On Fuzzy γI-Continuity and γI-Irresoluteness via K-Fuzzy γI-Open Sets

1. Introduction

The concept of a fuzzy set of a nonempty set Z is a mapping $\rho :Z\to I$ (where $I=[0,1]$). This concept was first defined in 1965 by Zadeh [1]. The integration between fuzzy sets and some uncertainty approaches such as rough sets and soft sets has been investigated in [2–4]. The concept of a fuzzy topology was presented in 1968 by Chang [5]. Several authors have successfully generalized the theory of general topology to the fuzzy setting with crisp methods. According to Šostak [6], the notion of a fuzzy topology being a crisp subclass of the class of fuzzy sets and fuzziness in the notion of openness of a fuzzy set have not been considered, which seems to be a drawback in the process of fuzzification of a topological space. Therefore, Šostak [6] defined a novel definition of a fuzzy topology as the concept of openness of fuzzy sets. It is an extension of a fuzzy topology defined by Chang [5]. Thereafter, many researchers (Ramadan [7], Chattopadhyay et. al. [8], El Gayyar et. al. [9], Höhle and Šostak [10], Ramadan et. al. [11], Kim et. al. [12], Abbas [13,14], Kim and Abbas [15], Aygun and Abbas [16,17], Li and Shi [18,19], Shi and Li [20], Fang and Guo [21], El-Dardery et. al. [22], Kalaivani and Roopkumar [23], Solovyov [24], Minana and Šostak [25]) have redefined the same notion and investigated fuzzy topological spaces ($\mathcal{FTS}s$) being unaware of Šostak’s work.

The generalizations of fuzzy open sets plays an effective role in a fuzzy topology through their ability to improve on many results, or to open the door to explore and discuss several fuzzy topological notions such as fuzzy continuity [7,8], fuzzy connectedness [8], fuzzy compactness [8,9], fuzzy separation axioms [18], etc. Overall, the notions of k-fuzzy pre-open (k-FP-open) sets, k-fuzzy semi-open (k-FS-open) sets, k-fuzzy $\beta$-open (k-F$\beta$-open) sets, and k-fuzzy $\alpha$-open (k-F$\alpha$-open) sets were presented and investigated by the authors of [12,14] in $\mathcal{FTS}s$ based on Šostak’s sense [6]. Also, Kim et al. [12] defined and discussed some weaker forms of fuzzy continuity, called FS-continuity (resp. FP-continuity and F$\alpha$-continuity) between $\mathcal{FTS}s$ based on Šostak’s sense. Abbas [14] explored and characterized the concepts of F$\beta$-continuous (resp. F$\beta$-irresolute) mappings between $\mathcal{FTS}s$ in the sense of Šostak. Also, Kim and Abbas [15] defined some new types of k-fuzzy compactness on $\mathcal{FTS}s$ in the sense of Šostak. Furthermore, the notions of k-fuzzy $\gamma$-open (k-F$\gamma$-open) sets and k-fuzzy $\gamma$-closed (k-F$\gamma$-closed) sets were defined and discussed by the authors of [26] on $\mathcal{FTS}s$ in the sense of Šostak [6].

A novel concept of fuzzy local function, called k-fuzzy local function was presented and investigated by Taha and Abbas [27] in an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ based on Šostak’s sense [6]. Moreover, the concepts of fuzzy lower (resp. upper) weakly and almost $\mathcal{I}$-continuous multifunctions were displayed and investigated by Taha and Abbas [27]. Also, Taha [28–30] introduced the notions of k-FS$\mathcal{I}$-open sets, k-FP$\mathcal{I}$-open sets, k-F$\alpha \mathcal{I}$-open sets, k-F$\beta \mathcal{I}$-open sets, k-FS$\beta \mathcal{I}$-open sets, k-F$\delta \mathcal{I}$-open sets, and k-GF$\mathcal{I}$-closed sets in an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ based on Šostak’s sense. Overall, Taha [29–31] presented the notions of fuzzy upper (resp. lower) generalized $\mathcal{I}$-continuous (resp. pre-$\mathcal{I}$-continuous, semi-$\mathcal{I}$-continuous, $\alpha$-$\mathcal{I}$-continuous, $\delta$-$\mathcal{I}$-continuous, and strong $\beta$-$\mathcal{I}$-continuous) multifunctions via fuzzy ideals [32].

The purpose of this study is as follows. Section 2 contains many basic results and notions that help in understanding the obtained results. In Section 3, we present and study a novel class of fuzzy sets, called k-F$\gamma \mathcal{I}$-open sets in $\mathcal{FITS}s$ based on Šostak’s sense. This class is contained in the class of k-FS$\beta \mathcal{I}$-open sets and contains all k-F$\alpha \mathcal{I}$-open sets, k-FP$\mathcal{I}$-open sets, and k-FS$\mathcal{I}$-open sets. We also define and discuss the closure and interior operators with respect to the classes of k-F$\gamma \mathcal{I}$-open sets and k-F$\gamma \mathcal{I}$-closed sets. Furthermore, we introduce new types of fuzzy $\mathcal{I}$-separation axioms using k-F$\gamma \mathcal{I}$-closed sets, called k-F$\gamma \mathcal{I}$-regular spaces and k-F$\gamma \mathcal{I}$-normal spaces, and study some properties of them. In Section 4, we present and investigate the concept of F$\gamma \mathcal{I}$-continuous mappings using k-F$\gamma \mathcal{I}$-open sets. Also, we display and characterize the concepts of FA$\gamma \mathcal{I}$-continuous and FW$\gamma \mathcal{I}$-continuous mappings, which are weaker forms of F$\gamma \mathcal{I}$-continuous mappings. In Section 5, we explore and discuss some new F$\gamma \mathcal{I}$-mappings using k-F$\gamma \mathcal{I}$-open sets and k-F$\gamma \mathcal{I}$-closed sets, called F$\gamma \mathcal{I}$-open mappings, F$\gamma \mathcal{I}$-closed mappings, F$\gamma \mathcal{I}$-irresolute mappings, F$\gamma \mathcal{I}$-irresolute open mappings, and F$\gamma \mathcal{I}$-irresolute closed mappings. In the last section, we close this work with proposed future articles and conclusions.

2. Preliminaries

In this study, non-empty sets will be denoted by Z, Y, X, etc. On Z, $I^Z$ is the class of all fuzzy sets. For any fuzzy set $\omega \in I^Z$, ${\omega }^c\left(z\right)=1-\omega \left(z\right)$, for each $z\in Z$. Also, for $s\in I,$$\underline{s}\left(z\right)=s$, for each $z\in Z$.

A fuzzy point $z_s$ on Z is a fuzzy set, is defined as follows: $z_s\left(v\right)=s$ if $v=z$, and $z_s\left(v\right)=0$ for any $v\in Z-\left\{z\right\}$. Moreover, we say that $z_s$ belongs to $\omega \in I^Z$ ($z_s\in \omega$), if $s\le \omega \left(z\right)$. On Z, $P_s\left(Z\right)$ is the class of all fuzzy points.

On Z, a fuzzy set $\nu \in I^Z$ is a quasi-coincident with $\rho \in I^Z$ ($\nu \mathcal{Q}\rho$), if there is $z\in Z$, with $\nu \left(z\right)+\rho \left(z\right)>1$. Otherwise, $\nu$ is not a quasi-coincident with $\rho$ ($\nu \overline{\mathcal{Q}}\rho$).

The difference between $\nu ,\rho \in I^Z$ [27] is defined as follows:

$$\nu \overline{\wedge }\rho =\left\{\begin{array}{c}\underline{0},\mathrm{if}\nu \le \rho ,\hfill \\ \nu \wedge {\rho }^c,\mathrm{otherwise}.\hfill \end{array}\right.$$

Lemma 2.1. 

$\left[33\right]$Let $\omega ,\rho \in I^Z$. Thus,

(1) $\omega \mathcal{Q}\rho$ iff there is $z_s\in \omega$ such that $z_s\mathcal{Q}\rho$,

(2) if $\omega \mathcal{Q}\rho$, then $\omega \wedge \rho \ne \underline{0}$,

(3) $\omega \overline{\mathcal{Q}}\rho$ iff $\omega \le {\rho }^c$,

(4) $\omega \le \rho$ iff $z_s\in \omega$ implies $z_s\in \rho$ iff $z_s\mathcal{Q}\omega$ implies $z_s\mathcal{Q}\rho$ iff $z_s\overline{\mathcal{Q}}\rho$ implies $z_s\overline{\mathcal{Q}}\omega$,

(5) $z_s\overline{\mathcal{Q}}{\bigvee }_{i\in \Gamma }{\omega }_i$ iff there is $i_{\circ }\in \Gamma$ such that $z_s\overline{\mathcal{Q}}{\omega }_{i_{\circ }}$.

Definition 2.1. 

$[6,7]$ A mapping $\zeta :I^Z⟶I$ is called a fuzzy topology on Z if it satisfies the following conditions:

(1) $\zeta \left(\underline{1}\right)=\zeta \left(\underline{0}\right)=1.$

(2) $\zeta (\omega \wedge \rho )\ge \zeta \left(\omega \right)\wedge \zeta \left(\rho \right),$ for each $\omega ,\rho \in I^Z.$

(3) $\zeta \left({\bigvee }_{i\in \Gamma }{\omega }_i\right)\ge {\bigwedge }_{i\in \Gamma }\zeta \left({\omega }_i\right),$ for each ${\omega }_i\in I^Z.$

Thus, $(Z,\zeta )$ is called a fuzzy topological space ($\mathcal{FTS}$) based on Šostak’s sense.

Definition 2.2. 

$[7,12]$ A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im )$ is called

(1) fuzzy continuous if $\zeta \left({\mathbb{P}}^{-1}\left(\rho \right)\right)\ge \Im \left(\rho \right)$, for every $\rho \in I^Y$;

(2) fuzzy open if $\Im \left(\mathbb{P}\right(\omega \left)\right)\ge \zeta \left(\omega \right)$, for every $\omega \in I^Z$;

(3) fuzzy closed if $\Im \left({\left(\mathbb{P}\left(\omega \right)\right)}^c\right)\ge \zeta \left({\omega }^c\right)$, for every $\omega \in I^Z$.

Definition 2.3. 

$[8,11]$ In an $\mathcal{FTS}$$(Z,\zeta )$, for each $\omega \in I^Z$ and $k\in I_{\circ }$ (where $I_{\circ }=(0,1]$), we define fuzzy operators $C_{\zeta }$ and $I_{\zeta }$$:I^Z\times I_{\circ }\to I^Z$ as follows:

$$C_{\zeta }(\omega ,k)=\bigwedge \{\nu \in I^Z:\omega \le \nu ,\zeta \left({\nu }^c\right)\ge k\}.$$

$$I_{\zeta }(\omega ,k)=\bigvee \{\nu \in I^Z:\nu \le \omega ,\zeta \left(\nu \right)\ge k\}.$$

Definition 2.4. 

$[12,14,26]$ Let $(Z,\zeta )$ be an $\mathcal{FTS}$ and $k\in I_{\circ }$. A fuzzy set $\omega \in I^Z$ is called

(1) k-F-open if $\omega =I_{\zeta }(\omega ,k)$;

(2) k-FP-open if $\omega \le I_{\zeta }(C_{\zeta }(\omega ,k),k)$;

(3) k-FS-open if $\omega \le C_{\zeta }(I_{\zeta }(\omega ,k),k)$;

(4) k-FR-open if $\omega =I_{\zeta }(C_{\zeta }(\omega ,k),k)$;

(5) k-F$\alpha$-open if $\omega \le I_{\zeta }(C_{\zeta }(I_{\zeta }(\omega ,k),k),k)$;

(6) k-F$\beta$-open if $\omega \le C_{\zeta }(I_{\zeta }(C_{\zeta }(\omega ,k),k),k)$;

(7) k-F$\gamma$-open if $\omega \le C_{\zeta }(I_{\zeta }(\omega ,k),k)\vee I_{\zeta }(C_{\zeta }(\omega ,k),k)$.

Remark 2.1. 

$[12,14,26]$From the previous definitions, we have the following diagram.

$$\mathbf{k}-\mathbf{FP}-\mathbf{open}\mathbf{set}$$

$$\nearrow \searrow$$

$$\mathbf{k}-\mathbf{F}-\mathbf{open}\mathbf{set}⟶\mathbf{k}-\mathbf{F}\alpha -\mathbf{open}\mathbf{set}⟶\mathbf{k}-\mathbf{F}\gamma -\mathbf{open}\mathbf{set}⟶\mathbf{k}-\mathbf{F}\beta -\mathbf{open}\mathbf{set}$$

$$\searrow \nearrow$$

$$\mathbf{k}-\mathbf{FS}-\mathbf{open}\mathbf{set}$$

Definition 2.5. 

$[12,14,26]$ A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im )$ is called FS-continuous (resp. FP-continuous, F$\alpha$-continuous, F$\beta$-continuous, and F$\gamma$-continuous) if ${\mathbb{P}}^{-1}\left(\omega \right)$ is an k-FS-open (resp. k-FP-open, k-F$\alpha$-open, k-F$\beta$-open, and k-F$\gamma$-open) set, for every $\omega \in I^Y$ with $\Im \left(\omega \right)\ge k$ and $k\in I_{\circ }$.

Definition 2.6. 

$\left[26\right]$ In an $\mathcal{FTS}$$(Z,\zeta )$, for each $\omega \in I^Z$ and $k\in I_{\circ }$, we define fuzzy operators $\gamma C_{\zeta }$ and $\gamma I_{\zeta }$$:I^Z\times I_{\circ }\to I^Z$ as follows:

$$\gamma C_{\zeta }(\omega ,k)=\bigwedge \{\nu \in I^Z:\omega \le \nu ,\nu \mathrm{is}k-\mathrm{F}\gamma -\mathrm{closed}\}.$$

$$\gamma I_{\zeta }(\omega ,k)=\bigvee \{\nu \in I^Z:\nu \le \omega ,\nu \mathrm{is}k-\mathrm{F}\gamma -\mathrm{open}\}.$$

Definition 2.7. 

$\left[32\right]$ A fuzzy ideal $\mathcal{I}$ on Z, is a map $\mathcal{I}:I^Z⟶I$ that satisfies the following:

(1) ∀$\omega ,\nu \in I^Z$ and $\omega \le \nu$⇒$\mathcal{I}\left(\nu \right)\le \mathcal{I}\left(\omega \right)$.

(2) ∀$\omega ,\nu \in I^Z$⇒$\mathcal{I}(\omega \vee \nu )\ge \mathcal{I}\left(\omega \right)\wedge \mathcal{I}\left(\nu \right)$.

Moreover, ${\mathcal{I}}_0$ is the simplest fuzzy ideal on Z, and is defined as follows:

$${\mathcal{I}}_0\left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu =\underline{0},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Definition 2.8. 

$\left[27\right]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $k\in I_{\circ }$, and $\omega \in I^Z$. Then the k-fuzzy local function ${\omega }_k^{*}$ of $\omega$ is defined as follows:

$${\omega }_k^{*}=\bigwedge \{\rho \in I^Z:\mathcal{I}\left(\omega \overline{\wedge }\rho \right)\ge k,\zeta \left({\rho }^c\right)\ge k\}.$$

Remark 2.2. 

$\left[27\right]$If we take $\mathcal{I}={\mathcal{I}}_0$, for each $\omega \in I^Z$ we have:

$${\omega }_k^{*}=\bigwedge \{\rho \in I^Z:\omega \le \rho ,\zeta \left({\rho }^c\right)\ge k\}=C_{\zeta }(\omega ,k).$$

Definition 2.9. 

$\left[27\right]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $k\in I_{\circ }$, and $\omega \in I^Z$. Then we define fuzzy operator $C_{\zeta }^{*}$$:I^Z\times I_{\circ }\to I^Z$ as follows:

$$C_{\zeta }^{*}(\omega ,k)=\omega \vee {\omega }_k^{*}.$$

Now if, $\mathcal{I}={\mathcal{I}}_0$ then $C_{\zeta }^{*}(\omega ,k)=\omega \vee {\omega }_k^{*}=\omega \vee C_{\zeta }(\omega ,k)=C_{\zeta }(\omega ,k)$ for each $\omega \in I^Z$.

Theorem 2.1. 

$\left[27\right]$Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $k\in I_{\circ }$, and $\omega ,\rho \in I^Z$. The operator $C_{\zeta }^{*}$$:I^Z\times I_{\circ }\to I^Z$ satisfies the following properties:

(1) $C_{\zeta }^{*}(\underline{0},k)=\underline{0}$.

(2) $\omega \le C_{\zeta }^{*}(\omega ,k)\le C_{\zeta }(\omega ,k)$.

(3) If $\omega \le \rho$, then $C_{\zeta }^{*}(\omega ,k)\le C_{\zeta }^{*}(\rho ,k)$.

(4) $C_{\zeta }^{*}(\omega \vee \rho ,k)=C_{\zeta }^{*}(\omega ,k)\vee C_{\zeta }^{*}(\rho ,k)$.

(5) $C_{\zeta }^{*}(\omega \wedge \rho ,k)\le C_{\zeta }^{*}(\omega ,k)\wedge C_{\zeta }^{*}(\rho ,k)$.

(6) $C_{\zeta }^{*}(C_{\zeta }^{*}(\omega ,k),k)=C_{\zeta }^{*}(\omega ,k)$.

Definition 2.10. 

$[28,30]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$ and $k\in I_{\circ }$. A fuzzy set $\omega \in I^Z$ is called

(1) k-FS$\mathcal{I}$-open if $\omega \le C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k)$;

(2) k-FP$\mathcal{I}$-open if $\omega \le I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)$;

(3) k-F$\alpha \mathcal{I}$-open if $\omega \le I_{\zeta }(C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k),k)$;

(4) k-F$\beta \mathcal{I}$-open if $\omega \le C_{\zeta }(I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k),k)$;

(5) k-FS$\beta \mathcal{I}$-open if $\omega \le C_{\zeta }^{*}(I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k),k)$;

(6) k-FR$\mathcal{I}$-open if $\omega =I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)$.

Definition 2.11. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is called F$\alpha \mathcal{I}$-continuous (resp. FP$\mathcal{I}$-continuous, FS$\mathcal{I}$-continuous, and FS$\beta \mathcal{I}$-continuous) if ${\mathbb{P}}^{-1}\left(\omega \right)$ is an k-F$\alpha \mathcal{I}$-open (resp. k-FP$\mathcal{I}$-open, k-FS$\mathcal{I}$-open, and k-FS$\beta \mathcal{I}$-open) set, for each $\omega \in I^Y$ with $\Im \left(\omega \right)\ge k$ and $k\in I_{\circ }$.

Some basic notations and results that we need in the sequel are found in [7-9,27-31].

3. On $\mathbf{k}$-Fuzzy $\mathbf{\gamma }\mathcal{I}$-Open Sets

Definition 3.1. 

Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$ and $k\in I_{\circ }$. A fuzzy set $\rho \in I^Z$ is called an k-F$\gamma \mathcal{I}$-open set if $\rho \le C_{\zeta }^{*}(I_{\zeta }(\rho ,k),k)\vee I_{\zeta }(C_{\zeta }^{*}(\rho ,k),k)$.

Remark 3.1. 

The complement of k-F$\gamma \mathcal{I}$-open sets are k-F$\gamma \mathcal{I}$-closed sets.

Lemma 3.1. 

Every k-F$\gamma \mathcal{I}$-open set is k-F$\gamma$-open [26].

Proof. 

The proof follows from Definitions 2.4, 3.1, and Theorem 2.1(2). □

Remark 3.2. 

If we take $\mathcal{I}={\mathcal{I}}_0$, then k-F$\gamma \mathcal{I}$-open set and k-F$\gamma$-open set [26] are equivalent.

Remark 3.3. 

The converse of Lemma 3.1 fails as Example 3.1 will show.

Example 3.1. 

Define $\zeta ,\mathcal{I}:I^Z⟶I$ as follows:

$$\zeta \left(\rho \right)=\left\{\begin{array}{c}1,\mathrm{if}\rho \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\rho =\underline{0.7},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\rho =\underline{0.3},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\nu \le \underline{0.6},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\underline{0.6}$ is an ${\displaystyle \frac{1}{3}}$-F$\gamma$-open set, but it is not ${\displaystyle \frac{1}{3}}$-F$\gamma \mathcal{I}$-open.

Proposition 3.1. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega \in I^Z$ and $k\in I_{\circ }$. Then

(1) each k-FP$\mathcal{I}$-open set [28] is k-F$\gamma \mathcal{I}$-open;

(2) each k-F$\gamma \mathcal{I}$-open set is k-FS$\beta \mathcal{I}$-open [30];

(3) each k-FS$\mathcal{I}$-open set [28] is k-F$\gamma \mathcal{I}$-open.

Proof.

(1) If $\omega$ is an k-FP$\mathcal{I}$-open set. Then

$$\omega \le I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)$$

$$\le I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)\vee I_{\zeta }(\omega ,k)$$

$$\le I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)\vee C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k).$$

Thus, $\omega$ is k-F$\gamma \mathcal{I}$-open.

(2) If $\omega$ is an k-F$\gamma \mathcal{I}$-open set. Then

$$\omega \le C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k)\vee I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)$$

$$\le C_{\zeta }^{*}(I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k),k)\vee I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k)$$

$$\le C_{\zeta }^{*}(I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k),k).$$

Thus, $\omega$ is k-FS$\beta \mathcal{I}$-open.

(3) If $\omega$ is an k-FS$\mathcal{I}$-open set. Then

$$\omega \le C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k)$$

$$\le C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k)\vee I_{\zeta }(\omega ,k)$$

$$\le C_{\zeta }^{*}(I_{\zeta }(\omega ,k),k)\vee I_{\zeta }(C_{\zeta }^{*}(\omega ,k),k).$$

Thus, $\omega$ is k-F$\gamma \mathcal{I}$-open. □

Remark 3.4. 

From the previous discussions and definitions, we have the following diagram.

$$\mathbf{k}-\mathbf{FP}\mathcal{I}-\mathbf{open}\mathbf{set}$$

$$\nearrow \downarrow$$

$$\mathbf{k}-\mathbf{F}\alpha \mathcal{I}-\mathbf{open}\mathbf{set}⟶\mathbf{k}-\mathbf{F}\gamma \mathcal{I}-\mathbf{open}\mathbf{set}⟶\mathbf{k}-\mathbf{FS}\beta \mathcal{I}-\mathbf{open}\mathbf{set}$$

$$\searrow \uparrow$$

$$\mathbf{k}-\mathbf{FS}\mathcal{I}-\mathbf{open}\mathbf{set}$$

Remark 3.5. 

The converse of the above diagram fails as Examples 3.2, 3.3, and 3.4 will show.

Example 3.2. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\rho ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.3}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$, $\lambda =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.5}}\}$. Define $\zeta ,\mathcal{I}:I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\nu =\rho ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\omega ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.3},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\lambda$ is an ${\displaystyle \frac{1}{4}}$-F$\gamma \mathcal{I}$-open set, but it is not ${\displaystyle \frac{1}{4}}$-FP$\mathcal{I}$-open.

Example 3.3. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\rho ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.3}},{\displaystyle \frac{z_2}{0.2}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.7}},{\displaystyle \frac{z_2}{0.8}}\},$$\lambda =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$. Define $\zeta ,\mathcal{I}:I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\nu =\omega ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\rho ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.5},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\lambda$ is an ${\displaystyle \frac{1}{3}}$-F$\gamma \mathcal{I}$-open set, but it is neither ${\displaystyle \frac{1}{3}}$-FS$\mathcal{I}$-open nor ${\displaystyle \frac{1}{3}}$-F$\alpha \mathcal{I}$-open.

Example 3.4. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$, $\lambda =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.5}}\}$. Define $\zeta ,\mathcal{I}:I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\omega ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.4},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, $\lambda$ is an ${\displaystyle \frac{1}{3}}$-FS$\beta \mathcal{I}$-open set, but it is not ${\displaystyle \frac{1}{3}}$-F$\gamma \mathcal{I}$-open.

Corollary 3.1. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ and $k\in I_{\circ }$. Then

(1) the union of k-F$\gamma \mathcal{I}$-open sets is k-F$\gamma \mathcal{I}$-open;

(2) the intersection of k-F$\gamma \mathcal{I}$-closed sets is k-F$\gamma \mathcal{I}$-closed.

Proof. 

This is easily proved by Definition 3.1 and Remark 3.1. □

Corollary .3.2. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each k-F$\gamma \mathcal{I}$-open set $\omega \in I^Z$.

(1) If $\omega$ is an k-FR$\mathcal{I}$-open set, then $\omega$ is k-FS$\mathcal{I}$-open.

(2) If $\omega$ is an k-FR$\mathcal{I}$-closed set, then $\omega$ is k-FP$\mathcal{I}$-open.

(3) If $I_{\zeta }(\omega ,k)=\underline{0}$, then $\omega$ is k-FP$\mathcal{I}$-open.

(4) If $C_{\zeta }^{*}(\omega ,k)=\underline{0}$, then $\omega$ is k-FS$\mathcal{I}$-open.

Proof. 

The proof follows by Definitions 2.10 and 3.1. □

Corollary 3.3. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each k-F$\gamma \mathcal{I}$-closed set $\omega \in I^Z$.

(1) If $\omega$ is an k-FR$\mathcal{I}$-open set, then $\omega$ is k-FP$\mathcal{I}$-closed.

(2) If $\omega$ is an k-FR$\mathcal{I}$-closed set, then $\omega$ is k-FS$\mathcal{I}$-closed.

(3) If $I_{\zeta }(\omega ,k)=\underline{0}$, then $\omega$ is k-FS$\mathcal{I}$-closed.

(4) If $C_{\zeta }^{*}(\omega ,k)=\underline{0}$, then $\omega$ is k-FP$\mathcal{I}$-closed.

Proof. 

The proof follows by Definition 2.10 and Remark 3.1. □

Definition 3.2. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega \in I^Z$ and $k\in I_{\circ }$, we define a fuzzy $\gamma$-$\mathcal{I}$-closure operator $\gamma C_{\zeta }^{*}$$:I^Z\times I_{\circ }⟶I^Z$ as follows:

$$\gamma C_{\zeta }^{*}(\omega ,k)=\bigwedge \{\rho \in I^Z:\omega \le \rho ,\rho \mathrm{is}k-\mathrm{F}\gamma \mathcal{I}-\mathrm{closed}\}.$$

Proposition 3.2. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega \in I^Z$ and $k\in I_{\circ }$. A fuzzy set $\omega$ is k-F$\gamma \mathcal{I}$-closed iff $\gamma C_{\zeta }^{*}(\omega ,k)=\omega$.

Proof. 

This is easily proved from Definition 3.2. □

Theorem 3.1. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega ,\rho \in I^Z$ and $k\in I_{\circ }$. A fuzzy $\gamma$-$\mathcal{I}$-closure operator $\gamma C_{\zeta }^{*}$$:I^Z\times I_{\circ }⟶I^Z$ satisfies the following properties.

(1) $\gamma C_{\zeta }^{*}(\underline{0},k)=\underline{0}$.

(2) $\omega \le \gamma C_{\zeta }^{*}(\omega ,k)\le C_{\zeta }(\omega ,k)$.

(3) $\gamma C_{\zeta }^{*}(\omega ,k)\le \gamma C_{\zeta }^{*}(\rho ,k)$ if $\omega \le \rho$.

(4) $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)=\gamma C_{\zeta }^{*}(\omega ,k)$.

(5) $\gamma C_{\zeta }^{*}(\omega \vee \rho ,k)\ge \gamma C_{\zeta }^{*}(\omega ,k)\vee \gamma C_{\zeta }^{*}(\rho ,k)$.

Proof. (1), (2), and (3) are easily proved by Definition 3.2.

(4) From (2) and (3), $\gamma C_{\zeta }^{*}(\omega ,k)\le \gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)$. Now, we show $\gamma C_{\zeta }^{*}(\omega ,k)\ge \gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)$. If $\gamma C_{\zeta }^{*}(\omega ,k)$ does not contain $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)$, there is $z\in Z$ and $s\in (0,1)$ with

$$\gamma C_{\zeta }^{*}(\omega ,k)\left(z\right)<s<\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)\left(z\right).\left(\mathcal{N}\right)$$

Since $\gamma C_{\zeta }^{*}(\omega ,k)\left(z\right)<s$, by Definition 3.2, there is $\mu \in I^Z$ as an k-F$\gamma \mathcal{I}$-closed set and $\omega \le \mu$ with $\gamma C_{\zeta }^{*}(\omega ,k)\left(z\right)\le \mu \left(z\right)<s$. Since $\omega \le \mu$, then $\gamma C_{\zeta }^{*}(\omega ,k)\le \mu$. Again, by the definition of $\gamma C_{\zeta }^{*}$, then $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)\le \mu$. Hence, $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)\left(z\right)\le \mu \left(z\right)<s$, which is a contradiction for $\left(\mathcal{N}\right)$. Thus, $\gamma C_{\zeta }^{*}(\omega ,k)\ge \gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)$. Therefore, $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,k),k)=\gamma C_{\zeta }^{*}(\omega ,k)$.

(5) Since $\omega$$\le \omega \vee \rho$ and $\rho$$\le \omega \vee \rho$, hence by (3), $\gamma C_{\zeta }^{*}(\omega ,k)\le \gamma C_{\zeta }^{*}(\omega \vee \rho ,k)$ and $\gamma C_{\zeta }^{*}(\rho ,k)\le \gamma C_{\zeta }^{*}(\omega \vee \rho ,k)$. Thus, $\gamma C_{\zeta }^{*}(\omega \vee \rho ,k)\ge \gamma C_{\zeta }^{*}(\omega ,k)\vee \gamma C_{\zeta }^{*}(\rho ,k)$. □

Definition 3.3. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega \in I^Z$ and $k\in I_{\circ }$, we define a fuzzy $\gamma$-$\mathcal{I}$-interior operator $\gamma I_{\zeta }^{*}$$:I^Z\times I_{\circ }⟶I^Z$ as follows: $\gamma I_{\zeta }^{*}(\omega ,k)=\bigvee \{\rho \in I^Z:\rho \le \omega ,\rho \mathrm{is}r-\mathrm{F}\gamma \mathcal{I}-\mathrm{open}\}.$

Proposition 3.3. 

Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $\omega \in I^Z$, and $k\in I_{\circ }$. Then

(1) $\gamma C_{\zeta }^{*}({\omega }^c,k)={\left(\gamma I_{\zeta }^{*}(\omega ,k)\right)}^c$;

(2) $\gamma I_{\zeta }^{*}({\omega }^c,k)={\left(\gamma C_{\zeta }^{*}(\omega ,k)\right)}^c$.

Proof. (1) For each $\omega \in I^Z$, we have $\gamma C_{\zeta }^{*}({\omega }^c,k)=\bigwedge \{\rho \in I^Z:{\omega }^c\le \rho ,\rho \mathrm{is}k-\mathrm{F}\gamma \mathcal{I}-\mathrm{closed}\}={[\bigvee \{{\rho }^c\in I^Z:{\rho }^c\le \omega ,{\rho }^c\mathrm{is}k-\mathrm{F}\gamma \mathcal{I}-\mathrm{open}\}]}^c$ = ${\left(\gamma I_{\zeta }^{*}(\omega ,k)\right)}^c$.

(2) This is similar to that of (1). □

Proposition 3.4. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega \in I^Z$ and $k\in I_{\circ }$. A fuzzy set $\omega$ is k-F$\gamma \mathcal{I}$-open iff $\gamma I_{\zeta }^{*}(\omega ,k)=\omega$.

Proof. 

This is easily proved from Definition 3.3. □

Theorem 3.2. 

In an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$, for each $\omega ,\rho \in I^Z$ and $k\in I_{\circ }$. A fuzzy $\gamma$-$\mathcal{I}$-interior operator $\gamma I_{\zeta }^{*}$$:I^Z\times I_{\circ }⟶I^Z$ satisfies the following properties.

(1) $\gamma I_{\zeta }^{*}(\underline{1},k)=\underline{1}$.

(2) $I_{\zeta }(\omega ,k)\le \gamma I_{\zeta }^{*}(\omega ,k)\le \omega$.

(3) $\gamma I_{\zeta }^{*}(\omega ,k)\le \gamma I_{\zeta }^{*}(\rho ,k)$ if $\omega \le \rho$.

(4) $\gamma I_{\zeta }^{*}(\gamma I_{\zeta }^{*}(\omega ,k),k)=\gamma I_{\zeta }^{*}(\omega ,k)$.

(5) $\gamma I_{\zeta }^{*}(\omega ,k)\wedge \gamma I_{\zeta }^{*}(\rho ,k)\ge \gamma I_{\zeta }^{*}(\omega \wedge \rho ,k)$.

Proof. 

The proof is similar to that of Theorem 3.1. □

Definition 3.4. 

Let $z_s\in P_s\left(Z\right)$, $\omega \in I^Z$, and $k\in I_{\circ }$. An $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ is said to be an k-F$\gamma \mathcal{I}$-regular space if $z_s\overline{\mathcal{Q}}\omega$ for each k-F$\gamma \mathcal{I}$-closed set $\omega$, there is ${\mu }_i\in I^Z$ with $\zeta \left({\mu }_i\right)\ge k$ for $i=1,2$, such that $z_s\in {\mu }_1$, $\omega \le {\mu }_2$, and ${\mu }_1\overline{\mathcal{Q}}{\mu }_2$.

Definition 3.5. 

Let $\omega ,\rho \in I^Z$ and $k\in I_{\circ }$. An $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ is said to be an k-F$\gamma \mathcal{I}$-normal space if $\omega \overline{\mathcal{Q}}\rho$ for each k-F$\gamma \mathcal{I}$-closed sets $\omega$ and $\rho$, there is ${\mu }_i\in I^Z$ with $\zeta \left({\mu }_i\right)\ge k$ for $i=1,2$, such that $\omega \le {\mu }_1$, $\rho \le {\mu }_2$, and ${\mu }_1\overline{\mathcal{Q}}{\mu }_2$.

Theorem 3.3. 

Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $z_s\in P_s\left(Z\right)$, $\omega \in I^Z$, and $k\in I_{\circ }$. The following statements are equivalent.

(1) $(Z,\zeta ,\mathcal{I})$ is an k-F$\gamma \mathcal{I}$-regular space.

(2) If $z_s\in \omega$ for each k-F$\gamma \mathcal{I}$-open set $\omega$, there is $\rho \in I^Z$ with $\zeta \left(\rho \right)\ge k$, and

$$z_s\in \rho \le C_{\zeta }(\rho ,k)\le \omega .$$

(3) If $z_s\overline{\mathcal{Q}}\omega$ for each k-F$\gamma \mathcal{I}$-closed set $\omega$, there is ${\mu }_i\in I^Z$ with $\zeta \left({\mu }_i\right)\ge k$ for $i=1,2$, such that $z_s\in {\mu }_1$, $\omega \le {\mu }_2$, and $C_{\zeta }({\mu }_1,k)\overline{\mathcal{Q}}{C}_{\zeta }({\mu }_2,k)$.

Proof. (1) ⇒ (2) Let $z_s\in \omega$ for each k-F$\gamma \mathcal{I}$-open set $\omega$, then $z_s\overline{\mathcal{Q}}{\omega }^c$. Since $(Z,\zeta ,\mathcal{I})$ is k-F$\gamma \mathcal{I}$-regular, then there is $\rho ,\nu \in I^Z$ with $\zeta \left(\rho \right)\ge k$ and $\zeta \left(\nu \right)\ge k$, such that $z_s\in \rho$, ${\omega }^c\le \nu$, and $\rho \overline{\mathcal{Q}}\nu$. Thus, $z_s\in \rho \le {\nu }^c\le \omega$, so $z_s\in \rho \le C_{\zeta }(\rho ,k)\le \omega$.

(2) ⇒ (3) Let $z_s\overline{\mathcal{Q}}\omega$ for each k-F$\gamma \mathcal{I}$-closed set $\omega$, then $z_s\in {\omega }^c$. By (2), there is $\nu \in I^Z$ with $\zeta \left(\nu \right)\ge k$ and $z_s\in \nu \le C_{\zeta }(\nu ,k)\le {\omega }^c$. Since $\zeta \left(\nu \right)\ge k$, then $\nu$ is an k-F$\gamma \mathcal{I}$-open set and $z_s\in \nu$. Again, by (2), there is $\mu \in I^Z$ such that $\zeta \left(\mu \right)\ge k$, and $z_s\in \mu \le C_{\zeta }(\mu ,k)\le \nu \le C_{\zeta }(\nu ,k)\le {\omega }^c$. Hence, $\omega \le {\left(C_{\zeta }(\nu ,k)\right)}^c=I_{\zeta }({\nu }^c,k)\le {\nu }^c$. Set $\lambda =I_{\zeta }({\nu }^c,k)$, and thus $\zeta \left(\lambda \right)\ge k$. Then, $C_{\zeta }(\lambda ,k)\le {\nu }^c\le {\left(C_{\zeta }(\mu ,k)\right)}^c$. Therefore, $C_{\zeta }(\mu ,k)\overline{\mathcal{Q}}{C}_{\zeta }(\lambda ,k)$.

(3) ⇒ (1) This is easily proved by Definition 3.4. □

Theorem 3.4. 

Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $\omega ,\rho \in I^Z$, and $k\in I_{\circ }$. The following statements are equivalent.

(1) $(Z,\zeta ,\mathcal{I})$ is an k-F$\gamma \mathcal{I}$-normal space.

(2) If $\rho \le \omega$ for each k-F$\gamma \mathcal{I}$-closed set $\rho$ and k-F$\gamma \mathcal{I}$-open set $\omega$, there is $\nu \in I^Z$ with $\zeta \left(\nu \right)\ge k$, and $\rho \le \nu \le C_{\zeta }(\nu ,k)\le \omega$.

(3) If $\omega \overline{\mathcal{Q}}\rho$ for each k-F$\gamma \mathcal{I}$-closed sets $\omega$ and $\rho$, there is ${\mu }_i\in I^Z$ with $\zeta \left({\mu }_i\right)\ge k$ for $i=1,2$, such that $\omega \le {\mu }_1$, $\rho \le {\mu }_2$, and $C_{\zeta }({\mu }_1,k)\overline{\mathcal{Q}}{C}_{\zeta }({\mu }_2,k)$.

Proof. 

The proof is similar to that of Theorem 3.3. □

4. On Fuzzy $\mathbf{\gamma }\mathcal{I}$-Continuity

Definition 4.1. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is called F$\gamma \mathcal{I}$-continuous if ${\mathbb{P}}^{-1}\left(\omega \right)$ is an k-F$\gamma \mathcal{I}$-open set, for any $\omega \in I^Y$ with $\Im \left(\omega \right)\ge k$ and $k\in I_{\circ }$.

Lemma 4.1. 

Every F$\gamma \mathcal{I}$-continuity is an F$\gamma$-continuity [26].

Proof. 

The proof follows from Definitions 2.5, 4.1, and Lemma 3.1. □

Remark 4.1. 

If we take $\mathcal{I}={\mathcal{I}}_0$, then F$\gamma \mathcal{I}$-continuity and F$\gamma$-continuity [26] are equivalent.

Remark 4.2. 

The converse of Lemma 4.1 fails as Example 4.1 will show.

Example 4.1. 

Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\rho \right)=\left\{\begin{array}{c}1,\mathrm{if}\rho \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\rho =\underline{0.7},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\rho =\underline{0.3},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\nu \le \underline{0.6},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\theta =\underline{0.6},\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is F$\gamma$-continuous, but it is not F$\gamma \mathcal{I}$-continuous.

Remark 4.3. 

From the previous definitions, we have the following diagram.

$$\mathbf{FP}\mathcal{I}-\mathbf{continuity}$$

$$\nearrow \downarrow$$

$$\mathbf{F}\alpha \mathcal{I}-\mathbf{continuity}⟶\mathbf{F}\gamma \mathcal{I}-\mathbf{continuity}⟶\mathbf{FS}\beta \mathcal{I}-\mathbf{continuity}$$

$$\searrow \uparrow$$

$$\mathbf{FS}\mathcal{I}-\mathbf{continuity}$$

Remark 4.4. 

The converse of the above diagram fails as Examples 4.2, 4.3, and 4.4 will show.

Example 4.2. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\rho ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.3}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$, $\lambda =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.5}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\mu =\rho ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\mu =\omega ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\nu <\underline{0.3},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is F$\gamma \mathcal{I}$-continuous, but it is not FP$\mathcal{I}$-continuous.

Example 4.3. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\rho ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.3}},{\displaystyle \frac{z_2}{0.2}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.7}},{\displaystyle \frac{z_2}{0.8}}\},$$\lambda =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\nu =\omega ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\rho ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.5},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is F$\gamma \mathcal{I}$-continuous, but it is neither FS$\mathcal{I}$-continuous nor F$\alpha \mathcal{I}$-continuous.

Example 4.4. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$, $\lambda =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.5}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\omega ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.4},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is FS$\beta \mathcal{I}$-continuous, but it is not F$\gamma \mathcal{I}$-continuous.

Theorem 4.1. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is F$\gamma \mathcal{I}$-continuous iff for any $z_s\in P_s\left(Z\right)$ and any $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$, there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le \rho$ and $k\in I_{\circ }$.

Proof. (⇒) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$, and then ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$. Since $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$, then we obtain $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)=\omega$ (say). Hence, $\omega \in I^Z$ is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le \rho$.

(⇐) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$. According to the assumption there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le \rho$. Hence, $z_s\in \omega \le {\mathbb{P}}^{-1}\left(\rho \right)$ and $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$. Thus, ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$, so ${\mathbb{P}}^{-1}\left(\rho \right)$ is an k-F$\gamma \mathcal{I}$-open set. Then, $\mathbb{P}$ is F$\gamma$-$\mathcal{I}$-continuous. □

Theorem 4.2. 

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping and $k\in I_{\circ }$. Then the following statements are equivalent for every $\omega \in I^Z$ and $\rho \in I^Y$:

(1) $\mathbb{P}$ is F$\gamma \mathcal{I}$-continuous.

(2) ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-closed, for every $\rho \in I^Y$ with $\Im \left({\rho }^c\right)\ge k$.

(3) $\mathbb{P}\left(\gamma C_{\zeta }^{*}(\omega ,k)\right)\le C_{\Im }(\mathbb{P}\left(\omega \right),k)$.

(4) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$.

(5) ${\mathbb{P}}^{-1}\left(I_{\Im }(\rho ,k)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$.

Proof. (1) ⇔ (2) The proof follows by ${\mathbb{P}}^{-1}\left({\rho }^c\right)={\left({\mathbb{P}}^{-1}\left(\rho \right)\right)}^c$ and Definition 4.1.

(2) ⇒ (3) Let $\omega \in I^Z$. By (2), we have ${\mathbb{P}}^{-1}\left(C_{\Im }(\mathbb{P}\left(\omega \right),k)\right)$ is k-F$\gamma \mathcal{I}$-closed. Thus,

$$\gamma C_{\zeta }^{*}(\omega ,k)\le \gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\mathbb{P}\left(\omega \right)\right),k)\le \gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\mathbb{P}\left(\omega \right),k)\right),k)={\mathbb{P}}^{-1}\left(C_{\Im }(\mathbb{P}\left(\omega \right),k)\right).$$

Therefore, $\mathbb{P}\left(\gamma C_{\zeta }^{*}(\omega ,k)\right)\le C_{\Im }(\mathbb{P}\left(\omega \right),k)$.

(3) ⇒ (4) Let $\rho \in I^Y$. By (3), $\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\right)\le C_{\Im }(\mathbb{P}\left({\mathbb{P}}^{-1}\left(\rho \right)\right),k)$$\le C_{\Im }(\rho ,k)$. Thus, $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\right)\right)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$.

(4) ⇔ (5) The proof follows by ${\mathbb{P}}^{-1}\left({\rho }^c\right)={\left({\mathbb{P}}^{-1}\left(\rho \right)\right)}^c$ and Proposition 3.3.

(5) ⇒ (1) Let $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$. By (5), we obtain ${\mathbb{P}}^{-1}\left(\rho \right)={\mathbb{P}}^{-1}\left(I_{\Im }(\rho ,k)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\rho \right)$. Then, $\gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)={\mathbb{P}}^{-1}\left(\rho \right)$. Thus, ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-open, so $\mathbb{P}$ is F$\gamma \mathcal{I}$-continuous. □

Definition 4.2. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is called FA$\gamma \mathcal{I}$-continuous if ${\mathbb{P}}^{-1}\left(\omega \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\omega ,k),k)\right),k)$, for any $\omega \in I^Y$ with $\Im \left(\omega \right)\ge k$ and $k\in I_{\circ }$.

Lemma 4.2. 

Every F$\gamma \mathcal{I}$-continuity is an FA$\gamma \mathcal{I}$-continuity.

Proof. 

The proof follows by Definitions 4.1 and 4.2. □

Remark 4.5. 

The converse of Lemma 4.2 fails as Example 4.5 will show.

Example 4.5. 

Let $Z=\{z_1,z_2,z_3\}$ and define $\omega ,\rho ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.4}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.5}},{\displaystyle \frac{z_3}{0.4}}\},$$\lambda =\{{\displaystyle \frac{z_1}{0.3}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.6}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{2}{3}},\mathrm{if}\nu =\omega ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\rho ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu \le \underline{0.6},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is FA$\gamma \mathcal{I}$-continuous, but it is not F$\gamma \mathcal{I}$-continuous.

Theorem 4.3. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is FA$\gamma \mathcal{I}$-continuous iff for any $z_s\in P_s\left(Z\right)$ and any $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$, there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le I_{\Im }(C_{\Im }(\rho ,k),k)$ and $k\in I_{\circ }$.

Proof. (⇒) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$, and then

$${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right),k).$$

Since $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$, then $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right),k)=\omega \left(\mathrm{say}\right).$ Therefore, $\omega \in I^Z$ is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le I_{\Im }(C_{\Im }(\rho ,k),k)$.

(⇐) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ such that $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$. According to the assumption there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le I_{\Im }(C_{\Im }(\rho ,k),k)$. Hence, $z_s\in \omega \le {\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right)$ and

$$z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right),k).$$

Thus, ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right),k)$. Therefore, $\mathbb{P}$ is FA$\gamma \mathcal{I}$-continuous. □

Theorem 4.4. 

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping, $\rho \in I^Y$, and $k\in I_{\circ }$. Then the following statements are equivalent:

(1) $\mathbb{P}$ is FA$\gamma \mathcal{I}$-continuous.

(2) ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-open, for every k-FR-open set $\rho$.

(3) ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-closed, for every k-FR-closed set $\rho$.

(4) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$, for every k-F$\gamma$-open set $\rho$.

(5) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$, for every k-FS-open set $\rho$.

Proof. (1) ⇒ (2) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ be an k-FR-open set with $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$. Hence, by (1), there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open with $z_s\in \omega$ and $\mathbb{P}\left(\omega \right)\le I_{\Im }(C_{\Im }(\rho ,k),k)$. Thus, $\omega \le {\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right)={\mathbb{P}}^{-1}\left(\rho \right)$ and $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$. Therefore, ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$, so ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-open.

(2) ⇒ (3) If $\rho \in I^Y$ is k-FR-closed, then by (2), ${\mathbb{P}}^{-1}\left({\rho }^c\right)={\left({\mathbb{P}}^{-1}\left(\rho \right)\right)}^c$ is k-F$\gamma \mathcal{I}$-open. Thus, ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-closed.

(3) ⇒ (4) If $\rho \in I^Y$ is k-F$\gamma$-open and since $C_{\Im }(\rho ,k)$ is k-FR-closed, then by (3), ${\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$ is k-F$\gamma \mathcal{I}$-closed. Since ${\mathbb{P}}^{-1}\left(\rho \right)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$, hence

$$\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right).$$

(4) ⇒ (5) The proof follows from the fact that any k-FS-open set is k-F$\gamma$-open.

(5) ⇒ (3) If $\rho \in I^Y$ is k-FR-closed, and then $\rho$ is k-FS-open. By (5),

$$\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)={\mathbb{P}}^{-1}\left(\rho \right).$$

Hence, ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-closed.

(3) ⇒ (1) If $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ such that $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$, and then $z_s\in {\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right)$. Since ${\left[I_{\Im }(C_{\Im }(\rho ,k),k)\right]}^c$ is k-FR-closed, by (3), ${\mathbb{P}}^{-1}\left({\left[I_{\Im }(C_{\Im }(\rho ,k),k)\right]}^c\right)$ is k-F$\gamma \mathcal{I}$-closed. Hence, ${\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right)$ is k-F$\gamma \mathcal{I}$-open and $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right),k).$ Thus, ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }(\rho ,k),k)\right),k).$ Therefore, $\mathbb{P}$ is FA$\gamma \mathcal{I}$-continuous. □

Definition 4.3. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is called FW$\gamma \mathcal{I}$-continuous if ${\mathbb{P}}^{-1}\left(\omega \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\omega ,k)\right),k)$, for any $\omega \in I^Y$ with $\Im \left(\omega \right)\ge k$ and $k\in I_{\circ }$.

Lemma 4.3. 

Every F$\gamma \mathcal{I}$-continuity is an FW$\gamma \mathcal{I}$-continuity.

Proof. 

The proof follows by Definitions 4.1 and 4.3. □

Remark 4.6. 

The converse of Lemma 4.3 fails as Example 4.6 will show.

Example 4.6. 

Let $Z=\{z_1,z_2,z_3\}$ and define $\omega ,\rho ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.4}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.4}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.5}},{\displaystyle \frac{z_3}{0.4}}\},$ $\lambda =\{{\displaystyle \frac{z_1}{0.3}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.6}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\nu =\omega ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\rho ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu \le \underline{0.6},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is FW$\gamma \mathcal{I}$-continuous, but it is not F$\gamma \mathcal{I}$-continuous.

Theorem 4.5. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is FW$\gamma \mathcal{I}$-continuous iff for any $z_s\in P_s\left(Z\right)$ and any $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$, there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le C_{\Im }(\rho ,k)$ and $k\in I_{\circ }$.

Proof. (⇒) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ containing $\mathbb{P}\left(z_s\right)$, and then

$${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k).$$

Since $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$, then $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k)=\omega$ (say). Hence, $\omega \in I^Z$ is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le C_{\Im }(\rho ,k)$.

(⇐) Let $z_s\in P_s\left(Z\right)$ and $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$ such that $z_s\in {\mathbb{P}}^{-1}\left(\rho \right)$. According to the assumption there is $\omega \in I^Z$ that is k-F$\gamma \mathcal{I}$-open containing $z_s$ with $\mathbb{P}\left(\omega \right)\le C_{\Im }(\rho ,k)$. Hence, $z_s\in \omega \le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$ and $z_s\in \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k)$. Thus, ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k)$. Therefore, $\mathbb{P}$ is FW$\gamma$-$\mathcal{I}$-continuous. □

Theorem 4.6. 

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping, $\rho \in I^Y$, and $k\in I_{\circ }$. Then the following statements are equivalent:

(1) $\mathbb{P}$ is FW$\gamma \mathcal{I}$-continuous.

(2) ${\mathbb{P}}^{-1}\left(\rho \right)\ge \gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(\rho ,k)\right),k)$, if $\Im \left({\rho }^c\right)\ge k$.

(3) $\gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k)\ge {\mathbb{P}}^{-1}\left(I_{\Im }(\rho ,k)\right)$.

(4) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(\rho ,k)\right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right)$.

Proof. (1) ⇔ (2) The proof follows by Proposition 3.3 and Definition 4.3.

(2) ⇒ (3) Let $\rho \in I^Y$. Hence by (2),

$$\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }(C_{\Im }({\rho }^c,k),k)\right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }({\rho }^c,k)\right).$$

Thus, ${\mathbb{P}}^{-1}\left(I_{\Im }(\rho ,k)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k)$.

(3) ⇔ (4) The proof follows from Proposition 3.3.

(4) ⇒ (1) Let $\rho \in I^Y$ with $\Im \left(\rho \right)\ge k$. Hence by (4), $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(I_{\Im }({\rho }^c,k)\right),k)\le {\mathbb{P}}^{-1}\left(C_{\Im }({\rho }^c,k)\right)={\mathbb{P}}^{-1}\left({\rho }^c\right)$. Thus, ${\mathbb{P}}^{-1}\left(\rho \right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\rho ,k)\right),k)$, so $\mathbb{P}$ is FW$\gamma \mathcal{I}$-continuous. □

Lemma 4.4. 

Every FA$\gamma \mathcal{I}$-continuity is an FW$\gamma \mathcal{I}$-continuity.

Proof. 

The proof follows by Definitions 4.2 and 4.3. □

Remark 4.7. 

The converse of Lemma 4.4 fails as Example 4.7 will show.

Example 4.7. 

Let $Z=\{z_1,z_2,z_3\}$ and define $\omega ,\lambda ,\rho \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.6}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.4}}\}$, $\lambda =\{{\displaystyle \frac{z_1}{0.3}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.5}}\},$$\rho =\{{\displaystyle \frac{z_1}{0.3}},{\displaystyle \frac{z_2}{0.2}},{\displaystyle \frac{z_3}{0.4}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\nu =\omega ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\rho ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu \le \underline{0.5},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{4}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is FW$\gamma \mathcal{I}$-continuous, but it is not FA$\gamma \mathcal{I}$-continuous.

Remark 4.8. 

From the previous discussions and definitions, we have the following diagram.

$$\mathbf{F}\gamma \mathcal{I}-\mathbf{continuity}⟶\mathbf{FA}\gamma \mathcal{I}-\mathbf{continuity}⟶\mathbf{FW}\gamma \mathcal{I}-\mathbf{continuity}$$

Proposition 4.1. 

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(X,\eta )$ and $\mathbb{Y}:(X,\eta )⟶(Y,\Im )$ be two fuzzy mappings. Then the composition $\mathbb{Y}\circ \mathbb{P}$ is FA$\gamma \mathcal{I}$-continuous if $\mathbb{P}$ is F$\gamma \mathcal{I}$-continuous and $\mathbb{Y}$ is fuzzy continuous.

Proof. 

The proof follows by the previous definitions. □

5. On Fuzzy $\mathbf{\gamma }\mathcal{I}$-Irresoluteness

Definition 5.1. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ is called F$\gamma \mathcal{I}$-irresolute if ${\mathbb{P}}^{-1}\left(\omega \right)$ is an k-F$\gamma \mathcal{I}$-open set, for any k-F$\gamma$-open set $\omega \in I^Y$ and $k\in I_{\circ }$.

Lemma 5.1. 

Every F$\gamma \mathcal{I}$-irresolute mapping is F$\gamma \mathcal{I}$-continuous.

Proof. 

The proof follows from Definitions 4.1, 5.1, and Remark 2.1. □

Remark 5.1. 

The converse of Lemma 5.1 fails as Example 5.1 will show.

Example 5.1. 

Let $Z=\{z_1,z_2\}$ and define $\lambda ,\rho \in I^Z$ as follows: $\lambda =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.5}}\}$, $\rho =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$. Define $\zeta ,\mathcal{I},\Im :I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\nu =\rho ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.5},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{3}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Z,\Im )$ is F$\gamma \mathcal{I}$-continuous, but it is not F$\gamma \mathcal{I}$-irresolute.

Theorem 5.1. 

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping and $k\in I_{\circ }$. Then the following statements are equivalent for every $\omega \in I^Z$ and $\rho \in I^Y$:

(1) $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute.

(2) ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-closed, for every k-F$\gamma$-closed set $\rho$.

(3) $\mathbb{P}\left(\gamma C_{\zeta }^{*}(\omega ,k)\right)\le \gamma C_{\Im }(\mathbb{P}\left(\omega \right),k)$.

(4) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\rho ,k)\right)$.

(5) ${\mathbb{P}}^{-1}\left(\gamma I_{\Im }(\rho ,k)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)$.

Proof. (1) ⇔ (2) The proof follows by ${\mathbb{P}}^{-1}\left({\rho }^c\right)={\left({\mathbb{P}}^{-1}\left(\rho \right)\right)}^c$ and Definition 5.1.

(2) ⇒ (3) Let $\omega \in I^Z$. By (2), we have ${\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\mathbb{P}\left(\omega \right),k)\right)$ is k-F$\gamma \mathcal{I}$-closed. Thus,

$$\gamma C_{\zeta }^{*}(\omega ,k)\le \gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\mathbb{P}\left(\omega \right)\right),k)\le \gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\mathbb{P}\left(\omega \right),k)\right),k)={\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\mathbb{P}\left(\omega \right),k)\right).$$

Therefore, $\mathbb{P}\left(\gamma C_{\zeta }^{*}(\omega ,k)\right)\le \gamma C_{\Im }(\mathbb{P}\left(\omega \right),k)$.

(3) ⇒ (4) Let $\rho \in I^Y$. By (3), $\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\right)\le \gamma C_{\Im }(\mathbb{P}\left({\mathbb{P}}^{-1}\left(\rho \right)\right),k)$$\le \gamma C_{\Im }(\rho ,k)$. Thus, $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\right)\right)\le {\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\rho ,k)\right)$.

(4) ⇔ (5) The proof follows by ${\mathbb{P}}^{-1}\left({\rho }^c\right)={\left({\mathbb{P}}^{-1}\left(\rho \right)\right)}^c$ and Proposition 3.3.

(5) ⇒ (1) Let $\rho \in I^Y$ be an k-F$\gamma$-open set. By (5),

$${\mathbb{P}}^{-1}\left(\rho \right)={\mathbb{P}}^{-1}\left(\gamma I_{\Im }(\rho ,k)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\rho \right).$$

Thus, $\gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\rho \right),k)={\mathbb{P}}^{-1}\left(\rho \right)$. Therefore, ${\mathbb{P}}^{-1}\left(\rho \right)$ is k-F$\gamma \mathcal{I}$-open, so $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute. □

Proposition 5.1. 

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(X,\eta )$ and $\mathbb{Y}:(X,\eta )⟶(Y,\Im )$ be two fuzzy mappings. Then the composition $\mathbb{Y}\circ \mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute (resp. F$\gamma \mathcal{I}$-continuous) if $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute and $\mathbb{Y}$ is F$\gamma$-irresolute (resp. fuzzy continuous).

Proof. 

The proof follows by the previous definitions. □

Definition 5.2. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ is called F$\gamma \mathcal{I}$-open if $\mathbb{P}\left(\omega \right)$ is an k-F$\gamma \mathcal{I}$-open set, for any $\omega \in I^Z$ with $\zeta \left(\omega \right)\ge k$ and $k\in I_{\circ }$.

Definition 5.3. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ is called F$\gamma \mathcal{I}$-irresolute open if $\mathbb{P}\left(\omega \right)$ is an k-F$\gamma \mathcal{I}$-open set, for any k-F$\gamma$-open set $\omega \in I^Z$ and $k\in I_{\circ }$.

Lemma 5.2. 

Each F$\gamma \mathcal{I}$-irresolute open mapping is F$\gamma \mathcal{I}$-open.

Proof. 

The proof follows from Definitions 5.2, 5.3, and Remark 2.1. □

Remark 5.2. 

The converse of Lemma 5.2 fails as Example 5.2 will show.

Example 5.2. 

Let $Z=\{z_1,z_2\}$ and define $\omega ,\lambda \in I^Z$ as follows: $\omega =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.5}}\}$, $\lambda =\{{\displaystyle \frac{z_1}{0.5}},{\displaystyle \frac{z_2}{0.4}}\}$. Define $\zeta ,\Im ,\mathcal{I}:I^Z⟶I$ as follows:

$$\zeta \left(\nu \right)=\left\{\begin{array}{c}1,\mathrm{if}\nu \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\mathrm{if}\nu =\omega ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.5},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

$$\Im \left(\theta \right)=\left\{\begin{array}{c}1,\mathrm{if}\theta \in \{\underline{1},\underline{0}\},\hfill \\ {\displaystyle \frac{1}{5}},\mathrm{if}\theta =\lambda ,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the identity fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Z,\Im ,\mathcal{I})$ is F$\gamma \mathcal{I}$-open, but it is not F$\gamma \mathcal{I}$-irresolute open.

Theorem 5.2. 

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $k\in I_{\circ }$. Then the following statements are equivalent for every $\omega \in I^Z$ and $\rho \in I^Y$:

(1) $\mathbb{P}$ is F$\gamma \mathcal{I}$-open.

(2) $\mathbb{P}\left(I_{\zeta }(\omega ,k)\right)\le \gamma I_{\Im }^{*}(\mathbb{P}\left(\omega \right),k)$.

(3) $I_{\zeta }({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}(\rho ,k)\right)$.

(4) For every $\rho$ and every $\omega$ with $\zeta \left({\omega }^c\right)\ge k$ and ${\mathbb{P}}^{-1}\left(\rho \right)\le \omega$, there is $\mu \in I^Y$ is k-F$\gamma \mathcal{I}$-closed with $\rho \le \mu$ and ${\mathbb{P}}^{-1}\left(\mu \right)\le \omega$.

Proof. (1) ⇒ (2) Since $\mathbb{P}\left(I_{\zeta }(\omega ,k)\right)\le \mathbb{P}\left(\omega \right)$, hence by (1), $\mathbb{P}\left(I_{\zeta }(\omega ,k)\right)$ is k-F$\gamma \mathcal{I}$-open. Thus,

$$\mathbb{P}\left(I_{\zeta }(\omega ,k)\right)\le \gamma I_{\Im }^{*}(\mathbb{P}\left(\omega \right),k).$$

(2) ⇒ (3) Set $\omega ={\mathbb{P}}^{-1}\left(\rho \right)$, and hence by (2), $\mathbb{P}\left(I_{\zeta }({\mathbb{P}}^{-1}\left(\rho \right),k)\right)\le \gamma I_{\Im }^{*}(\mathbb{P}\left({\mathbb{P}}^{-1}\left(\rho \right)\right),k)\le \gamma I_{\Im }^{*}(\rho ,k)$. Thus, $I_{\zeta }({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}(\rho ,k)\right).$

(3) ⇒ (4) Let $\rho \in I^Y$ and $\omega \in I^Z$ with $\zeta \left({\omega }^c\right)\ge k$ such that ${\mathbb{P}}^{-1}\left(\rho \right)\le \omega$. Since ${\omega }^c\le {\mathbb{P}}^{-1}\left({\rho }^c\right)$, ${\omega }^c=I_{\zeta }({\omega }^c,k)\le I_{\zeta }({\mathbb{P}}^{-1}\left({\rho }^c\right),k)$. Hence by (3), ${\omega }^c\le I_{\zeta }({\mathbb{P}}^{-1}\left({\rho }^c\right),k)\le {\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}({\rho }^c,k)\right)$. Then, we have

$$\omega \ge {\left({\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}({\rho }^c,k)\right)\right)}^c={\mathbb{P}}^{-1}\left(\gamma C_{\Im }^{*}(\rho ,k)\right).$$

Thus, $\gamma C_{\Im }^{*}(\rho ,k)\in I^Y$ is k-F$\gamma \mathcal{I}$-closed with $\rho \le \gamma C_{\Im }^{*}(\rho ,k)$ and ${\mathbb{P}}^{-1}\left(\gamma C_{\Im }^{*}(\rho ,k)\right)\le \omega .$

(4) ⇒ (1) Let $\nu \in I^Z$ with $\zeta \left(\nu \right)\ge k$. Set $\rho ={\left(\mathbb{P}\left(\nu \right)\right)}^c$ and $\omega ={\nu }^c$, ${\mathbb{P}}^{-1}\left(\rho \right)={\mathbb{P}}^{-1}\left({\left(\mathbb{P}\left(\nu \right)\right)}^c\right)\le \omega$. Hence by (4), there is $\mu \in I^Y$ is k-F$\gamma \mathcal{I}$-closed with $\rho \le \mu$ and ${\mathbb{P}}^{-1}\left(\mu \right)\le \omega ={\nu }^c$. Thus, $\mathbb{P}\left(\nu \right)\le \mathbb{P}\left({\mathbb{P}}^{-1}\left({\mu }^c\right)\right)\le {\mu }^c$. On the other hand, since $\rho \le \mu$, $\mathbb{P}\left(\nu \right)={\rho }^c\ge {\mu }^c$. Hence, $\mathbb{P}\left(\nu \right)={\mu }^c$, so $\mathbb{P}\left(\nu \right)$ is an k-F$\gamma \mathcal{I}$-open set. Therefore, $\mathbb{P}$ is F$\gamma \mathcal{I}$-open. □

Theorem 5.3. 

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $k\in I_{\circ }$. Then the following statements are equivalent for every $\omega \in I^Z$ and $\rho \in I^Y$:

(1) $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute open.

(2) $\mathbb{P}\left(\gamma I_{\zeta }(\omega ,k)\right)\le \gamma I_{\Im }^{*}(\mathbb{P}\left(\omega \right),k)$.

(3) $\gamma I_{\zeta }({\mathbb{P}}^{-1}\left(\rho \right),k)\le {\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}(\rho ,k)\right)$.

(4) For every $\rho$ and every $\omega$ is an k-F$\gamma$-closed set with ${\mathbb{P}}^{-1}\left(\rho \right)\le \omega$, there is $\mu \in I^Y$ is k-F$\gamma \mathcal{I}$-closed with $\rho \le \mu$ and ${\mathbb{P}}^{-1}\left(\mu \right)\le \omega$.

Proof. 

The proof is similar to that of Theorem 5.2. □

Definition 5.4. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ is called F$\gamma \mathcal{I}$-closed if $\mathbb{P}\left(\omega \right)$ is an k-F$\gamma \mathcal{I}$-closed set, for any $\omega \in I^Z$ with $\zeta \left({\omega }^c\right)\ge k$ and $k\in I_{\circ }$.

Definition 5.5. 

A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ is called F$\gamma \mathcal{I}$-irresolute closed if $\mathbb{P}\left(\omega \right)$ is an k-F$\gamma \mathcal{I}$-closed set, for any k-F$\gamma$-closed set $\omega \in I^Z$ and $k\in I_{\circ }$.

Lemma 5.3. 

Each F$\gamma \mathcal{I}$-irresolute closed mapping is F$\gamma \mathcal{I}$-closed.

Proof. 

The proof follows from Definitions 5.4 and 5.5. □

Theorem 5.4. 

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $k\in I_{\circ }$. Then the following statements are equivalent for every $\omega \in I^Z$ and $\rho \in I^Y$:

(1) $\mathbb{P}$ is F$\gamma \mathcal{I}$-closed.

(2) $\gamma C_{\Im }^{*}(\mathbb{P}\left(\omega \right),k)\le \mathbb{P}\left(C_{\zeta }(\omega ,k)\right)$.

(3) ${\mathbb{P}}^{-1}\left(\gamma C_{\Im }^{*}(\rho ,k)\right)\le C_{\zeta }({\mathbb{P}}^{-1}\left(\rho \right),k)$.

(4) For every $\rho$ and every $\omega$ with $\zeta \left(\omega \right)\ge k$ and ${\mathbb{P}}^{-1}\left(\rho \right)\le \omega$, there is $\mu \in I^Y$ is k-F$\gamma \mathcal{I}$-open with $\rho \le \mu$ and ${\mathbb{P}}^{-1}\left(\mu \right)\le \omega$.

Proof. 

The proof is similar to that of Theorem 5.2. □

Theorem 5.5. 

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $k\in I_{\circ }$. Then the following statements are equivalent for every $\omega \in I^Z$ and $\rho \in I^Y$:

(1) $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute closed.

(2) $\gamma C_{\Im }^{*}(\mathbb{P}\left(\omega \right),k)\le \mathbb{P}\left(\gamma C_{\zeta }(\omega ,k)\right)$.

(3) ${\mathbb{P}}^{-1}\left(\gamma C_{\Im }^{*}(\rho ,k)\right)\le \gamma C_{\zeta }({\mathbb{P}}^{-1}\left(\rho \right),k)$.

(4) For every $\rho$ and every $\omega$ is an k-F$\gamma$-open set with ${\mathbb{P}}^{-1}\left(\rho \right)\le \omega$, there is $\mu \in I^Y$ is k-F$\gamma \mathcal{I}$-open with $\rho \le \mu$ and ${\mathbb{P}}^{-1}\left(\mu \right)\le \omega$.

Proof. 

The proof is similar to that of Theorem 5.2. □

Proposition 5.2. 

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a bijective fuzzy mapping. Then $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute open iff $\mathbb{P}$ is F$\gamma \mathcal{I}$-irresolute closed.

Proof. 

The proof follows from:

$${\mathbb{P}}^{-1}\left(\gamma C_{\Im }^{*}(\nu ,k)\right)\le \gamma C_{\zeta }({\mathbb{P}}^{-1}\left(\nu \right),k)⟺{\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}({\nu }^c,k)\right)\le \gamma I_{\zeta }({\mathbb{P}}^{-1}\left({\nu }^c\right),k).$$

□

6. Conclusions

In this work, a novel class of fuzzy sets, called k-F$\gamma \mathcal{I}$-open sets, has been introduced in $\mathcal{FITS}s$ based on Šostak’s sense. Some characterizations of k-F$\gamma \mathcal{I}$-open sets along with their mutual relationships have been investigated with the help of some examples. Moreover, the notions of F$\gamma \mathcal{I}$-interior operators and F$\gamma \mathcal{I}$-closure operators have been presented and discussed. Also, we defined and investigated new types of fuzzy $\mathcal{I}$-separation axioms,called k-F$\gamma \mathcal{I}$-regular spaces and k-F$\gamma \mathcal{I}$-normal spaces using k-F$\gamma \mathcal{I}$-closed sets. After that, the notion of F$\gamma \mathcal{I}$-continuity has been explored and discussed. Additionally, the notions of FA$\gamma \mathcal{I}$-continuous mappings and FW$\gamma \mathcal{I}$-continuous mappings, which are weaker forms of F$\gamma \mathcal{I}$-continuous mappings, have been defined and characterized. Finally, we defined and studied some new fuzzy $\gamma \mathcal{I}$-mappings via k-F$\gamma \mathcal{I}$-open sets and k-F$\gamma \mathcal{I}$-closed sets, called F$\gamma \mathcal{I}$-open mappings, F$\gamma \mathcal{I}$-closed mappings, F$\gamma \mathcal{I}$-irresolute mappings, F$\gamma \mathcal{I}$-irresolute open mappings, and F$\gamma \mathcal{I}$-irresolute closed mappings. In the next works, we intend to explore the following topics:

• Defining fuzzy upper and lower $\gamma \mathcal{I}$-continuous multifunctions and k-fuzzy $\gamma \mathcal{I}$-connected sets.

• Extending these notions given here in the frame of fuzzy soft topological (k-minimal) spaces as defined in [34–39].

• Finding a use for these notions given here to include double fuzzy topological spaces as defined in [40,41].