On a new generalization of ditopological texture spaces

Abstract

Texture spaces were introduced by L.M. Brown as a means of representing fuzzy sets in a point-based setting. The concept of ditopological texture space was defined on a texture as a generalization of topological, bitopological and fuzzy topological spaces. This paper provides an introduction to the concept of diframe. In particular, we describe the morphisms of the category of diframes and establish a connection between texture spaces and frame theory.

Keywords

Diframe frame coframe colocale conuclei

1 Introduction

The idea of studying spaces in terms of the relations between the open sets began to emerge in the 1930s. In 1936, Stone [19] presented his representation theorem which establishes a connection between topological spaces and Boolean algebras. On the other hand, the idea of “topology without points”.ppeared in the studies of the authors such as Wallmann [6], Menger [15], McKinsey and Tarski [7]. Then, in the late fifties, the concept of frame was encountered in Ehresmann’s work [2]. In 1972 Isbell [11] studied the opposite of the category of frames and introduced the term “locale”. Locales can be regarded as generalized topological spaces, and hence many topological concepts (such as compactness, paracompactness, connectedness, etc.) can be extended to these generalized spaces. Finally, in 1983 the concept of biframe [1] was introduced as a point-free counterpart of bitopological spaces.

Point-free topology (locale theory) has many applications in such varied areas of mathematics as topos theory, logic, non-commutative rings and C ^∗-algebras, and even in theoretical computer science. In locale theory, there are many facts which can be proved constructively (that is, without axiom of choice and law of excluded middle). Hence, these proofs are valid in every topos. Further, the idea of viewing datatypes as “domains”.ed the researchers to represent them as locales because of their constructive nature [22].

For more details about frame and locale theory we advise the reader to see the classical monograph of Johnstone [20], which is still used as a standard reference book, and the comprehensive book of Picado and Pultr [8].

In topological spaces, the notion of open set is taken as primitive with that of closed set being auxiliary. This is fine if we have a set complement as in the classical case, or more generally an order reversing involution in the case of lattice-values topologies. However, such a complementation may not exist or may be irrelevant. Then we consider a topological structure consisting of unrelated families of open and closed sets. These structures were referred to as dichotomous topology, or ditopology for short. It should be stressed that a ditopology is considered as a single structure, with the open and closed sets playing an equal role. This is in contrast to a bitopology where there are two distinct topologies, complete with their open and closed sets. For motivation and background on textures and ditopological texture spaces the reader is referred to [16] and [17].

The aim of this study is to generalize ditopological spaces to diframes. We focus on the problem of constructing the category of diframes. The theory of diframes is significant because we obtain a larger family of lattices by dropping the complete distributivity requirement which makes a texture a spatial frame (that is, a frame isomorphic to the lattice of open sets, Ω (X), of a set X). Moreover, diframes provide a frame-theoretical perspective on the theory of ditopological spaces, which enables them to be applicable both in computer science and in many areas of mathematics.

2 Preliminaries

In this section we present some definitions and results which enable the reader to follow the general ideas.

Ditopological Texture Spaces: Let us recall some basic definitions and results from [16, 17].

A texturing on a set S is a point separating, complete, completely distributive lattice $S$ of subsets of S with respect to inclusion, which contains S and ∅, and for which arbitrary meet coincides with intersection and finite joins coincide with union. The pair $(S, S)$ is known as a texture space, or shortly a texture.

For a texture $(S, S)$ , most properties are conveniently defined in terms of the p-sets $P_{s} = \cap {A \in S : s \in A}$ and the q-sets $Q_{s} = \lor {A \in S : s \notin A} = \lor {P_{u} : u \in S, s \notin P_{u}} .$

The properties given in the following theorem are significant since they are commonly used in the theory of texture spaces.

Theorem 2.1. In any texture space $(S, S)$ , the following statements hold:

If A ⊈.B then there exists an s ∈.S with A ⊈.Q _s and P _s ⊈.B.

A = ⋂.Q _s : P _s ⊈.A} for all A ∈. $S$ .

A = ⋁.P _s : A ⊈.Q _s} for all A ∈. $S$

If A ⊈.Q _s then P _s ⊆.A for all A ∈. $S$ .

To make the concept clearer, we present some basic examples.

Example 2.2.

For any set X, (X, $P$ (X)) is called a discrete texture. Clearly, P _x = {x} and Q _x = X - {x} for all x ∈.X.

Let $(L,')$ be a fuzzy lattice, i.e., a complete, completely distributive lattice with an order reversing involution ^′. Denote the set of all join-irreducible elements of $L$ by L. For $a \in L$ , we define the set $L = {φ (a) : a \in L}$ where φ (a) = {m ∈.L : m ≤.a}. Then $(L, L)$ is a texture space.

Recall that, if (S _j, $S$ _j), j ∈.J are textures, $S = \prod_{j \in J} S_{j}$ and A _k ∈. $S$ _k for some k ∈.J we write $E (k, A_{k}) = \prod_{j \in J} Y_{i}$ where $Y_{j} = {\begin{array}{l} A_{j} & i f j = k \\ S_{j} & o t h e r w i s e \end{array}$

Then the product texturing $S$ = ⊗_j∈J $S$ _j of S consists of arbitrary intersections of elements of the set $ε = {\underset{j \in J_{1}}{\cup} E (j, A_{j}) : J_{1} \subseteq J, A_{j} \in S_{j} for j \in J_{1}}$ The p-sets and q-sets of the product texture (S ×.T, $P$ (S) ⊗. $T$ ) is denoted by ${\bar{P}}_{(s, t)}$ and ${\bar{Q}}_{(s, t)}$ , respectively.

Proposition 2.3. In a texture (S ×.T, $P$ (S) ⊗. $T$ ) we have ${\bar{P}}_{(s, t)} ⊈ \bar{Q} (s^{'}, t^{'}) \Leftrightarrow s = s^{'} and P_{t} ⊈ Q_{t^{'}}$

Direlations represent an appropriate generalization of the classical notion of relation.

Let (S, $S$ ) and (T, $T$ ) be texture spaces. Then:

$r \in (S) \otimes T$ is called a relation from (S, $S$ ) to (T, $T$ ) if it satisfies

$r ⊈ {\bar{Q}}_{(s, t)}, P_{s^{'}} ⊈ Q_{s} \Rightarrow r ⊈ {\bar{Q}}_{(s^{'}, t)},$

$r ⊈ {\bar{Q}}_{(s, t)} \Rightarrow \exists s^{'} \in S$ such that P _s ⊈.Q _s′ and $r ⊈ {\bar{Q}}_{(s^{'}, t)} .$

[(CR2)]

$R \in P (S) \otimes T$ is called a corelation from (S, $S$ ) to (T, $T$ ) if it satisfies

${\bar{P}}_{(s, t)} ⊈ R, P_{s} ⊈ Q_{s^{'}} \Rightarrow {\bar{P}}_{(s^{'}, t)} ⊈ R,$

${\bar{P}}_{(s, t)} ⊈ R \Rightarrow \exists s^{'} \in S$ such that P _s′ ⊈.Q _s and ${\bar{P}}_{(s^{'}, t)} ⊈ R .$

A pair (r, R) consisting of a relation r and a corelation R is called a direlation.

One of the most useful notions in the theory of texture spaces is that of a difunction. A difunction can be considered as a special type of direlation.

Let (f, F) be a direlation from (S, $S$ ) to (T, $T$ ). Then (f, F) is called a difunction if it satisfies

For s, s′.#x2208;.S, P _s ⊈.Q _s′ ⇒.#x2203;.t ∈.T with $f ⊈ {\bar{Q}}_{(s, t)}$ and ${\bar{P}}_{(s^{'}, t)} ⊈ F,$

For t, t′.#x2208;.T and s ∈.S, $f ⊈ {\bar{Q}}_{(s, t)}$ and ${\bar{P}}_{(s, t^{'})} ⊈ F \Rightarrow P_{t^{'}} ⊈ Q_{t}$ .

The category of texture spaces and direlations is denoted by drTex [18].

Now let (r, R) be a direlation from $S$ to $T$ . Then the inverse of (r, R) is defined by ${(r, R)}^{\leftarrow} = (R^{\leftarrow}, r^{\leftarrow}) : (T, T) \to (S, S)$ where $\begin{matrix} r^{\leftarrow} = \cap {{\bar{Q}}_{(t, s)} : r ⊈ {\bar{Q}}_{(s, t)}} and \\ R^{\leftarrow} = \lor {{\bar{P}}_{(t, s)} : {\bar{P}}_{(s, t)} ⊈ R} \end{matrix}$ .

For a direlation (r, R) : (S, $S$ ) →.T, $T$ ), $r^{\leftarrow} A = \cap {Q_{t} : \forall s, r ⊈ {\bar{Q}}_{(s, t)} \Rightarrow A \subseteq Q_{s}} \in T$ is called the A-section of r, and $R^{\leftarrow} A = \lor {P_{t} : \forall s, {\bar{P}}_{(s, t)} ⊈ R \Rightarrow P_{s} \subseteq A} \in T$ is called the A-section of R.

For B ∈. $T$ , the B-section of the corelation r ^← is called a B-presection of r (denoted by r ^← B) and, dually, the B-section of the relation R ^← is called a B-presection of R (denoted by R ^← B).

Proposition 2.4. Let (r, R) : (S, $S$ ) →.T, $T$ ) be a direlation. Then

$r ⊈ {\bar{Q}}_{(s, t)} \Leftrightarrow r^{\to} P_{s} ⊈ Q_{t} .$

(r ^→, r ^←) and (R ^←, R ^→) are adjoint pairs. (For definition, see the section “Frames and Locales”.elow.)

Finally we recall that a dichotomous topology, or ditopology for short, on a texture (S, $S$ ) is a pair (τ, κ) of subsets of $S$ where the set of open sets τ satisfies

S, ∅.#x2208;.τ,

G ₁, G ₂ ∈.τ ⇒.G ₁ ∩.G ₂ ∈.τ,

G _i ∈.τ, i ∈.I ⇒.#x22C1;_i∈I G _i ∈.τ,

and the set of closed sets κ satisfies

S, ∅.#x2208;.κ,

K ₁, K ₂ ∈.κ ⇒.K ₁ ∪.K ₂ ∈.κ,

K _i ∈.κ, i ∈.I ⇒.#x22C2;_i∈I K _i ∈.κ.

Hence a ditopology is topological structure in which there is no priori relation between the open and closed sets.

Frames and Locales: The concepts and results on frame and locale theory are taken from [8].

Recall that two monotone maps f : L →.M and g : M →.L between posets are said to be (Galois) adjoint if and only if the following condition is satisfied: $f (x) \leq y \Leftrightarrow x \leq g (y)$ In this case, f is called a left adjoint and g is called a right adjoint, written as f = g ^* or g = f _*.

Proposition 2.5. The left (resp., right) adjoints preserve all suprema (resp. infima). Further, if L and M are complete lattices and g : M →.L is a monotone map preserving all infima then it has a left adjoint f : L →.M defined by f (x) = ⋀.y : x ≤.g (y)}.

A complete lattice L is called a frame if it satisfies the infinite distributive law $b \land (\lor A) = \lor {b \land a : a : \in A}$ for any subset A ⊆.L and any b ∈.L.

A frame homomorphism is a map between frames preserving arbitrary joins and finite meets. The resulting category is denoted by Frm. Note that the opposite category of Frm is called the category of locales and denoted by Loc.

Example 2.6. The lattice of open sets of a topological space X, Ω (X), is a frame. On the other hand, a continuous map f : X →.Y between topological spaces induces a frame homomorphism Ω (f) : Ω (Y) →.Ω (X) defined by Ω (f) (G) = f ^-1 (G).

A Heyting algebra is a bounded lattice L equipped with a binary operation →. L ×.L ⟶.L satisfying $c \leq a \to b \Leftrightarrow c \land a \leq b$ for all a, b, c ∈.L. Note that each frame carries a Heyting operation.

A subset S ⊆.L satisfying the following properties is called a sublocale

⋀M ∈.S for all M ⊆.S,

a →.s ∈.S for every s ∈.S and a ∈.L.

Here is an alternative characterization of sublocales:

A mapping v : L →.L is called a nucleus if it satisfies:

a ≤.v (a),

a ≤.b ⇒.v (a) ≤.v (b),

v (v (a)) = v (a),

v (a ∧.b) = v (a) ∧.v (b).

The following result characterizes sublocales of a locale L in terms of nuclei defined on L:

Proposition 2.7. If S ⊆.L is a sublocale then v _S (a) = ⋀.s ∈.S : a ≤.s} is a nucleus. Conversely, for a nucleus v : L →.L, v (L) is a sublocale. Moreover, there is a one-one correspondence between two characterizations of sublocales.

Proposition 2.8. [9] Let L be a frame. Then, if M ⊆.L is a subframe (i.e., a subset closed under arbitrary join and finite meets) and S ⊆.L is a sublocale of L then v _S (M) is a subframe of S.

coFrames and coLocales: Now we start by briefly recalling the definitions of coframes and coHeyting algebras. Rasiowa and Sikorski [5] studied coHeyting algebras, as well as Heyting algebras, and they provided a wealth of information about these lattices.

A complete lattice L is called a coframe if it satisfies the infinite distributive law $b \lor (\land A) = \land {b \lor a : a \in A}$ for any subset A ⊆.L and any b ∈.L.

A coHeyting algebra is a bounded lattice L equipped with a binary operation ←. L ×.L ⟶.L satisfying $a \leftarrow b \leq c \Leftrightarrow a \leq b \lor c$ for all a, b, c ∈.L.

It is worth noting that every coframe is a complete coHeyting algebra, and vice versa. The following are elementary properties of a coHeyting operation:

Proposition 2.9. [5, 8] The following conditions hold in a complete coHeyting algebra M:

a ≤.b iff a ←.b = 0. In particular, a ←.a = 0.

a ←.b ≤.a. It is obvious since a ≤.a ∨.b.

b ←.a = (a ∨.b) ←.a.

a ≤.b implies a ←.c ≤.b ←.c, that is, the operation ←.s monotone in the first variable.

a ←. = a for all a ∈.M. Indeed, a ←. ≤.a ←. iff a ≤. ∨.a ←.) iff a ≤.a ←.. On the other hand, a ≤. ∨.a iff a ←. ≤.a.

(⋁_i∈I a _i) ←.b = ⋁_i∈I (a _i ←.b). Obviously, the operation _ ←.x : M →.M, y ↦.y ←.x) preserves all joins since it is a left adjoint.

a ∨.b = a ∨.c iff b ←.a = c ←.a.

x ←.#x22C0;_i∈I a _i = ⋁_i∈I (x ←.a _i).

a = (a ∧.b) ∨.a ←.b) for all a ∈.M.

(b ←.a) ←.a = b ←.a for all a, b ∈.M.

c ←.a ∨.b) = (c ←.b) ←.a for all a, b, c ∈.M.

(b ←.a) ∨.a = a ∨.b for all a, b ∈.M.

a ≤.b implies c ←.b ≤.c ←.a for all a, b, c ∈.M.

Example 2.10.

The family of closed sets of a topological space X, ( $C$ (X), ⊆), is a coframe, in which arbitrary meet and finite join coincide with the intersection and finite union, respectively. Notice that the arbitrary join is not the union, but it can be defined as the closure of the union, namely, $\underset{i \in I}{\lor} F_{i} = c1 (\underset{i \in I}{\cup} F_{i})$

A complete Boolean algebra is both a frame and a coframe.

Recall that an open set U is called regular open if it equals to the interior of its closure, that is, U = int (cl (U)). The family of regular open sets of a topological space, Ω _reg (X), is a complete Boolean algebra. The join and meet of regular open sets are defined, respectively, as ⋁_i∈I A _i = int ((cl (⋃_i∈I A _i)) and ⋀_i∈I A _i = int (⋂_i∈I A _i).

Thus, Ω _reg (X) is both a frame and a coframe.

A texturing $S$ or, in general, a complete, completely distributive lattice is both a frame and a coframe. However, the converse may not be true. For instance, Ω _reg (X) is a complete Boolean algebra but it is not a completely distributive lattice.

Note that a coframe homomorphism [10] is a mapping between coframes preserving arbitrary meet and finite joins.

Now we introduce some new concepts which will play an important role in our future studies on separation axioms in diframes. These concepts are discussed in this section because all the definitions and properties presented here can be obtained by mimicking the arguments in [8].

First, note that we will denote the category of coframes and coframe homomorphisms by coFrm, and the opposite category of coFrm by coLoc.

Now recall from [10] that a subcolocale is a subset S ⊆.M satisfying the following properties:

⋁L ∈.S for all L ⊆.S,

s ←.a ∈.S for all s ∈.S and a ∈.M.

Note that the subcolocales can be considered as generalized subspaces.

Proposition 2.11. Let M be a colocale. Then S ⊆.M is a subcolocale if and only if S is a colocale in the induced order and the embedding i _c : S →.M is a colocalic map.

The Lattice of Subcolocales: Let Scl (M) be the family of all subcolocales of a colocale M. Then Scl (M) is a complete lattice with the inclusion relation. Clearly, arbitrary meet coincides with the intersection, and the join is given by $\underset{i \in I}{\lor} S_{i} = {\lor L : L \subseteq \underset{i \in I}{\cup} S_{i}} .$

Proposition 2.12. (Scl (M), ⊆) is a coframe, that is, T ∨.⋂_i∈I S _i) = ⋂_i∈I (S _i ∨.T) for all T, S _i ∈.Scl (M).

Remark 2.13. The bottom element and top element of Scl (M) are 0_Scl(M) = {0} and 1_Scl(M) = M, respectively.

Open and Closed Subcolocales: Now we present two examples of subcolocales which can be regarded as generalized open and closed subspaces of a topological space. Let F ∈. $C$ (X). Then, K ⊆.F is closed in F if and only if it is closed in X. So we have $C$ (F) = ↓.F.

Based on the previous discussion, we can define the closed subcolocales of M as the subsets $c_{C} (a) = ↓ a \subseteq M$ . Indeed, ↓a ⊆.M is a subcolocale for all a ∈.M. It is obviously closed under arbitrary join. On the other hand, given s ∈.#x2193;.a and x ∈.M, we have s ←.x ≤.a since s ≤.a ∨.x.

We further define the open subcolocale $o_{C} (a)$ by $o_{C} (a) = {x \leftarrow a : x \in M} = {x \in M : x \leftarrow a = x} .$

The open subcolocale $o_{C} (a)$ is the complement of the closed subcolocale $o_{C} (a)$ in Scl (M). Namely, we have $o_{C} (a) \land c_{C} (a) = {0}$ and $o_{C} (a) \lor c_{C} (a) = M$ . Moreover, this complementation entails the following implications: $a \leq b \Leftrightarrow c_{C} (a) \subseteq c_{C} (b) \Leftrightarrow o_{C} (b) \subseteq o_{C} (a) .$

We shall use the following properties of the open and closed subcolocales:

Proposition 2.14.

$⋂_{i \in I} c_{C} (a_{i}) = c_{C} (⋀_{i \in I} a_{i})$

$c_{C} (a) \lor c_{C} (b) = c_{C} (a \lor b)$

$⋁_{i \in I} o_{C} (a_{i}) = o_{C} (⋀_{i \in I} a_{i})$

$o_{C} (a) \cap o_{C} (b) = o_{C} (a \lor b)$

Proof. (1) and (2) can be directly shown by using the definition of a closed subcolocale.

(3) Since ⋀_i∈I a _i ≤.a _i for all i, we have $o_{C} (a_{i}) \subseteq o_{C} (⋀_{i \in I} a_{i})$ , and hence $o_{C} (⋀_{i \in I} a_{i})$ is an upper bound of $A = {o_{C} (a_{i}) : i \in I}$ .

Now let S ⊆.M be another upper bound of $A$ and $y \in o_{C} (⋀_{i \in I} a_{i}) .$ Then $y = x \leftarrow \underset{i \in I}{\land} a_{i} = \underset{i \in I}{\lor} (x \leftarrow a_{i})$ Now, using (cs1) we obtain $y \in o_{C} (a_{i}) \subseteq S$ . Thus $\lor A = \underset{i \in I}{\lor} o_{C} (a_{i}) = o_{C} (\underset{i \in I}{\land} a_{i}) .$ (4) For x ∈.M, we have $x \leftarrow (a \lor b) = (x \leftarrow b) \leftarrow a = (x \leftarrow a) \leftarrow b,$ whence $x \leftarrow (a \lor b) \in o_{C} (a) \cap o_{C} (b) .$ On the other hand, if $k \in o_{C} (a) \cap o_{C} (b)$ then there exist x, y ∈.M such that k = x ←.a = y ←.b. We have, $k = x \leftarrow a = (x \leftarrow a) \leftarrow a = k \leftarrow a$ and $k = y \leftarrow b = (y \leftarrow b) \leftarrow b = k \leftarrow b$ Thus, k ←.a ∨.b) = (k ←.b) ←.a = (k ←.a) ←.a = k ←.a = k, and hence $k \in o_{C} (a \lor b)$ .□

Remark 2.15. Recall de Morgan’s second law stating that (⋀_i∈I a _i) ^∗ = ⋁_i∈I a _i ^∗ holds in a coframe whenever ⋀_i∈I a _i exists. Note that, in the previous proposition, (3) (resp., (4)) can be seen directly by (1) (resp., (2)) together with de Morgan’s second law. (Here, ^∗ indicates the pseudocomplement of an element.)

Another Representation of Subcolocales: It will be useful at this point to recall the concept of a conucleus [10].

Let M be a coframe. Then the mapping t : M →.M is called a conucleus if

t satisfies the dual of the extensiveness property, i.e., t (a) ≤.a for all a ∈.M.

t is a monotone map.

t is idempotent, i.e., t (t (a)) = t (a) for all a ∈.M.

t preserves finite joins.

The subcolocale generated by the conucleus t is S _t = t (M). On the other hand, for a subcolocale S ⊆.M, the corresponding conuclei t _S : M →.M is defined by

t_{S} (a) = i_{c_{*}} (a) = \lor {s \in S : s \leq a}

where i _c : S →.M is the embedding.

Example 2.16. Let L = $C$ (X). Then, for F ∈. $C$ (X), S = ↓.F is a subcolocale of L. Further, the conucleus corresponding to S is given by $\begin{matrix} t_{↓ F} (K) & = ⋁ {H \in ↓ F : H \subseteq K} \\ = cl (⋃ {H \in ↓ F : H \subseteq K}) \\ = F \cap H . \end{matrix}$

Proposition 2.17. There is a one to one correspondence between the subcolocales of M and the conuclei defined on M.

Proposition 2.18. Let M be a coframe. Then if M′.#x2286;.M is a subcoframe (that is, a subset closed under arbitrary meets and finite joins) and S ⊆.M is a subcolocale of M then t _S (M′) is a subcoframe of S.

Facts 2.19.

If S ₁, S ₂ ⊆.M are subcolocales and S ₁ ⊆.S ₂ then t _S
₁ ≤.t _S
₂.

The conuclei corresponding to the closed subcolocale $c_{C} (a)$ is $t c_{C} (a) (x) = \lor {s \in c_{C} (a) = ↓ a : s \leq x} = a \land x .$

The conuclei corresponding to the open subcolocale $o_{C} (a)$ is $t_{o_{C} (a)} (y) = y \leftarrow a$ . Indeed: We have $y \leftarrow a \leq t_{o_{C} (a)} (y)$ since y ←.a ≤.y. On the other hand, if x ←.a ≤.y then $x \leftarrow a \leq (x \leftarrow a) \leftarrow a \leq y \leftarrow a .$ We also know that $\begin{matrix} t_{o_{C} (a)} (y) & = ⋁ {s \in o_{C} (a) : s \leq y} \\ = ⋁ {x \leftarrow a : x \leftarrow a \leq y} . \end{matrix}$ Thus we have $t_{o_{C} (a)} (y) \leq y \leftarrow a$ .

3 The link between (co)frames and textures

In this section, we provide a link between morphisms of texture spaces and (co)frames. To discover the relevant morphisms, let (S, $S$ ), (T, $T$ ) be textures and (r, R) be a direlation between them. Then, by virtue of [16, Corollary 2.12], φ _r (A) = r ^→ A is a map preserving arbitrary joins, and ψ _R (A) = R ^→ A is a map preserving arbitrary meets. The following theorem shows that these results actually characterize the direlations.

Theorem 3.1. Let (S, $S$ ), (T, $T$ ) be textures.

If r is a relation from (S, $S$ ) to (T, $S$ →. $T$ defined by φ _r (A) = r ^→ A preserves arbitrary joins.

Conversely, if φ : $S$ →. $P$ (S) ⊗. $T$ defined by $\begin{array}{l} r_{φ} = \lor {{\bar{P}}_{(s, t)} : \exists u \in S, v \in T with P_{s} ⊈ Q_{u}, \\ P_{v} ⊈ Q_{t}; φ (B) \subseteq Q_{v} \Rightarrow B \subseteq Q_{u}, \forall B \in S} \end{array}$ is a relation from (S, $S$ ) to (T, $T$ ). Moreover, φ _{r
_φ} = φ and r _{φ
_r} = r.

If R is a corelation from (S, $S$ ) to (T, $S$ →. $T$ defined by ψ _R (A) = R ^→ A preserves arbitrary intersections.

Conversely, if ψ : $S$ →. $P$ (S) ⊗. $T$ defined by $\begin{array}{l} R_{ψ} = \cap {{\bar{Q}}_{(s, t)} : \exists u \in S, v \in T with P_{u} ⊈ Q_{s}, \\ P_{t} ⊈ Q_{v}; P_{v} \subseteq ψ (B) \Rightarrow P_{u} \subseteq B, \forall B \in S} \end{array}$ is a corelation from (S, $S$ ) to (T, $T$ ). Moreover ψ _{R
_ψ} = ψ and R _{ψ
_R} = R.

Proof. We first note that this theorem is a special case of [21, Proposition 2.2], hence we omit the details of the proof.

(2) Suppose that ψ : $S$ →. $T$ preserves arbitrary intersections. First, let us show that R _ψ is a corelation:

(CR1) Let ${\bar{P}}_{(s, t)} R_{ψ}$ and P _s ⊈.Q _s′. Suppose, by way of contradiction that, ${\bar{P}}_{(s^{'}, t)} \subseteq R_{ψ}$ . Then, by definition of R _ψ, we have a t′.#x2208;.T with ${\bar{P}}_{(s, t)} {\bar{Q}}_{(s, t^{'})}$ , and u ∈.S, v ∈.T satisfying P _u ⊈.Q _s, P _t′ ⊈.Q _v and $P_{v} \subseteq ψ (B) \Rightarrow P_{u} \subseteq B, \forall B \in S$ Now ${\bar{P}}_{(s, t)} ⊈ {\bar{Q}}_{(s, t^{'})}$ together with Proposition 2.3 gives ${\bar{P}}_{(s^{'}, t)} ⊈ {\bar{Q}}_{(s^{'}, t^{'})}$ . Moreover, if P _v ⊆.ψ (B) for B ∈. $S$ then P _u ⊆.B, whence B ⊈.Q _s. So we have P _s ⊆.B by Theorem 2.1. Now we obtain ${\bar{P}}_{(s^{'}, t)} ⊈ {\bar{Q}}_{(s^{'}, t^{'})}$ , P _s ⊈.Q _s′, P _t′ ⊈.Q _v and for B ∈. $S$ , “P _v ⊆.ψ (B) ⇒.P _s ⊆.B”, and hence ${\bar{P}}_{(s^{'}, t)} \subseteq R_{ψ} \subseteq {\bar{Q}}_{(s^{'}, t^{'})}$ . But this gives ${\bar{P}}_{(s, t)} \subseteq {\bar{Q}}_{(s, t^{'})}$ , which is a contradiction.

The condition (CR2) can be proved in a similar manner.

We now show that ψ _{R
_ψ} = ψ. Assume first that there is an A ∈. $S$ such that ψ _{R
_ψ} (A) ⊈.ψ (A). Then there exists a t ∈.T with ψ _{R
_ψ} (A) ⊈.Q _t and P _t ⊈.ψ (A). By definition of ψ _{R
_ψ} (A) = R _ψ ^→ (A), we have a t′.#x2208;.T with P _t′ ⊈.Q _t for which ${\bar{P}}_{(s, t^{'})} R_{ψ} \Rightarrow P_{s} \subseteq A .$ (1)Now P _t′ ⊈.Q _t together with Theorem 2.1 gives v, v′.#x2208;.T such that P _t′ ⊈.Q _v′, P _v′ ⊈.Q _v and P _v ⊈.Q _t.

Setting B ₀ = ⋀.B ∈. $S$ : P _v ⊆.ψ (B)} we obtain $P_{t} \subseteq P_{v} \subseteq ψ (B_{0}) = \land {ψ (B) \in T : P_{v} \subseteq ψ (B)},$ whence ψ (B ₀) ⊈.ψ (A). Since ψ is order-preserving, B ₀ ⊈.A, and hence there exists a s ∈.S such that B ₀ ⊈.Q _s and P _s ⊈.A. Moreover P _s ⊈.A gives an u ∈.S with P _s ⊈.Q _u and P _u ⊈.A.

Now we obtain P _(u,t′) ⊈.Q _(u,v′), P _s ⊈.Q _u, P _v′ ⊈.Q _v and “P _v ⊆.ψ (B) ⇒.P _s ⊆.B ₀ ⊆.B”.or all B ∈. $S$ , and thus P _(u,t′) ⊈.R _ψ. But this gives, by condition (1), that P _u ⊆.A, which is a contradiction.

The converse inclusion can be proved by using a similar technique.

Finally, replacing ψ by ψ _R in the equality ψ _{R
_ψ} = ψ, we get R _{ψ
_R} ^→ (A) = ψ _R (A) = R ^→ (A) for all A ∈. $S$ . Thus R _{ψ
_R} = R, as required.□

Corollary 3.2. The morphisms of the category drTex can be replaced by the pairs (φ, ψ), φ, ψ : $S$ →. $T$ , where φ preserves arbitrary joins and ψ preserves arbitrary intersections (meets).

When we come to drTex^op we are dealing with φ _{R
^←}, ψ _{r
^←} where φ _{R
^←} (B) = (R ^←) ^→ B = R ^← B, ψ _{r
^←} (B) = (r ^←) ^→ B = r ^← B, B ∈. $T$ , the presections for the original direlation. By [16, Theorem 2.11] and an argument similar to that in [21, Proposition 2.4] we have:

Theorem 3.3. Let (r, R) : (S, $S$ ) →.T, $S$ , $T$ as categories in the usual way. Then

(ψ _R) ^∗ = φ _{R
^←} : $S$ is a left adjoint and satisfies $(ψ_{R}) * (B) = R^{\leftarrow} B = \cap {A \in S : B \subseteq ψ_{R} (A)}$ for all B ∈. $T$ .

(φ _r) _∗ = ψ _{r
^←} : $S$ is a right adjoint and satisfies ${(φ_{r})}_{*} (B) = r^{\leftarrow} B = \lor {A \in S : φ_{r} (A) \subseteq B}$ for all B ∈. $T$ .

Proof. (1) By Proposition 2.4 (2), (R ^←, R ^→) = (φ _{R
^←}, ψ _R) is an adjoint pair, and hence (ψ _R) ^∗ = φ _{R
^←}. Moreover, by Proposition 2.5, we have $(ψ_{R}) * (B) = \cap {A \in S : B \subseteq ψ_{R} (A)}$ (2) It can be proved in a similar manner by using the adjoint pair (r ^→, r ^←) = (φ _r, ψ _{r
^←}).□

Corollary 3.4. The morphisms of the category drTex^op can be replaced by the pairs ((ψ _R) ^∗, (φ _r) _∗) where φ _r and ψ _R preserve arbitrary joins and arbitrary meets, respectively.

To provide a link with frames (resp., coframes), we need to consider mappings preserving finite meets (resp., finite joins) as well as arbitrary joins (resp., arbitrary meets).

Theorem 3.5. Let (r, R) : (S, $S$ ) →.T, $T$ ) be a direlation. Then

The following are equivalent for r:

φ _r preserves finite intersections.

Given t, t′.#x2208;.T with P _t ⊈.Q _t′, and s ₁, s ₂ ∈.S for which $r {\bar{Q}}_{(s_{1}, t)}$ and $r {\bar{Q}}_{(s_{2}, t)}$ , there exists an s ∈.S such that $r {\bar{Q}}_{(s, t^{'})}$ and (P _s
₁ ∩.P _s
₂) ⊈.Q _s.

The following are equivalent for R:

ψ _R preserves finite unions.

Given t, t′.#x2208;.T with P _t ⊈.Q _t′, and s ₁, s ₂ ∈.S for which ${\bar{P}}_{(s_{1}, t)} R$ and ${\bar{P}}_{(s_{2}, t)} R$ , there exists an s ∈.S such that ${\bar{P}}_{(s, t^{'})} R$ and P _s ⊈.Q _s
₁ ∪.Q _s
₂).

Proof. We only prove (1) and leave the proof of (2) to the interested reader.

Let (a) hold and take t, t′, s ₁, s ₂ with the stated properties. Then, by Proposition 2.4(1), we have r ^→ P _{s
₁} ⊈.Q _t and r ^→ P _{s
₂} ⊈.Q _t. Hence, by Theorem 2.1(4), we obtain P _t ⊆.r ^→ P _{s
₁} ∩.r ^→ P _{s
₂}, whence (r ^→ P _{s
₁} ∩.r ^→ P _{s
₂}) ⊈.Q _t′. Now by (a) $\begin{matrix} r^{\to} P_{s_{1}} \cap r^{\to} P_{s_{2}} & = φ_{r} (P_{s_{1}}) \cap φ_{r} (P_{s_{2}}) \\ = φ_{r} (P_{s_{1}} \cap P_{s_{2}}) \\ = r^{\to} (P_{s_{1}} \cap P_{s_{2}}) \end{matrix}$ Thus, by definition of the section of r, we obtain an s ∈.S satisfying $r ⊈ {\bar{Q}}_{(s, t^{'})}$ and (P _{s
₁} ∩.P _{s
₂}) ⊈.Q _s, as required.

Conversely, let (b) hold and take any A, B ∈. $S$ . Since the operation r ^→ preserves inclusion, we clearly have r ^→ (A ∩.B) ⊆.r ^→ A) ∩.r ^→ B).

Suppose, on the contrary, that (r ^→ A) ∩.r ^→ B) ⊈.r ^→ (A ∩.B). Then we have a t ∈.T with $(r^{\to} A) \cap (r^{\to} B) ⊈ Q_{t} and P_{t} ⊈ r^{\to} (A \cap B)$ and hence a t′.#x2208;.T with P _t ⊈.Q _t′ and $r {\bar{Q}}_{(z, t^{'})} \Rightarrow A \cap B \subseteq Q_{z}, \forall z \in S$ (2)by definition of the section. Now r ^→ A ⊈.Q _t and r ^→ B ⊈.Q _t give s ₁, s ₂ ∈.S with $r ⊈ {\bar{Q}}_{(s_{1}, t)}, A ⊈ Q_{s_{1}} and r ⊈ {\bar{Q}}_{(s_{2}, t)}, B ⊈ Q_{s_{2}},$ respectively. Hence, by the hypothesis, there exists an s ∈.S with $r ⊈ {\bar{Q}}_{(s_{1}, t^{'})} and (P_{s_{1}} \cap P_{s_{2}}) ⊈ Q_{s},$ Further, A ⊈.Q _{s
₁} and B ⊈.Q _{s
₂} give P _{s
₁} ⊆.A and P _{s
₂} ⊆.B, respectively, whence (A ∩.B) ⊈.Q _s. Applying the implication (2) with z = s gives rise to a contradiction. Thus $φ_{r} (A \cap B) = φ_{r} (A) \cap φ_{r} (B)$ and hence φ _r preserves finite intersections.□

We refer to a relation (resp., corelation) satisfying the equivalent conditions of Lemma 3.5(1) (resp., Lemma 3.5(2)) as a fr-relation (fr-corelation). A pair consisting of a fr-relation and a fr-corelation is called an fr-direlation.

Corollary 3.6. A composition of fr-relations (fr-corelations, fr-direlations) is a fr-relation (fr-corelation, fr-direlation).

Proof. These results can be established directly from the textural characterizations, or by noting that the composition of functions preserving finite joins (intersections) clearly has the same property.□

Proposition 3.7. The inverse of a di-function is a fr-direlation.

Proof. Let (f, F) be a difunction and let (r, R) = (f, F) ^←. Then, by [16, Theorem 2.24], r ^→ A = (F ^←) ^→ A = F ^← A = f ^← A = (f ^←) ^→ A = R ^→ A. It follows that φ _r = ψ _R preserves arbitrary joins and intersections, whence certainly (r, R) is a fr-direlation.□

Let frTex (resp., frcoTex) denote the category of textures and fr-relations (resp., fr-corelations). We observe that frTex (resp., frcoTex) is a full subcategory of Frm (resp., coFrm).

Remarks 3.8.

The property of being a fr-direlation is generally not preserved under taking the inverse: Let (f, F) be a difunction which is not a fr-direlation. Then, by Proposition 3.7, (r, R) = (f, F) ^← is a fr-direlation. Thus, by hypothesis, (r, R) ^← = (f, F) is not a fr-direlation.

We define a functor $I : drTex \to {drTex}^{op}$ such that $J ((S, S) \overset{(r, R)}{\to} (T, T)) = \overset{{(r, R)}^{\leftarrow}}{\to} (S, S) .$

A corresponding functor $G : {drTex}^{op} \to drTex$ satisfying $I \circ G = {id}_{{drTex}^{op}}$ and $G \circ I = {id}_{drTex}$ can be defined, and hence $I$ is an isomorphism. In particular, the categories drTex and drTex^op are isomorphic.

We conclude from (1) and (2) that $I$ does not restrict to an isomorphism between frTex and frTex^op.

Remarks 3.9.

Recall from [3] that a Hutton dispace is a triple (L, τ, κ) where L is a complete, completely distributive lattice and (τ, κ) is a ditopology. The category of Hutton dispaces and mappings φ : (L ₁, τ ₁, κ ₁) →.L ₂, τ ₂, κ ₂) preserving arbitrary meets and joins, and satisfying φ (τ ₁) ⊆.τ ₂, φ (κ ₁) ⊆.κ ₂ is denoted by diH.

Let (S, $S$ , τ _S, κ _S) and (T, $T$ , τ _T, κ _T) be ditopological texture spaces and let $(φ_{r}, ψ_{R}) : (S, S, τ_{S}, κ_{S}) \to (T, T, τ_{T}, κ_{T})$ be a fr-direlation satisfying φ _r (τ _S) ⊆.τ _T and ψ _R (κ _S) ⊆.κ _T. The resulting category is denoted by frDitop. Further we have a contravariant functor $F : dfDitop \to frDitop$ given by $\begin{array}{l} F ((S, S, τ_{S}, κ_{S}) \overset{(f, F)}{\to} (T, T, τ_{T}, κ_{T})) \\ = (T, T, τ_{T}, κ_{T}) \overset{(F^{\leftarrow}, f^{\leftarrow})}{\to} (S, S, τ_{S}, κ_{S}) \end{array}$

4 Diframes

In this section, we introduce the concepts of a diframe and a dilocale. As mentioned before, the main idea here is to replace the complete distributivity with the frame and coframe distributive laws. This enables us to obtain a larger family of lattices.

Definition 4.1. A diframe is a triple L = (L _e, L _fr, L _cf) with the following properties:

The lattice L _e is simultaneously a frame and a coframe.

L _fr ⊆.L _e is closed under arbitrary joins and finite meets.

L _cf ⊆.L _e is closed under arbitrary meets and finite joins.

(Here L _e is called the envelope of L.)

Example 4.2.

Given a topological space X, L _X = ( $P$ (X), Ω(X), $C$ (X)) is a diframe.

For a ditopological texture space (S, $S$ , τ, κ), the triple ( $S$ , τ, κ) is a diframe. Thus, the category of diframes contains more objects than the category of ditopological texture spaces, and hence diframes can be considered as generalized ditopological texture spaces.

Let $L_{e} = Ω_{reg} (ℝ)$ ( $ℝ$ endowed with usual topology).

(a) Let $L_{fr} = {(- \infty, a) : a \in ℝ} \cup {\emptyset, ℝ}$ and $L_{cf} = Ω_{reg} (ℝ)$ . Then L _fr is closed under arbitrary joins since $⋁_{n \in ℕ} (- \infty, a - \frac{1}{n}) = (- \infty, a)$ , and it is obviously closed under finite meets. Thus, L = (L _e, L _fr, L _cf) is a diframe.

(b) Let $L_{fr} = {(- \infty, a) : a \in ℝ} \cup {\emptyset, ℝ}$ and $L_{cf} = {(a, \infty) : a, b \in ℝ} \cup {\emptyset, ℝ}$ . Then L _cf is closed under arbirary meets since $\underset{n \in ℕ}{\land} (a - \frac{1}{n}, \infty) = int (\underset{n \in ℕ}{\cap} (a - \frac{1}{n}, \infty)) = (a, \infty) .$ Clearly, it is closed under finite joins. Therefore, L = (L _e, L _fr, L _cf) is a diframe.

Definition 4.3. Let L and M be diframes. Then a pair of mappings (φ, ψ) : L →.M is called a diframe homomorphism if

φ : L _e →.M _e is a frame homomorphism satisfying φ (L _fr) ⊆.M _fr,

ψ : L _e →.M _e is a coframe homomorphism satisfying ψ (L _cf) ⊆.M _cf.

We denote the category of diframes and diframe homomorphisms by diFrm. The opposite category of diFrm is denoted by diLoc, and it is called the category of dilocales.

Example 4.4. Given a continuous map f : X →.Y, (f ^-1, f ^-1) : L _Y →.L _X is obviously a diframe homomorphism.

Remark 4.5. The morphisms of the category diLoc are the pairs (f, g) : L →.M satisfying the following properties:

f preserves all suprema and g preserves all infima.

f _* : M _e →.L _e preserves finite joins and g ^* : M _e →.L _e preserves finite meets.

f _* (M _cf) ⊆.L _cf and g ^* (M _fr) ⊆.L _fr.

Observe that (g ^*, f _*) : M →.L is a diframe homomorphism.

Facts 4.6.

Recall the category dfDitop of ditopological texture spaces and bicontinuous difunctions [17]. Using Proposition 3.7 we define a covariant functor $E : dfDitop \to diLoc$ by setting $\begin{array}{l} E ((S_{1}, S_{1}, τ_{1}, κ_{1}) \overset{(f, F)}{\to} (S_{2}, S_{2}, τ_{2}, κ_{2}) \\ = (S_{1}, τ_{1}, κ_{1}) \overset{(φ f, ψ F)}{\to} (S_{2}, τ_{2}, κ_{2}) \end{array}$ where the second arrow represents the diLoc morphism corresponding to the diframe homomorphism $(S_{1}, τ_{1}, κ_{1}) \overset{((φ F^{\leftarrow}, ψ f^{\leftarrow}) = (ψ F) *, {(φ f)}_{*})}{\to} (S_{2}, τ_{2}, κ_{2})$

Let φ : (L ₁, τ ₁, κ ₁) →.L ₂, τ ₂, κ ₂) be a morphism of the category diH. Then (φ, φ) is obviously a diframe homomorphism. Now we obtain a covariant functor $H : diH \to diFrm$ given by $\begin{array}{l} ℌ ((L_{1}, τ_{1}, κ_{1}) \overset{φ}{\to} (L_{2}, τ_{2}, κ_{2})) \\ = (L_{1}, τ_{1}, κ_{1}) \overset{(φ, φ)}{\to} (L_{2}, τ_{2}, κ_{2}) \end{array}$ .

Corollary 4.7.

dfDitop is a non-full subcategory of diLoc.

diH is a non-full subcategory of diFrm.

Definition 4.8. Let L = (L _e, L _fr, L _cf) be a diframe.

A subset β ⊆.L _fr is called a base of L if for every a ∈.L _fr there exists a β _a ⊆.β such that a = ⋁.β _a.

A subset β ⊆.L _cf is called a cobase of L if for every k ∈.L _cf there exists a β _k ⊆.β such that k = ⋀.β _k.

Example 4.9. Let $L_{e} = L_{fr} = Ω_{reg} (ℝ)$ and $L_{cf} = {(a, \infty) : a, b \in ℝ} \cup {\emptyset, ℝ}$ . Then the family $β = {(a, b) : a, b \in ℝ}$ is a base for L = (L _e, L _fr, L _cf): Given U ∈.L _fr and x ∈.U, there is an r >. such that (x - r, x + r) ⊆.U, and hence U = ⋃_x∈U (x - r, x + r). From here we have $\begin{matrix} U & = int (cl (U)) \\ = int (cl (⋃_{x \in U} (x - r, x + r))) \\ = ⋁_{x \in U} (x - r, x + r) \end{matrix}$ The following proposition gives necessary and sufficient conditions for a subset of L _e to be a (co)base for a diframe with envelope L _e.

Proposition 4.10. Let L _e be a frame and a coframe, and β ⊆.L _e. Then the following statements hold:

β is a base for a diframe with the envelope L _e if and only if the following conditions are satisfied:

⋁β = 1

For every b ₁, b ₂ ∈.β there exists a β _a ⊆.β such that b ₁ ∧.b ₂ = ⋁.β _a.

β is a cobase for a diframe with the envelope L _e if and only if the following conditions are satisfied:

⋀β = 0

For every b ₁, b ₂ ∈.β there exists a β _k ⊆.β such that b ₁ ∨.b ₂ = ⋀.β _k.

Proof. (1) If β is a base for a diframe then the properties β ₁ and β ₂ are clear.

Conversely, suppose β ⊆.L _e is a subset with the properties stated above. Let L _fr = {⋁.A : A ⊆.β}. Then L _fr is obviously closed under arbitrary joins. Moreover if, ⋁A, ⋁B ∈.L _fr then, by frame distributivity, $\begin{matrix} (⋁ A) \land (⋁ B) = ⋁_{a \in A} (a \land (⋁ B)) \\ = ⋁_{a \in A} ⋁_{b \in B} (a \land b) . \end{matrix}$ Now, by (b), a ∧.b can be expressed as a join of elements of β. Thus (⋁.A) ∧.⋁.B) ∈.L _fr, and hence L _fr is closed under finite meets.

Now we conclude that L = (L _e, L _fr, L _e) is a diframe and β is a base for L.□

The following proposition is easy to check.

Proposition 4.11. Let L = (L _e, L _fr, L _cf) be a diframe.

If β is a base for L then, for any a ∈.L _fr and x ∈.L _e with anleqx, there exists a b ∈.β such that b ≤.a and bnleqx.

If β is a cobase for L then, for any k ∈.L _cf and x ∈.L _e with xnleqk, there exists a b ∈.β such that k ≤.b and xnleqb.

Definition 4.12. Let L be a diframe. Then a subset δ ⊆.L _fr (resp., δ ⊆.L _cf) is called a subbase (resp., subcobase) of L if the set of finite meets (resp., joins) of δ is a base (resp., cobase) for L.

We end this section by giving the definition of a subdilocale which will be useful in our future studies.

Definition 4.13. Let L = (L _e, L _fr, L _cf) be a diframe. A subdilocale of L is a triple S = (S _e, S _fr, S _cf) where S _e ⊆.L _e is both a sublocale and a subcolocale of L _e, S _fr = v _{S
_e} (L _fr) and S _cf = t _{S
_e} (L _cf).

Proposition 4.14. S = (S _e, S _fr, S _cf) is a diframe.

Proof. Since L _fr ⊆.L _e and L _cf ⊆.L _e, we have v _{S
_e} (L _fr) ⊆.v _{S
_e} (L _e) = S _e and t _{S
_e} (L _cf) ⊆.t _{S
_e} (L _e) = S _e. Moreover, v _{S
_e} (L _fr) is a subframe of L _e by Proposition 2.8, and t _{S
_e} (L _cf) is a subcoframe of L _e by Proposition 2.18. Thus S is a subdilocale, as required.□

5 Conclusion

In this paper we provide a link between morphisms of the category of texture spaces (drTex) and the category frames (Frm) and then obtained a full subcategory of Frm (resp., coFrm) which is denoted by frTex (resp., frcoTex). Then we construct the category diFrm of diframes and diframe homomorphisms by using this connection. As a future work, we plan to study topological and bitopological structures such as separation axioms, compactness, join compactness and connectedness, etc. on diframes. Another potential direction for the future work is to consider soft topological spaces. Ditopological texture spaces also provide a suitable framework for the soft variation of topology [23]. On the other hand, as pointed out in [12 –14] and [24], there is a strong connection between rough sets and soft sets in application, and these concepts are useful for decision making. By using this connections, the field of application of diframes may be extended to more applied areas, such as data analysis and information sciences.

Footnotes

Acknowledgement

This paper is dedicated the memory of Dr. L. Michael Brown. The authors thank the referees and associate editor for their constructive comments and suggestions.

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