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ABSTRACT
In this paper, we study soft homomorphisms on soft groups and investigate their properties. Given a soft mapping from G to Gʹ, we obtain an ordinary map
from the set
of soft elements of G to the set
of soft elements of Gʹ, and show that
is a soft homomorphism (respectively, soft isomorphism) if and only if
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
. 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
.
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 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 consisting of a nonempty set G equipped with a soft binary operation
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 , where
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
.
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
Definition 2.1.
(Molodtsov, Citation1999) A soft set over X is a pair where
is a mapping from A into power set of X. In other words, it is a parameterized family of subsets of
Definition 2.2.
(Maji et al., Citation2002) A soft set over G is said to be the absolute soft set over G if
for all
By a soft relation from X to Y we mean a soft set over X × Y. In this case, for
and
we write
to mean
(Zhang & Yuan, Citation2014).
Definition 2.3.
(Addis et al., Citation2022) A soft mapping from X to Y is a soft relation from X to Y such that:
(1) | for each | ||||
(2) | for each |
Definition 2.4.
(Addis et al., Citation2022) A soft mapping from X to Y is said to be:
(1) | injective if for each | ||||
(2) | surjective if for each | ||||
(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 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 where G is nonempty set and
is a soft binary operation on G satisfying the following conditions:
⟨SG1⟩ | Associativity: For all | ||||
⟨SG2⟩ | Existence of identity: For each | ||||
⟨SG3⟩ | Existence of inverse: For each |
Definition 2.7.
(Weldetekle et al., Citation2024) By a soft element in G we mean a soft set over G such that Card
for all
. That is,
is a single element set for each
. Let us denote by
the collection of all soft elements of G with a set of parameter A.
Definition 2.8.
(Weldetekle et al., Citation2024) Let be a soft group. A soft set
over G is said to be a soft subgroup of G if for each
, and
:
(1) |
| ||||
(2) |
| ||||
(3) |
|
Theorem 2.9.
(Weldetekle et al., Citation2024) Let be a soft group. A soft set
over G is a soft subgroup of G if and only if the following two conditions hold for each
and all
:
(1) |
| ||||
(2) | if |
Definition 2.10.
(Weldetekle et al., Citation2024) Let be a soft subgroup of a soft group
and
. Define a soft set
over G by:
We call a left soft coset of
corresponding to a. Right soft cosets can be defined in a dual manner.
Definition 2.11.
(Weldetekle et al., Citation2024) A soft subgroup of a soft group
is called normal if
for all
Theorem 2.12.
(Weldetekle et al., Citation2024) For a soft subgroup of
the following are equivalent:
(1) |
| ||||
(2) | for |
Definition 2.13.
(Weldetekle et al., Citation2024) We say that a soft group is abelian if for each
and any
,
if and only if
Definition 2.14.
(Weldetekle et al., Citation2024) Let be a soft group. The center of G denoted by
is a parameterized soft set over G defined by for each
:
3. Soft homomorphisms
Definition 3.1.
Let and
be soft groups. A soft mapping
from G to Gʹ is called a soft homomorphism, if for each
and
,
,
and
implies
Example 3.2.
Let be the set of real numbers,
the set of positive real numbers and
the set of natural numbers. Define soft binary operations
and
on
and
respectively as follows:
and
are soft groups. Moreover, define
Then
is a soft homomorphism from
to
Example 3.3.
With the notation of Example 3.2, define Then
is a soft homomorphism from
to
.
Proposition 3.4.
Let be a soft homomorphism from
to
. Then
(1) |
| ||||
(2) |
|
Proof.
(1) | Let | ||||
(2) | Let |
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 be a soft homomorphism from G to Gʹ and
be a soft homomorphism from Gʹ to
Then
is a soft homomorphism from G to
.
Definition 3.6.
A soft homomorphism from G to Gʹ 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 and
If there is a soft isomorphism from G to Gʹ, we say that G and Gʹ are soft isomorphic and write
symbolically. Note also that
forms an equivalence relation on the class of soft groups.
Recall from Weldetekle et al., (Citation2024) that, if is a soft group, then the set
of all soft elements of G is an ordinary group together with the induced binary operation
, 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 be a soft mapping from G to Gʹ. Define a mapping
as follows: for each
and
:
Theorem 3.8.
A soft mapping from G to Gʹ is a soft homomorphism if and only if
is a homomorphism.
Proof.
Suppose that is a soft homomorphism. Let
and
be arbitrary. Assume that
Then,
. If
, then
and
. On the other hand if
and
, then as
is a soft homomorphism, we get
and hence
Therefore,
is a group homomorphism. Conversely suppose that
is a group homomorphism. Let
,
and
such that
,
,
and
Let
such that
. We claim to show that u = z. Define soft elements
in G and
in Gʹ as follows: for each
:
Then, one can easily show that for each :
Since is a homomorphism, it should be the case that
for all
. In particular, it works for
and hence which would give z = u. Therefore,
is a soft homomorphism.
Theorem 3.9.
A soft homomorphism from G to Gʹ is a soft isomorphism if and only if
is a group isomorphism.
Proof.
Suppose that is soft isomorphism. We first show that
is injective. Let
such that
. Then
for all
. That is, for each
and any
we have:
Since f is injective, it must be true that for all
. Thus,
and hence
is injective. Next we show that
is surjective. Let
be any soft element in Gʹ. Since
is soft isomorphism, for ech
there is a unique element let say
such that
for all
. Noe define a sofet element
over G by
for all
. Then we must have
. Thus,
is surjective and hence an isomorphism. The converse can be proved using similar procedure.
Definition 3.10.
Let be a soft homomorphism from
to
(a) | If is the image of | ||||
(b) | If is the inverse image of |
Lemma 3.11.
Let be a soft homomorphism from
to
. If
is a soft subgroup of G then
is a soft subgroup of
Proof.
Let Since
and
, we have
Let and
such that
We need to prove that
As
there exists
such that
and
Since
we have
Let
such that
Since
is a soft mapping there exist
such that
As
is a soft homomorphism it follows that w = z. Since
is a soft subgroup of G, we have
Therefore
Lemma 3.12.
Let be a soft homomorphism from
to
If
is a soft subgroup of Gʹ then
is a soft subgroup of
Proof.
Let Since
and
, we get
Let and
such that
We need to prove that
As
there exist
such that
and
As
it holds that
Let
such that
Since
is a soft mapping there exist
such that
Since
is a soft homomorphism from G to
it holds that
Since
is a soft subgroup of Gʹ we get
Therefore
Hence
is a soft subgroup of
Theorem 3.13.
Let be a soft homomorphism from
to
(1) | If | ||||
(2) | If |
Proof.
(1) | By Lemma 3.11 | ||||
(2) | By Lemma 3.12 |
Definition 3.14.
Let be a soft homomorphism from G to
The kernel of
is the soft set
over G defined as:
where is an identity element of
Example 3.15.
Let and
be soft groups.
Let
be a soft mapping from G to Gʹ defined by:
then
is a soft homomorphism and
is the absolute soft set over
Let
be a soft mapping from G to G defined by:
then
is a soft homomorphism and
is the trivial soft subgroup of G.
Lemma 3.16.
For any soft homomorphism from
to
, the kernel
is a normal soft subgroup of
Proof.
For each , we have
Therefore the proof follows directly from Theorem 3.13.
Lemma 3.17.
A soft homomorphism from
to
is a soft monomorphism if and only if
is the trivial soft subgroup of
Proof.
Suppose is a soft monomorphism. We need to show that
for all
Let
This implies that
, for all
Since
, for all
we have
Therefore
Conversely, suppose that
and
Let
such that
Then
It follows that
So
Therefore
This implies that
Using the fact
and the cancellation law we get
Proposition 3.18.
Let and
be soft homomorphisms from
to
. Define a soft set
over G by:
for all . Then
is a soft subgroup of
Proof.
Since and
, we have
Let
and
such that
Since
and
are soft mapping there exist
such that
and
This implies that Again
is a soft mapping then there exist
such that
Since
is a soft homomorphism then we get that
Since
is a soft homomorphism and
, we have
Hence
is a soft subgroup of
Recall from (Weldetekle et al., Citation2024) that for a soft set over G, we define the subset
of
by:
In the following theorem, we establish a relation between the kernel of and the soft kernel of
.
Theorem 3.19.
For any soft homomorphism from G to Gʹ we have
Proof.
We know that
and
Therefore
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 be a normal soft subgroup of a soft group
. Put
where each is a left soft coset of
. Define a soft coset multiplication
on G/H by:
for some
with
Then,
is a soft group.
Proof.
First we shall prove that is well defined. Let
and
such that
and
. Suppose
and
This implies that
and
for some
with
and
Let
such that
Since
,
Let
such that
Since
,
Let
such that
and
It follows that
Let
such that
and
Since
is a normal soft subgroup of G then we have
Let
such that
,
and
Since
is a soft subgroup of G then we get
Using the associative property of
and
This implies that
Thus
is well defined. It remains to show that
satisfies the soft group axioms which are straightforward.
Definition 4.2.
If is a normal soft subgroup of
, then the soft group
defined as in Theorem 4.1 is called quotient soft group.
Definition 4.3.
We say that a normal soft subgroup has the property (Q) provided that for each
,
and
;
Lemma 4.4.
Let be a normal soft subgroup of
and
be the quotient soft group. Define a soft mapping
from G to
by
Then is a soft epimorphism from G to
such that
Moreover, the equality holds, whenever
has the property (Q).
Proof.
Define a soft mapping from G to
by
Clearly,
is surjective. Since
is normal in G then
is a soft homomorphism. Now, for each
and
:
Therefore Moreover, if
has the property (Q), then one can easily verify that
for all
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 and
be soft groups. If
is a soft epimorphism from G to Gʹ, then
.
Proof.
Define a soft mapping from
to Gʹ by
if and only if
We first show that
is well defined. Let
, and
such that
Let
such that
and
It follows that
and
Since
, we have
for all
If
then
for all
This implies
Since
is a soft homomorphism
Then, we get
Therefore
is well defined. Since
is an epimorphism for each
there exists
such that
Which implies that
Therefore
is surjective. It remains to show that
is injective. Suppose that
and
implies
and
Let
and
such that
and
Then we have
Which implies that
That is
Since α is arbitrary we can see that
for all
Thus
Therefore
is injective. Hence
Corollary 4.6.
Let be a soft epimorphism from
to
and
be a normal soft subgroup of
Then,
Proof.
Define a soft mapping from G to
by:
if and only if
for some
with
We show that
is a soft homomorphism. Let
,
and
such that
,
,
and
Then, there exist
such that
,
,
and
,
,
Since
is a soft homomorphism and
We get that
in
Then
in
Thus
is a soft homomorphism from G to
As
is surjective it is straightforward that
is also surjective. Now for each
consider the following:
Thus Therefore by the first soft isomorphism theorem we get that
Proposition 4.7.
Let and
be soft groups and
be a soft homomorphism from G to
Then, for any soft set
over G,
; where
is the centralizer of
. Moreover, the equality holds if and only if
is a soft isomorphism.
Theorem 4.8.
(The Third Soft isomorphism theorem) Let and
be normal soft subgroups of
such that
, and
has the property (Q). Define a soft set with
over G/M by
Then,
is a normal soft subgroup of
;
.
Proof.
First observe that if and only if
It is clear from the definition of
that
implies
Let
Then
for some
Which implies that for each
, and
, if
, then
In particular,
and since
we get that
Define a soft mapping
from
to
by
Then we first show that is well defined. Suppose
This implies that
, for all
Since
is a normal soft subgroup, if
then
Since
,
This implies that
, for all
Therefore
Hence
is well-defined. Next we show that
is a soft homomorphism. Let
,
such that
,
,
and
in
. Then
,
and
and
in G. Thus
in the quotient
Hence
is a soft homomorphism and it is clear that
is a soft epimorphism. Now consider its kernel:
Therefore By the first soft isomorphism theorem
.
Lemma 4.9.
Let ,
and
be soft normal subgroups of G such that
Then
if and only if
Proof.
Suppose that Then
for all
Now for any
, consider
This implies that
It follows that
Thus
Conversely, suppose that
Then
for all
Now for any
It follows that
Hence
Therefore
Thus
and hence
Corollary 4.10.
Under the assumption of Lemma 4.9, if and only if
.
Lemma 4.11.
If is soft isomorphic to
then G is abelian if and only if Gʹ is abelian.
Proof.
Let such that
Since
is surjective there exist
such that
and
Let
such that
Since
is soft homomorphism. Now,
Since G is abelian we have
Again since
is a soft homomorphism we get
Therefore Gʹ is abelian. Conversely, let
such that
Since
is a soft mapping, we have
and
for some
Let
such that
Since
is a soft homomorphism then we have
. Let
such that
. By the assumption Gʹ is abelian we get
Let
such that
Again since
is a soft homomorphism we have
Since
is injective we have
Therefore G is abelian.
Theorem 4.12.
(The correspondence theorem) Let be a normal soft subgroup of
Then there is a one to one correspondence between the set of all normal soft subgroups of G containing
and the set of all normal soft subgroups of
Proof.
Define to be the set of all normal soft subgroups of G containing
i.e
and let be the set of all normal soft subgroups of
. Define
by
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
be any soft normal subgroups of
For each
define
Then one can easily check that
is a soft normal subgroup of G containing
such that
Therefore g is onto and hence it is the required one to one correspondence.
Theorem 4.13.
Let be a normal soft subgroup of G having the property (Q). If
is a soft epimorphism from G to Gʹ, with
, then
Proof.
Since is a normal soft subgroup of G and
be a soft epimorphism it follows that
is a normal soft subgroup of Gʹ and hence the quotient soft group
is defined. Now, define a soft mapping
from G to
by
where
for all
Let
and
such that
,
and
where
,
and
Since
be a soft homomorphism and then we get that
in
It follows that
in
and therefore
be a soft homomorphism. Let
since
be a soft epimorphism, there exists
such that
and therefore
This implies that
be a soft epimorphism. Further, for any
Therefore Thus by the first soft isomorphism theorem,
Theorem 4.14.
Let and
be normal soft subgroups of G satisfying the property (Q) such that
is absolute soft set over G. Then,
Proof.
Define a soft mapping from G to
by
where
for
Let
,
and
such that
,
,
and
in G where
,
and
Since
and
are normal soft subgroups of G then we have
in
and
in
From the definition of direct product of a soft group and
then we have
in
and hence
is a soft homomorphism. Next we shall to prove that
is surjective. Let
where
and
Since
is an absolute soft set
So if
, then
for some
and
with
for some
and
Now, let
such that
Then,
for some
and hence
for all
Also, since
then we have
for some
and hence
Therefore
where
This implies that
Thus
is a soft epimorphism.
Now,
Thus Therefore, by the first soft isomorphism theorem
Corollary 4.15.
Any soft homomorphism of soft groups can be expressed as a composition of soft epimorphism and monomorphism.
Theorem 4.16.
Let ,
and
be soft groups. Then
if and only if there are normal soft subgroups
and
of G such that
is the absolute soft set over G,
is a trivial soft subgroup of
and
.
Proof.
For let eα,
and
be identities in G, G1 and G2 respectively. Suppose that
and let
be a soft isomorphism from G to
Put
and
Then
and
are normal soft subgroups of
Now, put
and
Then
and
are normal soft subgroups of G. For any
, we have
where
Choose x1 and x2 in G such that
and
where
,
Then,
,
and
Since
is a soft isomorphism then we have
and hence
Therefore
for all
Thus
is absolute soft set. Let
This implies that
,
It follows that
and
Then
Since
is soft homomorphism then we have
and
This implies that
Therefore,
for all
Thus
is a trivial soft subgroup of
Next,define a soft mapping
from G to G1 and
from G to G2 by
and
if
where
Then
and
are soft epimorphism and
and
For
and
where and
Therefore,
and
Conversely, suppose that
and
are normal soft subgroups of G such that
for all
,
and
Let
be a soft isomorphism from
to G1 and let
be a soft isomorphism from
to G2 and
be a soft homomorphism from G to
and
be a soft homomorphism from G to
Now define a soft mapping
from G to
as follows: for
,
and
:
Since and
are soft isomorphisms,
will also be a soft epimorphism. Put
. Then for any
, consider the following:
Therefore for all
Hence
is a soft monomorphism and hence a soft isomorphism. Therefore,
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 a bijective soft homomorphism of G onto itself is called a soft automorphism of
We denote by SAut(G) the collection of all soft automorphisms of G.
Example 5.2.
Consider the soft groups and
where
is the set of positive real numbers,
is the set of natural numbers and
and
are defined by:
and
If we define
, then
is a soft automorphism of
Lemma 5.3.
Let be a soft isomorphism from
to
Then
is also a soft isomorphism from
to
Lemma 5.4.
For any soft group G, is a group; where
is the composition of soft mappings.
Theorem 5.5.
is isomorphic to a subgroup of
.
Proof.
We show that the map is an embedding of SAut(G) into
where for each soft map
from to G to Gʹ,
is a soft mapping from
to
given in Definition 3.7. It is proved in Theorem 3.9 that if
is a soft automorphism of
then
is an automorphism of
Claim 1: is one to one. Let
such that
Then
for all
Then
for all
That is, for any
it holds that
if and only if
Implying that,
Claim 2:
Let and
such that
Then,
, and so there is some
such that
and
Thus
and
Conversely, if we are assuming that
then it can be shown that
Thus
for all
Therefore,
for all
Hence
Therefore the map
is a monomorphism hence an embedding.
Theorem 5.6.
For any soft groups and
. If
, then
Proof.
Suppose and
be a soft isomorphism from G to
For any soft automorphism
of G, one can easily check that
is a soft automorphism of Gʹ. Moreover, the map
is an ordinary group isomorphism of SAut(G) onto
.
Lemma 5.7.
Let be a soft group. Define a soft mapping
from G to G by
for any
Then
is a soft automorphism of G if and only if
is an abelian soft group.
Proof.
Suppose is a soft automorphism of G. Let
and
such that
Since
is a soft homomorphism of G. It follows that
Then we have
Therefore
is abelian soft group. Conversely, suppose that
is abelian soft group. We first show that
is a soft homomorphism. Let
and
such that
and
It follows that
Since G is abelian soft group we have
Therefore
is a soft homomorphism. Next we show that
is injective. Let
such that
and
This implies that
and
It follows that
Therefore
is injective. Finlay it remains to show that
is surjective. Let
then
It follows that
Thus
is surjective. Hence
is a soft automorphism.
Proposition 5.8.
Let be a soft automorphism of
Let
be a soft set over G define by
then is a soft subgroup of
Proof.
Since , we have
Let
then
and
Let
such that
and
Since
is a soft automorphism then we have
This implies that
Therefore
Let
,
such that
and
As
it follows that
So
Therefore
Hence
is a soft subgroup of
Proposition 5.9.
Let be a soft group and let
be a soft set over G defined by
for all . Then
is a normal soft subgroup of
Proof.
Since Let
and
such that
We have
and
Let
such that
Since
is a soft homomorphism of G to G and
, we have
and so
Therefore
Thus
Hence
is a soft subgroup of
Let
and
Suppose
and
We shall prove that
Let
such that
,
and
. Since
is a soft homomorphism and
, it holds that
As
it follows that
Again since
is a soft homomorphism and
, it follows from the cancellation law that z = n. As
is injective we get y = n. Then by transitivity we get
Therefore
Thus
Hence
is a normal soft subgroup of
Theorem 5.10.
Let be a soft group and
Define a soft mapping
from G to G as follows , for each
:
if and only if there is some
such that
and
Then
is a soft automorphism on
Proof.
We first show that is a soft homomorphism. Let
such that
,
,
and
Then there exists
such that
and
and
and
Using the fact that
and by associative property of
we get that
Thus
is a soft homomorphism. In addition for any
and any
let
such that
and
If
such that
then it can be verified that
Which implies that
Therefore, u is an element of G such that
and
Thus
and hence
is surjective. Finally, it remains to show that
is injective. Let
and
such that
and
Then there exist y1 and
such that
,
and
. Applying the cancellation law on equations
and
we have
. Again using
and applying the cancellation law on equations
and
gives us
Thus
is injective and therefore it is a soft automorphism.
Definition 5.11.
Let be a soft group and
The soft automorphism
of G given in the above theorem is called the inner soft automorphism of G corresponding to a. We denote by
the set of all inner soft isomorphisms of G with the given set of parameters A.
Define a soft binary operation on SIA by; for
,
if and only if
for some
with
. Then we have the following theorem.
Theorem 5.12.
For any soft group is a soft group.
Proof.
We first show that is well defined. Let
such that
Then we have for any and
if and only if
Similarly
if and only if
Again then we have
and
Claim:
For any and
Suppose that
Then there exists
such that
and
Since
there exist
such that
and
together implying
and there is also
such that
and
implying that
Therefore we can find
such that
and
Since
we get that for some
and
and therefore
that is
Since α is arbitrary we get
Similarly it can be shown that
and hence the equality holds.
Therefore the soft binary operation is well-defined . It remains to show that
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 is soft isomorphic with the quotient soft group
in the following theorem.
Theorem 5.14.
For any soft group we have
Proof.
Define a soft mapping from G to
by:
Then it is clear that is a soft epimorphism.
Claim: for any
Let
Then
i.e for
it holds that
if and only if
equivalently, if
such that
and
then z = x and
and hence
Since
is arbitrary we have
Conversely, for
suppose that
Then for all
it is the case that
if and only if
For any
and
if
then
Now we show that
for
let
which implies that there exist
such that
and
Then it follows that
and hence
i.e
Also if
we get that z = x. So that for any
if
then
So that
and hence
Therefore
Thus,
Therefore,
Hence by the first soft isomorphism theorem
Disclosure statement
No potential conflict of interest was reported by the author(s).
Supplementary material
Supplemental data for this article can be accessed online at https://doi.org/10.1080/27684830.2024.2314354.
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