Full article: Soft homomorphisms on soft groups

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ABSTRACT

In this paper, we study soft homomorphisms on soft groups and investigate their properties. Given a soft mapping $⟨ f, A ⟩$ from G to G^ʹ, we obtain an ordinary map $\overset{⌣}{f}$ from the set $S E_{A} (G)$ of soft elements of G to the set $S E_{A} (G^{'})$ of soft elements of G^ʹ, and show that $⟨ f, A ⟩$ is a soft homomorphism (respectively, soft isomorphism) if and only if $\overset{⌣}{f}$ is an ordinary group homomorphism (respectively, isomorphism). We apply this concept to study soft isomorphism theorems on soft groups. Moreover, we study those soft automorphisms of soft groups and the particular class of soft inner automorphisms. We establish an embedding of the group of soft automorphisms of a soft group G into the group of automorphisms of $S E_{A} (G)$ . Furthermore, it is shown that for every soft group G, the soft group of its inner automorphisms is soft isomorphic with the quotient of G by its center $Z_{A} (G)$ .

KEYWORDS:

MATHEMATICS SUBJECT CLASSIFICATION:

1. Introduction

Research works in soft set theory and its application in various field have been progressing rapidly. In spite of the introduction of soft set theory in 1999 by Molodtsov, there have been a very few studies focusing on soft groups (Molodtsov, Citation1999). In 2007 Akta and Cagman initiated the study of soft groups and discussed various properties (Aktaş & Çağman, Citation2007). In 2010 Acar et. al. defined soft rings and their properties (Acar et al., Citation2010). Some years later in 2016, Ghosh et al., (Citation2017) came up with a new idea of soft groups using the concept of soft elements. As continuation of these ideas, G. Yaylalı et. al. extended this notion and define soft groups as a nonempty collection of soft elements $⟨ F, A ⟩$ over G together with a binary operation satisfying all the defining properties of a classical group (Yaylalı et al., Citation2019).

It is evident that all the above discussed soft groups were defined based on classical binary operation which is the central unit determining all the algebraic properties of the structure. Taking this into account, we introduce the concept of soft groups based on soft binary operation in the paper (Weldetekle et al., Citation2024). In this scenario, a soft group is defined to be a triple $⟨ G, *, A ⟩$ consisting of a nonempty set G equipped with a soft binary operation $⟨ *, A ⟩$ respecting all the group axioms in a soft setting; whereby a soft binary operation on a set G, we mean a soft mapping in the sense of (Addis et al., Citation2022) from G × G to G.

The main purpose of this paper is to study the basic notion of soft homomorphisms on soft groups based on soft binary operation. Moreover, we state and prove several soft isomorphism theorems on soft groups. We further study the soft automorphisms of soft groups and particularly those soft inner automorphisms. We establish an embedding of the group of soft automorphisms of a soft group G into the group of automorphisms of $S E_{A} (G)$ , where $S E_{A} (G)$ is the collection of all soft elements of G with a set A of parameters. Finally, it is shown that for every soft group G, the soft group of its inner automorphisms is soft isomorphic with the quotient of G by its center $Z_{A} (G)$ .

The remainder of the paper is organized as follows. In Section 2, a brief preliminary account of relevant definitions along with some theorems is presented. Section 3 contains a new definition of soft homomorphism which is illustrated with examples and some results. In Section 4, soft isomorphism theorems are discussed and finally in Section 5, a discussion of soft automorphisms is presented.

2. Preliminaries

In this Section, we give some basic definitions which will be used in this paper, mainly following Addis et al., (Citation2022) and Weldetekle et al., (Citation2024). Throughout this section, unless stated otherwise X is a nonempty set which is referred to as an initial universe set, A is the set of all convenient parameters for X, and G denote the soft group $⟨ G, *, A ⟩ .$

Definition 2.1.

(Molodtsov, Citation1999) 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., Citation2002) A soft set $⟨ F, A ⟩$ over G is said to be the absolute soft set over G if $F (α) = G$ for all $α \in A .$

By a soft relation from X to Y we mean a soft set $⟨ R, A ⟩$ over X × Y. In this case, for $α \in A$ and $(x, y) \in X \times Y$ we write $⟨ α, x, y ⟩ \in R$ to mean $⟨ x, y ⟩ \in R (α)$ (Zhang & Yuan, Citation2014).

Definition 2.3.

(Addis et al., Citation2022) A soft mapping from X to Y is a soft relation $⟨ f, A ⟩$ from X to Y such that:

(1)	for each $α \in A$ and each $x \in X$ there exists some $y \in Y$ such that $⟨ α, x, y ⟩ \in f$ ;
(2)	for each $α \in A$ , $x \in X$ and $y_{1}, y_{2} \in Y$ , $⟨ α, x, y_{1} ⟩ \in f$ and $⟨ α, x, y_{2} ⟩ \in f$ implies $y_{1} = y_{2}$ .

Definition 2.4.

(Addis et al., Citation2022) A soft mapping $⟨ f, A ⟩$ from X to Y is said to be:

(1)	injective if for each $α \in A$ , $x_{1}, x_{2} \in X$ and $y \in Y$ ; $⟨ α, x_{1}, y ⟩ \in f$ and $⟨ α, x_{2}, y ⟩ \in f$ together imply $x_{1} = x_{2} .$
(2)	surjective if for each $α \in A$ and each $y \in Y$ there exists $x \in X$ such that $⟨ α, x, y ⟩ \in f .$
(3)	bijective if it is both injective and surjective.

Definition 2.5.

(Weldetekle et al., Citation2024) Let G be a nonempty set. By a soft binary operation on G we mean a soft mapping $⟨ *, A ⟩$ from G × G to G, where A is a set of parameters.

Definition 2.6.

(Weldetekle et al., Citation2024) A soft group is a triple $⟨ G, *, A ⟩$ where G is nonempty set and $⟨ *, A ⟩$ is a soft binary operation on G satisfying the following conditions:

⟨SG1⟩	Associativity: For all $α \in A$ and all $a, b, c, x, y, u, v \in G$ , if $⟨ α, a, b, x ⟩ \in $ , $⟨ α, x, c, y ⟩ \in $ , $⟨ α, b, c, u ⟩ \in $ and $⟨ α, a, u, v ⟩ \in $ then y = v.
⟨SG2⟩	Existence of identity: For each $α \in A$ there exists $e_{α} \in G$ such that $⟨ α, a, e_{α}, a ⟩ \in $ and $⟨ α, e_{α}, a, a ⟩ \in $ for all $a \in G$ ;
⟨SG3⟩	Existence of inverse: For each $α \in A$ and all $a \in G$ , there is an element of G denoted by $\exists a^{- α}$ such that $⟨ α, a, a^{- α}, e_{α} ⟩ \in $ and $⟨ α, a^{- α}, a, e_{α} ⟩ \in $ .

Definition 2.7.

(Weldetekle et al., Citation2024) By a soft element in G we mean a soft set $⟨ F, A ⟩$ over G such that Card $(F (α)) = 1$ for all $α \in A$ . That is, $F (α)$ is a single element set for each $α \in A$ . Let us denote by $S E_{A} (G)$ the collection of all soft elements of G with a set of parameter A.

Definition 2.8.

(Weldetekle et al., Citation2024) Let $⟨ G, *, A ⟩$ be a soft group. A soft set $⟨ H, A ⟩$ over G is said to be a soft subgroup of G if for each $α \in A$ , and $a, b, x \in G$ :

(1)	$e_{α} \in H (α)$ ;
(2)	$⟨ α, a, b, x ⟩ \in * \Rightarrow x \in H (α)$ for all $a, b \in H (α)$ ;
(3)	$a \in H (α) \Rightarrow a^{- α} \in H (α)$ .

Theorem 2.9.

(Weldetekle et al., Citation2024) Let $⟨ G, *, A ⟩$ 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 $α \in A$ and all $a, b, x \in G$ :

(1)	$e_{α} \in H (α)$ ;
(2)	if $a, b \in H (α)$ and $⟨ α, a, b^{- α}, x ⟩ \in *$ , then $x \in H (α)$ .

Definition 2.10.

(Weldetekle et al., Citation2024) Let $⟨ H, A ⟩$ be a soft subgroup of a soft group $⟨ G, *, A ⟩$ and $a \in G$ . Define a soft set $⟨^{a} H, A ⟩$ over G by:

\begin{array}{l} ^{a} H (α) = & {x \in G : ⟨ α, a, b, x ⟩ \in * for\ some b \in H (α)} \\ for\ each α \in A . \end{array}

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

Definition 2.11.

(Weldetekle et al., Citation2024) A soft subgroup $⟨ N, A ⟩$ of a soft group $⟨ G, *, A ⟩$ is called normal if $⟨^{a} N, A ⟩ \overset{⌣}{=} ⟨ N^{a}, A ⟩$ for all $a \in G .$

Theorem 2.12.

(Weldetekle et al., Citation2024) For a soft subgroup $⟨ N, A ⟩$ of $G,$ the following are equivalent:

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

Definition 2.13.

(Weldetekle et al., Citation2024) We say that a soft group $⟨ G, *, A ⟩$ is abelian if for each $α \in A$ and any $a, b, x \in G$ , $⟨ α, a, b, x ⟩ \in *$ if and only if $⟨ α, b, a, x ⟩ \in * .$

Definition 2.14.

(Weldetekle et al., Citation2024) Let $⟨ G, *, A ⟩$ 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 $α \in A$ :

\begin{array}{l} Z_{A} (G) (α) = & {a \in G : ⟨ α, a, x, y ⟩ \in * \Leftrightarrow ⟨ α, x, a, y ⟩ \in * \\ forallx, y \in G} \end{array}

3. Soft homomorphisms

Definition 3.1.

Let $⟨ G, *, A ⟩$ and $⟨ G^{'}, Δ, A ⟩$ be soft groups. A soft mapping $⟨ f, A ⟩$ from G to G^ʹ is called a soft homomorphism, if for each $α \in Aa, b, c \in G$ and $x, y, z \in G^{'},$ $⟨ α, a, x ⟩ \in f$ , $⟨ α, b, y ⟩ \in f$ , $⟨ α, c, z ⟩ \in f$ and $⟨ α, a, b, c ⟩ \in *$ implies $⟨ α, x, y, z ⟩ \in Δ .$

Example 3.2.

Let $R$ be the set of real numbers, $R^{+}$ the set of positive real numbers and $N$ the set of natural numbers. Define soft binary operations $⟨ *, N ⟩$ and $⟨ Δ, N ⟩$ on $R$ and $R^{+}$ respectively as follows:

\begin{array}{l} ⟨ α, a, b, c ⟩ \in * \Leftrightarrow c = α + a + band \\ ⟨ α, a, b, c ⟩ \in Δ \Leftrightarrow c = (α + 1)^{α} ab . \end{array}

$⟨ R, *, N ⟩$ and $⟨ R^{+}, Δ, N ⟩$ are soft groups. Moreover, define $f = {(α, x, y) : y = (α + 1)^{x}} .$ Then $⟨ f, N ⟩$ is a soft homomorphism from $R$ to $R^{+} .$

Example 3.3.

With the notation of Example 3.2, define $f^{'} = {(α, x, y) : y = \log_{α + 1} x} .$ Then $⟨ f^{'}, N ⟩$ is a soft homomorphism from $⟨ R^{+}, Δ, N ⟩$ to $⟨ R, *, N ⟩$ .

Proposition 3.4.

Let $⟨ f, A ⟩$ be a soft homomorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, △, A ⟩$ . Then

(1)	$⟨ α, e_{α}, e_{α}^{'} ⟩ \in f$ , for all $α \in A$ where e_α and $e_{α}^{'}$ are identity elements of G and G^ʹ respectively;
(2)	$⟨ α, a, y ⟩ \in f \Rightarrow ⟨ α, a^{- α}, y^{- α} ⟩ \in f$ , for all $α \in A, a \in G$ and $y \in G^{'} .$

Proof.

(1)	Let $x \in G^{'}$ Such that $⟨ α, e_{α}, x ⟩ \in f .$ Since $⟨ α, x, e_{α}^{'}, x ⟩ \in Δ$ and $⟨ α, x, x, x ⟩ \in Δ$ , by cancellation law it holds that $x = e_{α}^{'} .$ Therefore $⟨ α, e_{α}, e_{α}^{'} ⟩ \in f .$
(2)	Let $α \in A, a \in G$ and $x, y \in G^{'}$ Such that $⟨ α, a, y ⟩ \in f$ and $⟨ α, a^{- α}, x ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism, it follows from (1) and the condition $⟨ α, a, a^{- α}, e_{α} ⟩ \in *$ that $⟨ α, y, x, e_{α}^{'} ⟩ \in Δ .$ Again from the fact $⟨ α, y, y^{- α}, e_{α}^{'} ⟩ \in Δ$ and cancellation law we get $x = y^{- α} .$ Therefore $⟨ α, a^{- α}, y^{- α} ⟩ \in f .$

Using the idea of the composition of soft mappings given in Addis et al., (Citation2022), it is proved in the following proposition that the composition of two soft homomorphisms is a soft homomorphism.

Proposition 3.5.

Let $⟨ f, A ⟩$ be a soft homomorphism from G to G^ʹ and $⟨ g, A ⟩$ be a soft homomorphism from G^ʹ to $G′′ .$ Then $⟨ g \circ f, A ⟩$ is a soft homomorphism from G to $G′′$ .

Definition 3.6.

A soft homomorphism $⟨ f, A ⟩$ from G to G^ʹ is said to be:

(1)	a soft monomorphism if it is injective;
(2)	a soft epimorphism if it is surjective;
(3)	a soft isomorphism if it is bijective.

Note: Given soft groups $⟨ G, *, A ⟩$ and $⟨ G^{'}, Δ, A ⟩ .$ If there is a soft isomorphism from G to G^ʹ, we say that G and G^ʹ are soft isomorphic and write $G \overset{⌣}{≅} G^{'}$ symbolically. Note also that $\overset{⌣}{≅}$ forms an equivalence relation on the class of soft groups.

Recall from Weldetekle et al., (Citation2024) that, if $⟨ G, *, A ⟩$ is a soft group, then the set $S E_{A} (G)$ of all soft elements of G is an ordinary group together with the induced binary operation $\bar{*}$ , that can be used as a model to represent soft groups. With the idea of extending soft homomorphisms to classical group homomorphism, we define the following.

Definition 3.7.

Let $⟨ f, A ⟩$ be a soft mapping from G to G^ʹ. Define a mapping $\overset{⌣}{f} : S E_{A} (G) \to S E_{A} (G^{'})$ as follows: for each $\overset{⌣}{a} \in S E_{A} (G)$ and $α \in A$ :

\begin{array}{l} \overset{⌣}{f} (\overset{⌣}{a}) (α) = {b} if and only if ⟨ α, \overset{⌣}{a} (α), b ⟩ \in f . \end{array}

Theorem 3.8.

A soft mapping $⟨ f, A ⟩$ from G to G^ʹ is a soft homomorphism if and only if $\overset{⌣}{f}$ is a homomorphism.

Proof.

Suppose that $⟨ f, A ⟩$ is a soft homomorphism. Let $\overset{⌣}{a}, \overset{⌣}{b} \in S E_{A} (G)$ and $α \in A$ be arbitrary. Assume that $\overset{⌣}{f} (\overset{⌣}{a} \bar{*} \overset{⌣}{b}) (α) = {z} .$ Then, $⟨ α, (\overset{⌣}{a} \bar{*} \overset{⌣}{b}) (α), z ⟩ \in f$ . If $(\overset{⌣}{a} \bar{*} \overset{⌣}{b}) (α) = {c}$ , then $⟨ α, \overset{⌣}{a} (α), \overset{⌣}{b} (α), c ⟩ \in *$ and $⟨ α, c, z ⟩ \in f$ . On the other hand if $⟨ α, \overset{⌣}{a} (α), x ⟩ \in f$ and $⟨ α, \overset{⌣}{b} (α), y ⟩ \in f$ , then as $⟨ f, A ⟩$ is a soft homomorphism, we get $⟨ α, x, y, z ⟩ \in Δ$ and hence $(\overset{˘}{f} (\overset{˘}{a}) \bar{Δ} \overset{˘}{f} (\overset{˘}{b})) (α) = {z} .$ Therefore, $\overset{⌣}{f}$ is a group homomorphism. Conversely suppose that $\overset{⌣}{f}$ is a group homomorphism. Let $α \in A$ , $a, b, c \in G$ and $x, y, z \in G^{'}$ such that $⟨ α, a, x ⟩ \in f$ , $⟨ α, b, y ⟩ \in f$ , $⟨ α, c, z ⟩ \in f$ and $⟨ α, a, b, c ⟩ \in * .$ Let $u \in G^{'}$ such that $⟨ α, x, y, u ⟩ \in Δ$ . We claim to show that u = z. Define soft elements $\overset{⌣}{a}, \overset{⌣}{b}, \overset{⌣}{c}$ in G and $\overset{⌣}{x}, \overset{⌣}{y}, \overset{⌣}{z}$ in G^ʹ as follows: for each $λ \in A$ :

\begin{array}{l} \overset{⌣}{a} (λ) = {\begin{cases} {a} & ifλ = α \\ {e_{λ}} & ifλ \neq α \end{cases} \end{array}

\begin{array}{l} \overset{⌣}{b} (λ) = {\begin{cases} {b} & ifλ = α \\ {e_{λ}} & ifλ \neq α \end{cases} \end{array}

\begin{array}{l} \overset{⌣}{c} (λ) = {\begin{cases} {c} & ifλ = α \\ {e_{λ}} & ifλ \neq α \end{cases} \end{array}

Then, one can easily show that for each $λ \in A$ :

\begin{array}{l} \overset{⌣}{f} (\overset{⌣}{a} \bar{*} \overset{⌣}{b}) (λ) = {\begin{cases} {z} & ifλ = α \\ {e_{λ}^{'}} & ifλ \neq α \end{cases} and \end{array}

\begin{array}{l} (\overset{⌣}{f} (\overset{⌣}{a}) \bar{Δ} \overset{⌣}{f} (\overset{⌣}{b})) (λ) = {\begin{cases} {u} & ifλ = α \\ {e_{λ}^{'}} & ifλ \neq α \end{cases} \end{array}

Since $\overset{⌣}{f}$ is a homomorphism, it should be the case that $\overset{⌣}{f} (\overset{⌣}{a} \bar{*} \overset{⌣}{b}) (λ) = (\overset{⌣}{f} (\overset{⌣}{a}) \bar{Δ} \overset{⌣}{f} (\overset{⌣}{b})) (λ)$ for all $λ \in A$ . In particular, it works for $λ = α$ and hence which would give z = u. Therefore, $⟨ f, A ⟩$ is a soft homomorphism.

Theorem 3.9.

A soft homomorphism $⟨ f, A ⟩$ from G to G^ʹ is a soft isomorphism if and only if $\overset{⌣}{f}$ is a group isomorphism.

Proof.

Suppose that $⟨ f, A ⟩$ is soft isomorphism. We first show that $\overset{⌣}{f}$ is injective. Let $\overset{⌣}{a}, \overset{⌣}{b} \in S E_{A} (G)$ such that $\overset{⌣}{f} (\overset{⌣}{a}) = \overset{⌣}{f} (\overset{⌣}{b})$ . Then $\overset{⌣}{f} (\overset{⌣}{a}) (λ) = \overset{⌣}{f} (\overset{⌣}{b}) (λ)$ for all $λ \in A$ . That is, for each $λ \in A$ and any $x \in G^{'}$ we have:

\begin{array}{l} ⟨ λ, \overset{⌣}{a} (λ), x ⟩ \in f if and only if ⟨ λ, \overset{⌣}{b} (λ), x ⟩ \in f . \end{array}

Since f is injective, it must be true that $\overset{⌣}{a} (λ) = \overset{⌣}{b} (λ)$ for all $λ \in A$ . Thus, $\overset{⌣}{a} = \overset{⌣}{b}$ and hence $\overset{⌣}{f}$ is injective. Next we show that $\overset{⌣}{f}$ is surjective. Let $\overset{⌣}{y}$ be any soft element in G^ʹ. Since $⟨ f, A ⟩$ is soft isomorphism, for ech $α \in A$ there is a unique element let say $x_{α} \in G$ such that $⟨ α, x_{α}, \overset{⌣}{y} (α) ⟩ \in f$ for all $α \in A$ . Noe define a sofet element $\overset{⌣}{x}$ over G by $\overset{⌣}{x} (α) = {x_{α}}$ for all $α \in A$ . Then we must have $\overset{⌣}{f} (\overset{⌣}{x}) = \overset{⌣}{y}$ . Thus, $\overset{⌣}{f}$ is surjective and hence an isomorphism. The converse can be proved using similar procedure.

Definition 3.10.

Let $⟨ f, A ⟩$ be a soft homomorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩ .$

(a)

If $⟨ H, A ⟩$ is a soft subgroup of $G,$ then the soft set $⟨ f (H), A ⟩$ over G^ʹ defined as:

\begin{array}{l} f (H) (α) = & {y \in G^{'} : ⟨ α, x, y ⟩ \in fforsome \\ x \in H (α)} \end{array}

is the image of $⟨ H, A ⟩$ under f.

(b)

If $⟨ H^{'}, A ⟩$ is a soft subgroup of G^ʹ, then the soft set $⟨ f^{- 1} (H^{'}), A ⟩$ over G defined as:

\begin{array}{l} f^{- 1} (H^{'}) (α) = & {x \in G : ⟨ α, x, y ⟩ \in fforsome \\ y \in H^{'} (α)} \end{array}

is the inverse image of $⟨ H^{'}, A ⟩$ under f.

Lemma 3.11.

Let $⟨ f, A ⟩$ be a soft homomorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩$ . If $⟨ H, A ⟩$ is a soft subgroup of G then $⟨ f (H), A ⟩$ is a soft subgroup of $G .$

Proof.

Let $α \in A .$ Since $e_{α} \in H (α)$ and $⟨ α, e_{α}, e_{α}^{'} ⟩ \in f$ , we have $e_{α}^{'} \in f (H) (α) .$

Let $x, y \in f (H) (α)$ and $z \in G^{'}$ such that $< α, x, y^{- α}, z >\in Δ .$ We need to prove that $z \in f (H) (α) .$ As $x, y \in f (H) (α)$ there exists $a, b \in H (α)$ such that $⟨ α, a, x ⟩ \in f$ and $⟨ α, b, y ⟩ \in f .$ Since $⟨ α, b, y ⟩ \in f$ we have $⟨ α, b^{- α}, y^{- α} ⟩ \in f .$ Let $c \in G$ such that $⟨ α, a, b^{- α}, c ⟩ \in * .$ Since $⟨ f, A ⟩$ is a soft mapping there exist $w \in G^{'}$ such that $< α, c, w >\in f .$ As $⟨ f, A ⟩$ is a soft homomorphism it follows that w = z. Since $⟨ H, A ⟩$ is a soft subgroup of G, we have $c \in H (α) .$ Therefore $z \in f (H) (α) .$

Lemma 3.12.

Let $⟨ f, A ⟩$ be a soft homomorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩ .$ If $⟨ H^{'}, A ⟩$ is a soft subgroup of G^ʹ then $⟨ f^{- 1} (H^{'}), A ⟩$ is a soft subgroup of $G .$

Proof.

Let $α \in A .$ Since $e_{α}^{'} \in H^{'} (α)$ and $⟨ α, e_{α}, e_{α}^{'} ⟩ \in f$ , we get $e_{α} \in f^{- 1} (H^{'}) (α) .$

Let $a, b \in f^{- 1} (H^{'}) (α)$ and $c \in G$ such that $⟨ α, a, b^{- α}, c ⟩ \in * .$ We need to prove that $c \in f^{- 1} (H^{'}) (α) .$ As $a, b \in f^{- 1} (H^{'}) (α)$ there exist $x, y \in H^{'} (α)$ such that $⟨ α, a, x ⟩ \in f$ and $⟨ α, b, y ⟩ \in f .$ As $⟨ α, b, y ⟩ \in f$ it holds that $⟨ α, b^{- α}, y^{- α} ⟩ \in f .$ Let $c \in G$ such that $⟨ α, a, b^{- α}, c ⟩ \in * .$ Since $⟨ f, A ⟩$ is a soft mapping there exist $z \in G^{'}$ such that $⟨ α, c, z ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism from G to $G^{'},$ it holds that $⟨ α, x, y^{- α}, z ⟩ \in Δ .$ Since $⟨ H^{'}, A ⟩$ is a soft subgroup of G^ʹ we get $z \in H^{'} (α) .$ Therefore $c \in f^{- 1} (H^{'}) (α) .$ Hence $⟨ f^{- 1} (H^{'}), A ⟩$ is a soft subgroup of $G .$

Theorem 3.13.

Let $⟨ f, A ⟩$ be a soft homomorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩ .$

(1)	If $⟨ f, A ⟩$ is surjective and $⟨ N, A ⟩$ is a normal soft subgroup of G then $⟨ f (N), A ⟩$ is a normal soft subgroup of $G^{'} .$
(2)	If $⟨ N^{'}, A ⟩$ is a normal soft subgroup of G^ʹ then $⟨ f^{- 1} (N^{'}), A ⟩$ is a normal soft subgroup of $G .$

Proof.

(1)	By Lemma 3.11 $⟨ f (N), A ⟩$ is a soft subgroup of $G^{'} .$ Let $a \in f (N) (α) .$ Then $⟨ α, x, a ⟩ \in f$ for some $x \in N (α) .$ Let $b \in G^{'} .$ Since $⟨ f, A ⟩$ is surjective, there exist $y \in G$ such that $⟨ α, y, b ⟩ \in f$ . Let $z \in G$ such that $⟨ α, y, x, z ⟩ \in * .$ There exists $c \in G^{'}$ such that $⟨ α, z, c ⟩ \in f$ because $⟨ f, A ⟩$ is a soft homomorphism. Moreover, $⟨ α, b, a, c ⟩ \in Δ .$ Again let $d \in G$ such that $⟨ α, z, y^{- α}, d ⟩ \in * .$ Since $⟨ f, A ⟩$ is a soft mapping, there exists $d^{'} \in G^{'}$ such that $⟨ α, d, d^{'} ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism, $⟨ α, c, b^{- α}, d^{'} ⟩ \in Δ .$ As $⟨ N, A ⟩$ is a normal soft subgroup of G, $d \in N (α) .$ This implies that $d^{'} \in f (N) (α) .$ Therefore $⟨ f (N), A ⟩$ is a normal soft subgroup of $G^{'} .$
(2)	By Lemma 3.12 $⟨ f^{- 1} (N^{'}), A ⟩$ is a soft subgroup of $G .$ Let $a \in f^{- 1} (N^{'}) (α) .$ Then $⟨ α, a, x ⟩ \in f$ for some $x \in N^{'} (α) .$ Let $b, c \in G$ and $x_{1}, z_{1} \in G^{'}$ such that $⟨ α, b, x_{1} ⟩ \in f$ and $⟨ α, c, z_{1} ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism from G to G^ʹ and $⟨ α, b, a, c ⟩ \in * .$ Then we have $⟨ α, x_{1}, x, z_{1} ⟩ \in Δ .$ Let $d \in G$ and $z_{2} \in G$ such that $⟨ α, d, z_{2} ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism from G to G^ʹ and $⟨ α, c, b^{- α}, d ⟩ \in *, ⟨ α, z_{1}, x_{1}^{- α}, z_{2} ⟩ \in Δ .$ It follows that $d \in f^{- 1} (N^{'}) (α) .$ Therefore $⟨ f^{- 1} (N^{'}), A ⟩$ is a normal soft subgroup of $G^{'} .$

Definition 3.14.

Let $⟨ f, A ⟩$ be a soft homomorphism from G to $G^{'} .$ The kernel of $⟨ f, A ⟩$ is the soft set $⟨ K_{f}, A ⟩$ over G defined as:

\begin{array}{l} K_{f} (α) = {x \in G : ⟨ α, x, e_{α}^{'} ⟩ \in f for each α \in A} \end{array}

where $e_{α}^{'}$ is an identity element of $G^{'} .$

Example 3.15.

Let $⟨ G, *, A ⟩$ and $⟨ G^{'}, △, A ⟩$ be soft groups.

Let $⟨ f, A ⟩$ be a soft mapping from G to G^ʹ defined by:
$\begin{array}{l} f = {(α, x, e_{α}^{'}) : α \in A, x \in G} \end{array}$
then $⟨ f, A ⟩$ is a soft homomorphism and $⟨ K_{f}, A ⟩$ is the absolute soft set over $G .$
Let $< f, A >$ be a soft mapping from G to G defined by:
$\begin{array}{l} f = {(α, x, x) : α \in Ax, \in G} \end{array}$
then $⟨ f, A ⟩$ is a soft homomorphism and $⟨ K_{f}, A ⟩$ is the trivial soft subgroup of G.

Lemma 3.16.

For any soft homomorphism $⟨ f, A ⟩$ from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩$ , the kernel $⟨ K_{f}, A ⟩$ is a normal soft subgroup of $G .$

Proof.

For each $α \in A$ , we have

\begin{array}{l} K_{f} (α) = {x \in G : ⟨ α, x, e_{α} ⟩ \in f} = f^{- 1} ({\overset{⌣}{e}}^{'}) (α) \end{array}

Therefore the proof follows directly from Theorem 3.13.

Lemma 3.17.

A soft homomorphism $⟨ f, A ⟩$ from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩$ is a soft monomorphism if and only if $⟨ K_{f}, A ⟩$ is the trivial soft subgroup of $G .$

Proof.

Suppose $⟨ f, A ⟩$ is a soft monomorphism. We need to show that $K_{f} (α) = {e_{α}}$ for all $α \in A .$ Let $x \in K_{f} (α) .$ This implies that $⟨ α, x, e_{α}^{'} ⟩ \in f$ , for all $α \in A .$ Since $⟨ α, e_{α}, e_{α}^{'} ⟩ inf$ , for all $α \in A$ we have $x = e_{α} .$ Therefore $K_{f} (α) = {e_{α}} .$ Conversely, suppose that $⟨ α, x_{1}, y ⟩ \in f$ and $⟨ α, x_{2}, y ⟩ \in f .$ Let $z \in G$ such that $⟨ α, x_{1}, x_{2}^{- α}, z ⟩ \in * .$ Then $⟨ α, z, e_{α}^{'} ⟩ \in f .$ It follows that $z \in K_{f} (α) .$ So $z = e_{α} .$ Therefore $⟨ α, x_{1}, x_{2}^{- α}, e_{α} ⟩ \in * .$ This implies that $⟨ α, e_{α}, x_{2}, x_{1} ⟩ \in * .$ Using the fact $⟨ α, e_{α}, x_{1}, x_{1} ⟩ \in *$ and the cancellation law we get $x_{1} = x_{2} .$

Proposition 3.18.

Let $⟨ f, A ⟩$ and $⟨ g, A ⟩$ be soft homomorphisms from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩$ . Define a soft set $⟨ H, A ⟩$ over G by:

\begin{array}{l} H (α) = & {a \in G : ⟨ α, a, x ⟩ \in f \Leftrightarrow ⟨ α, a, x ⟩ \in g \\ for any x \in G^{'}} \end{array}

for all $α \in A$ . Then $⟨ H, A ⟩$ is a soft subgroup of $G .$

Proof.

Since $⟨ α, e_{α}, e_{α}^{'} ⟩ \in f$ and $⟨ α, e_{α}, e_{α}^{'} ⟩ \in g$ , we have $e_{α} \in H (α) .$ Let $a, b \in H (α)$ and $x \in G$ such that $⟨ α, a, b^{- α}, x ⟩ \in * .$ Since $⟨ f, A ⟩$ and $⟨ g, A ⟩$ are soft mapping there exist $y_{1}, y_{2} \in G^{'}$ such that

\begin{array}{l} ⟨ α, a, y_{1} >\in f \Leftrightarrow< α, a, y_{1} ⟩ \in g \end{array}

and

\begin{array}{l} ⟨ α, b, y_{2} ⟩ \in f \Leftrightarrow ⟨ α, b, y_{2} ⟩ \in g \end{array}

This implies that $⟨ α, b^{- α}, y_{2}^{- α} ⟩ \in f \Leftrightarrow ⟨ α, b^{- α}, y_{2}^{- α} ⟩ \in g .$ Again $⟨ f, A ⟩$ is a soft mapping then there exist $y_{3} \in G^{'}$ such that $⟨ α, x, y_{3} ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism then we get that $⟨ α, y_{1}, y_{2}^{- α}, y_{3} ⟩ \in Δ .$ Since $⟨ g, A ⟩$ is a soft homomorphism and $⟨ α, x, y_{3} ⟩ \in g$ , we have $x \in H (α) .$ Hence $⟨ H, A ⟩$ is a soft subgroup of $G .$

Recall from (Weldetekle et al., Citation2024) that for a soft set $⟨ H, A ⟩$ over G, we define the subset $\hat{H}$ of $S E_{A} (G)$ by:

\begin{array}{l} \hat{H} = {\overset{⌣}{a} \in S E_{A} (G) : \overset{⌣}{a} (α) \subseteq H (α) for all α \in A} \end{array}

In the following theorem, we establish a relation between the kernel of $\overset{⌣}{f}$ and the soft kernel of $⟨ f, A ⟩$ .

Theorem 3.19.

For any soft homomorphism $⟨ f, A ⟩$ from G to G^ʹ we have

\begin{array}{l} \ker (\overset{⌣}{f}) = \hat{K_{f}} \end{array}

Proof.

We know that

\begin{array}{l} \hat{K_{f}} = {\overset{⌣}{a} \in S E_{A} (G) : \overset{⌣}{a} (α) \subseteq K_{f} (α) for all α \in A} \end{array}

and

\begin{array}{l} \ker (\overset{⌣}{f}) = {\overset{⌣}{a} \in S E_{A} (G) : \overset{⌣}{f} (\overset{⌣}{a}) (λ) = {e_{α}} for all λ \in A} . \end{array}

\begin{array}{l} Now, wehave \overset{⌣}{a} \in \ker (\overset{⌣}{f}) & \Leftrightarrow \overset{⌣}{f} (\overset{⌣}{a}) (λ) = {e_{λ}} \\ \Leftrightarrow ⟨ α, \overset{⌣}{a} (λ), e_{λ} ⟩ \in f \\ \Leftrightarrow \overset{⌣}{a} (λ) \in K_{f} (λ) for all λ \in A \\ \Leftrightarrow \overset{⌣}{a} \in \hat{K_{f}} . \end{array}

Therefore $\ker (\overset{⌣}{f}) = \hat{K_{f}} .$

4. Soft isomorphism theorems

In this Section, we will see a method for checking whether two soft groups defined in different ways are structurally the same or not.

Theorem 4.1.

Let $⟨ H, A ⟩$ be a normal soft subgroup of a soft group $⟨ G, *, A ⟩$ . Put

\begin{array}{l} G / H = {^{a} H : a \in G} \end{array}

where each $^{a} H$ is a left soft coset of $⟨ H, A ⟩$ . Define a soft coset multiplication $⊛$ on G/H by: $⟨ α,^{a} H,^{b} H,^{c} H ⟩ \in ⊛ \Leftrightarrow ⟨ α, a, b, x ⟩ \in *$ for some $x \in G$ with $^{c} H =^{x} H .$ Then, $⟨ G / H, ⊛, A ⟩$ is a soft group.

Proof.

First we shall prove that $⊛$ is well defined. Let $n, m, d \in G$ and $α \in A$ such that $^{a} H =^{d} H$ and $^{b} H =^{n} H$ . Suppose $⟨ α,^{a} H,^{b} H,^{c} H ⟩ \in ⊛$ and $⟨ α,^{d} H,^{n} H,^{m} H ⟩ \in ⊛ .$ This implies that $⟨ α, a, b, z_{1} ⟩ \in *$ and $⟨ α, d, n, z_{2} ⟩ \in *$ for some $z_{1}, z_{2} \in G$ with $^{c} H =^{z_{1}} H$ and $^{m} H =^{z_{2}} H .$ Let $k_{1} \in G$ such that $⟨ α, c^{- α}, z_{1}, k_{1} ⟩ \in * .$ Since $^{c} H =^{z_{1}} H$ , $k_{1} \in H (α) .$ Let $k_{2} \in G$ such that $⟨ α, m^{- α}, z_{2}, k_{2} ⟩ \in * .$ Since $^{m} H =^{z_{2}} H$ , $k_{2} \in H (α) .$ Let $x_{1}, x_{2} \in G$ such that $⟨ α, a^{- α}, d, x_{1} ⟩ \in *$ and $⟨ α, b^{- α}, n, x_{2} ⟩ \in * .$ It follows that $x_{1}, x_{2} \in H (α) .$ Let $k_{3}, y \in G$ such that $⟨ α, n^{- α}, x_{1}, y ⟩ \in *$ and $⟨ α, y, n, k_{3} ⟩ \in * .$ Since $⟨ H, A ⟩$ is a normal soft subgroup of G then we have $k_{3} \in H (α) .$ Let $k_{4}, k_{5}, k_{6} \in G$ such that $⟨ α, k_{1}, x_{2}, k_{4} ⟩ \in *$ , $⟨ α, k_{4}, k_{3}, k_{5} ⟩ \in *$ and $⟨ α, k_{5}, k_{2}^{- α}, k_{6} ⟩ \in * .$ Since $⟨ H, A ⟩$ is a soft subgroup of G then we get $k_{4}, k_{5}, k_{6} \in H (α) .$ Using the associative property of $⟨ \in *, A ⟩$ and $⟨ α, c^{- α}, m, k_{6} ⟩ \in * .$ This implies that $^{c} H =^{m} H .$ Thus $⊛$ is well defined. It remains to show that $⟨ G / H, ⊛, A ⟩$ satisfies the soft group axioms which are straightforward.

Definition 4.2.

If $⟨ H, A ⟩$ is a normal soft subgroup of $⟨ G, *, A ⟩$ , then the soft group $⟨ G / H, ⊛, A ⟩$ defined as in Theorem 4.1 is called quotient soft group.

Definition 4.3.

We say that a normal soft subgroup $⟨ N, A ⟩$ has the property (Q) provided that for each $α, β \in A$ , $a \in N (α)$ and $b \in G$ ;

\begin{array}{l} ⟨ β, a^{- β}, e_{α}, b ⟩ \in * \Rightarrow b \in N (β) . \end{array}

Lemma 4.4.

Let $⟨ N, A ⟩$ be a normal soft subgroup of $⟨ G, *, A ⟩$ and $⟨ G / N, ⊛, A ⟩$ be the quotient soft group. Define a soft mapping $⟨ f, A ⟩$ from G to $G / N$ by

\begin{array}{l} f = {⟨ α, a,^{a} N ⟩ : α \in A, a \in G} . \end{array}

Then $⟨ f, A ⟩$ is a soft epimorphism from G to $G / N$ such that $⟨ K_{f}, A ⟩ \overset{⌣}{\subseteq} ⟨ N, A ⟩ .$ Moreover, the equality holds, whenever $⟨ N, A ⟩$ has the property (Q).

Proof.

Define a soft mapping $⟨ f, A ⟩$ from G to $G / N$ by $⟨ α, a,^{a} N ⟩ \in f .$ Clearly, $⟨ f, A ⟩$ is surjective. Since $⟨ N, A ⟩$ is normal in G then $⟨ f, A ⟩$ is a soft homomorphism. Now, for each $α \in A$ and $a \in G$ :

\begin{array}{l} a \in K_{f} (α) & \Rightarrow & ⟨ α, a,^{e_{α}} N ⟩ \in f \\ \Rightarrow & ^{a} N =^{e_{α}} N \\ \Rightarrow & ^{a} N (β) =^{e_{α}} N (β) ∀β \in A \\ \Rightarrow & ^{a} N (α) =^{e_{α}} N (α) \\ \Rightarrow & a \in N (α) . \end{array}

Therefore $⟨ K_{f}, A ⟩ \overset{⌣}{\subseteq} ⟨ N, A ⟩ .$ Moreover, if $⟨ N, A ⟩$ has the property (Q), then one can easily verify that $K_{f} (α) = N (α)$ for all $α \in A$ and hence the equality holds.

Note: The soft mapping defined in Lemma 4.4 is called soft canonical epimorphism.

Theorem 4.5.

(The first soft isomorphism theorem) Let $⟨ G, *, A ⟩$ and $⟨ G^{'}, Δ, A ⟩$ be soft groups. If $⟨ f, A ⟩$ is a soft epimorphism from G to G^ʹ, then $G / K_{f} \overset{⌣}{≅} G^{'}$ .

Proof.

Define a soft mapping $⟨ g, A ⟩$ from $G / K_{f}$ to G^ʹ by $⟨ α,^{a} K_{f}, y ⟩ \in g$ if and only if $⟨ α, a, y ⟩ \in f .$ We first show that $⟨ g, A ⟩$ is well defined. Let $α \in A$ , and $a, b, \in G$ such that $^{a} K_{f} =^{b} K_{f} .$ Let $y_{1}, y_{2} \in G^{'}$ such that $⟨ α,^{a} K_{f}, y_{1} ⟩ \in g$ and $⟨ α,^{b} K_{f}, y_{2} ⟩ \in g .$ It follows that $⟨ α, a, y_{1} ⟩ \in f$ and $⟨ α, b, y_{2} ⟩ \in f .$ Since $^{a} K_{f} =^{b} K_{f}$ , we have $^{a} K_{f} (α) =^{b} K_{f} (α)$ for all $α \in A .$ If $⟨ α, b^{- α}, a, x_{1} ⟩ \in *$ then $x_{1} \in K_{f} (α)$ for all $α \in A .$ This implies $⟨ α, x_{1}, e_{α}^{'} ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism $⟨ α, y_{2}^{- α}, y_{1}, e_{α}^{'} ⟩ \in Δ .$ Then, we get $y_{1} = y_{2} .$ Therefore $⟨ g, A ⟩$ is well defined. Since $⟨ f, A ⟩$ is an epimorphism for each $y \in G^{'}$ there exists $a \in G$ such that $⟨ α, a, y ⟩ \in f .$ Which implies that $⟨ α,^{a} K_{f}, y ⟩ \in g .$ Therefore $⟨ g, A ⟩$ is surjective. It remains to show that $⟨ g, A ⟩$ is injective. Suppose that $⟨ α,^{a} K_{f}, y ⟩ \in g$ and $⟨ α,^{b} K_{f}, y ⟩ \in g$ implies $⟨ α, a, y ⟩ \in f$ and $⟨ α, b, y ⟩ \in f .$ Let $x \in G$ and $z \in G^{'}$ such that $⟨ α, b^{- α}, a, x ⟩ \in *$ and $⟨ α, x, z ⟩ \in f .$ Then we have $⟨ α, y^{- α}, y, z ⟩ \in △ .$ Which implies that $z = e_{α}^{- α} .$ That is $⟨ α, x, e_{α}^{'} ⟩ \in f \Rightarrow x \in K_{f} (α) .$ Since α is arbitrary we can see that $⟨ α, b^{- α}, a, x ⟩ \in * \Rightarrow x \in K_{f} (α)$ for all $α \in A .$ Thus $^{a} K_{f} (α) =^{b} K_{f} (α) .$ Therefore $⟨ g, A ⟩$ is injective. Hence $G / K_{f} \overset{⌣}{≅} G^{'} .$

Corollary 4.6.

Let $⟨ f, A ⟩$ be a soft epimorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩$ and $⟨ M, A ⟩$ be a normal soft subgroup of $G^{'} .$ Then,

\begin{array}{l} G / f^{- 1} (M) \overset{⌣}{≅} G^{'} / M . \end{array}

Proof.

Define a soft mapping $⟨ g, A ⟩$ from G to $G^{'} / M$ by: $⟨ α, a, x_{M} ⟩ \in g$ if and only if $⟨ α, a, y ⟩ \in f$ for some $y \in G^{'}$ with $^{x} M =^{y} M .$ We show that $⟨ g, A ⟩$ is a soft homomorphism. Let $α \in A$ , $a, b, c \in G$ and $x, y, z \in G^{'}$ such that $⟨ α, a, x_{M} ⟩ \in g$ , $⟨ α, b, y_{M} ⟩ \in g$ , $⟨ α, c, z_{M} ⟩ \in g$ and $⟨ α, a, b, c ⟩ \in * .$ Then, there exist $x^{'}, y^{'}, z^{'} \in G^{'}$ such that $x_{M} = x_{M}^{'}$ , $y_{M} = y_{M}^{'}$ , $z_{M} = z_{M}^{'}$ and $⟨ α, a, x^{'} ⟩ \in f$ , $⟨ α, b, y^{'} ⟩ \in f$ , $⟨ α, c, z^{'} ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism and $⟨ α, a, b, c ⟩ \in * .$ We get that $⟨ α, x^{'}, y^{'}, z^{'} ⟩ \in Δ$ in $G^{'} .$ Then $⟨ α, x_{M}^{'}, y_{M}^{'}, z_{M}^{'} ⟩ \in \bar{Δ}$ in $G^{'} / M .$ Thus $⟨ g, A ⟩$ is a soft homomorphism from G to $G^{'} / M .$ As $⟨ f, A ⟩$ is surjective it is straightforward that $⟨ g, A ⟩$ is also surjective. Now for each $α \in A$ consider the following:

\begin{array}{l} K_{g} (α) & = & {a \in G : ⟨ α, a, {e_{α}}_{M} ⟩ \in g} \\ = & {a \in G : ⟨ α, a, x ⟩ \in fforsomex \in G^{'} with \\ x_{M} = {e_{α}}_{M}} \\ = & {a \in G : ⟨ α, a, x ⟩ \in fforsomex \in M (α)} \\ = & f^{- 1} (M) (α) \end{array}

Thus $⟨ K_{g}, A ⟩ \overset{⌣}{=} ⟨ f^{- 1} (M), A ⟩ .$ Therefore by the first soft isomorphism theorem we get that

\begin{array}{l} G / f^{- 1} (M) \overset{⌣}{≅} G^{'} / M . \end{array}

Proposition 4.7.

Let $⟨ G, *, A ⟩$ and $⟨ G^{'}, Δ, A ⟩$ be soft groups and $⟨ ϕ, A ⟩$ be a soft homomorphism from G to $G^{'} .$ Then, for any soft set $⟨ F, A ⟩$ over G, $⟨ ϕ (C_{F}), A ⟩ \overset{⌣}{\subseteq} ⟨ C_{ϕ (F)}, A ⟩$ ; where $⟨ C_{F}, A ⟩$ is the centralizer of $⟨ F, A ⟩$ . Moreover, the equality holds if and only if $⟨ ϕ, A ⟩$ is a soft isomorphism.

Theorem 4.8.

(The Third Soft isomorphism theorem) Let $⟨ M, A ⟩$ and $⟨ N, A ⟩$ be normal soft subgroups of $G,$ such that $⟨ M, A ⟩ \overset{⌣}{\subseteq} ⟨ N, A ⟩$ , and $< N, A >$ has the property (Q). Define a soft set with $⟨ N / M, A ⟩$ over G/M by $N / M (α) = {^{a} M : a \in N (α)} .$ Then,

$⟨ N / M, A ⟩$ is a normal soft subgroup of $G / M$ ;
$(G / M) / (N / M) \overset{⌣}{≅} G / N$ .

Proof.

First observe that $a_{M} \in N / M (α)$ if and only if $a \in N (α) .$ It is clear from the definition of $N / M$ that $a \in N (α)$ implies $a_{M} \in N / M (α) .$ Let $a_{M} \in N / M (α) .$ Then $a_{M} = b_{M}$ for some $b \in N (α) .$ Which implies that for each $β \in A$ , and $x \in G$ , if $⟨ β, b^{- β}, a, x ⟩ \in *$ , then $x \in M (β) \subseteq N (β) .$ In particular, $x \in N (α)$ and since $b \in N (α)$ we get that $a \in N (α) .$ Define a soft mapping $⟨ g, A ⟩$ from $G / M$ to $G / N$ by

\begin{array}{l} g = {⟨ α, a_{M}, a_{N} ⟩ : α \in A, a \in G} . \end{array}

Then we first show that $⟨ g, A ⟩$ is well defined. Suppose $^{a} M =^{b} M .$ This implies that $^{a} M (α) =^{b} M (α)$ , for all $\in A .$ Since $⟨ M, A ⟩$ is a normal soft subgroup, if $⟨ α, b^{- α}, a, x ⟩ \in *$ then $x \in M (α) .$ Since $M (α) \subseteq N (α)$ , $x \in N (α) .$ This implies that $^{a} N (α) =^{b} N (α)$ , for all $\in A .$ Therefore $^{a} N =^{b} N .$ Hence $⟨ g, A ⟩$ is well-defined. Next we show that $⟨ g, A ⟩$ is a soft homomorphism. Let $α \in A$ , $a, b, x, y, z \in G$ such that $⟨ α, a_{M}, x_{N} ⟩ \in g$ , $⟨ α, b_{M}, y_{N} ⟩ \in g$ , $⟨ α, c_{M}, z_{N} ⟩ \in g$ and $⟨ α, a_{M}, b_{M}, c_{M} ⟩ \in ⊛$ in $G / M$ . Then $x_{N} = a_{N}$ , $y_{N} = b_{N}$ and $c_{N} = z_{N}$ and $⟨ α, a, b, c ⟩ \in *$ in G. Thus $⟨ α, a_{N}, b_{N}, c_{N} ⟩ \in ⊛$ in the quotient $G / N .$ Hence $⟨ g, A ⟩$ is a soft homomorphism and it is clear that $⟨ g, A ⟩$ is a soft epimorphism. Now consider its kernel:

\begin{array}{l} K_{g} (α) & = & {a_{M} \in G / M : ⟨ α, a_{M}, {e_{α}}_{N} ⟩ \in g} \\ = & {a_{M} \in G / M :^{a} N =^{e_{α}} N} \\ = & {a_{M} \in G / M :^{a} N (β) =^{e_{α}} N (β) ∀β \in A} \\ = & {a_{M} \in G / M : a \in N (α)} \\ = & N / M (α) \end{array}

Therefore $⟨ K_{g}, A ⟩ \overset{⌣}{=} ⟨ N / M, A ⟩ .$ By the first soft isomorphism theorem $(G / M) / (N / M) \overset{⌣}{≅} G / N$ .

Lemma 4.9.

Let $⟨ M, A ⟩$ , $⟨ N_{1}, A ⟩$ and $⟨ N_{2}, A ⟩$ be soft normal subgroups of G such that $⟨ M, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{1}, A ⟩ \overset{⌣}{\cap} ⟨ N_{2}, A ⟩ .$ Then $⟨ N_{1} / M, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{2} / M, A ⟩$ if and only if $⟨ N_{1}, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{2}, A ⟩$

Proof.

Suppose that $⟨ N_{1}, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{2}, A ⟩ .$ Then $N_{1} (α) \subseteq N_{2} (α)$ for all $α \in A .$ Now for any $α \in A$ , consider $a_{M} \in N_{1} / M (α) .$ This implies that $a \in N_{1} (α) \subseteq N_{2} (α) .$ It follows that $a_{M} \in N_{2} / M (α) .$ Thus $⟨ N_{1} / M, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{2} / M, A ⟩ .$ Conversely, suppose that $⟨ N_{1} / M, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{2} / M, A ⟩ .$ Then $N_{1} / M (α) \subseteq N_{2} / M (α)$ for all $α \in A .$ Now for any $α \in A, a \in N_{1} (α) .$ It follows that $a_{M} \in N_{1} / M (α) .$ Hence $a_{M} \in N_{2} / M (α) .$ Therefore $a \in N_{2} (α) .$ Thus $N_{1} (α) \subseteq N_{2} (α)$ and hence $⟨ N_{1}, A ⟩ \overset{⌣}{\subseteq} ⟨ N_{2}, A ⟩ .$

Corollary 4.10.

Under the assumption of Lemma 4.9, $⟨ N_{1} / M, A ⟩ \overset{⌣}{=} ⟨ N_{2} / M, A ⟩$ if and only if $⟨ N_{1}, A ⟩ \overset{⌣}{=} ⟨ N_{2}, A ⟩$ .

Lemma 4.11.

If $⟨ G, *, A ⟩$ is soft isomorphic to $⟨ G^{'}, Δ, A ⟩$ then G is abelian if and only if G^ʹ is abelian.

Proof.

Let $c, d, y \in G^{'}$ such that $⟨ α, c, d, y ⟩ \in Δ .$ Since $⟨ f, A ⟩$ is surjective there exist $a, b \in G$ such that $⟨ α, a, c ⟩ \in f$ and $⟨ α, b, d ⟩ \in f .$ Let $x \in G$ such that $⟨ α, a, b, x ⟩ \in * .$ Since $⟨ f, A ⟩$ is soft homomorphism. Now, $⟨ α, c, d, y ⟩ \in Δ \Leftrightarrow ⟨ α, x, y ⟩ \in f .$ Since G is abelian we have $⟨ α, b, a, x ⟩ \in * .$ Again since $⟨ f, A ⟩$ is a soft homomorphism we get $⟨ α, d, c, y ⟩ \in Δ .$ Therefore G^ʹ is abelian. Conversely, let $a, b, c \in G$ such that $⟨ α, a, b, c ⟩ \in * .$ Since $⟨ f, A ⟩$ is a soft mapping, we have $⟨ α, a, c ⟩ \in f$ and $⟨ α, b, d ⟩ \in f$ for some $c, d \in G^{'} .$ Let $y \in G^{'}$ such that $⟨ α, c, d, y ⟩ \in Δ .$ Since $⟨ f, A ⟩$ is a soft homomorphism then we have $⟨ α, c, y ⟩ \in f \Leftrightarrow ⟨ α, c, d, y ⟩ \in Δ$ . Let $y \in G^{'}$ such that $⟨ α, d, c, y_{1} ⟩ \in Δ$ . By the assumption G^ʹ is abelian we get $y = y_{1} .$ Let $x \in G$ such that $⟨ α, b, a, x ⟩ \in * .$ Again since $⟨ f, A ⟩$ is a soft homomorphism we have $⟨ α, d, c, y ⟩ \in Δ \Leftrightarrow ⟨ α, x, y ⟩ \in f .$ Since $⟨ f, A ⟩$ is injective we have $c = x .$ Therefore G is abelian.

Theorem 4.12.

(The correspondence theorem) Let $⟨ M, A ⟩$ be a normal soft subgroup of $G .$ Then there is a one to one correspondence between the set of all normal soft subgroups of G containing $⟨ M, A ⟩$ and the set of all normal soft subgroups of $G / M .$

Proof.

Define $[⟨ M, A ⟩, G]$ to be the set of all normal soft subgroups of G containing $⟨ f, A ⟩ .$ i.e

\begin{array}{l} [M, G]_{A} & = & {⟨ N, A ⟩ \in S N_{A} (G) : ⟨ M, A ⟩ \subseteq ⟨ N, A ⟩} \end{array}

and let $S N_{A} (G / M)$ be the set of all normal soft subgroups of $G / M$ . Define $g : [M, G]_{A} ⟶ S N_{A} (G / M)$ by $g (⟨ N, A ⟩) = ⟨ N / M, A ⟩ \lor ⟨ N, A ⟩ \in [M, G]_{A}$ Then it is clear that g is really well-defined. Moreover,it follows from Lemma 4.9 that g is one to one. It remains to show that g is on to. Let $⟨ J, A ⟩$ be any soft normal subgroups of $G / M .$ For each $α \in A$ define $N (α) = {a \in G : a_{M} \in J (α)} .$ Then one can easily check that $⟨ N, A ⟩$ is a soft normal subgroup of G containing $⟨ M, A ⟩$ such that $⟨ N / M, A ⟩ \overset{⌣}{=} ⟨ J, A ⟩ .$ Therefore g is onto and hence it is the required one to one correspondence.

Theorem 4.13.

Let $⟨ N, A ⟩$ be a normal soft subgroup of G having the property (Q). If $⟨ f, A ⟩$ is a soft epimorphism from G to G^ʹ, with $⟨ K_{f}, A ⟩ \overset{⌣}{\subseteq} ⟨ N, A ⟩$ , then

\begin{array}{l} G / N \overset{⌣}{≅} G^{'} / f (N) . \end{array}

Proof.

Since $⟨ N, A ⟩$ is a normal soft subgroup of G and $⟨ f, A ⟩$ be a soft epimorphism it follows that $⟨ f (N), A ⟩$ is a normal soft subgroup of G^ʹ and hence the quotient soft group $G^{'} / f (N)$ is defined. Now, define a soft mapping $⟨ g, A ⟩$ from G to $G^{'} / f (N)$ by $⟨ α, a,^{x} f (N) ⟩ \in g$ where $⟨ α, a, x ⟩ \in f$ for all $a \in G .$ Let $x, y, z \in G$ and $a, b, c \in G^{'}, α \in A$ such that $⟨ α, x,^{a} f (N) ⟩ \in g$ , $⟨ α, y,^{b} f (N) ⟩ \in g ⟨ α, z,^{c} f (N) ⟩ \in g$ and $⟨ α, x, y, z ⟩ \in *$ where $⟨ α, x, a ⟩ \in f$ , $⟨ α, y, b ⟩ \in f$ and $⟨ α, z, c ⟩ \in f .$ Since $⟨ f, A ⟩$ be a soft homomorphism and then we get that $⟨ α, a, b, c ⟩ \in Δ$ in $G^{'} .$ It follows that $⟨ α,^{a} f (N),^{b} f (N),^{c} f (N) ⟩ \in \bar{Δ}$ in $G^{'} / f (N)$ and therefore $⟨ g, A ⟩$ be a soft homomorphism. Let $z \in G^{'}$ since $⟨ f, A ⟩$ be a soft epimorphism, there exists $a \in G$ such that $⟨ α, a, z ⟩ \in f$ and therefore $⟨ α, a,^{z} f (N) ⟩ \in g .$ This implies that $⟨ g, A ⟩$ be a soft epimorphism. Further, for any $α \in A :$

\begin{array}{l} K_{g} (α) = & {a \in G : ⟨ α, a,^{e_{α}^{'}} f (N) ⟩ \in g} \\ = & {a \in G :^{x} f (N) =^{e_{α}^{'}} f (N) forsomex \in G^{'} \\ with ⟨ α, a, x ⟩ \in f} \\ = & {a \in G : x \in f (N) (α) forsomex \in G^{'} with \\ ⟨ α, a, x ⟩ \in f} \\ = & {a \in G : ⟨ α, a, x ⟩ \in fand ⟨ α, b, x ⟩ \in ffor \\ someb \in N (α)} \\ = & {a \in G : ⟨ α, c, e_{α}^{'} ⟩ \in fand ⟨ α, b^{- α}, a, c ⟩ \in * \\ forsomeb \in N (α)} \\ = & {a \in G : c \in K_{f} (α) \subseteq N (α) andb \in N (α)} \\ = & N (α) . \end{array}

Therefore $⟨ K_{f}, A ⟩ \overset{⌣}{=} ⟨ N, A ⟩ .$ Thus by the first soft isomorphism theorem,

\begin{array}{l} G / N \overset{⌣}{≅} G^{'} / f (N) . \end{array}

Theorem 4.14.

Let $⟨ N, A ⟩$ and $⟨ M, A ⟩$ be normal soft subgroups of G satisfying the property (Q) such that $⟨ MN, A ⟩$ is absolute soft set over G. Then,

\begin{array}{l} G / M \cap N \overset{⌣}{≅} G / N \times G / M . \end{array}

Proof.

Define a soft mapping $⟨ f, A ⟩$ from G to $G / N \times G / M$ by $⟨ α, a, x ⟩ \in f$ where $x = (^{a} N,^{a} M)$ for $a \in G .$ Let $α \in A$ , $a, b, c \in G$ and $x_{1}, x_{2}, x_{3} \in G / N \times G / M$ such that $⟨ α, a, x_{1} ⟩ \in f$ , $⟨ α, b, x_{2} ⟩ \in f$ , $⟨ α, c, x_{3} ⟩ \in f$ and $⟨ α, a, b, c ⟩ \in *$ in G where $x_{1} = (^{a} N,^{a} M)$ , $x_{2} = (^{b} N,^{b} M)$ and $x_{3} = (^{c} N,^{c} M) .$ Since $⟨ N, A ⟩$ and $⟨ M, A ⟩$ are normal soft subgroups of G then we have $⟨ α,^{a} N,^{b} N,^{c} N ⟩ \in Δ$ in $G / N$ and $⟨ α,^{a} M,^{b} M,^{c} M ⟩ \in \bar{Δ}$ in $G / M .$ From the definition of direct product of a soft group and $⟨ α, c, x_{3} ⟩ \in f$ then we have $⟨ α, x_{1}, x_{2}, x_{3} ⟩ \in ⊛$ in $\in G / N \times G / M$ and hence $⟨ f, A ⟩$ is a soft homomorphism. Next we shall to prove that $⟨ f, A ⟩$ is surjective. Let $z = (^{x} N,^{y} M) \in G / N \times G / M .$ where $x, y \in G$ and $α \in A .$ Since $⟨ NM, A ⟩$ is an absolute soft set $[NM] (α) = G .$ So if $x, y \in [NM] (α)$ , then $⟨ α, r, s, x ⟩ \in *$ for some $r \in N (α)$ and $s \in M (α)$ with $⟨ α, t, u, y ⟩ \in *$ for some $t \in N (α)$ and $u \in M (α) .$ Now, let $a \in G$ such that $⟨ α, s, t, a ⟩ \in * .$ Then, $⟨ α, a^{- α}, x, n ⟩ \in *$ for some $n \in N (α)$ and hence $^{a} N (α) =^{x} N (α)$ for all $x \in A .$ Also, since $s, u, \in M (α)$ then we have $⟨ α, a^{- α}, y, m ⟩ \in *$ for some $m \in M (α)$ and hence $^{a} M (α) =^{x} M (α) .$ Therefore $⟨ α, a, x ⟩ \in f$ where $x = (^{a} N,^{a} M) .$ This implies that $⟨ α, a, z ⟩ \in f .$ Thus $⟨ f, A ⟩$ is a soft epimorphism.

Now,

\begin{array}{l} K_{f} (α) & = {a \in G : ⟨ α, a, z ⟩ \in fwherez = (^{e_{α}} N,^{e_{α}} M)} \\ = {a \in G :^{a} N =^{e_{α}} Nand^{a} M =^{e_{α}} M} \\ = {a \in G :^{a} N (β) =^{e_{α}} N (β) and \\ ^{a} M (β) =^{e_{α}} M (β) ∀β \in A} \\ = {a \in G : a \in N (α) anda \in M (α)} \\ = (M \cap N) (α) . \end{array}

Thus $⟨ K_{f}, A ⟩ \overset{⌣}{≅} ⟨ M \cap N, A ⟩ .$ Therefore, by the first soft isomorphism theorem

\begin{array}{l} G / M \cap N \overset{⌣}{≅} G / N \times G / M . \end{array}

Corollary 4.15.

Any soft homomorphism of soft groups can be expressed as a composition of soft epimorphism and monomorphism.

Theorem 4.16.

Let $⟨ G, *, A ⟩$ , $⟨ G_{1}, \circ, A ⟩$ and $⟨ G_{2}, △, A ⟩$ be soft groups. Then $G \overset{⌣}{≅} G_{1} \times G_{2}$ if and only if there are normal soft subgroups $⟨ N_{1}, A ⟩$ and $⟨ N_{2}, A ⟩$ of G such that $⟨ N_{1} N_{2}, A ⟩$ is the absolute soft set over G, $⟨ N_{1} \cap N_{2}, A ⟩$ is a trivial soft subgroup of $G,$ $G_{1} \overset{⌣}{≅} G / N_{1}$ and $G_{2} \overset{⌣}{≅} G / N_{2}$ .

Proof.

For $α \in A$ let e_α, $e_{α}^{'}$ and $e_{α}^{″}$ be identities in G, G₁ and G₂ respectively. Suppose that $G \overset{⌣}{≅} G_{1} \times G_{2}$ and let $⟨ f, A ⟩$ be a soft isomorphism from G to $G_{1} \times G_{2} .$ Put $⟨ M, A ⟩ \overset{⌣}{=} ⟨ {\overset{⌣}{e}}_{α}^{'} \times 1_{G_{2}}, A ⟩$ and $⟨ N, A ⟩ \overset{⌣}{=} ⟨ 1_{G_{1}} \times {\overset{⌣}{e}}_{α}^{″}, A ⟩ .$ Then $⟨ M, A ⟩$ and $⟨ N, A ⟩$ are normal soft subgroups of $G_{1} \times G_{2} .$ Now, put $N_{1} (α) = f^{- 1} (M) (α)$ and $N_{2} (α) = f^{- 1} (N) (α) .$ Then $⟨ N_{1}, A ⟩$ and $⟨ N_{2}, A ⟩$ are normal soft subgroups of G. For any $x \in G$ , we have $⟨ α, x, a ⟩ \in f$ where $a = (a_{1}, a_{2}) \in G_{1} \times G_{2} .$ Choose x₁ and x₂ in G such that $⟨ α, x_{1}, a^{'} ⟩ \in f$ and $⟨ α, x_{2}, a^{″} ⟩ \in f$ where $a^{'} = (e_{α}^{'}, a_{2})$ , $a^{″} = (a_{1}, e_{α}^{″}) .$ Then, $x_{1} \in f^{- 1} (M) (α)$ , $x_{2} \in f^{- 1} (N) (α)$ and $⟨ α, x, y ⟩ \in f \Leftrightarrow ⟨ α, a^{'}, a^{″}, y ⟩ \in * .$ Since $⟨ f, A ⟩$ is a soft isomorphism then we have $⟨ α, x_{1}, x_{2}, x ⟩ \in *$ and hence $x \in (N_{1} N_{2}) (α) .$ Therefore $(N_{1} N_{2}) (α) = G$ for all $α \in A .$ Thus $⟨ N_{1} N_{2}, A ⟩$ is absolute soft set. Let $x \in (N_{1} \cap N_{2}) (α) .$ This implies that $x \in f^{- 1} (M) (α)$ , $x \in f^{- 1} (N) (α) .$ It follows that $a \in M (α)$ and $a \in N (α) .$ Then $a \in M (α) \cap N (α) = (M \cap N) (α) .$ Since $⟨ f, A ⟩$ is soft homomorphism then we have $a = (e_{α}^{'}, e_{α}^{″})$ and $⟨ α, e_{α}, a ⟩ \in f .$ This implies that $x = e_{α} .$ Therefore, $(N_{1} \cap N_{2}) (α) = {e_{α}}$ for all $α \in A .$ Thus $⟨ N_{1} \cap N_{2}, A ⟩$ is a trivial soft subgroup of $G .$ Next,define a soft mapping $⟨ f_{1}, A ⟩$ from G to G₁ and $⟨ f_{2}, A ⟩$ from G to G₂ by $⟨ α, x, a_{1} ⟩ \in f_{1}$ and $⟨ α, x, a_{2} ⟩ \in f_{2}$ if $⟨ α, x, a ⟩ \in f$ where $a = (a_{1}, a_{2}) .$ Then $⟨ f_{1}, A ⟩$ and $⟨ f_{2}, A ⟩$ are soft epimorphism and $⟨ K_{f_{1}}, A ⟩ \overset{⌣}{=} ⟨ N_{1}, A ⟩$ and $⟨ K_{f_{2}}, A ⟩ \overset{⌣}{=} ⟨ N_{2}, A ⟩ .$ For

\begin{array}{l} x \in N_{1} (α) \Leftrightarrow a \in M (α) \Leftrightarrow a = a^{'} \Leftrightarrow a_{1} = e_{α}^{'} \end{array}

and

\begin{array}{l} x \in N_{2} (α) \Leftrightarrow a \in M (α) \Leftrightarrow a = a^{″} \Leftrightarrow a_{2} = e_{α}^{″} \end{array}

where $a^{'} = (e_{α}^{'}, a_{2})$ and $a^{″} = (a^{'}, e_{α}^{″}) .$ Therefore, $G_{1} \overset{⌣}{≅} G / N_{1}$ and $G_{2} \overset{⌣}{≅} G / N_{2} .$ Conversely, suppose that $⟨ N_{1}, A ⟩$ and $⟨ N_{2}, A ⟩$ are normal soft subgroups of G such that $(N_{1} N_{2}) (α) = G, (N_{1} \cap N_{2}) (α) = {e_{α}}$ for all $\in α \in A$ , $G_{1} \overset{⌣}{≅} G / N_{1}$ and $G_{2} \overset{⌣}{≅} G / N_{2} .$ Let $⟨ g_{1}, A ⟩$ be a soft isomorphism from $G / N_{1}$ to G₁ and let $⟨ g_{2}, A ⟩$ be a soft isomorphism from $G / N_{2}$ to G₂ and $⟨ h_{1}, A ⟩$ be a soft homomorphism from G to $G / N_{1}$ and $⟨ h_{2}, A ⟩$ be a soft homomorphism from G to $G / N_{2} .$ Now define a soft mapping $⟨ θ, A ⟩$ from G to $G_{1} \times G_{2}$ as follows: for $α \in A$ , $a \in G, z_{1} \in G_{1}$ and $z_{2} \in G_{2}$ :

\begin{array}{l} ⟨ α, a, ⟨ z_{1}, z_{2} ⟩ ⟩ \in θ if and only if ⟨ α,^{a} N_{1}, z_{1} ⟩ \in g_{1} and \\ ⟨ α,^{a} N_{2}, z_{2} ⟩ \in g_{2} \end{array}

Since $⟨ g_{1}, A ⟩$ and $⟨ g_{2}, A ⟩$ are soft isomorphisms, $⟨ θ, A ⟩$ will also be a soft epimorphism. Put $e_{α} = (e_{α}^{'}, e_{α}^{″})$ . Then for any $a \in G$ , consider the following:

\begin{array}{l} ⟨ α, a, e_{α} ⟩ \in θ & \Leftrightarrow ⟨ α,^{a} N_{1}, e_{α}^{'} ⟩ \in g_{1} and ⟨ α,^{a} N_{2}, e_{α}^{″} ⟩ \in g_{2} \\ \Leftrightarrow a \in (N_{1} \cap N_{2}) (α) = {e_{α}} . \end{array}

Therefore $K_{θ} (α) = {e_{α}}$ for all $α \in A .$ Hence $⟨ θ, A ⟩$ is a soft monomorphism and hence a soft isomorphism. Therefore, $G \overset{⌣}{≅} G_{1} \times G_{2} .$

5. Soft automorphisms

This section is devoted to present some fundamental results on soft automorphisms on soft groups.

Definition 5.1.

For any soft group $⟨ G, *, A ⟩$ a bijective soft homomorphism of G onto itself is called a soft automorphism of $G .$ We denote by SAut(G) the collection of all soft automorphisms of G.

Example 5.2.

Consider the soft groups $⟨ R^{+}, *, N ⟩$ and $< R^{+}, Δ, N >,$ where $R^{+}$ is the set of positive real numbers, $N$ is the set of natural numbers and $⟨ *, N ⟩$ and $< Δ, N >$ are defined by: $⟨ α, a, b, c ⟩ \in * \Leftrightarrow c = αab$ and $⟨ α, a, b, c ⟩ \in Δ \Leftrightarrow c = \frac{ab}{\sqrt{α}} .$ If we define $f = {(α, x, y) : y = α \sqrt{x}}$ , then $⟨ f, N ⟩$ is a soft automorphism of $R^{+} .$

Lemma 5.3.

Let $⟨ f, A ⟩$ be a soft isomorphism from $⟨ G, *, A ⟩$ to $⟨ G^{'}, Δ, A ⟩ .$ Then $⟨ f^{- 1}, A ⟩$ is also a soft isomorphism from $⟨ G^{'}, Δ, A ⟩$ to $⟨ G, *, A ⟩ .$

Lemma 5.4.

For any soft group G, $⟨ SAut (G), \circ ⟩$ is a group; where $\circ$ is the composition of soft mappings.

Theorem 5.5.

$⟨ SAut (G), \circ ⟩$ is isomorphic to a subgroup of $⟨ Aut (S E_{A} (G)), \circ ⟩$ .

Proof.

We show that the map $f ⟼ \overset{⌣}{f}$ is an embedding of SAut(G) into $Aut (S E_{A} (G))$ where for each soft map $⟨ f, A ⟩$ from to G to G^ʹ, $\overset{⌣}{f}$ is a soft mapping from $S E_{A} (G)$ to $S E_{A} (G)$ given in Definition 3.7. It is proved in Theorem 3.9 that if $⟨ f, A ⟩$ is a soft automorphism of $G,$ then $\overset{⌣}{f}$ is an automorphism of $S E_{A} (G) .$

Claim 1: $f ⟼ \overset{⌣}{f}$ is one to one. Let $f, g \in SAut (G)$ such that $\overset{⌣}{f} = \overset{⌣}{g} .$ Then $\overset{⌣}{f} (\overset{⌣}{a}) = \overset{⌣}{g} (\overset{⌣}{a})$ for all $\overset{⌣}{a} \in S E_{A} (G) .$ Then $\overset{⌣}{f} (\overset{⌣}{a}) (α) = \overset{⌣}{g} (\overset{⌣}{a}) (α)$ for all $α \in A .$ That is, for any $x \in G$ it holds that $⟨ α, \overset{⌣}{a} (α), x ⟩ \in f$ if and only if $⟨ α, \overset{⌣}{a} (α), x ⟩ \in g .$ Implying that, $⟨ f, A ⟩ \overset{⌣}{=} ⟨ g, A ⟩ .$

Claim 2: $\overset{⌣}{(f \circ g)} = \overset{⌣}{f} \circ \overset{⌣}{g}$

Let $\overset{⌣}{a} \in S E_{A} (G), α \in A$ and $x \in G$ such that $\overset{⌣}{(f \circ g)} (\overset{⌣}{a}) (α) = {x} .$ Then, $⟨ α, \overset{⌣}{a} (α), x ⟩ \in f \circ g$ , and so there is some $b \in G$ such that $⟨ α, \overset{⌣}{a} (α), b ⟩ \in g,$ and $⟨ α, b, x ⟩ \in f .$ Thus $\overset{⌣}{g} (\overset{⌣}{a}) (α) = {b}$ and $\overset{⌣}{f} (\overset{⌣}{g} (\overset{⌣}{a})) (α) = {x} .$ Conversely, if we are assuming that $\overset{⌣}{f} (\overset{⌣}{g} (\overset{⌣}{a})) (α) = {x},$ then it can be shown that $\overset{⌣}{(f \circ g)} (\overset{⌣}{a}) (α) = {x} .$ Thus $\overset{⌣}{(f \circ g)} (\overset{⌣}{a}) (α) = \overset{⌣}{f} (\overset{⌣}{g} (\overset{⌣}{a}) (α)$ for all $α \in A .$ Therefore, $\overset{⌣}{f \circ g} (\overset{⌣}{a}) = \overset{⌣}{f} (\overset{⌣}{g} (\overset{⌣}{a})$ for all $\overset{⌣}{a} \in S E_{A} (G) .$ Hence $\overset{⌣}{f \circ g} = \overset{⌣}{f} \circ \overset{⌣}{g} .$ Therefore the map $f ⟼ \overset{⌣}{f}$ is a monomorphism hence an embedding.

Theorem 5.6.

For any soft groups $⟨ G, *, A ⟩$ and $⟨ G^{'}, Δ, A ⟩$ . If $G \overset{⌣}{≅} G^{'}$ , then $SAut (G) \overset{⌣}{≅} SAut (G^{'}) .$

Proof.

Suppose $G \overset{⌣}{≅} G^{'}$ and $⟨ ϕ, A ⟩$ be a soft isomorphism from G to $G^{'} .$ For any soft automorphism $⟨ f, A ⟩$ of G, one can easily check that $⟨ ϕ \circ f \circ ϕ^{- 1}, A ⟩$ is a soft automorphism of G^ʹ. Moreover, the map $f \mapsto ϕ \circ f \circ ϕ^{- 1}$ is an ordinary group isomorphism of SAut(G) onto $SAut (G^{'})$ .

Lemma 5.7.

Let $⟨ G, *, A ⟩$ be a soft group. Define a soft mapping $⟨ f, A ⟩$ from G to G by $⟨ α, a, a^{- α} ⟩ \in f$ for any $a \in G .$ Then $⟨ f, A ⟩$ is a soft automorphism of G if and only if $⟨ G, *, A ⟩$ is an abelian soft group.

Proof.

Suppose $⟨ f, A ⟩$ is a soft automorphism of G. Let $a, b, x \in G$ and $α \in A$ such that $⟨ α, a, b, x ⟩ \in * .$ Since $⟨ f, A ⟩$ is a soft homomorphism of G. It follows that $⟨ α, a^{- α}, b^{- α}, x^{- α} ⟩ \in * .$ Then we have $⟨ α, b, a, x ⟩ \in * .$ Therefore $⟨ G, *, A ⟩$ is abelian soft group. Conversely, suppose that $⟨ G, *, A ⟩$ is abelian soft group. We first show that $⟨ f, A ⟩$ is a soft homomorphism. Let $a, b, c \in G$ and $α \in A$ such that $⟨ α, a, a^{- α} ⟩ \in f, ⟨ α, b, b^{- α} ⟩ \in f, ⟨ α, c, c^{- α} ⟩ \in f$ and $⟨ α, a, b, c ⟩ \in * .$ It follows that $⟨ α, b^{- α}, a^{- α}, c^{- α} ⟩ \in * .$ Since G is abelian soft group we have $⟨ α, a^{- α}, b^{- α}, c^{- α} ⟩ \in * .$ Therefore $⟨ f, A ⟩$ is a soft homomorphism. Next we show that $⟨ f, A ⟩$ is injective. Let $x_{1}, x_{2}, y \in G$ such that $⟨ α, x_{1}, y ⟩ \in f$ and $⟨ α, x_{2}, y ⟩ \in f .$ This implies that $y = x_{1}^{- α}$ and $y = x_{2}^{- α} .$ It follows that $x_{1} = x_{2} .$ Therefore $⟨ f, A ⟩$ is injective. Finlay it remains to show that $⟨ f, A ⟩$ is surjective. Let $h \in G$ then $g = h^{- α} \in G .$ It follows that $⟨ α, g, h ⟩ \in f .$ Thus $⟨ f, A ⟩$ is surjective. Hence $⟨ f, A ⟩$ is a soft automorphism.

Proposition 5.8.

Let $⟨ f, A ⟩$ be a soft automorphism of $G .$ Let $⟨ H, A ⟩$ be a soft set over G define by

\begin{array}{l} H (α) = {g \in G : ⟨ α, g, g ⟩ \in f \circ f} \end{array}

then $⟨ H, A ⟩$ is a soft subgroup of $G .$

Proof.

Since $⟨ α, e_{α}, e_{α} ⟩ \in f \circ f$ , we have $e_{α} \in H (α) .$ Let $g, h \in H (α)$ then $⟨ α, g, g ⟩ \in f \circ f$ and $⟨ α, h, h ⟩ \in f \circ f .$ Let $x, z \in G$ such that $⟨ α, g, h, x ⟩ \in *$ and $⟨ α, x, z ⟩ \in f \circ f .$ Since $⟨ f, A ⟩$ is a soft automorphism then we have $⟨ α, g, h, z ⟩ \in * .$ This implies that $z = x .$ Therefore $x \in H (α) .$ Let $g \in H (α)$ , $y \in G$ such that $⟨ α, g, g ⟩ \in f \circ f$ and $⟨ α, g^{- α}, y ⟩ \in f \circ f .$ As $⟨ α, g, g ⟩ \in f \circ f$ it follows that $⟨ α, g^{- α}, g^{- α} ⟩ \in f \circ f .$ So $y = g^{- α} .$ Therefore $g^{- α} \in H (α) .$ Hence $⟨ H, A ⟩$ is a soft subgroup of $G .$

Proposition 5.9.

Let $⟨ G, *, A ⟩$ be a soft group and let $⟨ H, A ⟩$ be a soft set over G defined by

\begin{array}{l} H (α) = {a \in G : ⟨ α, a, a ⟩ \in f, ∀f \in SAut (G)} \end{array}

for all $α \in A$ . Then $⟨ H, A ⟩$ is a normal soft subgroup of $G .$

Proof.

Since $⟨ α, e_{α}, e_{α} ⟩ \in f, e_{α} \in H (α) .$ Let $a, b \in H (α)$ and $x \in G$ such that $⟨ α, a, b^{- α}, x ⟩ \in * .$ We have $⟨ α, a, a ⟩ \in f$ and $⟨ α, b, b ⟩ \in f .$ Let $y \in G$ such that $⟨ α, x, y ⟩ \in f .$ Since $⟨ f, A ⟩$ is a soft homomorphism of G to G and $⟨ α, a, b^{- α}, x ⟩ \in *$ , we have $⟨ α, a, b^{- α}, y ⟩ \in *$ and so $y = x .$ Therefore $⟨ α, x, x ⟩ \in f .$ Thus $x \in H (α) .$ Hence $⟨ H, A ⟩$ is a soft subgroup of $G .$ Let $α \in A, a, x, y \in G$ and $n \in H (α) .$ Suppose $⟨ α, a, n, x ⟩ \in *$ and $⟨ α, x, a^{- α}, y ⟩ \in * .$ We shall prove that $y \in H (α) .$ Let $y_{1}, y_{2}, z \in G$ such that $⟨ α, a, y_{1} ⟩ \in f$ , $⟨ α, y, z ⟩ \in f$ and $⟨ α, x, y_{2} ⟩ \in f$ . Since $⟨ f, A ⟩$ is a soft homomorphism and $⟨ α, a, y, x ⟩ \in *$ , it holds that $⟨ α, y_{1}, z, y_{2} ⟩ \in * .$ As $n \in H (α)$ it follows that $⟨ α, n, n ⟩ \in f .$ Again since $⟨ f, A ⟩$ is a soft homomorphism and $⟨ α, a, n, x ⟩ \in *, ⟨ α, y_{1}, n, y_{2} ⟩ \in *$ , it follows from the cancellation law that z = n. As $⟨ f, A ⟩$ is injective we get y = n. Then by transitivity we get $y = z .$ Therefore $⟨ α, y, y ⟩ \in f .$ Thus $y \in H (α) .$ Hence $⟨ H, A ⟩$ is a normal soft subgroup of $G .$

Theorem 5.10.

Let $⟨ G, *, A ⟩$ be a soft group and $a \in G .$ Define a soft mapping $⟨ T_{a}, A ⟩$ from G to G as follows , for each $α \in A$ : $⟨ α, x, z ⟩ \in T_{a}$ if and only if there is some $y \in G$ such that $⟨ α, a, x, y ⟩ \in *$ and $⟨ α, y, a^{- α}, z ⟩ \in * .$ Then $⟨ T_{a}, A ⟩$ is a soft automorphism on $G .$

Proof.

We first show that $< T_{a}, A >$ is a soft homomorphism. Let $α \in A, x_{1}, x_{2}, x_{3}, z_{1}, z_{2}, z_{3} \in G$ such that $⟨ α, x_{1}, z_{1} ⟩ \in T_{a}$ , $⟨ α, x_{2}, z_{2} ⟩ \in T_{a}$ , $⟨ α, x_{3}, z_{3} ⟩ \in T_{a}$ and $⟨ α, x_{1}, x_{2}, x_{3} ⟩ \in * .$ Then there exists $y_{1}, y_{2}, y_{3} \in G$ such that $⟨ α, a, x_{1}, y_{1} ⟩ \in *$ and $⟨ α, y_{1}, a^{- α}, z_{1} ⟩ \in *,$ $⟨ α, a, x_{2}, y_{2} ⟩ \in *$ and $⟨ α, y_{2}, a^{- α}, z_{2} ⟩ \in *,$ $⟨ α, a, x_{3}, y_{3} ⟩ \in *$ and $⟨ α, y_{3}, a^{- α}, z_{3} ⟩ \in * .$ Using the fact that $⟨ α, a^{- α}, a, e_{α} ⟩ \in *$ and by associative property of $*$ we get that $⟨ α, z_{1}, z_{2}, z_{3} ⟩ \in * .$ Thus $⟨ T_{a}, A ⟩$ is a soft homomorphism. In addition for any $α \in A$ and any $z \in G,$ let $x, y \in G$ such that $⟨ α, a^{- α}, z, y ⟩ \in *$ and $⟨ α, y, a, x ⟩ \in * .$ If $u \in G$ such that $⟨ α, a, x, u ⟩ \in *,$ then it can be verified that $⟨ α, z, a, u ⟩ \in * .$ Which implies that $⟨ α, u, a^{- α}, z ⟩ \in * .$ Therefore, u is an element of G such that $⟨ α, a, x, u ⟩ \in *,$ and $⟨ α, u, a^{- α}, z ⟩ \in * .$ Thus $⟨ α, x, z ⟩ \in T_{α}$ and hence $⟨ T_{a}, A ⟩$ is surjective. Finally, it remains to show that $⟨ T_{a}, A ⟩$ is injective. Let $α \in A,$ and $x_{1}, x_{2}, z \in G$ such that $⟨ α, x_{1}, z ⟩ \in T_{a}$ and $⟨ α, x_{2}, z ⟩ \in T_{a} .$ Then there exist y₁ and $y_{2} \in G$ such that $⟨ α, a, x_{1}, y_{1} ⟩ \in *$ , $⟨ α, y_{1}, a^{- α}, z ⟩ \in *$ and $⟨ α, y_{2}, a^{- α}, z ⟩ \in *$ . Applying the cancellation law on equations $⟨ α, y_{1}, a^{- α}, z ⟩ \in *$ and $⟨ α, y_{2}, a^{- α}, z ⟩ \in *$ we have $y_{1} = y_{2}$ . Again using $y_{1} = y_{2}$ and applying the cancellation law on equations $⟨ α, a, x_{1}, y_{1} ⟩ \in *$ and $⟨ α, a, x_{2}, y_{2} ⟩ \in *$ gives us $x_{1} = x_{2} .$ Thus $⟨ T_{a}, A ⟩$ is injective and therefore it is a soft automorphism.

Definition 5.11.

Let $⟨ G, *, A ⟩$ be a soft group and $a \in G .$ The soft automorphism $⟨ T_{a}, A ⟩$ of G given in the above theorem is called the inner soft automorphism of G corresponding to a. We denote by $S I_{A} (G)$ the set of all inner soft isomorphisms of G with the given set of parameters A.

Define a soft binary operation $⟨ \circ, A ⟩$ on SI_A by; for $α \in A$ , $a, b, c \in G ⟨ α, T_{a}, T_{b}, T_{c} ⟩ \in \circ$ if and only if $⟨ α, a, b, d ⟩ \in *$ for some $d \in G$ with $T_{c} = T_{d}$ . Then we have the following theorem.

Theorem 5.12.

For any soft group $⟨ G, *, A ⟩, ⟨ S I_{A} (G), \break \circ, A ⟩$ is a soft group.

Proof.

We first show that $⟨ \circ, A ⟩$ is well defined. Let $a, b, c, d, e, f \in G$ such that

\begin{array}{l} ⟨ T_{a}, A ⟩ \overset{⌣}{=} ⟨ T_{d}, A ⟩, ⟨ T_{b}, A ⟩ \overset{⌣}{=} ⟨ T_{e}, A ⟩ \end{array}

\begin{array}{l} ⟨ α, T_{a}, T_{b}, T_{c} ⟩ \in \circ and ⟨ α, T_{d}, T_{e}, T_{f} ⟩ \in \circ . \end{array}

Then we have for any $x, z \in G$ and $α \in A, ⟨ α, x, z ⟩ \in T_{a}$ if and only if $⟨ α, x, z ⟩ \in T_{d} .$ Similarly $⟨ α, x, z ⟩ \in T_{b}$ if and only if $⟨ α, x, z ⟩ \in T_{e} .$ Again then we have $⟨ α, a, b, c ⟩ \in *$ and $⟨ α, d, e, f ⟩ \in * .$

Claim: $⟨ T_{c}, A ⟩ \overset{⌣}{=} ⟨ T_{f}, A ⟩ .$

For any $α \in A,$ and $x, z \in G .$ Suppose that $⟨ α, x, z ⟩ \in T_{c} .$ Then there exists $y \in G$ such that $⟨ α, c, x, y ⟩ \in *$ and $⟨ α, y, c^{- α}, z ⟩ \in * .$ Since $⟨ α, a, b, c ⟩ \in *$ there exist $u, w \in G$ such that $⟨ α, b, x, u ⟩ \in *$ and $⟨ α, u, b^{- α}, w ⟩ \in *$ together implying $⟨ α, x, w ⟩ \in T_{b} = T_{e}$ and there is also $v \in G$ such that $⟨ α, a, w, v ⟩ \in *$ and $⟨ α, v, a^{- α}, z ⟩ \in *$ implying that $⟨ α, w, z ⟩ \in T_{a} = T_{d} .$ Therefore we can find $u_{1}, w_{1}, v_{1} \in G$ such that $⟨ α, d, w_{1}, v_{1} ⟩ \in *, ⟨ α, v_{1}, d^{- α}, z ⟩ \in *,$ $⟨ α, e, x, u_{1} ⟩ \in *$ and $⟨ α, u_{1}, e^{- α}, w_{1} ⟩ \in * .$ Since $⟨ α, d, e, f ⟩ \in *$ we get that for some $g \in G ⟨ α, f, x, g ⟩ \in *$ and $⟨ α, g, f^{- α}, z ⟩ \in *$ and therefore $⟨ α, x, z ⟩ \in T_{f}$ that is $T_{c} (α) \subseteq T_{f} (α) .$ Since α is arbitrary we get $⟨ T_{c}, A ⟩ \overset{⌣}{\subseteq} ⟨ T_{f}, A ⟩ .$ Similarly it can be shown that $⟨ T_{f}, A ⟩ \overset{⌣}{\subseteq} ⟨ T_{c}, A ⟩$ and hence the equality holds.

Therefore the soft binary operation $⟨ \circ, A ⟩$ is well-defined . It remains to show that $⟨ S I_{A} (G), \circ, A ⟩$ satisfies the soft group axioms which are straightforward.

Corollary 5.13.

For any abelian soft group G, an identity soft mapping is the only soft inner automorphism of G.

We conclude this paper by showing that $S I_{A} (G)$ is soft isomorphic with the quotient soft group $G / Z_{A} (G)$ in the following theorem.

Theorem 5.14.

For any soft group $⟨ G, *, A ⟩$ we have $G / Z_{A} (G) \overset{⌣}{=} S I_{A} (G) .$

Proof.

Define a soft mapping $⟨ π, A ⟩$ from G to $S I_{A} (G)$ by:

\begin{array}{l} π = {⟨ α, a, T_{a} ⟩ : α \in Aanda \in G} . \end{array}

Then it is clear that $⟨ π, A ⟩$ is a soft epimorphism.

Claim: $⟨ K_{π}, A ⟩ \overset{⌣}{=} ⟨ Z_{A} (G), A ⟩$ for any $α \in A .$ Let $a \in K_{π} (α) .$ Then $T_{a} = T_{e_{α}}$ i.e for $x, y \in G$ it holds that $⟨ α, x, z ⟩ \in T_{a}$ if and only if $⟨ α, x, z ⟩ \in T_{e_{α}}$ equivalently, if $y \in G$ such that $⟨ α, a, x, y ⟩ \in *$ and $⟨ α, y, a^{- α}, z ⟩ \in *$ then z = x and $⟨ α, x, a, y ⟩ \in *$ and hence $a \in Z_{A} (G) (α) .$ Since $α \in A$ is arbitrary we have $⟨ K_{π}, A ⟩ \overset{⌣}{\subseteq} ⟨ Z_{A} (G), A ⟩ .$ Conversely, for $α \in A,$ suppose that $a \in Z_{A} (G) (α) .$ Then for all $x, y \in G$ it is the case that $⟨ α, a, x, y ⟩ \in *$ if and only if $⟨ α, x, a, y ⟩ \in * .$ For any $x \in G$ and $y \in G$ if $⟨ α, a, x, y ⟩ \in *,$ then $⟨ α, y, a^{- α}, x ⟩ \in * .$ Now we show that $T_{a} = T_{e_{α}}$ for $x, y \in G,$ let $⟨ α, x, z ⟩ \in T_{α}$ which implies that there exist $y \in G$ such that $⟨ α, a, x, y ⟩ \in *,$ and $⟨ α, y, a^{- α}, z ⟩ \in * .$ Then it follows that $z = x,$ and hence $⟨ α, x, x ⟩ \in T_{e_{α}}$ i.e $T_{a} \subseteq T_{e_{α}} .$ Also if $⟨ α, x, z ⟩ \in T_{e_{α}}$ we get that z = x. So that for any $y \in G,$ if $⟨ α, a, x, y ⟩ \in *,$ then $⟨ α, y, a^{- α}, x ⟩ \in * .$ So that $⟨ α, x, x ⟩ \in T_{a}$ and hence $T_{e_{α}} \subseteq T_{a} .$ Therefore $T_{a} = T_{e_{α}} .$ Thus, $a \in K_{π} (α) .$ Therefore, $⟨ K_{π}, A ⟩ \overset{⌣}{=} ⟨ Z_{A} (G), A ⟩ .$ Hence by the first soft isomorphism theorem $G / Z_{A} (G) \overset{⌣}{=} S I_{A} (G) .$

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Soft homomorphisms on soft groups

ABSTRACT

1. Introduction

2. Preliminaries

3. Soft homomorphisms

4. Soft isomorphism theorems

5. Soft automorphisms

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Soft homomorphisms on soft groups

ABSTRACT

1. Introduction

2. Preliminaries

3. Soft homomorphisms

4. Soft isomorphism theorems

5. Soft automorphisms

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Supplementary material

References

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