A study on soft Z -congruence relations over hemirings

Abstract

Soft set theory was first proposed by Molodtsov in 1999 as a general mathematical tool for dealing with uncertainties. In this paper, we introduce the notion of soft Z-congruence relations and investigate several related properties. Furthermore, we obtain a one-to-one correspondence between soft Z-congruence relations and strong h-idealistic soft hemirings and a one-to-one correspondence between soft Z-congruence relations and soft strong h-ideals. Moreover, we show that the relationships between fuzzy congruence relations and soft congruence relations. Finally, we show that soft homomorphisms of soft hemirings and establish homomorphism theorems for soft hemirings by using soft Z-congruence relations.

Keywords

soft homomorphism

1 Introduction

With the rapid development of science and technology, we know that in the real world, there are many problems where uncertainty is a part the data associated with them. Facing so many uncertain data, methods in classical mathematics may not be be successful in dealing with them due to the fact is that such tools are designed for certain situations. In order to describe and extract the useful information hidden in uncertain data, researchers in mathematics, computer science and related areas have proposed a number of theories such as the theories of probabilities, fuzzy sets [28], rough sets [22] and other mathematical tools. All these theories, with their own focus, are very useful for solving some uncertain problems. Specifically, fuzzy set theory places emphasis on the truth degree, rough set theory focus on granular, the fact is that the actual situations under consideration are often very complex, thus all these theories have their inherent difficulties which were pointed out in [20]. In order to overcome these difficulties, in 1999, Molodtsov [20] put forward the concept of soft sets, which can be seen as a new mathematical tool for dealing with vagueness and uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields, including the smoothness of functions, game theory, operation researches, Riemann integrations, probability theory, measurement theory and biologic resources protection. Most of these applications have already been demonstrated in Molodtsov’s book, which is refereed to [21].

Currently, the research on soft sets is progressing rapidly, Maji [16] defined several operations on soft sets. In 2009, Ali [3] gave some new operations on soft sets, Sezgin [24] obtained the basic properties of operations on soft sets. Feng [5] discussed attribute analysis of information systems based on elementary soft implications. The notion of soft groups was defined by Aktaş and Çağman [2]. They also derived some related properties. Furthermore, Jun [14] applied soft sets to the theory of BCK/BCI-algebras. The applications of soft sets in ideal theory of BCK/BCI-algebras were investigated by Jun and Park [15]. We also notice that Feng et al. [7] defined soft semirings and several related notions to establish a connection between soft sets and semirings. In 2010, Sezgin [23] proposed soft union set theory to rings. Further, some characterizations of hemirings by soft set theory were investigated by Ma, Zhan and others [1 , 31]. In particular, Feng et al. [8, 9] proposed rough soft sets by combining Pawlak rough sets and soft sets. It is pointed out that rough soft sets can be regarded as a collection of rough sets sharing a common Pawlak approximation space. In particular, Zhan investigated the ideal theory on hemirings, most of relevant conclusions have been already demonstrated in Zhan’s book, which is referred to [29].

On the other hand, as far as known that semirings are also a useful tool for dealing with problems in different areas of applied mathematics and information sciences and have the same properties as rings except that (S, +) is assumed to be a semigroup. In the study of semirings, we found that the ideal of semirings plays an important role in the structure theory, but ideals in semirings don’t coincide with the usual rings ideals. Consequently, some more restricted concepts of ideals in hemirings have been given by Henriksen [10], which is called a k-ideal. After that, a still more restricted concepts of ideals in hemirings have been given by Iizuka [11], it is called an h-ideal. The properties of k-ideals and h-ideals of hemirings were completely investigated by La Torre [25]. In particular, fuzzy h-ideals and fuzzy strong h-ideals of hemirings were studied by Jun and Yin [12 , 27].

Recently, Ali [3] constructed algebraic structures of soft sets associated with new operations. Xin [26] investigated soft congruence relations over rings and obtained some properties. The main purpose of this paper is to show that a novel soft Z-congruence relation, and we show that the connections between soft Z-congruence relations and soft strong h-ideals over soft hemirings. This paper is organized as follows: we recall some concepts and results on hemirings and soft sets in Section 2. In Section 3, we investigate the relationships between soft Z-congruence relations and strong h-idealistic soft hemirings. In Section 4, the relationships between soft Z-congruence relations and soft strong h-ideals are investigated. Moreover, we show that the relationships between fuzzy congruence relations and soft congruence relations. Finally, we show that soft homomorphisms of soft hemirings and establish homomorphism theorem for soft hemirings using soft Z-congruence relations in Section 5.

2 Preliminaries

In this section, we shall review some basic notions about hemirings and soft sets.

By a zero of a semiring (S, + , ·), we mean an element 0 ∈ S such that 0 · x = x · 0 =0 and 0 + x = x + 0 = x for all x ∈ S. A semiring with zero and a commutative semigroup (S, +) is called a hemiring. For the sake of simplicity, we shall write ab for a · b (a, b ∈ S). Throughout this paper, S is a hemiring.

A non-empty subset A in S is called a subhemiring of S if A is closed under addition and multiplication. A non-empty subset A in S is called a left (resp. right) ideal of S if A is closed under addition and SA ⊆ A (resp. AS ⊆ A). Further, A is called an ideal of S if it is both a left ideal and a right ideal of S.

An ideal I of S is called a k-ideal of S if x ∈ S, a, b ∈ I and x + a = b imply x ∈ I. An ideal I of S is called an h-ideal if x, z ∈ S, a, b ∈ I and x + a + z = b + z imply x ∈ I. An ideal I of S is called a strong h-ideal if x, y, z ∈ S, a, b ∈ I and x + a + z = y + b + z imply x ∈ y + I [12 , 34].

Let ρ be a congruence relation on S, that is, ρ is an equivalence relation on S such that (a, b) ∈ ρ and (c, d) ∈ ρ in S implies (a + c, b + d) ∈ ρ and (ac, bd) ∈ ρ for all a, b, c, d ∈ S.

Definition 2.1. [20] A pair $S = (F, A)$ is called a soft set over U, where A ⊆ E and F : A → 𝒫 (U) is a set-valued mapping.

For a soft set (F, A), the set Supp (F, A) = {x ∈ A|F (x) ≠ ∅} is called a soft support of (F, A). Thus a null soft set is indeed a soft set with an empty support, and we say that a soft set (F, A) is non-null if Supp (F, A)≠ ∅.

Definition 2.2. [7, 34] Let (F, A) be a non-null soft set over S. Then

(F, A) is called a soft hemiring over S if F (x) is a subhemiring of S for all x ∈ Supp (F, A);

(F, A) is called a strong h-idealistic soft hemiring over S if F (x) is a strong h-ideal of S for all x ∈ Supp (F, A).

Definition 2.3. [8] Let (F, A) and (G, B) be two soft sets over a common universe U. The inclusion symbol “⊆” of (F, A) and (G, B), denoted by (F, A) ⊆ (G, B), is defined as

A ⊆ B;

F (x) ⊆ G (x) for all x ∈ A.

Definition 2.4. [7, 33] Let (η, A) be a soft hemiring over S. A soft set (γ, I) over S is called a soft ideal of (η, A), denote by $(γ, I) \tilde{◃} (η, A)$ , if it satisfies:

I ⊆ A;

γ (x) is an ideal of η (x) for all x ∈ Supp (γ, I).

(ii) Let (F, A) be a soft hemiring over S. A soft set (G, B) over S is called a soft strong h-ideal of (F, A), denote by

(G, B) {\tilde{◃}}_{h} (F, A)

, if it satisfies:

B ⊆ A;

G (x) is a strong h-ideal of F (x) for all x ∈ B.

3 The relationships between soft Z-congruence relations and strong h-idealistic soft hemirings

In this section, we introduce soft Z-congruence relations over S. We show that the connections between soft Z-congruence relations and strong h-idealistic soft hemirings over hemirings.

Definition 3.1. A non-null soft set (ρ, A) over S × S is called a soft congruence relation over S if ρ (α) is a congruence relation on S for all α ∈ Supp (ρ, A). If α∈ Supp (ρ, A) = ∅, then (ρ, A) is said to be a null soft congruence relation over S, denoted by $\emptyset_{A}^{2}$ .

Example 3.2. Let $S = ℕ_{0}$ be a hemiring and $A = ℕ^{+}$ . Consider the set-valued function ρ : A → φ (S × S) given by ρ (α) = {(x, y) ∈ S × S|x ≡ y (modα)} for all α ∈ Supp (ρ, A); that is, ρ (α) is a congruence on modulo α. It is clear that ρ (α) is a congruence relation on S. Hence (ρ, A) is a soft congruence relation over S.

Definition 3.3. Let I be an ideal of S. We define a congruence relation ρ on S by (x, y) ∈ ρ if and only if there exist a, b ∈ I and z ∈ S such that x + a + z = y + b + z. Then it is called a Z-congruence relationon S.

Example 3.4. Let $S = ℤ_{6} = {0, 1, 2, 3, 4, 5}$ be a hemiring of non-negative integers modulo 6 and ρ be a congruence relation on S. Define I =<3 > and [4] _ρ = {1, 4, 7, . . .}, then I is an ideal of S and we can check that ρ is a Z-congruence relation on S.

Lemma 3.5. [32] Let ρ be a congruence relation on S. For any elements a, b, x ∈ S, if (a, b) ∈ ρ, then we have a + [x] _ρ = b + [x] _ρ.

Remark 3.6. [32] In the above Lemma, we can obtain [a] + [x] _ρ = a + [x] _ρ.

Lemma 3.7. (1) Let I be an ideal of S and ρ be a Z-congruence relation on S. If I = [0] _ρ, then I is a strong h-ideal of S.

(2) Let I be a strong h-ideal of S and ρ be a relation on S. Define (x, y) ∈ ρ ⇔ ∃ a, b ∈ I and z ∈ S such that x + a + z = y + b + z, then ρ is a Z-congruence relation on S and [0] _ρ = I.

Proof. (1) Let x + a + z = y + b + z, a, b ∈ I, x, y, z ∈ S. Since ρ is a Z-congruence relation on S and by Remark 3.6 we have (x, y) ∈ ρ ⇒ x ∈ [y] _ρ = [y + 0] _ρ ⊆ [y] _ρ + [0] _ρ = y + [0] _ρ, hence I = [0] _ρ is a strong h-ideal of S.

(2) Let I be a strong h-ideal of S and ρ be a relation on S. For any x ∈ S, we have x + 0 +0 = x + 0 +0, that is, (x, x) ∈ ρ, and thus ρ is reflexive. Obviously ρ is symmetric. Now let (x, y) ∈ ρ and (y, z) ∈ ρ. Then there exist a₁, a₂, b₁, b₂ ∈ I, z₁, z₂ ∈ S such that x + a₁ + z₁ = y + b₁ + z₁, y + a₂ + z₂ = z + b₂ + z₂. Since I is a strong h-ideal of S, we have x ∈ y + I, y ∈ z + I, hence x ∈ z + I + I = z + I. This means that there exists a ∈ I such that x = z + a, that is, x + 0 +0 = z + a + 0, hence (x, z) ∈ ρ. Therefore, ρ is an equivalence relation on S. Let (x, y) ∈ ρ, (a, b) ∈ ρ. Obviously, (x + a, y + b) ∈ ρ, (xa, yb) ∈ ρ for all x, y, a, b ∈ S. And by Definition 3.3, we know that ρ is a Z-congruence relation on S.

Now let x ∈ [0] _ρ, ∀ x ∈ S, then (x, 0) ∈ ρ. Since ρ is a Z-congruence relation on S, there exist a, b ∈ I, z ∈ S such that x + a + z = 0 + b + z. By the definition of strong h-ideals, we have x ∈ 0 + I = I, that is, [0] _ρ ⊆ I. Conversely, if x ∈ I, then x + 0 +0 = 0 + x + 0, since ρ is a Z-congruence relation on S, we have (x, 0) ∈ ρ, that is, x ∈ [0] _ρ, so I ⊆ [0] _ρ. Consequently, I = [0] _ρ. □

Definition 3.8. A non-null soft set (ρ, A) over S × S is called a soft Z-congruence relation over S if ρ (α) is a Z-congruence relation on S for all α ∈ Supp (ρ, A).

Example 3.9. Let $S = ℕ_{0}$ be a hemiring and $A = ℕ_{0}$ . For any x ∈ A, let F (x) = {y|yρx ⇔ y = 2xa, a ∈ A}. Then we can check that (F, A) is a strong h-idealistic soft hemirings over S. Let us consider the set-valued function ρ : A → φ (S × S) define by ρ (α) = {(x, y) |x + a + z = y + b + z} for some x, y, z ∈ S, a, b ∈ F (α), α ∈ A. Then it is clear that ρ (α) is a Z-congruence relation on S, hence (ρ, A) is a soft Z-congruence relation over S.

The next two theorems show that the connections between soft Z-congruence relations and soft strong h-idealistic soft hemirings over hemirings. We denote (F, A) = [0] _(ρ,A) by F (α) = [0] _ρ(α) for all x ∈ Supp (ρ, A).

Theorem 3.10. (1) Let (F, A) be an idealistic soft hemiring over S and (ρ, A) be a soft Z-congruence relation over S. If (F, A) = [0] _(ρ,A), then (F, A) is a strong h-idealistic soft hemiring over S.

(2) Let (F, A) be a strong h-idealistic soft hemiring over S and consider the set-valued function ρ : A → φ (S × S) define by ρ (α) = {(x, y) |x + a + z = y + b + z} for some x, y, z ∈ S, a, b ∈ F (α), α ∈ A. Then (ρ, A) is a soft Z-congruence relation over S and [0] _(ρ,A) = (F, A).

Proof. (1) By Definition 3.8, we know that, for all α ∈ Supp (ρ, A), ρ (α) is a Z-congruence relation on S. And according to Lemma 3.7 (1), we have F (α) = [0] _ρ(α) is a strong h-ideal of S. Since (F, A) = [0] _(ρ,A) is a non-null soft set and by Definition 2.2, we have that (F, A) is a strong h-idealistic soft hemiring over S.

(2) By Definition 2.2, we know that, for all α ∈ Supp (F, A), F (α) is a strong h-ideal of S. By Lemma 3.7 (2), we have that ρ (α) is a Z-congruence relation on S and [0] _(ρ,A) = F (α) for all α ∈ Supp (F, A), hence (ρ, A) is a soft Z-congruence relation over S and [0] _(ρ,A) = (F, A). □

Here, we obtain that any soft Z-congruence relation (ρ, A) over S can be represented by the soft strong h-idealistic soft hemiring generated by (ρ, A). Also, we observe that any soft strong h-idealistic soft hemirings (F, A) of S is the soft Z-congruence class of 0 with respect to the soft Z-congruence relation (ρ, A) generated by (F, A).

Summarizing Theorem 3.10, we have the following theorem.

Denote by SZC (S) ^E the set of all soft Z-congruence relations and by SIS (S) ^E the set of all soft strong h-ideals of a soft hemirings over S. We can establish the following two mapping:

ψ : SZC (S) ^E → SIS (S) ^E, ψ ((ρ, A)) = (F, A), where (F, A) = [0] _(ρ,A);

φ : SIS (S) ^E → SZC (S) ^E, φ ((F, A)) = (ρ, A), where for all α ∈ A we define ρ (α) = {(x, y) |x + a + z = y + b + z} for some x, y, z ∈ S, a, b ∈ F (α).

Theorem 3.11. The above two mappings ψ and φ are inverse mappings. Then there is one-to-one correspondence between SZC (S) ^E and SIS (S) ^E over S.

4 The relationships between soft Z-congruence relations and soft strong h-ideals

In this section, we show that the connections between soft Z-congruence relations and soft strong h-ideals of soft hemirings.

Definition 4.1. Let (F, A) be a soft hemiring over S. A non-null soft set (ρ, B) over S × S is called a soft binary relation of (F, A) if it satisfies the following:

B ⊆ A;

ρ (α) is a binary relation of F (α) for all α ∈ B.

Definition 4.2. Let (F, A) be a soft hemiring over S. A soft binary relation (ρ, B) of (F, A) is called a soft Z-congruence relation of (F, A) if ρ (α) is a Z-congruence relation on F (α) for all α ∈ Supp (ρ, B).

Example 4.3. Let $ℤ_{6} = {0, 1, 2, 3, 4, 5}$ be a hemiring and (F, A) be a soft set over $ℤ_{6}$ , where $A = ℤ_{6}$ and $F : A \to 𝒫 (ℤ_{6})$ is a set-valued function defined by $F (x) = {y \in ℤ_{6} | x θ y \Leftrightarrow xy \in {0, 2, 4}}$ for all x ∈ A. It is clear that (F, A) is a soft hemiring over $ℤ_{6}$ . Let (G, B) be a soft strong h-ideal of (F, A) and define a soft binary (ρ, B) over (F, A) by ρ (α) = {(x, y) |x + a + z = y + b + z} for some x, y, z ∈ F (α) , a, b ∈ G (α), α ∈ B. We can check that ρ (α) is a Z-congruence relation on F (α) for all α ∈ B. Then (ρ, B) is a soft Z-congruence relation over (F, A).

The next theorem shows that the connections between soft Z-congruence relations and soft strong h-ideals of soft hemirings. We denote (G, B) = [0] _(ρ,B) by G (α) = [0] _ρ(α) for all x ∈ Supp (ρ, B).

Theorem 4.4. Let (F, A) be a soft hemiring over S. Then we have the following theorem:

Let (G, B) be a soft ideal of (F, A) and (ρ, B) be a soft Z-congruence relation over (F, A). If (G, B) = [0] _(ρ,B), then (G, B) is a soft strong h-ideal of (F, A).

Let (G, B) be a soft strong h-ideal of (F, A) and define a soft binary relation (ρ, B) over (F, A) by ρ (α) = {(x, y) |x + a + z = y + b + z} for some x, y, z ∈ F (α) , a, b ∈ G (α), α ∈ B. Then (ρ, B) is a soft Z-congruence relation over (F, A) and [0] _(ρ,B) = (G, B).

Proof. (1) By Definition 4.2, we know that, for all α ∈ Supp (ρ, B), ρ (α) is a Z-congruence relation over a subhemiring F (α). Since (G, B) = [0] _(ρ,B), we have G (α) = [0] _ρ(α) is an ideal of F (α) for all α ∈ Supp (ρ, B). By Lemma 3.7 (1), we have $(G, B) {\tilde{◃}}_{h} (F, A)$ .

(2) By Definition 2.4, we know that for all α ∈ B, G (α) is a strong h-ideal of F (α). By hypothesis, we know that ρ (α) is a Z-congruence relation on F (α) and [0] _ρ(α) = G (α) by Lemma 3.7 (2). Then (ρ, B) is a soft Z-congruence relation over (F, A) and [0] _(ρ,B) = (G, B). □

Here, we obtain that any soft Z-congruence relation (ρ, B) over a soft hemiring (F, A) can be represented by the soft strong h-ideal generated by (ρ, B). Also, we observe that any soft strong h-ideal (G, B) of (F, A) is the soft Z-congruence class of 0 with respect to the soft Z-congruence relation (ρ, B) generated by (G, B).

Summarizing Theorem 4.4, we have the following theorem.

Denote by SZC ((F, A)) ^E the set of all soft Z-congruence relations and by SSC ((F, A)) ^E the set of all soft strong h-ideals of a soft hemirings (F, A). We can establish the following two mapping:

ψ : SZC ((F, A)) ^E → SSC ((F, A)) ^E, ψ ((ρ, B)) = (G, B), where (G, B) = [0] _(ρ,B);

φ : SSC ((F, A)) ^E → SZC ((F, A)) ^E, φ ((G, B)) = (ρ, B), where for all α ∈ B we define ρ (α) = {(x, y) |x + a + z = y + b + z} for some x, y, z ∈ F (α) , a, b ∈ G (α).

Theorem 4.5. The above two mappings ψ and φ are inverse mappings. Then there is one-to-one correspondence between SZC ((F, A)) ^E and SSC ((F, A)) ^E of (F, A).

The concept of fuzzy relations was introduced by Zadeh in [28]. Fuzzy relations, especially fuzzy equivalence/congruence relations play a central role in the structure theory. Now, we show that every fuzzy congruence relation may be considered as a soft congruence relation on S.

A fuzzy relation R on S is regarded as a fuzzy subset μ_R of S, where μ_R is a mapping from S × S to [0, 1]. The mapping μ_R associates the grade of membership μ_R (x, y) to each (x, y) ∈ S × S in S.

Definition 4.6. [28] Let R be a fuzzy relation on S. For any α ∈ [0, 1], an α-level set of R is denoted by R_α and defined as R_α = {(x, y) : μ_R (x, y) ≥ α} . The notion of α-level sets provides a connection between a fuzzy relation and a family of crisp binary relations. Further, we have the following proposition.

Proposition 4.7. Every fuzzy congruence relation may be considered as a soft congruence relation on S.

Proof. Let R be a fuzzy congruence relation on S and μ_R be the membership function of R. We consider the family {R_α : α ∈ [0, 1]} consisting of α level sets of the fuzzy congruence relation R. The fuzzy congruence relation R can be reconstructed from the family of α-level sets according to the following formula μ_R (x, y) = ⋁ {α : (x, y) ∈ R_α} for all (x, y) ∈ S × S. Now, let us define a soft congruence relation (δ_R, [0, 1]) over S such that δ_R (α) = R_α for all α ∈ [0, 1]. Then we see that the fuzzy congruence relation can be identified with the soft congruence relation (δ_R, [0, 1]) over S. Consequently, every fuzzy congruence relation may be considered as a soft congruence relation on S. □

5 Soft homomorphisms and soft quotient hemirings

In this section, we show that soft homomorphisms of soft hemirings and establish homomorphism theorem for soft hemirings by using soft Z-congruence relations.

Definition 5.1. [7] Let (F, A) and (G, B) be soft hemirings over two hemirings S and T, respectively. Let f : S → T and g : A → B be two functions. Then the pair (f, g) is called a soft hemiring homomorphism if it satisfies the following conditions:

f is an epimorphism of hemirings.

g is a surjective mapping.

f (F (x)) = G (g (x)) for all x ∈ A.

If there exists a soft hemiring homomorphism between (F, A) and (G, B), we say that (F, A) is soft homomorphic to (G, B), which is denoted by (F, A) ∼ (G, B). Moreover, if f is an isomorphism of hemirings and g is a bijective mapping, then (f, g) is called a soft hemiring isomorphism. In this case, we say that (F, A) is soft isomorphic to (G, B), which is denoted by (F, A) ≃ (G, B).

Example 5.2. Denote by $ℤ$ and $ℤ_{n}$ the hemiring of integers and the hemiring of integers modulo (a positive integer) n, respectively. Let $f : ℤ \to ℤ_{n}$ be the natural mapping defined by f (x) = [x] for all $x \in ℤ$ . Evidently, f is an epimorphism of hemirings. Let $ℤ^{+}$ be the set of positive integers and define a mapping $g : ℤ^{+} \to ℤ_{n}$ by g (x) = [x] for all $x \in ℤ^{+}$ . Then it is easy to see that the mapping g is surjective. Let $(α, ℤ^{+})$ be a soft set over $ℤ$ , where $α : ℤ \to 𝒫 (ℤ)$ is a set-valued function defined by $α (x) = {3 xk | k \in ℤ}$ for all $x \in ℤ^{+}$ . One can easily verify that $α (x) = 3 x ℤ$ is a subhemiring of $ℤ$ for all $x \in ℤ^{+}$ . Thus $(α, ℤ^{+})$ is a soft hemiring over $ℤ$ . Let $(β, ℤ_{n})$ be a soft set over $ℤ_{n}$ , where $β : ℤ_{n} \to 𝒫 (ℤ_{n})$ is a set-valued function given by $β ([x]) = {[3 xk] | k \in ℤ}$ for all $[x] \in ℤ_{n}$ . Then one can also prove that $(β, ℤ_{n})$ is a soft hemiringover $ℤ_{n}$ . Moreover, since $f (α (x)) = f (3 x ℤ) = {[3 xk] | k \in ℤ}$ and $β (g (x)) = β ([x]) = {[3 xk] | k \in ℤ}$ for all $x \in ℤ^{+}$ , we deduce that f (α (x)) = β (g (x)) for all $x \in ℤ^{+}$ . Hence (f, g) is a soft hemiring homomorphism and $(α, ℤ^{+}) \sim (β, ℤ_{n})$ .

Next, we introduce the concept of soft quotient structures of S with respect to soft Z-congruence relations on S.

Theorem 5.3. Let (F, A) be an idealistic soft hemiring over S and (ρ, A) be a soft Z-congruence relation over S, and S/ρ (α) = {[x] _ρ(α) : x ∈ S}, where α ∈ A. Then for any α ∈ A, S/ρ (α) is a hemiring under the binary operation induced by S, which is given by $\begin{matrix} [x]_{ρ (α)} + [y]_{ρ (α)} = [x + y]_{ρ (α),} \\ [x]_{ρ (α)} [y]_{ρ (α)} = [xy]_{ρ (α)} \end{matrix}$ for all x, y ∈ S. Moreover, S is homomorphic to S/ρ (α) .

Proof. First of all we check the binary operations are well defined. Consider [a] _ρ(α) = [a′] _ρ(α) and [b] _ρ(α) = [b′] _ρ(α) for all a, a′, b, b′ ∈ S, this implies that (a, a′) ∈ ρ (α) and (b, b′) ∈ ρ (α) for all α ∈ A. Since (F, A) is an idealistic soft hemiring over S and (ρ, A) is a soft Z-congruence relation over S, we know that for all α ∈ A, F (α) is an ideal of S and ρ (α) is a Z-congruence relation on S. Then there exist a₁, a₂, b₁, b₂ ∈ F (α) and z₁, z₂ ∈ S such that $\begin{matrix} a + a_{1} + z_{1} = a^{'} + b_{1} + z_{1}, \\ b + a_{2} + z_{2} = b^{'} + b_{2} + z_{2} \end{matrix}$ and we have $\begin{matrix} a + b + a_{1} + a_{2} + z_{1} + z_{2} \\ = a^{'} + b^{'} + b_{1} + b_{2} + z_{1} + z_{2}, \end{matrix}$ ab + aa₂ + az₂ + a₁b + a₁a₂ + a₁z₂ + z₁ (b + a₂ + z₂) = a′b′ + a′b₂ + a′z₂ + b₁b′ + b₁b₂ + b₁z₂ + z₁ (b′ + b₂ + z₂) , that is $a + b + a_{1} + a_{2} + z^{'} = a^{'} + b^{'} + b_{1} + b_{2} + z^{'},$ where z′ = z₁ + z₂, and

ab + aa₂ + a₁b + a₁a₂ + az₂ + a₁z₂ + z₁z₂ + z₁ (b + a₂ + z₂) = a′b′ + a′b₂ + b₁b′ + b₁b₂ + a′z₂ + b₁z₂ + z₁z₂ + z₁ (b′ + b₂ + z₂) , so we have

ab + aa₂ + a₁b ++ a₁a₂ + (a + a₁ + z₁) z₂ + z₁ (b + a₂ + z₂) = a′b′ + a′b₂ + b₁b′ + b₁b₂ + (a′ + b₁ + z₁) z₂ + z₁ (b′ + b₂ + z₂) , that is $\begin{matrix} ab + {aa}_{2} + a_{1} b + a_{1} a_{2} + z^{″} \\ = a^{'} b^{'} + a^{'} b_{2} + b_{1} b^{'} + b_{1} b_{2} + z^{″}, \end{matrix}$ where z″ = (a + a₁ + z₁) z₂ + z₁ (b + a₂ + z₂)^=(a′ + b₁ + z₁) z₂ + z₁ (b′ + b₂ + z₂) .

Since a₁, a₂, b₁, b₂ ∈ F (α) and F (α) is an ideal of S, we have a₁ + a₂ ∈ F (α) and b₁ + b₂ ∈ F (α) and so (a + b, a′ + b′) ∈ ρ (α). In a similar way, aa₂, a₁b, a₁a₂, a′b₂, b₁b′, b₁b₂ ∈ F (α), and so aa₂ + a₁b + a₁a₂ ∈ F (α) , a′b₂ + b₁b′ + b₁b₂ ∈ F (α), thus (ab, a′b′) ∈ ρ (α). Hence we have [a + b] _ρ(α) = [a′ + b′] _ρ(α) and [ab] _ρ(α) = [a′b′] _ρ(α). Now it is easy to check that S/ρ (α) is a hemiring for all α ∈ A.

Above discussion shows that for all α ∈ A, we have a homomorphism f_α : S → S/ρ (α) defined as f_α (x) = [x] _ρ(α) for x ∈ S. □

Let ρ be a Z-congruence over S. Let (F, A) be a soft hemiring over S. Denote the (F, A)/ρ by (K, A) where K (α) = {[a] _ρ : a ∈ F (α)} for all α ∈ A. Since F (α) is a subhemring of S, it is clear that K (α) is a subhemring of S/ρ for all α ∈ A. Hence (F, A)/ρ is a soft hemiring over S/ρ.

In the next paper, T is also a hemiring.

Definition 5.4. Let (F, A) and (G, B) be two soft hemirings over S and T. Let (f, g) be a soft homomorphism from (F, A) to (G, B). We define ker (f, g) as the soft binary relation (δ, A) over S, where δ (α) = {(a, b) ∈ F (α) × F (α) : f (a) = f (b)} = (kerf) |_F(α), for allα ∈ A.

Theorem 5.5. Let (F, A) and (G, B) be two soft hemirings over S and T. If (f, g) : (F, A) → (G, B) is a soft homomorphism, and ρ be a Z-congruence on S such that ρ ⊆ kerf. Then there exists a unique soft homomorphism (h, g) : (F, A)/ρ → (G, B) such that h (F (α)/ρ) = G (g (α)) for all α ∈ A.

Proof. Since (f, g) : (F, A) → (G, B) is a soft homomorphism, it is obtained that f : S → T is an epimorphism of hemirings and g : A → B is a surjective map such that f (F (α)) = G (g (α)) for all α ∈ A. Let h : S/ρ → T be a map defined by h ([a] _ρ) = f (a), where a ∈ S. Since $[a]_{ρ} = [b]_{ρ} \Rightarrow (a, b) \in ρ \subseteq kerf \Rightarrow f (a) = f (b),$ we immediately deduce that h is well-defined. Moreover, one can verify that h : S/ρ → T is an epimorphism of hemirings. In fact, let a, b ∈ S.Clearly, $\begin{matrix} \begin{matrix} h ([a]_{ρ} + [b]_{ρ}) & = h ([a + b]_{ρ}) \\ = f (a + b) \\ = f (a) + f (b) \\ = h ([a]_{ρ}) + h ([b]_{ρ}), \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} h ([a]_{ρ} [b]_{ρ}) & = h ([ab]_{ρ}) \\ = f (ab) \\ = f (a) f (b) \\ = h ([a]_{ρ}) h ([b]_{ρ}), \end{matrix} \end{matrix}$ whence h is a homomorphism from S/ρ to T. Also, it is easy to see that h is surjective, by hypothesis, f : S → T is an epimorphism. Denote the soft set (F, A)/ρ by (K, A), where K (α) = F (α)/ρ for all a ∈ A. Then we have $\begin{matrix} \begin{matrix} h (K (α)) & = h (F (α) / ρ) \\ = {h ([a]_{ρ}) : a \in F (α)} \\ = {f (a) : a \in F (a)} \\ = f (F (α)) \\ = G (g (α)), \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} h (K (α)) & = h (F (α) / ρ) \\ = {h ([a]_{ρ}) : a \in F (α)} \\ = {f (a) : a \in F (a)} \\ = f (F (α)) \\ = G (g (α)), \end{matrix} \end{matrix}$ for all α ∈ A. Therefore, we conclude that (h, g) is a soft homomorphism from (F, A)/ρ to (G, B). □

6 Conclusions

In the present paper, we characterize soft Z-congruence relations over hemirings and describe relationships between soft Z-congruence relations and strong h-idealistic soft hemirings over hemirings. In the light of these results, our future work on this topic will be considered the relationships between fuzzy congruence relations and soft Z-congruence relations. Maybe It would be served as a foundation of information science, biologic resources protection and soft computing, and so on.

Acknowledgments

The authors are very thankful for the reviewers to give some valuable comments to improve this paper.

This research is partially supported by the Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province PKLHB1523), Key Subject of Hubei Province (Forestry Sciences) and National Natural Science Foundation of China (11461025).

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