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 and each there exists some such that ; | ||||
(2) | for each , and , and implies . |
Definition 2.4.
(Addis et al., Citation2022) A soft mapping from X to Y is said to be:
(1) | injective if for each , and ; and together imply | ||||
(2) | surjective if for each and each there exists such that | ||||
(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 and all , if , , and then y = v. | ||||
⟨SG2⟩ | Existence of identity: For each there exists such that and for all ; | ||||
⟨SG3⟩ | Existence of inverse: For each and all , there is an element of G denoted by such that and . |
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) | for all ; | ||||
(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 and , then . |
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) | is normal; | ||||
(2) | for any a,x,y and any and together imply . |
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) | , for all where eα and are identity elements of G and Gʹ respectively; | ||||
(2) | , for all and |
Proof.
(1) | Let Such that Since and , by cancellation law it holds that Therefore | ||||
(2) | Let and Such that and Since is a soft homomorphism, it follows from (1) and the condition that Again from the fact and cancellation law we get Therefore |
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 a soft subgroup of then the soft set over Gʹ defined as: is the image of under f. | ||||
(b) | If is a soft subgroup of Gʹ, then the soft set over G defined as: is the inverse image of under f. |
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 is surjective and is a normal soft subgroup of G then is a normal soft subgroup of | ||||
(2) | If is a normal soft subgroup of Gʹ then is a normal soft subgroup of |
Proof.
(1) | By Lemma 3.11 is a soft subgroup of Let Then for some Let Since is surjective, there exist such that . Let such that There exists such that because is a soft homomorphism. Moreover, Again let such that Since is a soft mapping, there exists such that Since is a soft homomorphism, As is a normal soft subgroup of G, This implies that Therefore is a normal soft subgroup of | ||||
(2) | By Lemma 3.12 is a soft subgroup of Let Then for some Let and such that and Since is a soft homomorphism from G to Gʹ and Then we have Let and such that Since is a soft homomorphism from G to Gʹ and It follows that Therefore is a normal soft subgroup of |
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.
References
- Acar, U., Koyuncu, F., & Tanay, B. (2010). Soft sets and soft rings. Computers & Mathematics with Applications, 59(11), 3458–3463. https://doi.org/10.1016/j.camwa.2010.03.034
- Addis, G. M., Engidaw, D. A., & Davvaz, B. (2022). Soft mappings: A new approach. Soft Computing, 26(8), 3589–3599. https://doi.org/10.1007/s00500-022-06814-5
- Aktaş, H., & Çağman, N. (2007). Soft sets and soft groups. Information Sciences, 177(13), 2726–2735. https://doi.org/10.1016/j.ins.2006.12.008
- Ghosh, J., Mandal, D., & Samanta, T. (2017). Soft structures of groups and rings. International Journal of Scientific World, 5(2), 117–125. https://doi.org/10.14419/ijsw.v5i2.8012
- Maji, P., Roy, A. R., & Biswas, R. (2002). An application of soft sets in a decision making problem. Computers & Mathematics with Applications, 44(8–9), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X
- Molodtsov, D. (1999). Soft set theory—first results. Computers & Mathematics with Applications, 37(4–5), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
- Weldetekle, T. D., Belayneh, B. B., Wale, Z. T., Addis, G. M., & Liu, L. (2024). A new approach to soft groups based on soft binary operations. Research in Mathematics, 11(1), 2289733. https://doi.org/10.1080/27684830.2023.2289733
- Yaylalı, G., Polat, N. Ç., & Tanay, B. (2019). A completely new approach for the theory of soft groups and soft rings. Journal of Intelligent & Fuzzy Systems, 36(3), 2963–2972. https://doi.org/10.3233/JIFS-171083
- Zhang, Y.-H., & Yuan, X.-H. (2014). Soft relation and fuzzy soft relation. In Fuzzy Information & Engineering and Operations Research & Management (pp. 205–213). Springer-Verlag Berlin Heidelberg. https://doi.org/10.1007/978-3-642-38667-1_21