# A Note on the Soft Group Category
## 2 Preliminaries  
In this section, we will recall some basic definitions and results related to soft set theory and soft group theory that will be used throughout the paper.  
Definition 1: Let \(U\), \(P(U)\), and \(E\) be a universal set, the power set of \(U\), and a set of parameters, respectively. If \(F:A\to P(U)\) is a function, then the pair \((F,A)\) is said to be a soft set over \(U\), where \(A\subseteq E\)[1].  
Definition 2: Let \((F,A)\) and \((G,B)\) be any two soft sets over the same universal set \(U\). If the following conditions are satisfied, then \((F,A)\) is called a soft subset of \((G,B)\), and we denote it by \((F,A)\sqsubseteq(G,B)\)[6]:  
1. \(A\subseteq B\)
2. \(F(a)\) and \(G(a)\) are two identical predictions for every \(a\in A\).  
See [1,3,6] for more detailed information about soft sets.  
**Definition 3**.: _Let \(G\) be any group, and let \((F,A)\) be a soft set over \(G\). If \(F(a)\) is a subgroup of \(G\) for each \(a\in A\), then the pair \((F,A)\) is called a soft group over \(G\)[3]._  
As a convention, throughout this paper, we denote by \((F,A)_{G}\) the soft group \((F,A)\) over the group \(G\).  
**Definition 4**.: _Let \((F,A)_{G}\) be a soft group. If \(F(x)=\{e\}\) for all \(x\in A\), then \((F,A)_{G}\) is said to be a trivial or identity soft group, where \(e\) stands for the identity element of \(G\)[3]._  
**Definition 5**.: _Let \((F,A)_{G}\) be a soft group. If \(F(x)=G\) for all \(x\in A\), then \((F,A)_{G}\) is called a completely soft group [3]._  
Inspired by [7], we will define a soft group homomorphism between any two soft groups as follows:  
**Definition 6**.: _Let \((F,A)_{G}\) and \((H,B)_{K}\) be two soft groups. If there is a group homomorphism \(f:G\to K\) and a function \(p:A\to B\) such that_  
\[f\circ F=H\circ p\]  
_then the pair \((f,p)\) is said to be a soft group homomorphism from \((F,A)_{G}\) to \((H,B)_{K}\), where \(f\) and \(f\) are identical on the power set \(P(G)\)._  
In other words, the pair \((f,p)\) is a soft group homomorphism if and only if the following diagram is commutative:  
(1)  
**Definition 7**.: _Let the pair \((f,p)\) be a soft group homomorphism from \((F,A)_{G}\) to \((H,B)_{K}\). If \(f\) is a group isomorphism and \(p\) is a bijection, then the soft groups \((F,A)_{G}\) and \((H,B)_{K}\) are called isomorphic and written \((F,A)_{G}\cong(H,B)_{K}\)._  
The proof of the following theorem is immediately seen from Equation (1) and the associative property of composition of functions.  
**Theorem 1**.: _The composition of two soft group homomorphisms is a soft group homomorphism._  
**Definition 8** ([3]).: _Let \((F,A)_{G}\) and \((H,B)_{K}\) be two soft groups. The soft product of the soft groups \((F,A)_{G}\) and \((H,B)_{K}\) is defined as_  
\[U(x,y)=F(x)\times H(y)\]  
_for all \((x,y)\in A\times B\), and it is represented by_  
\[(F,A)_{G}\hat{X}(H,B)_{K}=(U,A\times B)_{G\times K}.\]  
This concept can be generalized to three or finitely many soft groups in the following way:  
**Definition 9**.: _Let \((F_{1},A_{1})_{G_{1}},(F_{2},A_{2})_{G_{2}},\cdots,\ (F_{n},A_{n})_{G_{n}}\) be soft groups. The soft product of these soft groups is defined as_  
\[U(x_{1},x_{2},\cdots,x_{n})=F_{1}(x_{1})\times F_{2}(x_{2})\cdots\times F_{n}(x _{n})\]  
_for all \((x_{1},x_{2},\cdots,x_{n})\in A_{1}\times A_{2}\cdots\times A_{n}\) and is denoted by \((F_{1},A_{1})_{G_{1}}\hat{\times}(F_{2},A_{2})_{G_{2}}\hat{\times}\cdots\hat{ \times}(F_{n},A_{n})_{G_{n}}\)\(=\ (U,A_{1}\times A_{2}\cdots A_{n})_{G_{1}\times G_{2}\to\times G_{n}}\)._