A completely new approach for the theory of Soft Groups and Soft Rings

Abstract

A new soft group definition is given by Ghosh and Samanta by using the definition of soft element which is introduced by Wardowski. In this paper we study on binary operation on a soft set by using the soft element definition. Considering this binary operation which is the generalization of the soft group definition of Ghosh and Samanta we gave the notions such as soft groupoid, soft group, soft ring and some related results are obtained.

Keywords

Soft Set Soft Element Soft Group Soft Ring

1 Introduction

In the real world, most of problems in economics, engineerings and environmental areas can be solved approximately by using different mathematical theories. Besides these theories, a new approach, the Soft Set Theory, that approximates the initial universe with respect to parameters for the problems, was introduced by Molodtsov [4]. In the Soft Set Theory, one can use any parametrization with the help of words, adjectives, sentences, real numbers, functions, mappings, and so on. Maji et al. [5] introduced many operators for soft set theory. Furthermore, some structures such as cartesian product, soft set relation and orderings on soft sets were introduced by Babitha and Sunil [1, 2]. Aktaş and Çağman [3] defined soft group by taking images of parametrized set valued functions as a subgroup of the initial universe which was also a group. According to this study many researchers studied on algebraic structures on soft sets such as soft groups, soft rings, soft ideals, normalistic soft group [9 , 11–13].

Furthermore, Wardowski [6] gave the notion of soft element and strengthen the definition of soft function, which is reported firstly by Babitha and Sunil [1], by using the soft element. Moreover, Ghosh et al. [10] introduced a new definition of soft group by defining a binary operation on the set of all nonempty soft elements of a given soft set with the help of a binary operation defined in a parameter set as well as in a universal set.

In this paper, we approached to construct a binary operation on the soft set using the notion of the soft element without having any binary operation on the parameter set and also on the universal set. Considering our new definition, we stated some notions such as soft groupoid, soft group and soft ring and showed that this version is a generalization of soft group definition of Ghosh and Samanta [10]. Then, we illustrated some examples which show that a soft group in the sense of our definition is not necessarily a soft group in the sense of Aktaş and Çağman [3] and vice versa. Moreover in the 4 th section we report the notion of soft ring with an operation on soft sets by using the soft element definition and obtain some results. Also we introduce soft ideal of a soft ring and soft homomorphism with some theoretical results.

2 Preliminaries and basic definitions

In this section, we recall some basic concepts which are necessary for our study.

Definition 1. [4] Let U be an initial universe, E be the set of parameters. Let $P (U)$ be the set of all subsets of U and A be a subset of E. A pair F_A is called a soft set over U where $F : A ⟶ P (U)$ is a set-valued function.

In some studies a soft set F_A was expressed as F_A = {(a, F (a)) |a ∈ A}, but in some studies F (a) was written instead of (a, F (a)) just as a notation for making it shorter.

The collection of all soft sets on U will be denoted by S (U).

Definition 2. [5] For two soft sets F_A and G_B over a common universe U, we say that F_A is a soft subset of G_B and is denoted by $F_{A} \tilde{\subset} G_{B}$ if

(i) A ⊆ B and,

(ii) ∀ϵ ∈ A, F (ϵ) ⊆ G (ϵ)

F_A is said to be a soft super set of G_B, if G_B is a soft subset of F_A.

Definition 3. [5] Union of two soft sets F_A and G_B over the common universe U is the soft set H_C, where C = A ∪ B, and ∀e ∈ C, $H (e) = {\begin{matrix} F (e), & if e \in A - B, \\ G (e), & if e \in B - A, \\ F (e) \cup G (e), & if e \in A \cap B \end{matrix}$ Union of two soft sets F_A and G_B is denoted by $F_{A} \tilde{\cup} G_{B} = H_{C}$ .

Definition 4. [8] Intersection of two soft sets F_A and G_B over a common universe U is the soft set H_C, where C = A ∩ B and ∀e ∈ C, H (e) = F (e) ∩ G (e).

Intersection of two soft sets F_A and G_B is denoted by $F_{A} \tilde{\cap} G_{B} = H_{C}$ .

Definition 5.

i) [9] For a soft set F_A, the set SuppF_A = {e ∈ A : F (e) ≠ ∅} is called the support of the soft set F_A;

ii) [5] A soft set F_A over U is said to be a Null soft set and denote by Φ, if for every ϵ ∈ A, F (ϵ) =∅;

iii) [9] A soft set F_A is said to be non-null if SuppF_A≠ ∅;

iv) [10] A soft set F_A is said to be full soft set if SuppF_A = A.

The collection of all full soft sets on U will be denoted by S_f (U).

Definition 6. [1] Let F_A and G_B be two soft sets over U, then the cartesian product of F_A and G_B is defined as, F_A × G_B = H_A×B where $H \to P (U \times U)$ and H (a, b) = F (a) × G (b), where (a, b) ∈ A × B i.e. H (a, b) = {(h_i, h_j) |h_i ∈ F (a) , h_j ∈ F (b)}

Definition 7. [1] Let R be a soft set relation from F_A to G_B. Then the domain of R is defined as the soft set (D, A₁) where

A₁ = {a ∈ A : H (a, b) ∈ R for some b ∈ B} and D (a₁) = F (a₁), ∀a₁ ∈ A.

The range of R is defined as the soft set (RG, B₁), where B₁ ⊂ B and B₁ = {b ∈ B : H (a, b) ∈ R for some a ∈ A} and RG (b₁) = G (b₁), ∀b₁ ∈ B₁.

Definition 8. [1] Let F_A and G_B be two soft sets over U, then a soft set relation R from F_A to G_B is a soft subset of F_A × G_B.

In other words, a soft set relation R from F_A to G_B is of the form R = H_S where S ⊂ A × B and H_S (a, b) = H (a, b) for all (a, b) ∈ S where H_A×B = F_A × G_B.

Definition 9. [1] Let F_A and G_B be two non-empty soft sets. Then a soft set relation f from F_A and G_B is called a soft set function if every element in domain has a unique element in the range. If F (a) fG (b) then we write f (F (a)) = G (b).

Definition 10. [6] Let F_A ∈ S (U). We say that α = (p, {u}) is a nonempty soft element of F_A if p ∈ E and u ∈ F (p). The pair (p, ∅), where p ∈ E, will be called an empty soft element of F_A. Nonempty soft elements of F_A and empty soft elements of F_A will be called the soft elements of F_A. If a soft element α = (p, {u}) is in F_A, this fact will be denoted by $α \tilde{\in} F_{A}$ or $(p, {u}) \tilde{\in} F_{A}$ . The set of all nonempty soft elements of F_A will be denoted by $F_{A}^{•}$ . The soft set $F_{A}^{•}$ is called finite if it has finite number of soft element.

Note that A soft element α = (p, u) of a soft set F_A can be considered as a soft set (H_u, B) on U where B = {p} ⊆ E and H_u (p) = {u} ⊆ U. Then for a soft set F_A = {(p, F (p)) | p ∈ A}, we can deduce for each p ∈ A and F (p) ⊆ U that ${(p, F (p))} = {\tilde{U}}_{u \in F (p)} (H_{u}, {p})$ Indeed ${\tilde{U}}_{u \in F (p)} (H_{u}, {p}) = (K, C)$ where C = ⋃ {p} and K (p) = ⋃ _u∈F(p) {u} = F (p).

Proposition 1. [6] For each F_A ∈ S (U), the following holds: $F_{A} = {\tilde{U}}_{(e_{i}, {u_{j}}) \tilde{\in} F_{A}} {(e_{i}, {u_{j}})}$

Note that For each $F_{A}^{•} \in S_{f} (U)$ , the following also holds: $F_{A}^{•} = {\tilde{U}}_{(e_{i}, {u_{j}}) \tilde{\in} {F_{A}}^{•}} {(e_{i}, {u_{j}})}$

Definition 11. [6] Let F_A, G_B ∈ S (U). A soft set relation $T \tilde{\subseteq} F_{A} \times G_{B}$ is called a soft mapping from F_A to G_B, which is denoted by $T : F_{A} \tilde{\to} G_{B}$ , if the following two conditions are satisfied:

SM1 for each soft element $α \tilde{\in} F_{A}$ , there exists only one soft element $β \tilde{\in} G_{B}$ such that αTβ (which will be noted as T (α) = β);

SM2 for each empty soft element $α \tilde{\in} F_{A}$ , T (α) is an empty soft element of G_B.

It can be seen from the Definition 9 and Definition 11 that the notion "soft mapping" introduced by Wardowski in [6] is different and more effective from the notion of "soft function" introduced by Babitha and Sunil in [1]. Definition 11 is strictly related with the soft element, and enables us to obtain a natural (similar to classical mappings) behavior of soft mappings. See the following example:

Example 1. Let E = {a, b, c} and U = {u₁, u₂, u₃} be a parameter set and an universal set respectively. Choose A = {a}, B = {b, c} as subsets of E, define following soft sets:

F : A ⟶ P (U), F (a) = {u₁, u₂}, (i.e., F_A = {(a, {u₁}) , (a, {u₂})}), in the sense of soft element.

G : B ⟶ P (U), G (b) = {u₁}, G (c) = {u₂, u₃} (i.e., G_B = {(b, {u₁}) , (c, {u₂}) , (c, {u₃})}), in the sense of soft element. Now, we can define eight different "soft mappings" from F_A to G_B. One of these "soft mappings" is as following: $T ((a, {u_{1}})) = (b, {u_{1}}), T (a, {u_{2}}) = (c, {u_{3}}) .$ If we rewrite these soft sets as F_A = {(a, F (a) = {u₁, u₂})} and G_B = {(b, G (b) = {u₁}) , (c, G (c) = {u₃})}, it is easily seen that only two "soft function" can be defined between these soft sets as follows: f₁ (F (a)) = G (b) and f₂ (F (a)) = G (c).

The notion of soft group was firstly defined by Çağman et. al. [3] in 2007.

Definition 12. [3] Let G be a group and F_A be a soft set over G. The F_A is said to be a soft group over G if and only if F (x) is a subgroup of G, for all x ∈ A .

A different approach for the soft algebraic structures are given by Ghosh et. al. in [10].

Definition 13. [10] Let (E, ∘) and (U, ★) be two groupoids and A ⊆ E. Also let F_A ∈ S_f (U), i.e., F_A be a full soft set on U, i.e., for each parameter e ∈ A, there exists at least one nonempty soft element of F_A. We define a binary composition $\tilde{★}$ on $F_{A}^{•}$ by $(e_{i}, {u_{k}}) \tilde{★} (e_{j}, {u_{l}}) = (e_{i} \circ e_{j}, {u_{k} ★ u_{l}})$ for all $(e_{i}, {u_{k}}), (e_{j}, {u_{l}}) \tilde{\in} F_{A}^{•}$ .

$F_{A}^{•}$ is said to be closed under the binary composition $\tilde{★}$ if and only if $(e_{i} \circ e_{j}, {u_{k} ★ u_{l}}) \tilde{\in} F_{A}^{•}$ for all $(e_{i}, {u_{k}}), (e_{j}, {u_{l}}) \tilde{\in} F_{A}^{•}$ i.e., if and only if e_i ∘ e_j ∈ A and u_k ★ u_l ∈ F (e_i ∘ e_j) for all $(e_{i}, {u_{k}}), (e_{j}, {u_{l}}) \tilde{\in} F_{A}^{•}$ .

Definition 14. [10] If $F_{A}^{•}$ is closed under the binary composition $\tilde{★}$ , then the algebraic system $(F_{A}^{•}, \tilde{★})$ is said to be a soft groupoid over (E, U).

Definition 15. [10] Let $(F_{A}, \tilde{★})$ be a soft groupoid over (E, U), where the binary composition $\tilde{★}$ is defined in Definition 13. Then $\tilde{★}$ is said to be

commutative if $(e_{i}, {u_{j}}) \tilde{★} (e_{k}, {u_{l}}) = (e_{k}, {u_{l}}) \tilde{★} (e_{i}, {u_{j}})$ , for all $(e_{i}, {u_{j}}), (e_{k}, {u_{l}}) \tilde{\in} F_{A}^{•}$ .

associative if $[(e_{i}, {u_{j}}) \tilde{★} (e_{k}, {u_{l}})] \tilde{★} (e_{m}, {u_{n}}) = (e_{i}, {u_{j}}) \tilde{★} [(e_{k}, {u_{l}}) \tilde{★} (e_{m}, {u_{n}})]$ for all $(e_{i}, {u_{j}}), (e_{k}, {u_{l}}), (e_{m}, {u_{n}}) \tilde{\in} F_{A}^{•}$ .

Definition 16. [10] A soft element $(e, {u}) \tilde{\in} F_{A}^{•}$ is said to be a soft identity element in a soft groupoid $(F_{A}^{•}, \tilde{★})$ iff for all $(e_{i}, {u_{j}}) \tilde{\in} F_{A}^{•}$ , $(e, {u}) \tilde{★} (e_{i}, {u_{j}}) = (e_{i}, {u_{j}}) = (e_{i}, {u_{j}}) \tilde{★} (e, {u}) .$

Definition 17. [10] Let $(F_{A}, \tilde{★})$ be a soft groupoid with a soft identity element (e, {u}). A soft element $(e_{i}, {u_{j}}) \tilde{\in} F_{A}^{•}$ is said to be invertible if there exists a soft element $(e_{i}^{'}, {u_{j}^{'}}) \tilde{\in} F_{A}^{•}$ such that $(e_{i}, {u_{j}}) \tilde{★} (e_{i}^{'}, {u_{j}^{'}}) = (e, {u}) = (e_{i}^{'}, {u_{j}^{'}}) \tilde{★} (e_{i}, {u_{j}})$ Then $(e_{i}^{'}, {u_{j}^{'}})$ is called the soft inverse of (e_i, {u_j}).

The soft inverse of a soft element $(e_{i}, {u_{j}}) \tilde{\in} F_{A}^{•}$ is denoted by - (e_i, {u_j}).

Definition 18. [10]

(i) A soft groupoid $(F_{A}^{•}, \tilde{★})$ is said to be a soft semigroup if $\tilde{★}$ is associative;

(ii) A soft semigroup $(F_{A}^{•}, \tilde{★})$ containing soft identity element is said to be a soft monoid.

Definition 19. [10] Let (E, ∘) , (U, ★) be two groups, A ⊆ E and F_A ∈ S_f (U). A soft groupoid $(F_{A}, \tilde{★})$ over (E, U) is said to be a soft group if

$\tilde{★}$ is associative,

there exists a soft element $(e, {u}) \tilde{\in} F_{A}^{•}$ such that $(e, {u}) \tilde{★} (e_{i}, {u_{j}}) = (e_{i}, {u_{j}}) \tilde{★} (e, {u}) = (e_{i}, {u_{j}})$ for all $(e_{i}, {u_{j}}) \tilde{\in} F_{A}^{•}$ ,

for each soft element $(e_{i}, {u_{j}}) \tilde{\in} F_{A}^{•}$ , there exists a soft element $(e_{i}^{'}, {u_{j}^{'}}) \tilde{\in} F_{A}^{•}$ such that $(e_{i}, {u_{j}}) \tilde{★} (e_{i}^{'}, {u_{j}^{'}}) = (e_{i}^{'}, {u_{j}^{'}}) \tilde{★} (e_{i}, {u_{j}}) = (e, {u})$ .

Here (e, {u}) is said to be the soft identity element and the soft element

(e_{i}^{'}, {u_{j}^{'}})

is said to be the soft inverse of (e_i, {u_j}).

Example 2. Let the parameter set A be the set ${\bar{0}, \bar{1}, \bar{2}, \bar{3}} = Z_{4}$ and the universe (U, ★) be the group (Z₄, +) and the set valued function is defined as; $F (\bar{0}) = {\bar{0}}, F (\bar{3}) = {\bar{0}}, F (\bar{1}) = {\bar{0}, \bar{2}}, F (\bar{2}) = {\bar{0}, \bar{2}} .$ Since F (x) is a subgroup of G for all x ∈ A, then F_A is a soft group over U according to the Definition 12. Let we adapt this example to the Definition 19 as follows: (E, ∘) = (U, ★) = (Z₄, +). The binary operation $\tilde{★}$ defined on $F_{A} = {(\bar{0}, {\bar{0}}), (\bar{1}, {\bar{0}}), (\bar{1}, {\bar{2}}), (\bar{2}, {\bar{0}}), (\bar{2}, {\bar{2}}), (\bar{3}, {\bar{0}})}$ is not closed on F_A. Since $(\bar{1}, {\bar{2}}) + (\bar{2}, {\bar{0}}) = (\bar{1} + \bar{2}, {\bar{3} + \bar{0}}) = (\bar{2}, {\bar{3}})$ . But $(\bar{2}, {\bar{0}}) = (\bar{3}, {\bar{2}}) \notin F_{A}$ . So it is not a soft group according to the Definition 19.

It can be deduced from this example that the Definition 12 and 19 do not imply each other.

Theorem 1. [10] Let $(F_{A}^{•}, \tilde{★})$ be a soft groupoid over (E, U).

(i) If the composition ∘ on A and the composition ★ on U are associative (commutative) then the composition $\tilde{★}$ on $F_{A}^{•}$ is associative (commutative).

(ii) If $F_{A}^{•}$ contains the soft identity element (e, {u}) then e is the identity element of A and u is the identity element of ⋃_{e_i∈A}F (e_i).

It is also noted by Ghosh and Samanta [10] that, condition (i) of Definition 19 can be omitted, since $\tilde{★}$ is an associative binary operation on $F_{A}^{•}$ . Therefore the soft groupoid $(F_{A}^{•}, \tilde{★})$ is said to form a soft group if and only if the conditions (ii) and (iii) hold.

3 New approach for Soft Groups based on Soft Element

The notion of Soft Group (Groupoid) that was given by Ghosh et. al. [10] is based on the concept of Soft Element, with the help of assuming the parameter set E and universal set U be groups (groupoids) with respect to given certain operations on E and U. In this paper, we generalize these notions by redefining the soft binary operation based on the soft element without assuming any further requirements and show that the definitions and results given in [10] about these concepts are the special cases of our generalized version. Also we introduce some basic definitions and theorems on Soft Groups in the generalized form.

Definition 20. A soft mapping $\tilde{★}$ from $F_{A}^{•} \times F_{A}^{•} \to F_{A}^{•}$ such that $\begin{matrix} \tilde{★} : F_{A}^{•} \times F_{A}^{•} & \to F_{A}^{•} \\ α \tilde{★} β & \to γ \end{matrix}$

is called a binary operation on $F_{A}^{•}$ , where α, β, γ are soft elements of $F_{A}^{•}$ .

Since the image of $\tilde{★}$ is soft subset of $F_{A}^{•}$ , we say that $F_{A}^{•}$ is closed under $\tilde{★}$ .

Definition 21. Let $\tilde{★}$ be a binary operation on $F_{A}^{•}$ then $(F_{A}^{•}, \tilde{★})$ is called soft groupoid.

Definition 22. Let $(F_{A}^{•}, \tilde{★})$ be a soft groupoid. Then for all $α, β, γ \tilde{\in} F_{A}^{•}$ ;

i) The Binary operation $\tilde{★}$ is called commutative if $α \tilde{★} β = β \tilde{★} α$ ,

ii) The Binary operation $\tilde{★}$ is called associative if $α \tilde{★} (β \tilde{★} γ) = (β \tilde{★} α) \tilde{★} γ$ .

Definition 23. Let $(F_{A}^{•}, \tilde{★})$ be a soft groupoid. A soft element $α \tilde{\in} F_{A}^{•}$ is called soft identity element with respect to binary operation $\tilde{★}$ (shortly identity element), denoted by $e_{\tilde{★}}$ iff for all $β \tilde{\in} F_{A}^{•}$ , $α \tilde{★} β = β = β \tilde{★} α$ .

Theorem 2. A soft identity element (if it exists) of a soft groupoid $(F_{A}^{•}, \tilde{★})$ is unique. Let $α, β \tilde{\in} F_{A}^{•}$ be soft identities in $(F_{A}^{•}, \tilde{★})$ then $α = α \tilde{★} β = β$ .

Example 3. Let A = {a, b} be a parameter set, U = {h₁, h₂} be a universal set and F_A = {(a, {h₁, h₂}) , (b, {h₂})} be a soft set. The non-empty soft elements of F_A are {(a, {h₁}) , (a, {h₂}) , (b, {h₂})}. Now, let us define a binary operation $\tilde{★}$ as follows:

Table 1
Table1

$\tilde{★}$ (a, {h₁}) (a, {h₂}) (b, {h₂})

(a, {h₁}) (a, {h₁}) (a, {h₂}) (b, {h₂})

(a, {h₂}) (a, {h₂}) (b, {h₂}) (b, {h₂})

(b, {h₂}) (b, {h₂}) (b, {h₂}) (a, {h₂})

$\tilde{★}$	(a, {h₁})	(a, {h₂})	(b, {h₂})
(a, {h₁})	(a, {h₁})	(a, {h₂})	(b, {h₂})
(a, {h₂})	(a, {h₂})	(b, {h₂})	(b, {h₂})
(b, {h₂})	(b, {h₂})	(b, {h₂})	(a, {h₂})

$(F_{A}^{•}, \tilde{★})$ is a soft groupoid and (a, {h₁}) is a soft identity element in $(F_{A}^{•}, \tilde{★})$ .

Definition 24. Let $(F_{A}^{•}, \tilde{★})$ be a soft groupoid with soft identity element $e_{\tilde{★}}$ . A soft element $β \tilde{\in} F_{A}^{•}$ is said to be invertible with respect to binary operation $\tilde{★}$ (shortly invertible) if there exists a soft element $β^{'} \tilde{\in} F_{A}^{•}$ such that $β \tilde{★} β^{'} = e_{\tilde{★}} = β^{'} \tilde{★} β$ . Then β′ is called soft inverse of β and denoted by $β_{\tilde{★}}^{- 1}$ .

Definition 25. i) A soft groupoid $(F_{A}^{•}, \tilde{★})$ is said to be a soft semigroup if the binary operation $\tilde{★}$ is associative. ii) A soft semigroup $(F_{A}^{•}, \tilde{★})$ containing a soft identity element is called a soft monoid.

Example 4. The soft groupoid $(F_{A}^{•}, \tilde{★})$ in Example 3 is a soft semigroup since $\tilde{★}$ is associative and it is soft monoid since it contains the soft identity element which is (a, {h₁}).

Definition 26. soft group is an ordered pair $(F_{A}^{•}, \tilde{★})$ , where $F_{A}^{•}$ is a nonnull soft set and $\tilde{★}$ is a binary operation on $F_{A}^{•}$ such that the following properties hold.

i) The binary operation $\tilde{★}$ is associative.

ii) There exists an identity element with respect to the binary operation $\tilde{★}$ .

iii) For each $β \tilde{\in} F_{A}^{•}$ , there exists inverse element $β_{\tilde{★}}^{- 1}$ .

Example 5. Let A = {a, b, c} be a parameter set, U = {h₁, h₂, h₃} be a universal set and F_A = {(a, {h₁}) , (b, {h₁, h₂}) , (c, {h₃})} be a soft set. The non-empty soft elements of F_A are {(a, {h₁}) , (b, {h₁}) , (b, {h₂}) , (c, {h₃})}. Let the binary operation $\tilde{★}$ be defined as follows:

Table 2

Table2

$\tilde{★}$	(a, {h₁})	(b, {h₁})	(b, {h₂})	(c, {h₃})
(a, {h₁})	(b, {h₂})	(c, {h₃})	(a, {h₁})	(b, {h₁})
(b, {h₁})	(c, {h₃})	(b, {h₂})	(b, {h₁})	(a, {h₁})
(b, {h₂})	(a, {h₁})	(b, {h₁})	(b, {h₂})	(c, {h₃})
(c, {h₃})	(b, {h₁})	(a, {h₁})	(c, {h₃})	(b, {h₂})

The identity soft element is $e_{\tilde{★}} = (b, {h_{2}})$ and it is easy to check that $(F_{A}^{•}, \tilde{★})$ is a soft group.

Theorem 3.

Let (E, ∘) , (U, ★) be two groups, A ⊆ E and $F_{A}^{•} \in S_{f} (U)$ . If we define a binary operation $\tilde{★}$ on $F_{A}^{•}$ for all $α = (e_{i}, {u_{k}}), β = {e_{j}, {u_{l}}} \tilde{\in} F_{A}^{•}$ as $(e_{i}, {u_{k}}) \tilde{★} (e_{j}, {u_{l}}) = (e_{i} \circ e_{j}, {u_{k} ★ u_{l}})$ where the following conditions are satisfied,

i) u_k ★ u_l ∈ F (e_i ∘ e_j),

ii) there exists an identity element with respect to $\tilde{★}$ ,

iii) For each $β \tilde{\in} F_{A}^{•}$ , there exists an inverse element $β_{\tilde{★}}^{- 1} \tilde{\in} F_{A}^{•}$ ,

then $(F_{A}^{•}, \tilde{★})$ is a soft group. It is obvious from Theorem 1 and Definition 26.

Note that The Theorem 3 states that soft group with respect to the definition of Ghosh and Samanta is also a soft group according to Definition 26, i.e. Definition 26 is a generalization of the definition of Ghosh and Samanta. But converse is not generally true. See the following example.

Example 6. In Example 5, there is no group structure on the parameter set A and on the universal set U. So it is not a soft group in the sense of definition given in 19.

Theorem 4. Let $(F_{A}^{•}, \tilde{★})$ be a soft group. Then followings are true:

i) The soft identity element is unique.

ii) The inverse element $β_{\tilde{★}}^{- 1}$ for each $β \tilde{\in} F_{A}^{•}$ , is unique.

Proof. i) Since soft group is a soft groupoid, it is clear by Theorem 2 and the second condition in the Definition 26.

ii) Let $β \tilde{\in} F_{A}^{•}$ . By the third condition of Definition 26, there exists $β^{'} \tilde{\in} F_{A}^{•}$ such that $β \tilde{★} β^{'} = e_{\tilde{★}} = β^{'} \tilde{★} β$ . Assume there exists $β^{″} \tilde{\in} F_{A}^{•}$ such that $β \tilde{★} β^{″} = e_{\tilde{★}} = β^{″} \tilde{★} β$ . Then $\begin{matrix} β^{'} & = β^{'} \tilde{★} e_{\tilde{★}} \\ = β^{'} \tilde{★} (β \tilde{★} β^{″}) \\ = (β^{'} \tilde{★} β) \tilde{★} β^{″} \\ = e_{\tilde{★}} \tilde{★} β^{″} \\ = β^{″} \end{matrix}$ Thus β′ is unique.

Definition 27. Let $(F_{A}^{•}, \tilde{★})$ be a soft group. If the binary operation $\tilde{★}$ is commutative then $(F_{A}^{•}, \tilde{★})$ is called commutative or Abelian soft group. Otherwise it is called noncommutative soft group.

Theorem 5. Let $(F_{A}^{•}, \tilde{★})$ be a soft group.

i) $(α_{\tilde{★}}^{- 1})_{\tilde{★}}^{- 1} = α$ for all α ∈ F_A.

ii) $(α \tilde{★} β)_{\tilde{★}}^{- 1} = (β_{\tilde{★}}^{- 1}) \tilde{★} (α_{\tilde{★}}^{- 1})$ for all $α, β \tilde{\in} F_{A}^{•}$ .

iii) For $α, β, γ \tilde{\in} F_{A}^{•}$ , $α \tilde{★} γ = β \tilde{★} γ$ , then α = β (right cancellation law).

iv) For $α, β, γ \tilde{\in} F_{A}^{•}$ , $γ \tilde{★} α = γ \tilde{★} β$ , then α = β (left cancellation law).

Proof. i) Let $α \in F_{A}^{•}$ . Since $(α_{\tilde{★}}^{- 1}) \tilde{★} α = \tilde{0} = α \tilde{★} (α_{\tilde{★}}^{- 1})$ , α is the inverse of $(α_{\tilde{★}}^{- 1})$ . Since inverse element is unique then $(α_{\tilde{★}}^{- 1})_{\tilde{★}}^{- 1} = α$ .

ii) Let $α, β \tilde{\in} F_{A}^{•}$ . Then $\begin{matrix} (α \tilde{★} β) \tilde{★} ((β_{\tilde{★}}^{- 1}) \tilde{★} (α_{\tilde{★}}^{- 1})) & = ((α \tilde{★} β) \tilde{★} (β_{\tilde{★}}^{- 1})) \tilde{★} (α_{\tilde{★}}^{- 1}) \\ = (α \tilde{★} (β \tilde{★} (β_{\tilde{★}}^{- 1})) \tilde{★} (α_{\tilde{★}}^{- 1}) \\ = (α \tilde{★} e_{\tilde{★}}) \tilde{★} (α_{\tilde{★}}^{- 1}) \\ = α \tilde{★} (α_{\tilde{★}}^{- 1}) \\ = e_{\tilde{★}} \end{matrix}$

Similarly we have $((β_{\tilde{★}}^{- 1}) \tilde{★} (α_{\tilde{★}}^{- 1})) \tilde{★} (α \tilde{★} β) = e_{\tilde{★}}$ and then $(α \tilde{★} β)_{\tilde{★}}^{- 1} = (β_{\tilde{★}}^{- 1}) \tilde{★} (α_{\tilde{★}}^{- 1})$ .

iii) Let $α, β, γ \tilde{\in} F_{A}^{•}$ and suppose that $α \tilde{★} γ = β \tilde{★} γ$ . Then,

$(α \tilde{★} γ) \tilde{★} (γ_{\tilde{★}}^{- 1}) = (β \tilde{★} γ) \tilde{★} (γ_{\tilde{★}}^{- 1})$ implies

$α \tilde{★} (γ) \tilde{★} (γ_{\tilde{★}}^{- 1})) = β \tilde{★} (γ \tilde{★} (γ_{\tilde{★}}^{- 1}))$ . Hence $α \tilde{★} e_{\tilde{★}} = β \tilde{★} e_{\tilde{★}}$ then α = β.

iv) Similar to iii).

Definition 28. A soft group $(F_{A}^{•}, \tilde{★})$ is called finite soft group if $F_{A}^{•}$ has finite number of soft element. The order of the soft group $F_{A}^{•}$ is the number of the soft elements of $F_{A}^{•}$ and it is denoted by $∣ F_{A}^{•} ∣$

A soft group with an infinite number of soft elements is referred as an infinite soft group.

Note that If the initial universe U and the parameter set A has m elements and n elements respectively then order of a soft group F_A is less than or equal to m × n, i.e. $∣ F_{A}^{•} ∣ \leq m . n$

Definition 29. Let $(F_{A}^{•}, \tilde{★})$ be a soft group and $α \tilde{\in} F_{A}^{•} .$ If there exist a positive integer n such that $α^{n} = e_{\tilde{★}}$ where $α^{n} = \underset{ntimes}{\underset{︸}{α \tilde{★} \dots \tilde{★} α}}$ , then the smallest such positive integer is called the order of α. If no such positive integer n exists, then we say that α is of infinite order.

Example 7. Let consider the soft element (b, {h₁}) of $F_{A}^{•}$ in Example 5. The order of (b, {h₁}) is 2 and the order of $F_{A}^{•}$ is 4.

Definition 30. Let $(F_{A}^{•}, \tilde{★})$ be a soft group and $G_{B}^{•}$ be a nonnull soft subset of $F_{A}^{•}$ . Then $(G_{B}^{•}, {\tilde{★}}_{G_{B}})$ is called a soft subgroup of $(F_{A}^{•}, \tilde{★})$ and denoted by $G_{B}^{•} \tilde{\leq} F_{A}^{•}$ if $(G_{B}^{•}, {\tilde{★}}_{G_{B}})$ is a soft group itself where ${\tilde{★}}_{G_{B}}$ is the induced binary operation from $\tilde{★}$ .

Example 8. Let $(F_{A}^{•}, \tilde{★})$ be a soft group given in Example 5. Let B = {a, b} be a parameter set, U = {h₁, h₂, h₃} be the universal set and G_B = {(a, {h₁}) , (b, {h₂})} be a soft subset of $F_{A}^{•}$ . Then we obtain the table of binary operation ${\tilde{★}}_{G_{B}}$ on G_B as follows:

Table 3

${\tilde{★}}_{G_{B}}$	(a, {h₁})	(b, {h₂})
(a, {h₁})	(b, {h₂})	(a, {h₁})
(b, {h₂})	(a, {h₁})	(b, {h₂})

$(G_{B}^{•}, {\tilde{★}}_{G_{B}})$ is a soft subgroup of $(F_{A}^{•}, \tilde{★})$ .

Definition 31. If $(F_{A}^{•}, \tilde{★})$ is a soft group, then the subgroup consisting of $F_{A}^{•}$ itself is the improper soft subgroup of $F_{A}^{•}$ . All the other soft subgroups are proper soft subgroups. The soft subgroup ${e_{\tilde{★}}}$ is the trivial soft subgroup of $F_{A}^{•}$ . All the other soft subgroups are nontrivial.

Theorem 6. Let $(F_{A}^{•}, \tilde{★})$ be a soft group and $G_{B} \tilde{\subset} F_{A}$ . $(G_{B}^{•}, \tilde{★})$ is a soft subgroup of $(F_{A}^{•}, \tilde{★})$ if and only if for all $α, β \tilde{\in} G_{B}$ $α \tilde{★} (β_{\tilde{★}}^{- 1}) \tilde{\in} G_{B}$

If G_B is a soft subgroup of F_A, it is obvious that $α \tilde{★} (β_{\tilde{★}}^{- 1}) \tilde{\in} G_{B}$ is satisfied.

Conversely let for all $α, β \tilde{\in} G_{B}$ , $α \tilde{★} β_{\tilde{★}}^{- 1} \tilde{\in} G_{B}$ , is satisfied. Since G_B is nonnull there exists $α \tilde{\in} G_{B}$ . Therefore $e_{\tilde{★}} = α \tilde{★} α_{\tilde{★}}^{- 1} \tilde{\in} G_{B}$ . Now for all $β \tilde{\in} G_{B}$ , $β_{\tilde{★}}^{- 1} = e_{\tilde{★}} \tilde{★} β_{\tilde{★}}^{- 1} \tilde{\in} G_{B}$ . Thus for all $α, β \tilde{\in} G_{B}$ , $α, β_{\tilde{★}}^{- 1} \tilde{\in} G_{B}$ . So $α \tilde{★} β = α \tilde{★} (β_{\tilde{★}}^{- 1})^{- 1} \tilde{\in} G_{B} .$ Also associativity holds for G_B. Hence G_B is a soft group so G_B is a soft subgroup of F_A.

Theorem 7. Let $(F_{A}^{•}, \tilde{★})$ be a soft group and G_B, H_C be two soft subgroups of $F_{A}^{•}$ . Then followings are true.

i) $G_{B} \tilde{\cap} H_{C}$ is a soft subgroup of $F_{A}^{•}$ .

ii) $G_{B} \tilde{\cup} H_{C}$ is a soft subgroup of $F_{A}^{•}$ if $G_{B} \tilde{\subset} H_{C}$ or $H_{C} \tilde{\subset} G_{B} .$

Straightforward.

Definition 32. Let $(F_{A}^{•}, \tilde{★_{1}})$ and $(G_{B}^{•}, \tilde{★_{2}})$ be a soft groups and T be a soft mapping from $F_{A}^{•}$ to $G_{B}^{•}$ . Then T is called a soft homomorphism of $F_{A}^{•}$ into $G_{B}^{•}$ if for all $α, β \tilde{\in} F_{A}^{•}$ ; $T (α \tilde{★_{1}} β) = T (α) \tilde{★_{2}} T (β) .$ Note that. There always exists a soft homomorphism between the soft groups $F_{A}^{•}$ and $G_{B}^{•}$ . Indeed, to show this, let $e_{\tilde{★_{2}}}$ be the soft identity element of the soft group $(G_{B}^{•}, \tilde{★_{2}})$ .

Define $T : F_{A}^{•} ⟶ G_{B}^{•}$ by $T (α) = e_{\tilde{★_{2}}}$ for all $α \tilde{\in} F_{A}^{•} .$ Since

$T (α \tilde{★_{1}} β) = e_{\tilde{★_{2}}} = e_{\tilde{★_{2}}} \tilde{★_{2}} e_{\tilde{★_{2}}} = T (α) \tilde{★_{2}} T (β)$ for all $α, β \tilde{\in} F_{A}^{•},$ T is a soft homomorphism from $F_{A}^{•}$ to $G_{B}^{•}$ . This soft homomorphism is called the soft trivial homomorphism.

Definition 33. Let T be soft homomorphism of a soft group $F_{A}^{•}$ into a soft group $G_{B}^{•}$ . Then the kernel of T, written by KerT, is defined to be the soft set $KerT = {α \tilde{\in} F_{A}^{•} : T (α) = e_{\tilde{★_{2}}}} .$ where $e_{\tilde{★_{2}}}$ is identity soft element of $G_{B}^{•}$

Theorem 8. Let T be soft homomorphism of a soft group $(F_{A}^{•}, \tilde{★_{1}})$ into soft group $(G_{B}^{•}, \tilde{★_{2}})$ . Then following assertions hold.

i) $T (e_{\tilde{★_{1}}}) = e_{\tilde{★_{2}}}$ , where $e_{\tilde{★_{1}}}, e_{\tilde{★_{2}}}$ are soft identity elements of $F_{A}^{•}$ and $G_{B}^{•}$ respectively.

ii) T (α^-1) = T (α) ^-1, for all $α \tilde{\in} F_{A}^{•} .$

iii) If $H_{C}^{•}$ is a soft subgroup of $F_{A}^{•}$ then $T (H_{C}^{•}) = {T (α) : α \tilde{\in} H_{C}^{•}}$ is a soft subgroup of $G_{B}^{•} .$

iv) If $F_{A}^{•}$ is commutative, then $T (F_{A}^{•})$ is commutative.

v) $KerT \tilde{\leq} F_{A}^{•}$ .

Proof. i) Since T is a soft homomorphism, $T (e_{\tilde{★_{1}}}) \tilde{★_{2}} T (e_{\tilde{★_{1}}}) = T (e_{\tilde{★_{1}}} \tilde{★_{1}} e_{\tilde{★_{1}}}) = T (e_{\tilde{★_{1}}}) = T (e_{\tilde{★_{1}}}) \tilde{★_{2}} e_{\tilde{★_{2}}}$ .

This implies that $T (e_{\tilde{★_{2}}}) = e_{\tilde{★_{2}}}$ by the cancellation law.

ii) Let $α \tilde{\in} F_{A}^{•}$ . Then $T (α) \tilde{★_{2}} T (α_{\tilde{★}}^{- 1}) = T (α \tilde{★_{1}} (α_{\tilde{★}}^{- 1})) = T (e_{\tilde{★_{1}}}) = e_{\tilde{★_{2}}}$ . Similarly $T (α_{\tilde{★}}^{- 1}) \tilde{★_{2}} T (α) = e_{\tilde{★_{2}}} .$ Since T (α) has a unique inverse, $T (α_{\tilde{★}}^{- 1}) = T (α)_{\tilde{★}}^{- 1}$ .

iii) Let $H_{C}^{•}$ be a soft subgroup of $F_{A}^{•}$ . Then $e_{\tilde{★_{1}}} \tilde{\in} H_{C}^{•}$ and by i) $T (e_{\tilde{★_{1}}}) = e_{\tilde{★_{2}}}$ . Thus $e_{\tilde{★_{2}}} = T (e_{\tilde{★_{1}}}) \tilde{\in} T (H_{C}^{•})$ and so $T (H_{C}^{•})$ is non null soft set. Let $T (α), T (β) \tilde{\in} T (H_{C}^{•})$ , where $α, β \tilde{\in} H_{C}^{•}$ . Since $H_{C}^{•}$ is soft subgroup, $α \tilde{★_{1}} (β_{\tilde{★}}^{- 1}) \tilde{\in} H_{C}^{•} .$ Thus $T (α) \tilde{★_{2}} (T (β)^{- 1}) = T (α) \tilde{★_{2}} T ((β)^{- 1})) = T (α \tilde{★_{1}} (β)_{\tilde{★}}^{- 1}) \tilde{\in} T (H_{C}^{•})$ . Hence by Theorem 3, $T (H_{C}^{•})$ is a soft subgroup of $G_{B}^{•} .$

iv) Suppose that $F_{A}^{•}$ is commutative. Let $T (α), T (β) \tilde{\in} T (F_{A}^{•}) .$ Then $T (α) \tilde{★_{2}} T (β) = T (α \tilde{★_{1}} β) = T (β \tilde{★_{1}} α) = T (β) \tilde{★_{2}} T (α)$ . Hence $T (F_{A}^{•})$ is commutative.

4 Soft Rings based on soft element

Definition 34. A soft ring is an ordered triple $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ with two binary operations $\tilde{★}$ and $\tilde{\circ}$ , defined on $F_{A}^{•}$ satisfying the following axioms.

i) $(F_{A}^{•}, \tilde{★})$ is commutative soft group.

ii) The binary operation $\tilde{\circ}$ is associative.

iii) For all $α, β, γ \tilde{\in} F_{A}^{•}$ , $α \tilde{\circ} (β \tilde{★} γ) = (α \tilde{\circ} β) \tilde{★} (α \tilde{\circ} γ)$ (right distributive law).

iv) For all $α, β, γ \tilde{\in} F_{A}^{•}$ , $(β \tilde{★} γ) \tilde{\circ} α = (β \tilde{\circ} α) \tilde{★} (γ \tilde{\circ} α)$ (left distributive law).

The identity element $e_{\tilde{★}}$ of the soft group $(F_{A}^{•}, \tilde{★})$ is called the zero soft element(or zero, shortly) of the soft ring $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ and denoted by ${\tilde{0}}_{F_{A}^{•}}$ .

Example 9. If we consider the binary operation $\tilde{\circ}$ given in the table below. The soft group $(F_{A}^{•}, \tilde{★})$ given in Example 5 becomes a soft ring.

Table 4

$\tilde{\circ}$ (a, {h₁}) (b, {h₁}) (b, {h₂}) (c, {h₃})

(a, {h₁}) (a, {h₁}) (b, {h₂}) (b, {h₂}) (a, {h₁})

(b, {h₁}) (b, {h₁}) (b, {h₂}) (b, {h₂}) (b, {h₁})

(b, {h₂}) (b, {h₂}) (b, {h₂}) (b, {h₂}) (b, {h₂})

(c, {h₃}) (c, {h₃}) (b, {h₂}) (b, {h₂}) (c, {h₃})

$\tilde{\circ}$	(a, {h₁})	(b, {h₁})	(b, {h₂})	(c, {h₃})
(a, {h₁})	(a, {h₁})	(b, {h₂})	(b, {h₂})	(a, {h₁})
(b, {h₁})	(b, {h₁})	(b, {h₂})	(b, {h₂})	(b, {h₁})
(b, {h₂})	(b, {h₂})	(b, {h₂})	(b, {h₂})	(b, {h₂})
(c, {h₃})	(c, {h₃})	(b, {h₂})	(b, {h₂})	(c, {h₃})

Example 10. Let E be a parameter set, U be universal set and R = (Z₆, + ,) be a ring. Suppose that E = R, U = R and define a soft set F_A over U and binary operations on $F_{A}^{•}$ as follows:

$F_{A}^{•} = {(\bar{0}, {\bar{0}}), (\bar{1}, {\bar{1}}), (\bar{2}, {\bar{2}}), (\bar{3}, {\bar{3}}), (\bar{4}, {\bar{4}}),$ $(\bar{5}, {\bar{5}})}$

$\tilde{★} : F_{A} \times F_{A} ⟶ F_{A}, (e_{1}, {u_{1}}) \tilde{★} (e_{2}, {u_{2}}) = (e_{1} + e_{2}, {u_{1} + u_{2}})$ and its table is

Table 5

$\tilde{★}$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$
$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$
$(\bar{1}, {\bar{1}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$	$(\bar{0}, {\bar{0}})$
$(\bar{2}, {\bar{2}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$
$(\bar{3}, {\bar{3}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$
$(\bar{4}, {\bar{4}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{2}, {\bar{2}})$
$(\bar{5}, {\bar{5}})$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$

\tilde{\circ} : F_{A} \times F_{A} ⟶ F_{A}, (e_{1}, {u_{1}}) \tilde{\circ} (e_{2}, {u_{2}}) = (e_{1} \cdot e_{2}, {u_{1} \cdot u_{2}})

and its table is

Table 6

$\tilde{\circ}$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$
$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$	$(\bar{0}, {\bar{0}})$
$(\bar{1}, {\bar{1}})$	$(\bar{0}, {\bar{0}})$	$(\bar{1}, {\bar{1}})$	$(\bar{2}, {\bar{2}})$	$(\bar{3}, {\bar{3}})$	$(\bar{4}, {\bar{4}})$	$(\bar{5}, {\bar{5}})$
$(\bar{2}, {\bar{2}})$	$(\bar{0}, {\bar{0}})$	$(\bar{2}, {\bar{2}})$	$(\bar{4}, {\bar{4}})$	$(\bar{0}, {\bar{0}})$	$(\bar{2}, {\bar{2}})$	$(\bar{4}, {\bar{4}})$
$(\bar{3}, {\bar{3}})$	$(\bar{0}, {\bar{0}})$	$(\bar{3}, {\bar{3}})$	$(\bar{0}, {\bar{0}})$	$(\bar{3}, {\bar{3}})$	$(\bar{0}, {\bar{0}})$	$(\bar{3}, {\bar{3}})$
$(\bar{4}, {\bar{4}})$	$(\bar{0}, {\bar{0}})$	$(\bar{4}, {\bar{4}})$	$(\bar{2}, {\bar{2}})$	$(\bar{0}, {\bar{0}})$	$(\bar{4}, {\bar{4}})$	$(\bar{2}, {\bar{2}})$
$(\bar{5}, {\bar{5}})$	$(\bar{0}, {\bar{0}})$	$(\bar{5}, {\bar{5}})$	$(\bar{4}, {\bar{4}})$	$(\bar{3}, {\bar{3}})$	$(\bar{2}, {\bar{2}})$	$(\bar{1}, {\bar{1}})$

So $F_{A}^{•}$ is an abelian soft group over (E, U) and the conditions given in Definition 34 are satisfied. Hence $F_{A}^{•}$ is a soft ring.

Definition 35. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring. Then it is called a commutative soft ring if the binary operation $\tilde{\circ}$ is commutative.

Theorem 9. (E, ★ ₁, ∘ ₁) and (U, ★ ₂, ∘ ₂) be two rings, A ⊆ E and $F_{A}^{•} \in S_{f} (U)$ . If we define $α \tilde{★} β$ and $α \tilde{\circ} β$ for $α = (e_{i}, {u_{m}}), β = (e_{j}, {u_{n}}) \tilde{\in} F_{A}^{•}$ speacially as $(e_{i}, {u_{m}}) \tilde{★} (e_{j}, {u_{n}}) = (e_{i} ★_{1} e_{j}, {u_{m} ★_{2} u_{n}})$ $(e_{i}, {u_{m}}) \tilde{\circ} (e_{j}, {u_{n}}) = (e_{i} \circ_{1} e_{j}, {u_{m} \circ_{2} u_{n}})$ where $(F_{A}^{•}, \tilde{★})$ is an abelian soft group and u_m ∘ ₂u_n ∈ F (e_i ∘ ₁e_j), then $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ is a soft ring.

i) First condition of Definition 34 is obvious since $(F_{A}^{•}, \tilde{★})$ is an abelian soft group.

ii) It is clear from Theorem 1.

iii) Let $(e_{i}, {u_{m}}), (e_{j}, {u_{n}}), (e_{k}, {u_{p}}) \in F_{A}^{•}$ .

$\begin{matrix} (e_{i}, {u_{m}}) \tilde{\circ} ((e_{j}, {u_{n}}) \tilde{★} (e_{k}, {u_{p}})) \\ = (e_{i}, {u_{m}}) \tilde{\circ} (e_{j} ★_{1} e_{k}, {u_{n} ★_{2} u_{p}}) \\ = (e_{k} \circ_{1} (e_{j} ★_{1} e_{k}), {u_{m} \circ_{2} (u_{n} ★_{2} u_{p})}) \\ = ((e_{k} \circ_{1} e_{j}) ★_{1} (e_{k} \circ_{1} e_{k}), {(u_{m} \circ_{2} u_{n}) ★_{2} (u_{m} \circ_{2} u_{p})}) \\ = ((e_{k} \circ_{1} e_{j}), {u_{m} \circ_{2} u_{n}}) \tilde{★} ((e_{k} \circ_{1} e_{k}), {u_{m} \circ_{2} u_{p}}) \\ = ((e_{i}, {u_{m}}) \tilde{\circ} (e_{j}, {u_{n}})) \tilde{★} ((e_{i}, {u_{m}}) \tilde{\circ} (e_{k}, {u_{p}})) \end{matrix}$

iv) Similar to (iii).

Definition 36. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring. If $(F_{A}^{•}, \tilde{\circ})$ has the soft identity element then $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ is called soft ring with soft identity element. We will use the notation ${\tilde{1}}_{F_{A}^{•}}$ for the soft identity element with respect to the binary operation $\tilde{\circ}$ .

Definition 37. Let $F_{A}^{•}$ be a soft ring with the soft operations $\tilde{★}, \tilde{\circ}$ . If α, β are two nonzero soft elements of soft ring $F_{A}^{•}$ such that $α \tilde{\circ} β = {\tilde{0}}_{F_{A}^{•}}$ , then α and β are called soft zero divisors.

Example 11. Consider the soft ring given in Example 10. $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ is a commutative soft ring and ${\tilde{0}}_{F_{A}^{•}} = (\bar{0}, {\bar{0}})$ , ${\tilde{1}}_{F_{A}^{•}} = (\bar{1}, {\bar{1}})$ . Soft zero divisors of F_A are $(\bar{2}, {\bar{2}}), (\bar{4}, {\bar{4}}), (\bar{3}, {\bar{3}})$ .

Theorem 10. The cancellation laws hold in a soft ring $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ if and only if $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ has no soft zero divisors of ${\tilde{0}}_{F_{A}^{•}}$ .

Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring which the cancellation laws hold, and suppose $α \tilde{\circ} β = {\tilde{0}}_{F_{A}^{•}}$ . We must show that either α or β is ${\tilde{0}}_{F_{A}^{•}}$ . If $α \neq {\tilde{0}}_{F_{A}^{•}}$ , then $α \tilde{\circ} β = α \tilde{\circ} {\tilde{0}}_{F_{A}^{•}}$ implies that $β = {\tilde{0}}_{F_{A}^{•}}$ by cancellation laws.

Conversely, suppose that $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ has no soft zero divisors of ${\tilde{0}}_{F_{A}^{•}}$ , and suppose that $α \tilde{\circ} β = α \tilde{\circ} γ$ with $α \neq {\tilde{0}}_{F_{A}^{•}}$ . Then

$α \tilde{\circ} β \tilde{★} α_{\tilde{★}}^{- 1} \tilde{\circ} γ = α \tilde{\circ} (β \tilde{★} γ_{\tilde{★}}^{- 1}) = {\tilde{0}}_{F_{A}^{•}}$ .

Since $α \neq {\tilde{0}}_{F_{A}^{•}}$ , and since $F_{A}^{•}$ has no zero divisor of ${\tilde{0}}_{F_{A}^{•}}$ , we must have $β \tilde{★} γ_{\tilde{★}}^{- 1} = {\tilde{0}}_{F_{A}^{•}},$ so γ = β . A similar argument shows that $β \tilde{\circ} α = γ \tilde{\circ} α$ with $α \neq {\tilde{0}}_{F_{A}^{•}}$ implies β = γ .

Definition 38. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring and $G_{B} \tilde{\subseteq} F_{A}^{•}$ . Then $(G_{B}^{•}, \tilde{★}, \tilde{\circ})$ is called a soft subring of $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ if

i) $(G_{B}, \tilde{★})$ is a soft subgroup of $(F_{A}^{•}, \tilde{★})$ ,

ii) $α \tilde{\circ} β \tilde{\in} G_{B}^{•}$ , for all $α, β \tilde{\in} G_{B}^{•}$ .

Example 12. Consider the soft ring given in Example 10. Let G_B be a soft subset of F_A defined as $B = {\bar{0}, \bar{2}, \bar{4}}$ and $G (\bar{0}) = F (\bar{0})$ , $G (\bar{2}) = F (\bar{2})$ , $G (\bar{4}) = F (\bar{4})$ . Then G_B is a soft subring of F_A.

Theorem 11. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring and $G_{B} \tilde{\subset} F_{A}$ . $(G_{B}^{•}, \tilde{★}, \tilde{\circ})$ is a soft subring of $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ if and only if for all $α, β \in G_{B}^{•}$ ,

i) $α \tilde{★} β \in G_{B}^{•}$ ,

ii) $α_{\tilde{★}}^{- 1} \in G_{B}^{•}$ ,

iii) $α \tilde{\circ} β \in G_{B}^{•}$ .

If $G_{B}^{•}$ is a soft subring of $F_{A}^{•}$ , it is obvious that Conditions i), ii), iii) are satisfied.

Conversely let $G_{B}^{•}$ satisfies Conditions i), ii), iii). Since for all $α, β \in G_{B}^{•}$ ,

$α \tilde{★} β$ , $α_{\tilde{★}}^{- 1}$ , $α \tilde{\circ} β \in G_{B}^{•}$ , $G_{B}^{•}$ is closed under the operations $\tilde{★}, \tilde{\circ}$ . Also it is obvious that conditions ii) and iii) are satisfied. Hence $G_{B}^{•}$ is a soft subring of $F_{A}^{•}$ .

Theorem 12. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring and $(G_{B}^{•}, \tilde{★}, \tilde{\circ})$ , $(H_{C}^{•}, \tilde{★}, \tilde{\circ})$ are a soft subrings of $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ , then $G_{B}^{•} \tilde{\cap} H_{C}^{•}$ is a soft subrings of $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ .

Let $α, β \tilde{\in} G_{B}^{•} \tilde{\cap} H_{C}^{•}$ . Then $α, β \tilde{\in} G_{B}^{•}$ and $α, β \tilde{\in} H_{C}^{•}$ . Since $G_{B}^{•}$ , $H_{C}^{•}$ soft subrings $α \tilde{★} β \tilde{\in} G_{B}^{•}$ , $- α \tilde{\in} G_{B}^{•}$ and $α \tilde{\circ} β \tilde{\in} G_{B}^{•}$ , $α \tilde{★} β \tilde{\in} H_{C}^{•}$ , $- α \tilde{\in} H_{C}^{•}$ and $α \tilde{\circ} β \tilde{\in} H_{C}^{•}$ . So $α \tilde{★} β \tilde{\in} G_{B}^{•} \tilde{\cap} H_{C}^{•}$ , $- α \tilde{\in} G_{B}^{•} \tilde{\cap} H_{C}^{•}$ and $α \tilde{\circ} β \tilde{\in} G_{B}^{•} \tilde{\cap} H_{C}^{•}$ .

Definition 39. Let $(F_{A}^{•}, \tilde{★_{1}}, \tilde{\circ_{1}})$ and $(G_{B}^{•}, \tilde{★_{2}}, \tilde{\circ_{2}})$ be soft rings and T be a soft mapping from $F_{A}^{•}$ to $G_{B}^{•}$ . Then T is called a soft ring homomorphism of $F_{A}^{•}$ into $G_{B}^{•}$ if for all $α, β \tilde{\in} F_{A}^{•}$ ; $T (α \tilde{★_{1}} β) = T (α) \tilde{★_{2}} T (β) .$ $T (α \tilde{\circ_{1}} β) = T (α) \tilde{\circ_{2}} T (β) .$

Definition 40. Let T be soft ring homomorphism of a soft ring $F_{A}^{•}$ into a soft ring $G_{B}^{•}$ . Then the kernel of T, written by KerT, is defined to be the soft set $KerT = {α \tilde{\in} F_{A}^{•} : T (α) = {\tilde{0}}_{G_{B}^{•}}} .$ where ${\tilde{0}}_{G_{B}^{•}}$ is zero soft element of $G_{B}^{•}$

Theorem 13. Let T be soft ring homomorphism of $F_{A}^{•}$ into $G_{B}^{•}$ . Then following assertions hold.

i) $KerT \tilde{\leq} F_{A}^{•}$ .

ii) $T (F_{A}^{•}) = {T (α) : α \tilde{\in} F_{A}^{•}}$ is a soft subring of $G_{B}^{•} .$

iii) If $F_{A}^{•}$ is commutative, then $T (F_{A}^{•})$ is commutative.

Suppose that $F_{A}^{•}$ has a soft identity and $T (F_{A}^{•}) = G_{B}^{•}$ . Then,

iv) $G_{B}^{•}$ has a soft identity, namely $T (\tilde{1})$ .

v) If $α \tilde{\in} F_{A}^{•}$ is invertible soft element, then T (α) is an invertible element in $G_{B}^{•}$ and (T (α)) ^-1 = T (α^-1) .

The proof is similar to the proof of Theorem 3.

4.1 Soft Ideals of Soft Ring

Definition 41.

Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring and $(G_{B}^{•}, \tilde{★})$ be a soft subgroup of $(F_{A}^{•}, \tilde{★})$ . For all $α \tilde{\in} G_{B}^{•}$ and $β \tilde{\in} F_{A}^{•}$ ;

i) If $α \tilde{\circ} β \tilde{\in} G_{B}^{•}$ , then $G_{B}^{•}$ is called soft right ideal of $F_{A}^{•}$ .

ii) If $β \tilde{\circ} α \tilde{\in} G_{B}^{•}$ , then $G_{B}^{•}$ is called soft left ideal of $F_{A}^{•}$ .

iii) If $G_{B}^{•}$ is both soft left ideal and soft right ideal of $F_{A}^{•}$ , then $G_{B}^{•}$ is called soft ideal of $F_{A}^{•}$ .

Example 13. Suppose that $F_{A}^{•}$ be the soft ring given in Example 10. If we consider the subgroup $G_{B}^{•} = {(\bar{0}, {\bar{0}}), (\bar{2}, {\bar{2}}), (\bar{4}, {\bar{4}})}$ of $F_{A}^{•}$ with the operation $\tilde{★}$ , $G_{B}^{•}$ is an ideal of the soft ring $F_{A}^{•}$ .

Example 14. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring. The soft subsets ${\tilde{0}}$ and $F_{A}^{•}$ of $F_{A}^{•}$ are soft (left, right) ideals. These soft ideals are called trivial soft ideals. All other soft (left, right) ideals are called nontrivial.

An soft ideal $G_{B}^{•}$ of a soft ring $F_{A}^{•}$ is called a soft proper ideal if $G_{B}^{•} \neq F_{A}^{•}$

Theorem 14. Let $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ be a soft ring and $G_{B}^{•}$ be a non null soft subset. Then $G_{B}^{•}$ is a soft ideal if and only if followings hold for all $α, β \tilde{\in} G_{B}^{•}$ and for all $γ \tilde{\in} F_{A}^{•}$ ;

i) $α \tilde{★} β \tilde{\in} G_{B}^{•}$ ,

ii) $α_{\tilde{★}}^{- 1} \tilde{\in} G_{B}^{•}$ ,

iii) $γ \tilde{\circ} α \tilde{\in} G_{B}^{•}$ ,

iv) $α \tilde{\circ} γ \tilde{\in} G_{B}^{•}$ .

If $G_{B}^{•}$ is a soft ideal of $F_{A}^{•}$ it is obvious that the conditions are satisfied.

Conversely it can be deduced from $α \tilde{★} β \tilde{\in} G_{B}^{•}$ , $α_{\tilde{★}}^{- 1} \tilde{\in} G_{B}^{•}$ that $G_{B}^{•}$ is commutative soft subgroup according to the Theorem 6. Also $G_{B}^{•}$ is a soft ideal since for all $α \tilde{\in} G_{B}^{•}$ and for all $γ \tilde{\in} F_{A}^{•}$ , $γ \tilde{\circ} α$ , $α \tilde{\circ} γ \tilde{\in} F_{A}^{•}$ .

Theorem 15. Let $F_{A}^{•}$ be a soft ring and $(G_{B}^{•}, \tilde{★}, \tilde{\circ})$ , $(H_{C}^{•}, \tilde{★}, \tilde{\circ})$ are soft ideals of $(F_{A}^{•}, \tilde{★}, \tilde{\circ})$ , then $G_{B}^{•} \tilde{\cap} H_{C}^{•}$ is a soft ideal of $F_{A}^{•}$ .

Straightforward.

Theorem 16. Let T be a soft homomorphism of a soft ring $F_{A}^{•}$ into a soft ring $G_{B}^{•}$ . Then KerT is an soft ideal of $F_{A}^{•}$ .

Straightforward.

5 Conclusion

In this study, the binary operation given by Ghosh et al., on a soft set was generalized by using the definition of soft element without having any binary operation on the parameter set and on the universal set. Considering this generalized definition, we gave some basic soft algebraic notions such as soft group and soft ring, then generalization and relations among the soft algebraic structures especially soft groups are stated and illustrated by examples. To extend this work which is based on completely new definition, one can study the soft algebraic structures such as fields, vector spaces and algebras that are very important in many research areas of mathematics.

Footnotes

Acknowledgement

This work is supported by the Scientific Research Project of Muğla Sıtkı Koçman University, SRPO (no:16/001) and the Scientific Research Project of Muğla Sıtkı Koçman University, SRPO (no: 16/073).

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A completely new approach for the theory of Soft Groups and Soft Rings

Abstract

Keywords

1 Introduction

2 Preliminaries and basic definitions

3 New approach for Soft Groups based on Soft Element

Table 1 Table1 ★ ˜ (a, {h1}) (a, {h2}) (b, {h2}) (a, {h1}) (a, {h1}) (a, {h2}) (b, {h2}) (a, {h2}) (a, {h2}) (b, {h2}) (b, {h2}) (b, {h2}) (b, {h2}) (b, {h2}) (a, {h2})

Table 4 ∘ ˜ (a, {h1}) (b, {h1}) (b, {h2}) (c, {h3}) (a, {h1}) (a, {h1}) (b, {h2}) (b, {h2}) (a, {h1}) (b, {h1}) (b, {h1}) (b, {h2}) (b, {h2}) (b, {h1}) (b, {h2}) (b, {h2}) (b, {h2}) (b, {h2}) (b, {h2}) (c, {h3}) (c, {h3}) (b, {h2}) (b, {h2}) (c, {h3})

5 Conclusion

Footnotes

Acknowledgement

References

Table 1
Table1

$\tilde{★}$ (a, {h₁}) (a, {h₂}) (b, {h₂})

(a, {h₁}) (a, {h₁}) (a, {h₂}) (b, {h₂})

(a, {h₂}) (a, {h₂}) (b, {h₂}) (b, {h₂})

(b, {h₂}) (b, {h₂}) (b, {h₂}) (a, {h₂})