Some new results on fuzzy soft $r$-minimal spaces

1.   Introduction

Zadeh ^[30] introduced the basic idea of a fuzzy set as an extension of classical set theory. The basic notions of fuzzy sets have been improved and applied in different directions. Along this direction, we can refer to ^{[16,20,22,24,25,26]}. The concept of soft set theory was initiated by Molodtsov ^[19] in 1999 as a general mathematical tool for modeling uncertainties. Maji et al. ^[17] introduced the concept of a fuzzy soft set, which combines fuzzy sets ^[30] and soft sets ^[19]. Soft set and fuzzy soft set theories have a rich potential for applications in several directions. So far, lots of spectacular and creative studies about the theories of soft sets and fuzzy soft sets have been considered by some scholars (see ^{[2,3,4,6,9,11,12,13]}). Also, Aygünoğlu et al. ^[8] studied the topological structure of fuzzy soft sets based on fuzzy topologies in the sense of Šostak ^[21]. The concept of fuzzy $r$-minimal structure was introduced by Yoo et al. ^[29] as an extension of fuzzy topology introduced by Šostak ^[21]. Also, the concepts of a fuzzy $r$-minimal space, fuzzy $r$-minimal continuity, and fuzzy $r$-minimal compactness were introduced in ^[15,29]. Later, Taha ^[23] introduced the concept of fuzzy soft $r$-minimal structure, which is an extension of fuzzy soft topology introduced by Aygünoğlu et al. ^[8]. Also, the concept of fuzzy soft $r$-minimal continuity and several types of fuzzy soft $r$-minimal compactness were introduced in ^[23].

We lay out the remainder of this article as follows. Section 2 contains some basic definitions and results that help in understanding the obtained results. In Section 3, we introduce and study a weaker form of fuzzy soft $r$-minimal continuous mappings. Additionally, we show that fuzzy soft $r$-minimal continuity ^[23] $\Rightarrow$ fuzzy soft almost $r$-minimal continuity $\Rightarrow$ fuzzy soft weakly $r$-minimal continuity, but the converse need not be true. In Section 4, we introduce a concept of continuity in a very general setting under the name "fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous mappings". We prove that if $\mathcal{A}$ and $\mathcal{B}$ are operators on $(X, \widetilde{M})$, and $\mathcal{C}$, $\mathcal{C}^*$ and $\mathcal{D}$ are operators on $(Y, \widetilde{M}^*)$, then $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}\sqcap\mathcal{C}^*, \mathcal{D})$-continuous iff it is both fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous and fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous. Finally, Section 5 gives some conclusions and suggests some future works.

2.   Preliminary assertions

In this section, we present the basic definitions which we need in the next sections. Throughout this paper, $X$ refers to an initial universe, $E$ is the set of all parameters for $X$ and $A \subseteq E$, the family of all fuzzy sets in $X$ is denoted by $I^{X}$ (where $I_{\circ} = (0, 1], \; I = [0, 1]$), and for $t\in I,$ $\underline{t}(x) = t$, for all $x\in X.$

Definition 2.1. ^[1,8,17] A fuzzy soft set $f_A$ over $X$ is a mapping from $E$ to $I^X$ such that $f_A(e)$ is a fuzzy set on $X$, for each $e\in A$ and $f_A(e) = \underline{0}$, if $e\not\in A$, where $\underline{0}$ is zero function on $X$. The fuzzy set $f_A(e)$, for each $e\in E$, is called an element of the fuzzy soft set $f_A$. $\widetilde{(X, E)}$ denotes the collection of all fuzzy soft sets on $X$ and is called a fuzzy soft universe.

Definition 2.2. ^[18,28] A fuzzy soft point $e_{x_t}$ over $X$ is a fuzzy soft set over $X$ defined as follows:

where $x_{t}$ is a fuzzy point in $X$. A fuzzy soft point $e_{x_t}$ is said to belong to a fuzzy soft set $f_A$, denoted by $e_{x_t}\tilde{\in} f_A$, if $t\le f_A(e)(x)$. The family of all fuzzy soft points in $X$ is denoted by $\widetilde{P_t(X)}$.

Definition 2.3. ^[8] A mapping $\tau: E \longrightarrow [0, 1]^{\widetilde{ {(X, E)}}}$ is called a fuzzy soft topology on $X$ if it satisfies the following conditions for each $e\in E$.

(i) $\tau_{e}(\Phi) = \tau_{e}(\widetilde{E}) = 1.$

(ii) $\tau_{e}(f_A \sqcap g_B)\geq \tau_{e}(f_A)\wedge \tau_{e} (g_B), \; \; \forall f_{A}, g_{B}\in \widetilde{(X, E)}.$

(iii) $\tau_{e}(\bigsqcup_{i\in \Delta}(f_A)_i)\ge \bigwedge_{i\in\Delta}\tau_{e}((f_A)_i), \forall (f_A)_i\in \widetilde{(X, E)}, i\in \Delta.$

Then, the pair $(X, \tau_E)$ is called a fuzzy soft topological space (FSTS, for short).

Definition 2.4. ^[23] Let $X$ be a nonempty set and $r\in I_{\circ}$. A fuzzy soft mapping $\widetilde{M}: E \longrightarrow [0, 1]^{\widetilde{ {(X, E)}}}$ on $X$ is said to be a fuzzy soft $r$-minimal structure if the family $\widetilde{M}_{e, r} = \{f_A \in \widetilde{(X, E)}\; | \; \widetilde{M}_{e}(f_A)\ge r\}$ for each $e\in E$ contains $\Phi$ and $\widetilde{E}$. Then $(X, \widetilde{M})$ is called a fuzzy soft $r$-minimal space (simply, $r$-$\widetilde{FMS}$). Every member of $\widetilde{M}_{e, r}$ is called a fuzzy soft $r$-minimal open set.

Definition 2.5. ^[23] Let $(X, \widetilde{M})$ be an $r$-$\widetilde{FMS}$, $e\in E$ and $r\in I_{\circ}$. The fuzzy soft $r$-minimal interior and fuzzy soft $r$-minimal closure of $f_A$, denoted by $I_{m}(e, f_A, r)$ and $C_{m}(e, f_A, r)$, resp., are defined as $I_{m}(e, f_A, r) = \bigsqcup\{g_B \in \widetilde{(X, E)}\; :g_B\sqsubseteq f_A, \; g_B \in \widetilde{M}_{e, r}\}$ and $C_{m}(e, f_A, r) = \sqcap\{g_B \in \widetilde{(X, E)}\; :f_A\sqsubseteq g_B, \; g_B^c \in \widetilde{M}_{e, r}\}.$

Definition 2.6. ^[23] Let $(X, \widetilde{M})$ and $(Y, \widetilde{M}^*)$ be $r$-$\widetilde{FMS}$s. Then, a fuzzy soft mapping $\varphi_\psi$ from $\widetilde{(X, E)}$ into $\widetilde{(Y, F)}$ is called fuzzy soft $r$-minimal continuous if $\varphi_\psi^{-1}(g_B) \in \widetilde{M}_{e, r}$ for every $g_B \in \widetilde{M}^*_{k, r}$, $e\in E$ and $(\psi(e) = k)\in F$.

Definition 2.7. ^[23] Let $X$ be a nonempty set and $\widetilde{M}: E \longrightarrow[0, 1]^{\widetilde{ {(X, E)}}}$. Then, $\widetilde{M}$ is said to have property ($P$) if

for $(f_A)_j \in \widetilde{(X, E)}$, $j\in J$ and $e\in E$.

Definition 2.8. ^[23] Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, $e\in E$ and $r \in I_{\circ}$. Then, $\varphi_\psi$ is called fuzzy soft $r$-minimal open if $\varphi_\psi(f_A) \in \widetilde{M}^*_{k, r}$ for every $f_A \in \widetilde{M}_{e, r}$. Also, $\varphi_\psi$ is called fuzzy soft $r$-minimal closed if $(\varphi_\psi(f_A))^c \in \widetilde{M}^*_{k, r}$ for every $f^c_A \in \widetilde{M}_{e, r}$.

Definition 2.9 ^[23] Let $(X, \widetilde{M})$ be an $r$-$\widetilde{FMS}$, $g_B \in \widetilde{(X, E)}$, $e\in E$ and $r \in I_{\circ}$. Then, $g_B$ is called fuzzy soft $r$-minimal compact (resp. fuzzy soft $r$-minimal almost compact and fuzzy soft $r$-minimal nearly compact) iff for every family $\{(f_A)_i \in \widetilde{(X, E)}\; | \; (f_A)_i\in \widetilde{M}_{e, r}\}_{i\in \Gamma}$ such that $g_B \sqsubseteq \bigsqcup_{i\in \Gamma} (f_A)_i$, there exists a finite subset $\Gamma_{\circ}$ of $\Gamma$ such that $g_B \sqsubseteq \bigsqcup_{i\in \Gamma_{\circ}} (f_A)_i$ (resp. $g_B \sqsubseteq \bigsqcup_{i\in \Gamma_{\circ}} C_{m}(e, (f_A)_i, r)$ and $g_B \sqsubseteq \bigsqcup_{i\in \Gamma_{\circ}} I_{m}(e, C_{m}(e, (f_A)_i, r), r)$).

Main properties of fuzzy soft sets and soft topology are found in ^{[1,8,10,14,17,27]}.

3.   Weaker forms of fuzzy soft $r$-minimal continuity

In this section, we introduce a weaker form of fuzzy soft $r$-minimal continuity called fuzzy soft almost (resp. weakly) $r$-minimal continuous mappings and investigate some properties of these mappings. Also, we show that fuzzy soft $r$-minimal continuity ^[23] $\Rightarrow$ fuzzy soft almost $r$-minimal continuity $\Rightarrow$ fuzzy soft weakly $r$-minimal continuity, but the converse need not be true.

Definition 3.1. A fuzzy soft mapping $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ is called fuzzy soft almost (resp. weakly) $r$-minimal continuous if, for fuzzy soft point $e_{x_t}$ over $X$ and each $g_B \in \widetilde{M}^*_{k, r}$ containing $\varphi_\psi(e_{x_t})$, there is $f_A \in \widetilde{M}_{e, r}$ containing $e_{x_t}$ such that $\varphi_\psi(f_A)\sqsubseteq I_{m^*}(k, C_{m^*}(k, g_B, r), r)$ (resp. $\varphi_\psi(f_A)\sqsubseteq C_{m^*}(k, g_B, r)$).

Theorem 3.1. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping. Suppose that one of the following properties holds:

(i) $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r)), r)$, if $g_B \in \widetilde{M}^*_{k, r}$.

(ii) $C_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, I_{m^*}(k, g_B, r), r)), r)\sqsubseteq \varphi_\psi^{-1}(g_B)$, if $g^c_B \in \widetilde{M}^*_{k, r}$.

Then, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Proof. (i) $\Rightarrow$ (ii) Let $g^c_B \in \widetilde{M}^*_{k, r}$. Then, from (i), it follows

Hence, $C_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, I_{m^*}(k, g_B, r), r)), r)\sqsubseteq \varphi_\psi^{-1}(g_B)$.

Similarly, we get (ii) $\Rightarrow$ (i).

Suppose that (i) holds. Let $e_{x_t}\in \widetilde{P_t(X)}$, and $g_B \in \widetilde{M}^*_{k, r}$ containing $\varphi_\psi(e_{x_t})$. Then, by (i),

and so there exists $f_A \in \widetilde{M}_{e, r}$ containing $e_{x_t}$ such that $f_A\sqsubseteq \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r))$. It follows that $\varphi_\psi(f_A)$ $\sqsubseteq I_{m^*}(k, C_{m^*}(k, g_B, r), r)$. Hence, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

In a similar way, one can prove the following corollary.

Corollary 3.1. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $Y$ has the property ($P$). Suppose that one of the following properties holds for $g_B \in \widetilde{(Y, F)}$, $e\in E$ and $r \in I_{\circ}$:

(i) $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r)), r)$, if $g_B \in \widetilde{M}^*_{k, r}$.

(ii) $\varphi_\psi^{-1}(I_{m^*}(k, g_B, r)) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, I_{m^*}(k, g_B, r), r), r)), r)$.

(iii) $C_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, I_{m^*}(k, C_{m^*}(k, g_B, r), r), r)), r)\sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$.

Then, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Lemma 3.1. Every fuzzy soft $r$-minimal continuous mapping ^[23] is fuzzy soft almost $r$-minimal continuous.

Proof. It follows from Theorem 3.1.

In general, the converse of Lemma 3.1 is not true, as shown by Example 3.1.

Example 3.1 Let $X = \{x, y \}$ and $E = \{e_1, e_2 \}$ be the parameter set of X. Define $f_E$ and $g_E$ $\in\widetilde{(X, E)}$ as follows: $f_E = \{(e_1, \{\frac x{0.2}, \frac y{0.4}\}), (e_2, \{\frac x{0.2}, \frac y{0.4}\})\}$, $g_E = \{(e_1, \{\frac x{0.1}, \frac y{0.5}\}), (e_2, \{\frac x{0.1}, \frac y{0.5}\})\}$. Define fuzzy soft $r$-minimal structures $\widetilde{M}_E, \widetilde{W}_E : E \longrightarrow [0, 1]^{\widetilde{ {(X, E)}}}$ as follows: $\forall \; e\in E$,

Then, the identity fuzzy soft mapping $\varphi_\psi:(X, \widetilde{M})\longrightarrow (X, \widetilde{W})$ is fuzzy soft almost $\frac{1}{3}$-minimal continuous, but it is not fuzzy soft $\frac{1}{3}$-minimal continuous.

Definition 3.2. Let $(X, \widetilde{M})$ be an $r$-$\widetilde{FMS}$, $f_A\in \widetilde{(X, E)}$, $e\in E$ and $r \in I_{\circ}$. Then,

(i) $f_A$ is fuzzy soft $r$-minimal semiopen if $f_A \sqsubseteq C_{m}(e, I_{m}(e, f_A, r), r)$,

(ii) $f_A$ is fuzzy soft $r$-minimal preopen if $f_A \sqsubseteq I_{m}(e, C_{m}(e, f_A, r), r)$,

(iii) $f_A$ is fuzzy soft $r$-minimal regularly open if $f_A = I_{m}(e, C_{m}(e, f_A, r), r)$,

(iv) $f_A$ is fuzzy soft $r$-minimal $\beta$-open if $f_A \sqsubseteq C_{m}(e, I_{m}(e, C_{m}(e, f_A, r), r), r)$.

A fuzzy soft set $f_A$ is called a fuzzy soft $r$-minimal semiclosed (resp., fuzzy soft $r$-minimal preclosed, fuzzy soft $r$-minimal regularly closed and fuzzy soft $r$-minimal $\beta$-closed) set if the complement of $f_A$ is a fuzzy soft $r$-minimal semiopen (resp., fuzzy soft $r$-minimal preopen, fuzzy soft $r$-minimal regularly open and fuzzy soft $r$-minimal $\beta$-open) set.

Theorem 3.2. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $Y$ has the property ($P$). Suppose that one of the following properties holds for $g_B \in \widetilde{(Y, F)}$, $e\in E$ and $r \in I_{\circ}$:

(i) $\varphi_\psi^{-1}(g_B) = C_{m}(e, \varphi_\psi^{-1}(g_B), r)$, if $g_B$ is fuzzy soft $r$-minimal regularly closed.

(ii) $\varphi_\psi^{-1}(g_B) = I_{m}(e, \varphi_\psi^{-1}(g_B), r)$, if $g_B$ is fuzzy soft $r$-minimal regularly open.

Then, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Proof. (i) $\Rightarrow$ (ii) is obvious.

Suppose that (ii) holds. Let $e_{x_t}\in \widetilde{P_t(X)}$ and $g_B \in \widetilde{M}^*_{k, r}$ containing $\varphi_\psi(e_{x_t})$. Since $I_{m^*}(k, C_{m^*}(k, g_B, r), r)$ is fuzzy soft $r$-minimal regularly open, then by (ii),

there is $f_A \in \widetilde{M}_{e, r}$ containing $e_{x_t}$ such that $f_A \sqsubseteq \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r))$. This implies $\varphi_\psi(f_A) \sqsubseteq I_{m^*}(k, C_{m^*}(k, g_B, r), r)$. Hence, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

In a similar way, one can prove the following corollary.

Corollary 3.2. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Suppose that one of the following properties holds for $g_B \in \widetilde{(Y, F)}$, $e\in E$ and $r \in I_{\circ}$:

(i) $\varphi_\psi^{-1}(g_B) \in \widetilde{M}_{e, r}$, if $g_B$ is fuzzy soft $r$-minimal regularly open.

(ii) $(\varphi_\psi^{-1}(g_B))^c \in \widetilde{M}_{e, r}$, if $g_B$ is fuzzy soft $r$-minimal regularly closed.

Then, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Theorem 3.3. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Suppose that one of the following properties holds for $g_B \in \widetilde{(Y, F)}$, $e\in E$ and $r \in I_{\circ}$:

(i) $C_{m}(e, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$, if $g_B$ is fuzzy soft $r$-minimal $\beta$-open.

(ii) $C_{m}(e, \varphi_\psi^{-1}(g_B), r) \sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$, if $g_B$ is fuzzy soft $r$-minimal semiopen.

(iii) $\varphi_\psi^{-1}(g_B)\sqsubseteq I_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r)), r)$, if $g_B$ is fuzzy soft $r$-minimal preopen.

(iv) $C_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, I_{m^*}(k, C_{m^*}(k, g_B, r), r), r)), r) \sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$, if $g_B$ is fuzzy soft $r$-minimal preopen.

Then, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Proof. (i) $\Rightarrow$ (ii) Since every fuzzy soft $r$-minimal semiopen set is fuzzy soft $r$-minimal $\beta$-open, it is obvious.

Suppose that (ii) holds. Let $g_B$ be a fuzzy soft $r$-minimal regular closed set. Then, $g_B$ is fuzzy soft $r$-minimal semiopen, and so from (ii), we have

This implies $\varphi_\psi^{-1}(g_B) = C_{m}(e, \varphi_\psi^{-1}(g_B), r)$, and hence from Theorem 3.2, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Suppose that (iii) holds. Let $g_B$ be a fuzzy soft $r$-minimal regular open set. Then, $g_B$ is fuzzy soft $r$-minimal preopen, and so from (iii), we have

This implies $\varphi_\psi^{-1}(g_B) = I_{m}(e, \varphi_\psi^{-1}(g_B), r)$, and hence by Theorem 3.2, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Suppose that (iv) holds. Let $g_B$ be a fuzzy soft $r$-minimal regular closed set. Then, $I_{m^*}(k, g_B, r)$ is fuzzy soft $r$-minimal preopen. From hypothesis and $g_B = C_{m^*}(k, I_{m^*}(k, g_B, r), r)$, it follows that

This implies $\varphi_\psi^{-1}(g_B) = C_{m}(e, \varphi_\psi^{-1}(g_B), r)$. Hence, by Theorem 3.2, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Theorem 3.4. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping. Suppose that one of the following properties holds:

(i) $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, g_B, r)), r)$, if $g_B \in \widetilde{M}^*_{k, r}$.

(ii) $C_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, g_B, r)), r)\sqsubseteq \varphi_\psi^{-1}(g_B)$, if $g^c_B \in \widetilde{M}^*_{k, r}$.

Then, $\varphi_\psi$ is fuzzy soft weakly $r$-minimal continuous.

Proof. (i) $\Leftrightarrow$ (ii) Let $g^c_B \in \widetilde{M}^*_{k, r}$. Then, from (i), it follows that

Hence, $C_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, g_B, r)), r)\sqsubseteq \varphi_\psi^{-1}(g_B)$. Similarly, we get (ii) $\Rightarrow$ (i).

Suppose that (i) holds. Let $e_{x_t}\in \widetilde{P_t(X)}$ and $g_B \in \widetilde{M}^*_{k, r}$ containing $\varphi_\psi(e_{x_t})$. Then, by (i),

and so there exists $f_A \in \widetilde{M}_{e, r}$ containing $e_{x_t}$ such that $f_A\sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$. Thus, $\varphi_\psi(f_A)\sqsubseteq C_{m^*}(k, g_B, r)$. Hence, $\varphi_\psi$ is fuzzy soft weakly $r$-minimal continuous.

In a similar way, one can prove the following corollary.

Corollary 3.3. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $Y$ has the property ($P$). Suppose that one of the following properties holds for $g_B \in \widetilde{(Y, F)}$, $e\in E$ and $r \in I_{\circ}$:

(i) $C_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, g_B, r)), r)\sqsubseteq \varphi_\psi^{-1}(g_B)$, if $g^c_B \in \widetilde{M}^*_{k, r}$.

(ii) $C_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r)), r)\sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$.

(iii) $\varphi_\psi^{-1}(I_{m^*}(k, g_B, r)) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, I_{m^*}(k, g_B, r), r)), r)$.

Then, $\varphi_\psi$ is fuzzy soft weakly $r$-minimal continuous.

Lemma 3.2. Every fuzzy soft almost $r$-minimal continuous mapping is fuzzy soft weakly $r$-minimal continuous.

Proof. It follows from Definition 3.1.

In general, the converse of Lemma 3.2 is not true, as shown by Example 3.2.

Example 3.2. Let $X = \{x, y \}$ and $E = \{e_1, e_2 \}$ be the parameter set of X. Define $f_E$ and $g_E$ $\in\widetilde{(X, E)}$ as follows: $f_E = \{(e_1, \{\frac x{0.5}, \frac y{0.5}\}), (e_2, \{\frac x{0.5}, \frac y{0.5}\})\}$, $g_E = \{(e_1, \{\frac x{0.4}, \frac y{0.2}\}), (e_2, \{\frac x{0.4}, \frac y{0.2}\})\}$. Define fuzzy soft $r$-minimal structures $\widetilde{M}_E, \widetilde{W}_E : E \longrightarrow [0, 1]^{\widetilde{ {(X, E)}}}$ as follows: $\forall \; e\in E$,

Then, the identity fuzzy soft mapping $\varphi_\psi:(X, \widetilde{M})\longrightarrow (X, \widetilde{W})$ is fuzzy soft weakly $\frac{1}{2}$-minimal continuous, but it is not fuzzy soft almost $\frac{1}{2}$-minimal continuous.

The following implications are obtained:

Theorem 3.5. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Then, $\varphi_\psi$ is fuzzy soft weakly $r$-minimal continuous if $C_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r)), r) \sqsubseteq \varphi_\psi^{-1}(C_{m^*}(k, g_B, r))$ for each $g_B \in \widetilde{(Y, F)}$ that is a fuzzy soft $r$-minimal semiopen set, $e\in E$ and $r \in I_{\circ}$.

Proof. Let $g_B \in \widetilde{M}^*_{k, r}$. Then, $g_B$ is a fuzzy soft $r$-minimal semiopen set. From hypothesis and $g_B\sqsubseteq I_{m^*}(k, C_{m^*}(k, g_B, r), r)$, it follows that

Hence, by Theorem 3.3, $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous. This implies $\varphi_\psi$ is fuzzy soft weakly $r$-minimal continuous.

4.   Fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous mappings

In this section, we introduce a concept of continuity in a very general setting called fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous mappings, and we investigate some properties of them.

First of all, let us introduce a concept of continuity in a very general setting. Let $(X, \widetilde{M})$ and $(Y, \widetilde{M}^*)$ be $r$-$\widetilde{FMS}$s, $\mathcal{A}$ and $\mathcal{B}$ $: E \times\widetilde{(X, E)}\times I_{\circ}\rightarrow I^X$ be operators on $(X, \widetilde{M})$, and $\mathcal{C}$ and $\mathcal{D}$ $: F \times\widetilde{(Y, F)}\times I_{\circ}\rightarrow I^Y$ be operators on $(Y, \widetilde{M}^*)$, respectively. The difference between two fuzzy soft sets $f_A$ and $g_B$ is a fuzzy soft set, denoted by $f_A \; \overline{\sqcap}\; g_B$, where

Definition 4.1. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Then, $\varphi_\psi$ is called fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous if

for each $g_B \in \widetilde{M}^*_{k, r}$, $e\in E$ and $(\psi(e) = k)\in F$.

In 2021, Taha ^[23] defined the concept of fuzzy soft $r$-minimal continuous mappings: $\varphi_\psi^{-1}(g_B) \in \widetilde{M}_{e, r}$ for each $g_B \in \widetilde{M}^*_{k, r}$, $e\in E$ and $(\psi(e) = k)\in F$. We can see that the above definition generalizes the concepts of fuzzy soft $r$-minimal continuous mappings, when we choose $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = identity operator and $\mathcal{D}$ = identity operator.

Let us give a historical justification of the above definition:

I. In Section 3, we defined the concept of fuzzy soft almost $r$-minimal continuous mappings: $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(I_{m^*}(k, C_{m^*}(k, g_B, r), r)), r)$ for each $g_B \in \widetilde{M}^*_{k, r}$. Here, $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = interior closure operator and $\mathcal{D}$ = identity operator.

II. In Section 3, we defined the concept of fuzzy soft weakly $r$-minimal continuous mappings: $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(C_{m^*}(k, g_B, r)), r)$ for each $g_B \in \widetilde{M}^*_{k, r}$. Here, $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = closure operator and $\mathcal{D}$ = identity operator.

Definition 4.2. A fuzzy soft mapping $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ is called fuzzy soft $r$-minimal $\circledS$-continuous iff $\varphi_\psi^{-1}(g_B) \in \widetilde{M}_{e, r}$ $\forall$ $g_B \in \widetilde{M}^*_{k, r}$ satisfies property $\circledS$.

Let $\mathcal{C}_{\circledS}: F \times\widetilde{(Y, F)}\times I_{\circ}\rightarrow I^Y$ be an operator on $(Y, \widetilde{M}^*)$ defined as follows:

Theorem 4.1. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Then, $\varphi_\psi$ is fuzzy soft $r$-minimal $\circledS$-continuous iff it is fuzzy soft $r$-minimal $(id, I_{m}, \mathcal{C}_{\circledS}, id)$-continuous.

Proof. $(\Rightarrow)$ Let $\varphi_\psi$ be a fuzzy soft $r$-minimal $\circledS$-continuous and $g_B \in \widetilde{M}^*_{k, r}$.

Case 1. If $g_B$ satisfies property $\circledS$, $\mathcal{C}_{\circledS} (k, g_B, r) = g_B$, and $\varphi_\psi^{-1}(g_B) \in \widetilde{M}_{e, r}$. Thus, we obtain $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(g_B), r) = I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}_{\circledS}(k, g_B, r)), r).$ Then, $\varphi_\psi^{-1}(g_B) \; \overline{\sqcap}\; I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}_{\circledS}(k, g_B, r)), r) = \Phi$. Hence, $\varphi_\psi$ is fuzzy soft $r$-minimal $(id, I_{m}, \mathcal{C}_{\circledS}, id)$-continuous.

Case 2. If $g_B$ does not satisfy property $\circledS$, $\mathcal{C}_{\circledS} (k, g_B, r) = \widetilde{E}$. Thus, we obtain $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(\widetilde{E}), r)$ $= I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}_{\circledS}(k, g_B, r)), r)$. Then, $\varphi_\psi^{-1}(g_B) \; \overline{\sqcap}\;$ $I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}_{\circledS}(k, g_B, r)), r) = \Phi$. Hence, $\varphi_\psi$ is fuzzy soft $r$-minimal $(id, I_{m}, \mathcal{C}_{\circledS}, id)$-continuous.

$(\Leftarrow)$ Suppose that $\varphi_\psi^{-1}(g_B) \; \overline{\sqcap}\; I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}_{\circledS}(k, g_B, r)), r) = \Phi$ for each $g_B \in \widetilde{M}^*_{k, r}$. Then, $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}_{\circledS}(k, g_B, r)), r).$ If $g_B \in \widetilde{M}^*_{k, r}$ satisfies property $\circledS$, $\mathcal{C}_{\circledS} (k, g_B, r) = g_B$, and hence $\varphi_\psi^{-1}(g_B) \sqsubseteq I_{m}(e, \varphi_\psi^{-1}(g_B), r).$ Thus, $\varphi_\psi^{-1}(g_B) \in \widetilde{M}_{e, r}$. Then, $\varphi_\psi$ is fuzzy soft $r$-minimal $\circledS$-continuous.

Definition 4.3. If $\mathcal{A}$ and $\mathcal{B}$ are operators on $(X, \widetilde{M})$, the intersection operator $\mathcal{A} \sqcap \mathcal{B}$ is defined as follows: $(\mathcal{A} \sqcap \mathcal{B})(e, f_A, r) = \mathcal{A}(e, f_A, r) \sqcap \mathcal{B}(e, f_A, r),$ $\; \forall\; f_A\in \widetilde{(X, E)}$ and $e\in E$. $\mathcal{A}$ and $\mathcal{B}$ are called mutually dual if $\mathcal{A}\sqcap\mathcal{B}$ is the identity operator.

Theorem 4.2. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Let $\mathcal{A}$ and $\mathcal{B}$ be operators on $(X, \widetilde{M})$, and $\mathcal{C}$, $\mathcal{C}^*$ and $\mathcal{D}$ be operators on $(Y, \widetilde{M}^*)$. Then, $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}\sqcap\mathcal{C}^*, \mathcal{D})$-continuous iff it is both fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous and fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous.

Proof. If $\varphi_\psi$ is both fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous and fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous, then for each $g_B \in \widetilde{M}^*_{k, r}$ we have

Hence,

However,

Thus, $\mathcal{A} [e, \varphi_\psi^{-1}(\mathcal{D} (k, g_B, r)), r] \; \overline{\sqcap}\; \mathcal{B} [e, \varphi_\psi^{-1}((\mathcal{C}\sqcap\mathcal{C}^*)(k, g_B, r)), r] = \Phi$. Then, $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}\sqcap\mathcal{C}^*, \mathcal{D})$-continuous.

Conversely, if $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}\sqcap\mathcal{C}^*, \mathcal{D})$-continuous, and $g_B \in \widetilde{M}^*_{k, r}$, then

Now, by the above equalities, we get that

Then, we have

and

which means that $\varphi_\psi$ is both fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous and fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous.

Corollary 4.1. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be fuzzy soft almost $r$-minimal and fuzzy soft $r$-minimal $(id, I_{m}, \mathcal{G}, id)$-continuous where $\mathcal{G}$ and $I_{m^*}(C_{m^*})$ are mutually dual operators on $Y$ such that $\mathcal{G}(k, g_B, r) = g_B\sqcup(I_{m^*}(k, C_{m^*}(k, g_B, r), r))^c$ for each $g_B \in \widetilde{M}^*_{k, r}$ and $e\in E$. Then, $\varphi_\psi$ is fuzzy soft $r$-minimal continuous iff $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous, and $\varphi_\psi^{-1}(g_B) \; \overline{\sqcap}\; I_{m}(e, \varphi_\psi^{-1}(\mathcal{G}(k, g_B, r)), r) = \Phi.$

Proof. Fuzzy soft almost $r$-minimal continuous is equal to fuzzy soft $r$-minimal $(id, I_{m}, I_{m^*}(C_{m^*}), id)$-continuous. Since $\mathcal{G}$ and $I_{m^*}(C_{m^*})$ are mutually dual operators on $Y$, then the result follows from Theorem 4.2.

Definition 4.4. Let $\mathcal{A}$ and $\mathcal{B}$ be operators on $(X, \widetilde{M})$. Then, $\mathcal{A}\sqsubseteq\mathcal{B}$ iff $\mathcal{A}(e, f_A, , r)$ $\sqsubseteq\mathcal{B}(e, f_A, r), \; \forall\; f_A\in \widetilde{(X, E)}$ and $e\in E$.

Theorem 4.3. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Let $\mathcal{A}$ and $\mathcal{B}$ be operators on $(X, \widetilde{M})$, and $\mathcal{C}$, $\mathcal{C}^*$ and $\mathcal{D}$ be operators on $(Y, \widetilde{M}^*)$ with $\mathcal{C}\sqsubseteq\mathcal{C}^*$. If $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous, then it is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous.

Proof. If $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous, and $g_B \in \widetilde{M}^*_{k, r}$, thus we obtain

Now, we know $\mathcal{C}\sqsubseteq\mathcal{C}^*$, for each $g_B \in \widetilde{M}^*_{k, r}$, $\mathcal{B} [e, \varphi_\psi^{-1}(\mathcal{C} (k, g_B, r)), r]$ $\sqsubseteq \mathcal{B} [e, \varphi_\psi^{-1}(\mathcal{C}^* (k, g_B, r)), r]$. Therefore,

Then, $\mathcal{A} [e, \varphi_\psi^{-1}(\mathcal{D}(k, g_B, r)), r] \; \overline{\sqcap}\; \mathcal{B} [e, \varphi_\psi^{-1}(\mathcal{C}^*(k, g_B, r)), r] = \Phi$. Hence, $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous.

Definition 4.5. An operator $\mathcal{B}$ on $(X, \widetilde{M})$ induces another operator $I_{m}(\mathcal{B})$ defined as follows: $I_{m}(\mathcal{B})(e, f_A, r) = I_{m}(e, \mathcal{B}(e, f_A, r), r), \; \forall\; f_A\in \widetilde{(X, E)}.$ Observe that $I_{m}(\mathcal{B})\sqsubseteq \mathcal{B}$.

Definition 4.6. A fuzzy soft mapping $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ satisfies the openness condition with respect to the operator $\mathcal{B}$ on $X$ $\text{if} \; \mathcal{B} [e, \varphi_\psi^{-1}(f_A), r]$ $\sqsubseteq \mathcal{B} [e, \varphi_\psi^{-1}(I_{m^*}(k, f_A, r)), r]$, $\forall\; f_A\in \widetilde{(X, E)}.$

Theorem 4.4. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Let $\mathcal{A}$ and $\mathcal{B}$ be operators on $(X, \widetilde{M})$, and $\mathcal{C}$ and $\mathcal{D}$ be operators on $(Y, \widetilde{M}^*)$. If $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous and satisfies the openness condition with respect to the operator $\mathcal{B}$, then $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, I_{m^*}(\mathcal{C}), \mathcal{D})$-continuous.

Proof. If $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous, and $g_B \in \widetilde{M}^*_{k, r}$, thus

Since $\varphi_\psi$ satisfies the openness condition with respect to the operator $\mathcal{B}$, then $\mathcal{B} [e, \varphi_\psi^{-1}(\mathcal{C} (k, g_B, r)), r]\sqsubseteq \mathcal{B} [e, \varphi_\psi^{-1}(I_{m^*}(k, \mathcal{C}(k, g_B, r), r)), r]$, and it follows that

Thus, we obtain

Then, $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, I_{m^*}(\mathcal{C}), \mathcal{D})$-continuous.

Corollary 4.2. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft mapping, and $X$ has the property ($P$). Let $\mathcal{A}$ and $\mathcal{B}$ be operators on $(X, \widetilde{M})$, and $\mathcal{C}$ and $\mathcal{D}$ be operators on $(Y, \widetilde{M}^*)$. If $\varphi_\psi$ is fuzzy soft weakly $r$-minimal continuous and satisfies the openness condition with respect to the operator $\mathcal{B}$, then $\varphi_\psi$ is fuzzy soft almost $r$-minimal continuous.

Proof. Let $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = closure operator and $\mathcal{D}$ = identity operator. Then, the result follows from Theorem 4.4.

Theorem 4.5. If $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ is a fuzzy soft $r$-minimal $(\mathcal{A}, I_{m}, \mathcal{C}, \mathcal{D})$-continuous mapping, $f_A\sqsubseteq \mathcal{A}(e, f_A, r)$, and $g_B\sqsubseteq \mathcal{D}(k, g_B, r)$ $\forall$ $f_A\in \widetilde{(X, E)}$, $g_B\in \widetilde{(Y, F)}$, then $\varphi_\psi(f_A)$ is fuzzy soft $r$-minimal $\mathcal{C}$-compact if $f_A$ is fuzzy soft $r$-minimal compact.

Proof. Given $\{(g_B)_i \in \widetilde{(Y, F)} | \; (g_B)_i \in \widetilde{M}^*_{k, r}\}_{i\in \Gamma}$ with $\varphi_\psi(f_A) \sqsubseteq \bigsqcup_{i\in \Gamma} (g_B)_i$, then $f_A \sqsubseteq \bigsqcup_{i\in \Gamma} \varphi_\psi^{-1}((g_B)_i)$. Since $\varphi_\psi$ is fuzzy soft $r$-minimal $(\mathcal{A}, I_{m}, \mathcal{C}, \mathcal{D})$-continuous, then for each $g_B \in \widetilde{M}^*_{k, r}$,

Then, $f_A \sqsubseteq \bigsqcup_{i\in \Gamma} I_{m}(e, \varphi_\psi^{-1}(\mathcal{C}(k, g_B, r)_i), r)$. Since $f_A$ is fuzzy soft $r$-minimal compact, there exists a finite subset $\Gamma_{\circ}$ of $\Gamma$ such that

Thus, $\varphi_\psi(f_A) \sqsubseteq \bigsqcup_{i\in \Gamma_{\circ}} \mathcal{C}(k, (g_B)_i, r)$. Hence, $\varphi_\psi(f_A)$ is fuzzy soft $r$-minimal $\mathcal{C}$-compact, as required.

The following corollaries are direct results.

Corollary 4.3. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft $r$-minimal continuous mapping. Then, $\varphi_\psi(f_A)$ is fuzzy soft $r$-minimal compact if $f_A\in \widetilde{(X, E)}$ is fuzzy soft $r$-minimal compact.

Proof. Let $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = identity operator and $\mathcal{D}$ = identity operator. Then, the result follows from Theorem 4.5.

Corollary 4.4. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft almost $r$-minimal continuous mapping. Then, $\varphi_\psi(f_A)$ is fuzzy soft $r$-minimal nearly compact if $f_A\in \widetilde{(X, E)}$ is fuzzy soft $r$-minimal compact.

Proof. Let $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = interior closure operator and $\mathcal{D}$ = identity operator. Then, the result follows from Theorem 4.5.

Corollary 4.5. Let $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ be a fuzzy soft weakly $r$-minimal continuous mapping. Then, $\varphi_\psi(f_A)$ is fuzzy soft $r$-minimal almost compact if $f_A\in \widetilde{(X, E)}$ is fuzzy soft $r$-minimal compact.

Proof. Let $\mathcal{A}$ = identity operator, $\mathcal{B}$ = interior operator, $\mathcal{C}$ = closure operator and $\mathcal{D}$ = identity operator. Then, the result follows from Theorem 4.5.

5.   Conclusions

In this paper, we have defined weaker forms of fuzzy soft $r$-minimal continuity called fuzzy soft almost $r$-minimal continuity and fuzzy soft weakly $r$-minimal continuity. We have investigated the master properties of these continuous forms and provided some illustrative examples to show the relationships between them. Then, we have introduced the concept of fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous mappings and given some characterizations of them. Moreover, we have proved that if $\mathcal{A}$ and $\mathcal{B}$ are operators on $(X, \widetilde{M})$, and $\mathcal{C}$, $\mathcal{C}^*$ and $\mathcal{D}$ are operators on $(Y, \widetilde{M}^*)$, then $\varphi_\psi:(X, \widetilde{M})\longrightarrow (Y, \widetilde{M}^*)$ is fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}\sqcap\mathcal{C}^*, \mathcal{D})$-continuous iff it is both fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D})$-continuous and fuzzy soft $r$-minimal $(\mathcal{A}, \mathcal{B}, \mathcal{C}^*, \mathcal{D})$-continuous.

In the upcoming work, we will define some new separation axioms in fuzzy soft $r$-minimal spaces. Also, we shall discuss the concepts given herein in the frames of infra soft topologies ^[5] and infra fuzzy topologies ^[7]. We hope that this work will contribute to fuzzy soft $r$-minimal structure studies.

Conflict of interest

The author declares that there is no conflict of interest.