A new approach to soft groups based on soft binary operations

ABSTRACT

In this paper, we propose a new definition for soft groups based on soft binary operations. The idea is to bring the archetype of “softness” into the spectrum of algebraic structures using soft binary operations parametrized by a given set of suitable parameters. One of our achievement is that we obtain an ordinary group model representing our soft group. The existing classical group serves as a model to describe and characterize the overall internal properties of our soft groups. In this vein, we further investigate the soft subgroups (respectively, normal soft subgroups) and proved some structural theorems.

KEYWORDS:

• Soft mappings
• soft binary operations
• soft groups based on soft binary operations
• soft subgroups
• normal soft subgroups

MATHEMATICS SUBJECT CLASSIFICATION:

• 06D72
• 20N99

1. Introduction

Soft set theory was initiated by Molodtsov in 1999 as a general mathematical tool for dealing with uncertainty (Molodtsov, 1999). In recent years, the concept of ”softness” has gained significant attention in various fields, including mathematics and computer science. Soft computing techniques have been successfully applied to solve complex problems that involve uncertainty and imprecision. Several researchers have been extensively working on the development of the theory of soft sets. For instance, Maji & Biswas (2003) and Maji et al. (2002) introduced several operations on soft sets and applied it to decision-making problems by defining some fundamental operations such as equality of two soft sets, subset and super set of a soft set, etc. As continuation of these ideas, many hybrid structures involving soft sets were proposed. Some of them are the following: fuzzy N-soft sets by (Akram et al., 2018), intuitionistic fuzzy soft set by Xu et al. (Xu et al., 2010), hesitant fuzzy soft sets by (Wang et al., 2014), rough soft sets by (Roy & Bera, 2015) and so on.

Traditional algebraic structures, such as groups, rings, and fields, have been extensively studied and utilized to model and analyze various mathematical phenomena. However, these structures often assume precise and deterministic values, which may not adequately capture the inherent uncertainty and imprecision present in real-world scenarios. Soft computing, on the other hand, provides a framework to handle such uncertainties by incorporating fuzzy logic, probability theory, and other mathematical tools. With the idea of integrating soft computing techniques to the algebraic structures (Rosenfeld, 1971) have introduced the concept of fuzzy subgroups of a group, Biswas (1994) applied the idea of roughness in group theory and (Aktaş & Çağman, 2007). initiated the study of soft groups. According to Aktaş & Çağman (2007) soft groups over a given ordinary group G are defined as a soft set $<F,A>$ over G for which the values F(e) are subgroups of G for all $e\in A$. After a decade, in 2016, GHOSH & SAMANTA (2016) came up with a new idea of soft groups using the concept of soft elements as defined by (Wardowski, 2013). Given a group $<G,\ast >$ as an initial set of universe and a group $<E,\circ >$ as a set of parameters, they define a soft group to be the collection of nonempty soft elements of a soft set $<F,A>$ over G together with the binary operation induced by the binary operations $\ast$ and $\circ$ of G and E respectively, and satisfying all the classical group axioms. Some years later, Gozde Yaylalı & Tanay (2019) et al. extended this notion in some extent and define soft groups as the collection of nonempty soft elements of a soft set $<F,A>$ over G together with a binary operation satisfying all the defining properties of a classical group, where in this case G and E are not assumed neither to be a group nor to have binary operations.

One can easily observe that all the above discussed soft groups are defined based on classical binary operation which is the central unit determining all the algebraic properties of the structure. Taking this into consideration, in this paper, we propose a new approach to define soft groups based on soft binary operations. A soft group is defined as a triple $<G,\ast ,A>$ consisting of a nonempty set G equipped with a soft binary operation $<\ast ,A>$ respecting all the group axioms in a soft setting. These soft binary operations are parameterized by a given set of suitable parameters, allowing us to model and analyze the inherent imprecision and uncertainty in group operations. By doing so, we bridge the gap between traditional algebraic structures and the realm of soft computing.

One of the key achievements of our research is we develop an ordinary group model that represents our soft group. This model serves as a foundation for describing and characterizing the internal properties of our soft groups. By leveraging the existing classical group theory, we can establish a solid theoretical framework for our soft groups, enabling us to study their properties and behavior in a rigorous manner. Furthermore, we delve into the study of soft subgroups and normal soft subgroups within our proposed framework. Soft subgroups are soft subsets of a soft group that retain the essential properties of a group. Soft normal subgroups, on the other hand, possess additional properties that make them particularly interesting for applications in various areas, such as cryptography and data analysis. We present several structural theorems that shed light on the properties and relationships between soft subgroups and normal soft subgroups, providing valuable insights into the structure and behavior of our soft groups.

The significance of our research lies in its potential applications in various domains. Soft groups can serve as a powerful tool for modeling and analyzing complex systems that involve uncertainty and imprecision. By incorporating soft binary operations, we can capture and manipulate imprecise data, allowing us to make informed decisions in real-world scenarios. Moreover, the study of soft subgroups and normal soft subgroups opens up new avenues for exploring the interplay between classical group theory and soft computing techniques.

2. Preliminaries

In this section, we give some basic definitions which will be used in this paper, mainly following (Addis et al., 2022). Throughout this section, X and Y are assumed to be nonempty sets considered as an initial universe set and A is the set of all convenient parameters.

Definition 2.1.

(Molodtsov, 1999) A soft set over X is a pair $<F,A>$ where $F:A⟶P(X)$ is a mapping from A into power set of X. In other words, it is a parameterized family of subsets of X.

Definition 2.2.

(Maji et al., 2002) Given two soft sets $<F,A>$ and $<E,A>$ over X: we say that

(1)	$<F,A>$ is included in $<E,A>,$ written as $<F,A>\stackrel{⌣}{\subseteq }<E,A>,$ provided that $(\mathrm{\forall }x)[x\in F(\alpha )\Rightarrow x\in E(\alpha )]$ for all $\alpha \in A$;
(2)	$<F,A>$ and $<E,A>$ are equal provided that $(\mathrm{\forall }x)[x\in F(\alpha )\iff x\in E(\alpha )]$ for all $\alpha \in A$. We write $(F,A)\stackrel{⌣}{=}(E,A)$ to say that $<F,A>$ and $<E,A>$ are equal.

Definition 2.3.

(Maji et al., 2002) Given that $<F,A>$ and $<E,A>$ are soft sets over X.

(1)	their union denoted by $<F,A>\stackrel{⌣}{\cup }<E,A>,$ is a soft set $<H,A>$ over X defined as follows: for all $\alpha \in A$ and $x\in X$: $x\in H(\alpha )$ if and only if either $x\in F(\alpha )$ or $x\in E(\alpha )$;
(2)	their intersection denoted by $<F,A>\stackrel{⌣}{\cap }<E,A>,$ is a soft set $<H,A>$ over X defined as follows: for all $\alpha \in A$ and $x\in X$: $x\in H(\alpha )$ if and only if $x\in F(\alpha )$ and $x\in E(\alpha )$.

Definition 2.4.

(Maji et al., 2002) A null soft set over X is a soft set denoted by $<0_X,A>$ such that for each $\alpha \in A$, $0_X(\alpha )=\mathrm{\varnothing }$. Moreover, the absolute soft set over X is a soft set denoted by $<1_X,A>$ such that $1_X(\alpha )=X$ for all $\alpha \in A$.

By a soft relation from X to Y we mean a soft set $<R,A>$ over X × Y; i.e. $R:A\to P(X\times Y)$ such that $R(\alpha )\subseteq X\times Y$ for all $\alpha \in A$. For $\alpha \in A$ and $x,y\in X\times Y$ we write $<\alpha ,x,y>\in R$ to say that $<x,y>\in R(\alpha )$ (Zhang & Yuan, 2013).

Definition 2.5.

(Addis et al., 2022) A soft mapping from X to Y is a soft relation $<f,A>$ from X to Y such that:

(1)	for each $\alpha \in A$ and each $x\in X$ there exists some $y\in Y$ such that $<\alpha ,x,y>\in f$;
(2)	for each $\alpha \in A$, $x\in X$ and $y_1,y_2\in Y$, $<\alpha ,x,y_1>\in f$ and $<\alpha ,x,y_2>\in f$ implies $y_1=y_2$.

Definition 2.6.

(Addis et al., 2022) A soft mapping $<f,A>$ from X to Y is said to be :

(1)	injective if for each $\alpha \in A$, $x_1,x_2\in X$ and $y\in Y$; $<\alpha ,x_1,y>\in f$ and $<\alpha ,x_2,y>\in f$ together imply $x_1=x_2.$
(2)	surjective if for each $\alpha \in A$ and each $y\in Y$ there exists $x\in X$ such that $<\alpha ,x,y>\in f.$
(3)	bijective if it is both injective and surjective.

Definition 2.7.

A soft equivalence relation on X is a soft set $<\theta ,A>$ over X × X such that

(1)	$<\alpha ,x,x>\in \theta$   $\mathrm{\forall }\alpha \in A$ and $x\in X$;
(2)	for any $\alpha \in A$ and $x,y\in X$; $<\alpha ,x,y>\in \theta \Rightarrow <\alpha ,y,x>\in \theta$;
(3)	for any $\alpha \in A$ and $x,y,z\in X$; $<\alpha ,x,y>\in \theta ,<\alpha ,y,z>\in \theta \Rightarrow <\alpha ,x,z>\in \theta$.

3. Soft groups

Definition 3.1.

Let G be a nonempty set. By a soft binary operation on G, we mean a soft mapping $<\ast ,A>$ from G × G to G, where A is a set of parameters.

Definition 3.2.

A soft group is a triple $<G,\ast ,A>$ where G is nonempty set and $<\ast ,A>$ is a soft binary operation on G satisfying the following conditions:

<SG1>	For any $\alpha \in A$ and $a,b,c,x,y,u,v\in G$ it holds that $<\alpha ,a,b,x>\in \ast$, $<\alpha ,x,c,y>\in \ast$ and $<\alpha ,b,c,u>\in \ast$, $<\alpha ,a,u,v>\in \ast$ implies y=v. That is $<\ast ,A>$ is associative
<SG2>	Existence of identity for each $\alpha \in A,\mathrm{\exists }{e}_{\alpha }\in G$ such that $<\alpha ,a,e_{\alpha },a>\in \ast$ and $<\alpha ,e_{\alpha },a,a>\in \ast$ for all $a\in G$.
<SG3>	Existence of inverse for each $\alpha \in A$ and all $a\in G$, there is an element of G denoted by $\mathrm{\exists }{a}^{-\alpha }$ such that $<\alpha ,a,a^{-\alpha },e_{\alpha }>\in \ast$ and $<\alpha ,a^{-\alpha },a,e_{\alpha }>\in \ast$.

Example 3.3.

Let $\mathbb{R}$ be the set of all real numbers and $\mathbb{Z}$ the set of all integers.

Define $\ast \subseteq \mathbb{Z}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ by $<\alpha ,a,b,c>\in \ast \iff \text{break}c=a+b+\alpha$, $\mathrm{\forall }\alpha \in \mathbb{Z}$, $a,b,c\in \mathbb{R}$ then $<\mathbb{R},\ast ,\mathbb{Z}>$ is a soft group.

Example 3.4.

Let X be a nonempty set. Put $G=P(X)$ the power set of X and A = X be our set of parameters. Define a soft binary operation $<\ast ,A>$ on G by $\ast \subseteq A\times G\times G\times G$ such that $<x,B,C,D>\in \ast$ if and only if

${\displaystyle D=\{\begin{array}{ll}(B\oplus C)\cup \{x\}& ifx\notin B\oplus C\\ (B\oplus C)-\{x\}& ifx\in B\oplus C\end{array}}$

where $\oplus$ is the symmetric difference of sets. Then $<G,\ast ,A>$ is a soft group.

Example 3.5.

Let $M_2(\mathbb{R})$ be given by:

$M_2(\mathbb{R})=\{\left[\begin{array}{ll}1& a_1\\ 0& a_2\end{array}\right]:a_1,a_2\in \mathbb{R},a_2\ne 0\}.$

Put $A=\mathbb{R}\mathrm{\setminus }\{0\}$ and define a soft binary operation $<\ast ,A>$ on $M_2(\mathbb{R})$ as follows: for $\alpha \in A$ and matrices $B=\left[\begin{array}{ll}1& b_1\\ 0& b_2\end{array}\right]$, $C=\left[\begin{array}{ll}1& c_1\\ 0& c_2\end{array}\right]$ and $D=\left[\begin{array}{ll}1& d_1\\ 0& d_2\end{array}\right]$ in $M_2(\mathbb{R})$;

$\begin{array}{l}<\alpha ,B,C,D>\in \ast ifandonlyifD=\left[\begin{array}{ll}1& c_1+b_1{c}_2\alpha \\ 0& b_2{c}_2\alpha \end{array}\right].\end{array}$

Therefore $<M_2(\mathbb{R}),\ast ,A>$ is a soft group.

Example 3.6.

Let X be any set with at least two distinct elements and B(X) be the set of all bijective functions from X onto X. Put

$A=\{<x,y>\in X\times X:x\ne y\}$

and define a soft binary operation $<⊛,A>$ on B(X) as follows. For any pair $<x,y>\in A$ and $f,g,h\in B(X)$:

$\begin{array}{ll}& <,<x,y>f,g,h\in ⊛>\text{if and only if ;}\\ & h(z)=\{\begin{array}{ll}g(f(z))& \text{if }z\notin \{x,y\}\\ g(f(y))& \text{if }z=x\\ g(f(x))& \text{if }z=y\end{array}\mathrm{\forall }z\in X.\end{array}$

Then $<B(X),⊛,A>$ is a soft group.

Lemma 3.7.

Let $<G,\ast ,A>$ be a soft group. Then, the following conditions hold for all $\alpha \in A,$ and $a,b,c\in G$:

(1)	$<\alpha ,a,b,e_{\alpha }>\in \ast$ $\iff$ $b=a^{-\alpha }$ and $a=b^{-\alpha }$;
(2)	$<\alpha ,a,b,c>\in \ast$ $\iff$ $<\alpha ,a^{-\alpha },c,b>\in \ast$;
(3)	$<\alpha ,a,b,c>\in \ast$ $\iff$ $<\alpha ,c,b^{-\alpha },a>\in \ast$;
(4)	$<\alpha ,a,b,c>\in \ast$ $\iff$ $<\alpha ,b^{-\alpha },a^{-\alpha },c^{-\alpha }>\in \ast .$

Proof.

(1)	Suppose $<\alpha ,a,b,e_{\alpha }>\in \ast .$ As $<\alpha ,a^{-\alpha },e_{\alpha },a^{-\alpha }\text{break}>\in \ast$, $<\alpha ,a^{-\alpha },a,e_{\alpha }>\in \ast$, and $<\alpha ,e_{\alpha },b,b>\in \text{break}\ast$ it follows then from the associativity of $<\ast ,A>$ that $b=a^{-\alpha }.$ Similarly we get $a=b^{-\alpha }.$ The converse directly follows from the definition.
(2)	Suppose that $<\alpha ,a,b,c>\in \ast .$ Let $d\in G$ such that $<\alpha ,a^{-\alpha },c,d>\in \ast$. Then, we have $<\alpha ,a,b,c>\in \ast$, $<\alpha ,a^{-\alpha },c,d>\in \ast$, $<\alpha ,a^{-\alpha },a,e_{\alpha }>\in \ast$, and $<\alpha ,e_{\alpha },b,b>\in \ast .$ By associativity it holds that $b=d.$ Therefore $<\alpha ,a^{-\alpha },c,b>\in \ast .$ Conversely, suppose that $<\alpha ,a^{-\alpha },c,b>\in \ast .$ Then we have $<\alpha ,a^{-\alpha },c,b>\in \ast$,$<\alpha ,a,b,d>\in \ast$, $<\alpha ,a,a^{-\alpha },e_{\alpha }>\in \ast$, and $<\alpha ,e_{\alpha },c,c>\in \ast .$ So by associativity of $<\ast ,A>$ we get $d=c.$ Therefore $<\alpha ,a,b,c>\in \ast .$
(3)	The proof is similar to (2).
(4)	Suppose that $<\alpha ,a,b,c>\in \ast .$ We need to prove that $<\alpha ,b^{-\alpha },a^{-\alpha },c^{-\alpha }>\in \ast .$ Let $d\in G$ such that $<\alpha ,b^{-\alpha },a^{-\alpha },d>\in \ast$. Then, it can be shown that $<\alpha ,c,d,e_{\alpha }>\in \ast .$ Using the fact $<\alpha ,c,c^{-\alpha },e_{\alpha }>\in \ast$, we get $d=c^{-\alpha }.$ Therefore $<\alpha ,b^{-\alpha },a^{-\alpha },c^{-\alpha }>\in \ast .$ Conversely suppose that $<\alpha ,b^{-\alpha },a^{-\alpha },c^{-\alpha }>\in \ast$ and let $x\in G$ such that $<\alpha ,a,b,x>\in \ast$. Then $<\alpha ,c^{-\alpha },x,e_{\alpha }>\in \ast .$ Also, as $<\alpha ,c^{-\alpha },c,e_{\alpha }>\in \ast$, it holds that $x=c.$ Thus, $<\alpha ,a,b,c>\in \ast .$

Theorem 3.8.

Cancellation laws Let $<G,\ast ,A>$ be a soft group, $\alpha \in A$, and $a,b,c\in G.$ Then,

(1)	if $<\alpha ,a,b,x>\in \ast$ and $<\alpha ,a,c,x>\in \ast$ then b=c and
(2)	if $<\alpha ,b,a,x>\in \ast$ and $<\alpha ,c,a,x>\in \ast$ then b=c.

Proof.

(1)	Suppose $<\alpha ,a,b,x>\in \ast$ and $<\alpha ,a,c,x>\in \ast .$ Let $y\in G$ such that $<\alpha ,a^{-\alpha },x,y>\in \ast$. Then, by applying $<SG1>$ it can be shown that y = b. Now we have $<\alpha ,a,c,x>\in \ast$, $<\alpha ,a^{-\alpha },x,b>\in \ast$ and $<\alpha ,a^{-\alpha },a,e_{\alpha }>\in \ast$, $<\alpha ,e_{\alpha },c,c>\in \ast$. Again by applying $<SG1>$ we have get b = c.
(2)	The proof is similar to (1).

Definition 3.9.

Let G be a nonempty set and $<\ast ,A>$ a soft binary operation on G. Then, a triple $<G,\ast ,A>$ satisfying $<SG1>$ only is called a soft semi-group.

Example 3.10.

Let $G=\mathbb{R}\mathrm{\setminus }\{0\}$ and $A=\mathbb{Z}$. Where $\mathbb{R}$ is the set of all real numbers and $\mathbb{Z}$ is the set of all integers. Define $\ast \subseteq \mathbb{Z}\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ by $<\alpha ,a,b,c>\in \ast \iff c=\frac{\alpha ab}{5}$, $\mathrm{\forall }\alpha \in \mathbb{Z}$ and $a,b,c\in \mathbb{R}$ then $<G,\ast ,A>$ is a soft semi-group.

In the following two theorems, we obtain equivalent conditions for a soft semi-group to be a soft group.

Theorem 3.11.

Let $<G,\ast ,A>$ be a soft semi-group. Then, $<G,\ast ,A>$ is a soft group if and only if the following conditions are satisfied:

(1)	For each $\alpha \in A$, $a\in G,$ there exists $e_{\alpha }\in G$ such that $<\alpha ,a,e_{\alpha },a>\in \ast$;
(2)	For each $\alpha \in A$, $a\in G,$ there exists $a^{-\alpha }\in G$ such that $<\alpha ,a,a^{-\alpha },e_{\alpha }>\in \ast$.

Proof.

By assumption, $<G,\ast ,A>$ is a soft group then clearly (1) and (2) are satisfied. Conversely, suppose (1) and (2) are satisfied. By (1), there exists $e_{\alpha }\in G$ such that $<\alpha ,a,e_{\alpha },a>\in \ast$ $\mathrm{\forall }a\in G.$ Let a be an arbitrary element in G. By (2), there exists a^−α in G such that $<\alpha ,a,a^{-\alpha },e_{\alpha }>\in \ast$. Let $y\in G$ such that $<\alpha ,a^{-\alpha },a,y>\in \ast .$ Then by (2) again there is $x\in G$ be such that $<\alpha ,y,x,e_{\alpha }>\in \ast .$ Using the associative property of $<\ast ,A>$ one can easily verify that $<\alpha ,y,y,y>\in \ast$. Moreover, since $<\alpha ,y,x,e_{\alpha }>\in \ast$, $<\alpha ,y,x,e_{\alpha }>\in \ast$, and $<\alpha ,y,e_{\alpha },y>\in \ast$, it follows from the associativity of $<\ast ,A>$ that $y=e_{\alpha }.$ This implies that $<\alpha ,a^{-\alpha },a,e_{\alpha }>\in \ast .$ Let $z\in G$ such that $<\alpha ,e_{\alpha },a,z>\in \ast .$ Since $<\alpha ,,a,a^{-\alpha },e_{\alpha }>\in \ast$, $<\alpha ,a^{-\alpha },a,e_{\alpha }>\in \ast$, and $<\alpha ,a,e_{\alpha },a>\in \ast$, it holds that z = a; i.e. $<\alpha ,e_{\alpha },a,a>\in \ast .$ Therefore $<G,\ast ,A>$ is a soft group.

Theorem 3.12.

A soft semi-group $<G,\ast ,A>$ is a soft group if and only if for any elements a and b in G and all $\alpha \in A$, the equations $<\alpha ,a,x,b>\in \ast$ and $<\alpha ,y,a,b>\in \ast$ are solvable in G.

Proof.

Suppose $<G,\ast ,A>$ is a soft group. Let $a,b,x\in G$ such that $<\alpha ,a,x,b>\in \ast$. As G is a soft group, there exists $a^{-\alpha }\in G$ such that $<\alpha ,a,a^{-\alpha },e_{\alpha }>\in \ast$. Let $c\in G$ such that $<\alpha ,a^{-\alpha },b,c>\in \ast$. Then by Lemma 3.7 it can be shown that $<\alpha ,a,c,b>\in \ast$ and hence c is a solution for the equation $<\alpha ,a,x,b>\in \ast$. Similarly, it can be verified that $<\alpha ,y,a,b>\in \ast$ has a solution in G. Conversely suppose that the equations $<\alpha ,a,x,b>\in \ast$ and $<\alpha ,y,a,b>\in \ast$ are solvable in G for all $a,b\in G$ and all $\alpha \in A$. Let $a\in G.$ Then there exists $e_{\alpha }\in G$ such that $<\alpha ,a,e_{\alpha },a>\in \ast .$ Let b be any arbitrary element in G. We show that $<\alpha ,a,e_{\alpha },a>\in \ast .$ Since the equation $<\alpha ,y,a,b>\in \ast$ are solvable in G we can choose an element s in G such that $<\alpha ,s,a,b>\in \ast .$ So we have $<\alpha ,s,a,b>\in \ast$, $<\alpha ,b,e_{\alpha },m>\in \ast$, $<\alpha ,a,e_{\alpha },a>\in \ast ,$ and $<\alpha ,s,a,b>\in \ast .$ By associativity of $<\ast ,A>$ we get m = b. This implies that $<\alpha ,b,e_{\alpha },b>\in \ast .$ Moreover, as $<\alpha ,a,x,e_{\alpha }>\in \ast$ is solvable in G, we get that for any $a\in G$ there exists $a^{-\alpha }\in G$ such that $<\alpha ,a,a^{-\alpha },e_{\alpha }>\in \ast .$ Thus, by Theorem 3.11 $<G,\ast ,A>$ is a soft group.

Definition 3.13.

By a soft element in G, we mean a soft set $<F,A>$ over G such that $Card(F(\alpha ))=1$ $\mathrm{\forall }\alpha \in A.$ That is, $F(\alpha )$ is a single element set for each $\alpha \in A$.

We denote soft elements using lower case letters like $<\stackrel{⌣}{a},A>$ ,$<\stackrel{⌣}{b},A>$ etc. We say that a soft element $\stackrel{⌣}{a}$ belongs to a soft set $<F,A>$ provided that $a_{\alpha }\in F(\alpha )$ $\mathrm{\forall }\alpha \in A$, where a_α is the unique element of $\stackrel{⌣}{a}(\alpha )=\{a_{\alpha }\}$. Let us denote by $S{E}_A(G)$ the collection of all soft elements of G with a set of parameter A.

Note : Every element $a\in G$ can be identified as a soft element denoted by $<\stackrel{⌣}{a},A>$ over G in the following way $\stackrel{⌣}{a}(\alpha )=\{a\}$ $\mathrm{\forall }\alpha \in A.$

Definition 3.14.

Given a soft binary operation $<\ast ,A>$ on G, define a binary operation $\overline{\ast }$ on $S{E}_A(G)$ by

$\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{b}=\stackrel{⌣}{c}\iff <\alpha ,a_{\alpha },b_{\alpha },c_{\alpha }>\in \ast \mathrm{\forall }\alpha \in A.$

Theorem 3.15.

Let $<\ast ,A>$ be a soft binary operation on G. Then $<G,\ast ,A>$ is a soft group if and only if $<S{E}_A(G),\overline{\ast }>$ is a group.

Proof.

Suppose that $<G,\ast ,A>$ is a soft group. We first show that $\overline{\ast }$ is well-defined. Let $\stackrel{⌣}{a},\stackrel{⌣}{b},\stackrel{⌣}{c},\stackrel{⌣}{x},\stackrel{⌣}{y},\stackrel{⌣}{z}$ be soft elements over G such that $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{b}=\stackrel{⌣}{c}$, $\stackrel{⌣}{x}\overline{\ast }\stackrel{⌣}{y}=\stackrel{⌣}{z}$ and $\stackrel{⌣}{a}=\stackrel{⌣}{x}$,$\stackrel{⌣}{b}=\stackrel{⌣}{y}.$ We need to show that $\stackrel{⌣}{c}=\stackrel{⌣}{z}.$ From $\stackrel{⌣}{a}=\stackrel{⌣}{x}$ and $\stackrel{⌣}{b}=\stackrel{⌣}{y}$ we have $\stackrel{⌣}{a}(\alpha )=\stackrel{⌣}{x}(\alpha )\mathrm{\forall }\alpha \in A,$ and $\stackrel{⌣}{b}(\alpha )=\stackrel{⌣}{y}(\alpha )\mathrm{\forall }\alpha \in A$ which implies that $a_{\alpha }=x_{\alpha }$ and $b_{\alpha }=y_{\alpha }\mathrm{\forall }\alpha \in A.$ Also since $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{b}=\stackrel{⌣}{c}$ and $\stackrel{⌣}{x}\overline{\ast }\stackrel{⌣}{y}=\stackrel{⌣}{z}$ we have $<\alpha ,a_{\alpha },b_{\alpha },c_{\alpha }>\in \ast$ and $<\alpha ,x_{\alpha },y_{\alpha },z_{\alpha }>\in \ast$ for all $\alpha \in A$. Moreover, as $\ast$ is well-defined it follows that $c_{\alpha }=z_{\alpha }\mathrm{\forall }\alpha \in A.$ This implies that $\stackrel{⌣}{c}=\stackrel{⌣}{z}.$ Now $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{b}=\stackrel{⌣}{x}$ and $\stackrel{⌣}{x}\overline{\ast }\stackrel{⌣}{c}=\stackrel{⌣}{y}.$ Which implies that $<\alpha ,a_{\alpha },b_{\alpha },x_{\alpha }>\in \ast$ and $<\alpha ,x_{\alpha },c_{\alpha },y_{\alpha }>\in \ast \mathrm{\forall }\alpha \in A.$ Again if we put $\stackrel{⌣}{b}\overline{\ast }\stackrel{⌣}{c}=\stackrel{⌣}{u}$ and $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{u}=\stackrel{⌣}{v}.$ Then $<\alpha ,b_{\alpha },c_{\alpha },u_{\alpha }>\in \ast$ and $<\alpha ,a_{\alpha },u_{\alpha },v_{\alpha }>\in \ast \mathrm{\forall }\alpha \in A.$ Then, since $<\ast ,A>$ is associative we get that $y_{\alpha }=v_{\alpha }\mathrm{\forall }\alpha \in A.$ This implies that $\stackrel{⌣}{y}=\stackrel{⌣}{v}$ That is $\overline{\ast }$ is associative. Next consider the soft element $\stackrel{⌣}{e}$ in G defined by $\stackrel{⌣}{e}(\alpha )=\{e_{\alpha }\}$ for all $\alpha \in A$ where each e_α is an identity element of $<G,\ast ,A>$ with respect to $\alpha .$ Then one can easily show that $\stackrel{⌣}{e}$ is an identity element in $S{E}_A(G).$ Also for every soft element $\stackrel{⌣}{a}$ over G, consider the soft element ${\stackrel{⌣}{a}}^{-1}$ of G defined by ${\stackrel{⌣}{a}}^{-1}(\alpha )=\{a_{\alpha }^{-\alpha }\}\mathrm{\forall }\alpha \in A.$ Then ${\stackrel{⌣}{a}}^{-1}$ is the inverse of $\stackrel{⌣}{a}$ in $S{E}_A(G).$ Thus, $<S{E}_A(G),\overline{\ast }>$ is a group. Conversely suppose that $<S{E}_A(G),\overline{\ast }>$ is a group. We need to show that $<G,\ast ,A>$ is a soft group. We first to show that $<\ast ,A>$ is associative. Let $\alpha \in A$ and $a,b,c,x,y,u,v\in G$ such that $<\alpha ,a,b,x>\in \ast$, $<\alpha ,x,c,y>\in \ast$ and $<\alpha ,b,c,u>\ast$, $<\alpha ,a,u,v>\in \ast$. Our aim is to show that $y=v.$ Consider the soft elements $\stackrel{⌣}{a},\stackrel{⌣}{b},\stackrel{⌣}{c},\stackrel{⌣}{x},\stackrel{⌣}{y},\stackrel{⌣}{u},\stackrel{⌣}{v}\in <S{E}_A(G)>$ defined as follows $\stackrel{⌣}{a}(\lambda )=\{a\}$, $\stackrel{⌣}{b}(\lambda )=\{b\}$ , $\stackrel{⌣}{c}(\lambda )=\{c\}$ $\mathrm{\forall }\lambda \in A.$

${\displaystyle \stackrel{⌣}{x}(\lambda )=\{\begin{array}{ll}\{x\}& if\lambda =\alpha \\ \{x_{\lambda }\}& if\lambda \ne \alpha \end{array}}$

where $x_{\lambda }\in G$ with $<\lambda ,a,b,x_{\lambda }>\in \ast$, and

${\displaystyle \stackrel{⌣}{y}(\lambda )=\{\begin{array}{ll}\{y\}& if\lambda =\alpha \\ \{y_{\lambda }\}& if\lambda \ne \alpha \end{array},}$

where $y_{\lambda }\in G$ with $<\lambda ,x_{\lambda },c,y_{\lambda }>\in \ast .$ Similarly, define

${\displaystyle \stackrel{⌣}{u}(\lambda )=\{\begin{array}{ll}\{u\}& if\lambda =\alpha \\ \{u_{\lambda }\}& if\lambda \ne \alpha \end{array},}$

where $u_{\lambda }\in G$ with $<\lambda ,b,c,u_{\lambda }>\in \ast$, and

${\displaystyle \stackrel{⌣}{v}(\lambda )=\{\begin{array}{ll}\{v\}& if\lambda =\alpha \\ \{v_{\lambda }\}& if\lambda \ne \alpha \end{array},}$

where $v_{\lambda }\in G$ such that $<\lambda ,a,u_{\lambda },v_{\lambda }>\in \ast \mathrm{\forall }\lambda \ne \alpha .$

Then we get that $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{b}=\stackrel{⌣}{x}$ , $\stackrel{⌣}{x}\overline{\ast }\stackrel{⌣}{c}=\stackrel{⌣}{y}$ and $\stackrel{⌣}{b}\overline{\ast }\stackrel{⌣}{c}=\stackrel{⌣}{u}$, $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{u}=\stackrel{⌣}{v}.$ Since $\overline{\ast }$ is associative we get $\stackrel{⌣}{y}=\stackrel{⌣}{v}.$ That is $\stackrel{⌣}{y}(\lambda )=\stackrel{⌣}{v}(\lambda )\mathrm{\forall }\lambda \in A.$ In particular $\stackrel{⌣}{y}(\alpha )=\stackrel{⌣}{v}(\alpha ).$ This implies that $y=v.$ Thus, $<\ast ,A>$ is associative. Secondly, as $<S{E}_A(G),\overline{\ast }>$ is a group it has an identity element let say $\stackrel{⌣}{e}.$ Now let $a\in G$ and $\alpha \in A.$ Then consider a soft element $\stackrel{⌣}{a}$ given by $\stackrel{⌣}{a}(\lambda )=\{a\}\mathrm{\forall }\lambda \in A.$ Then $\stackrel{⌣}{a}\overline{\ast }\stackrel{⌣}{e}=\stackrel{⌣}{a}.$ This implies that $<\lambda ,\stackrel{⌣}{a}(\lambda ),\stackrel{⌣}{e}(\lambda ),\stackrel{⌣}{a}(\lambda )>\in \ast \mathrm{\forall }\lambda \in A.$ In particular, for $\alpha \in A$ $<\alpha ,\stackrel{⌣}{a}(\alpha ),\stackrel{⌣}{e}(\alpha ),\stackrel{⌣}{a}(\alpha )>\in \ast .$ Which implies that $<\alpha ,a,e_{\alpha },a>\in \ast .$ Similarly it can be shown that $<\alpha ,e_{\alpha },a,a>\in \ast .$ Finally, as $<S{E}_A(G),\overline{\ast }>$ is a group every element $\stackrel{⌣}{a}$ in $S{E}_A(G)$ has an inverse let say ${\stackrel{⌣}{a}}^{-1}$. Now let $a\in G$ and $\alpha \in A.$ Then consider a soft element $\stackrel{⌣}{a}$ given by $\stackrel{⌣}{a}(\lambda )=\{a\}\mathrm{\forall }\lambda \in A.$ Then $\stackrel{⌣}{a}\overline{\ast }{\stackrel{⌣}{a}}^{-1}=\stackrel{⌣}{e}.$ This implies that $<\lambda ,\stackrel{⌣}{a}(\lambda ),{\stackrel{⌣}{a}}^{-1}(\lambda ),\stackrel{⌣}{e}(\lambda )>\in \ast \mathrm{\forall }\lambda \in A.$ In particular, for $\alpha \in A$ $<\alpha ,\stackrel{⌣}{a}(\alpha ),{\stackrel{⌣}{a}}^{-1}(\alpha ),\stackrel{⌣}{e}(\alpha )>\in \ast .$ Which implies that $<\alpha ,a,a^{-1},e_{\alpha }>\in \ast .$ Similarly it can be shown that $<\alpha ,a^{-1},a,e_{\alpha }>\in \ast .$ Therefore $<G,\ast ,A>$ is a soft group.

Note: The group $S{E}_A(G)$ obtained in the above theorem is a model representing the soft group $<G,\ast ,A>$, and is useful to transfer most of the important properties of classical groups to soft groups.

4. Soft subgroups

Definition 4.1.

Let $<G,\ast ,A>$ be a soft group. A soft set $<H,A>$ over G is said to be a soft subgroup of G if for each $\alpha \in A$, and $a,b,x\in G$:

(1)	eα $\in$ $H(\alpha )$;
(2)	$<\alpha ,a,b,x>\in \ast \Rightarrow x\in H(\alpha )$ for all $a,b\in H(\alpha )$;
(3)	$a\in H(\alpha )$ $\Rightarrow$ $a^{-\alpha }\in H(\alpha )$.

Notation. We denote by $S{S}_A(G)$ the collection of all soft subgroups of $<G,\ast ,A>$.

Example 4.2.

(1)	Given a soft group $<G,\ast ,A>$, the absolute soft set $<1_G,A>$ over G and the soft element $<\stackrel{⌣}{e},A>$ of G are both soft subgroups of G.
(2)	Let $G=\mathbb{Z}=$ the set of integers and $A=\mathbb{N}=$ the set of natural numbers. Let $\ast \subseteq A\times G\times G\times G$ be given by $\ast =\{<\alpha ,a,b,c>:c=a+b-6\alpha \}$. Define $H:A\to p(G)$ by $H(\alpha )=\{(2k)\alpha :k\in \mathbb{Z}\}$ then $<H,A>$ is a soft subgroup of G.

Theorem 4.3.

Let $<G,\ast ,A>$ be a soft group. A soft set $<H,A>$ over G is a soft subgroup of G if and only if the following two conditions hold for each $\alpha \in A$ and all $a,b,x\in G$:

(1)	eα $\in$ $H(\alpha )$;
(2)	if $a,b\in H(\alpha )$ and $<\alpha ,a,b^{-\alpha },x>\in \ast$, then $x\in H(\alpha )$.

Proof.

Suppose $<H,A>$ is a soft subgroup of $<G,\ast ,A>.$ Let $\alpha \in A$ and $a,b,\in H(\alpha ).$ Then $b^{-\alpha }\in H(\alpha )$. If $x\in G$ such that $<\alpha ,a,b^{-\alpha },x>\in \ast$, then it is immediate that $x\in H(\alpha ).$ Conversely suppose that (1) and (2) hold. We show that $<H,A>$ is a soft subgroup of G. Let $\alpha \in A$. By (1), e_α $\in H(\alpha )$. Let $a,b,\in H(\alpha ).$ Since $<\alpha ,e_{\alpha },b^{-\alpha },b^{-\alpha }>\in \ast$ it follows that $b^{-\alpha }\in H(\alpha ).$ Let $x\in G$ such that $<\alpha ,a,b,x>\in \ast$. Then, as $b=(b^{-\alpha }{)}^{-\alpha }$ we get that $x\in H(\alpha ).$ Therefore $<H,A>$ is a soft subgroup of G.

Definition 4.4.

For a soft subset $<H,A>$ over G, define a subset $\hat{H}$ of $S{E}_A(G)$ by:

$\hat{H}=\{\stackrel{⌣}{a}\in S{E}_A(G):\stackrel{⌣}{a}(\alpha )\subseteq H(\alpha )\text{for all }\alpha \in A\}$

Theorem 4.5.

A soft set $<H,A>$ over G is a soft subgroup of G if and only if $\hat{H}$ is a subgroup of $S{E}_A(G)$.

Proof.

Suppose that $<H,A>$ is a soft subgroup of G. Then for each $\alpha \in A$, $e_{\alpha }\in H(\alpha )$, so that the soft identity element $\stackrel{⌣}{e}$ belongs to $\hat{H}$. Let $\stackrel{⌣}{a},\stackrel{⌣}{b}$ be any soft elements in G belonging to $\hat{H}$. Then $\stackrel{⌣}{a}(\alpha )\subseteq H(\alpha )$ and $\stackrel{⌣}{b}(\alpha )\subseteq H(\alpha )$ for all $\alpha \in A$. Now let $\stackrel{⌣}{c}$ be a soft element of G with $\stackrel{⌣}{a}\overline{\ast }{\stackrel{⌣}{b}}^{-1}=\stackrel{⌣}{c}$. Then $<\alpha ,\stackrel{⌣}{a}(\alpha ),{\stackrel{⌣}{b}}^{-1}(\alpha ),\stackrel{⌣}{c}(\alpha )>\in \ast$ for all $\alpha \in A$.

Lemma 4.6.

Arbitrary intersection of soft subgroups is a soft subgroup.

Proof.

(1)	Let $<G,\ast ,A>$ be a soft group and $\alpha \in A.$ Let $\{<H_i,A>:i\in I\}$ be a family of soft subgroup of G. Since $e_{\alpha }\in H_i(\alpha )$ for all $i\in I.$ This implies that $e_{\alpha }\in \bigcap _{i\in I}{H}_i(\alpha ).$
(2)	Let $a,b\in \bigcap _{i\in I}{H}_i(\alpha )$ and $<\alpha ,a,b^{-\alpha },x>\in \ast .$ This implies that $a,b\in H_i(\alpha )$ $\mathrm{\forall }i\in I$ and $<\alpha ,a,b^{-\alpha },x>\in \ast .$ So $x\in H_i(\alpha )\mathrm{\forall }i\in I.$ Therefore $x\in \bigcap _{i\in I}{H}_i(\alpha ).$ Thus $<{\bigcap ^{⌣}}_{i\in I}{H}_i,A>$ is a soft subgroup of G.

Remark 4.7.

The union of two soft subgroups may not be a soft subgroup. This is verified in the following example.

Example 4.8.

Let $G=\mathbb{Z}=$ the set of integers and $A=\mathbb{N}=$ the set of natural numbers.

Let $\ast \subseteq A\times G\times G\times G$ be given by $\ast =\{<\alpha ,a,b,c>:c=a+b-6\alpha \}$. Define $H:A\to p(G)$ by $H(\alpha )=\{(2k)\alpha :k\in \mathbb{Z}\}$ and $K:A\to p(G)$ by $K(\alpha )=\{(3k)\alpha :k\in \mathbb{Z}\}$ then $<H\cup K,A>$ is not a soft subgroup of G.

Theorem 4.9.

Let $<G,\ast ,A>$ be a soft group and $<H,A>$ and $<K,A>$ are soft subgroups of G. If $<H,A>\stackrel{⌣}{\subseteq }<K,A>$ or $<K,A>\stackrel{⌣}{\subseteq }<H,A>$ then $<H\stackrel{⌣}{\cup }K,A>$ is a soft subgroup of G. But the converse is not true.

Example 4.10.

Let $G=\mathbb{Z}=$ the set of integers and $A=\mathbb{N}=$ the set of natural numbers.

Let $A\times G\times G\times G$ be given by $\ast =\{<\alpha ,a,b,c>:\text{break}c=a+b-6\alpha \}$.Define $H,K:A\to p(G)$ by

${\displaystyle H(\alpha )=\{\begin{array}{ll}\{2k\alpha :k\in \mathbb{Z}\}& if\alpha =1\\ \mathbb{Z}& if\alpha =2\\ \{6k\alpha :k\in \mathbb{Z}\}& if\alpha \ge 3\end{array}}$

and

${\displaystyle K(\alpha )=\{\begin{array}{ll}\mathbb{Z}& if\alpha =1\\ \{3k\alpha :k\in \mathbb{Z}\}& if\alpha =2\\ \{6k\alpha :k\in \mathbb{Z}\}& if\alpha \ge 3\end{array}}$

for all $\alpha \in A$. Then$<H\stackrel{⌣}{\cup }K,A>$ is a soft subgroup of G but $<H,A>\stackrel{⌣}{⊈}<K,A>$ and $<K,A>\stackrel{⌣}{⊈}<H,A>$.

Definition 4.11.

Let $<H,A>$ and $<K,A>$ be soft subgroups of $<G,\ast ,A>$. Define soft sets $<HK,A>$ and $<H^{-1},A>$ over G respectively as follows:

$\begin{array}{ll}HK(\alpha )=& \{x\in G:\mathrm{\exists }a\in H(\alpha )\text{and }b\in K(\alpha )\\ & \text{such that }<\alpha ,a,b,x>\in \ast \}\\ H^{-1}(\alpha )=& \{x\in G:x^{-\alpha }\in H(\alpha )\}\end{array}$

Theorem 4.12.

The product $<HK,A>$ is a soft subgroup of $<G,\ast ,A>$ if and only if $<HK,A>\stackrel{⌣}{=}<KH,A>$

Proof.

Suppose that $<HK,A>$ is a soft subgroup of $<G,\ast ,A>$. We show that $HK(\alpha )=KH(\alpha )$ for all $\alpha \in A$. Let $\alpha \in A$ and $a\in HK(\alpha )$ then $a^{-\alpha }\in HK(\alpha ).$ This implies that there exist $h\in H(\alpha ),$ $k\in K(\alpha )$ such that $<\alpha ,h,k,a^{-\alpha }>\in \ast$, and so $k^{-\alpha }\in k(\alpha )$, $h^{-\alpha }\in H(\alpha )$ and $<\alpha ,k^{-\alpha },h^{-\alpha },a>\in \ast .$ Thus $a\in KH(\alpha )$ and hence $HK(\alpha )\subseteq KH(\alpha )$. Similarly, it can be shown that $KH(\alpha )\subseteq HK(\alpha )$ and hence the equality holds. Conversely suppose that $HK(\alpha )=KH(\alpha )$ for all $\alpha \in A.$ Let $\alpha \in A$ be fixed. It is clear that $e_{\alpha }\in HK(\alpha ).$ Let $a,b\in HK(\alpha )$ and $x\in G$ such that $<\alpha ,a,b^{-\alpha },x>\in \ast .$ Then there exist $h_1,h_2\in H(\alpha )$ and $k_1,k_2\in K(\alpha )$ such that $<\alpha ,h_1,k_1,a>\in \ast$ and $<\alpha ,h_2,k_2,b>\in \ast .$ It implies that $<\alpha ,k_2^{-\alpha },h_2^{-\alpha },b^{-\alpha }>\in \ast .$ Now let x₁,x₂ and x₃ $\in G$ such that $<\alpha ,k_1,k_2^{-\alpha },x_1>\in \ast$, $<\alpha ,x_1,h_2^{-\alpha },x_2>\in \ast$ and $<\alpha ,h_1,x_2,x_3>\in \ast .$ Since k₁,$k_2\in k(\alpha )$, $x_1\in K(\alpha )$ and as $h_2\in H(\alpha )$, $x_2\in KH(\alpha )=HK(\alpha ).$ Which implies that $x_2\in HK(\alpha ).$ Then there exists $y_1\in H(\alpha )$ and $y_2\in k(\alpha )$ such that $<\alpha ,y_1,y_2,x_2>\in \ast .$ If $z\in G$ and that $<\alpha ,h_1,y_1,z>\in \ast$ then , $z\in H(\alpha )$ such that $<\alpha ,z,y_2,x_3>\in \ast .$ Also by associative property of $\ast ,$ we have $x=x_3.$ That is $<\alpha ,z,y_2,x>\in \ast ,$ where $z\in H(\alpha )$ and $y_2\in K(\alpha ).$ Thus, $x\in HK(\alpha ).$ Therefore, $<HK,A>$ is a soft subgroup of G.

In the following theorem, we characterize soft subgroups using the product and inverse operations defined on the class of soft sets over G.

Theorem 4.13.

A soft set $<H,A>$ over a soft group G is a soft subgroup of G if and only if

(1)	$e_{\alpha }\in H(\alpha )$ $\mathrm{\forall }\alpha \in A$;
(2)	$<HH,A>\stackrel{⌣}{\subseteq }<H,A>$;
(3)	$<H^{-1},A>\stackrel{⌣}{\subseteq }<H,A>$.

Proof.

Suppose that $<H,A>$ is a soft subgroup of G. Clearly $e_{\alpha }\in H(\alpha ).$ Let $x\in HH(\alpha )$ then there exist $h_1,h_2\in H(\alpha )$ such that $<\alpha ,h_1,h_2,x>\in \ast .$ It follows that $x\in H(\alpha ).$ Thus $<HH,A>\stackrel{⌣}{\subseteq }<H,A>.$ Let $x\in H^{-1}(\alpha )$ then $x^{-\alpha }\in H(\alpha ).$ Which implies that $x\in H(\alpha ).$ Therefore $<H^{-1},A>\stackrel{⌣}{\subseteq }<H,A>.$ Conversely we show that $<H,A>$ is a soft subgroup of G. By (1), $e_{\alpha }\in H(\alpha ).$ Let $x\in H(\alpha ).$ Which implies that $x^{-\alpha }\in H^{-1}(\alpha ).$ By (3), we get $x^{-\alpha }\in H(\alpha ).$ Let $a,b\in H(\alpha )$ and $<\alpha ,a,b,x>\in \ast .$ Which implies that $x\in HH(\alpha ),$ so by (2), $x\in H(\alpha ).$ Therefore $<H,A>$ is a soft subgroup of G.

Definition 4.14.

We say that a soft group $<G,\ast ,A>$ is abelian if for each $\alpha \in A$ and any $a,b,x\in G$, $<\alpha ,a,b,x>\in \ast$ if and only if $<\alpha ,b,a,x>\in \ast .$

Example 4.15.

Soft groups given in Example 3.3 and 3.4 are abelian, whereas soft groups given in Example 3.5 and 3.6 are non-abelian.

Lemma 4.16.

A soft group $<G,\ast ,A>$ is abelian if and only if for each $\alpha \in A$ and all $a,b,x,y\in G$ it holds that: $<\alpha ,a,b,x>\in \ast$ and $<\alpha ,b,a,y>\in \ast$ together imply x = y.

Lemma 4.17.

A soft group $<G,\ast ,A>$ is abelian if and only if the classical group $S{E}_A(G)$ is an abelian group.

Theorem 4.18.

Let $<H,A>$ and $<K,A>$ be soft subgroups of a soft group $<G,\ast ,A>$. If G is abelian, then $<HK,A>$ is the least soft subgroup of G containing both $<H,A>$ and $<K,A>$.

Proof.

Since G is abelian soft group, then $<HK,A>\stackrel{⌣}{=}<KH,A>.$ So $<HK,A>$ is a soft subgroup of G. Let $<M,A>$ be a soft subgroup of G containing both $<H,A>$ and $<K,A>$. We show that $<HK,A>\stackrel{⌣}{\subseteq }<M,A>$. Let $x\in HK(\alpha )$, $\mathrm{\exists }$ $h\in H(\alpha )$ and $k\in K(\alpha )$ such that $<\alpha ,h,k,x>\in \ast$. This implies that $x\in M(\alpha )$. Therefore, $<HK,A>$ is the least soft subgroup of G containing both $<H,A>$ and $<K,A>$.

Definition 4.19.

For any soft set $<F,A>$ over a soft group G, define a soft set $<C_F,A>$ over G by

$\begin{array}{ll}{C}_F(\alpha )=& \{a\in G:<\alpha ,a,x,y_1>\in \ast ,<\alpha ,x,a,\\ & y_2>\in \ast \Rightarrow y_1=y_2,\mathrm{\forall }x\in F(\alpha ),y_1,y_2\in G\}\end{array}$

We call $<C_F,A>$ a centralizer of $<F,A>.$

Lemma 4.20.

Let $<F,A>$ be a soft set over a soft group G, $\alpha \in A$ and $a\in G$. Then $a\in C_F(\alpha )$ is equivalent to the condition $<\alpha ,a,x,y>\in \ast$ if and only if $<\alpha ,x,a,y>\in \ast$ for all $x\in F(\alpha )$ and all $y\in G$.

Theorem 4.21.

For and soft set $<F,A>$ over a soft group G, $<C_F,A>$ is a soft subgroup of G.

Proof.

For any $\alpha \in A$, all $x\in F(\alpha )$ and all $y\in G$, it is true that $<\alpha ,e_{\alpha },x,y>\in \ast$ if and only if $<\alpha ,x,e_{\alpha },y>\in \ast .$ Which implies that $e_{\alpha }\in C_F(\alpha ).$ Let $a\in C_F(\alpha )$. Then for any $x\in F(\alpha )$ and all $y\in G$ it is the case that $<\alpha ,a,x,y>\in \ast$ if and only if $<\alpha ,x,a,y>\in \ast$. We show that $a^{-\alpha }\in C_F(\alpha )$. Let $x\in F(\alpha )$ and $y,z\in G$ such that $<\alpha ,a^{-\alpha },x,y>\in \ast$ and $<\alpha ,x,a,z>\in \ast$. Since $a\in C_F(\alpha )$ and $x\in F(\alpha )$ we have $<\alpha ,a,x,z>\in \ast$. Using the associativity of $<\ast ,A>$ we get $<\alpha ,y,a,x>\in \ast$, which implies that $<\alpha ,x,a^{-\alpha },y>\in \ast$. Therefore $a^{-\alpha }\in C_F(\alpha )$. Next let $a,b\in C_F(\alpha )$ and $c\in G$ such that $<\alpha ,a,b,c>\in \ast$. Let $x\in F(\alpha )$ and $y,z,w\in G$ such that: $<\alpha ,c,x,y>\in \ast$, $<\alpha ,a,x,z>\in \ast$ and $<\alpha ,b,x,w>\in \ast$. Since $b\in C_F(\alpha )$ it holds that $<\alpha ,x,b,w>\in \ast$ which implies that $<\alpha ,z,b,y>\in \ast$. Again as $a\in C_F(\alpha )$ it holds that $<\alpha ,x,a,z>\in \ast$ implying $<\alpha ,x,c,y>\in \ast$. Thus, $c\in C_F(\alpha )$ and this is true for all $\alpha \in A$. Therefore, $<C_F(\alpha ),A>$ is a soft subgroup of G.

Definition 4.22.

Let $<X,A>$ be a soft set over a soft group G. The smallest soft subgroup of G containing $<X,A>$ is called the soft subgroup of G generated by $<X,A>$ and is denoted by $S{g}_A(X)$. That is $(S{g}_A(X),A)=\bigcap ^{⌣}\{<H,A>:<H,A>$ is a soft subgroup of G such that $<X,A>\stackrel{⌣}{\subseteq }<H,A>\}.$

Theorem 4.23.

The soft subgroup of G generated by a soft set $<F,A>$ over G can be described as follows : for $\alpha \in A$, if $F(\alpha )=\mathrm{\varnothing }$, then $S{g}_A(F)(\alpha )=\{e_{\alpha }\}$. If $F(\alpha )\ne \mathrm{\varnothing }$, then $x\in S{g}_A(F)(\alpha )$ if and only if there exist $n\in \mathbb{N}$ and sequences $\{a_i{\}}_{i=1}^n$ and $\{x_i{\}}_{i=1}^n$ of elements of G such that $x_1=a_1,x_n=x$ and $<\alpha ,x_i,a_{i+1},x_{i+1}>\in \ast$ $\mathrm{\forall }i=1,2,3,...n-1$ where for each i, either $a_i\in F(\alpha )$ or $a_i^{-\alpha }\in F(\alpha )$

Proof.

Let $x,y\in S{g}_A(F)(\alpha ).$ Then there exist $n,m\in \mathbb{N}$ and sequences $\{a_i{\}}_{i=1}^n$ , $\{b_j{\}}_{j=1}^m$ and $\{x_i{\}}_{i=1}^n$ , $\{y_j{\}}_{j=1}^m$ of element of G such that $x_1=a_1,x_n=x$ and $<\alpha ,x_i,a_{i+1},x_{i+1}>\in \ast$,$y_1=b_1,y_m=y$ and $<\alpha ,y_j,b_{j+1},y_{j+1}>\in \ast$ for all $i=1,2,3,...n-1$ and $j=1,2,3,...m-1.$ Let $z\in G$ be such that $<\alpha ,x,y^{-\alpha },z>\in \ast .$ We need to show that $z\in S{g}_A(F)(\alpha ).$ Consider the sequence $\{c_k{\}}_{k=1}^{n+1}$ of elements of G defined by $c_k=a_k$ for $1⩽k⩽n$ and $c_{n+k}=b_{m-k+1}^{-\alpha }$ for $1⩽k⩽m.$ Then, for each $k=1,2,3,...n+m$ either $c_k\in F(\alpha )$ or $c_k^{-\alpha }\in F(\alpha ).$ Moreover, if we define a sequence $\{z_k{\}}_{k=1}^{n+m}$ by $z_1=c_1=a_1,=x_1$ and $<\alpha ,z_i,c_{i+1},z_{i+1}>\in \ast$ $\mathrm{\forall }i=1,2,3,...n+m-1.$ Then, using associativity of $\ast$, it can be shown that $z_{n+m}=z.$ Thus $z\in S{g}_A(F)(\alpha ).$ Therefore $<S{g}_A(F),A>$ is a soft subgroup of G. Next we show that $<F,A>\stackrel{⌣}{\subseteq }<S{g}_A(F),A>.$ Let $\alpha \in A$ and $x\in F(\alpha )$. Now considering sequences $\{x_i{\}}_{i=1}^n$ and $\{a_i{\}}_{i=1}^n$ taking n = 1, where $a_1=x=x_1.$ Then it is vacuously true that $<\alpha ,x_i,a_{i+1},x_{i+1}>\in \ast$ for all $1⩽i⩽n$ and hence $x\in S{g}_A(F)(\alpha ).$ Thus $<F,A>\stackrel{⌣}{\subseteq }<S{g}_A(F),A>.$ Now let $<H,A>$ be any other soft subgroup of G such that $<F,A>\stackrel{⌣}{\subseteq }<H,A>.$ Then $F(\alpha )\subseteq H(\alpha )\mathrm{\forall }\alpha \in A.$ We next show that $S{g}_A(F)(\alpha )\subseteq H(\alpha )\mathrm{\forall }\alpha \in A.$ Let $\alpha \in A$ and $x\in S{g}_A(F)(\alpha ).$ Then there exist $n\in \mathbb{N}$ and sequences $\{a_i{\}}_{i=1}^n$ and $\{x_i{\}}_{i=1}^n$ of elements of G such that $x_1=a_1,x_n=x$ and $<\alpha ,x_i,a_{i+1},x_{i+1}>\in \ast .$ $\mathrm{\forall }i=1,2,3,...n-1$ where for each i,$1⩽i⩽n$ either $a_i\in F(\alpha )$ or $a_i^{-\alpha }\in F(\alpha ).$ Since $F(\alpha )\subseteq H(\alpha )$ and $<H,A>$ is a soft subgroup of G. We get that $a_i\in H(\alpha )$, $\mathrm{\forall }1⩽i⩽n.$ As $x_1=a_1$, and $<\alpha ,x_1,a_2,x_2>\in \ast$ we get that $x_2\in H(\alpha ).$ Similarly, from $<\alpha ,x_2,a_3,x_3>\in \ast$ we get that $x_3\in H(\alpha ).$ Continuing this process until all x_i are exhausted we get finally $x_n\in H(\alpha ).$ That is $x\in H(\alpha ).$ Thus $S{g}_A(F)(\alpha )\subseteq H(\alpha )$. As $\alpha \in A$ is arbitrary it can be concluded that $<S{g}_A(F),A>\stackrel{⌣}{\subseteq }<H,A>.$ Therefore $<S{g}_A(F),A>$ is the smallest soft subgroup of G containing $<F,A>.$

Theorem 4.24.

The soft subgroup of G generated by the soft element $<\stackrel{⌣}{a},A>$ can be described as follows: for $x\in G$, $x\in S{g}_A(\stackrel{⌣}{a})(\alpha )$ if and only if there is $n\in \mathbb{N}$ and a sequence $x_1,x_2,...x_n$ in G such that $<\alpha ,x_i,x_1,x_{i+1}>\in \ast$ $\mathrm{\forall }i=1,2,3,...n-1$; where $x_1\in \{a_{\alpha },a_{\alpha }^{-\alpha }\}$ and $x_n=x$.

Definition 4.25.

We call $<S{g}_A(\stackrel{⌣}{a}),A>$ the cyclic soft subgroup of G generated by the soft element $<\stackrel{⌣}{a},A>$ over $G.$

Lemma 4.26.

For any soft sets $<F,A>$ and $<E,A>$ over G, the following hold:

(1)	$<F,A>\stackrel{⌣}{\subseteq }<S{g}_A(F),A>$;
(2)	$<F,A>\stackrel{⌣}{\subseteq }<E,A>\Rightarrow <S{g}_A(F),A>\stackrel{⌣}{\subseteq }<S{g}_A(E),A>$
(3)	$<S{g}_A(S{g}_A(F)),A>\stackrel{⌣}{=}<S{g}_A(F),A>$.

The results in Lemma 4.26 justifies that the map $<F,A>\mapsto <S{g}_A(F),A>$ forms a closure operator on the class of all soft sets over G with a fixed set A of parameters, and the closed elements with respect to this closure operator are precisely those soft subgroups of G.

Theorem 4.27.

Let $<G,\ast ,A>$ be a soft group and $S{S}_A(G)$ the class of all soft subgroups of G. Then, $<S{S}_A(G),\wedge ,\vee >$ is a complete lattice; where for soft subgroups $<H,A>$ and $<K,A>$ of G:

$<H,A>\wedge <K,A>=<H,A>\stackrel{⌣}{\cap }<K,A>$

and

$<H,A>\vee <K,A>=<S{g}_A(H\stackrel{⌣}{\cup }K),A>$

Proof.

It is given in Example 4.2 the absolute soft set $<1_G,A>$ over G is a soft subgroup of G and hence $S{S}_A(G)$ has the largest element. It is also proved in Lemma 4.6 that $S{S}_A(G)$ is closed under the arbitrary intersection of soft sets, and hence closed under arbitrary infima. Therefore, it is a complete lattice.

Theorem 4.28.

The lattice $S{S}_A(G)$ can be embedded into the lattice of all subgroups of the classical group $S{E}_A(G)$.

Proof.

It is enough to show that the map sending each soft group $<H,A>$ of G to a subgroup $\hat{H}$ of $S{E}_A(G)$ is an injective order homomorphism.

5. Normal soft subgroups

Definition 5.1.

Let $<H,A>$ be a soft subgroup of a soft group $<G,\ast ,A>$ and $a\in G$. Define a soft set $<{}^{a}H,A>$ over G by:

$\begin{array}{l}{}^{a}H(\alpha )=\{x\in G:<\alpha ,a,b,x>\in \ast forsomeb\in H(\alpha )\}.\end{array}$

We call $<{}^{a}H,A>$ a left coset of $<H,A>$ corresponding to a. Right cosets can be defined in a dual manner.

Theorem 5.2.

Let $<H,A>$ be a soft subgroup of G and let $a,b\in G$.

(1)	$<{}^{a}H,A>\stackrel{⌣}{\subseteq }<{}^{b}H,A>$ if and only if for $x\in G,$ $<\alpha ,b^{-\alpha },a,x>\in \ast \Rightarrow x\in H(\alpha )$;
(2)	$<{}^{a}H,A>\stackrel{⌣}{=}<H,A>$ if and only if $a\in H(\alpha )$ for all α in A;
(3)	Either $<{}^{a}H,A>$ and $<{}^{b}H,A>$ are weakly disjoint or $<{}^{a}H,A>\stackrel{⌣}{=}<{}^{b}H,A>$.

Proof.

(1)	Suppose that $<{}^{a}H,A>\stackrel{⌣}{\subseteq }<{}^{b}H,A>.$ Then, $[{}^{a}H](\alpha )\stackrel{⌣}{\subseteq }[{}^{b}H](\alpha )$ $\mathrm{\forall }\alpha \in A.$ Let $x\in G$ such that $<\alpha ,b^{-\alpha },a,x>\in \ast .$ Then, (1) $<\alpha ,b,x,a>\in \ast .$(1) Since $e_{\alpha }\in H(\alpha )$ and $<\alpha ,a,e_{\alpha },a>\in \ast .$ We have $a\in [{}^{a}H](\alpha )=[{}^{b}H](\alpha ).$ Which implies that $a\in [{}^{b}H](\alpha ).$ So there is some $y\in H(\alpha )$ such that (2) $<\alpha ,b,y,a>\in \ast .$(2) Applying the cancellation law on (1(1) $<\alpha ,b,x,a>\in \ast .$(1) ) and (2(2) $<\alpha ,b,y,a>\in \ast .$(2) ), we get $x=y\in H(\alpha ).$ Conversely suppose that for any $x\in G$ it is the case that $<\alpha ,b^{-\alpha },a,x>\text{break}\in \ast \Rightarrow x\in H(\alpha ).$ Let $\alpha \in A$ and $x\in [{}^{a}H](\alpha )$ then there exist $y\in H(\alpha )$ such that $<\alpha ,a,y,x>\in \ast .$ Let $x_1,x_2,x_3\in G$ such that $<\alpha ,b^{-\alpha },a,x_1>\in \ast$,$<\alpha ,x_1,y,x_2>\in \ast$ and $<\alpha ,b^{-\alpha },x,x_3>\in \ast .$ Then by associative property of $\ast$, we get $x_2=x_3.$ Moreover it follows from our hypothesis that $x_1\in H(\alpha ).$ Since $x_1\in H(\alpha )$ and $y\in H(\alpha ).$ We get $x_2\in H(\alpha ).$ That is $x_3\in H(\alpha ).$ Further since $<\alpha ,b^{-\alpha },x,x_3>\in \ast .$ We have $<\alpha ,b,x_3,x>\in \ast$ and $x_3\in H(\alpha ).$ So that $x\in [{}^{b}H](\alpha ).$ Therefore $[{}^{a}H](\alpha )\stackrel{⌣}{\subseteq }[{}^{b}H](\alpha ).$ Since α is arbitrary, we get $<{}^{a}H,A>\stackrel{⌣}{\subseteq }<{}^{b}H,A>.$
(2)	Let $\alpha \in A$ and $a\in G.$ Since $e_{\alpha }\in H(\alpha )$ such that $<\alpha ,a,e_{\alpha },a>\in \ast .$ This implies that $a\in H(\alpha ).$ Conversely suppose that $a\in H(\alpha ).$ Let $\alpha \in A$ and $h\in H(\alpha )$ such that $<\alpha ,a,a^{-\alpha },h>\in \ast .$ This implies that $h\in {}^{a}H(\alpha ).$ Therefore $H(\alpha )\subseteq {}^{a}H(\alpha ).$ Let $x\in {}^{a}H(\alpha )$ then there exist $h\in H(\alpha )$ such that $<\alpha ,a,h,x>\in \ast .$ This implies that $x\in H(\alpha ).$ It follows that ${}^{a}H(\alpha )\subseteq H(\alpha ).$ Thus, $H(\alpha )={}^{a}H(\alpha ).$ Therefore $<{}^{a}H,A>\stackrel{⌣}{=}<H,A>$
(3)	Suppose that $<{}^{a}H,A>$ and $<{}^{b}H,A>$ are not weakly disjoint. That is ${}^{a}H(\alpha )\stackrel{⌣}{\cap }{}^{b}H(\alpha )\ne \mathrm{\varnothing }$ $\mathrm{\forall }\alpha \in A.$ This implies that there exist at least one $x\in {}^{a}H(\alpha )\cap {}^{b}H(\alpha ).$ This implies that $x\in {}^{a}H(\alpha )$ and $x\in {}^{b}H(\alpha ),\mathrm{\forall }\alpha \in A.$ It follows that there exist $h_1,h_2\in H(\alpha )$ such that $<\alpha ,a,h_1,x>\in \ast$ and $<\alpha ,b,h_2,x>\in \ast .$ Let $y\in G$ such that $<\alpha ,x,h_1^{-\alpha },y>\in \ast .$ Consider $<\alpha ,h_1,h_1^{-\alpha },e_{\alpha }>\in \ast$, $<\alpha ,a,e_{\alpha },a>\in \ast$ and $<\alpha ,a,h_1,x>\in \ast$, $<\alpha ,x,h_1^{-\alpha },y>\in \ast .$ Which implies that $y=a.(1)$ Let $z\in G$ and $h_3\in H(\alpha )$ such that $<\alpha ,b,h_2,x>\in \ast$, $<\alpha ,x,h_1^{-\alpha },y>\in \ast$ and $<\alpha ,h_2,h_1^{-\alpha },h_3>\in \ast$, $<\alpha ,b,h_3,z>\in \ast .$ Which implies that $y=z.(2)$ From (1) and (2) we have $a=z.$ Let $x_1\in {}^{a}H(\alpha ).$ Which implies that there exist $h_4\in H(\alpha )$ such that $<\alpha ,a,h_4,x_1>\in \ast .$ Since $<H,A>$ is a normal soft subgroup. If $<\alpha ,h_3,h_4,h_5>\in \ast$ then $h_5\in H(\alpha ).$ This implies that $<\alpha ,b,h_5,x_1>\in \ast .$ It follows that $x_1\in {}^{b}H(\alpha ).$ Therefore ${}^{a}H(\alpha )\subseteq {}^{b}H(\alpha ).$ Similarly ${}^{b}H(\alpha )\subseteq {}^{a}H(\alpha ).$ Hence ${}^{a}H(\alpha )={}^{b}H(\alpha ).$ This concludes our argument that $<{}^{a}H,A>\stackrel{⌣}{=}<{}^{b}H,A>$.

Definition 5.3.

A soft subgroup $<N,A>$ of a soft group $<G,\ast ,A>$ is called normal if $<{}^{a}N,A>\stackrel{⌣}{=}<N^a,A>$ for all $a\in G.$

Notation: We denote by $S{N}_A(G)$ the collection of all normal soft subgroups of G with the set of parameters of A.

Theorem 5.4.

For a soft subgroup $<N,A>$ of G, the following are equivalent:

(1)	$<N,A>$ is normal;
(2)	for $\alpha \in A,$ any a,x,y $\in G$ and any $n\in N(\alpha ),<\alpha ,a,n,x>\in \ast$ and $<\alpha ,x,a^{-\alpha },y>\in \ast$ together imply $y\in N(\alpha )$.

Proof.

$(1\Rightarrow 2)$ Suppose that $<N,A>$ is normal. That is ${}^{a}N(\alpha )=N^a(\alpha ),$ $\mathrm{\forall }\alpha \in A.$ Let $\alpha \in A$ be fixed, $a,x,y\in G$ and $n\in N(\alpha )$ such that $<\alpha ,a,n,x>\in \ast$ and $<\alpha ,x,a^{-\alpha },y>\in \ast .$ Then $x\in {}^{a}N(\alpha )$ and $<\alpha ,y,a,x>\in \ast .$ Since ${}^{a}N(\alpha )=N^a(\alpha )$ there is some $n_1\in N(\alpha )$ such that $<\alpha ,n_1,a,x>\in \ast$. Then, by the cancellation law we get $y=n_1\in N(\alpha ).$

$(2\Rightarrow 1)$ Let $a,y\in G$ and $x\in {}^{a}N(\alpha )$. Then, there exists $n\in N(\alpha )$ such that $<\alpha ,a,n,x>\in \ast .$ By assumption $<\alpha ,a,n,x>\in \ast$ and $<\alpha ,x,a^{-\alpha },y>\in \ast$ which implies that $y\in N(\alpha ).$ As $<\alpha ,x,a^{-\alpha },y>\text{break}\in \ast \Rightarrow <\alpha ,y,a,x>\in \ast .$ Since $y\in N(\alpha )$ then $x\in N^a(\alpha ).$ Therefore ${}^{a}N(\alpha )\subseteq N^a(\alpha ).$ From the assumption $<\alpha ,a,n,x>\in \ast$ and $<\alpha ,x,a^{-\alpha },y>\in \ast$ we get $x\in {}^{a}N(\alpha ).$ Thus, $N^a(\alpha )\subseteq {}^{a}N(\alpha ).$ Therefore $<N,A>$ is normal soft subgroup of G.

Theorem 5.5.

A soft subgroup $<N,A>$ of G is normal if and only if $\hat{N}$ is a normal subgroup of $S{E}_A(G)$.

Proof.

The proof is similar to that of Theorem 4.5.

Definition 5.6.

Let $<G,\ast ,A>$ be a soft group. The center of G denoted by $<Z_A(G),A>$ is a parameterized soft set over G defined by for each $\alpha \in A$:

$\begin{array}{ll}{Z}_A(G)(\alpha )=& \{a\in G:<\alpha ,a,x,y>\in \ast \iff <\alpha ,x,\\ & a,y>\in \ast forallx,y\in G\}\end{array}$

Theorem 5.7.

For any soft group $<G,\ast ,A>$ its center $<Z_A(G),A>$ is a normal soft subgroup of G.

Proof.

We first show that $<Z_A(G),A>$ is a soft subgroup of G. For any $\alpha \in A$ and $x,y\in G.$ $<\alpha ,e_{\alpha },x,y>\in \ast .$ This implies that y = x and we have $<\alpha ,x,e_{\alpha },x>\in \ast .$ Thus $e_{\alpha }\in Z_A(G)(\alpha ).$ Now let $a,b\in Z_A(G)(\alpha )$ and $c\in G$ such that $<\alpha ,a,b^{-\alpha },c>\in \ast .$ Claim: $c\in Z_A(G)(\alpha ).$ We first show that $b^{-\alpha }\in Z_A(G)(\alpha ).$ Let $x,y\in G$ such that $<\alpha ,b^{-\alpha },x,y>\in \ast .$ Then $<\alpha ,x^{-\alpha },b,y^{-\alpha }>\in \ast .$ Since $b\in Z_A(G)(\alpha )$ we get that $<\alpha ,b,x^{-\alpha },y^{-\alpha }>\in \ast .$ This implies that $<\alpha ,x,b^{-\alpha },y>\in \ast .$ Similarly it can be shown that $<\alpha ,x,b^{-\alpha },y>\in \ast$ this implies that $<\alpha ,b^{-\alpha },x,y>\in \ast$ and hence $b^{-\alpha }\in Z_A(G)(\alpha ).$ Let $x,y,z\in G$ such that $<\alpha ,c,x,y>\in \ast$ and $<\alpha ,b^{-\alpha },x,z>\in \ast .$ Then using the associative property of $<\ast ,A>$ we get that $<\alpha ,a,z,y>\in \ast .$ Since $a,b^{-\alpha }\in Z_A(G)(\alpha )$,it follows that $<\alpha ,x,b^{-\alpha },z>\in \ast$ and $<\alpha ,z,a,y>\in \ast .$ If $u\in G$ such that $<\alpha ,b^{-\alpha },a,u>\in \ast ,$ then by the associativity of $\ast$ we get $<\alpha ,x,u,y>\in \ast .$ Moreover, as $a\in Z_A(G)(\alpha )$ it holds that $<\alpha ,a,b^{-\alpha },u>\in \ast .$ Thus u = c and hence $<\alpha ,x,c,y>\in \ast$. That is, the implication: $<\alpha ,c,x,y>\in \ast \Rightarrow <\alpha ,x,c,y>\in \ast$ holds $\mathrm{\forall }x,y\in G.$ By symmetry, the other side of the implication holds and then $c\in Z_A(G)(\alpha ).$ Therefore $<Z_A(G),A>$ is a soft subgroup of G. Next we show normality. Let $a\in Z_A(G)(\alpha )$ and $g,x,y\in G$ such that

(3) $\begin{array}{l}<\alpha ,g,a,x>\in \ast \end{array}$(3)

and

(4) $\begin{array}{l}<\alpha ,x,g^{-\alpha },y>\in \ast \end{array}$(4)

Claim: $y\in Z_A(G)(\alpha ).$ It is enough to show that $y=a.$ As $a\in Z_A(G)(\alpha ),$ it follows from (3(3) $\begin{array}{l}<\alpha ,g,a,x>\in \ast \end{array}$(3) ) we get that

(5) $\begin{array}{l}<\alpha ,a,g,x>\in \ast \end{array}$(5)

Then (4(4) $\begin{array}{l}<\alpha ,x,g^{-\alpha },y>\in \ast \end{array}$(4) ) and (5(5) $\begin{array}{l}<\alpha ,a,g,x>\in \ast \end{array}$(5) ) together imply that $y=a\in Z_A(G)(\alpha )$ (using associative of $<\ast ,A>$). Therefore $<Z_A(G),A>$ is a normal soft subgroup of G.

6. Soft congruences

Definition 6.1.

A soft equivalence relation $<\theta ,A>$ on a soft group $<G,\ast ,A>$ is called a soft congruence relation if for each $\alpha \in A$ and any $a,b,c,d,x,y\in G$ with $<\alpha ,a,c,x>\in \ast$ and $<\alpha ,b,d,y>\in \ast$; $<a,b>$, $<c,d>\in \theta (\alpha )$ imply $<x,y>\in \theta (\alpha ).$

Notation: The collection of all soft congruence relations on G with the set of parameters A will be denoted by $SCo{n}_A(G)$ .

Let $<\theta ,A>$ be a soft congruence relation on G. Define the soft equivalence class ${\theta }_a:A\to P(G)$ of θ determined by $a\in G$ as follows:

${\theta }_a(\alpha )=\{x\in G:<a,x>\in \theta (\alpha )\}.$

Then we have the following properties.

Lemma 6.2.

Let $<\theta ,A>$ be a soft congruence relation on G and $a,b\in G.$ Then,

(1)	$<{\theta }_a,A>\stackrel{⌣}{=}<{\theta }_b,A>$ if and only if $<a,b>\in \theta (\alpha )$ $\mathrm{\forall }\alpha \in A.$
(2)	Either $<{\theta }_a,A>\stackrel{⌣}{=}<{\theta }_b,A>$ or $<{\theta }_a,A>$ and $<{\theta }_b,A>$ are weakly disjoint. i.e $\mathrm{\exists }\alpha \in A$ such that ${\theta }_a(\alpha )\cap {\theta }_b(\alpha )=\mathrm{\varnothing }$

Proof.

(1)

Suppose $<{\theta }_a,A>\stackrel{⌣}{=}<{\theta }_b,A>.$ Then ${\theta }_a(\alpha )={\theta }_b(\alpha )$, $\mathrm{\forall }\alpha \in A.$ Since $e_{\alpha }\in {\theta }_a(\alpha )$ we have $<a,e_{\alpha }>\in \theta (\alpha )$ and $e_{\alpha }\in {\theta }_b(\alpha )$ we have $<b,e_{\alpha }>\in \theta (\alpha ).$ As $<b,e_{\alpha }>\in \theta (\alpha )\Rightarrow <e_{\alpha },b>\in \theta (\alpha ).$ From $<a,e_{\alpha }>\in \theta (\alpha )$ and $<e_{\alpha },b>\in \theta (\alpha )$ we get $<a,b>\in \theta (\alpha ),$ $\mathrm{\forall }\alpha \in A.$ Conversely $<a,b>\in \theta (\alpha )$ $\mathrm{\forall }\alpha \in A.$ We need to show that $<{\theta }_a,A>\stackrel{⌣}{=}<{\theta }_b,A>.$ Let $x\in {\theta }_a(\alpha )$. Then $<a,x>\in \theta (\alpha ).$ Since $<a,b>\in \theta (\alpha )$, we get $<b,x>\in \theta (\alpha )$ and hence $x\in {\theta }_b(\alpha ).$ Therefore ${\theta }_a(\alpha )\subseteq {\theta }_b(\alpha ).$ Similarly, it can be shown that ${\theta }_b(\alpha )\subseteq {\theta }_a(\alpha ).$ Thus $<{\theta }_a,A>\stackrel{⌣}{=}<{\theta }_b,A>.$

(2)

Suppose $<{\theta }_a,A>$ and $<{\theta }_b,A>$ are not weakly disjoint. This means that $\mathrm{\forall }\alpha \in A,$ ${\theta }_a(\alpha )\cap {\theta }_b(\alpha )\ne \mathrm{\varnothing }.$ It follows that there exist at least one $x\in {\theta }_a(\alpha )\cap {\theta }_b(\alpha ).$ This implies that $x\in {\theta }_a(\alpha )$ and $x\in {\theta }_b(\alpha ).$ So $<a,x>\in \theta (\alpha )$ and $<b,x>\in \theta (\alpha )$ implying that $<a,b>\in \theta (\alpha )$ $\mathrm{\forall }\alpha \in A.$ Therefore, by (1) we get $<{\theta }_a,A>\stackrel{⌣}{=}<{\theta }_b,A>.$

For a soft congruence $<\theta ,A>$ on G define a soft set $<{\theta }_0,A>$ over G by

${\theta }_0(\alpha )=\{x\in G:<x,e_{\alpha }>\in \theta (\alpha )\}$

Theorem 6.3.

$<{\theta }_0,A>$ is a normal soft subgroup of G.

Proof.

Clearly, $e_{\alpha }\in {\theta }_0(\alpha ).$ Let $a,b\in {\theta }_0(\alpha )$. This implies that $<a,e_{\alpha }>\in \theta (\alpha )$ and $<b,e_{\alpha }>\in \theta (\alpha ).$ As $<b,e_{\alpha }>\in \theta (\alpha )$ which implies that $<b^{-\alpha },e_{\alpha }>\in \theta (\alpha ).$ Let $x\in G$ such that $<\alpha ,a,b^{-\alpha },x>\in \ast .$ Since $<a,e_{\alpha }>\in \theta (\alpha )$ and $<b^{-\alpha },e_{\alpha }>\in \theta (\alpha )$ we have $<x,e_{\alpha }>\in \theta (\alpha )$,$\mathrm{\forall }\alpha \in A.$ This implies that $x\in {\theta }_0(\alpha ).$ Let $a,x,y\in G$ and $n\in {\theta }_0(\alpha ).$ Suppose $<\alpha ,a,n,x>\in \ast$ and $<\alpha ,x,a^{-\alpha },y>\in \ast .$ As $n\in {\theta }_0(\alpha ),$ this implies that $<n,e_{\alpha }>\in \theta (\alpha ).$ Suppose $<x,a>\in \theta (\alpha )$ and $<a^{-\alpha },n>\in \theta (\alpha ).$ This implies that $<y,x>\in \theta (\alpha ).$ Since $<y,x>\in \theta (\alpha )$ and $<x,e_{\alpha }>\in \theta (\alpha )$, we have $<y,e_{\alpha }>\in \theta (\alpha ).$ It follows that $y\in {\theta }_0(\alpha ).$ Therefore $<{\theta }_0,A>$ is a normal soft subgroup of G.

Theorem 6.4.

Given a normal soft subgroup $<N,A>$ of G. Define a soft relation denoted by $<{\theta }^N,A>$ over G by:

${\theta }^N(\alpha )=\{<x,y>\in G\times G:<\alpha ,y^{-\alpha },x,z>\in \ast \Rightarrow z\in N(\alpha )\}.$

Then $<{\theta }^N,A>$ is a soft congruence relation on G.

Proof.

First we show that $<{\theta }^N,A>$ a soft equivalence relation. Let $\alpha \in A$ and $x\in G.$ Since $<\alpha ,x^{-\alpha },x,z>\in \ast .$ This implies that $z=e_{\alpha }\in N(\alpha ).$ It follows that $<x,x>\in {\theta }^N(\alpha ).$ Therefore $<{\theta }^N,A>$ is reflexive. Let $\alpha \in A$ and $x,y\in G.$ Suppose $<x,y>\in {\theta }^N(\alpha )$ such that $<\alpha ,y^{-\alpha },x,z>\in \ast$, we have $z\in N(\alpha ).$ Since $<\alpha ,x^{-\alpha },y,z^{-\alpha }>\in \ast$ and $z^{-\alpha }\in N(\alpha ).$ This implies that $<y,x>\in {\theta }^N(\alpha ).$ Therefore $<{\theta }^N,A>$ is symmetric. Let $\alpha \in A$ and $x,y,z\in G.$ Suppose $<x,y>\in {\theta }^N(\alpha )$ and $<y,z>\in {\theta }^N(\alpha ).$ Which implies that $<\alpha ,y^{-\alpha },x,z_1>\in \ast$, $<\alpha ,z^{-\alpha },y,z_2>\in \ast$ for some $z_1,z_2\in N(\alpha ).$ Let $z_3\in G$ such that $<\alpha ,z_2,z_1,z_3>\in \ast .$ This implies $z_3\in N(\alpha ).$ Since $<\alpha ,z^{-\alpha },x,z_3>\in \ast$ and $z_3\in N(\alpha )$, it follows that $<x,z>\in {\theta }^N(\alpha ).$ Therefore $<{\theta }^N,A>$ is transitive. Thus, $<{\theta }^N,A>$ a soft equivalence relation. Finally, let $\alpha \in A$ and $a,b,c,x,y\in G$ such that $<\alpha ,a,c,x>\in \ast$ and $<\alpha ,b,d,y>\in \ast .$ Suppose $<a,b>$,$<c,d>\in {\theta }^N(\alpha ).$ We need to prove that $<x,y>\in {\theta }^N(\alpha )$. Let $z\in G$ with the property $<\alpha ,y^{-\alpha },x,z>\in \ast$. As $<a,b>\in {\theta }^N(\alpha )$ we have $<\alpha ,b^{-\alpha },a,z_1>\in \ast$ for some $z_1\in N(\alpha ).$ Again as $<c,d>\in {\theta }^N(\alpha )$, there is some $z_2\in N(\alpha )$ such that $<\alpha ,d^{-\alpha },c,z_2>\in \ast$. Let $z_3,z_4,z_5,w\in G$ such that $<\alpha ,d^{-\alpha },z_1,z_4>\in \ast$, $<\alpha ,z_4,d,z_5>\in \ast$ and $<\alpha ,z_5,z_2,w>\in \ast$. Then, using the facts: $z_1,z_2\in N(\alpha )$ and $<N,A>$ is normal, one can easily check that $w\in N(\alpha )$. Moreover, by the associativity of $\ast$, we have z = w and hence $<x,y>\in {\theta }^N(\alpha ).$ Therefore $<{\theta }^N,A>$ is a soft congruence relation on G.

Theorem 6.5.

There is a lattice isomorphism between $S{N}_A(G)$ and $SCo{n}_A(G).$

Proof.

Let $g:S{N}_A(G)⟶SCo{n}_A(G)$ and $H:SCo{n}_A(G)⟶S{N}_A(G)$ be mapping defined by $g:<N,A>\mapsto <{\theta }^N,A>$ and $H:<\theta ,A>\mapsto <{\theta }_0,A>.$ We first show that g and h are mutually inverse to each other; that is, we show that $<[{\theta }^N{]}_0,A>\stackrel{⌣}{=}<N,A>$ and $<{\theta }^{[{\theta }_0]},A>\stackrel{⌣}{=}<\theta ,A>$ for all $<N,A>\in S{N}_A(G)$ and all $<\theta ,A>\in SCo{n}_A(G)$.

(1)

For each $\alpha \in A$ ; $\begin{array}{ll}{\left[{\theta }^N\right]}_o(\alpha )=& \{x\in G:<x,e_{\alpha }>\in {\theta }^N(\alpha )\}\\ =& \{x\in G:<\alpha ,e_{\alpha }^{-1},x,z>\ast \\ & forsomez\in N(\alpha )\}\\ =& \{x\in G:x=z\in N(\alpha )\}=N(\alpha )\end{array}$ Thus $<[{\theta }^N{]}_o,A>\stackrel{⌣}{=}<N,A>.$

(2)

For any $\alpha \in A;$ $<x,y>\in {\theta }^{[{\theta }_o]}(\alpha )$ if and only if $<\alpha ,y^{-\alpha },x,z>\ast forsomez\in {\theta }_0(\alpha )$. Equivalently,$<x,y>\in \theta (\alpha )$ showing that $<{\theta }^{[{\theta }_o]},A>\stackrel{⌣}{=}<\theta ,A>.$ Thus g and h are inverse to each other and hence both are one to one correspondences. It remains to show that both g and h are order homomorphisms. In other words, it is enough to show that $<N_1,A>\stackrel{⌣}{\subset }<N_2,A>$ if and only if $<{\theta }^{N_1},A>\stackrel{⌣}{\subset }<{\theta }^{N_2},A>.$ Suppose that $N_1(\alpha )\subseteq N_2(\alpha )$ for all $\alpha \in A$. Let $\alpha \in A$ and $(x,y)\in {\theta }^{N_1}(\alpha ).$ Then $<\alpha ,x,y^{-\alpha },z_1>\ast$ for some $z_1\in N_1(\alpha )$. That is, $z_1\in N_2(\alpha )$ and $<\alpha ,x,y^{-\alpha },z_1>\ast$. So $(x,y)\in {\theta }^{N_2}(\alpha ).$ Since α is arbitrary in A, we get $<{\theta }^{N_1},A>\stackrel{⌣}{\subset }<{\theta }^{N_2},A>.$ Conversely, suppose ${\theta }^{N_1}(\alpha )\subseteq {\theta }^{N_2}(\alpha )$ for all $\alpha \in A$. Let $\alpha \in A$ and $x\in N_1(\alpha )$. Then, it is clear that $<x,e_{\alpha }>\in {\theta }^{N_1}(\alpha )\subseteq {\theta }^{N_2}(\alpha )$ and hence $<x,e_{\alpha }>\in {\theta }^{N_2}(\alpha )$. Then, there exists some $\in N_2(\alpha )$ such that $<\alpha ,e_{\alpha },x,z>\in \ast$. Whence $x=z\in N_2(\alpha )$ and so $<N_1,A>\stackrel{⌣}{\subseteq }<N_2(\alpha ),A>$. Hence proved.

7. Direct products

Lemma 7.1.

Let $<G_1,{\ast }_1,A>,\dots <G_n,{\ast }_n,A>$ be soft groups with the same set of parameters A, and G be the set

$G=\prod _{i=1}^n{G}_i=\{<a_1,\dots ,a_n>:a_i\in G_i\mathrm{\forall }i=1,2,\dots ,n\}.$

Define a soft binary operation $<\ast ,A>$ on G by: $\ast \subseteq A\times G\times G\times G$ given by $<\alpha ,\overline{a},\overline{b},\overline{c}>\in \ast$ if and only if $<\alpha ,a_i,b_i,c_i>\in {\ast }_i$ for all $1⩽i⩽n.$ Where $\overline{a}=<a_1,\dots ,a_n>$, $\overline{b}=<b_1,\dots ,b_n>$ and $\overline{c}=<c_1,\dots ,c_n>.$ Then $<G,\ast ,A>$ is a soft group.

Definition 7.2.

Given soft groups $<G_1,{\ast }_1,A>,\text{break}\dots ,<G_n,{\ast }_n,A>$ with a fixed set A of parameters. The group $<G,\ast ,A>$ obtained in Lemma 7.1 is called the direct product of $<G_1,{\ast }_1,A>,\dots ,<G_n,{\ast }_n,A>$.

In general, for any indexed family $\{<G_i,{\ast }_i,A>{\}}_{i\in I}$ of soft groups, the underlying set for their direct product is given by:

$\prod _{i\in I}{G}_i=\{a:I⟶\bigcup _{i\in I}(G_i):a(i)\in G_i\mathrm{\forall }i\in I\}.$

Define a soft binary operation $<\ast ,A>$ on $\prod _{i\in I}{G}_i$ as follows $<\alpha ,a,b,c>\in \ast$ if and only if $<\alpha ,a(i),b(i),c(i)>\in {\ast }_i$ for all $i\in I$. Then $<\prod _{i\in I}{G}_i,\ast ,A>$ is a soft group.

Theorem 7.3.

Let $<G_1,{\ast }_1,A>$ and $<G_2,{\ast }_2,A>$ be soft groups. Let $<H_1,A>$ and $<H_2,A>$ be soft subgroups of G₁ and G₂ respectively. Then, their product $<H_1\times H_2,A>$ is a soft subgroup of $G_1\times G_2.$ Moreover, any soft subgroup of $G_1\times G_2$ is of the form $<H_1\times H_2,A>$ for some soft subgroups $<H_1,A>$ and $<H_2,A>$ of G₁ and G₂, respectively.

Proof.

Suppose that $<H_1,A>$ is soft a subgroup of G₁ and $<H_2,A>$ a soft subgroup of $G_2.$ For $\alpha \in A,$ we have $e_{\alpha }^1\in H_1(\alpha )$ and $e_{\alpha }^2\in H_2(\alpha ).$ Which implies that $<e_{\alpha }^1,e_{\alpha }^2>\in H_1(\alpha )\times H_2(\alpha ).$ Let $\overline{a}=<a_1,a_2>$,$\overline{b}=<b_1,b_2>\in H_1\times H_2(\alpha )$ and $\overline{c}=<c_1,c_2>\in G_1\times G_2$ such that $<\alpha ,\overline{a},{\overline{b}}^{-\alpha },\overline{c}>\in \ast .$ Then $<\alpha ,a_1,b_1^{-\alpha },c_1>\in {\ast }_1$ and $<\alpha ,a_2,b_2^{-\alpha },c_2>\in {\ast }_2.$ Since $<H_1,A>$ is a soft subgroup of G₁ and $<H_2,A>$ is a soft subgroup of G₂, we get that $c_1\in H_1(\alpha )$ and $c_2\in H_2(\alpha )>.$ So that $\overline{c}=<c_1,c_2>\in H_1\times H_2(\alpha ).$ That is $<H_1\times H_2,A>$ is a soft subgroup of $G_1\times G_2.$ Conversely suppose that $<H,A>$ is a soft subgroup of $G_1\times G_2.$ Define soft subsets $<H_1,A>$ and $<H_2,A>$ of G₁ and G₂ respectively by:

$H_1(\alpha )=\{a\in G_1:<a,e_{\alpha }^2>\in H(\alpha )\}$

and

$H_2(\alpha )=\{a\in G_2:<e_{\alpha }^1,a>\in H(\alpha )\}.$

Then we show that $<H_1,A>$ is a soft subgroup of G₁. For each $\alpha \in A$, since $<e_{\alpha }^1,e_{\alpha }^2>\in H(\alpha )$ we have $e_{\alpha }^1\in H_1(\alpha ).$ Let $a,b\in H_1(\alpha ).$ Then $<a,e_{\alpha }^2>,$ $<b,e_{\alpha }^2>\in H(\alpha ).$ Let $c\in G_1$ such that $<\alpha ,a,b^{-\alpha },c>\in {\ast }_1.$ Then $<\alpha ,<a,e_{\alpha }^2>,<b^{-\alpha },e_{\alpha }^2>,<c,e_{\alpha }^2>>\in \ast$ and since $<H,A>$ is a soft subgroup of $G_1\times G_2$, we get that $<c,e_{\alpha }^2>\in H_1\times H_2(\alpha )$, which implies that $c\in H_1(\alpha ).$ Thus, $<H_1,A>$ is a soft subgroup of $G_1.$ Similarly, it can be shown that $<H_2,A>$ is a soft subgroup of $G_2.$ Moreover, one can easily check that $<H_1\times H_2,A>\stackrel{⌣}{=}<H,A>.$ Hence proved.

8. Conclusion

This manuscript presents a new approach to soft groups based on soft binary operations aiming to incorporate the concept of ”softness” into the realm of algebraic structures. By proposing a novel definition for soft groups and utilizing soft binary operations parameterized by suitable parameters, we have successfully introduced a framework to model and analyze algebraic structures that capture uncertainty and imprecision.

Moreover, we construct an ordinary group model that represents our soft group. This model is important to describe and characterize the overall internal structure of soft groups through the existing classical group theories. The study of soft subgroups and normal soft subgroups further enhances our understanding of the internal structure of soft groups. We believe that our research will contribute to the growing field of soft computing and pave the way for new applications in various domains.

Authors contribution

All the authors contributed equally to the writing of this manuscript. They also read and approved the final manuscript

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

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Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2023.2289733

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