A Short Note on Fuzzy Characteristic Interior Ideals of Po-Gamma-Semigroups

A semigroup is an algebraic structure consisting of a non-empty set S together with an associative binary operation [12]. The formal study of semigroups began in the early 20th century. Semigroups are important in many areas of mathematics, for example, coding and language theory, automata theory, combinatorics and mathematical analysis. The concept of fuzzy sets was introduced by Lofti Zadeh [33] in his classic paper in 1965. Azirel Rosenfeld [21] used the idea of fuzzy set to introduce the notions of fuzzy subgroups. Nobuaki Kuroki [16,17,18,20] is the pioneer of fuzzy ideal theory of semigroups. The idea of fuzzy subsemigroup was also introduced by Kuroki [16,18]. In [17], Kuroki characterized several classes of semigroups in terms of fuzzy left, fuzzy right and fuzzy bi-ideals. Others who worked on fuzzy semigroup theory, such as X.Y. Xie [31,32], Y.B. Jun [14], are mentioned in the bibliography. X.Y. Xie [31] introduced the idea of extensions of fuzzy ideals in semigroups.

The notion of a Γ-semigroup was introduced by Sen and Saha [26] as a generalization of semigroups and ternary semigroup. Γ-semigroup have been analyzed by lot of mathematicians, for instance by Chattopadhyay [1,2], Dutta and Adhikari [6,7], Hila [10,11], Chinram [3], Saha [24], Sen et al. [25,26,24], Seth [27]. S.K. Sardar and S.K. Majumder [8,9,22,23] have introduced the notion of fuzzification of ideals, prime ideals, semiprime ideals and ideal extensions of Γ-semigroups and studied them via its operator semigroups. Authors who worked on fuzzy partially ordered Γ-semigroup theory are Y.I. Kwon and S.K. Lee [19], Chinram [4], P. Dheena and B. Elavarasan [5], M. Siripitukdet and A. Iampan [28,29,30]. In [15], A.Khan, T. Mahmmod and M.I. Ali introduced the concept of fuzzy interior ideals in Po-Γ-semigroups. In this paper the concept of fuzzy characteristic ideals of a Po-Γ-semigroup has been introduced and it is observed that it satisfies level subset criterion as well as characteristic function criterion.

In this section we discuss some elementary definitions that we use in the sequel. Definition 2.1. [13] Let S and Γ be two non-empty sets. S is called a Γ-semigroup if there exist mapping from S × Γ × S to S, written as (a, α, b) → aαb satisfying the identity (aαb)βc = aα(bβc) for all a, b, c ∈ S and for all α, β ∈ Γ. Definition 2.6. [22] Let µ be a fuzzy subset of a non-empty set X. Then the set µ t = {x ∈ X : µ(x) ≥ t} for t ∈ [0, 1], is called the t-cut of µ.

In what follows Aut(S) denote the set of all automorphisms of the Po-Γ-semigroup S. Definition 3.1. An interior ideal A of a Po-Γ-semigroup S is called a characteristic interior ideal of S if f (A) = A ∀f ∈ Aut(S). Definition 3.2. A fuzzy interior ideal µ of a Po-Γ-semigroup S is called a fuzzy characteristic interior ideal of S if µ(f (x)) = µ(x) ∀x ∈ S and ∀f ∈ Aut(S). Theorem 3.3. A non-empty fuzzy subset µ of a Po-Γ-semigroup S is a fuzzy characteristic interior ideal of S if and only if the t-cut of µ is a characteristic interior ideal of S for all t ∈ [0, 1], provided µ t is non-empty.

Proof. Let µ be a fuzzy interior ideal of S and t ∈ [0, 1] be such that µ t is non-empty. Let x, y ∈ µ t and γ ∈ Γ. Then µ(x) ≥ t and µ(y) ≥ t. Now since µ is a fuzzy interior ideal, it is a fuzzy subsemigroup of S and hence µ(xγy) ≥ min{µ(x), µ(y)}. Thus we see that µ(xγy) ≥ t. Consequently, xγy ∈ µ t . Hence µ t is a subsemigroup of S. Now let x, y ∈ S; β, δ ∈ Γ and a ∈ µ t . Then µ(xβaδy) ≥ µ(a) ≥ t and so xβaδy ∈ µ t .

Let x, y ∈ S be such that y ≤ x. Let x ∈ µ t . Then µ(x) ≥ t. Since µ is a fuzzy interior ideal of S, so µ(y) ≥ µ(x) ≥ t. Consequently, y ∈ µ t . Hence we conclude that µ t is an interior ideal of S.

In order to prove the converse, we have to show that µ satisfies the condition of Definition 2.7 and the condition of Definition 2.8. If the condition of Definition 2.7 is false, then there exist x 0 , y 0 ∈ S, γ ∈ Γ such that µ(x 0 γy 0 ) < min{µ(x 0 ), µ(y 0 )}. Taking t 0 := 1 2 [µ(x 0 γy 0 ) + min{µ(x 0 ), µ(y 0 )}], we see that µ(x 0 γy 0 ) < t 0 < min{µ(x 0 ), µ(y 0 )}. This implies that x 0 , y 0 ∈ µ t 0 and x 0 γy 0 / ∈ µ t 0 , which is a contradiction. Hence the condition of Definition 2.7 is true. Similarly we can prove the other condition also.

Let x, y ∈ S be such that x ≤ y. Let µ(y) = t, then y ∈ µ t . Since µ t is an interior ideal of S, so x ∈ µ t . Then µ(x) ≥ t = µ(y). Consequently, µ is a fuzzy interior ideal of S.

Theorem 3.4. Let A be a non-empty subset of a Po-Γ-semigroup S. Then A is a characteristic interior ideal of S if and only if its characteristic function χ A is a fuzzy characteristic interior ideal of S.

Proof. Let A be a characteristic interior ideal of S. Then A is an interior ideal of S. Hence by Lemma 1 [15], χ A is a fuzzy interior ideal of S. Let

Conversely, let χ A is a fuzzy characteristic interior ideal of S. Then χ A is a fuzzy interior ideal of S. Hence by Lemma 1 [15], A is an interior ideal of S.

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