L. M. Brown in 1992 at a conference on Fuzzy systems and artificial intelligence held in Trabzon introduced the notion of a texture space under the name fuzzy structure. Textures first arose in representation of connection of lattices of *L*−*fuzzy sets* and Hutton algebras under a point based settings. This representation provided a fruitful atmosphere to study complement free concepts in mathematics. We now recall the definition of the texture space as follows.

**Texture space:** [12] If 𝓢 is a set, a texturing ℑ ⊆ *P*(𝓢) is complete, point separating, completely distributive lattice containing S and ∅, and, for which finite join ⋁ coincides with union ⋃ and arbitrary meet ⋀ coincides with intersection ⋃. Then the pair (S, ℑ) is called the texture space.

A mapping *σ* : ℑ → ℑ satisfying *σ*^{2}(*A*) = *A*, for each *A* ∈ ℑ and *A* ⊆ *B* implies *σ*(*B*) ⊆ *σ*(*A*), ∀ *A*, *B* ∈ ℑ is called a complementation on (𝓢, ℑ) and (𝓢, ℑ, *σ*) is then said to be a complemented texture [12]. The sets $\begin{array}{}{P}_{s}=\bigcap \{A\in \mathrm{\Im}|s\in A\}\end{array}$ and $\begin{array}{}{Q}_{s}=\bigvee \{{P}_{t}|t\in S,\phantom{\rule{thickmathspace}{0ex}}s\notin {P}_{t}\}\end{array}$ defines conveniently most of the properties of the texture space and are known as *p-sets* and *q-sets* respectively.

For *A* ∈ ℑ the core *A*^{b}of *A* is defined by *A*^{b}={*s* ∈ 𝓢 ∣ *A* ⊈ *Q*_{s}}. The set *A*^{b}does not necessarily belong to ℑ.

If $\begin{array}{}(S,P(s)),(\mathcal{L},{\mathrm{\Im}}_{2})\end{array}$ are textures, then the product texture of (𝓢, *P*(𝓢)) and $\begin{array}{}(\mathcal{L},{\mathrm{\Im}}_{2})\end{array}$ is *P*(𝓢) ⊗ℑ_{2} for which *P*_{(s,t)} and *Q*_{(s,t}) denotes the *p-sets* and *q-sets* respectively. For *s* ∈ 𝓢, *t* ∈ 𝓛 we have p-sets and q-sets in the product space as following :

$$\begin{array}{}{\overline{P}}_{(s,t)}=\{s\}\times {P}_{t}\\ {\overline{Q}}_{(s,t)}=(S\mathrm{\setminus}\{s\}\times T)\cup (S\times {Q}_{t}).\end{array}$$

**Direlation:** [13] Let (*S*, ℑ_{1}), (𝓛, ℑ_{2}) be textures. Then for $\begin{array}{}r\in \mathcal{P}(S)\otimes {\mathrm{\Im}}_{2}\end{array}$ satisfying:

(R_{1}) *r* ⊈ Q_{(s,t)} and *P*_{ś} ⊈ *Q*_{s} implies *r* ⊈ Q_{(ś,t)},

(R_{2}) *r* ⊈ Q_{(s,t)} then there is ś ∈ 𝓢 such that *P*_{s} ⊈ *Q*_{ś} and *r* ⊈ Q_{(ś,t)}, is called *relation* and for $\begin{array}{}R\in \mathcal{P}(S)\otimes {\mathrm{\Im}}_{2}\end{array}$ such that:

(CR_{1}) $\begin{array}{}{\overline{P}}_{(s,t)}\u2288R\end{array}$ *R* and *P*_{s} ⊈ *Q*_{ś} implies $\begin{array}{}{\overline{P}}_{(\stackrel{\xb4}{s},t)}\u2288R\end{array}$,

(CR_{2}) If $\begin{array}{}{\overline{P}}_{(s,t)}\u2288R\end{array}$ then there exists ś ∈ 𝓢 such that *P*_{ś} ⊈ *Q*_{s} and $\begin{array}{}{\overline{P}}_{(\stackrel{\xb4}{s},t)}\u2288R\end{array}$, is called a *corelation* from (𝓢, *P*(𝓢)) to (𝓛, ℑ_{2}). The pair (*r*, *R*) together is a direlation from (𝓢, ℑ_{1}) to (𝓛, ℑ_{2}).

#### Lemma 2.1

([13]) Let (*r*, *R*) be a direlation from (𝓢, ℑ_{1}) to (*T*, ℑ_{2}), *J* be an index set, $\begin{array}{}{A}_{j}\in {\mathrm{\Im}}_{1}\text{,}\mathrm{\forall}j\in J\end{array}$ and *B*_{j} ∈ ℑ_{2},∀*j* ∈ *J*. Then:

(1) $\begin{array}{}{r}^{\leftarrow}(\bigcap _{j\in J}{B}_{j})=\bigcap _{j\in J}{r}^{\leftarrow}{B}_{j}and{R}^{\to}(\bigcap _{j\in J}{A}_{j})=\bigcap _{j\in J}{R}^{\to}{A}_{j}\end{array}$,

(2) $\begin{array}{}{r}^{\to}(\underset{j\in J}{\bigvee}{A}_{j})=\underset{j\in J}{\bigvee}{r}^{\to}{A}_{j}\phantom{\rule{thickmathspace}{0ex}}and\phantom{\rule{thickmathspace}{0ex}}{R}^{\leftarrow}(\underset{j\in J}{\bigvee}{B}_{j})=\underset{j\in J}{\bigvee}{R}^{\leftarrow}{B}_{j}\end{array}$.

#### Difunction

Let (*f*, *F*) be a direlation from (𝓢, ℑ_{1}) to 𝓛, ℑ_{2}). Then (*f*, *F*): (𝓢, ℑ_{1}) → (𝓛, ℑ_{2}) is a difunction if it satisfies the following two conditions
:

(DF1) For $\begin{array}{}s,\stackrel{\xb4}{s}\in S,{P}_{s}\u2288{Q}_{\stackrel{\xb4}{s}}\u27f9t\in \mathcal{L}\end{array}$ with $\begin{array}{}f\u2288{\overline{Q}}_{(s,t)}\end{array}$ and $\begin{array}{}{\overline{P}}_{(\stackrel{\xb4}{s},t)}\u2288F.\end{array}$

(DF2) For $\begin{array}{}t,{t}^{\mathrm{\prime}}\in \mathcal{L}\end{array}$ and *s* ∈ *S*, *f* ⊈ *Q*_{(s, t)} and $\begin{array}{}{\overline{P}}_{(s,{t}^{\mathrm{\prime}})}\u2288F\u27f9{P}_{{t}^{\mathrm{\prime}}}\u2288{Q}_{t.}\end{array}$

#### Definition 2.2

[13] *Let* (*f*, *F*): (*S*, ℑ_{1}) → (𝓛, ℑ_{2}) *be a difunction. For A* ∈ ℑ_{1}, *the image f*^{→}(*A*) *and coimage F*^{→}(*A*) *are defined as*:

$$\begin{array}{}{f}^{\text{}\u27f6}(A)=\bigcap \{{Q}_{t}:\mathrm{\forall}s,\text{}f\u2288{\overline{Q}}_{(s,t)}\u27f9A\subseteq {Q}_{s}\},\end{array}$$

$$\begin{array}{}{F}^{\u27f6}(A)=\bigvee \{{P}_{t}:\mathrm{\forall}s,\text{}{\overline{P}}_{(s,t)}\u2288F\u27f9{P}_{s}\subseteq A\},\end{array}$$

*and for B* ∈ ℑ_{2}, *the inverse image f*^{⇐}(*B*) *and inverse coimage F*^{⇐}(*B*) *are defined as*:

$$\begin{array}{}f{\text{}}^{\u27f5}(B)=\bigvee \{{P}_{s}:\mathrm{\forall}t,\text{}f\text{}\u2288{\overline{Q}}_{(s,t)}\u27f9{P}_{t}\subseteq B\},\end{array}$$

$$\begin{array}{}F{\text{}}^{\u27f5}(B)=\bigcap \{{Q}_{s}:\mathrm{\forall}t,\text{}{\overline{P}}_{(s,t)}\u2288F\u27f9B\subseteq {Q}_{t}\}.\end{array}$$

For a difunction, the inverse image and the inverse coimage are equal, but the image and coimage are usually not.

#### Lemma 2.3

[13] *For a direlation* (*f*, *F*) *from* (*S*, ℑ_{1}) *to* (*T*, ℑ_{2}) *the following are equivalent:*

(*f*, *F*) *is a direlation.*

*The following inclusion holds:*

*f* ^{←}(*F*^{→}(*A*))⊆ *A* ⊆ *F*^{←}(*f*^{→}(*A*)); ∀ *A* ∈ ℑ_{1}, *and*

*f*^{←}(*F*^{←}(*B*))⊆ *B* ⊆ *F*^{→}(*f* ^{←}(*B*)); ∀ *B* ∈ ℑ_{1}

*f* ^{←}(*B*)=*F* ^{←}(*B*); ∀ *B* ∈ ℑ_{2}.

#### Definition 2.4

[13] *Let* (*f*, *F*): (*S*, ℑ_{1}) → (𝓛, ℑ_{2}) *be a difunction. Then* (*f*, *F*) *is called surjective if it satisfies the condition*:

*(SUR) For* $\begin{array}{}t,{t}^{\mathrm{\prime}}\in \mathcal{L},{P}_{t}\u2288{Q}_{{t}^{\mathrm{\prime}}}\u27f9\mathrm{\exists}s\in S,\u2288{Q}_{(s,{t}^{\mathrm{\prime}})}\end{array}$ *and* $\begin{array}{}{\overline{P}}_{(s,t)}\u2288F\end{array}$. *Similarly*, (*f*, *F*) *is called injective if it satisfies the condition (INJ) For s, ś ∈ S, and t* ∈ 𝓛 *with f* ⊈ *Q*_{(s,t)} *and* $\begin{array}{}{\overline{P}}_{(\stackrel{\xb4}{s},t)}\u2288F\u27f9{P}_{s}\u2288{Q}_{\stackrel{\xb4}{s}}\end{array}$.

We now recall the notion of ditopology on texture spaces.

#### Definition 2.5

[14] *A pair* (*τ*, *κ*) *of subsets of* ℑ *is said to be a ditopology on a texture space* (*S*, ℑ), *if τ* ⊆ ℑ *satisfies*:

*S*, ∅ ∈ *τ*.

*G*_{1}, *G*_{2} ∈ *τ implies G*_{1} ∩ *G*_{2} ∈ *τ and*

*G*_{α} ∈ *τ*, *α* ∈ *I implies* $\begin{array}{}\underset{\alpha}{\bigvee}{G}_{\alpha}\in \tau \end{array}$,

*and κ* ⊆ ℑ *satisfies*:

*S*, ∅ ∈ *κ*.

*F*_{1}, *F*_{2} ∈ *κ implies F*_{1}∪ *F*_{2} ∈ *κ and*

*F*_{α} ∈ *κ*, *α* ∈ *I implies* ⋂ *F*_{α} ∈ *κ*,

where the members of *τ* are called open sets and members of *κ* are closed sets. Also *τ* is called topology, *κ* is called cotopology and (*τ*, *κ*) is called ditopology. If (*τ*, *κ*) is a ditopology on (*S*, ℑ) then (*S*, ℑ, *τ*, *κ*) is called a ditopological texture space.

Note that in general we assume no relation between the open and closed sets in ditopology. In case of complemented texture space (*S*, ℑ, *σ*), *τ* and *κ* are connected by the relation *κ*=*σ*(*τ*), where *σ* is a complementation on (*S*, ℑ), that is an inclusion reversing involution *σ* :ℑ → ℑ, then we call (*τ*, *κ*) a complemented ditopology on (*S*, ℑ). A complemented ditopological texture space is denoted by (*S*, ℑ, *σ*, *τ*, *κ*). In this case we have σ (*A*) = (*σ*(*A*))° and σ(*A*°) = (σ (*A*)), where ()° denotes the interior and ( ) denotes the closure. Recall that for a ditopology (*τ*, *κ*) on (*S*, ℑ), for *A* ∈ ℑ the closure of *A* for the ditopology (*τ*, *κ*) is denoted by (*A*) and defined by

$$\begin{array}{}(\overline{A})=\bigcap \{F\in \kappa :A\subseteq F\},\end{array}$$

and the interior of *A* is denoted by (*A*)° and defined by

$$\begin{array}{}(A{)}^{\circ}=\bigvee \{G\in \tau :G\subseteq A\}\end{array}$$

For terms not defined here, the reader is referred to see [6, 13, 15].

The idea of semi-open sets in topological spaces was first introduced by Norman Levine in 1963 in [16]. Ş Dost extended this concept of semi-open sets from topological spaces to ditopological texture spaces in 2012 in [17].

It is known from [17] that in a ditopological texture space (*S*, ℑ, *τ*, *κ*):

*A* ∈ ℑ is semi-open if and only if there exists a set *G* ∈ *O*(*S*) such that *G* ⊆ *A* ⊆ *G*.

*B* ∈ ℑ is semi-closed if and only if there exists a set *F* ∈ *C*(*S*) such that (*F*)°⊆ *B* ⊆ *F*.

*O*(*S*)⊆ *SO* (*S*) and *C*(*S*)⊆ *SC* (*S*). The collection of all semi-open (resp. semi-closed) sets in ℑ is denoted by *SO* (*S*, ℑ,*τ*, *κ*) or simply *SO* (*S*) (resp. *SC* (*S*, ℑ, *τ*, *κ*) or simply *SC* (*S*)). *SR* (*S*) is the collection of all the semi-regular sets in S. A set *A* is semi-regular if *A* is semi-open as well as semi-closed in *S*.

Arbitrary join of semi-open sets is semi-open.

Arbitrary intersection of semi-closed sets is semi-closed.

If *A* is semi-open in ditopoloical texture space (*S*, ℑ, *τ*, *κ*) then its complement may not be *semi-closed*. Every open set is semi-open, whereas a semi-open set may not be open. The intersection of two semi-open sets may not be semi-open, but intersection of an open set and a semi-open set is always semi-open.

In general there is no connection between the semi-open and semi-closed sets, but in case of complemented ditopological texture space (*S*, ℑ, *σ*, *τ*, *κ*), *A* ∈ ℑ is semi-open if and only if *σ*(*A*) is semi-closed. Where ()_{∘} denotes the semi-Interior and ( ) denotes the semi-closure.

#### Definition 2.6

[18] Let (*S*, ℑ, *τ*, *κ*) be a ditopological texture space and *A* ∈ ℑ. We define:

*(i) The semi-closure* (*A*) of *A* under (*τ*, *κ*) by

$$\begin{array}{}(\underset{\_}{A})=\bigcap \{B:B\in SC(S)\text{, and}A\subseteq B\}\end{array}$$

*(ii) The semi-interior* (*A*)_{∘} of *A* under (*τ*, *κ*) by

$$(A{)}_{\circ}=\bigvee \{B:B\in SO(S)\text{, and}B\subseteq A\}.$$

#### Lemma 2.7

[18] *Let* (*S*, ℑ, *τ*, *κ*) *be a ditopological texture space. A set A* ∈ ℑ *is called*:

*(a) semi-open if and only if A* ⊆ (*A*°)

*(b) semi-closed if and only if* (*A*)° ⊆ *A*.

A difunction (*f*, *F*): (*S*, ℑ_{1}, *τ*_{S,}*κ*_{S}) → (*T*, ℑ_{2},*τ*_{T}, *κ*_{T}) is:

continuous [17]; if *F*^{←}(*G*) ∈ *τ*_{S} where *G* ∈ *τ*_{T};

cocontinuous [17]; if *f*^{←}(*K*) ∈ *κ*_{S} where *K* ∈ *κ*_{T};

bicontinuous [17]; if it is continuous and cocontinuous.

#### Definition 2.8

*[17] Let* (*S*_{i}, ℑ_{i}, *τ*_{i}, *κ*_{i}), *i*=1, 2 *be ditoplogical texture spaces. A difunction* (*f*, *F*):(*S*_{1}, ℑ_{1}) → (*S*_{2}, ℑ_{2}) *is said to be*:

*semi-continuous (semi-irresolute) if for each open (resp. semi-open) set A* ∈ ℑ_{2,} *the inverse image F*^{←}(*A*) ∈ ℑ_{1} *is a semi-open set*.

*semi-cocontinuous (semi-co-irresolute) if for each closed (resp. semi-closed) set B* ∈ ℑ_{2,} *the inverse image f* ^{←}(*B*) ∈ ℑ_{1} *is a semi-closed set*.

*semi-bicontinuous if it semi-continuous and semi-cocontinuous*.

*semi-bi-irresolute if it is semi-irresolute and semi-co-irresolute*

Throughout this paper a space *S* is an infinite ditopological texture space (*S*, ℑ, *τ*, *κ*) on which no separation axioms are assumed unless otherwise stated.

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