticing that $\overline{\left({\bigwedge }_{i\in I}\left(A_i\right)\right)}\le {\bigwedge }_{i\in I}\left({\overline{A}}_i\right)$, we have the following series of (in)equalities

$$\begin{gathered} {}^{\sim }\mathcal{S}_{\sigma }^{-}\left({\bigwedge }_{i\in I}\left(A_i\right)\right)=\overline{\left({\bigwedge }_{i\in I}\left(A_i\right)\right)}\hookrightarrow ̸\left({\bigwedge }_{i\in I}\left(A_i\right)\right){\le }_{Prop.4\left(2\right)}\left({\bigwedge }_{i\in I}\left({\overline{A}}_i\right)\right)\hookrightarrow ̸\left({\bigwedge }_{i\in I}\left(A_i\right)\right){\le }_{Prop.4\left(4\right)} \\ {\bigvee }_{i\in I}\left({\overline{A}}_i\hookrightarrow ̸B_i\right)={\bigvee }_{i\in I}{}^{\sim }\mathcal{S}_{\sigma }^{-}\left(A_i\right). \end{gathered}$$

Thus, ${\mathcal{S}}_{\sigma }^{-}\left({\bigwedge }_{i\in I}{A}_i\right)\ge {\bigwedge }_{i\in I}{\mathcal{S}}_{\sigma }^{-}\left(A_i\right)$ and hence ${\mathcal{S}}_{\sigma }^{-}:L^X\to L^{-}$ also satisfies the third axiom of an $L^{-}$-fuzzy co-topology. □

4.3. Bipolar Fuzzy Co-Topology on a L-Co-Topological Space

Based on the fuzzy co-topologies ${\mathcal{S}}_{\sigma }^{+}:L^X\to L$ and ${\mathcal{S}}_{\sigma }^{-}:L^X\to L^{-}$ introduced above, we define a mapping ${\mathcal{S}}_{\sigma }:L^X\to \mathcal{L}$ on a set X by setting ${\mathcal{S}}_{\sigma }\left(A\right)=({\mathcal{S}}_{\sigma }^{+}\left(A\right),{\mathcal{S}}_{\sigma }^{-}\left(A\right))\in \mathcal{L} \mathrm{for} \mathrm{every} A\in L^X.$

Theorem 10. 

Mapping ${\mathcal{S}}_{\sigma }:L^X\to \mathcal{L}$, thus defined is an $(L,\mathcal{L})$-fuzzy cotopology on the set X.

Proof. 

(1) ${\mathcal{S}}_{\sigma }\left({\mathbf{0}}_X\right)=({\mathcal{S}}_{\sigma }^{+}\left({\mathbf{0}}_X\right),{\mathcal{S}}_{\sigma }^{-}\left({\mathbf{0}}_X\right))=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}}$;

${\mathcal{S}}_{\sigma }\left({\mathbf{1}}_X\right)=({\mathcal{S}}_{\sigma }^{+}\left({\mathbf{1}}_X\right),{\mathcal{S}}_{\sigma }^{-}\left({\mathbf{1}}_X\right))=(\mathbf{1},{\mathbf{0}}^{-})={\top }_{\mathcal{L}}$.

(2) Given $A,B\in L^X$, we have the following:

$$\begin{gathered} {\mathcal{S}}_{\sigma }(A\vee B)=\left({\mathcal{S}}_{\sigma }^{+}(A\vee B),{\mathcal{S}}_{\sigma }^{-}(A\vee B)\right)⪰\left(({\mathcal{S}}_{\sigma }^{+}\left(A\right)\wedge {\mathcal{S}}_{\sigma }^{+}\left(B\right),({\mathcal{S}}_{\sigma }^{-}\left(A\right)\wedge {\mathcal{S}}_{\sigma }^{-}\left(B\right)\right)= \\ \left(({\mathcal{S}}_{\sigma }^{+}\left(A\right),({\mathcal{S}}_{\sigma }^{-}\left(A\right)\right)\wedge \left({\mathcal{S}}_{\sigma }^{+}\left(B\right),{\mathcal{S}}_{\sigma }^{-}\left(B\right)\right)={\mathcal{S}}_{\sigma }\left(A\right)\wedge {\mathcal{S}}_{\sigma }\left(B\right). \end{gathered}$$

(3) Given a family $\{A_i\mid i\in I\}$, we have the following:

$$\begin{gathered} {\mathcal{S}}_{\sigma }\left({\bigwedge }_i{A}_i\right)=({\mathcal{S}}_{\sigma }^{+}\left({\bigwedge }_i{A}_i\right),{\mathcal{S}}_{\sigma }^{-}\left({\bigwedge }_i{A}_i\right))⪰({\bigwedge }_i({\mathcal{S}}_{\sigma }^{+}\left(A_i\right),{\bigwedge }_i{\mathcal{S}}_{\sigma }^{-}\left(A_i\right))= \\ {\bigwedge }_i({\mathcal{S}}_{\sigma }^{+}\left(A_i\right),{\mathcal{S}}_{\sigma }^{-}\left(A_i\right))={\bigwedge }_i{\mathcal{S}}_{\sigma }\left(A_i\right). \end{gathered}$$

□

As in the case of $(L,\mathcal{L})$-fuzzy topology wishing to emphasize the role of bipolarity in our constructions, we sometimes refer to $(L,\mathcal{L})$-fuzzy co-topology $\mathcal{T}:L^X\to \mathcal{L}$ as the bipolar $\mathcal{L}$-fuzzy co-topology, extending the L-co-topology $\tau \subseteq L^X$. Compare also our bipolar $\mathcal{L}$-fuzzy co-topology with the definition of a bipolar closedness gradation introduced in [17], see Definition 11.

5. Bipolar $\mathcal{L}$-Fuzzy Extension of an L-Topology in Case of a Girard Monoid

5.1. Girard Monoids as the Context of Our Studies

The definition of an L-topology depends only on the lattice structure of L. On the other hand, the model of the extension of an L-topology to an $(L,\mathcal{L})$-fuzzy topology is based on the extra operations $\ast :L\times L\to L$, $\mapsto :L\times L\to L$ and ^c $:L\to L$. Therefore, the properties of the extended bipolar $\mathcal{L}$-fuzzy topologies and their relations with the underlying L-topologies depend on the choice of the operators $\ast :L\times L\to L$, $\mapsto :L\times L\to L$ and ^c $:L\to L$. In Definition 2.3, we have already recalled the concept of a residuated lattice. In residuated lattices, the first two operations — conjunction and implications — have all the properties that are required for these operations in this work. In addition, they are closely interconnected by the Galois connection and satisfy the following two important properties.

Proposition 5. 

[∗-transitivity of the residuum] In a residuated lattice

$(L,\le ,\wedge ,\vee ,\ast ,\mapsto )$

$$(a\mapsto b)\ast (b\mapsto c)\le a\mapsto c \mathit{for} \mathit{all} a,b,c\in L.$$

Proposition 6. 

[Exchange of ∗ and ↦.] In a residuated lattice

$(L,\le ,\wedge ,\vee ,\ast ,\mapsto )$

$$a\mapsto (b\mapsto c)=(a\ast b)\mapsto c \mathit{for} \mathit{all} a,b,c\in L.$$

In order to connect these operations with the last one, complementation, we have to take one step further and restrict ourselves to a special kind of residuated lattices, the so called Girard monoids.

Definition 15 

([46]). A residuated lattice $(L,\le ,\wedge ,\vee ,\ast ,\mapsto )$ is called a Girard monoid if

$$(a\mapsto \mathbf{0})\mapsto \mathbf{0}=a for every a\in L.$$

Based on this property, we can define complementation in a Girard monoid in a unique way by setting $a^c=a\mapsto \mathbf{0}$ for every $a\in L.$

Thus, while relations between $\ast :\times L\to L$ and $\mapsto :L\times L\to L$ are fixed already by the assumption that $(L,\le ,\wedge ,\vee ,\ast ,\mapsto )$ is the residuated lattice, by setting $a^c=a\mapsto 0$ in case of a Girard monoid, we have connected all elements of our original object, denoted as $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$.

In our work we will need the special relations between elements of Girard monoids described in the following two propositions. For completeness, we include their easy proofs.

Proposition 7 

([47]). In a Girard monoid $a\mapsto b={\left(a\ast b^c\right)}^c$ for any $a,b\in L$.

Proof. 

${\left(a\ast b^c\right)}^c=(a\ast (b\mapsto 0))\mapsto 0{=}_{Prop.6}a\mapsto ((b\mapsto 0)\mapsto 0)=a\mapsto b$. □

Proposition 8 

([47]). In a Girard monoid, $a\mapsto b=b^c\mapsto a^c$ for any $a,b\in L$

Proof. 

$a\mapsto b={\left(a\ast b^c\right)}^c={\left(b^c\ast a\right)}^c={\left(b^c\ast {\left(a^c\right)}^c\right)}^c{=}_{Prop.7}{b}^c\mapsto a^c.$□

5.2. Relations between Fuzzy Topologies ${\mathcal{T}}_{\tau }^{+}$ and ${\mathcal{T}}_{\tau }^{-}$

The next theorem and two of its corollaries show the mutual dependence between the positive and the negative parts of the bipolar fuzzy extension of an L-topology in the case when L is a Girard monoid. Further, in Section 6, this theorem will allow important connections between the category of L-topological spaces and the category of extended bipolar L-fuzzy topological spaces to be revealed.

Theorem 11. 

If $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid, then ${\mathcal{T}}_{\tau }^{+}={{(}^{\sim }{\mathcal{T}}_{\tau }^{-})}^c$.

Proof. 

Let $(X,\tau )$ be an L-topological space, where $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid. We have to prove that ${\mathcal{T}}_{\tau }^{+}\left(A\right)={\left({}^{\sim }\mathcal{T}_{\tau }^{-}\left(A\right)\right)}^c$ for every $A\in L^X.$

Given any L-fuzzy sets $A,B\in L^X$, from Proposition 7, we have $A\left(x\right)\mapsto B\left(x\right)={\left(A\left(x\right)\ast B^c\left(x\right)\right)}^c$ for every $x\in X.$ Taking infimum over $x\in X$ on both sides of the equality and applying Proposition 1, we have

$${\bigwedge }_{x\in X}(A\left(x\right)\mapsto B\left(x\right)={\bigwedge }_{x\in X}\left({\left(A\left(x\right)\ast B^c\left(x\right)\right)}^c\right)={\left({\bigvee }_{x\in X}\left(A\left(x\right)\ast B^c\left(x\right)\right)\right)}^c.$$

Now, taking in the above equality $B=A^o$, we get the required equality:

$${\mathcal{T}}_{\tau }^{+}\left(A\right)={\bigwedge }_{x\in X}(A\left(x\right)\mapsto A^o\left(x\right))={\left({\bigvee }_{x\in X}A\left(x\right)\ast {\left(A^o\right)}^c\right)}^c={{(}^{\sim }{\mathcal{T}}_{\tau }^{-}\left(A\right))}^c.$$

□

Corollary 1. 

In a Girard monoid, $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ ${}^{\sim }\mathcal{T}_{\tau }^{-}\left(A\right)={\left({\mathcal{T}}_{\tau }^{+}\left(A\right)\right)}^c$ for every $A\in L^X.$

Corollary 2. 

In Łukasiewicz algebra, $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ ${\mathcal{T}}^{+}\left(A\right){+}^{\sim }{\mathcal{T}}^{-}\left(A\right)=\mathbf{1}$ for every $A\in L^X$.

Remark 1. 

In the case when $L=\{0,{\displaystyle \frac{1}{2}},1\}$, we have the following tautology from the previous theorem:

• a fuzzy set is open if and only if it is not non open;
• a fuzzy set is half open if and only if it is half non open

Corollary 3. 

The $(L,\mathcal{L})$-fuzzy topology ${\mathcal{T}}_{\tau }$ induced by an L-topology τ on a set X in the case of a Girard monoid $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is defined by

$${\mathcal{T}}_{\tau }\left(A\right)=\left({\mathcal{T}}_{\tau }^{+}\left(A\right){,}^{\sim }{\left({\mathcal{T}}_{\tau }^{+}\left(A\right)\right)}^c\right) for every A\in L^X.$$

In particular, if $(L,\le ,\wedge ,\vee ,\mapsto )$ is Łukasiewicz algebra, then

$${\mathcal{T}}_{\tau }\left(A\right)=\left({\mathcal{T}}_{\tau }^{+}\left(A\right){,}^{\sim }(1-{\mathcal{T}}_{\tau }^{+}\left(A\right))\right).$$

5.3. Relations between Closedness and Non-Closedness Operators ${\mathcal{S}}^{+}$ and ${\mathcal{S}}^{-}$

Results similar to the ones proved above for bipolar $(L,\mathcal{L})$-fuzzy topology ${\mathcal{T}}_{\tau }$ induced by an L-topology $\tau$ are valid also for the $(L,\mathcal{L})$-fuzzy co-topology ${\mathcal{S}}_{\sigma }$ induced by an L-co-topology $\sigma$. The proofs are analogous to the proofs of the corresponding statements for $(L,\mathcal{L})$-fuzzy topology and are omitted. Note also that such results can be obtained directly from their topological counterparts by applying Theorems 13 and 14 from the next section.

Theorem 12. 

If $(X,\sigma )$ is an L-co-topological space where $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid, then ${\mathcal{S}}_{\sigma }^{+}={{(}^{\sim }{\mathcal{S}}_{\sigma }^{-})}^c$.

Corollary 4. 

Given an L-co-topological space $(X,\sigma )$ where $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid, then ${}^{\sim }\mathcal{S}_{\sigma }^{-}\left(A\right)={\left({\mathcal{S}}_{\sigma }^{+}\left(A\right)\right)}^c$ for every $A\in L^X.$

Corollary 5. 

If $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is the Łukasiewicz algebra, then ${\mathcal{S}}_{\sigma }^{+}\left(A\right){+}^{\sim }{\mathcal{S}}_{\sigma }^{-}\left(A\right)=1$ for every $A\in L^X.$

Corollary 6. 

The $(L,\mathcal{L})$-fuzzy co-topology induced by an L-co-topology σ on X in the case of a Girard monoid $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is defined by ${\mathcal{S}}_{\sigma }\left(A\right)=({\mathcal{S}}_{\sigma }^{+}\left(A\right){,}^{\sim }\left({\mathcal{S}}_{\sigma }^{+}{\left(A\right)}^c)\right)$ for every $A\in L^X.$ In particular, if $(L,\le ,\wedge ,\vee ,\mapsto {,}^c)$ is the Łukasiewicz algebra, then ${\mathcal{S}}_{\sigma }\left(A\right)=({\mathcal{S}}_{\sigma }^{+}\left(A\right),1-{\mathcal{S}}^{+}\left(A\right)).$

5.4. Relations between Bipolar $\mathcal{L}$-Fuzzy Topology and Bipolar $\mathcal{L}$-Fuzzy Co-Topology

Notice first that in case of a Girard monoid $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$, it is possible (thanks to the double negation law ${\left(a^c\right)}^c=a\forall a\in L$) to consider an L-co-topology as a structure dual to the L-topology. Namely, given an L-topology $\tau \subseteq L^X$ on a set X, the corresponding L-co-topology ${\tau }^c{=}_{def}\sigma$ is defined as $\sigma =\{A\in L^X\mid A^c\in \tau \}$.

This correspondence between L-topology and L-co-topology also allows us to establish the following known (see, e.g., Ref. [40]) and easy provable connections between closure and interior operators in an L-topological space in case of a Girard monoid $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$.

Proposition 9. 

Given an L-topological space $(X,\tau )$ and an L-fuzzy set $A\in L^X$, the following relations hold between the interior and closure operators (1) $\overline{A}={\left({\left(A^c\right)}^o\right)}^c$, (2) ${\left(\overline{A}\right)}^c={\left(A^c\right)}^o$, (3) $A^o={\left(\overline{A^c}\right)}^c$ (4) ${\left(A^o\right)}^c=\overline{A^c}.$

Referring to this statement, we can establish connections between $(L,\mathcal{L})$-fuzzy topology and $(L,\mathcal{L})$-fuzzy co-topology induced by an L-topology $\tau$ on a set X.

Theorem 13. 

Let $(X,\tau )$ be an L-topological space, where $(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid. Then ${\mathcal{T}}_{\tau }^{+}\left(A\right)={\mathcal{S}}_{\sigma }^{+}\left(A^c\right)$ for every $A\in L^X$.

Proof. 

We have to show that for every $A\in L^X$

$${\mathcal{T}}_{\tau }^{+}\left(A\right)={\bigwedge }_{x\in X}(A\left(x\right)\mapsto A^o\left(x\right))={\bigwedge }_{x\in X}(\overline{A^c}\left(x\right)\mapsto A^c\left(x\right))={\mathcal{S}}_{\sigma }^{+}\left(A^c\right).$$

Obviously, it is enough to show that $A\left(x\right)\mapsto A^o\left(x\right)=\overline{A^c}\left(x\right)\mapsto A^c\left(x\right)$ for each $x\in X.$

We fix $x\in X$ and denote $A\left(x\right)=a,A^o\left(x\right)=b,\overline{A^c}\left(x\right)=c$ and $A^c\left(x\right)=d$. Now, the provable equality can be rewritten as $a\mapsto b=c\mapsto d.$ Notice that $d=a^c$ and from Proposition 9(4), we have $c=\overline{A^c}\left(x\right)={\left(A^o\right)}^c\left(x\right)=b^c$. Therefore, the provable equality can be rewritten as $a\mapsto b=b^c\mapsto a^c.$ However, this is just the statement of Proposition 8. □

In the case of the Girard monoid, we can also establish connections between extended bipolar L-fuzzy topologies and extended bipolar L-fuzzy cotopologists — a fact whose prototype in classical topology is probably known to every mathematician.

Theorem 14. 

Let $(X,\tau )$ be an L-topological space, where $(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid. Then ${\mathcal{T}}_{\tau }^{-}\left(A\right)={\mathcal{S}}_{\sigma }^{-}\left(A^c\right)$ for every $A\in L^X$.

Proof. 

We have to prove

$${}^{\sim }\mathcal{T}_{\tau }^{-}\left(A\right)={\bigvee }_{x\in X}\left(A\left(x\right)\ast {\left(A^o\right)}^c\left(x\right)\right)={\bigvee }_{x\in X}\left({\overline{A}}^c\left(x\right)\ast A\left(x\right)\right){=}^{\sim }{\mathcal{S}}_{\sigma }^{-}\left(A^c\right).$$

Obviously, it is enough to show $A\left(x\right)\ast {\left(A^o\right)}^c\left(x\right)={\overline{A}}^c\left(x\right)\ast A\left(x\right)$ for each $x\in X.$ However, this follows from Proposition 9(4). □

Corollary 7. 

Let $(X,\tau )$ be a L-topological space, where $(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ is a Girard monoid. Then the bipolar $\mathcal{L}$-fuzzy topology ${\mathcal{T}}_{\tau }:L^X\to \mathcal{L}$ and the $(L,\mathcal{L})$-fuzzy co-topology ${\mathcal{S}}_{\sigma }:L^X\to L$ generated by σ are defined respectively by the following:

$${\mathcal{T}}_{\tau }\left(A\right)=\left({\mathcal{T}}_{\tau }^{+}\left(A\right),{\mathcal{T}}_{\tau }^{-}\left(A\right)\right)and{\mathcal{S}}_{\sigma }A)=\left({\mathcal{T}}_{\tau }^{+}\left(A^c\right),{\mathcal{T}}_{\tau }^{-}\left(A^c\right)\right) for every A\in L^X.$$

6. Categorical Issues of the Extension Procedure of L-Topologies

To get a broader understanding of the construction of bipolar extension of L-topologies developed in our work, in this section, we consider some categorical issues of this construction. The section consists of two subsections: in the first, we will limit ourselves to the positive part of the extension, i.e., the extension of L-topological spaces to L-fuzzy topological spaces. In the second, under the assumption that L is a Girard monoid, we trace both parts of the extension; that is, the extension of L-topological spaces to $(L,\mathcal{L})$-fuzzy topological spaces.

6.1. Extension of L-Topologies to L-Fuzzy Topologies from the Point of View of Category Theory

Let $\tau \subseteq L^X$ be an L-topology and let the mapping ${\mathcal{T}}_{\tau }^{+}$ be defined as it is in Section 5, i.e., ${\mathcal{T}}_{\tau }^{+}\left(A\right)=A\hookrightarrow A^o$ for every $A\in L^X$. By Theorem 2, we know that ${\mathcal{T}}_{\tau }^{+}$ is an L-fuzzy topology. Further, from Theorem 3, it follows that if $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ is a continuous mapping of L-topological spaces, then $f:(X,{\mathcal{T}}_{{\tau }_X}^{+})\to (Y,{\mathcal{T}}_{{\tau }_Y}^{+})$ is a continuous mapping of L-fuzzy topological spaces. Thus, by assigning to an L-topological space $(X,\tau )$ the L-fuzzy topological space $\Phi (X,{\tau }_X)=(X,{\mathcal{T}}_{{\tau }_X}^{+})$ and viewing a continuous mapping $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ as the mapping $\Phi \left(f\right):(X,{\mathcal{T}}_{{\tau }_X}^{+})\to (Y,{\mathcal{T}}_{{\tau }_Y}^{+})$ of the corresponding L-fuzzy topological spaces, we obtain a functor $\Phi :L-\mathbf{TOP}\to L-\mathbf{FTOP}$ from the category $L-\mathbf{TOP}$ of L-topological spaces and continuous mappings into the category $L-\mathbf{FTOP}$ of L-fuzzy topological spaces and their continuous mappings. Obviously, functor $\Phi$ is injective on objects and on morphisms; hence, it is an embedding functor of the category $L-\mathbf{TOP}$ into the category $L-\mathbf{FTOP}$.

Further, given an L-fuzzy topological space $(X,\mathcal{T})$, let ${\tau }_{\mathcal{T}}=\{A\in L^X\mid \mathcal{T}\left(A\right)=\mathbf{1}\}$. It is easy to notice that ${\tau }_{\mathcal{T}}\subseteq L^X$ is an L-topology and the continuity of a mapping $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ implies the continuity of the mapping $f:(X,{\tau }_{{\mathcal{T}}_{{\tau }_X}})\to (Y,{\tau }_{{\mathcal{T}}_{{\tau }_Y}})$. Hence, by assigning to an L-fuzzy topological space $(X,\mathcal{T})$ the L-topological space $\Psi (X,\mathcal{T})=(X,{\tau }_{\mathcal{T}})$ and viewing a continuous mapping $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ as (obviously, continuous) mapping $\Psi \left(f\right):(X,{\tau }_{{\mathcal{T}}_X})\to (Y,{\tau }_{{\mathcal{T}}_Y})$ of the corresponding L- topological spaces, we obtain a functor $\Psi :L-\mathbf{FTOP}\to L-\mathbf{TOP}$. Further, from the definitions, it is clear that ${\tau }_{{\mathcal{T}}_{\tau }}=\tau$ for every L-topology $\tau$; hence, functor $\Psi :L-\mathbf{FTOP}\to L-\mathbf{TOP}$ is right inverse to the functor $\Phi \left(f\right):(X,{\mathcal{T}}_{{\tau }_X})\to (Y,{\mathcal{T}}_{{\tau }_Y})$; that is, $\Psi \circ \Phi {=}_{def}{{\rm Y}}_{L-\mathbf{TOP}}:L-\mathbf{TOP}\to L-\mathbf{TOP}$ is the identity functor.

As for the opposite composition $\Phi \circ \Psi :L-\mathbf{FTOP}\to L-\mathbf{FTOP}$, there are no specific relations with the identity functor ${{\rm Y}}_{L-\mathbf{FTOP}}L-\mathbf{FTOP}\to L-\mathbf{FTOP}$. Let us illustrate this statement with the following examples.

Example 1. 

Let X be a non-empty set, $L=[0,1]$ and let $a\in (0,1)$. We define $\mathcal{T}:L^X\to L$ by setting

$$\mathcal{T}\left(B\right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{a}{b}}& \mathrm{if} B=b_X \mathrm{and} b\ge a\hfill \\ \hfill 0& \mathrm{otherwise}.\hfill \end{array}\right.$$

Then $\mathcal{T}$ is an L-fuzzy topology. Indeed, notice first that $\mathcal{T}\left(0_X\right)=\mathcal{T}\left(1_X\right)=\mathbf{1}$.

To verify the validity of the second and the third axiom it is sufficient to consider only constant fuzzy sets $b\ge a$ since $\mathcal{T}\left(A\right)=0$ for all other fuzzy sets A.

Let $b^1,b^2\ge a$. Then $\mathcal{T}(b_X^1\wedge b_X^2)={\displaystyle \frac{a}{b^1\wedge b^2}}={\displaystyle \frac{a}{b^1}}\vee {\displaystyle \frac{a}{b^2}}=\mathcal{T}\left(b_X^1\right)\vee \mathcal{T}\left(b_X^2\right)\ge \mathcal{T}\left(b_X^1\right)\wedge \mathcal{T}\left(b_X^2\right)$.

Let $\{b^i\mid i\in I\}$ be a family of constants such that $b^i\ge a$ for all $i\in I$. Then $\mathcal{T}\left({\bigvee }_{i\in I}{b}_X^i\right)={\displaystyle \frac{a}{{\bigvee }_{i\in I}{b}^i}}={\bigwedge }_{i\in I}{\displaystyle \frac{a}{b^i}}={\bigwedge }_{i\in I}\mathcal{T}\left(b_X^i\right)$

From the definition of functor $\Phi$, it is clear that ${\tau }_{\mathcal{T}}=\{0_X,a_X,1_X\}$. Now, from the definition of functor $\Psi$, we have

$${\mathcal{T}}_{{\tau }_{\mathcal{T}}}\left(A\right)=\left\{\begin{array}{cc}\hfill \mathbf{1}& \mathrm{if} A=0_X,a_X,1_X\hfill \\ \hfill \mathbf{0}& \mathrm{if} infA{=}_{\mathrm{def}}{A}_0<a\hfill \\ \hfill A_0\mapsto a& \mathrm{if} A_0\ge a.\hfill \end{array}\right.$$

Further, we consider three cases of implications used in the definition of the original L-fuzzy topology $\mathcal{T}:L^X\to L$. They are residiums ${\mapsto }_{Ł}$, ${\mapsto }_P$ and ${\mapsto }_{\wedge }$ corresponding to the Łukasiewicz, the product and the minimum t-norm, respectively. In the result for $b>a$, we have

${\mathcal{T}}_{{\tau }_{\mathcal{T}}}^{Ł}\left(b\right)=1-a+b<{\mathcal{T}}_{{\tau }_{\mathcal{T}}}^P\left(b\right)={\displaystyle \frac{a}{b}}=\mathcal{T}\left(b\right)<b={\mathcal{T}}_{{\tau }_{\mathcal{T}}}^{\wedge }\left(b\right)\square$

6.2. Extension of L-Topologies to Bipolar $\mathcal{L}$-Fuzzy Topologies in the Case of a Girard Monoid from the Point of View of Category Theory

The categorical viewpoint on the extension of an L-topology to an L-fuzzy topology developed above together with the additional tools available in the case of a Girard monoid will allow us to consider also the “full-body” extension of an L-topology to $(L,\mathcal{L})$-fuzzy topology from the point of of view of the category theory.

Let $\mathfrak{L}=(L,\le ,\wedge ,\vee ,\ast ,\mapsto {,}^c)$ be a Girard monoid and let $\tau \subseteq L^X$ be an L-topology on a set X. Further, let ${\mathcal{T}}_{\tau }^{+}:L^X\to L$ be the L-fuzzy topology induced by the L-topology $\tau$, see Section 6.1. Further, let ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ be defined by ${\mathcal{T}}_{\tau }^{-}\left(A\right){=}^{\sim }{\left({\mathcal{T}}_{\tau }\left(A\right)\right)}^c$. From Section 6.1, we know that ${\mathcal{T}}_{\tau }^{-}:L^X\to L^{-}$ is an $(L,L^{-})$-fuzzy topology; hence, ${\mathcal{T}}_{\tau }=({\mathcal{T}}_{\tau },{\mathcal{T}}_{\tau }^{-}):L^X\to \mathcal{L}$ is an $(L,\mathcal{L})$-fuzzy topology.

Further, let $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ be a continuous function. Then, according to Section 6.1, the function $f:(X,{\mathcal{T}}_{{\tau }_X}^{+})\to (Y,{\mathcal{T}}_{{\tau }_Y}^{+})$ is also continuous. However, this means that ${\mathcal{T}}_{{\tau }_Y}^{+}\left(B\right)\le {\mathcal{T}}_{{\tau }_X}^{+}\left(f^{-1}\left(B\right)\right)$ for every $B\in L^Y$. Therefore,

$${\left({\mathcal{T}}_{{\tau }_Y}^{+}\right)}^c\left(B\right)\ge {\left({\mathcal{T}}_{{\tau }_X}^{+}\right)}^c(f^{-1}\left(B\right)$$

and hence,

$${\mathcal{T}}_{{\tau }_Y}^{-}\left(B\right)\le {\mathcal{T}}_{{\tau }_X}^{-}\left(f^{-1}\left(B\right)\right).$$

Thus,

$${\mathcal{T}}_{{\tau }_X}{f}^{-1}\left(B\right)=({\mathcal{T}}_{{\tau }_X}^{+}{f}^{-1}\left(B\right),{\mathcal{T}}_{{\tau }_X}^{-}{f}^{-1}\left(B\right))\ge {\mathcal{T}}_{{\tau }_Y}\left(B\right)=({\mathcal{T}}_{{\tau }_Y}^{+}\left(B\right),{\mathcal{T}}_{{\tau }_Y}^{-}\left(B\right))$$

for every $B\in L^X$; hence, $f:(X,{\mathcal{T}}_{{\tau }_X}\to (Y,{\mathcal{T}}_{{\tau }_Y})$ is continuous.

We can summarise the above results as follows.

Theorem 15. 

Assigning to an L-topological space $(X,\tau )$ the $(L,\mathcal{L})$-fuzzy topological space $\Phi (X,\tau )=(X,({\mathcal{T}}_{\tau }^{+},{\mathcal{T}}_{\tau }^{-}))$ and viewing a continuous mapping $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ as a (continuous) mapping

$$\Phi \left(f\right):(X,\left({\mathcal{T}}_{{\tau }_X}^{+},{\mathcal{T}}_{{\tau }_X}^{-})\right)\to (Y,\left({\mathcal{T}}_{{\tau }_Y}^{+},{\mathcal{T}}_{{\tau }_Y}^{-}),\right)$$

we obtain an embedding functor $\Phi :L-\mathbf{TOP}\to (L-\mathcal{L})-\mathbf{FTOP}$ of the category $L-\mathbf{TOP}$ into the category $(L,\mathcal{L})-\mathbf{FTOP}$ of $(L,\mathcal{L})$-fuzzy topological spaces.

Moreover, recalling the construction $\mathcal{T}\Rightarrow {\tau }_{\mathcal{T}}\Rightarrow {\mathcal{T}}_{{\tau }_{\mathcal{T}}}$ considered in the previous subsection and taking into account relations between openness and non-openness degrees of L-fuzzy sets in an L-fuzzy topological space in the case when L is a Girard monoid, we have also the following result:

Theorem 16. 

Let L be a Girard monoid and let $\mathcal{T}=({\mathcal{T}}^{+},{\mathcal{T}}^{-}):L^X\to \mathcal{L}$ be an $(L,\mathcal{L})$-fuzzy topology on a set X. Further, let ${\tau }_{{\mathcal{T}}^{+}}=\{A\in L^X\mid {\mathcal{T}}^{+}\left(A\right)=\mathbf{1}\}$. Then by setting $\Psi (X,({\mathcal{T}}_X^{+},{\mathcal{T}}_X^{-}))=(X,{\tau }_{{\mathcal{T}}_X^{+}})$ for an $(L,\mathcal{L}$)-fuzzy topological space $(X,({\mathcal{T}}_X^{+},{\mathcal{T}}_X^{-}))$ and interpreting a continuous mapping $f:(X,\left({\mathcal{T}}_X^{+},{\mathcal{T}}_X^{-})\right)\to \left(Y,({\mathcal{T}}_Y^{+},{\mathcal{T}}_Y^{-})\right)$ as a (continuous) mapping $\Psi \left(f\right):(X,{\tau }_{{\mathcal{T}}_X^{+}})\to (Y,{\tau }_{{\mathcal{T}}_Y^{+}})$, we obtain a functor $\Psi :(L,\mathcal{L})-\mathbf{FTOP}\to L-\mathbf{TOP}$. This functor is the right inverse of the embedding functor $\Phi :L-\mathbf{TOP}\to (L,\mathcal{L})-\mathbf{FTOP}$, i.e., $\Psi \circ \Phi {=}_{def}{{\rm Y}}_{L-\mathbf{TOP}}:L-\mathbf{TOP}\to L-\mathbf{TOP}$ is the identity functor.

7. Conclusions

As we noted in the introduction, in this work, we focus on studying the situation when some object (in our case, a fuzzy topological space or its subset) may, to a certain extent, have some property P along with the opposite property not -P. A clear example of such a situation is the property of connectedness: a fuzzy topological space may have some signs of connectedness and at the same time signs of, say, total disconnectedness. And in principle, there should be no certain relationships between them. Natural tools for studying problems of this type are bipolar fuzzy sets and bipolar fuzzy lattices, introduced and carefully studied by Zhang, see [9,10,12] et al.

Having started a programme for analysing fuzzy topological properties by means of bipolar fuzzy lattices, in this article, we limited ourselves to the openness and closedness properties of fuzzy sets in L-topological spaces. As the result of such analysis, we presented a model for fuzzification of L-topological spaces. The essence of this model is the transition from the “topological” structure, where sets are only allowed to be fuzzy, to the case where the topological structure itself becomes fuzzy, incorporating a bipolar scale of values.

In our work, this model is presented in two versions that differ in the degree of generality. The first, presented in Section 3, Section 4 and Section 6.1, develops the model in the most general context, as we see namely in a situation where no connections are assumed between the operators ∗, ↦ and ^c on the original lattice L. As a result, the degrees of openness and non-openness of fuzzy sets are independent quantities in the positive and negative parts, respectively, of a bipolar fuzzy lattice. Therefore, the results presented in these sections are of a very general nature and allow for further specification in order to obtain more valuable and specifically motivated results. On the other hand, in Section 5 and Section 6.2, we limit the scope of our study to the case where L is a Girard monoid. In this case, thanks to the double negation property in a Girard monoid the “sum” of the degrees of openness and non-openness can be identified with the top element of the original lattice L. Consequently, in this case, the bipolar lattice $\mathcal{L}$ can be interpreted as an “intuitionistc-type” lattice. (For a detailed analysis of relations between bipolat fuzzy sets (lattices) and intuitionistic fuzzy sets (lattices), see, e.g., Ref. [7].) Additionally, the connection between bipolar $\mathcal{L}$-fuzzy topology and the corresponding bipolar $\mathcal{L}$-fuzzy co-topology in this case becomes dual.

As for plans for further work, one of the first we see is the adaptation of the model presented here for the study of specific topological properties, and first of all, the for the properties of compactness, of connectedness (particularly in relation with various kinds of disconnectedness), and for the properties related to separation axioms.

The second type of question we are interested in is that of further exploring the categorical properties of our model. In particular, the question of preserving basic topological constructions (products, coproducts, quotients, etc.) by the functors defined in Section 6 seems not only interesting in itself, but also useful for extracting some specific information from the fuzzification scheme developed in our work. However, another actual problem is to investigate the existing relations between the category $(L-\mathcal{L})-\mathbf{FTOP}$ of extended bipolar L-fuzzy topological spaces and the category of bipolar graded fuzzy topological spaces considered in [17].

We plan also to adapt our model for the bipolar extension of L-fuzzy uniform structures, and L-fuzzy proximities. These are our ideas regarding the application of the model presented here to the study of topology-related issues. However, we believe that this model can also be useful when adapted for bipolar fuzzification of other mathematical structures, in particular algebraic ones.