Some Results on Fuzzy Implicative Hyper GR-ideals

The implicative hyper GR-ideals, the fuzzy implicative hyper GR-ideals of type 1 and the fuzzy implicative hyper GR-ideals of type 2 are introduced, and several properties are investigated. Characterizations of fuzzy implicative hyper GR-ideals of type 1 are established using level subsets of fuzzy sets. 2010 Mathematics Subject Classifications: 20N20, 06F3, 03G25, 03E72, 03B52


Introduction
Hyperstructure theory was introduced in 1934 by F. Marty [11] at the 8th Congress of Scandinavian Mathematics.It is studied from the theoritical point of view and for their applications to many areas of pure and applied mathematics.Y.B. Jun et al. applied this concept to BCK-algebras [8] and X.X.Long introduced hyper BCI-algebras [10] as a generalization of BCI-algebras.Different types of hyper BCI-ideals are also defined in [10].After the introduction of the concept of hyper BCI-algebras, several researches were conducted.Among these studies are fuzzy hyper BCK-ideals of hyper BCK-algebras [6], fuzzy ideals in hyper BCI-algebras [13], fuzzy implicative hyper BCK-ideals of hyper BCK-algebras [7], bi-polar-valued fuzzy hyper subalgebras of a hyper BCI-algebra [12], intuitionistic fuzzy hyper BCK-ideals of hyper BCK-algebras [2], and intuitionistic fuzzy ideals in hyper BCI-algebras [14] where fuzzy sets are applied to hyper BCK-algebras and hyper BCI-algebras.By following these hyperstructures, Indangan et al. introduced hyper GR-algebras [4].They established some results on hyper GR-ideals and hyper homomorphic properties on hyper GR-algebras [5].Fuzzy set was introduced by L.A. Zadeh [17].This concept of fuzzy sets are extremely useful for many people involved in research and development including engineers, mathematicians, computer software developers and researchers, natural scientists, medical researchers, social scientists, public policy analysts, business analysts, and jurists [15].With this, several researches investigated on the generalization of the notion of fuzzy sets.Other studies where fuzzy sets are applied are hyper K-subalgebras based on fuzzy points [9] and on fuzzy hyper B-ideals of hyper B-algebras [16].
In this paper, we introduce an implicative hyper GR-ideal and apply fuzzy sets on this notion.The definition of implicative hyper GR-ideal is somewhat alike with the definition of weak implicative hyper BCK-ideal [1].However, some examples in section 3 are not hyper BCI-algebras.These examples show that implicative hyper GR-ideals and weak implicative hyper BCK-ideals are not equivalent.Other than implicative hyper GRideals, two types of fuzzy implicative hyper GR-ideals are introduced and investigated.
Using the notion of level subsets of a fuzzy set, we give a characterization of a fuzzy implicative hyper GR-ideal of type 1.

Preliminaries
Let P (H) be the power set of H. Let H be a nonempty set and be a hyperoperation on H. Then (H; , 0) is called a hyper GR-algebra if it contains a constant 0 ∈ H and it satisfies the following conditions, for all x, y, z ∈ H: x y; (HGR2) (x y) z = (x z) y; (HGR3) x x; (HGR4) 0 (0 x) x, x = 0; and For the sake of simplicity, we also call H a hyper GR-algebra.
It can be verified that H is a hyper GR-algebra.
Definition 2.3.[4] Let H be a hyper GR-algebra and S be a subset of H containing 0.
If S is a hyper GR-algebra with respect to the hyperoperation on H, then we say that S is a hyper subGR-algebra on H.
Theorem 2.4.[4](Hyper SubGR-algebra Criterion) Let H be a hyper GR-algebra and S be a non-empty subset of H. Then S is a hyper subGR-algebra of H if and only if (x y) ⊆ S for all x, y ∈ S.
Definition 2.5.[4] Let I be a subset of a hyper GR-algebra H. Then I is said to be a hyper GR-ideal of H if i) 0 ∈ I; and ii) for all x, y ∈ H, x y ⊆ I and y ∈ I imply that x ∈ I.
[8] By a hyper BCK-algebra we mean a non-empty set H endowed with a hyperoperation "• and a constant 0 satisfying the following axioms: Definition 2.9.[3] Let µ be a fuzzy set in M .For a fixed t ∈ [0, 1], the set µ t = {x ∈ M |µ(x) ≥ t} is a subset of M , called a level subset of µ.
Remark 3.5.Example 3.4 shows that {0} is not always an implicative hyper GR-ideal and is not always a hyper GR-ideal of a hyper GR-algebra H.
In Example 3.4, we can see that 1 / ∈ 1 0 and not all subset I of H containing 0 is an implicative hyper GR-ideal of H. Example 3.6.Consider a set H = {0, 1, 2} with the Cayley table below.
It can be verified that H is a hyper GR-algebra.Clearly, x ∈ x y for all x, y ∈ H.By routine calculations, we can show that any subset of I of H containing 0 is an implicative hyper GR-ideal of H.
The next proposition will give a generalization when x ∈ x y for all x and y in a hyper GR-algebra H. Proposition 3.7.Let H be a hyper GR-algebra such that x ∈ x y for any x, y ∈ H. Then any subset I of H containing 0 is an implicative hyper GR-ideal of H.
Proof.Let I ⊆ H and 0 ∈ I. Let x, y, z ∈ H such that (x z) (y x) ⊆ I and z ∈ I.By hypothesis, x ∈ x z.Then x w ⊆ I for any w ∈ y x.It follows from the hypothesis that x ∈ x w ⊆ I. Hence, I is an implicative hyper GR-ideal of H. Example 3.9.Consider a set H = {0, 1, 2} with the Cayley table below.
Example 3.11.Consider the hyper GR-algebra H in Example 2.2.By routine calculations, it can be shown that the following are true: • {0, 1, 2} is a hyper GR-ideal of H.
The preceding example is generalized in the following proposition.Proposition 3.12.Let I be a hyper GR-ideal of H.If x (y x) ⊆ I implies x ∈ I, then I is implicative hyper GR-ideal of H.
HK4) x y and y x imply x = y, for all x, y, z ∈ H where x y is defined by 0 ∈ x • y and for every A, B ⊆ H, A B is defined by for all a ∈ A, there exist b ∈ B such that a b.Definition 2.7.[1] Let I be a non-empty subset of H and 0 ∈ H. Then I is called a weak implicative hyper BCK-ideal of H if, (x • z) • (y • x) ⊂ I and z ∈ I imply x ∈ I, for all x, y, z ∈ H. Definition 2.8.[17] Let M be a nonempty set.A fuzzy set µ in M is a function µ : M → [0, 1].

Lemma 3 . 8 .
If I is a hyper GR-ideal of a hyper GR-algebra H, then A B ⊆ I and B ⊆ I imply A ⊆ I. Proof.Suppose A B ⊆ I and B ⊆ I. Let a ∈ A. Then a b ⊆ I for any b ∈ B. Since I is a hyper GR-ideal of H, a ∈ I. Hence, A ⊆ I.
Definition 3.1.A nonempty subset I of a hyper GR-algebra H is called an implicative hyper GR-ideal of H if for any x, y, z ∈ H (IH1) 0 ∈ I, and (IH2) (x z) (y x) ⊆ I and z ∈ I imply x ∈ I.
2 µ in H .It can be seen that this is true by routine calculation.Example 4.5.Consider the hyper GR-algebra H in Example 3.6.Define a fuzzy set µ in H by µ(1) = 0.1, µ(2) = 0.2 and µ(0) = 0.3.By routine calculations, we see that µ is a fuzzy implicative hyper GR-ideal of type 2 in H. Remark 4.6.From Definitions 4.1 and 4.3, it shows that every fuzzy implicative hyper GR-ideal of type 2 is a fuzzy implicative hyper GR-ideal of type 1. Example 4.4 shows that not all fuzzy implicative hyper GR-ideal of type 1 is a fuzzy implicative hyper GR-ideal of type 2.
Theorem 4.7.A fuzzy set µ in a hyper GR-algebra H is a fuzzy implicative hyper GRideal of type 1 if and only if µ t is an implicative hyper GR-ideal of H whenever µ t = φ and t ∈ [0, 1].