Roughness in S_|U| -Submodules

Numerous mathematical concepts are delivered exclusively using set theory as a key method of presenting all of mathematics. Rough sets were introduced by Pawlak [1]. With the help of lower and upper approximations, this theory can approximate a subset of a universe. The two main approaches for developing rough set theory are constructive and axiomatic. Using constructive methods, primitive notions, such as binary relations on universes, partitions of universes and neighborhood systems, are used to construct lower and upper approximation operators. In contrast, the axiomatic approach focuses on the primitive notions of upper and lower approximation operators, which are appropriate for examining rough set algebras.

An algebraic structure provides a rigorous framework for analyzing and manipulating rough sets by formalizing operations and relationships. The concept of an approximation space is a fundamental algebraic structure in rough set theory, which consists of a universe of discourse, attributes, and binary relations that define the indiscernibility of objects. Furthermore, algebraic structures, including semigroups, monoids, and groups, have been employed in the study of rough sets, particularly within the context of algebraic approaches. By means of these algebraic systems, it becomes possible to study the properties and relationships of rough sets and their connections with other mathematical ideas. Thus, the algebraic structures of rough set theory provide a useful mathematical framework for studying the properties, relationships, and operations within rough sets, offering insights into the nature of uncertainty and approximation in data analysis and knowledge discovery.

The algebraic aspects of rough set theory are proposed by Bonikowski [2]. The structures of the lower and upper approximations based on arbitrary binary relations was given by Liu and Zhu [3]. Rough sets and their properties were applied to rings, modules, semi-groups, groups, ideals, and graphs [4-8]

According to Pomykala and Pomykala [9], rough sets form Stone algebra. Comer [10] discussed several algebras related to algebraic logic, including stone algebras and relation algebras. By considering the upper approximation, Biswas and Nanda [11] gave the concept rough subgroups. A rough ideal into a semigroup was introduced by Kuroki [12] in 1997, containing rough left and right ideals with appropriate examples. With respect to normal subgroups Kuroki and Wang [13] explored lower and upper approximations. The roughness of gamma subsemigroups and ideals in gamma-semigroups were discussed by Jun [14]. In addition, rough ideals are discussed as a generalization of ideals in BCK-algebras by Jun [15].

A topological approach was given by Al-Shami [16] to generate new rough set models. Through ideals, Guler et al. [17] provided rough approximations based on different topologies. The concept of generalized rough approximation spaces based on maximal neighborhoods and ideals was discussed by Hosny et al. [18]. A power set approximation operator for a given set was defined by Mordeson in 2001 [19] using covers of the universal set. The roughness based on fuzzy ideals given by Davvaz [20]. An introduction to rough prime ideals and rough fuzzy prime ideals in a semigroup was made by Xiao and Zhang in 2006 [21]. Sangeetha and Sathish [22] defined rough groups using upper and lower approximations to rough sets within a finite universe. Bağırmaz et al. [23] introduced the notion of topological rough groups, and Altassan et al. [24] introduced rough action on topological rough groups.

In this paper, we present a rough action by a symmetric group $S_{|U|}$ acting on all rough sets in an approximation space with a finite universe. Moreover, we prove that number of $S_{|U|}$ orbits in rough sets is 1. This led us to introduce a rough $S_{|U|}$ submodules and prove results related to some homomorphisms. We have also provided an example of using rough action to find missing values in cancer sample data.

A brief review of rough set theory and rough groups is provided in Section 2 of this paper. In Section 3, we presented rough actions and their properties along with suitable examples. Our discussion of rough $S_{|U|}$ submodules cover Section 4. An example of rough action is provided in Section 5. Conclusion explains the relevance of this work.

This section introduces rough action, rough stabilizer, rough orbit, and rough homomorphism.

Definition 3.1 [25] G-Set

Let $G$ be a group and $J$ be any set. $G$ acts on $J$ if : $G \times J \rightarrow$ $J$ given by $(g, j) \rightarrow g \cdot j \in J$, also,

$e \cdot j=j, \forall j \in J ; g \cdot(h \cdot j)=(g \cdot h) \cdot j$

Then $J$ is said to be $G$-Set.

Definition 3.2

Let $(U, R)$ be an approximation space. $S_{|U|}$ be symmetric group on $|U|$ elements. $R o(U)$ be the set of all rough sets. We say $S_{|U|}$ acts on $R o(U)$ if: $S_{|U|} \times R o(U) \rightarrow R o(U)$ where $(\sigma, W) \rightarrow \sigma\{W\}=\left\{\sigma\left(w_n\right)\right\} \in R o(U)$, such that:

$\begin{gathered}e \cdot A_i=A_i, \forall A_i \in \operatorname{Ro}(U) \\ \sigma_1 \cdot\left(\sigma_2 \cdot A_i\right)=\left(\sigma_1 \sigma_2\right) \cdot A_i\end{gathered}$

Definition 3.3 Rough $\boldsymbol{\sigma}$-Stabilizer

$R o\left(U_1\right)$, set of all rough sets in $U_1$ with respect to $R_1$ and $S_{|U|}$ acts on $\operatorname{Ro}\left(U_1\right)$. Let $X_n \in R o\left(U_1\right)$ then $X^\sigma=\left\{X_n \in{Ro}\left(U_1\right) \mid \sigma\left(x_n\right)=x_n, \forall x_n \in X_n\right\}$.

Definition 3.4

If $W \in \operatorname{Ro}\left(U_1\right) \quad$ then, $\quad \sum_{g \in S_{\left|U_1\right|}}\left|W^g\right|=\left|S_{\left|U_1\right|}\right| . \quad$ As demonstrated in the following example, this result is true.

Example 3.1

Given $U_1=\{1,2,3\}$ and $R$ be any equivalence relation on $U_1$.

Equivalence class of $U_1$ with respect to $R$ is given by:

$\begin{gathered}U_1 / R=\{\{1,2\},\{3\}\} \\ 2^{U_1}=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}, \phi\}\end{gathered}$

represented as $X_i$, where $i=1,2,3, \ldots 8$.

The rough sets are given by $\operatorname{Ro}(U)=\left\{X_1, X_2, X_5, X_6\right\}$, where,

$X_1=\{1\}, X_2=\{2\}, X_5=\{1,3\}, X_6=\{2,3\}$

$\bullet$ $R^{X_1}=\{1,2\} \& R_{X_1}=\phi$

$\bullet$ $R^{X_2}=\{1,2\} \& R_{X_2}=\phi$

$\bullet$ $R^{X_5}=\{1,2,3\} \& R_{X_5}=\{3\}$

$\bullet$ $R^{X_6}=\{1,2,3\} \& R_{X_6}=\{3\}$

$\overline{R(U)} \cup \underline{R(U)}=\{\phi,\{3\},\{1,2\},\{1,2,3\}\}$

where,

$\begin{gathered}R(U)=\{\{1\},\{2\},\{1,3\},\{2,3\}\}, \\ \overline{R(U)}=\{\{1,2\},\{1,2,3\}\} \\ \underline{R(U)}=\{\phi,\{3\}\}\end{gathered}$

$(\overline{R(U)} \cup \underline{R(U)}, \triangle)$ forms subgroup of $\left(2^{|U|}, \triangle\right)$.

Table 1. Cayley table of $R\left(U_1\right)$

Hence $R(U)$ is a rough group (Table 1).

Define $\sigma \cdot X_n=1$, if $\sigma \cdot x=x, \forall x \in X_n$ otherwise 0 (Table 2).

Table 2. Rough action on R(U₁)

$\bullet$ $X^e=\left\{X_1, X_2, X_6, X_7\right\}$

$\bullet$  $X^{\sigma_1}=\phi$

$\bullet$ $X^{\sigma_2}=\left\{X_2\right\}$

$\bullet$ $X^{\sigma_3}=\left\{X_1\right\}$

$\bullet$ $X^{\sigma_4}=\phi, X^{\sigma_5}=\phi$

$\bullet$ $\sum_{g \in S_{|U|}}\left|X^g\right|=6=S_{|U|}$​

Then we have the following result: The number of $S_{|U|}$ orbits in $R(U)$ is 1 since:

$\frac{\sum_{g \in S_{|U|}\left|X^g\right|}}{S_{|U|}}=\frac{6}{6}=1$

Example 3.2

Let $U=\{1,2,3,4\}, R$, an equivalence on $U$.

$U / R=\{\{1,2\},\{3\},\{4\}\}$

The possible subsets of $U$ are {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}, {} represented as $W_i$ where i=1,2,…16.

The upper and lower approximation of $W_i$ are given by:

$\begin{gathered}R^{W_1}=\{1,2\} \& R_{W_1}=\{\} \\ R^{W_2}=\{1,2\} \& R_{W_2}=\{\} \\ R^{W_3}=\{3\} \& R_{W_3}=\{3\} \\ R^{W_4}=\{4\} \& R_{W_4}=\{4\} \\ R^{W_5}=\{1.2\} \& R_{W_5}=\{1,2\} \\ R^{W_6}=\{1,2,3\} \& R_{W_6}=\{3\} \\ R^{W_7}=\{1,2,4\} \& R_{W_7}=\{4\} \\ R^{W_8}=\{1,2,3\} \& R_{W_8}=\{3\} \\ R^{W_9}=\{1,2,4\} \& R_{W_1}=\{4\} \\ R^{W_{10}}=\{3,4\} \& R_{W_{10}}=\{3,4\} \\ R^{W_{11}}=\{1,2,3\} \& R_{W_1}=\{1,2,3\} \\ R^{W_{12}}=\{1,2,4\} \& R_{W_{12}}=\{1,2,4\} \\ R^{W_{13}}=\{1,2,3,4\} \& R_{W_{13}}=\{3,4\} \\ R^{W_{14}}=\{1,2,3,4\} \& R_{W_{14}}=\{3,4\} \\ R^{W_{15}}=\{1,2,3,4\} \& R_{W_{15}}=\{1,2,3,4\} \\ R^{W_{16}}=\{\} \& R_{W_{13}}=\{\}\end{gathered}$

The Rough sets are given by:

$\begin{gathered}R o(U)=\left\{W_1, W_2, W_6, W_7, W_8, W_9, W_{13}, W_{14}\right\} \\ W_1=\{1\}, W_2=\{2\}, W_6=\{1,3\}, W_7=\{1,4\} \\ W_8=\{1,2,3\}, W_9=\{2,4\}, W_{13}=\{1,3,4\}, W_{14}=\{2,3,4\} \\ \overline{R(U)}=\{\{1,2\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\} \\ \underline{R(U)}=\{\phi,\{3\},\{4\},\{3,4\}\} \\ \overline{R(U)} \cup \underline{R(U)}=\{\phi,\{3\},\{4\},\{3,4\},\{1,2\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\}\}\end{gathered}$

$(\overline{R(U)} \cup \underline{R(U)}, \triangle)$ forms subgroup of $\left(2^{|U|}, \triangle\right)$ (Table 3).

Table 3. Cayley table of rough group $R\left(U_2\right)$

Define $\sigma . W_n=s$, if $\sigma \cdot x=x, \forall x \in W_n$, otherwise (Table $4)$.

Table 4. Rough action on $R\left(U_2\right)$

$W^e=\left\{W_1, W_2, W_6, W_7, W_8, W_9, W_{13}, W_{14}\right\}$

$W^{\operatorname{per}_1}, W^{\operatorname{per}_3}, W^{\operatorname{per}_4}, W^{\text {per }_5}$,$W^{\operatorname{per}_6}, W^{\operatorname{per}_7}, W^{\operatorname{per}_{10}}, W^{\text {per }_{12}}$,$W^{\operatorname{per}_{14}}, W^{\operatorname{per}_{16}}, W^{\operatorname{per}_{22}}  \& W^{\text {per }_{23}}$,$=\phi$

$W^{\operatorname{per}_2}=\left\{W_2\right\}$, $W^{\operatorname{per}_9}=\left\{W_2\right\}$, $W^{\operatorname{per}_{11}}=\left\{W_1\right\}$, $W^{\operatorname{per}_{13}}=\left\{W_2, W_8\right\}$, $W^{\operatorname{per}_{15}}=\left\{W_1\right\}$, $W^{\operatorname{per}_{17}}=\left\{W_1\right\}$,

$W^{p e r_{18}}=\left\{W_1, W_2, W_7\right\}$, $W^{\operatorname{per}_{19}}=\left\{W_1, W_6\right\}$, $W^{\operatorname{per}_{20}}=\left\{W_2, W_8\right\}$, $W^{p e r_{21}}=\left\{W_2, W_9\right\}$

$\sum_{g \in S_{|U|}}\left|X^g\right|=24=S_{|U|}$

Theorem 3.1

Every action of $S_{|U|}$ on $R o(U)$ induces a homomorphism from $S_{|U|} \rightarrow \operatorname{Sym}(R o(U))$

Proof:

Let $S_{|U|}$ acts on $\operatorname{Ro}(U), .: S_{|U|} \times \operatorname{Ro}(U) \rightarrow \operatorname{Ro}(U)$, where $(\sigma, W) \rightarrow \sigma\{W\} \in \operatorname{Ro}(U)$ such that: $e . W=W, \forall W \in \operatorname{Ro}(U) ; \sigma_1 \cdot\left(\sigma_2 \cdot W\right)=\left(\sigma_1 \sigma_2\right) . W$.

Consider the map $\phi: S_{|U|} \rightarrow \operatorname{Sym}(\operatorname{Ro}(U))$, where, $\operatorname{Sym}(\operatorname{Ro}(U)): \operatorname{Ro}(U) \rightarrow \operatorname{Ro}(U)$ is defined as:

$\begin{gathered}\Sigma_\sigma(W)=\sigma(W) \\ \text { Let } \Sigma_\sigma(X)=\Sigma_\sigma(Y) \Rightarrow \sigma\{X\}=\sigma\{Y\} \\ \left\{\sigma\left(x_n\right)\right\}=\left\{\sigma\left(y_n\right)\right\}, \forall x_n \in X \& \forall y_n \in Y \\ \sigma^{-1}\left\{\sigma\left(x_n\right)\right\}=\sigma^{-1}\left\{\sigma\left(y_n\right)\right\},\left\{\sigma^{-1} \sigma\left(x_n\right)\right\}=\left\{\sigma^{-1} \sigma\left(y_n\right)\right\} \\ X=Y \Rightarrow \text { is } 1-1 . \\ \forall Y \in R o(U), \exists \sigma^{-1}(\{Y\}) \in R o(U) \text { such that } \\ \Sigma_\sigma \sigma^{-1}(\{Y\})=\sigma \sigma^{-1}(\{Y\})=Y, \Rightarrow \text {. is onto }\end{gathered}$

Hence $\Sigma_\sigma \in \operatorname{Sym}(\operatorname{Ro}(U))$.

Define $\phi: S_{|U|} \rightarrow \operatorname{Sym}(\operatorname{Ro}(U))$ as $\phi\left(\sigma_1 \sigma_2\right)=\Sigma_{\sigma_1 \sigma_2}$, $\Sigma_{\sigma_1 \sigma_2}(X)=\left\{\sigma_1 \sigma_2(X)\right\}=\sigma_1\left\{\sigma_2(X)\right\}=\Sigma_{\sigma_1} \Sigma_{\sigma_2}$.

Conversely,

Let $\phi: S_{|U|} \rightarrow$ Sym $(\operatorname{Ro}(U))$ be a homomorphism where $\phi(\sigma)=\Sigma_\sigma$.

Define.$: S_{|U|} \times \operatorname{Ro}(U) \rightarrow R o(U)$ as: $(\sigma, W)=\Sigma_\sigma(W)$

It defines a rough action since:

$\begin{gathered}(e, W)=\Sigma_e(W)=e(W)=W \\ 5 \sigma_1\left(\sigma_2(W)\right)=\sigma_1\left(\Sigma_{\sigma_2}(W)\right) \\ \Sigma_{\sigma_1} \cdot \Sigma_{\sigma_2}(W)=\Sigma_{\sigma_1 \sigma_2}=\sigma_1 \sigma_2(W)\end{gathered}$

So, . defines the R Action.

Definition 3.5 Stabilizer of R Action

Approximation space is composed of a finite set $U \neq \phi$, has $n$ elements. Let $S_{|U|}$ be symmetric group on $\mathrm{n}$ elements and $\operatorname{Ro}(U)$ collection of all rough sets. If $S_{|U|}$ acts on $\operatorname{Ro}(U)$ then

$\boldsymbol{s t a b}()=.\left\{\sigma \in \boldsymbol{S}_{|U|} \mid \sigma\left(x_n\right)=x_n, \forall x_n \in X, \forall X \in \boldsymbol{R o}(U)\right\}$

from above examples we have stab(.) $=\{e\}$.

So, in an $R$ Action, stab(.) is a trivial subgroup of $S_{|U|}$.

Result 3.1

If $S_{|U|}$ acts on $R o(U)$ then $\sigma \cdot \underline{X} \subseteq \underline{\sigma \cdot X}$ and $\overline{\sigma \cdot X} \subseteq \sigma \cdot \bar{X}$

Example 3.3

Given $U_1=\{1,2,3\}$ and $R$ be any equivalence relation on $U_1$.

Equivalence class of $U_1$ with respect to $R$ is given by:

$\begin{gathered}U_1 / R=\{\{1,2\},\{3\}\} \\ 2^{U_1}=\{\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}, \phi\}\end{gathered}$

represented as $X_i$, where $i=1,2,3, \ldots 8$. The rough sets are given by $R o(U)=$ $\left\{X_1, X_2, X_5, X_6\right\}$,

$X_1=\{1\}, X_2=\{2\}, X_5=\{1,3\}, X_6=\{2,3\}$

$\bullet$ $R^{X_1}=\{1,2\} \& R_{X_1}=\phi$

$\bullet$ $R^{X_2}=\{1,2\} \& R_{X_2}=\phi$

$\bullet$ $R^{X_5}=\{1,2,3\} \& R_{X_5}=\{3\}$

$\bullet$ $R^{X_6}=\{1,2,3\} \& R_{X_6}=\{3\}$$S_3=\{e,(12),(13),(23),(123),(132)\}$

Represented as $\sigma_0, \sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5$. Consider $X_6=\{2,3\}$ where $\underline{X_6}=\{3\} \& \overline{X_6}=\{1,2,3\}$.

$\begin{gathered}\sigma_0 \cdot \underline{X_6}=\{3\}, \sigma_0 \cdot \overline{X_6}=\{1,2,3\} \\ \sigma_1 \cdot \underline{X_6}=\{3\}, \sigma_1 \cdot \overline{X_6}=\{1,2,3\} \\ \sigma_2 \cdot \underline{X_6}=\{1\}, \sigma_2 \cdot \overline{X_6}=\{1,2,3\} \\ \sigma_3 \cdot \underline{X_6}=\{2\}, \sigma_3 \cdot \overline{X_6}=\{1,2,3\} \\ \sigma_4 \cdot \underline{X_6}=\{1\}, \sigma_4 \cdot \overline{X_6}=\{1,2,3\} \\ \sigma_5 \cdot \underline{X_6}=\{2\}, \sigma_5 \cdot \overline{X_6}=\{1,2,3\} \\ \underline{\sigma_0 \cdot X_6}=\{2,3\}, \overline{\sigma_0 \cdot X_6}=\{1,2,3\} \\ \underline{\sigma_1 \cdot X_6}=\{1,3\}, \overline{\sigma_1 \cdot X_6}=\{1,2,3\} \\ \underline{\sigma_2 \cdot X_6}=\{2,1\}, \overline{\sigma_2 \cdot X_6}=\{1,2,3\} \\ \underline{\sigma_3 \cdot X_6}=\{3,2\}, \overline{\sigma_3 \cdot X_6}=\{1,2\} \\ \underline{\sigma_4 \cdot X_6}=\{3,1\}, \overline{\sigma_4 \cdot X_6}=\{1,2,3\} \\ \underline{\sigma_5 \cdot X_6}=\{1,2\}, \overline{\sigma_5 \cdot X_6}=\{1,2\}\end{gathered}$

From above, $\sigma_i . \underline{X_6} \subseteq \underline{\sigma_i . X_6}$, and $\overline{\sigma_i \cdot X_6} \subseteq \sigma_i \cdot \overline{X_6}$ for all $i=0,1,2,3,4,5$.

A rough action example is given in this section to find missing cancer data values.

Example 5.1

Consider a finite universe with five patients and attributes Age, Gender, Treatment, and Survival Status with missing values in Age and Gender. The missing values are determined by applying a rough group action rule as follows in Table 5.

Table 5. Given information system with missing values

The following decision are the decision rules with respect to AGE attribute:

• If Age≤45 & Gender “M” & Treatment “Surgery” then Survival Status is “Alive.”
• If Age≤60 & Treatment “Chemotherapy” then Survival Status is “Alive.”
• If Age≤40 & Gender “F” & Treatment “Surgery” then Survival Status is “Alive.”

Rough Group Action Rule

For each patient, if the “Age” attribute is missing, the “Treatment” attribute will be changed to “Surgery” If the “Gender” attribute is missing, the “Treatment” attribute will be changed to “Chemotherapy” as shown in Table 6.

Table 6. Rough action on information system

Using the above decision rules, P2 will be of age >60 P3 Gender will be “F” & P5 Age will be >45 (Table 7).

Table 7. Information system without missing values

As a result, we are able to find the missing values by using the Rough Group Action rule.