New types of soft rough sets in groups based on normal soft groups

Abstract

Hybridization of soft sets and rough sets is an important way to deal with uncertainties. This paper aims to study the concept of roughness in soft sets over groups. In this regard, a pair of two soft sets, viz. soft lower and soft upper approximation spaces, are introduced by applying the normal soft groups corresponding to each parameter. Some important results related to these soft approximation spaces over groups are studied with examples. Furthermore, this paper presents a relationship between the soft approximation spaces based on the soft image and soft pre-image of a normal soft group via group homomorphisms. This work can be applicable in the field of information technology to connect two information systems.

1 Introduction

In our modern society, handling of imprecision and uncertainties in the collected data is growing rapidly, especially in the fields of computer sciences, medical sciences and environmental sciences. Scientists and practitioners have proposed various useful mathematical theories to deal with such kind of problems, namely fuzzy set theory (Zadeh 1965), intuitionistic fuzzy set theory, probability theory, rough set theory (Pawlak 1982) and soft set theory (Molodtsov 1999).

One of the most successful theory amongst aforementioned theories is fuzzy set theory, proposed by Zadeh (1965). Fuzzy set theory is a generalization of set theory. In set theory, a set is uniquely defined by its elements, i.e. an element is either a member of a set or not. So, there is a membership function describing the belongingness of elements of the universe to the set. This function can attain only one value, either 0 or 1. For example, a man may be young or old, an object is expensive or cheap, a painting is beautiful or not, etc. In fuzzy set theory, membership function assigns the grade of membership to the elements of the universe in the unit interval [0, 1]. Let us take a painting as an illustration. In real world, one cannot classify all the paintings into two distinct classes, i.e. beautiful or not. Some paintings cannot be judged whether they are beautiful or not. Thus they remain in doubtful area. Similarly, we cannot say confidently that either a person is ill or not. Because a person’s disease may not be on its initial or last stage. In fuzzy set theory, a person who is very sick could have the degree of sickness near to 0.89. On contrary, a person who is having degree of sickness 0.12 indicates that he has nearly recovered from illness. Likewise, a painting having degree of beauty near to 1 representing that a painting is very beautiful and the one having degree of 0.2 indicates that a painting is somehow beautiful. However, assigning the grade of membership is also a problem sometimes.

The concept of rough set theory was given by a computer scientist Pawlak (1982) in 1982 to deal with granularity in information systems. Rough set theory is an extension of classical set theory which is defined by means of two precise sets instead of membership function. These sets are called the lower and upper approximations. These approximations are utilized in extraction of useful information hidden in data. Rough set theory provides us very simple algorithms to characterize the original objects having the same value of attributes in an information system. Rough set theory is based on the assumption that we have some additional information (data) about the elements of a set. Consider, as an example, a group of some patients suffering from malaria. To diagnose malaria, one must see various symptoms, e.g. headache, fever, fatigue, muscle pain, back pain, chills, sweating, dry cough, nausea and vomiting. The patients revealing the same symptoms are indiscernible with respect to the available information and form elementary granules of knowledge. Similarly, two acids with pH levels of 4.12 and 4.53 will, in many contexts, be perceived as so equally weak, that they are indiscernible with respect to this attribute. They are part of a rough set “weak acids” as compared to “strong” or “medium” or whatsoever other categories are relevant in this context.

Primarily, rough set theory is used to reveal useful information from an information system (i.e. data table, where columns are labeled by attributes and rows are labeled by objects) based on the indiscernibility relation which is an equivalence relation. The classes obtained from the indiscernibility relation containing similar elements in view of available information are the fundamental blocks of knowledge about the objects of the universe. The lower approximation of a set X is the set of all elements that surely belongs to the set X, whereas the upper approximation of a set X is the set of all elements that possibly belongs to X. Thus, based on the lower approximation certain information can be derived, while by using upper approximation partially certain information may be derived. The difference of the lower and upper approximation spaces of X is its boundary region. A set is rough if its boundary region is non-empty; otherwise, it is a crisp set.

The theory of rough sets has been successfully implemented to various complicated problems in computer sciences, cognitive sciences, data analysis, conflict analysis, artificial intelligence and machine learning. Since rough set theory is a new emerging theory, many mathematicians investigated interesting and useful connections between rough set theory and various algebraic structures (see Ali et al. 2012; Davvaz and Mahdavipour 2006; Kuroki and Wang 1996; Kuroki 1997; Zhan et al. 2015). At present, this work is mainly concerned with the roughness in a special algebraic structure groups. Foremost, Biswas and Nanda (1994) initiated the study of roughness in groups and introduced the concept of rough subgroups. However, the notion of rough subgroups provided by Biswas and Nanda (1994) depends only on the lower approximation space. Keeping this point in mind, Kuroki and Wang (1996) proposed the notion of lower and upper approximation spaces in groups based on normal subgroups and normal fuzzy subgroups. Later, some improvements of Kuroki and Wang’s work Kuroki and Wang (1996) have been proposed by Wang and Chen (2010), Wang and Shu (2012), Cheng et al. (2007). In Mahmood et al. (2018) studied the notion of roughness in fuzzy subgroups based on a congruence relation and explored a relationship between the approximation spaces of two different groups by utilizing a group homomorphism. The same authors Mahmood et al. (2017) investigated the concept of roughness in quotient groups and established several homomorphism theorems. In Chen et al. (2019), investigated the notion of rough modules of fractions and established a connection among the approximation spaces by maneuvering the module homomorphisms of two different modules of fractions.

In 1999, Molodtsov (1999) introduced the concept of soft set theory as a different approach from existing methods (Zadeh 1965; Pawlak 1982). A soft set is a family of subsets of the universe corresponding to each parameter, that is, a parameterized family of the subsets of the universe. Hence, it is also a generalization of the classical set theory. In this theory, the membership of objects is described by the suitable parameters. It has broad applications in various fields including game theory, probability theory, operational research and smoothness of functions. There is a detailed study on the fundamental properties and operations on soft set theory by various authors (see Aktaş and Çağman 2007; Ali et al. 2009; Çağman et al. 2012). Due to the theoretical enrichment of soft set theory, many researchers have found its applications in different algebraic systems. For instance, Aktaş and Çağman (2007) was the first to initiate the study of soft groups and have shown that fuzzy groups are special instance of soft groups. Later on, among many others, Sezgin and Atagun (2011) and Aslam and Qurashi (2012) have made some contributions to soft groups and introduced the notion of normal soft groups. In Feng et al. (2013), Feng et al. investigated the notion of soft semigroups using soft relations. Furthermore, Feng et al. (2008) investigated the notion of soft semirings and studied some of its related properties. In Feng and Li (2013), Feng et al. studied various types of soft subsets and soft relations. They also proved some important characterizations of different soft subsets. Çağman et al. (2012) studied the notion of soft intersection groups and carried a detailed theoretical study on soft intersection groups.

However, these theories are not distinct and can be unified in a constructive manner. In recent years, the theories of rough sets Pawlak (1982), soft sets Molodtsov (1999) and fuzzy sets Zadeh (1965) are important mathematical tools to cope with the issue of uncertainty. There is a large number of researchers who have investigated a relationship between them (see Feng et al. 2010; Moinuddin 2017; Shabir et al. 2013; Maji et al. 2001; Li et al. 2012). Applications of L-fuzzy soft sets Li et al. (2012) can be found in Shabir et al. (2017a), Shabir et al. (2017b), Mahmood et al. (2017), Ali and Shabir and Samina (2014). Among many others, there is a large number of authors who investigated the concept of soft rough sets based on the idea of rough fuzzy sets Dubios and Prade (1990), soft rough sets Feng et al. (2010) and modified soft rough sets (MSR-sets) Shabir et al. (2013) in groups (see Pan and Zhan 2016, 2017; Ghosh and Samanta 2013). Moreover, the authors in Ayub et al. (2019) introduced the concept of roughness in soft sets and found its applications in soft-intersection groups. Furthermore, they have established a connection between the soft approximation spaces via group homomorphism. In addition, Jiang et al. (2019), proposed four types of (I,T)-fuzzy rough set models and used them to solve a multi-attribute method decision making problems. In Ma et al. (2018), Ma et al. proposed two different methods to solve decision making problems using soft rough fuzzy sets (SRF-sets) and soft fuzzy rough sets (SFR-sets). Zhan et al. (2019) introduced covering-based multi-granulation(I,T)-fuzzy rough set models and gave some of its applications in multi-attribute group decision-making. Zhan and Xu investigated two types of coverings based multi-granulation rough fuzzy sets and suggested their applications to decision making (Zhan and Xu 2018). Zhang et al. (2019) investigated some new types of fuzzy soft $\beta -$coverings-based fuzzy rough sets and applied them to solve multi-criteria fuzzy group decision making problems. In Zhang et al. (2019), Zhang et al. established some covering-based general multi-granulation intuitionistic fuzzy rough sets and found its applications in multi-attribute group decision-making. Moreover, Zhang et al. (2019) proposed a novel approach to fuzzy rough set models and applied them to solve multi-criteria decision-making problems. Covering-based generalized IF-rough sets and their applications in multi-attribute decision-making have been investigated by Zhang et al. (2019). Later on, Zhang et al. (2019) introduced a topsis method based on a fuzzy covering approximation space and gave its application in biological nano-materials selection.

In the present paper, a new kind of soft rough sets are investigated over groups based on a normal soft group. The rest of paper is organized as follows: Sect. 2 consisting of some basic facts about soft sets, soft groups and normal soft groups. In Sect. 3, the concept of soft rough sets is introduced over groups by utilizing the concept of normal soft groups. Some important properties related to the soft lower and upper approximation spaces are proved with illustrative examples. In Sect. 4, connections between the soft approximation spaces of two different groups is studied by using the group homomorphisms. Finally Sect. 5 concludes the paper.

2 Preliminaries

This section presents some fundamental notions relevant to soft sets, soft groups and normal soft groups. In this paper, G denotes a multiplicative group with identity element e. To define the notion of a soft set, first we fix some notations: Let O be the set of objects, A a set of attributes (or parameters) of O and $\mho$ a subset of A. P(O) denotes the collection of all subsets of O.

Definition 2.1

(Molodtsov 1999) A pair $(\kappa ,\mho )$ is called a soft set over O, where $\kappa :\mho \rightarrow P(O)$ is a set-valued function.

Definition 2.2

(Feng et al. 2010) Let $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ be any two soft sets over O, where $\mho _1,\mho _2\subseteq A$. Then, $(\kappa _1,\mho _1)$ is called a soft subset of $(\kappa _2,\mho _2)$, if:

• (1) $\mho _1\subseteq \mho _2$.

• (2) $\kappa _1(r)\subseteq \kappa _2(r)$, for all $r\in \mho _1$.

We write it as $(\kappa _1,\mho _1)\widetilde{\subseteq }(\kappa _2,\mho _2)$. Two soft sets $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ are said to be equal, if $(\kappa _1,\mho _1)\widetilde{\subseteq }(\kappa _2,\mho _2)$ and $(\kappa _1,\mho _1)\widetilde{\supseteq }(\kappa _2,\mho _2)$. It will be denoted by $(\kappa _1,\mho _1)\widetilde{=}(\kappa _2,\mho _2)$.

Definition 2.3

Feng et al. (2008) Let $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ be two soft sets over O, where $\mho _1,\mho _2\subseteq A$. Then, their restricted union$(\kappa _1,\mho _1)\cup _{{\mathfrak {R}}}(\kappa _2,\mho _2)=(\lambda _1,\mho _3)$ and restricted intersection$(\kappa _1,\mho _1)\Cap (\kappa _2,\mho _2)=(\lambda _2,\mho _3)$, are defined as follows:

$$\begin{aligned} \lambda _1(r)=\kappa _1(r)\cup \kappa _2(r) \text { and } \lambda _2(r)=\kappa _1(r)\cap \kappa _2(r), \end{aligned}$$

for all $r\in \mho _3=\mho _1\cap \mho _2\ne \emptyset$.

From now on, it is assumed that $O=G$.

Definition 2.4

(Aktaş and Çağman 2007) A soft set $(\kappa ,\mho )$ is called a soft group over G, if $\kappa (r)$ is a subgroup of G, for all $r\in \mho$.

Definition 2.5

(Aktaş and Çağman 2007; Aslam and Qurashi 2012; Sezgin and Atagun 2011) A soft group $(\kappa ,\mho )$ over G is called a normal soft group over G, if $\kappa (r)$ is a normal subgroup of G, for all $r\in \mho$.

Definition 2.6

(Aslam and Qurashi 2012) Let $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ be two soft groups over G. Then, their restricted soft product is denoted by $(\kappa _1,\mho _1)\hat{\circ }(\kappa _2,\mho _2)=(\lambda ,\mho _3)$, and is defined by $\lambda (r)=\kappa _1(r)\kappa _2(r)$, for all $r\in \mho _3=\mho _1\cap \mho _2\ne \emptyset$.

To illustrate Definitions 2.4–2.6, we consider the following Example:

Example 2.7

Let $G={\mathbf {V}}_{4}=\{e,a,b,c\}$ be the Klien-4 group and $\mho =\{l,m\}$. Consider the following soft sets $\kappa _1:\mho \rightarrow P(G)$ and $\kappa _2:\mho \rightarrow P(G)$:

$$\begin{aligned} \kappa _1(r)= {\left\{ \begin{array}{ll} \{e,a\},&{} \text { if }r=l, \\ \{e,c\},&{} \text { if }r=m, \end{array}\right. } \quad \text { and }\quad \kappa _2(r)= {\left\{ \begin{array}{ll} \{e,b\},&{} \text { if }r=l, \\ \{e\},&{} \text { if }r=m, \end{array}\right. } \end{aligned}$$

for all $r\in \mho$. One can easily see that $\kappa _i(r)$ are normal subgroups of G, for all $r\in \mho$ and $i=1,2$. Thus, $(\kappa _i,\mho )$ are normal soft groups over G, for all $i=1,2$. Furthermore, from Definition 2.6, we have:

$$\begin{aligned} \lambda (r)= {\left\{ \begin{array}{ll} \{e,a,b,c\},&{} \text { if }r=l, \\ \{e,c\},&{} \text { if }r=m, \end{array}\right. } \end{aligned}$$

for all $r\in \mho$.

3 Lower and upper approximations in groups via normal soft groups

In this section, we initiate the concept of soft rough set over group. We introduce the notion of soft lower approximation space and soft upper approximation space over groups by using a normal soft group and study some related properties.

Definition 3.1

Let $(\kappa ,\mho )$ be a normal soft group over a group G and L be a non-empty subset of G. Define the soft lower approximation space $(\underline{\kappa }_{L},\mho )$ and the soft upper approximation space $(\overline{\kappa }^{L},\mho )$ of L in G with respect to $\kappa (r)$ as follows:

$$\begin{aligned} \underline{\kappa }_{L}(r)=\{g\in G:g\kappa (r)\subseteq L\} \quad \text { and }\quad \overline{\kappa }^{L}(r)=\{g\in G:g\kappa (r)\cap L\ne \emptyset \}, \end{aligned}$$

for each $r\in \mho$. Note that, $\underline{\kappa }_{L}:\mho \rightarrow P(G)$ and $\overline{\kappa }^{L}:\mho \rightarrow P(G)$ are soft sets over G.

To illustrate this notion, let us consider the following example.

Example 3.2

Consider a group of matrices $G=\{{\mathcal {I}},{\mathcal {A}},{\mathcal {B}},{\mathcal {C}}\}$, where

$$\begin{aligned} {\mathcal {I}}= \left[ {\begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} 1\\ \end{array} } \right] ,\quad {\mathcal {A}}= \left[ {\begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} -1\\ \end{array} } \right] ,\quad {\mathcal {B}}= \left[ {\begin{array}{c@{\quad }c} -1 &{} 0\\ 0 &{} 1\\ \end{array} } \right] \quad \text { and }\quad {\mathcal {C}}= \left[ {\begin{array}{c@{\quad }c} -1 &{} 0\\ 0 &{} -1\\ \end{array} } \right] , \end{aligned}$$

with the following multiplication table:

$\cdot$	${\mathcal {I}}$	${\mathcal {A}}$	${\mathcal {B}}$	${\mathcal {C}}$
${\mathcal {I}}$	${\mathcal {I}}$	${\mathcal {A}}$	${\mathcal {B}}$	${\mathcal {C}}$
${\mathcal {A}}$	${\mathcal {A}}$	${\mathcal {I}}$	${\mathcal {C}}$	${\mathcal {B}}$
${\mathcal {B}}$	${\mathcal {B}}$	${\mathcal {C}}$	${\mathcal {I}}$	${\mathcal {A}}$
${\mathcal {C}}$	${\mathcal {C}}$	${\mathcal {B}}$	${\mathcal {A}}$	${\mathcal {I}}$

Let $\mho =\{e_1,e_2\}$. Define a soft set $(\kappa ,\mho )$ by a mapping $\kappa :\mho \rightarrow P(G)$ as follows:

$$\begin{aligned} \kappa (r)= {\left\{ \begin{array}{ll} \{{\mathcal {I}},{\mathcal {A}}\},&{} \hbox { if}\ r=e_1\\ \{{\mathcal {I}},{\mathcal {C}}\},&{} \hbox { if}\ r=e_2 \end{array}\right. } \end{aligned}$$

It is clear that $(\kappa ,\mho )$ is a normal soft group over G. Let $L=\{{\mathcal {A}},{\mathcal {B}}\}$. Then, using the Definition 3.1 of soft lower approximation space, we obtain:

$$\begin{aligned} \underline{\kappa }_{L}(r)= {\left\{ \begin{array}{ll} \emptyset , &{}\hbox { if}\ r=e_1\\ \{{\mathcal {A}},{\mathcal {B}}\}, &{}\hbox { if}\ r=e_2 \end{array}\right. } \end{aligned}$$

for all $r\in \mho$. Using the Definition 3.1 of soft upper approximation space, we get the following soft set:

$$\begin{aligned} \overline{\kappa }^{L}(r)= {\left\{ \begin{array}{ll} G, &{}\hbox { if}\ r=e_1\\ \{{\mathcal {A}},{\mathcal {B}}\}, &{}\hbox { if}\ r=e_2 \end{array}\right. } \end{aligned}$$

for all $r\in \mho$.

From the Definition 3.1 of soft approximation spaces, one can obtain the following properties:

Proposition 3.3

Let $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ be two normal soft groups over G. Suppose that $L_1,L_2$ be any two non-empty subsets of G. Then

• (1) $\underline{\kappa _1}_{L_1}(r)\subseteq L_1\subseteq \overline{\kappa _1}^{L_1}(r)$, for all $r\in \mho _1$.

• (2) If $L_1\subseteq L_2$, then $(\underline{\kappa _1}_{L_1},\mho _1)\widetilde{\subseteq }(\underline{\kappa _1}_{L_2},\mho _1)$.

• (3) If $L_1\subseteq L_2$, then $(\overline{\kappa _1}^{L_1},\mho _1)\widetilde{\subseteq }(\overline{\kappa _1}^{L_2},\mho _1)$.

• (4) $(\underline{\kappa _1}_{L_1},\mho _1)\cup _{\mathfrak {R}}(\underline{\kappa _1}_{L_2},\mho _1)\widetilde{\subseteq }(\underline{\kappa _1}_{L_1\cup L_2},\mho _1)$.

• (5) $(\overline{\kappa _1}^{L_1\cap L_2},\mho _1)\widetilde{\subseteq }(\overline{\kappa _1}^{L_1},\mho _1)\Cap (\overline{\kappa _1}^{L_2},\mho _1)$.

• (6) $(\overline{\kappa _1}^{L_1\cup L_2},\mho _1)\widetilde{=}(\overline{\kappa _1}^{L_1},\mho _1)\cup _{{\mathfrak {R}}} (\overline{\kappa _1}^{L_2},\mho _1)$.

• (7) $(\underline{\kappa _1}_{L_1\cap L_2},\mho _1)\widetilde{=}(\underline{\kappa _1}_{L_1},\mho _1)\Cap (\underline{\kappa _1}_{L_2},\mho _1)$.

• (8) If $(\kappa _1,\mho _1)\widetilde{\subseteq }(\kappa _2,\mho _2)$, then $(\overline{\kappa _1}^{L_1},\mho _1)\widetilde{\subseteq } (\overline{\kappa _2}^{L_1},\mho _2)$.

• (9) If $(\kappa _1,\mho _1)\widetilde{\subseteq }(\kappa _2,\mho _2)$, then $(\underline{\kappa _1}_{L_1},\mho _1)\widetilde{\supseteq } (\underline{\kappa _2}_{L_1},\mho _2)$.

• (10) $(\underline{\lambda _2}_{L_1},\mho _3)\widetilde{\supseteq } (\underline{\kappa _1}_{L_1},\mho _1)\Cap (\underline{\kappa _2}_{L_1},\mho _2)$.

• (11) $(\overline{\lambda _2}^{L_1},\mho _3)\widetilde{\subseteq }(\overline{\kappa _1}^{L_1},\mho _1)\Cap (\overline{\kappa _2}^{L_1},\mho _2)$.

Here, $(\kappa _1,\mho _1)\Cap (\kappa _2,\mho _2)=(\lambda _2,\mho _3)$ and $\mho _3=\mho _1\cap \mho _2\ne \emptyset$ (see Definition 2.3).

Proof

The proof is obvious. $\square$

The following Examples illustrate that inclusions in (4), (5), (10) and (11) of Proposition 3.3 are strict:

Example 3.4

Let $G={\mathcal {D}}_{4}=<x,y:x^{4}=y^{2}=e,yx=x^{3}y>$ be the dihedral group and $\mho =\{a_{1},a_{2}\}$ be the set of parameters. Consider a set valued function $\kappa :\mho \rightarrow P(G)$ defined as follows:

$$\begin{aligned} \kappa (r)= {\left\{ \begin{array}{ll} \{e,x,x^{2},x^{3}\},&{} \text { if }r=a_{1}, \\ \{e,x^{2},y,x^{2}y\},&{} \text { if }r=a_{2}, \end{array}\right. } \end{aligned}$$

for all $r\in \mho$. Then $(\kappa ,\mho )$ is a normal soft group over G. Assume that $L_1=\{x,y,xy,x^{2}y\}$ and $L_2=\{e,x^{2},x^{3},x^{3}y\}$, then $L_1\cup L_2=G$. By simple calculations, we have:

$$\begin{aligned} \underline{\kappa }_{L_1}(a_1)=\underline{\kappa }_{L_2}(a_1)=\emptyset \quad \text { and } \quad \underline{\kappa }_{L_1\cup L_2}(a_1)=G. \end{aligned}$$

This shows that, $\underline{\kappa }_{L_1\cup L_2}(a_1)\nsubseteq \underline{\kappa }_{L_1}(a_1)\cup \underline{\kappa }_{L_2}(a_1)$. Let $L_1=\{x,y,xy,x^{2}y\}$ and $L_3=\{y,x^{3}\}$, we obtain:

$$\begin{aligned} \overline{\kappa }^{L_1}(a_2)=\overline{\kappa }^{L_3}(a_2)=G \quad \text { and }\quad \overline{\kappa }^{L_1\cap L_3}(a_2)=\{e,y,x^{2},x^{2}y\}. \end{aligned}$$

It implies that $\overline{\kappa }^{L_1}(a_2)\cap \overline{\kappa }^{L_3}(a_2)\nsubseteq \overline{\kappa }^{L_1\cap L_3}(a_2)$.

Example 3.5

Assume that the normal soft groups $(\kappa _i,\mho )$, $i=1,2$ are same as in Example 2.7. Let $(\lambda _2,\mho )=(\kappa _1,\mho )\Cap (\kappa _2,\mho )$. Then, by Definition 2.3, the soft set $\lambda _2:\mho \rightarrow P(G)$ can be written as follows:

$$\begin{aligned} \lambda _2(r)=\{e\},\quad \text { for all }\quad r\in \mho . \end{aligned}$$

Assume that $L=\{e,c\}$. Using the Definition 3.1 of the soft lower approximation space, the following results can be achieved:

$$\begin{aligned} \underline{\kappa _1}_{L}(r)= {\left\{ \begin{array}{ll} \emptyset ,&{} \hbox { if}\ r=l\\ \{e,c\},&{} \hbox { if}\ r=m \end{array}\right. },\quad \text { and }\quad \quad \underline{\kappa _2}_{L}(r)= {\left\{ \begin{array}{ll} \emptyset ,&{} \hbox { if}\ r=l\\ \{e,c\},&{} \hbox { if}\ r=m \end{array}\right. } \end{aligned}$$

for all $r\in \mho$. Also,

$$\begin{aligned} \underline{\lambda _2}_{L}(r)=L \quad \text { for all } \quad r\in \mho . \end{aligned}$$

Hence, $\underline{\lambda _2}_{L}(r)\nsubseteq \underline{\kappa _1}_{L}(r)\cap \underline{\kappa _2}_{L}(r)$ for all $r\in \mho$. Now, by using the Definition 3.1 of the soft upper approximation space, we obtain:

$$\begin{aligned} \overline{\kappa _1}^{L}(r)= {\left\{ \begin{array}{ll} G,&{} \hbox { if}\ r=l\\ \{e,c\},&{} \hbox { if}\ r=m \end{array}\right. },\quad \text { and }\quad \quad \overline{\kappa _2}^{L}(r)= {\left\{ \begin{array}{ll} G,&{} \hbox { if}\ r=l\\ \{e,c\},&{} \hbox { if}\ r=m \end{array}\right. } \end{aligned}$$

for all $r\in \mho$. Also,

$$\begin{aligned} \overline{\lambda _2}^{L}(r)=L \quad \text { for all } \quad r\in \mho . \end{aligned}$$

Thus $\overline{\kappa _1}^{L}(r)\cap \overline{\kappa _2}^{L}(r)\nsubseteq \overline{\lambda _2}^{L}(r)$ for all $r\in \mho$.

In the rest of paper, $(\kappa ,\mho )$ will denote a normal soft group over G.The following Theorem illustrates that if L is a subgroup of G, then the soft approximation spaces of a subgroup do not provide us any new information with a non-empty soft lower approximation space.

Theorem 3.6

Let L be a subgroup of G such that $\underline{\kappa }_{L}(r)\ne \emptyset$, for some $r\in \mho$. Then, the following equalities hold:

$$\begin{aligned} \underline{\kappa }_{L}(r)=L=\overline{\kappa }^{L}(r). \end{aligned}$$

for all $r\in \mho$.

Proof

We claim that $e\in \underline{\kappa }_{L}(r)$, $r\in \mho$. By assumption on $\underline{\kappa }_{L}(r)$, there exists $g\in \underline{\kappa }_{L}(r)$. Proposition 3.3 (1) follows that $g\in L$. From hypothesis:

$$\begin{aligned} \kappa (r)=g^{-1}.g\kappa (r)\subseteq L.L=L. \end{aligned}$$

(3.1)

Thus, $e\in \underline{\kappa }_{L}(r)$ and this proves the claim.

Note that $\underline{\kappa }_{L}(r)\subseteq L\subseteq \overline{\kappa }^{L}(r)$, for all $r\in \mho$ (see Proposition 3.3 (1)). Let $x\in \overline{\kappa }^{L}(r)$, where $r\in \mho$. There exists $u\in G$ such that $u\in x\kappa (r)\cap L$. Then $u\kappa (r)=x\kappa (r)$ and $u\in L$. We claim that $u\kappa (r)\subseteq L$. From Equation 3.1, we have $\kappa (r)\subseteq L$. Then $u\kappa (r)\subseteq uL= L$. This proves the claim. Hence, $x\in \underline{\kappa }_{L}(r)$ for all $r\in \mho$. Therefore, $\underline{\kappa }_{L}(r)=L=\overline{\kappa }^{L}(r)$, for all $r\in \mho$. $\square$

Theorem 3.7

Let W and N be normal subgroups of G. Suppose that $\underline{W}_{N}\ne \emptyset$. Then,

$$\begin{aligned} \underline{W}_{N}=W=\overline{W}^{N}, \end{aligned}$$

where $\underline{W}_{N}$ and $\overline{W}^{N}$ denote the lower and upper approximations of W with respect to N, respectively, defined at page 204 of Paper Kuroki and Wang (1996).

Proof

Since $\underline{W}_{N}\subseteq W\subseteq \overline{W}^{N}$ (see Proposition 2.1 (1) of Kuroki and Wang 1996). From hypothesis $\underline{W}_{N}\ne \emptyset$, there exists $q\in G$ such that $q\in \underline{W}_{N}$. By Proposition 2.1 (1) in Kuroki and Wang (1996), we have $q\in W$. Hence,

$$\begin{aligned} N=eN=(q.q^{-1})N=(qN).(q^{-1}N)\subseteq W.W\subseteq W \end{aligned}$$

(3.2)

Let $x\in \overline{W}^{N}$. From Definition of upper approximation in Kuroki and Wang (1996) at page 204, we have $y\in xN\cap W$ for some $y\in G$. Hence, Eq. 3.2 yields that

$$\begin{aligned} yN=(y.e)N=yN.eN\subseteq yN.W\subseteq W \end{aligned}$$

But $yN=xN$. Thus $xN\subseteq W$. Hence, $x\in \underline{W}_{N}$. This completes the proof. $\square$

In order to discover a relationship between soft approximation spaces of the product of two subsets of G and restricted soft product of approximations of these sets, the following results are presented:

Proposition 3.8

Let $L_1$ and $L_2$ be any non-empty subsets of G. Then,

• (1) $(\overline{\kappa }^{L_1L_2},\mho )\widetilde{=}(\overline{\kappa }^{L_1},\mho )\hat{\circ }(\overline{\kappa }^{L_2},\mho )$.

• (2) $(\underline{\kappa }_{L_1},\mho )\hat{\circ }(\underline{\kappa }_{L_2},\mho )\widetilde{\subseteq }(\underline{\kappa }_{L_1L_2},\mho )$.

where $\hat{\circ }$ represents the restricted soft product of two soft groups (see Definition 2.6).

Proof

(1) Suppose that $x\in \overline{\kappa }^{L_1}(r)\cdot \overline{\kappa }^{L_2}(r)$ and $r\in \mho$. Then $x=uv$, for some $u\in r\overline{\kappa }^{L_1}(r)$ and $v\in \overline{\kappa }^{L_2}(r)$. There exist $x_1,x_2\in G$ such that $x_1\in u\kappa (r)\cap L_1$ and $x_2\in v\kappa (r)\cap L_2$. It yields that $x_1x_2\in uv\kappa (r) \text { and }x_1x_2\in L_1L_2$. Thus $x_1x_2\in uv\kappa (r)\cap L_1L_2$. Therefore, $x=uv\in \overline{\kappa }^{L_1L_2}(r)$, for $r\in \mho$.

Conversely, assume that $z\in \overline{\kappa }^{L_1L_2}(r)$ with $r\in \mho$. This implies that, $g\in z\kappa (r)\cap L_1L_2$, for some $g\in G$. Since $(\kappa ,\mho )$ is a normal soft group over G, it follows that $z\in g\kappa (r)$ and $g=l_1l_2$, for some $l_1\in L_1$ and $l_2\in L_2$. Thus, $z\in (l_1l_2)\kappa (r)=(l_1\kappa (r))(l_2\kappa (r))$. Let $z=p_1p_2$, for some $p_1\in l_1\kappa (r)$ and $p_2\in l_2\kappa (r)$. Then, $l_1\in p_1\kappa (r)$ and $l_2\in p_2\kappa (r)$. Thus, $l_1\in p_1\kappa (r)\cap L_1 \text { and }l_2\in p_2\kappa (r)\cap L_2$, which yields that $p_1\in \overline{\kappa }^{L_1}(r)\text { and }p_2\in \overline{\kappa }^{L_2}(r)$. Hence, $z=p_1p_2\in \overline{\kappa }^{L_1}(r)\cdot \overline{\kappa }^{L_2}(r)$, for all $r\in \mho$.

(2) Let $z\in \underline{\kappa }_{L_1}(r)\cdot \underline{\kappa }_{L_2}(r)$, where $r\in \mho$. Then $z=uv, \text { for some }u\in \underline{\kappa }_{L_1}(r) \text { and }v\in \underline{\kappa }_{L_2}(r)$. Hence, $u\kappa (r)\subseteq L_1$ and $v\kappa (r)\subseteq L_2$. It implies that $(uv)\kappa (r)=(u\kappa (r))(v\kappa (r))\subseteq L_1L_2$. This proves that $z=uv\in \underline{\kappa }_{L_1L_2}(r)$, for all $r\in \mho$. $\square$

The following Example illustrates that the sign of soft inclusion $\widetilde{\subseteq }$ in above Proposition (2) cannot be replaced with the sign of soft equality $\widetilde{=}$:

Example 3.9

Let $G={\mathcal {S}}_{3}$ and $\mho =\{m,n\}$. Define a normal soft group $(\kappa ,\mho )$ over G as follows:

$$\begin{aligned} \kappa (r)= {\left\{ \begin{array}{ll} {\mathcal {S}}_{3},&{} \hbox { if}\ r=m,\\ \{e,(123),(132)\},&{} \hbox { if}\ r=n \end{array}\right. } \end{aligned}$$

for all $r\in \mho$. Assume that $L_1=\{e,(12)\}$ and $L_2=\{(12),(13),(23)\}$. Then $L_1L_2=G$. From Definition 3.1 of soft lower approximation, we have the following:

$$\begin{aligned} \underline{\kappa }_{L_1L_2}(n)=G, \underline{\kappa }_{L_1}(n)=\emptyset \text { and } \underline{\kappa }_{L_2}(n)=L_2. \end{aligned}$$

This shows that $\underline{\kappa }_{L_1}(n)\cdot \underline{\kappa }_{L_2}(n)\nsupseteq \underline{\kappa }_{L_1L_2}(n)$.

The following Lemma will be helpful to prove some results in the sequel:

Lemma 3.10

Let L be any non-empty subset of G. Then, $\overline{\kappa }^{L}(r)=L.\kappa (r)$ for all $r\in \mho$.

Proof

Let $x\in \overline{\kappa }^{L}(r)$, $r\in \mho$. Then, there exists $g\in G$ such that $g\in x.\kappa (r)\cap L$. Since $(\kappa ,\mho )$ is a normal soft group over G, we have $x\in g\kappa (r)\subseteq L.\kappa (r)$ for all $r\in \mho$. Thus $\overline{\kappa }^{L}(r)\subseteq L.\kappa (r)$ for all $r\in \mho$.

Conversely, assume that $g\in L.\kappa (r)$ for $r\in \mho$. Then $g=l.u$ for some $l\in L$ and $u\in \kappa (r)$. It follows that $l=gu^{-1}\in g\kappa (r)\text { and }l\in L$. Therefore, $l\in g\kappa (r)\cap L$. Thus $g\in \overline{\kappa }^{L}(r)$ for $r\in \mho$. This completes the proof. $\square$

Since the restricted soft product of two normal soft groups is a normal soft group (see Corollary 6.10 of Aslam and Qurashi 2012), the following results are obtained:

Proposition 3.11

Let $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ be two normal soft groups over G. Assume that L is a non-empty subset of G; then the following statements hold:

• (1) $(\overline{\kappa _1}^{L},\mho _1)\hat{\circ }(\overline{\kappa _2}^{L},\mho _2)\widetilde{=}(\overline{\lambda }^{L},\mho _3)\hat{\circ } (\overline{\lambda }^{L},\mho _3)$.

• (2) If L is a subgroup of G. Then

  $$\begin{aligned} (\overline{\lambda }^{L} ,\mho _3)\widetilde{=}(\overline{\kappa _1}^{L},\mho _1)\hat{\circ }(\overline{\kappa _2}^{L},\mho _2)\widetilde{=} (\overline{\kappa _2}^{L},\mho _2)\hat{\circ }(\kappa _1,\mho _1)\widetilde{=}(\overline{\kappa _1}^{L},\mho _1)\hat{\circ }(\kappa _2,\mho _2), \end{aligned}$$

where $(\lambda ,\mho _3)=(\kappa _1,\mho _1)\hat{\circ }(\kappa _2,\mho _2)$ and $\mho _3=\mho _1\cap \mho _2\ne \emptyset$ (See Definition 2.6).

Proof

(1) Let $r\in \mho _3$. Then,

$$\begin{aligned} \overline{\kappa _1}^{L}(r)\overline{\kappa _2}^{L}(r)&=(L\kappa _1(r))(L\kappa _2(r)), \quad \quad \quad \quad \quad \quad \quad \text { by Lemma }3.10 \\&=(L\kappa _1(r)\kappa _1(r))(L\kappa _2(r)\kappa _2(r)), \quad \quad \text { from Corollary 6.9 of }[5]\\&=(L\kappa _1(r)\kappa _1(r))(\kappa _2(r)L)\kappa _2(r) \\&=L\kappa _1(r)(\kappa _1(r)\kappa _2(r))L\kappa _2(r)\\&=L\kappa _1(r)(\kappa _2(r)\kappa _1(r))L\kappa _2(r),\quad \quad \quad \text { from Theorem 6.6 of } [5] \\&=(L\kappa _1(r)\kappa _2(r))(L\kappa _1(r)\kappa _2(r))\\&=(L\lambda (r))(L\lambda (r)), \quad \quad \quad \quad \quad \quad \quad \quad \text { using Definition }2.6\\&=\overline{\lambda }^{L}(r)\overline{\lambda }^{L}(r),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text { by Lemma }3.10 \end{aligned}$$

Hence, $(\overline{\kappa _1}^{L},\mho _1)\hat{\circ }(\overline{\kappa _2}^{L},\mho _2)\widetilde{=}(\overline{\lambda }^{L},\mho _3)\hat{\circ }(\overline{ \lambda }^{L},\mho _3)$.

(2) By Lemma 3.10, $\overline{\kappa _1}^{L}(r)\kappa _2(r)=L\kappa _1(r)\kappa _2(r)=L\lambda (r)=\overline{\lambda }^{L}(r)$. Similarly, $\overline{\kappa _2}^{L}(r)\kappa _1(r)=\overline{\lambda }^{L}(r)$. Therefore, $\overline{\lambda }^{L}(r)=\overline{\kappa _2}^{l}(r)\kappa _1(r)\cap \overline{\kappa _1}^{L}(r)\kappa _2(r)\text { for all }r\in \mho _3$. Thus,

$$\begin{aligned} (\overline{\lambda }^{L},\mho _3)\widetilde{=}(\overline{\kappa _2}^{L},\mho _2)\hat{\circ }(\kappa _1,\mho _1)\Cap (\overline{\kappa _1}^{L},\mho _1) \hat{\circ }(\kappa _2,\mho _2). \end{aligned}$$

This completes the proof. $\square$

Note that (2) in above Proposition is inconsistent with Proposition 3.5 and 3.6 of Kuroki and Wang (1996).

4 Connection between soft lower and upper approximation spaces

Let $\phi :G_{1}\rightarrow G_{2}$ be a group homomorphism. In this section, $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ will represent the normal soft groups over $G_{1}$ and $G_{2}$, respectively. In the following, the soft image and soft pre-image of $(\kappa _1,\mho _1)$ and $(\kappa _2,\mho _2)$ are defined under $\phi$, respectively:

Definition 4.1

Let $\phi$ be as defined above. Then,

• (i) The soft image $\phi (\kappa _1):\mho _1\rightarrow P(G_{2})$ of $(\kappa _1,\mho _1)$ is defined as $\phi (\kappa _1)(w)=\phi (\kappa _1(w)),\text { for all }w\in \mho _1$.

• (ii) The soft pre-image $\phi ^{-1}(\kappa _2):\mho _2\rightarrow P(G_{1})$ of $(\kappa _2,\mho _2)$ is defined as $\phi ^{-1}(\kappa _2)(r)=\phi ^{-1}(\kappa _2(r)),\text { for all }r\in \mho _2$.

Lemma 4.2

The soft pre-image $(\phi ^{-1}(\kappa _2),\mho _2)$ of $(\kappa _2,\mho _2)$ is a normal soft group over $G_{1}$ and $\ker \phi \subseteq \phi ^{-1}(\kappa _2)(r)$, for all $r\in \mho _2$.

Proof

Straightforward. $\square$

Theorem 4.3

With the above notion, let $L_1$ be a non-empty subset of $G_{1}$ and $L_2$ a non-empty subset of $G_{2}$ and $r\in \mho _2$. Then, the following implications are true:

$$\begin{aligned}&x\in \overline{\phi ^{-1}(\kappa _2)}^{L_1}(r)\Rightarrow \phi (x)\in \overline{\kappa _2}^{\phi (L_1)}(r). \\&x\in \overline{\phi ^{-1}(\kappa _2)}^{\phi ^{-1}(L_2)}(r)\Rightarrow \phi (x)\in \overline{\kappa _2}^{L_2}(r). \end{aligned}$$

Moreover, if $\phi$ is onto then the converse of above statements also hold.

Proof

Let $x\in \overline{\phi ^{-1}(\kappa _2)}^{L_1}(r)$. By Definition 3.1, there exists $g\in G_{1}$ such that $g\in x\cdot \phi ^{-1}[\kappa _2(r)]\cap L_1$. Since $\phi$ is a group homomorphism and $\phi (\phi ^{-1}(Z))\subseteq Z$, for any $Z\subseteq G_2$. It follows that

$$\begin{aligned} \phi (g)\in \phi (x\cdot \phi ^{-1}[\kappa _2(r)]\cap L_1)\subseteq \phi (x\cdot \phi ^{-1}[\kappa _2(r)])\cap \phi (L_1) \subseteq \phi (x)\kappa _2(r)\cap \phi (L_1). \end{aligned}$$

This completes the proof of first implication. The second implication can be proved in a similar way.

To prove the converse, assume that $\phi$ is onto and $\phi (x)\in \overline{\kappa _2}^{\phi (L_1)}(r)$. Then there exists $g'\in G_{2}$ such that $g'\in \phi (x)\kappa _2(r)\cap \phi (L_1)$. It follows that $g'=\phi (x)\cdot u=\phi (y)$, for some $u\in \kappa _2(r)$ and $y\in L_1$. Since $\phi$ is onto, so $u=\phi (v),\text { for some }v\in G_{1}$. Then, $\phi (y)=\phi (x)\phi (v)=\phi (xv)$. By Lemma 4.2, it follows that $y^{-1}(xv)\in \ker \phi \subseteq \phi ^{-1}(\kappa _2(r))$ and hence $x^{-1}y\in \phi ^{-1}(\kappa _2(r))$. Then, $y\in x\phi ^{-1}(\kappa _2(r))\cap L_1$. This proves the first implication. The second implication can be proved by following the same steps. $\square$

Theorem 4.4

Fix the notion of Theorem 4.3 and assume that $\phi$ is onto. Then the following statements hold:

$$\begin{aligned}&x\in \underline{\phi ^{-1}(\kappa _2)}_{\phi ^{-1}(L_2)}(r)\Rightarrow \phi (x)\in \underline{\kappa _2}_{L_2}(r). \\&x\in \underline{\phi ^{-1}(\kappa _2)}_{L_1}(r)\Rightarrow \phi (x)\in \underline{\kappa _2}_{\phi (L_1)}(r). \end{aligned}$$

Proof

Since $\phi$ is onto, we have $\phi (\phi ^{-1}(\kappa _2(r)))=\kappa _2(r)$. Then the assumption $x\in \underline{\phi ^{-1}(\kappa _2)}_{\phi ^{-1}(L_2)}(r)$ implies that $x(\phi ^{-1}(\kappa _2)(r))\subseteq \phi ^{-1}(L_2)$. It follows that $\phi (x)\kappa _2(r)\subseteq L_2$. Also, the assumption $x\in \underline{\phi ^{-1}(\kappa _2)}_{L_1}(r)$ implies that $x(\phi ^{-1}(\kappa _2)(r))\subseteq L_1$. This yields that $\phi (x)\kappa _2(r)\subseteq \phi (L_1)$. So, the claims are proved. $\square$

In the following Example, it is shown that the assertions of Theorem 4.4 fail, if $\phi$ is not onto.

Example 4.5

Let $G_{1}={\mathcal {S}}_{3}$ and $G_{2}={\mathbb {Z}}_{6}$. Consider a group homomorphism $\phi :{\mathcal {S}}_{3}\rightarrow {\mathbb {Z}}_{6}$ defined as follows:

$$\begin{aligned} \phi (x)= {\left\{ \begin{array}{ll} \overline{0},&{} \hbox { if}\ x=e,(123),(132),\\ \overline{3},&{} \hbox { if}\ x=(12),(13),(23), \end{array}\right. } \end{aligned}$$

for all $x\in G_{1}$. One can see that $\phi$ is not onto. Assume that $\mho _2={\mathbb {Z}}_{6}$. Define a normal soft group $\kappa _2:\mho _2\rightarrow P(G_{2})$ as follows:

$$\begin{aligned} \kappa _2(w)= {\left\{ \begin{array}{ll} {\mathbb {Z}}_{6},&{} \hbox { if}\ w=\overline{0},\\ \{\overline{0}\},&{} \hbox { if}\ w=\overline{1},\overline{5},\\ \{\overline{0},\overline{3}\},&{} \hbox { if}\ w=\overline{2},\overline{4},\\ \{\overline{0},\overline{2},\overline{4}\},&{} \text { if }w=\overline{3}, \end{array}\right. } \end{aligned}$$

for all $w\in \mho _2$. By Definition 4.1 of $\phi ^{-1}(\kappa _2)$, we get the following:

$$\begin{aligned} \phi ^{-1}(\kappa _2)(w)= {\left\{ \begin{array}{ll} {\mathcal {S}}_{3},&{} \hbox { if}\ w=\overline{0},\overline{2},\overline{4},\\ \{e,(123),(132)\},&{} \text { if }w=\overline{1},\overline{5},\overline{3}, \end{array}\right. } \end{aligned}$$

for all $w\in \mho _2$. Suppose that $L_2=\{\overline{0},\overline{1},\overline{2},\overline{3}\}$, $\phi ^{-1}(L_2)={\mathcal {S}}_{3}$. By simple calculations, we obtain the following:

$$\begin{aligned} \underline{\phi ^{-1}(\kappa _2)}_{\phi ^{-1}(L_2)}(\overline{0})={\mathcal {S}}_{3}\text { and }\underline{\kappa _2}_{L_2}(\overline{0})=\emptyset . \end{aligned}$$

Thus $x\in \underline{\phi ^{-1}(\kappa _2)}_{\phi ^{-1}(L_2)}(r)\nRightarrow \phi (x)\in \underline{\kappa _2}_{L_2}(r)$. Now, assume that $L_1=\{(12),(13),(23),(123)\}$, then $\phi (L_1)=\{\overline{0},\overline{3}\}$. From Definition 3.1 of the soft lower approximation, we achieve the following:

$$\begin{aligned} \underline{\phi ^{-1}(\kappa _2)}_{L_1}(\overline{3})=\{(12),(13),(23)\}\text { and }\underline{\kappa _2}_{\phi (L_1)}(\overline{3})=\emptyset . \end{aligned}$$

Hence, $x\in \underline{\phi ^{-1}(\kappa _2)}_{L_1}(r)\nRightarrow \phi (x)\in \underline{\kappa _2}_{\phi (L_1)}(r)$.

In the next result, converse of Theorem 4.4 is proved under some conditions.

Theorem 4.6

With the same assumptions as in Theorem 4.3, the following assertion holds:

$$\begin{aligned} \phi (x)\in \underline{\kappa _2}_{L_2}(r)\Rightarrow x\in \underline{\phi ^{-1}(\kappa _2)}_{\phi ^{-1}(L_2)}(r). \end{aligned}$$

Further, if $L_1$ is a subgroup of $G_{1}$ such that $\ker \phi \subseteq L_1$. Then,

$$\begin{aligned} \phi (x)\in \underline{\kappa _2}_{\phi (L_1)}(r)\Rightarrow x\in \underline{\phi ^{-1}(\kappa _2)}_{L_1}(r). \end{aligned}$$

Proof

Let $\phi (x)\in \underline{\kappa _2}_{L_2}(r)$. By Definition 3.1, $\phi (x)\kappa _2(r)\subseteq L_2$. Suppose that $v\in \phi ^{-1}(\kappa _2(r))$. Then, $\phi (v)\in \kappa _2(r)$. It induces that $\phi (xv)=\phi (x)\phi (v)\in L_2$. Then $xv\in \phi ^{-1}(L_2)$. This proves that $x\phi ^{-1}(\kappa _2(r))\subseteq \phi ^{-1}(L_2)$ and $x\in \underline{\phi ^{-1}(\kappa _2)}_{\phi ^{-1}(L_2)}(r)$.

Now, assume that $\phi (x)\in \underline{\kappa _2}_{\phi (L_1)}(r)$ and $L_1$ is a subgroup; then $\phi (x)\kappa _2(r)\subseteq \phi (L_1)$. It implies that

$$\begin{aligned} \phi (x)u\in \phi (L_1), \text { for all }u\in \kappa _2(r). \end{aligned}$$

(4.1)

Let $l\in x(\phi ^{-1}(\kappa _2)(r))$ be an arbitrary element. Then, $l=xv$ such that $v\in \phi ^{-1}(\kappa _2)(r)=\phi ^{-1}(\kappa _2(r))$. It implies that $\phi (v)\in \kappa _2(r)$ and $\phi (l)=\phi (xv)\in \phi (L_1)$, see Eq. (4.1). So, it can be written as $\phi (xv)=\phi (y)$, for some $y\in L_1$. Then, $y^{-1}(xv)\in \ker \phi \subseteq L_1$. It yields that $l=xv\in L_1$. Hence, $x\phi ^{-1}(\kappa _2(r)) \subseteq L_1$. $\square$

To obtain a relationship between the soft approximation spaces of two different groups with respect to the soft image $(\phi (\kappa _1),\mho _1)$, the following Lemma will be used:

Lemma 4.7

If $\phi$ is onto, then the soft image $(\phi (\kappa _1),\mho _1)$ of $(\kappa _1,\mho _1)$ is a normal soft group over $G_{2}$.

Proof

The proof is analogous to the proof of (Sezgin and Atagun 2011, Proposition 2.1). $\square$

Remark 4.8

In the view of Lemma 4.7, $\phi$ will be taken as an onto homomorphism in the remaining results of this section.

Theorem 4.9

With the same notion as in Theorem 4.3, let $w\in \mho _1$. Then, the following statements hold:

$$\begin{aligned}&x\in \overline{\kappa _1}^{L_1}(w)\Rightarrow \phi (x)\in \overline{\phi (\kappa _1)}^{\phi (L_1)}(w).\\&x\in \overline{\kappa _1}^{\phi ^{-1}(L_2)}(w)\Rightarrow \phi (x)\in \overline{\phi (\kappa _1)}^{L_2}(w). \end{aligned}$$

Proof

This proof is parallel to the proof of Theorem 4.3.$\square$

The converse of above Theorem is proved in the following result with some conditions:

Theorem 4.10

With the previous notion, suppose that $\ker \phi \subseteq \kappa _1(w)$. Then,

$$\begin{aligned}&\phi (x)\in \overline{\phi (\kappa _1)}^{\phi (L_1)}(w)\Rightarrow x\in \overline{\kappa _1}^{L_1}(w).\\&\phi (x)\in \overline{\phi (\kappa _1)}^{L_2}(w)\Rightarrow x\in \overline{\kappa _1}^{\phi ^{-1}(L_2)}(w). \end{aligned}$$

Proof

Let $\phi (x)\in \overline{\phi (\kappa _1)}^{\phi (L_1)}(w)$. Then, there exists $g\in G_{2}$ such that $g\in \phi (x)(\phi (\kappa _1)(w))\cap \phi (L_1)$. It follows that $g\in \phi (x)(\phi (\kappa _1(w)))$ and $g\in \phi (L_1)$. Therefore, $g=\phi (x)\phi (u)=\phi (xu)=\phi (y)$, for some $u\in \kappa _1(w)$ and $y\in L_1$. Then $y^{-1}(xu)\in \ker \phi \subseteq \kappa _1(w)$. Since $(\kappa _1,\mho _1)$ is a normal soft group, it implies that $y^{-1}x\in \kappa _1(w)$ and $y\in x\kappa _1(w)$. This proves that $y\in x\kappa _1(w)\cap L_1\ne \emptyset$ and $x\in \overline{\kappa _1}^{L_1}(w)$. Similarly, the second implication can be proved. $\square$

Example 4.11

Let $G_{1}={\mathcal {C}}_{4}=\{\pm 1,\pm i\}$ and $G_{2}={\mathcal {C}}_{2}=\{\pm 1\}$ such that $i^2=-1$. Define an onto group homomorphism $\phi :G_{1}\rightarrow G_{2}$ as follows:

$$\begin{aligned} \phi (x)= {\left\{ \begin{array}{ll} 1,&{} \hbox { if}\ x=1,-1\\ -1,&{} \hbox { if}\ x=i,-i \end{array}\right. } \end{aligned}$$

for all $x\in G_{1}$. Let $\mho _1=\{1,-1\}$ be a subset of $G_{1}$. Consider a normal soft group $\kappa _1:\mho _1\rightarrow P(G_{1})$ with $\kappa _1(1)=\kappa _1(-1)=\{1\}$. Then,

$$\begin{aligned} (\phi (\kappa _1))(1)=(\phi (\kappa _1))(-1)=\{1\}. \end{aligned}$$

Note that $\ker \phi =\{1,-1\}\nsubseteq \kappa _1(w)$, for all $w\in \mho _1$. Let $L_1=\{1,-1,i\}$. Then, $\phi (L_1)=G_{2}$. By simple calculations, it can be proved that

$$\begin{aligned} \overline{\phi (\kappa _1)}^{\phi (L_1)}(1)=\overline{\phi (\kappa _1)}^{\phi (L_1)}(-1)=\{1,-1\} \text { and }\overline{\kappa _1}^{L_1}(1)=\overline{\kappa _1}^{L_1}(-1)=\{1,-1,i\} \end{aligned}$$

It is clear that $\phi (-i)=-1\in \overline{\phi (\kappa _1)}^{\phi (L_1)}(w)$ but $-i\notin \overline{\kappa _1}^{L_1}(w)$, for all $w\in \mho _1$.

Theorem 4.12

With the same notion as in Theorem 4.9, the following implications hold:

$$\begin{aligned}&x\in \underline{\kappa _1}_{\phi ^{-1}(L_2)}(w)\Leftrightarrow \phi (x)\in \underline{\phi (\kappa _1)}_{L_2}(w).\\&x\in \underline{\kappa _1}_{L_1}(w)\Rightarrow \phi (x)\in \underline{\phi (\kappa _1)}_{\phi (L_1)}(w). \end{aligned}$$

The converse of second implication is also true, if $L_1$ is a subgroup of $G_{1}$ such that $\ker \phi \subseteq L_1$.

Proof

This can be proved by following the same methodology as in Theorems 4.4 and 4.6. $\square$

5 Concluding remarks

This paper mainly extended the classical concept of rough subgroups studied by Kuroki and Wang, and a connection between lower and upper approximations via group homomorphism investigated by Mahmood et al. to soft set theory. In this regard, the concept of soft lower and upper approximation spaces is introduced in groups by manipulating normal soft groups. Based on the soft image and soft pre-image of a normal soft group, some connections between the soft approximation spaces are built by maneuvering group homomorphisms. This work can be extended to other algebraic structures such as rings, fields and modules in the similar manner. In future this work can be extended to make it applicable to decision making.