New Einstein-Randers metrics on certain homogeneous manifolds arising from the generalized Wallach spaces

1.   Introduction

Pawlak introduced the rough set theory ^[1,2] as a conceptual framework designed to address the challenges posed by vagueness and uncertainty inherent in data analysis and information systems. Atef et al. ^[3] discussed the generalization of three types of rough set models based on j-neighborhood structure and explored some of their basic properties. Alcantud et al. ^[4] introduced a new model that combines multi-granularity, soft set, and rough set-based overlays. It strives to provide a hybrid model that captures the strengths of each theory and can be applied to a variety of industries. Another approximation space based on topological near open sets and the properties of these spaces is presented by Mareay et al. ^[5]. It also includes an algorithm to detect the side effects of the COVID-19 infection. Azzam et al. ^[6] discussed a proposed reduction method based on similarity relations and pretopology concepts, as well as new pretopological structures for creating information systems. Hu et al. ^[7] discussed the combination of kernel methods and rough sets in machine learning. It also proposes a fuzzy rough set model and a Gaussian kernel approximation algorithm for feature ranking and reduction. Rough sets and their applications have attracted many researchers in different fields. Stefania Boffa discussed "Sequences of Refinements of Rough Sets". The idea discussed in the thesis is known as "sequences of orthopairs" within the generalized hard set theory. This notion aims to establish operations between sequences of orthopairs and explore methods for generating them based on operations related to common rough sets ^[8]. The writer demonstrates multiple representation theorems for the class of finite centered Kleene algebras ^[9].

Covering based rough set involves the utilization of sets to approximate other sets ^[10,11,12]. Bonikowski et al. ^[13] put forth a new model of covering sets via minimal description concepts. Many other models of covering rough set depending on the neighborhood and the complementary neighborhood are proposed in ^[14,15,16]. An intuitionistic fuzzy set (IFS) on the structure of rough sets based on covering is introduced by using the notion of the neighborhood in ^[17]. It defines three models of IFS approximation structure based on covering. Interestingly, any covering can be linked to a tolerance relation, and vice versa. This technique leaves the upper approximations in rough set theory unchanged. An axiomatic description of the second type of covering higher approximations is also given. Tolerance relations and coverings are powerful tools for comprehending the structure and interactions within sets. Partitions and overlapping covers are mathematical constructions that assist in understanding similarity and discernment ^[18].

In 1990, Dubois and Prade ^[19] introduced rough fuzzy sets and fuzzy rough sets, which marked a significant development in the field. Scholars have extensively explored these concepts, as evidenced by studies conducted by various researchers ^{[20,21,22,23,24,25]}. Deng et al. ^[22] introduced fuzzy covering based on fuzzy relations in 2007, while Ma ^[23] devised two categories of fuzzy covering rough sets in 2016 using fuzzy $\beta$-neighborhood. In 2017, Yang and Hu ^[24] established various types of fuzzy covering-based rough sets through fuzzy $\beta$-neighborhood. Hu ^[26] conducted a comprehensive study in 2019, investigating four types of fuzzy neighborhood operators and their properties by introducing the concept of fuzzy $\beta$-minimal description. Deer et al. ^[27] delved into fuzzy neighborhoods based on fuzzy covering.

In the realm of soft computing, the synergy of soft sets, rough sets, and fuzzy sets has emerged as a pivotal area of research, providing a nuanced framework for handling uncertainties in diverse domains. The foundational principles of soft set theory were initially formulated by Molodtsov ^[28] as a versatile mathematical framework tailored to address vagueness and uncertainty. Subsequently, an expanding body of research has explored the properties and advantages of soft set theory ^[29,30].

$\mathcal SRFSs$ incorporate elements from three distinct mathematical frameworks: Rough sets, fuzzy sets, and soft sets. In this hybrid model, we examine uncertainty, ambiguity, and indiscernibility at the same time. SRFS enables us to manage imprecise data and make more flexible decisions. Researchers investigated many algebraic structures and operations within this paradigm ^[31]. The SRFS model builds on classical rough set theory by including fuzzy membership degrees. It addresses ambiguity and gentle transitions between different granularities. By combining rough set approximations with fuzzy membership functions, this paradigm improves our capacity to analyze complex data. Applications include data mining, pattern recognition, and decision-making ^[32]. Zhan ^[33] introduced the concept of soft fuzzy rough set-based covering through the notion of soft neighborhoods. Zhan's model for soft rough fuzzy coverings stands at the forefront of this convergence, offering a distinctive perspective on the interplay between soft, rough, and fuzzy characteristics. The relations between models for soft rough fuzzy covering sets and Zhan's model for SRF covering must be investigated in order to comprehend their synergy. These relations shed light on how various mathematical constructs can be combined or related. Researchers explore the implications of these connections and their practical applications in ^[34].

Throughout in this paper, we analyze and enhance Zhan's model for soft rough fuzzy coverings, emphasizing the interplay between soft, rough, and fuzzy characteristics and developing mathematical formulations and algorithms for accurate representation and manipulation. For Zhan's model, we increase the lower approximation while simultaneously reducing the upper approximation to make it more accurate. Furthermore, we introduce three novel models of $\mathcal SRFSs$, strategically grounded in the concept of covering via the core of the neighborhood concept. These additions are crafted to address specific challenges and intricacies encountered in real-world applications, where the management of uncertainties is paramount.

First, we present the basic concepts of rough sets and soft sets. Second, a new model of $\mathcal SRFSs$ based on covering is introduced in Section 3 by using the core of the neighborhood concept. We put forth new other two models of soft rough fuzzy covering $\mathcal SRFC$ based on neighborhood in Section 4 and established the relations between our models and Zhan's model. The conclusions are presented in Section 5.

2.   Basic concepts of rough sets and soft set theory

Along this section, consider $\mathcal{{R}}$ as an equivalence relation on a nonempty set $\mathcal{{U}}$. Hence, $\mathcal {{U}}/\mathcal {{R}} = \{\mathcal Y_1, \mathcal Y_2, \mathcal Y_3, ..., \mathcal Y_m\}$ is a partition on $\mathcal {{U}}$, where $\mathcal {{R}}$ is an equivalence relation that generates the classes of equivalence $\mathcal Y_1, \mathcal Y_2, \mathcal Y_3, ..., \mathcal Y_m$. For the soft set, consider $\mathcal {{U}}$ is a universe set, $\mathcal {{A}}$ is a set of parameters on $\mathcal {{U}}$, $\mathcal {P({U})}$ is the power set of $\mathcal {{U}}$, and we fix a soft set $\mathcal \mho_G = (\mathcal F, \mathcal {{A}})$ over $\mathcal {{U}}$.

Definition 2.1. ^[35] If $\mathcal X_1\subseteq \mathcal {{U}}$ and $\mathcal {{U}}\neq \phi$, then the set of approximation operators lower (upper) is defined as : $\underline{\mathcal {{R}}}(\mathcal X_1) = \bigcup \{\mathcal Y_i\in \mathcal {{U}}/\mathcal {{R}}: \mathcal Y_i\subseteq \mathcal X_1\}$. $\overline{\mathcal {{R}}}(\mathcal X_1) = \bigcup \{\mathcal Y_i\in \mathcal {{U}}/\mathcal {{R}}: \mathcal Y_i\cap \mathcal X_1\neq \emptyset\}$, respectively.

Proposition 2.1. ^[35] If $\mathcal K = (\mathcal {{U}}, \mathcal {{R}})$ is an approximation structure, then the following axioms hold for $\mathcal Q_1, \mathcal Q_2\subseteq \mathcal {{U}}$ :

(I) $\underline{\mathcal {{R}}}(\mathcal Q_1) = \mathcal Q_1$, $\overline{\mathcal {{R}}}(\mathcal Q_1) = \mathcal Q_1$;

(II) $\underline{\mathcal {{R}}}(\emptyset) = \emptyset$, $\overline{\mathcal {{R}}}(\emptyset) = \emptyset$;

(III) $\underline{\mathcal {{R}}}(\mathcal Q_1)\subseteq \mathcal Q_1\subseteq \overline{\mathcal {{R}}}\mathcal (Q_1)$;

(IV) $\underline{\mathcal {{R}}}(\mathcal Q_1\cap \mathcal Q_2) = \underline{\mathcal {{R}}}(\mathcal Q_1)\cap \underline{\mathcal {{R}}}(\mathcal Q_2)$;

(V) $\overline{\mathcal {{R}}}(\mathcal Q_1\cup \mathcal Q_2) = \overline{\mathcal {{R}}}(\mathcal Q_1)\cup \overline{\mathcal {{R}}}(\mathcal Q_2)$;

(VI) $\underline{\mathcal {{R}}}(\mathcal Q_1^c) = [\overline{\mathcal {{R}}}(\mathcal Q_1)]^c$, where $(\mathcal Q_1^c)$ is the complement of $\mathcal Q_1$;

(VII) $\underline{\mathcal {{R}}}(\underline{\mathcal {{R}}}(\mathcal Q_1)) = \underline{\mathcal {{R}}}(\mathcal Q_1)$;

(VIII) $\overline{\mathcal {{R}}}(\overline{\mathcal {{R}}}(\mathcal Q_1)) = \overline{\mathcal {{R}}}(\mathcal Q_1)$;

(IX) $\mathcal Q_1\subseteq \mathcal Q_2 \Rightarrow$ $\underline{\mathcal {{R}}}(\mathcal Q_1)\subseteq \underline{\mathcal {{R}}}(\mathcal Q_2)$ and $\overline{\mathcal {{R}}}(\mathcal Q_1)\subseteq \overline{\mathcal {{R}}}(\mathcal Q_2)$;

(X) $\underline{\mathcal {{R}}}(\underline{\mathcal {{R}}}(\mathcal Q_1))^c = (\underline{\mathcal {{R}}}(\mathcal Q_1))^c$, $\overline{\mathcal {{R}}}(\overline{\mathcal {{R}}}(\mathcal Q_1))^c = (\overline{\mathcal {{R}}}(\mathcal Q_1))^c$;

(XI) $\underline{\mathcal {{R}}}(\mathcal Q_1)\cup \underline{\mathcal {{R}}}(Q_2)\subseteq \underline{\mathcal {{R}}}(\mathcal Q_1\cup \mathcal Q_2)$;

(XII) $\overline{\mathcal {{R}}}(\mathcal Q_1\cap \mathcal Q_2)\subseteq\overline{\mathcal {{R}}}(\mathcal Y)\cap \overline{\mathcal {{R}}}(\mathcal Q_2)$.

Definition 2.2. ^[13] Consider $\mathcal{{C}}$ is a family of subsets of the universe $\mathcal{{U}}$. We call $\mathcal{{C}}$ a covering of $\mathcal{{U}}$ if $\cup$ $\mathcal {{C}} = \mathcal{{U}}$, where no subset in $\mathcal{{C}}$ is empty.

Definition 2.3. ^[13,36] Assume that $\mathcal {{C}}$ is a covering of the nonempty set $\mathcal{{U}}$, so the structure $\prec \mathcal{{U}}, \mathcal {{C}} \succ$ is a rough approximation structure based on covering.

Definition 2.4. ^[13,36] Suppose that $\prec \mathcal{{U}}, \mathcal {{C}} \succ$ is covering rough approximation structure and let $\mho_1 \in \mathcal{{U}}$. Hence, the set family $Md(\mho_1)$ is called the minimal description of $\mho_1$, since $Md(\mho_1) = \lbrace \mathcal \omega \in \mathcal {{C}}: \mho_1 \in \mathcal \omega \wedge (\forall \mathcal {{S}} \in \mathcal {{C}}\wedge \mho_1\in\mathcal {{S}} \wedge \mathcal {{S}} \subseteq \mathcal \omega \Rightarrow \mathcal \omega = \mathcal {{S}})\rbrace$.

Definition 2.5. ^[37] Let $\mathcal F:\mathcal {{A}}\rightarrow \mathcal P({U})$, so the structure $\mathcal \mho_G = \prec\mathcal F, \mathcal {{A}}\succ$ is called a soft set on $\mathcal{{U}}$. If $\bigcup_{e\in \mathcal {{A}}}\mathcal F(e) = \mathcal {{U}}$, then the soft set is full soft set.

Definition 2.6. ^[38] Let $\mathcal \mho_G = (\mathcal F, \mathcal {{A}})$ be a soft set over $\mathcal {{U}}$. The structure $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is called a soft covering approximation structure ($\mathcal SCAS$) based on $\mathcal S$.

Definition 2.7. ^[33] If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is a soft covering approximation structure, then the soft neighborhood of $x\in \mathcal {{U}}$ is defined as follows:

$\mathscr {N}_s(x) = \cap\{\mathcal F(e): x\in \mathcal F(e) \}$.

Definition 2.8. ^[33] If $\mathcal S$ is a soft covering approximation structure and $\mathscr A\in\mathscr F(\mathcal {{U}})$, then the two operators:

$\aleph ^{-0}(\mathscr A) (x) = \wedge\{\mathscr A(y):y\in \mathscr {N}_s(x)\}$,

$\aleph ^{+0}(\mathscr A) (x) = \vee\{\mathscr A(y):y\in \mathscr {N}_s(x)\}$, for all $x\in \mathcal {{U}}$

are called the soft fuzzy covering lower (upper) approximation structure $\mathcal SFCLA-0$ ($\mathcal SFCUA-0$), respectively.

Clearly, the set $\mathscr A$ is called a soft rough covering-based fuzzy set ($\mathcal SRCF-0$) if $\aleph ^{-0}(\mathscr A)\neq\aleph ^{+0}(\mathscr A)$. Otherwise, the set $\mathscr A$ is definable.

During this research, we will express that $\aleph ^{-i}(\mathscr A)$ ($\aleph ^{+i}(\mathscr A)$) is the $i$ type of $\mathcal SFCLA$ ($\mathcal SFCUA$) as $\mathcal SFCLA-i$ ($\mathcal SFCUA-i$), and if $\aleph ^{-i}(\mathscr A)\neq\aleph ^{+i}(\mathscr A)$, then the set $\mathscr A$ is called ($\mathcal SRCF-i$). Otherwise, the set $\mathscr A$ is definable.

3.   A new model of $\mathcal SRFSs$ based on covering

In this section, we introduce new models of $\mathcal SRFSs$ based on covering by the core of soft neighborhood. We present the properties of the new models along with some illustrative examples.

Definition 3.1. Consider that $(\mathcal {{U}}, \mathcal \mho_G)$ is a soft rough covering approximation structure ($\mathcal SRCAS$) where we fix the soft set $\mathcal \mho_G = \prec\mathcal F, \mathcal {{A}}\succ$, then $\forall x\in \mathcal {{U}}$ the core of the soft neighborhood is defined as $\mathscr {CN}_s(x) = \{y\in \mathcal {{U}}: \mathscr {N}_s(x) = \mathscr {N}_s(y)\}$.

Example 3.1. Consider that $({{U}}, \mathcal \mho_G)$ is a $\mathcal SRCAS$ where $\mathcal {{U}} = \{\mho_1, \mho_2, \mho_3, \mho_4, \mho_5, \mho_6\}$, $\mathcal {{C}} = \{\{\mho_1, \mho_2\}$, $\{\mho_1, \mho_2, \mho_3\}, \{\mho_1, \mho_2, \mho_4, \mho_5\}$, $\{\mho_3, \mho_4, \mho_5, \mho_6\}, , \{\mho_3, \mho_5, \mho_6\}\}$, and $\mathcal \mho_G = \prec\mathcal F, \mathcal {{A}}\succ$is a soft set defined in Table 1.

From Table 1, the soft neighborhood and the core of the soft neighborhood are computed as follows: $\mathscr {N}_s(\mho_1) = \{\mho_1, \mho_2\}$, $\mathscr {N}_s(\mho_2) = \{\mho_1, \mho_2\}$, $\mathscr {N}_s(\mho_3) = \{\mho_3\}$, $\mathscr {N}_s(\mho_4) = \{\mho_4, \mho_5\}$, $\mathscr {N}_s(\mho_5) = \{\mho_5\}$, $\mathscr {N}_s(\mho_6) = \{\mho_3, \mho_5, \mho_6\}$. Therefore, $\mathscr {CN}_s(\mho_1) = \{\mho_1, \mho_2\}$, $\mathscr {CN}_s(\mho_2) = \{\mho_1, \mho_2\}$, $\mathscr {CN}_s(\mho_3) = \{\mho_3\}$, $\mathscr {CN}_s(\mho_4) = \{\mho_4\}$, $\mathscr {CN}_s(\mho_5) = \{\mho_5\}$, $\mathscr {CN}_s(\mho_6) = \{\mho_6\}$.

Definition 3.2. Assume that $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is $\mathcal SCAS$ and $\mathscr A_1 \in \mathscr F(\mathcal {{U}})$. The two operators:

$\aleph ^{-1}(\mathscr A_1) (x) = \wedge\{\mathscr A_1(y):y\in \mathscr {CN}_s(x)\}$ for all $x\in \mathcal {{U}}\}$ is called $\mathcal SFCLA-1$,

$\aleph ^{+1}(\mathscr A_1) (x) = \vee\{\mathscr A_1(y):y\in \mathscr {CN}_s(x)\}$, for all $x\in \mathcal {{U}}\}$ is called $\mathcal SFCUA-1$.

Clearly, the set $\mathscr A_1$ is called ($\mathcal SRCF-1$) if $\aleph ^{-1}(\mathscr A_1)\neq\aleph ^{+1}(\mathscr A_1)$. Otherwise the set $\mathscr A_1$ is definable.

Example 3.2. If $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$. By using Example 3.1, we get the following:

$\aleph ^{-0}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.5)\}$,

$\aleph ^{+0}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.5), (\mho_5, 0.5), (\mho_6, 0.8)\}$,

$\aleph ^{-1}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$,

$\aleph ^{+1}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

Obviously, $\aleph ^{-0}(\mathscr A_1)\subseteq \aleph ^{-1}(\mathscr A_1)$ and $\aleph ^{+1}(\mathscr A_1\subseteq \aleph ^{+0}(\mathscr A_1)$.

Theorem 3.1. Assume that $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SRCAS$) and $\mathscr A_1, \mathscr A_2 \in \mathscr F(\mathcal {{U}})$, then $\forall \mho_1, \mho_2, \mho_3\in \mathcal {{U}}$ the following properties are satisfied:

(iL) If $\mathscr A_1\subseteq \mathscr A_2$, then $\aleph ^{-1}(\mathscr A_1)\subseteq \aleph ^{-1}(\mathscr A_2)$;

(iH) If $\mathscr A_1\subseteq \mathscr A_2$, then $\aleph ^{+1}(\mathscr A_1)\subseteq \aleph ^{+1}(\mathscr A_2)$;

(iiL) $\aleph ^{-1}(\mathscr A_1\cap \mathscr A_2) = \aleph ^{-1}(\mathscr A_1)\cap \aleph ^{-1}(\mathscr A_2)$;

(iiH) $\aleph ^{+1}(\mathscr A_1\cap \mathscr A_2)\subseteq \aleph ^{+1}(\mathscr A_1)\cap \aleph ^{+1}(\mathscr A_2)$;

(iiiL) $\aleph ^{-1}(\mathscr A_1)\cup \aleph ^{-1}(\mathscr A_2)\subseteq \aleph ^{-1}(\mathscr A_1\cup \mathscr A_2)$;

(iiiH) $\aleph ^{+1}(\mathscr A_1)\cup \aleph ^{+1}(\mathscr A_2) = \aleph ^{+1}(\mathscr A_1\cup \mathscr A_2)$;

(ivL) $\aleph ^{-1}(\mathscr A_1^c) = (\aleph ^{+1}(\mathscr A_1))^c$;

(ivH) $\aleph ^{+1}(\mathscr A_1^c) = (\aleph ^{-1}(\mathscr A_1))^c$;

(vL) $\aleph ^{-1}(\mathscr A_1) = \aleph ^{-1}(\aleph ^{-1}(\mathscr A_1))$;

(vH) $\aleph ^{+1}(\mathscr A_1) = \aleph ^{+1}(\aleph ^{+1}(\mathscr A_1))$;

(viLH) $\aleph ^{-1}(\mathscr A_1)\subseteq \mathscr A_1\subseteq \aleph ^{+1}(\mathscr A_1)$.

Proof. We will prove only $iL$, $iiL$, $iiiL$, $ivL$ and $ivL$ items. The proof of other items is similar:

(iL) If $\mathscr A_1\subseteq \mathscr A_2$ where $\mathscr A_1, \mathscr A_2 \in \mathscr F(\mathcal {{U}})$ and $\mho_1, \mho_2 \in \mathcal {{U}}$, hence, we get $\aleph ^{-1}(\mathscr A_1) (\mho_1) = \wedge\{\mathscr A_1(\mho_2)$:$\mho_2\in \mathscr {CN}_s(\mho_1)\}\leq \wedge\{\mathscr A_2(\mho_2)$:$\mho_2\in \mathscr {CN}_s(\mho_1)\} = \aleph ^{-1}(\mathscr A_2) (\mho_1)$;

(iiL) $\aleph ^{-1}(\mathscr A_1\cap \mathscr A_2)(\mho_1) = \wedge\{(\mathscr A_1\cap \mathscr A_2)(\mho_2)$:$\mho_2\in \mathscr {CN}_s(\mho_1)\} =$ $\wedge\{\mathscr A_1(\mho_2):\mho_2\in \mathscr {CN}_s(\mho_1)\}\cap \wedge\{\mathscr A_2(\mho_2)$:$\mho_2\in \mathscr {CN}_s(\mho_1)\} = \aleph ^{-1}(\mathscr A_1) (\mho_1)\cap \aleph ^{-1}(\mathscr A_2) (\mho_1)$;

(iiiL) Since $\mathscr A_1\subseteq \mathscr A_1\cup \mathscr A_2$, $\mathscr A_2\subseteq \mathscr A_1\cup \mathscr A_2$, then $\aleph ^{-1}(\mathscr A_1)\subseteq \aleph ^{-1}(\mathscr A_1\cup \mathscr A_2)$ and $\aleph ^{-1}(\mathscr A_2)\subseteq \aleph ^{-1}(\mathscr A_1\cup \mathscr A_2)$. Therefore, $\aleph ^{-1}(\mathscr A_1)\cup \aleph ^{-1}(\mathscr A_2)\subseteq \aleph ^{-1}(\mathscr A_1\cup \mathscr A_2)$;

(ivL) $\aleph ^{-1}(\mathscr A_1^c) = \wedge\{\mathscr A_1^c(\mho_2)$:$\mho_2\in \mathscr {CN}_s(\mho_1)\} = \wedge\{1-\mathscr A_1(\mho_2)$:$\mho_2\in \mathscr {CN}_s(\mho_1)\} = 1-\vee\{\mathscr A_1(\mho_2):\mho_2\in \mathscr {CN}_s(\mho_1)\} = (\aleph ^{+1}(\mathscr A_1))^c$;

(vL) $\aleph ^{-1}(\aleph ^{-1}(\mathscr A_1))(\mho_1) = \wedge\{\aleph ^{-1}(\mathscr A_1^c(\mho_2))$:$\mho_2\in \mathscr {CN}_s(\mho_1)\} = \wedge\{\wedge\{\mathscr A_1^c(\mho_3):\mho_3\in (\mathscr {CN}_s(\mho_2))\}:\mho_2\in \mathscr {CN}_s(\mho_1)\} = \wedge\{\mathscr A_1(\mho_3)$:$\mho_3\in \mathscr {CN}_s(\mho_2) \wedge \mho_2\in \mathscr {CN}_s(\mho_1)\} = \wedge\{\mathscr A_1(\mho_3)$:$\mho_3\in \mathscr {CN}_s(\mho_2)\subseteq \mathscr {CN}_s(\mho_1)\} = \wedge\{\mathscr A_1(\mho_3)$:$\mho_3\in \mathscr {CN}_s(\mho_1)\} = \aleph ^{-1}(\mathscr A_1)(\mho_1)$.

We define the first type of soft measure degree $(\mathcal SMD-1)$ as follows.

Definition 3.3. Assume that $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mho_1, \mho_2\in \mathcal {{U}}$, then $(\mathcal SMD-1)$ is defined as:

Clearly, $0\leq \mathscr D_s^1(\mho_1, \mho_2)\leq1$, $\mathscr D_s^1(\mho_1, \mho_2) = \mathscr D_s^1(\mho_2, \mho_1)$, and $\mathscr D_s^1(\mho_1, \mho_1) = 1$.

Example 3.3. Let us consider Example 3.1. The $(\mathcal SMD-1)$ between each two elements $\mho_i, \mho_j\in \mathcal {{U}}, i, j = 1, 2, ..., 6$ is calculated in Table 2.

Based on Definition 3.3, we define the first type of $SCRF$ based on $\lambda$-lower (upper) approximation $\{ \lambda-\mathcal SFCLA-1$, ($\lambda-\mathcal SFCUA-1 )\}$ as follows.

Definition 3.4. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr D_s^1(\mho_1, \mho_2)$ is $(\mathcal SMD-1)$ for $\mho_1, \mho_2\in \mathcal {{U}}$, for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, the $\lambda-\mathcal SFCLA-1$, ($\lambda-\mathcal SFCUA-1$) is defined as follows, respectively:

$\aleph_\lambda^{-1}(\mathscr A_1)(\mho_1) = \wedge\{\mathscr A_1)(\mho_2):\mathscr D_s^1(\mho_1, \mho_2) > \lambda\}$,

$\aleph_\lambda^{+1}(\mathscr A_1)(\mho_1) = \vee\{\mathscr A_1)(\mho_2):\mathscr D_s^1(\mho_1, \mho_2) > \lambda\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$.

If $\aleph_\lambda^{-1}(\mathscr A_1)\neq \aleph_\lambda^{+1}(\mathscr A_1)$, then $\mathscr A_1$ is called $\lambda-\mathcal SCRF-1$; otherwise $\mathscr A_1$ is called definable.

Example 3.4 Continued from Example 3.3 and $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, for $\lambda = 0.5$, we have the following approximations operators:

$\aleph_\lambda^{-1}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$,

$\aleph_\lambda^{+1}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

Theorem 3.2. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$), then for $\mathscr A_1, A_2\in \mathscr F(\mathcal {{U}})$, the following properties hold:

(iL) If $\mathscr A_1\subseteq \mathscr A_2$, then $\aleph_\lambda^{-1}(\mathscr A_1)\subseteq \aleph_\lambda^{-1}(\mathscr A_2)$;

(iH) If $\mathscr A_1\subseteq \mathscr A_2$, then $\aleph_\lambda^{+1}(\mathscr A_1)\subseteq \aleph_\lambda^{+1}(\mathscr A_2)$;

(iiL) $\aleph_\lambda^{-1}(\mathscr A_1\cap \mathscr A_2) = \aleph_\lambda^{-1}(\mathscr A_1)\cap \aleph_\lambda^{-1}(\mathscr A_2)$;

(iiH) $\aleph_\lambda ^{+1}(\mathscr A_1\cap \mathscr A_2)\subseteq \aleph_\lambda ^{+1}(\mathscr A_1)\cap \aleph_\lambda ^{+1}(\mathscr A_2)$;

(iiiL) $\aleph_\lambda ^{-1}(\mathscr A_1)\cup \aleph_\lambda ^{-1}(\mathscr A_2)\subseteq \aleph_\lambda ^{-1}(\mathscr A_1\cup \mathscr A_2)$;

(iiiH) $\aleph_\lambda ^{+1}(\mathscr A_1)\cup \aleph_\lambda ^{+1}(\mathscr A_2) = \aleph_\lambda ^{+1}(\mathscr A_1\cup \mathscr A_2)$;

(ivL) $\aleph_\lambda ^{-1}(\mathscr A_1^c) = (\aleph_\lambda ^{+1}(\mathscr A_1))^c$;

(ivH) $\aleph_\lambda ^{+1}(\mathscr A_1^c) = (\aleph_\lambda ^{-1}(\mathscr A_1))^c$;

(vL) If $\alpha_1\leq \alpha_2$, then $\aleph_\lambda ^{-1}(\mathscr A_1)\subseteq \aleph_\lambda ^{-1}(\aleph_\lambda ^{-1}(\mathscr A_2))$;

(vH) If $\alpha_1\leq \alpha_2$, then $\aleph_\lambda ^{+1}(\mathscr A_1)\subseteq\aleph_\lambda ^{+1}(\aleph_\lambda ^{+1}(\mathscr A_2))$;

(viLH) $\aleph_\lambda ^{-1}(\mathscr A_1)\subseteq \mathscr A_1\subseteq \aleph_\lambda ^{+1}(\mathscr A_1)$.

Proof. Similar to Theorem 3.1.

Definition 3.5. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr D_s^1(\mho_1, \mho_2)$ is $(\mathcal SMD-1)$ for $\mho_1, \mho_2\in \mathcal {{U}}$, then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, the first type of SFC $\mathscr D$-lower (upper) approximation ($\mathscr D-\mathcal SFCLA-1$), ($\mathscr D-\mathcal SFCUA-1$) is defined as follows, respectively:

$\aleph_\mathscr D^{-1}(\mathscr A_1)(\mho_1) = \underset{\mho_2\in \mathcal {{U}}}{\wedge}\{(1-\mathscr D_s^1)(\mho_1, \mho_2)\vee \mathscr A_1(\mho_2)\}$,

$\aleph_\mathscr D^{+1}(\mathscr A_1)(\mho_1) = \underset{\mho_2\in \mathcal {{U}}}{\vee}\{\mathscr D_s^1(\mho_1, \mho_2)\wedge \mathscr A_1(\mho_2)\}$, $\forall \mho_1\in \mathcal {{U}}$.

If $\aleph_\mathscr D^{-1}(\mathscr A_1)\neq \aleph_\mathscr D^{+1}(\mathscr A_1)$, then $\mathscr A_1$ is called $\mathscr D-\mathcal SCRF-1$; otherwise, $\mathscr A_1$ is called definable.

Example 3.5. Continued from Example 3.3 and $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, then we get the following approximations operators:

$\aleph_\mathscr D^{-1}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$,

$\aleph_\mathscr D^{+1}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

Theorem 3.3. Assume that $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr A_1, \mathscr A_2 \in \mathscr F(\mathcal {{U}})$, then $\forall \mho_1, \mho_2, \mho_3\in \mathcal {{U}}$, and the following properties are satisfied:

(iL) If $\mathscr A_1\subseteq \mathscr A_2$, then $\aleph_\mathscr D^{-1}(\mathscr A_1)\subseteq \aleph_\mathscr D^{-1}(\mathscr A_2)$;

(iH) If $\mathscr A_1\subseteq \mathscr A_2$, then $\aleph_\mathscr D^{+1}(\mathscr A_1)\subseteq \aleph_\mathscr D^{+1}(\mathscr A_2)$;

(iiL) $\aleph_\mathscr D^{-1}(\mathscr A_1\cap \mathscr A_2) = \aleph_\mathscr D^{-1}(\mathscr A_1)\cap \aleph_\mathscr D^{-1}(\mathscr A_2)$;

(iiH) $\aleph ^{+1}(\mathscr A_1\cap \mathscr A_2)\subseteq \aleph ^{+1}(\mathscr A_1)\cap \aleph ^{+1}(\mathscr A_2)$;

(iiiL) $\aleph_\mathscr D^{-1}(\mathscr A_1)\cup \aleph_\mathscr D^{-1}(\mathscr A_2)\subseteq \aleph_\mathscr D^{-1}(\mathscr A_1\cup \mathscr A_2)$;

(iiiH) $\aleph_\mathscr D ^{+1}(\mathscr A_1)\cup \aleph_\mathscr D^{+1}(\mathscr A_2) = \aleph_\mathscr D ^{+1}(\mathscr A_1\cup \mathscr A_2)$;

(ivL) $\aleph_\mathscr D^{-1}(\mathscr A_1^c) = (\aleph_\mathscr D ^{+1}(\mathscr A_1))^c$;

(ivH) $\aleph_\mathscr D ^{+1}(\mathscr A_1^c) = (\aleph_\mathscr D ^{-1}(\mathscr A_1))^c$;

(vL) $\aleph_\mathscr D^{-1}(\mathscr A_1) = \aleph_\mathscr D ^{-1}(\aleph_\mathscr D ^{-1}(\mathscr A_1))$;

(vH) $\aleph_\mathscr D ^{+1}(\mathscr A_1) = \aleph_\mathscr D ^{+1}(\aleph_\mathscr D ^{+1}(\mathscr A_1))$;

(viLH) $\aleph_\mathscr D ^{-1}(\mathscr A_1)\subseteq \mathscr A_1\subseteq \aleph_\mathscr D ^{+1}(\mathscr A_1)$.

Proof. Similar to Theorem 3.1.

3.1.   A simulation experiment for our model

Suppose that $\mathcal {U} = \{ \mho_1, \mho_2, \mho_3, \mho_4, \mho_5, \mho_6, \mho_7, \mho_8, \mho_9, \mho_{10}\}$ is a set of pilots. They are trained with respect to five attributes $\mathcal {A} = \{e_1, e_2, e_3, e_4, e_5\}$. They had been evaluated by an expert to determine whether they are sufficiently well trained according to these attributes or not, as shown in Table 3.

The core of soft neighborhood is: $\mathscr {CN}_s(\mho_1) = \mathscr {CN}_s(\mho_{10}) = \{\mho_1, \mho_{10}\}$, $\mathscr {CN}_s(\mho_2)$.

$= \mathscr {CN}_s(\mho_3) = \mathscr {CN}_s(\mho_9) = \{\mho_2, \mho_3, \mho_9\}$, $\mathscr {CN}_s(\mho_4) = \{\mho_4\}$, $\mathscr {CN}_s(\mho_5) = \mathscr {CN}_s(\mho_8) = \{\mho_5, \mho_8\}$, $\mathscr {CN}_s(\mho_6) = \{\mho_6\}$, $\mathscr {CN}_s(\mho_7) = \{\mho_7\}$.

Suppose $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8)$, $(\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7), (\mho_7, 0.9), (\mho_8, 0.3), (\mho_9, 0.9)$, $(\mho_{10}, 0.4)\}$

represents evaluation's degrees which are given by the expert. We can check the accuracy of this evaluation by our model which helps in decision making as follows:

$\aleph ^{-1}(\mathscr A_1) = \{(\mho_1, 0.1)$, $(\mho_2, 0.3), (\mho_3, 0.3), (\mho_4, 0.2), (\mho_5, 0.3), (\mho_6, 0.7), (\mho_7, 0.9)$, $(\mho_8, 0.3), (\mho_9, 0.9), (\mho_{10}, 0.1)\}$,

$\aleph ^{+1}(\mathscr A_1) = \{(\mho_1, 0.4), (\mho_2, 0.9), (\mho_3, 0.9)$, $(\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7), (\mho_7, 0.9), (\mho_8, 0.5), (\mho_9, 0.9)$, $(\mho_{10}, 0.4)\}$.

4.   New other two models of $\mathcal SCRF$ based on neighborhoods

We introduce new two models of $\mathcal SCRF$ based on merging core soft neighborhoods and soft neighborhoods. The second model of $\mathcal SCRF$ is denoted by $\mathcal SCRF-2$ and the third model is denoted by $\mathcal SCRF-3$.

4.1.   $\mathcal SCRF-2$-model

Definition 4.1. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$), then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$

$\aleph^{-2}(\mathscr A_1)(\mho_1) = {\wedge}\{\mathscr A_1(\mho_2):\mho_2\in (\mathscr {N}_s\cap \mathscr {CN}_s)(\mho_1)\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$ is called $\mathcal SFCLA-2$,

$\aleph^{+2}(\mathscr A_1)(\mho_1) = \vee \{\mathscr A_1(\mho_2):\mho_2\in (\mathscr {N}_s\cap \mathscr {CN}_s)(\mho_1)\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$ is called $\mathcal SFCUA-2$.

If $\aleph^{-2}(\mathscr A_1)\neq \aleph^{+1}(\mathscr A_1)$, then $\mathscr A_1$ is called $\mathcal SCRF-2$; otherwise, $\mathscr A_1$ is called definable.

Example 4.1 Continued from Example 3.1, $\mathscr {CN}_s (\mho_1)\cap \mathscr {N}_s(\mho_1) = \{\mho_1, \mho_2\}$, $\mathscr {CN}_s (\mho_2)\cap \mathscr {N}_s(\mho_2) = \{\mho_1, \mho_2\}$, $\mathscr {CN}_s (\mho_3)\cap \mathscr {N}_s(\mho_3) = \{\mho_3\}$, $\mathscr {CN}_s (\mho_4)\cap \mathscr {N}_s(\mho_4) = \{\mho_4\}$, $\mathscr {CN}_s (\mho_5)\cap \mathscr {N}_s(\mho_5) = \{\mho_5\}$, $\mathscr {CN}_s (\mho_6)\cap \mathscr {N}_s(\mho_6) = \{\mho_6\}$, $\mathscr {CN}_s (\mho_1)\cup \mathscr {N}_s(\mho_1) = \{\mho_1, \mho_2\}$, $\mathscr {CN}_s (\mho_2)\cup \mathscr {N}_s(\mho_2) = \{\mho_1, \mho_2\}$, $\mathscr {CN}_s (\mho_3)\cup \mathscr {N}_s(\mho_3) = \{\mho_3\}$, $\mathscr {CN}_s (\mho_4)\cup \mathscr {N}_s(\mho_4) = \{\mho_4, \mho_5\}$, $\mathscr {CN}_s (\mho_5)\cup \mathscr {N}_s(\mho_5) = \{\mho_5\}$, $\mathscr {CN}_s (\mho_6)\cup \mathscr {N}_s(\mho_6) = \{\mho_3, \mho_5, \mho_6\}$. Therefore, $\aleph^{-2}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, and $\aleph^{+2}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

Let us define the second type of soft measure degree $(\mathcal SMD-2)$ as follows.

Definition 4.2. Assume that $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mho_1, \mho_2\in \mathcal {{U}}$, then the $(\mathcal SMD-2)$ is defined as:

Clearly, $0\leq \mathscr D_s^2(\mho_1, \mho_2)\leq1$, $\mathscr D_s^2(\mho_1, \mho_2) = \mathscr D_s^2(\mho_2, \mho_1)$, and $\mathscr D_s^1(\mho_1, \mho_1) = 1$.

Example 4.2. According to Example 4.1, the values of $\mathcal SMD-2$ are shown in the following Table 4

Based on Definition 4.1, we define the second type of $SCRF$ based on $\lambda$-lower (upper) approximation ($\lambda-\mathcal SFCLA-2$ ($\lambda-\mathcal SFCUA-2$)) as follows.

Definition 4.3. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr D_s^2(\mho_1, \mho_2)$ is $(\mathcal SMD-2)$ for $\mho_1, \mho_2\in \mathcal {{U}}$, then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, the $\lambda-\mathcal SFCLA-2$ ($\lambda-\mathcal SFCUA-2$) is defined as follows, respectively:

$\aleph_\lambda^{-2}(\mathscr A_1)(\mho_1) = \wedge\{\mathscr A_1)(\mho_2):\mathscr D_s^2(\mho_1, \mho_2) > \lambda\}$,

$\aleph_\lambda^{+2}(\mathscr A_1)(\mho_1) = \vee\{\mathscr A_1)(\mho_2):\mathscr D_s^2(\mho_1, \mho_2) > \lambda\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$.

If $\aleph_\lambda^{-2}(\mathscr A_1)\neq \aleph_\lambda^{+2}(\mathscr A_1)$, then $\mathscr A_1$ is called $\lambda-\mathcal SCRF-2$; otherwise $\mathscr A_1$ is called definable.

Example 4.3. Consider Example 4.2 and $\mathscr A_1 = \{(\mho_1, 0.3), (\mho_2, 0.4), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, then for $\lambda = 0.5$, we have the following approximations operators:

$\aleph_\lambda^{-2}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$,

$\aleph_\lambda^{+2}(\mathscr A_1) = \{(\mho_1, 0.4), (\mho_2, 0.4), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

Definition 4.4. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr D_s^2(\mho_1, \mho_2)$ is $(\mathcal SMD-2)$ for $\mho_1, \mho_2\in \mathcal {{U}}$, then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, the second type of SFC $\mathscr D$-lower (upper) approximation $\mathscr D-\mathcal SFCLA-2$ ($\mathscr D-\mathcal SFCUA-2$) is defined as follows, respectively:

$\aleph_\mathscr D^{-2}(\mathscr A_1)(\mho_1) = \underset{\mho_2\in \mathcal {{U}}}{\wedge}\{(1-\mathscr D_s^2)(\mho_1, \mho_2)\vee \mathscr A_1(\mho_2)\}$,

$\aleph_\mathscr D^{+2}(\mathscr A_1)(\mho_1) = \underset{\mho_2\in \mathcal {{U}}}{\vee}\{\mathscr D_s^2(\mho_1, \mho_2)\wedge \mathscr A_1(\mho_2)\}$, $\forall \mho_1\in \mathcal {{U}}$.

Example 4.4. Consider Example 4.2 and $\mathscr A_1 = \{(\mho_1, 0.3), (\mho_2, 0.4), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, then we have the following approximations operators:

$\aleph_\mathscr D^{-2}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$

$\aleph_\mathscr D^{+2}(\mathscr A_1) = \{(\mho_1, 0.4), (\mho_2, 0.4), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

4.2.   $\mathcal SCRF-3$-model

Definition 4.5. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$), then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$:

$\aleph^{-3}(\mathscr A_1)(\mho_1) = {\wedge}\{\mathscr A_1(\mho_2):\mho_2\in (\mathscr {N}_s\cup \mathscr {CN}_s)(\mho_1)\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$ is called $\mathcal SFCLA-3$,

$\aleph^{+3}(\mathscr A_1)(\mho_1) = \vee \{\mathscr A_1(\mho_2):\mho_2\in (\mathscr {N}_s\cup \mathscr {CN}_s)(\mho_1)\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$ is called $\mathcal SFCUA-3$.

If $\aleph_\lambda^{-3}(\mathscr A_1)\neq \aleph_\lambda^{+3}(\mathscr A_1)$, then $\mathscr A_1$ is called $\lambda-\mathcal SCRF-3$; otherwise, $\mathscr A_1$ is called definable.

Example 4.5. From Example 4.1 and $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, we have the following approximations operators:

$\aleph^{-3}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.5)\}$,

$\aleph^{-3}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.5), (\mho_5, 0.5), (\mho_6, 0.8)\}.$

We define the third type of soft measure degree $(\mathcal SMD-3)$ as follows.

Definition 4.6. Assume that $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is $\mathcal SCAS$ and $\mho_1, \mho_2\in \mathcal {{U}}$, then the $(\mathcal SMD-3)$ is defined as:

Clearly, $0\leq \mathscr D_s^3(\mho_1, \mho_2)\leq1$, $\mathscr D_s^3(\mho_1, \mho_2) = \mathscr D_s^3(\mho_2, \mho_1)$, and $\mathscr D_s^3(\mho_1, \mho_1) = 1$.

Example 4.6. From Example 4.1, the values of $\mathcal SMD-3$ are shown in the following Table 5.

Based on Definition 4.6, we define the third type of SFC based on $\lambda$-lower(upper) approximation$\{ \lambda-\mathcal SFCLA-3$($\lambda-\mathcal SFCUA-3 )\}$ as follows.

Definition 4.7. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr D_s^3 (\mho_1, \mho_2)$ is $(\mathcal SMD-3)$ for $\mho_1, \mho_2\in \mathcal {{U}}$, then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, the $\{ \lambda-\mathcal SFCLA-3$($\lambda-\mathcal SFCUA-3 )\}$ is defined as:

$\aleph_\lambda^{-3}(\mathscr A_1)(\mho_1) = \wedge\{\mathscr A_1)(\mho_2):\mathscr D_s^3(\mho_1, \mho_2) > \lambda\}$,

$\aleph_\lambda^{+3}(\mathscr A_1)(\mho_1) = \vee\{\mathscr A_1)(\mho_2):\mathscr D_s^3(\mho_1, \mho_2) > \lambda\}$, $\forall \mho_1, \mho_2\in \mathcal {{U}}$.

If $\aleph_\lambda^{-3}(\mathscr A_1)\neq \aleph_\lambda^{+3}(\mathscr A_1)$, then $\mathscr A_1$ is called $\lambda-\mathcal SCRF-3$; otherwise, $\mathscr A_1$ is called definable.

Example 4.7. From Example 4.6 and $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, then for $\lambda = 0.2$, we have the following approximations operators:

$\aleph_\lambda^{-2}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.2), (\mho_6, 0.2)\}$,

$\aleph_\lambda^{+2}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.7), (\mho_5, 0.7), (\mho_6, 0.7)\}$.

Definition 4.8. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr D_s^3(\mho_1, \mho_2)$ is $(\mathcal SMD-3)$ for $\mho_1, \mho_2\in \mathcal {{U}}$, then for $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, the third type of SFC $\mathscr D$-lower (upper) approximation (($\mathscr D-\mathcal SFCLA-3$), ($\mathscr D-\mathcal SFCUA-3$)) is defined as follows, respectively:

$\aleph_\mathscr D^{-3}(\mathscr A_1)(\mho_1) = \underset{\mho_2\in \mathcal {{U}}}{\wedge}\{(1-\mathscr D_s^3)(\mho_1, \mho_2)\vee \mathscr A_1(\mho_2)\}$,

$\aleph_\mathscr D^{+3}(\mathscr A_1)(\mho_1) = \underset{\mho_2\in \mathcal {{U}}}{\vee}\{\mathscr D_s^3(\mho_1, \mho_2)\wedge \mathscr A_1(\mho_2)\}$, $\forall \mho_1\in \mathcal {{U}}$.

Example 4.8. Consider Example 4.6 and $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, then, we have the following approximations operators:

$\aleph_\mathscr D^{-3}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$,

$\aleph_\mathscr D^{+3}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.7), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

4.3.   Comparison between our $\mathcal SCRF$-models and Zhan's model

We set forth the relationship between our proposed $\mathcal SCRF$-models and Zhan's model for soft rough fuzzy approximation structure.

Theorem 4.1. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, then the following axioms are satisfied:

(iL) $\aleph^{-3}(\mathscr A_1)\subseteq \aleph^{-1}(\mathscr A_1)\subseteq \aleph^{-2}(\mathscr A_1)$;

(iH) $\aleph^{-3}(\mathscr A_1)\subseteq \aleph^{-0}(\mathscr A_1)\subseteq \aleph^{-2}(\mathscr A_1)$;

(iiL) $\aleph^{+2}(\mathscr A_1)\subseteq \aleph^{+1}(\mathscr A_1)\subseteq \aleph^{+3}(\mathscr A_1);$

(iiH) $\aleph ^{+2}(\mathscr A_1)\subseteq \aleph ^{+0}(\mathscr A_1)\subseteq \aleph ^{+3}(\mathscr A_1)$.

Proof. The proof comes from Definitions 3.2, 4.1, and 4.5.

Remark 4.1. From the previous theorem, the lower approximation of our model $\aleph^{-2}$ is bigger than Zhan's model $\aleph^{-0}$ while the upper approximation of our model $\aleph^{+2}$ is less than Zhan's model $\aleph^{+0}$. This leads to decreasing the boundary region and makes the soft rough fuzzy set more accurate in solving the uncertainty issues.

Example 4.9. If $\mathscr A_1 = \{(\mho_1, 0.1), (\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$, then by Example 3.1, $\aleph^{-0}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1)$, $(\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.5)\}$, $\aleph^{+0}(\mathscr A_1) = \{(\mho_1, 0.3), (\mho_2, 0.3)$, $(\mho_3, 0.8), (\mho_4, 0.5), (\mho_5, 0.5), (\mho_6, 0.8)\}$. $\aleph^{-2}(\mathscr A_1) = \{(\mho_1, 0.1), (\mho_2, 0.1), (\mho_3, 0.8), (\mho_4, 0.2)$, $(\mho_5, 0.5), (\mho_6, 0.7)\}$ and $\aleph^{+2}(\mathscr A_1) = \{(\mho_1, 0.3)$, $(\mho_2, 0.3), (\mho_3, 0.8), (\mho_4, 0.2), (\mho_5, 0.5), (\mho_6, 0.7)\}$.

Theorem 4.2. If $\mathcal S = (\mathcal {{U}}, \mathcal \mho_G)$ is ($\mathcal SCAS$) and $\mathscr A_1\in \mathscr F(\mathcal {{U}})$, then the following properties are satisfied:

(iL) $\aleph^{-2}(\mathscr A_1) = \aleph^{-0}(\mathscr A_1)\cup \aleph^{-1}(\mathscr A_1)$;

(iH) $\aleph^{+2}(\mathscr A_1) = \aleph^{-0}(\mathscr A_1)\cap \aleph^{-1}(\mathscr A_1)$;

(iiL) $\aleph^{-3}(\mathscr A_1) = \aleph^{-0}(\mathscr A_1)\cap \aleph^{-1}(\mathscr A_1)$

(iiH) $\aleph^{+3}(\mathscr A_1) = \aleph^{-0}(\mathscr A_1)\cup \aleph^{-1}(\mathscr A_1)$.

Proof. The proof is straightforward.

5.   Conclusions

Zhan's model for soft rough fuzzy covering stands as an innovative approach, though further exploration and refinement are warranted. Similarly, the exploration of soft covering-based rough fuzzy sets opens avenues for the integration of various mathematical structures. We introduced a combination of soft sets, fuzzy sets and rough sets. Three models of the approximation of $\mathcal SRFS$- based covering are presented. We deduced that our approximation is more refined than Zhan's model as we decreased the boundary region. The integration of soft sets with fuzzy logic in soft fuzzy covering and the discernment of upper approximation provide additional layers to the $\mathcal SCRF$ framework, offering a comprehensive solution to complex problem structures.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgement

This research is funded by Zarqa University-Jordan.

Conflict of interest

The authors declare no conflict of interest.