CHARACTERIZATION IN TERM OF CUBIC IDEAL IN Γ -SEMIRINGS

. In this paper we introduce the notion cubic ideal in Γ -semiring and we study basic properties of cubic ideal. Finally, we characterized in term of cubic ideal in Γ -semirings.


INTRODUCTION
Zadeh initiated the concept of fuzzy sets in 1965. In 1975, Zadeh made an extension concept of a fuzzy set by an interval-valued fuzzy set. A semigroup is an algebraic structure consisting of a non-empty sets together with an associative binary operation.Semirings which is a common generalization of rings and distributive lattices, was introduced by Vandiver [8]. It has been found very useful for solving problems in different areas of pure and applied mathematics, information sciences, etc., since the structure of a semiring provides an algebraic framework for modelling and studying the key factors in these applied areas. Ideals of semiring play a central role in the structure theory and useful for many purposes. The theory of Γ-semirings was introduced by [2]. Since then many researchers enriched this field. Many authors have studied semigroups in terms of fuzzy sets. Kuroki is the main contributor of this study. Kuroki introduced the notion of fuzzy ideals and fuzzy bi-ideals in semigroups. Atanassov introduced intuitionistic fuzzy set is characterized by a membership function and a non-membership function for each element in the Universe. In this paper we studied properties of cubic ideals of Γ-semirings. Furthermore we can show that the images or inverse images of a cubic ideal of an Γ-semiring become a cubic ideal.

PRELIMINARIES
2.1. Γ-Semiring. Now we review definition of some types Γ-semiring, which we use in the next section.
An Γ-semiring S is said to be regular if for each element a ∈ S, there exists an element x ∈ S and α, β ∈ Γ such that a = aαxβ a. An Γ-semigroup S is called intra-regular if for every a ∈ S there exist x, y ∈ S and α, β , γ ∈ Γ such that a = xαaβ aγy. A non-empty subset A of an Γ-semiring S is an Γ-subsemiring of S if A is a subsemigroup of (S, +) and AΓA ⊆ A. A nonempty subset A of an Γ-semiring S is called a left(right)ideal of S if A is a additive subsemigroup of S and SΓA ⊆ A(AΓS ⊆ A). An ideal of S is a non-empty subset which is both a left ideal and a right ideal of S. [2] CUBIC IDEAL IN Γ-SEMIRINGS 3 2.2. Fuzzy Γ-semiring and interval valued fuzzy set. Now we will give definition of a fuzzy subset and types of fuzzy Γ-subsemigroups. Let X be a non-empty set. A mapping ω : X → [0, 1] is a fuzzy subset of S. Definition 2.1. [5] Let S be an Γ-semiring. A fuzzy subset ω of S is said to be a fuzzy Γsubsemiring of S if ω(x+y) ≥ min{ω(x), ω(y)} and ω(xαy) ≥ min{ω(x), ω(y)}, for all x, y ∈ S and α ∈ Γ.
Definition 2.2. [5] Let S be a Γ-semiring. A fuzzy subset ω of S is said to be a fuzzy left (right) ideal of S if ω(x + y) ≥ min{ω(x), ω(y)} and ω(xαy) ≥ ω(y)(ω(xαy) ≥ ω(x)), for all x, y ∈ S and α ∈ Γ. A non-empty fuzzy subset of an Γ-semigroup S is a fuzzy ideal of S if it is a fuzzy left ideal and fuzzy right ideal of S. Definition 2.3. For a family {ω i | i ∈ I} of fuzzy sets in X, we define the join (∨) and meet (∧) operations as follows: respectively, for all x ∈ X. Now we will introduce a new relation of an interval.
The interval [a, a] is identified with the number a ∈ [0.1].
We consider two interval numbers a := [a − , a + ] and b : Definition 2.6. Let X be a set. An interval valued fuzzy set A on X is defined as where µ − and µ + are two fuzzy sets of X such that and ( i∈I µ i ) are defined as follows: respectively, for all x ∈ X where µ : X → D[0, 1].

Cubic Γ-semiring.
Definition 2.8. Let X be a non-empty set. A cubic set A in X is a structure of the form and denoted by A = µ, ω where µ is an interval valued fuzzy set (briefly. IVF) in X and ω is a fuzzy set in X. In this case we will use For all x ∈ X. Note that a cubic set is a generalization of an intuitionistic fuzzy set.
Definition 2.9. Let A be a subset of a non-empty set X. Then cubic set characteristic function and and we say it is a cubic level set of A = µ, ω .
A non-empty cubic set A = µ, ω of S is called cubic ideal of S if it is a cubic left ideal and a cubic right ideal of S.
Thus, A B is a cubic left ideal of S. Proof. Let {A i } i∈I be a family of cubic left ideal of S and x, y ∈ S, α ∈ Γ. Then Thus the condition of (1) in Defintion 3.2 is true. Consider Thus, i∈I A i is a cubic left ideal of S. Proof. Suppose that A is a left ideal of S and let x, y ∈ S with y ∈ A and α ∈ Γ. Then x + y ∈ A and xαy ∈ A. Thus µ χ A (x + y) = µ χ A (xαy) = [1,1] and ω χ A (x + y) = ω χ A (xαy) = 0. So If y / ∈ A, then µ χ A (y) = [0, 0] and ω χ A (y) = 1. Since A is a left ideal of S we have x + y ∈ A and xαy ∈ A. Thus µ χ A (x + y) = µ χ A (xαy) = [1,1] and ω χ A (x + y) = ω χ A (xαy) = 0.
We will show that A is a left ideal of S. Let x, y ∈ S with y ∈ A. Since and ω χ A (xαy) ≤ ω χ A (y).
Hence xαy ∈ A. Therefore A is a left ideal of S. Hence A = µ, ω is a cubic left ideal of S.

CHARACTERIZATIONS OF REGULAR AND INTRA-REGUALR Γ-SEMIRINGS BY CU-BIC IDEALS
In this section, we study characterizations of regular and intra-regualr Γ-semirings by cubic ideals.
Lemma 4.1. Let A and B be non-empty subsets of a semigroup S and Q be any non-empty subset. Then the following statements hold (2) (χ A • χ B ) = (χ AB ).
(1) S is regular  Proof. Suppose that S is regular. Then Lemma 3.5 and 3.6. Thus Conversely, let R and L be a right ideal and left ideal of S respectively. Then by Theorem 3.3, χ R = µ χ R , ω χ R is a cubic right ideal and χ L = µ χ L , ω χ L is a cubic left ideal of S respectively.
Conversely, let R and L be a right ideal and left ideal of S respectively. Then by Theorem 3.3, χ R = µ χ R , ω χ R is a cubic rignt ideal and χ L = µ χ L , ω χ L is a cubic left ideal of S respectively.