Some New Directions in Soft (Fuzzy) Hypermodules

ABSTRACT We introduce and study some new directions on soft hypermodules and soft fuzzy hypermodules. In this regard, we apply soft set theory to hypermodules to introduce the classes of soft hypermodules and soft fuzzy hypermodules and obtain their basic properties. In particular, we study the connection between soft hypermodules and soft fuzzy hypermodules by associated (fuzzy) hyperoperations and obtain some related basic results.

in connection with hypergroupoids [26], polygroups (isomorphism theorems) [27], semihyperrings (prime soft hyperideals, regularity criterion and relationship with m-systems) [28], and -hypermodules (isomorphism and fuzzy isomorphism theorems) [29], and also in topological spaces [30]. Now, in this paper, we introduce a new direction on soft algebras to hypermodules. We will proceed by introducing soft hypermodules and soft fuzzy hypermodules as a generalisation of soft modules as well as fuzzy soft modules. Some basic properties are studied. Moreover, we investigate the connection between soft hypermodules and soft fuzzy hypermodules by associated (fuzzy) hyperoperations and give some related basic results.

(Fuzzy) Hypermodules
We give some definitions of (fuzzy) algebraic hyperstructures which we need to develop our paper: Let H a nonempty set and let P * (H) ( (ii) (r s) · a = (r · a) + (s · a); A nonempty subset N of a the hypermodule M is called a subhypermodule of the hypermodule (M, +, ·), if (N, +) is a hypergroup and R · N ∈ P * (N). Also, a nonempty subset N of a fuzzy hypermodule M is called a subfuzzy hypermodule if for all x, y ∈ N and r ∈ R, we have: Example 2.1: Let R = [0, 1] and define the hyperoperation ⊕ max for all x, y ∈ R by Then, (R, ⊕ max , ·) is a hyperring (Krasner hyperring, see in Ref. [31]) where '·' is ordinary multiplication on real numbers. Also, where "+" is ordinary additive, and define the following hyperoperations on R/I as follows: Then, (R/I, , ) is a hypermodule over the hyperring (R, ⊕ max , •), by Ref. [31].

A Connection Between Hypermodules and Fuzzy Hypermodules
We recall a connection between fuzzy hypermodules and hypermodules, using the p-cuts of fuzzy sets for p ∈ [0, 1]. By Ref. [11], a structure (M, ⊕, ) is a fuzzy hypermodule over a fuzzy hyperring (R, , ) if and only if (M, ⊕ p , p ) is a hypermodule over the hyperring (R, p , p ), for all p ∈ [0, 1], where for all x, y ∈ M and r, s ∈ R.
On the other hand, let (M, +, ·) be a hypermodule over hyperring (R, , •). Consider the following fuzzy hyperoperations for a, b ∈ M and r, s ∈ R a ⊕ b = χ a+b , r a = χ r·a , r s = χ r s and r s = χ r•s .
We recall the following theorem regarding connection between subhypermodules and subfuzzy hypermodules and also homomorphism of hypermodules and fuzzy hypermodules in Ref. [11]:

Soft Sets
Now, we briefly review some notions concerning soft sets. Let X be an initial universe set and E be a set of parameters. P(X) denotes the power set of X and A ⊆ E. Then, F A is called a soft set over X, where F is a mapping given by F : A −→ P(X). In fact, a soft set over X is a parameterised family of subsets of the universe X. For e ∈ A, F(e) may be considered as the set of e-approximate elements of the soft set F A . Note to the following example: of subsets of the set X, and can be viewed as a collection of approximations:

Example 2.4 ([26]):
Consider two universes X and Y and an arbitrary relation,

This is denoted by F
For a soft set F A , the set Supp( Let F A and G B be two soft sets over X and X , respectively, and f : X −→ X and g : A −→ B be two functions. Then we say that pair The concept of soft homomorphism on groups was first introduced in Ref. [22].

Soft Hypermodules
In this section, we introduce soft hypermodules and investigate their basic properties. Also, soft homomorphisms of hypermodules are discussed, and some illustrative examples are given. Suppose that (M, +, ·) is a hypermodule over a hyperring (R, , •) and all soft sets are considered over the hypermodule M.     (2) Similar to the proof of (3), by hypothesis, for all (x, y) ∈ Supp(H A×B ) we have H(x, y) = F(x) or H(x, y) = G(y) which are subhypermodules of M. Therefore, H A×B = F A ∨ G B is a soft hypermodule of M.

be a strong homomorphism from F A to G B , two soft hypermodules over M and M , respectively. Then f (F) B is a soft hypermodule over M .
Proof: Let y ∈ Supp(f (F) B ). By Lemma 3.1 (i), there exists x ∈ Supp(F A ) such that y = g(x), and F(x) = ∅ is a subhypermodule of M. Also, since g is one to one, we have (F(x)). f is a strong homomorphism, f (F(x)) is a subhypermodule of M . It follows that f (F) B is a soft hypermodule over M .

is a soft (resp. strong) homomorphism of hypermodules.
Proof: It is straightforward.

is a soft subhypermodule of F A if it is non-null and i∈I
for all i, j ∈ I and a i ∈ A i , then i∈I (F i ) A i is a soft subhypermodule over M.

Proof:
The proof is similar to the proof of Theorems 3.1 and 3.2. (

Soft Fuzzy Hypermodules
In this section we introduce the notion of a soft fuzzy hypermodule and present some properties of them. In what follow, (M, ⊕, ) is a fuzzy hypermodule over a fuzzy hyperring (R, , ) and all soft sets are considered over M.   Proof: Let x, y ∈ N ∩ K, t ∈ M and (x ⊕ y)(t) > 0. Since, x, y ∈ N, x, y ∈ K, and N and K are subfuzzy hypermodules, then t ∈ N and t ∈ K, and so t ∈ N ∩ K. Similarly, (r x)(t) > 0 implies that t ∈ N ∩ K for all r ∈ R. Also, for z ∈ N ∩ K and p ∈ (0, 1], by associated hypermodule (M, ⊕ p , p ), we have Now, consider the associated hypermodule (M, +, ·). Then It follows that z ⊕ (N ∩ K) = χ N∩K . Therefore, the proof is complete.   (1) If f is strong, then f(N) is a subfuzzy hypermodule of M .
Consider the associated hypermodule (M, +, ·). By Theorem 2.2 (1) and Theorem 2.1, it implies that . Also, similar to the proof of Lemma 4.2, we can show that Similarly, (2) will be proved.
By Lemma 4.3, it can be seen that Theorems 3.3, 3.4, 3.5 and 3.7 are valid for soft fuzzy hypermodules similarly.

Connections Between Soft Hypermodules and Soft Fuzzy Hypermodules
In this section, the connections between soft hypermodules and soft fuzzy hypermodules and also their homomorphisms are investigated based on associated hyperoperations and fuzzy hyperoperations ( [11]).
The following theorems establish a similar result for soft hypermodules and soft fuzzy hypermodules.

Proof:
Since F A is soft fuzzy hypermodule, F(x) is a subfuzzy hypermodule of (M, ⊕, ) for all x ∈ A. By Theorem 2.2 (1), F(x) is a subhypermodule of associated hypermodule M, for all x ∈ A. Therefore, F A is a soft hypermodule over (M, +, ·).
If we denote by SHM the class of all soft hypermodules and by SFHM the class of all soft fuzzy hypermodules, then we can consider the map S : SFHM −→ SHM such that S ((F A , ⊕, )) = (F A , +, ·).
The next theorem shows that we can obtain a soft fuzzy hypermodule by a soft hypermodule.