On j-Fuzzy γI-Open Sets with Some Applications

1. Introduction

The concept of a fuzzy set was first defined in 1965 by Zadeh [1] as a suitable approach to address with uncertainty cases that we cannot be efficiently managed via classical techniques. Over the last decades, the researches of fuzzy sets have a vital role in mathematics and applied sciences and garnered significant attention due to its ability to handle uncertain and vague information in various real life applications such as control systems [2,3], artificial intelligence [4,5], image processing [6,7], decision-making [8,9], etc. The integration between fuzzy sets and some uncertainty approaches such as rough sets and soft sets has been discussed in [10-12]. The notion of a fuzzy topology was introduced in 1968 by Chang [13] and this development has led to the expansion and discussion of several classical topological concepts in the context of a fuzzy topology [14-17], providing more accurate and flexible models to address problems of uncertainty in various real life ears. Overall, according to Šostak [18], the concept of a fuzzy topology being a crisp subclass of the class of fuzzy sets and fuzziness in the concept 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. Thereafter, Šostak [18] introduced a new notion of a fuzzy topology as the notion of openness of fuzzy sets. It is an extension of a fuzzy topology defined by Chang [13]. Furthermore, several researchers (see [19-27]) have redisplayed the same concept 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 several results, or to open the door to explore and discuss many fuzzy topological concepts such as fuzzy compactness [20,21], fuzzy connectedness [20], fuzzy continuity [19,20], etc. Furthermore, the concepts of j-fuzzy pre-open (j-FP-open) sets, j-fuzzy semi-open (j-FS-open) sets, j-fuzzy $\beta$-open (j-F$\beta$-open) sets, and j-fuzzy $\alpha$-open (j-F$\alpha$-open) sets were presented and investigated by the authors of [24,26] in $\mathcal{FTS}s$ based on Šostak’s sense [18]. Kim et al. [24] displayed and investigated weaker forms of fuzzy continuity, called FS-continuity (resp. FP-continuity and F$\alpha$-continuity) between $\mathcal{FTS}s$ in the sense of Šostak. Abbas [26] defined and discussed the concepts of F$\beta$-continuous (resp. F$\beta$-irresolute) mappings. Also, Kim and Abbas [27] explored and characterized new types of j-fuzzy compactness. Overall, the notions of j-fuzzy $\gamma$-open (j-F$\gamma$-open) sets and j-fuzzy $\gamma$-closed (j-F$\gamma$-closed) sets were introduced and discussed by the authors of [28].

The notion of j-fuzzy local function was presented and investigated by Taha and Abbas [29] in an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ based on Šostak’s sense [18]. Moreover, the notions of fuzzy lower (resp. upper) weakly and almost $\mathcal{I}$-continuous multifunctions were displayed and investigated by Taha and Abbas [29]. Also, Taha [30-32] introduced the notions of j-FS$\mathcal{I}$-open sets, j-FP$\mathcal{I}$-open sets, j-F$\alpha \mathcal{I}$-open sets, j-F$\beta \mathcal{I}$-open sets, j-FS$\beta \mathcal{I}$-open sets, j-F$\delta \mathcal{I}$-open sets, and j-GF$\mathcal{I}$-closed sets in an $\mathcal{FITS}$$(Z,\zeta ,\mathcal{I})$ based on Šostak’s sense. Overall, Taha [31-33] presented the concepts of fuzzy lower (resp. upper) generalized $\mathcal{I}$-continuous (resp. semi-$\mathcal{I}$-continuous, pre-$\mathcal{I}$-continuous, $\delta$-$\mathcal{I}$-continuous, $\alpha$-$\mathcal{I}$-continuous, $\beta$-$\mathcal{I}$-continuous, and strong $\beta$-$\mathcal{I}$-continuous) multifunctions via fuzzy ideals [34].

The arrangement of this research is as follows.

(a) Section 2 provides fundamental results and concepts which we use them in our article.

(b) In Section 3, we introduce and study a new class of fuzzy sets, called j-F$\gamma \mathcal{I}$-open sets on $\mathcal{FITS}s$ in the sense of Šostak. We also define and discuss the interior and closure operators with respect to the classes of j-F$\gamma \mathcal{I}$-open sets and j-F$\gamma \mathcal{I}$-closed sets. Furthermore, we explore new types of fuzzy $\mathcal{I}$-separation axioms using j-F$\gamma \mathcal{I}$-closed sets, called j-F$\gamma \mathcal{I}$-regular spaces and j-F$\gamma \mathcal{I}$-normal spaces.

(c) In Section 4, we display and characterize the notion of F$\gamma \mathcal{I}$-continuous mappings using j-F$\gamma \mathcal{I}$-open sets. However, we present and discuss the notions of FA$\gamma \mathcal{I}$-continuous and FW$\gamma \mathcal{I}$-continuous mappings, which are weaker forms of F$\gamma \mathcal{I}$-continuous mappings.

(d) In Section 5, we explore and investigate new F$\gamma \mathcal{I}$-mappings via j-F$\gamma \mathcal{I}$-open sets and j-F$\gamma \mathcal{I}$-closed sets, called F$\gamma \mathcal{I}$-closed mappings, F$\gamma \mathcal{I}$-open mappings, F$\gamma \mathcal{I}$-irresolute mappings, F$\gamma \mathcal{I}$-irresolute open mappings, and F$\gamma \mathcal{I}$-irresolute closed mappings.

(e) In Section 6, we give some potential future studies and conclusions.

2. Preliminaries

In this research, non-empty sets will be denoted by Y, X, Z, etc. For any fuzzy set $\omega \in I^Z$ (where $I=[0,1]$ and $I^Z$ is the class of all fuzzy sets on 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 $z\in Z$.

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

A fuzzy set $\nu \in I^Z$ is a quasi coincident with $\mu \in I^Z$ ($\nu \mathcal{Q}\mu$) on Z, if there is $z\in Z$, with $\nu \left(z\right)+\mu \left(z\right)>1$. Also, $\nu$ is not a quasi coincident with $\mu$ ($\nu \overline{\mathcal{Q}}\mu$) otherwise.

The difference between $\psi ,\mu \in I^Z$ [29] is defined as follows:

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

Lemma 1.

$\left[35\right]$ Let $\omega ,\nu \in I^Z$. Thus,

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

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

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

(d) $\omega \le \nu$ iff $z_s\in \omega$ implies $z_s\in \nu$ iff $z_s\mathcal{Q}\omega$ implies $z_s\mathcal{Q}\nu$ iff $z_s\overline{\mathcal{Q}}\nu$ implies $z_s\overline{\mathcal{Q}}\omega$.

Definition 1.

$[18,19]$ A mapping $\zeta :I^Z⟶I$ is called a fuzzy topology on Z if it satisfies the following conditions:

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

(b) $\zeta (\omega \wedge \nu )\ge \zeta \left(\omega \right)\wedge \zeta \left(\nu \right),$ for any $\omega ,\nu \in I^Z.$

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

Thus, $(Z,\zeta )$ is called a fuzzy topological space ($\mathcal{FTS}$) in the sense of Šostak.

Definition 2.

$[19,24]$ A fuzzy mapping $\mathbb{P}:(Z,\zeta )⟶(Y,\Im )$ is called

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

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

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

Definition 3.

$[20,23]$ For any $\omega \in I^Z$ and $j\in I_{\circ }$ (where $I_{\circ }=(0,1]$) in an $\mathcal{FTS}$$(Z,\zeta )$, we define fuzzy operators $C_{\zeta }$ and $I_{\zeta }$$:I^Z\times I_{\circ }\to I^Z$ as follows:

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

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

Definition 4.

$[24,26,28]$ Let $(Z,\zeta )$ be an $\mathcal{FTS}$ and $j\in I_{\circ }$. A fuzzy set $\omega \in I^Z$ is called

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

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

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

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

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

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

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

Remark 1.

$[24,26,28]$ We have the following diagram from the previous definitions.

$$\mathit{j}-\mathbf{FP}-\mathbf{open}$$

$$\nearrow \searrow$$

$$\mathit{j}-\mathbf{F}-\mathbf{open}\mathbf{set}⟹\mathit{j}-\mathbf{Fff}-\mathbf{open}\mathit{j}-\mathbf{Ffl}-\mathbf{open}⟹\mathit{j}-\mathbf{Ffi}-\mathbf{open}$$

$$\searrow \nearrow$$

$$\mathit{j}-\mathbf{FS}-\mathbf{open}$$

Definition 5.

$[24,26,28]$ 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 a j-FS-open (resp. j-FP-open, j-F$\alpha$-open, j-F$\beta$-open, and j-F$\gamma$-open) set, for any $\omega \in I^Y$ with $\Im \left(\omega \right)\ge j$ and $j\in I_{\circ }$.

Definition 6.

$\left[28\right]$ For any $\omega \in I^Z$ and $j\in I_{\circ }$ in an $\mathcal{FTS}$$(Z,\zeta )$, we define fuzzy operators $\gamma C_{\zeta }$ and $\gamma I_{\zeta }$$:I^Z\times I_{\circ }\to I^Z$ as follows:

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

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

Definition 7.

$\left[34\right]$ A fuzzy ideal $\mathcal{I}$ on Z, is a map $\mathcal{I}:I^Z⟶I$ that satisfies the following:

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

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

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

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

Definition 8.

$\left[29\right]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $j\in I_{\circ }$, and $\omega \in I^Z$. Then the j-fuzzy local function ${\omega }_j^{*}$ of $\omega$ is defined as follows:

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

Remark 2.

$\left[29\right]$ If we take $\mathcal{I}={\mathcal{I}}_0$, for each $\omega \in I^Z$ we have:

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

Definition 9.

$\left[29\right]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $j\in I_{\circ }$, and $\omega \in I^Z$. Then we define fuzzy operator $C_{\zeta }^{*}$$:I^Z\times I_{\circ }\to I^Z$ as follows:

$$C_{\zeta }^{*}(\omega ,j)=\omega \vee {\omega }_j^{*}.$$

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

Theorem 1.

$\left[29\right]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $j\in I_{\circ }$, and $\omega ,\rho \in I^Z$. The operator $C_{\zeta }^{*}$$:I^Z\times I_{\circ }\to I^Z$ satisfies the following properties:

(a) $C_{\zeta }^{*}(\underline{0},j)=\underline{0}$.

(b) $\omega \le C_{\zeta }^{*}(\omega ,j)\le C_{\zeta }(\omega ,j)$.

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

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

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

(f) $C_{\zeta }^{*}(C_{\zeta }^{*}(\omega ,j),j)=C_{\zeta }^{*}(\omega ,j)$.

Definition 10.

$[30,32]$ Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$ and $j\in I_{\circ }$. A fuzzy set $\omega \in I^Z$ is called

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

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

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

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

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

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

Definition 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 a j-F$\alpha \mathcal{I}$-open (resp. j-FP$\mathcal{I}$-open, j-FS$\mathcal{I}$-open, and j-FS$\beta \mathcal{I}$-open) set, for any $\omega \in I^Y$ and $\Im \left(\omega \right)\ge j$ with $j\in I_{\circ }$.

Some basic results and concepts that we need in the sequel are found in [19-21,29-32].

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

Definition 12.

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

Remark 3.

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

Lemma 2.

Each j-F$\gamma \mathcal{I}$-open set is j-F$\gamma$-open [28].

Proof. 

The proof follows by Theorem 2.1(b) and by Definitions 2.4 and 3.1. □

Remark 4.

If we take $\mathcal{I}={\mathcal{I}}_0$, then j-F$\gamma \mathcal{I}$-open set and j-F$\gamma$-open set [28] are equivalent.

Remark 5.

The converse of Lemma 3.1 fails, as can be seen in Example 3.1.

Example 1.

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

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

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 1.

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

(a) each j-FP$\mathcal{I}$-open set [30] is j-F$\gamma \mathcal{I}$-open;

(b) each j-F$\gamma \mathcal{I}$-open set is j-FS$\beta \mathcal{I}$-open [32];

(c) each j-FS$\mathcal{I}$-open set [30] is j-F$\gamma \mathcal{I}$-open.

Proof.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

□

Remark 6.

We have the following diagram from the previous definitions and discussions.

$$\mathit{j}-\mathbf{FP}\mathcal{I}-\mathbf{open}$$

$$\nearrow \downarrow$$

$$\mathit{j}-\mathbf{Fff}\mathcal{I}-\mathbf{open}\mathbf{set}⟶\mathit{j}-\mathbf{Ffl}\mathcal{I}-\mathbf{open}⟶\mathit{j}-\mathbf{FSfi}\mathcal{I}-\mathbf{open}$$

$$\searrow \uparrow$$

$$\mathit{j}-\mathbf{FS}\mathcal{I}-\mathbf{open}$$

Remark 7.

The reverse implication of the above diagram does not hold, as demonstrated by the Examples 3.2, 3.3, and 3.4.

Example 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:

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

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.

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:

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

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 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:

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

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.

Definition 13.

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

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

Proposition 2.

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

Proof. 

This follows directly from Definition 3.2.

□

Theorem 2.

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

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

(b) $\omega \le \gamma C_{\zeta }^{*}(\omega ,j)\le C_{\zeta }(\omega ,j)$.

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

(d) $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,j),j)=\gamma C_{\zeta }^{*}(\omega ,j)$.

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

Proof. (a), (b), and (c) are easily proved by Definition 3.2.

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

$$\gamma C_{\zeta }^{*}(\omega ,j)\left(z\right)<s<\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,j),j)\left(z\right).\left(1\right)$$

Since $\gamma C_{\zeta }^{*}(\omega ,j)\left(z\right)<s$, by Definition 3.2, there is $\mu \in I^Z$ as a j-F$\gamma \mathcal{I}$-closed set and $\omega \le \mu$ with $\gamma C_{\zeta }^{*}(\omega ,j)\left(z\right)\le \mu \left(z\right)<s$. Since $\omega \le \mu$, then $\gamma C_{\zeta }^{*}(\omega ,j)\le \mu$. Again, by the definition of $\gamma C_{\zeta }^{*}$, then $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,j),j)\le \mu$. Hence, $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,j),j)\left(z\right)\le \mu \left(z\right)<s$, which is a contradiction for $\left(1\right)$. Thus, $\gamma C_{\zeta }^{*}(\omega ,j)\ge \gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,j),j)$. Therefore, $\gamma C_{\zeta }^{*}(\gamma C_{\zeta }^{*}(\omega ,j),j)=\gamma C_{\zeta }^{*}(\omega ,j)$.

(e) Since $\omega$$\le \omega \vee \rho$ and $\rho$$\le \omega \vee \rho$, then by (c), $\gamma C_{\zeta }^{*}(\omega ,j)\le \gamma C_{\zeta }^{*}(\omega \vee \rho ,j)$ and $\gamma C_{\zeta }^{*}(\rho ,j)\le \gamma C_{\zeta }^{*}(\omega \vee \rho ,j)$. Hence, $\gamma C_{\zeta }^{*}(\omega \vee \rho ,j)\ge \gamma C_{\zeta }^{*}(\omega ,j)\vee \gamma C_{\zeta }^{*}(\rho ,j)$.

□

Definition 14.

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

Proposition 3.

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

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

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

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

(b) This is similar to that of (a).

□

Proposition 4.

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

Proof. 

This is immediate from Definition 3.3.

□

Theorem 3.

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

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

(b) $I_{\zeta }(\omega ,j)\le \gamma I_{\zeta }^{*}(\omega ,j)\le \omega$.

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

(d) $\gamma I_{\zeta }^{*}(\gamma I_{\zeta }^{*}(\omega ,j),j)=\gamma I_{\zeta }^{*}(\omega ,j)$.

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

Proof. 

This can be proven using the same approach as in Theorem 3.1.

□

Definition 15.

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

Definition 16.

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

Theorem 4.

Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $z_s\in P_s\left(Z\right)$, $\omega \in I^Z$, and $j\in I_{\circ }$. Each of the following statements implies the others.

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

(b) If $z_s\in \omega$ for any j-F$\gamma \mathcal{I}$-open set $\omega$, there is $\mu \in I^Z$ with $\zeta \left(\mu \right)\ge j$, and

$$z_s\in \mu \le C_{\zeta }(\mu ,j)\le \omega .$$

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

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

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

(c) ⇒ (a) This is immediate from Definition 3.4.

□

Theorem 5.

Let $(Z,\zeta ,\mathcal{I})$ be an $\mathcal{FITS}$, $\omega ,\rho \in I^Z$, and $j\in I_{\circ }$. Each of the following statements implies the others.

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

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

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

Proof. 

This can be proven using the same approach as in Theorem 3.3.

□

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

Definition 17.

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 a j-F$\gamma \mathcal{I}$-open set, for any $\omega \in I^Y$ with $\Im \left(\omega \right)\ge j$ and $j\in I_{\circ }$.

Lemma 3.

Every F$\gamma \mathcal{I}$-continuity is an F$\gamma$-continuity [28].

Proof. 

The proof follows by Lemma 3.1 and by Definitions 2.5 and 4.1. □

Remark 8.

If we take $\mathcal{I}={\mathcal{I}}_0$, then F$\gamma \mathcal{I}$-continuity and F$\gamma$-continuity [28] are equivalent.

Remark 9.

The converse of Lemma 4.1 fails, as can be seen in Example 4.1.

Example 5.

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

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

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

Then, 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 10.

We have the following diagram from the previous definitions.

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

$$\nearrow \downarrow$$

$$\mathbf{Fff}\mathcal{I}-\mathbf{continuity}⟶\mathbf{Ffl}\mathcal{I}-\mathbf{continuity}⟶\mathbf{FSfi}\mathcal{I}-\mathbf{continuity}$$

$$\searrow \uparrow$$

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

Remark 11.

The reverse implication of the above diagram does not hold, as demonstrated by the Examples 4.2, 4.3, and 4.4.

Example 6.

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:

$$\zeta \left(\psi \right)=\left\{\begin{array}{c}1,\mathrm{if}\psi \in \{\underline{0},\underline{1}\},\hfill \\ {\displaystyle \frac{1}{5}},\mathrm{if}\psi =\rho ,\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\psi =\omega ,\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.\mathcal{I}\left(\mu \right)=\left\{\begin{array}{c}1,\mathrm{if}\mu =\underline{0},\hfill \\ {\displaystyle \frac{1}{2}},\mathrm{if}\underline{0}<\mu <\underline{0.3},\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right.$$

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

Then, 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 7.

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:

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

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

Then, 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 8.

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:

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

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

Then, 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 6.

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 $\psi \in I^Y$ with $\Im \left(\psi \right)\ge j$ containing $\mathbb{P}\left(z_s\right)$, there is $\omega \in I^Z$ that is j-F$\gamma \mathcal{I}$-open containing $z_s$ and $\mathbb{P}\left(\omega \right)\le \psi$ with $j\in I_{\circ }$.

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

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

□

Theorem 7.

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping and $j\in I_{\circ }$. Each of the following statements implies the others for any $\omega \in I^Z$ and $\psi \in I^Y$:

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

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

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

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

(e) ${\mathbb{P}}^{-1}\left(I_{\Im }(\psi ,j)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)$.

Proof. (a) ⇔ (b) The proof follows from Definition 4.1 and ${\mathbb{P}}^{-1}\left({\psi }^c\right)={\left({\mathbb{P}}^{-1}\left(\psi \right)\right)}^c$.

(b) ⇒ (c) Let $\omega \in I^Z$. By (b), we obtain ${\mathbb{P}}^{-1}\left(C_{\Im }(\mathbb{P}\left(\omega \right),j)\right)$ is j-F$\gamma \mathcal{I}$-closed. Hence,

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

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

(c) ⇒ (d) Let $\psi \in I^Y$. By (c), we obtain $\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\right)\le C_{\Im }(\mathbb{P}\left({\mathbb{P}}^{-1}\left(\psi \right)\right),j)$$\le C_{\Im }(\psi ,j)$. Hence, $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\le {\mathbb{P}}^{-1}\left(\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\right)\right)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\psi ,j)\right)$.

(d) ⇔ (e) The proof follows from Proposition 3.3 and ${\mathbb{P}}^{-1}\left({\psi }^c\right)={\left({\mathbb{P}}^{-1}\left(\psi \right)\right)}^c$.

(e) ⇒ (a) Let $\psi \in I^Y$ with $\Im \left(\psi \right)\ge j$. By (e), we have ${\mathbb{P}}^{-1}\left(\psi \right)={\mathbb{P}}^{-1}\left(I_{\Im }(\psi ,j)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\le {\mathbb{P}}^{-1}\left(\psi \right)$. Then, $\gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)={\mathbb{P}}^{-1}\left(\psi \right)$. Hence, ${\mathbb{P}}^{-1}\left(\psi \right)$ is j-F$\gamma \mathcal{I}$-open, so $\mathbb{P}$ is F$\gamma \mathcal{I}$-continuous. □

Definition 18.

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

Lemma 4.

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

Proof. 

This follows directly from Definitions 4.2 and 4.1.

□

Remark 12.

It is clear from Example 4.5 that the converse of Lemma 4.2 does not apply.

Example 9.

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:

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

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

Then, 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 8.

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 $\Im \left(\rho \right)\ge j$ containing $\mathbb{P}\left(z_s\right)$, there is $\omega \in I^Z$ that is j-F$\gamma \mathcal{I}$-open containing $z_s$ and $\mathbb{P}\left(\omega \right)\le I_{\Im }(C_{\Im }(\rho ,j),j)$ with $j\in I_{\circ }$.

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

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

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

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

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

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

□

Theorem 9.

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping, $\psi \in I^Y$, and $j\in I_{\circ }$. Each of the following statements implies the others.

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

(b) ${\mathbb{P}}^{-1}\left(\psi \right)$ is j-F$\gamma \mathcal{I}$-open, for each j-FR-open set $\psi$.

(c) ${\mathbb{P}}^{-1}\left(\psi \right)$ is j-F$\gamma \mathcal{I}$-closed, for each j-FR-closed set $\psi$.

(d) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\psi ,j)\right)$, for each j-F$\gamma$-open set $\psi$.

(e) $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\psi ,j)\right)$, for each j-FS-open set $\psi$.

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

(b) ⇒ (c) If $\psi \in I^Y$ is j-FR-closed, hence by (b), ${\mathbb{P}}^{-1}\left({\psi }^c\right)={\left({\mathbb{P}}^{-1}\left(\psi \right)\right)}^c$ is j-F$\gamma \mathcal{I}$-open. Thus, ${\mathbb{P}}^{-1}\left(\psi \right)$ is j-F$\gamma \mathcal{I}$-closed.

(c) ⇒ (d) If $\psi \in I^Y$ is j-F$\gamma$-open and since $C_{\Im }(\psi ,j)$ is j-FR-closed, then by (c), ${\mathbb{P}}^{-1}\left(C_{\Im }(\psi ,j)\right)$ is j-F$\gamma \mathcal{I}$-closed. Since ${\mathbb{P}}^{-1}\left(\psi \right)\le {\mathbb{P}}^{-1}\left(C_{\Im }(\psi ,j)\right)$, thus

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

(d) ⇒ (e) The proof follows by the fact that each j-FS-open set is j-F$\gamma$-open.

(e) ⇒ (c) If $\psi \in I^Y$ is j-FR-closed, and then $\psi$ is j-FS-open. By (e),

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

Thus, ${\mathbb{P}}^{-1}\left(\psi \right)$ is j-F$\gamma \mathcal{I}$-closed.

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

□

Definition 19.

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

Lemma 5.

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

Proof. 

This follows directly from Definitions 4.3 and 4.1.

□

Remark 13.

It is clear from Example 4.6 that the converse of Lemma 4.3 does not apply.

Example 10.

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:

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

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

Then, 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 10.

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 $\Im \left(\rho \right)\ge j$ containing $\mathbb{P}\left(z_s\right)$, there is $\omega \in I^Z$ that is j-F$\gamma \mathcal{I}$-open containing $z_s$ and $\mathbb{P}\left(\omega \right)\le C_{\Im }(\rho ,j)$ with $j\in I_{\circ }$.

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

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

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

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

□

Theorem 11.

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping, $\psi \in I^Y$, and $j\in I_{\circ }$. Each of the following statements implies the others.

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

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

(c) $\gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(C_{\Im }(\psi ,j)\right),j)\ge {\mathbb{P}}^{-1}\left(I_{\Im }(\psi ,j)\right)$.

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

Proof. (a) ⇔ (b) This follows directly from Definition 4.3 and Proposition 3.3.

(b) ⇒ (c) Let $\psi \in I^Y$. Then by (b),

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

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

(c) ⇔ (d) This follows directly from Proposition 3.3.

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

Lemma 6.

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

Proof. 

This follows directly from Definitions 4.3 and 4.2. □

Remark 14.

It is clear from Example 4.7 that the converse of Lemma 4.4 does not apply.

Example 11.

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:

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

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

Then, 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 15.

We have the following diagram from the previous definitions and discussions.

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

Proposition 5.

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. 

This follows directly from Definitions 2.2, 4.1, and 4.2. □

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

Definition 20.

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 a j-F$\gamma \mathcal{I}$-open set, for any j-F$\gamma$-open set $\omega \in I^Y$ with $j\in I_{\circ }$.

Lemma 7.

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

Proof. 

This follows directly from Definitions 4.1, 5.1, and Remark 2.1. □

Remark 16.

It is clear from Example 5.1 that the converse of Lemma 5.1 does not apply.

Example 12.

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:

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

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

Then, 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 12.

Let $\mathbb{P}:(Z,\zeta ,\mathcal{I})⟶(Y,\Im )$ be a fuzzy mapping and $j\in I_{\circ }$. Each of the following statements implies the others for any $\omega \in I^Z$ and $\psi \in I^Y$:

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

(b) ${\mathbb{P}}^{-1}\left(\psi \right)$ is j-F$\gamma \mathcal{I}$-closed, for each j-F$\gamma$-closed set $\psi$.

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

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

(e) ${\mathbb{P}}^{-1}\left(\gamma I_{\Im }(\psi ,j)\right)\le \gamma I_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)$.

Proof. (a) ⇔ (b) This follows directly from Definition 5.1 and ${\mathbb{P}}^{-1}\left({\psi }^c\right)={\left({\mathbb{P}}^{-1}\left(\psi \right)\right)}^c$.

(b) ⇒ (c) Let $\omega \in I^Z$. By (b), ${\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\mathbb{P}\left(\omega \right),j)\right)$ is j-F$\gamma \mathcal{I}$-closed. Then,

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

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

(c) ⇒ (d) Let $\psi \in I^Y$. By (c), $\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\right)\le \gamma C_{\Im }(\mathbb{P}\left({\mathbb{P}}^{-1}\left(\psi \right)\right),j)$$\le \gamma C_{\Im }(\psi ,j)$. Hence, $\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\le {\mathbb{P}}^{-1}\left(\mathbb{P}\left(\gamma C_{\zeta }^{*}({\mathbb{P}}^{-1}\left(\psi \right),j)\right)\right)\le {\mathbb{P}}^{-1}\left(\gamma C_{\Im }(\psi ,j)\right)$.

(d) ⇔ (e) This follows directly from Proposition 3.3 and ${\mathbb{P}}^{-1}\left({\psi }^c\right)={\left({\mathbb{P}}^{-1}\left(\psi \right)\right)}^c$.

(e) ⇒ (a) Let $\psi \in I^Y$ be a j-F$\gamma$-open set. By (e),

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

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

Proposition 6.

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. 

This follows directly from Definitions 2.2, 4.1, and 5.1. □

Definition 21.

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 a j-F$\gamma \mathcal{I}$-open set, for any $\omega \in I^Z$ with $\zeta \left(\omega \right)\ge j$ and $j\in I_{\circ }$.

Definition 22.

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 a j-F$\gamma \mathcal{I}$-open set, for any j-F$\gamma$-open set $\omega \in I^Z$ with $j\in I_{\circ }$.

Lemma 8.

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

Proof. 

This follows directly from Definitions 5.2, 5.3, and Remark 2.1. □

Remark 17.

It is clear from Example 5.2 that the converse of Lemma 5.2 does not apply.

Example 13.

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:

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

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

Then, 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 13.

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $j\in I_{\circ }$. Each of the following statements implies the others for any $\omega \in I^Z$ and $\psi \in I^Y$:

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

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

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

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

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

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

(b) ⇒ (c) Set $\omega ={\mathbb{P}}^{-1}\left(\psi \right)$, and hence by (b), $\mathbb{P}\left(I_{\zeta }({\mathbb{P}}^{-1}\left(\psi \right),j)\right)\le \gamma I_{\Im }^{*}(\mathbb{P}\left({\mathbb{P}}^{-1}\left(\psi \right)\right),j)\le \gamma I_{\Im }^{*}(\psi ,j)$. Then, $I_{\zeta }({\mathbb{P}}^{-1}\left(\psi \right),j)\le {\mathbb{P}}^{-1}\left(\gamma I_{\Im }^{*}(\psi ,j)\right).$

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

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

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

(d) ⇒ (a) Let $\sigma \in I^Z$ with $\zeta \left(\sigma \right)\ge j$. Set $\psi ={\left(\mathbb{P}\left(\sigma \right)\right)}^c$ and $\omega ={\sigma }^c$, ${\mathbb{P}}^{-1}\left(\psi \right)={\mathbb{P}}^{-1}\left({\left(\mathbb{P}\left(\sigma \right)\right)}^c\right)\le \omega$. By (d), there exists $\mu \in I^Y$ is j-F$\gamma \mathcal{I}$-closed with $\psi \le \mu$ and ${\mathbb{P}}^{-1}\left(\mu \right)\le \omega ={\sigma }^c$. Then, $\mathbb{P}\left(\sigma \right)\le \mathbb{P}\left({\mathbb{P}}^{-1}\left({\mu }^c\right)\right)\le {\mu }^c$. Since $\psi \le \mu$, $\mathbb{P}\left(\sigma \right)={\psi }^c\ge {\mu }^c$. Thus, $\mathbb{P}\left(\sigma \right)={\mu }^c$, so $\mathbb{P}\left(\sigma \right)$ is a j-F$\gamma \mathcal{I}$-open set. Hence, $\mathbb{P}$ is F$\gamma \mathcal{I}$-open.

□

Theorem 14.

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $j\in I_{\circ }$. Each of the following statements implies the others for any $\omega \in I^Z$ and $\psi \in I^Y$:

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

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

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

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

Proof. 

This can be proven using the same approach as in Theorem 5.2.

□

Definition 23.

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 a j-F$\gamma \mathcal{I}$-closed set, for any $\omega \in I^Z$ with $\zeta \left({\omega }^c\right)\ge j$ and $j\in I_{\circ }$.

Definition 24.

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 a j-F$\gamma \mathcal{I}$-closed set, for any j-F$\gamma$-closed set $\omega \in I^Z$ and $j\in I_{\circ }$.

Lemma 9.

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

Proof. 

This follows directly from Definitions 5.4 and 5.5. □

Theorem 15.

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $j\in I_{\circ }$. Each of the following statements implies the others for any $\omega \in I^Z$ and $\psi \in I^Y$:

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

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

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

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

Proof. 

This can be proven using the same approach as in Theorem 5.2.

□

Theorem 16.

Let $\mathbb{P}:(Z,\zeta )⟶(Y,\Im ,\mathcal{I})$ be a fuzzy mapping and $j\in I_{\circ }$. Each of the following statements implies the others for any $\omega \in I^Z$ and $\psi \in I^Y$:

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

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

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

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

Proof. 

This can be proven using the same approach as in Theorem 5.2.

□

Proposition 7.

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

Proof. 

This follows directly from:

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

□

6. Conclusions

In this research, a novel class of fuzzy sets, called j-F$\gamma \mathcal{I}$-open sets, has been defined and investigated on $\mathcal{FITS}s$ in the sense of Šostak. After that, the concepts of F$\gamma \mathcal{I}$-interior operators and F$\gamma \mathcal{I}$-closure operators have been introduced and studied. We also presented and investigated some types of fuzzy $\mathcal{I}$-separation axioms, called j-F$\gamma \mathcal{I}$-normal spaces and j-F$\gamma \mathcal{I}$-regular spaces via j-F$\gamma \mathcal{I}$-closed sets. Moreover, the notion of F$\gamma \mathcal{I}$-continuity has been defined and discussed. 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, have been introduced and studied. Finally, we explored and characterized some new fuzzy $\gamma \mathcal{I}$-mappings via j-F$\gamma \mathcal{I}$-open sets and j-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 upcoming paper, we intend to study the following topics: (a) extending these concepts given here to include fuzzy soft minimal (topological) spaces as introduced in [36,37]; (b) introducing fuzzy lower and upper $\gamma \mathcal{I}$-continuous multifunctions and j-fuzzy $\gamma \mathcal{I}$-connected sets; (c) finding a use for these concepts given here to include double fuzzy topological spaces as introduced in [38]; and (d) defining these concepts given here base on lattice valued fuzzy sets.