Generalized lower and upper approximations in hyperrings

1 Introduction

The theory of algebraic hyperstructures is a branch of algebraic theory which was first introduced by Marty [6]. Since then, various algebraic hyperstructures have been proposed [13]. Among this algebraic hyperstructures, hyperrings are very important algebraic hyperstructures. Hyperrings are algebraic structure similar to rings, while the composition of two elements is not a element but a set. A very important class of hyperring is the Krasner hyperring [8], which was introduced by and developed by lots of authors. Further results of hyperring could be referee [2–5]. Apart from their important algebraic properties, hyperrings also have many applications in chemistry and physics [5].

Rough set theory [18] was first presented by Pawlak. As well known, the most important way to developed rough set theory based on equivalence relation. Unfortunately, the equivalence relation is not easy to find in an environment of imperfect information since the vagueness and incompleteness of human reasoning, which lead to the applications of rough set model is very narrow. Therefore, lots of authors do their best to generlied of rough set. For example, they extend the equivalence relation to general binary relations in Pawlak rough set, which bring about certain new direction that consider the relation between rough sets and algebraic structures.

Roughness in semigroup and group were first introduced by Kuroki [10] and Wang [11]. After then, Davvaz concerned roughness on rings and the rough subrings and ideals respect to ideals of rings [2]. Based on fuzzy ideal, Kazanci and Davvaz investigated the rough fuzzy ideal of a ring and obtained some important results [12]. Moreover, Davvaz and Mahdavipour extend the roughness in modules and gave various meaning results of it [3]. Apart from this, roughness in various hyperstructures are discussed more and more. For example Leoreanu-Fotea [16, 17] present roughness in hypergroup and analyze approximations of subset respect to an invertible subhypergroup. Zhan [7; (Tables 1 and 2)] put forth the notions of a fuzzy rough hyperideal of rough hypernearrings and gave some interesting results. Davvaz [1] introduced the approximations in Hyperring. In order to extend the equivalence relation to general binary relations in rough set, Davvaz [4] put forth an very interesting method to expand the standard concept of rough set approximation space. He gave a new type of definition of the generalized lower and upper approximations and proved the lower and upper approximations is a particular case of them. Since then, Yamak et al. [14, 15] and Ali [9] investigated the generalized approximations in a ring, hemirings and module, respectively, and get some more meaning results from them.

In this paper we investigate the relationship between hyperring and rough sets based on set valued homomorphism. The results of this paper can be regards as generalizations of related results in [14].

2 Preliminaries

Let H be a non-empty set and P^∗ (H) be the set of all non-empty subsets of H. A hyperoperation is a map • : H × H ⟶ P ∗ ( H ) and the pair (H, •) is said to be a hypergroupoid. For any two non-empty subsets U and V of H and h ∈ H, we define U • V = ⋃ _u∈U,v∈Vu • v, U • h = u • {h} and h • V = {h} • V. A semihypergroup is a hypergroupoid (H, •) satisfies the condition (u • v) • w = u • (v • w) for all u, v, w ∈ H. A hypergroup is a semihypergroup satisfies the condition x • H = H • x = H, for any x ∈ H.([5, 13])

Definition 2.1. [5, 13] An algebraic hyperstructure (A, + , ·) of type (2, 2) is called a hyperring If the conditions are established:

(1) (A, +) is a commutative hypergroup, that is,

(i) a + (b + c) = (a + b) + c;

(ii) a + b = b + a;

(iii) there exists 0 ∈ R such that 0 + a = a;

(iv) there exists a unique element -a such that 0 ∈ a + (- a);

(v) c ∈ a + b implies b ∈ - a + c and a ∈ c - b.

(2) (A, ·) is a semigroup with 0 · a = a · 0 =0.

(3) the multiplication · is distributive over the hyperoperation +.

A hyperring (A, + , ·) is said to be commutative if (A, ·) is a commutative semigroup. A non-empty subset S of hyperring (A, + , ·) is a subhyperring if a + b ⊆ S, -a ∈ S and a · b ∈ S for all a, b ∈ S. A subhyperring I of A is called ahyperideal if ia, ai ∈ I for every a ∈ A and i ∈ I. A hyperideal P of A is said to be aprime hyperideal if P ≠ A and U, V ⊆ P be hyperideals of A and UV ⊆ P, and hence U ⊆ P or V ⊆ P. If A is a commutative hyperring, P is a prime hyperideal iff a, b ∈ A and ab ∈ P implies a ∈ P or b ∈ P. If ∅ ≠ U, V ⊆ A. The sumU + V is defined as U + V = {a|a ∈ u + v for some u ∈ U, v ∈ V} and the productUV is defined as UV= { a | a ∈ ∑ i = 1 n u i v i , u i ∈ U , v i ∈ V , n ∈ N }. From now on, we denote a · b replaced by ab, and by A we denote the universe of a ring (A, + , ·). (see [5, 13])

Definition 2.2. [5] A homomorphism from a hyperring (A, + , ·) into a hyperring (B, + ₁, · ₁) is a mapping h : A ⟶ B such that h (a + b) = h (a) + ₁h (b) and h (a · b) = h (a) · ₁h (b), for all a, b ∈ A.

Definition 2.3. [5] Let A be a non-empty set with U, V ⊆ A and θ be an equivalence relation on A. We write that U θ ¯ V iff ∀ u ∈ U, ∃ v ∈ V satisfies uθv and ∀ u ∈ V, ∃ v ∈ U satisfies uθv.

Definition 2.4. [3, 18] Let U≠ ∅ and θ be an equivalence relation on U. We called (U, θ) an approximation space.

Definition 2.5. [3, 9] Let (U, θ) be an approximation apace. Two mappings △ : P (U) ⟶ P (U) and ▽ : P (U) ⟶ P (U) are gave as for all V ∈ P (U), ▽ (V) = {x ∈ U| [x]  _θ  ⊆ V},△ (V) = {x ∈ U| [x]  _θ  ∩ V ≠ ∅}. We called ▽ (V)(△ (V)) a lower(upper) rough approximation of V in (U, θ).

Definition 2.6. [3, 9] Let (U, θ) be an approximation space. We called (W, Z) in P (U) × P (U) a rough set in (U, θ) if the condition (W, Z) = (▽ (V) , △ (V)) for some V ∈ P (U) holds.

Definition 2.7. [3, 9] Let A,B be two universes, τ : A ⟶ P (B) be a set valued map. We call (A, B, τ) a generalized approximation space. Every set-valued map defined as above can be determine a binary relation from A to B via putting θ _τ  = {(a, b) |b ∈ τ (a)}. Clearly, if θ is an arbitrary relation from A to B, and hence it can be determine a set valued map τ _θ  : U ⟶ P (B) by τ _θ  (a) = {b ∈ B| (a, b) ∈ θ} where a ∈ A. Given arbitrary set S ⊆ B, a pair of lower and upper approximations, U _τ  (S) and L _τ  (S), are defined as U _τ  (S) = {a ∈ B|τ (a) ∩ S ≠ ∅}; L _τ  (S) = {a ∈ B|τ (a) ⊆ S} Then (L _τ  (S) , U _τ  (S)) is called a generalized rough set.

Remark. [9] Based on the above analysis, we can easily obtain that L _{τ θ}  (V) = ▽ (V) and U _{τ θ}  (V) = △ (V). From this point, a rough set is a particular case of a generalized rough set.

3 Generalized approximations in hyperring

Inspired by Yamak, we present the set valued homomorphism in a hyperring and investigate certain basic properties of it.

Definition 3.1. Let A, B be two hyperrings and τ : A ⟶ P (B) be a set valued mapping (simply by SV-mapping). Then τ is called a set valued homomorphism (simply by SV-homomorphism) if

(i) τ (a + b) = τ (a) + τ (b);

(ii) τ (ab) = {uv|u ∈ τ (a) , v ∈ τ (b)};

(iii) τ (- a) = - τ (a).

for any a, b ∈ A.

Example 3.2. Let A, B be two two hyperrings, h : A ⟶ B be a homomorphism from A to B. Then the SV-mapping τ (a) = {h (a)} is a SV-homomorphism from A to P (B).

Example 3.3. Let A, B be two two hyperrings. Then the SV-mapping τ (a) = B is a SV-homomorphism from A to P (B).

Example 3.4. Let A = {x, y, z, w}, hyperaddition + and multiplication · as follows:

It is easy to check that (A, + , ·) is a hyperring. Define a mapping by τ (a) = x for all a ∈ A. It is easy to see that τ is a SV-homomorphism from A to P (A).

Proposition 3.5. Let A, B be two hyperrings and τ : A ⟶ P (B) be a SV-homomorphism. Suppose K is a subhyperring of B with L _τ  (K)≠ ∅, U _τ  (K)≠ ∅, then L _τ  (K) and U _τ  (K) are subhyperrings of B. Therefore, (L _τ  (K),U _τ  (K)) is a generalized rough subhyperring of B.

Proof. First, we will show that L _τ  (K) is a sub-hyperring of B. Assume that a, b ∈ L _τ  (K), thus τ (a) , τ (b) ⊆ K. From K is a subhyperring of B, we have τ (a - b) = τ (a + (-) b) = τ (a) - τ (b) ⊆ K - K ⊆ K,τ (ab) = {uv|u ∈ τ (a) , v ∈ τ (b)} ⊆ τ (a) τ (b)⊆KK ⊆ K, which implies that a - b ⊆ L _τ  (K) , ab ∈ L _τ  (K). Therefore L _τ  (K) is a subhyperring of B. Then, we will show that U _τ  (K) is a subhyperring of B. Assume that a, b ∈ U _τ  (K), thus τ (a)∩ K ≠ ∅,τ (b)∩ K ≠ ∅. Then there is u ∈ τ (x) ∩ K, v ∈ τ (y) ∩ K. Thus u - v ⊆ τ (a) - τ (b) = τ (a - b). By K is a subhyperring of B, we have a - b ⊆ K. So a - b ∈ τ (a - b) ∩ K and hence τ (a - b)∩ K ≠ ∅. Thus a - b ⊆ U _τ  (K). Moreover, uv ∈ {uv|u ∈ τ (a) , v ∈ τ (b)} = τ (ab) and ab ∈ U, thus ab ∈ U _τ  (K). Therefore, U _τ  (K) is a subhyperring of B. Combine them, (L _τ  (K),U _τ  (K)) is a generalized rough subhyperring of B.

Proposition 3.6. Let A, B be two hyperrings and τ : A ⟶ P (B) be a SV-homomorphism. If ∅ ≠ W ⊆ B, ∅ ≠ V ⊆ B, then

(1) L _τ  (W) L _τ  (V) ⊆ L _τ  (WV),

(2) L _τ  (W) + L _τ  (V) ⊆ L _τ  (W + V),

(3) U _τ  (W) U _τ  (V) ⊆ U _τ  (WV),

(4) U _τ  (W) + U _τ  (V) ⊆ U _τ  (W + V).

Proof. (1) Suppose that a ∈ L _τ  (W) L _τ  (V) and hence a = ∑ i = 1 n w i v i for w _i  ∈ L _τ  (W) and v _i  ∈ L _τ  (V). Thus τ (w _i ) ⊆ W and τ (v _i ) ⊆ V for 1 ≤ i ≤ n. Using the definition of τ and WV, one can obtain that ∑ i = 1 n τ ( w i ) τ ( v i ) ⊆ WV which implies that τ (a) ⊆ WV, so a ∈ L _τ  (WV). Therefore, L _τ  (W) L _τ  (V) ⊆ L _τ  (WV).

(2) If a is an element of L _τ  (W) + L _τ  (V). Then a ∈ b + c with b ∈ L _τ  (W) and c ∈ L _τ  (V) and hence τ (b)   ⊆  W and τ (c) ⊆ V. Thus, τ (b) + τ (c) ⊆ W + V. That is, τ (b + c) ⊆ W + V and so a ∈ L _τ  (W + V).

(3) It is similar to (1) and hence we omit it.

(4) It is similar to (2) and hence we omit it.

The containment of the above proposition is proper.

Example 3.7. Consider the Example 3.4. If W = {x}, V = {y}, then L _τ  (W) L _τ  (V) = ∅ , L _τ  (WV) = A. Thus L _τ  (W) L _τ  (V) ≠ L _τ  (WV). Further, if W = V= {z}, then L _τ  (W)+ L _τ  (V) = ∅, L _τ  (W + V) = A. Thus L _τ  (W) + L _τ  (V) ≠ L _τ  (W + V). Moreover,if W = {x} , V = {y}, then U _τ  (W) = A, U _τ  (V) = ∅ , U _τ  (WV) = A. Thus U _τ  (W) U _τ  (V) ≠ U _τ  (WV). If W = {y} , V = {y}, then U _τ  (W) = ∅ , U _τ  (V) = ∅ , U _τ (W + V) = A. Thus U _τ  (W) + U _τ  (V) ≠ U _τ  (W  +  V).

Proposition 3.8. Let A,B be two hyperrings and τ : A ⟶ P (B) be a SV-homomorphism. Suppose that I is a hyperideal of B, L _τ  (I)≠ ∅ and U _T  (I)≠ ∅, then (L _τ  (I),U _τ  (I)) is a generalized rough hyperidealof B.

Proof. By Proposition 3.4, we obtain that L _τ  (I) is a subhyperring of B. Now, we will show that L _τ  (I) is a hyperideal of B. Assume that a ∈ A and i ∈ L _τ  (I), and hence τ (i) ⊆ I. From I is a hyperideal of B, so we have τ (ai) = {uv|u∈τ (a) , v ∈ τ (i)} ⊆ τ (a) τ (i) ⊆ I, τ (ia)   =   {uv|u ∈ τ (a) , v ∈ τ (i)} ⊆ τ (a) τ (i) ⊆ I, that means that ai, ia ∈ L _τ  (I). In summary, L _τ  (I) is a hyperideal of B. On the other hand, we will show that U _τ  (I) is a hyperideal of B. Assume that a ∈ A and i ∈ U _τ  (I) and hence τ (a)∩ I ≠ ∅. Suppose u ∈ τ (a) ∩ I, and so u ∈ τ (a) and u ∈ I. Since I is a hyperideal of B, τ (a) u ⊆ I and τ (a) u ⊆ {uz|u ∈ τ (a) , z ∈ τ (i)} = τ (ai), that is to say τ (ai)∩ I ≠ ∅. So ai ∈ U _τ  (I). In the same way, we can prove that xr ∈ U _τ  (I). Therefore,U _τ  (I) is a hyperideal of B. Combine them, (L _τ  (I),U _τ  (I)) is a generalized rough hyperideal of B.

Proposition 3.9. Let A,B be two commutative hyperrings, τ : A ⟶ P (B) be a SV-homomorphism. Suppose I is a prime hyperideal of B, L _T  (I)≠ ∅ and U _T  (I)≠ ∅, then (L _τ  (I),U _τ  (I)) is a generalized rough prime hyperideal of B.

Proof. By Proposition 3.8, we obtain that L _τ  (I) is a hyperideal of S. Now, we will show that L _τ  (I) is a prime hyperideal of B. Assume that a, b ∈ A and ab ∈ L _τ  (I), thus {uv|u ∈ τ (a) , v ∈ τ (b)} = τ (ab) ⊆ I. There is u ∈ τ (a) , v ∈ τ (b) with uv ∈ I. Next, we will prove that τ (a) ⊆ I or τ (b) ⊆ I. Assume that τ (a) ⊆ I and τ (b) ⊆ I. There is x ∈ τ (a) with a ∉ I, y ∈ τ (b) with y ∉ I. From I is a prime hyperideal of B, one can obtain that xy ∉ I. While that is a contradiction. Thus a ∈ L _τ  (I) or b ∈ L _τ  (I). Combine them, we can get L _τ  (I) is a prime hyperideal of B. On the other hand, we will show that U _τ  (I) is a prime hyperideal of B. Assume that a, b ∈ A and ab ∈ U _τ  (I), hence τ (ab)∩ I ≠ ∅, {uv|u ∈ τ (a) , v ∈ τ (b)} = τ (ab). Then there exists x ∈ τ (a) , y ∈ τ (b) with xy ∈ I. From I is a prime hyperideal of B and hence x ∈ I or y ∈ I, thus τ (a)∩ I ≠ ∅ or τ (b)∩ I ≠ ∅. Thus a ∈ U _τ  (I) or b ∈ U _τ  (I). Based on above, one can obtain that B ≠ U _τ  (I) is a prime hyperideal of B. Combine them, (L _τ  (I),U _τ  (I)) is a generalized rough prime hyperidealof B.

In what following, we discuss the relation between SV-homomorphism and homomorphism onhyperring.

Proposition 3.10. Let τ : A ⟶ P (B) be a SV-homomorphism, h : C ⟶ A be a homomorphism. Then τ ∘ h : C ⟶ P (B) is a SV-homomorphism such that L_τ∘h (Y) = h^-1 ((L _τ  (Y)) and U_τ∘h (Y) = h^-1 (U _τ  (Y)), for any Y ∈ P (B).

Proof. The result is obvious.

Proposition 3.11. Let τ : A ⟶ P (B) be a SV-homomorphism, h : B ⟶ C be a homomorphism. Then τ ∘ h : A ⟶ P (C) is a SV-homomorphism defined by τ _h  (a) = h (τ (a)) such that (L_τ∘h (X) = h^-1 (L _τ  (X)) and U_τ∘h (X) = (U _τ  (h^-1X)), for all X ∈ P (C).

Proof. The result is obvious.

Let I be a hyperideal of A. Define the binary relation ≡ _I on A by u ≡  _I v if and only if u ∈ v + I. Obviously, ≡ _I is an equivalence relation on A. Denote A/I = {|u ∈ A} is the set of all equivalence class of A/I, where [u] = {u ∈ L|u ≡  _I v}. Define ⊕,⊗ on A/I as follows: [x] ⊕ [y] = {[z] |z ∈ x + y}, [u] ⊗ [v] = [u · v] = u · v + I for all [u] , [v] ∈ A/I, then (A/I, ⊕ , ⊗) is a hyperring. (see [5]).

Proposition 3.12. Let A ⟶ P (B) be a SV-homomorphism, I be a hyperideal of B. Define as τ _I  : A ⟶ P (B/I) by τ _I  = {x + I|x ∈ τ (a)}. Then τ _I is a SV-homomorphism from A to P (B/I).

Proof Now, we will show that τ _I  : A → P (B/I) is a SV-homomorphism. First, for any a, b ∈ A we have τ _I  (a + b) = {w + I|x ∈ τ (a + b)} = {w + I|w ∈ τ (a) + τ (b)} = {(t + m) + I|t ∈ τ (a) , m ∈ τ (b)} ={t  +  I|t ∈ τ (a)}   +   {m + I|m ∈ τ (b)}   =  τ _I  (a)   +  τ _I  (b).

Next, for any a, b ∈ A we have τ _I  (ab) = {w+ I|w ∈ τ (ab)} = {w + I|w ∈ {tm|t ∈ τ (a) , m ∈τ (z)}} = {w + I|w = mt, m ∈ τ (a) , t ∈ τ (b)} = {(mt) + I|m ∈ τ (a) , t ∈ τ (b)} = {m + I|m ∈ τ (a)}{t + I|t ∈ τ (b)} = τ _I  (a) τ _I  (b). Finally, for all a ∈ A we have τ _I  (- a) = {w + I|w ∈ τ (- a)} = {w + I|w ∈ - τ (a)} = {- w + I| - w ∈ τ (a)} =-τ _I  (a).

Combining them, we obtain that τ _I  : A → P (B/I) is a SV-homomorphism.

4 Generalized lower and upper approximation with respect to a hyperideal on hyperrings

Definition 4.1. Let (A, B, τ _I ) be a generalized approximation space (simply by GA-space) w.r.t a hyperideal I, ∅ ≠ ⊆ B. We called the L _{τ I}  (S) = {a ∈ A| (τ (a) + I) ⊆ S}, U _{τ I}  (S) = {a ∈ A|(τ (a) + I) ∩ S ≠ ∅} generalized lower and upper approximations of S w.r.t the hyperideal I, respectively.

Proposition 4.2. Let A, B be two hyperrings and W, V be two hyperideal of B. Let S be a subset of B and W ⊆ V. Then

(1) U _{τ W}  (S) ⊆ U _{τ V}  (S),

(2) L _{τ V}  (S) ⊆ L _{τ W}  (S).

Proof. (1) Suppose that a ∈ U _{τ W}  (S). Then (τ₍a)+ W) ∩ S ≠ ∅. Hence there is m ∈ (τ_(a) + W) ∩ S with m ∈ (T_(a) + W) and m ∈ S. That is exist t ∈ T (a) , s ∈ W such that m ∈ t + s. Since W ⊆ V, we have s ∈ V. Thus m ∈ t + s ⊆ (τ_(a) + V) and m ∈ S. So (τ_(a)+ V) ∩ S ≠ ∅. As a consequent, we obtain U _{τ W}  (S) ⊆ U _{τ V}  (S).

(2) It is similar to (1) and hence we omit it.

From the above proposition, we can get the next conclusion directly.

Corollary 4.3. Let A, B be two hyperrings and W, V be two hyperideal of B. Let S be a subset of B and W ⊆ V. Then

(1) U _{τ W∩V}  (S) ⊆ U _{τ W}  (S) ∩ U _{τ V}  (S),

(2) L _{τ V}  (S) ∩ L _{τ V}  (S) ⊆ L _{τ W∩V}  (S).

Proposition 4.4. Let (A, B, τ _I ) be a GA-space w.r.t hyperideal I, S be a subhyperring of B. Suppose τ : A ⟶ P (B) is a SV-homomorphism with L _{τ I}  (S)≠ ∅, U _{τ I}  (S)≠ ∅, then (L _{τ I}  (S) , U _{τ I}  (S)) is a generalized rough subhyperring w.r.t ahyperideal I.

Proof. Suppose that a, b ∈ U _{τ I}  (S)). Thus (τ (a) + I)∩S≠ ∅,(τ (b)+ I) ∩ S ≠ ∅. There is x∈ (τ (a) +I) ∩ S, y ∈ (τ (b) + I) ∩ S. From S is a subhyperring of B, we have x + y ⊆ S and xy ∈ S. On the other hand, x + y ⊆ (τ (a) + I) + (τ (b) + I) ⊆ (τ (a) + τ (y)) + I = τ (a + b) + I and xy⊆ (τ (a) +I) (τ (b) + I) ⊆ (τ (a) τ (b)) + I = τ (ab) + I. So τ (a+b)+ I ∩ S ≠ ∅ and τ (ab)+ I ∩ S ≠ ∅. Thus a + b⊆U _{τ I}  (S)) , ab ∈ U _{τ I}  (S)) and hence U _{τ I}  (S)) is sub-hyperring of B. Similarly, L _{τ I}  (S) is a subhyperring of A. Therefore, (L _{τ I}  (S) , U _{τ I}  (S)) is a generalized rough subhyperring w.r.t a hyperideal I.

Proposition 4.5. Let (A, B, τ _I ) be a GA-space w.r.t hyperideal I and S be a hyperideal of B. Suppose τ : A ⟶ P (B) is a SV-homomorphism, L _{τ I}  (S)≠ ∅, U _{τ I}  (S)≠ ∅, then (L _{τ I}  (S) , U _{τ I}  (S)) is a generalized rough hyperideal w.r.t a hyperideal I.

Proof. Using Proposition 4.4, U _{τ I}  (S) is a subhyperring of A. Let x ∈ U _{τ I}  (S) and a ∈ A and hence (τ (x)+ I) ∩ S ≠ ∅. So there is u ∈ (τ (x) + I) ∩ S. From U _{τ I}  (S) is non-empty set, one can chose y ∈ T (a). From S is a hyperideal of A, we have xu ∈ S. Further more, xu ∈ (τ (a) + I) + τ (c) ⊂ τ (a + c) + I. Thus, τ (a+ c) + I) ∩ S ≠ ∅ which implies xu ∈ U _{τ I}  (S). Similarly, ux ∈ U _{τ I}  (S). Therefore, U _{τ I}  (S) is a hyperideal of B. By the same way, one can prove that L _{τ I}  (S) is also a hyperideal of B. Hence, (L _{τ I}  (S) , U _{τ I}  (S)) is a generalized rough hyperideal w.r.t a hyperideal I.

Proposition 4.6. Let (A, B, τ _I ) be a GA-space w.r.t hyperideal I, S be a prime hyperideal of B. Suppose τ : A ⟶ P (B) is a SV-homomorphism, L _{τ I}  (S)≠ ∅, U _{τ I}  (S)≠ ∅, then (L _{τ I}  (S) , U _{τ I}  (S)) is a generalized rough prime hyperideal w.r.t a hyperideal I.

Proof. By Proposition 4.5, we have U _{τ I}  (S) is a hyperideal of B. Now, we will show that U _{τ I}  (S) is a prime hyperideal of B. Let a, b ∈ A such that ab ∈ U _{τ I}  (S) and so (τ (ab)+ I) ∩ S ≠ ∅, τ _ab  + I = (τ _a  + I) (τ _b  + I). Then there have m ∈ τ (a) + I, t ∈ τ (b) + I with mt ∈ S. From S is a prime hyperideal of B, thus m ∈ S or t ∈ S and (τ (a)+ I) ∩ S ≠ ∅ or (τ (b)+ I) ∩ S ≠ ∅. Hence a ∈ U _{τ I}  (S) or b ∈ U _{τ I}  (S). Combine this, one can obtain that U _{τ I}  (S) is a prime hyperideal of B. Similarly, we can prove that L _{T I}  (A) is a prime hyperideal of S. Therefore, (L _{τ I}  (S) , U _{τ I}  (S)) a generalized rough prime hyperideal w.r.t a hyperideal I.

Proposition 4.7. Let A,B be two hyperrings, I be a hyperideal of B and W, V ⊆ B. Suppose τ : A ⟶ P (B) is a SV-homomorphism. Then

(1) U _{τ I}  (W) U _{τ I}  (V) ⊆ U _{τ I}  (WV),

(2) U _{τ I}  (W) + U _{τ I}  (V) ⊆ U _{τ I}  (W + V),

(3) L _{τ I}  (W) L _{τ I}  (V) ⊆ L _{τ I}  (WV),

(4) L _{τ I}  (W) + L _{τ I}  (V) ⊆ L _{τ I}  (W + V).

Proof. The proof is similar to Proposition 3.6 and hence we omit this.

Proposition 4.8. Let A,B be two hyperrings, W, V be two hyperideals of B and S be subhyperring of B. If τ : A ⟶ P (B) is a SV-homomorphism. Then

(1) L _{τ W}  (S) + L _{τ V}  (S) = L _{τ W+V}  (S),

(2) U _{τ W}  (S) U _{τ V}  (S) ⊆ U _{τ W+V}  (S),

(3) U _{T W}  (S) + U _{τ V}  (S) ⊆ U _{τ W+V}  (S),

(4) L _{τ W}  (S) L _{τ V}  (S) = L _{τ W+V}  (S).

Proof. (1) Since W ⊆ W + V and V ⊆ W + V, then by Proposition 4.2(2), we obtain L _{τ W+V} (S) ⊆ L _{τ W}  (S) and L _{τ W+V}  (S) ⊆ L _{τ V}  (S) and L _{τ W+V} (S) ⊆ L _{τ W}  (S) + L _{τ V}  (S). Now, let a∈ L _{τ W}  (S) +L _{τ V}  (S), then a ∈ b + c for some b ∈ L _{τ W}  (S) and c ∈ L _{τ V}  (S). Hence, τ (b) + I ⊆ S and (τ (c) + V) ⊆ S. So τ (a) + W + V) ⊆ τ (b + c) + W + V = τ (b) + τ (c) + W + V = τ (b) + W + τ (c) + V ⊆ S + S ⊆ S, that is means a ∈ L _{τ W+V}  (S). Thus, we obtain L _{τ W}  (S) + L _{τ V}  (S) = L _{τ W+V}  (S).

(2) Let a be an element of U _{τ W}  (S) U _{τ V}  (S). So a = ∑ i = 1 n x i y i for x _i  ∈ U _{τ W}  (S) and y _i ∈U _{τ V}  (S). Hence, (τ (x _i )+ W) ∩ S ≠ ∅ and (τ (y _i )+ V) ∩ S ≠ ∅, and so there exist a _i ∈(τ (x _i ) + W) ∩ S and b _i  ∈ (τ (y _i ) + V) ∩ S for 1 ≤ i ≤ n. Since S is a subhyperring of B, we have ∑ i = 1 n a i b i ∈ ∑ i = 1 n τ ( x i ) τ ( y i ) + W + V and ∑ i = 1 n a i b i ∈ S. Using the definition of τ, we have ∑ i = 1 n τ ( x i ) τ ( y i ) + W + V ⊆ τ ( ∑ i = 1 n x i y i ) + W + V. Therefore ∑ i = 1 n a i b i ∈ ( τ ( a ) + W + V ) ∩ S. Thus (τ (a)+ W + V) ∩ S ≠ ∅ which implies that a ∈ U _{τ W+V}  (S), and so U _{τ W}  (S) U _{τ V}  (S) ⊆ U _{τ W+S}  (S).

(3) Since W ⊆ W + V and V ⊆ W + V, then by Proposition 4.2(2), we obtain U _{τ W}  (S) ⊆ U _{τ W+V}  (S) and U _{τ V}  (S) ⊆ U _{τ W+V}  (S). So, we obtain U _{τ W}  (S) + U _{τ V}  (S) ⊆ U _{τ W+V}  (S).

(4) Let a be an element of L _{τ W}  (S) L _{τ V}  (S). So a = ∑ i = 1 n x i y i for x _i  ∈ L _{τ W}  (S) and y _i  ∈ L _{τ V}  (S). Hence, (τ (x _i ) + W) ⊆ S and (τ (y _i ) + V) ⊆ S. Now, we have τ (x _i ) τ (y _i ) + W + V ⊆ S or ∑ i = 1 n x i y i + W + V ⊆ S. Then, by set valued homo-morphism τ, we have τ ( ∑ i = 1 n x i y i ) + W + V ⊆ S, that means that ∑ i = 1 n x i y i ∈ L τ W + V ( S ). Conversely, since W ⊆ W + V and V ⊆ W + V, then by Proposition 4.2(1), we have L _{τ W+V}  (S) ⊆ L _{τ W}  (S) and L _{τ W+V}  (S) ⊆ L _{τ V}  (S). Thus L _{τ W+V}  (S) L _{τ W+V}  (S) ⊆ L _{τ W}  (S) L _{τ V}  (S). Since L _{τ W+V}  (S) is s subhyperring of A, we obtain L _{τ W+V}  (S) ⊆ L _{τ W}  (S) L _{τ V}  (S). Therefore, L _{τ W}  (S) L _{τ V}  (S) = L _{τ W+V}  (S).

5 Conclusion

Motivating by the previous research of rings, we extended the concept of set valued homomorphism in hyperring. The related results in [14] are generalized by this paper. Since hyperring, hyperlattices and hyper MV-algebras are closely related, we will use the results of this paper to study related topic in the above hyperstructure in future.