ON INTERVAL VALUED FUZZY ALMOST (m,n)-BI-IDEAL IN SEMIGROUPS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we study the concept of an interval valued fuzzy almost (m,n)bi-ideals. We investigate properties of an interval valued fuzzy almost (m,n)-biideal in semigroups.


INTRODUCTION
The theory of fuzzy set was presented in 1965 by Zadeh [12]. The theory of fuzzy semigroups contained by Kuroki in 1979 [8]. Later the theory of interval valued fuzzy sets was introduced in 1975 by Zadeh [13], as a generalization of the notion of fuzzy sets. Interval valued fuzzy sets have various applications in several areas like medical science [3], image processing [2], decision making [14], etc. In 2006, Narayanan and Manikantan [7] developed the theory of interval valued fuzzy subsemigroup and studied types interval valued fuzzy ideals in semigroups. In 1961, Lajos [5] studied the concepts of (m, n)-ideals in semigroups which generalized of ideals of semigroups. The reseach of (m, n)-ideals of semigroups has interested many such as Akram et al. [1], N. Yaqoob and M. Aslam [10] and many others. In 2020 Ahsan et al. [6] extened 6658 THITI GAKETEM the idelas of (m, n)-ideals in semigroups to fuzzy sets in semigroup and they characterize the regular semigroup by using fuzzy (m, n)-ideals.
In this paper, we give the concept of an interval valued fuzzy almost (m, n)-bi-ideals. We prove properties of an interval valued fuzzy almost (m, n)-bi-ideal in semigroups.

PRELIMINARIES
In this section, we give some definition and theory helpful in later sections.
A non-empty subset L of a semigroup G is called (3) an ideal of a semigroup G we mean a left ideal and a right ideal of G, A non-empty subset L of a semigroup G. We denote the i.e., the smallest (m, n)-ideal, the smallest (m, 0)-ideal and the smallest (0, n)-ideal of G containing L, respectively.  (2) G([π] (0,n) ) n = Gπ n .
A fuzzy set of a non-empty set T is a function ω : L → [0, 1].
[6] A fuzzy set ω of a semigroup G is said to be for all e 1 , e 2 ∈ G, (2) a fuzzy left (right) ideal of G if ω(e 1 e 2 ) ≥ ω(e 2 ) (ω(e 1 e 2 ) ≥ ω(e 1 )) for all e 1 , e 2 ∈ G, (3) a fuzzy ideal of G if it is both a fuzzy left ideal and a fuzzy right ideal of G, Let Ω[0, 1] be the set of all closed subintervals of [0, 1], i.e., We note that If p q, we mean q p.

For each interval
For two IVF sets ω and ϖ of a non-empty set G, define (2) ω = ϖ ⇔ ω ϖ and ϖ ω, For two IVF sets ω and ϖ in a semigroup G, define the product ω • ϖ as follows : for all e ∈ G, Next, we shall give definitions of various types of IVF subsemigroups. (1) ω 0 := χ g and ω 0 • χ G • ω 0 := χ G , The following theorem we can easy to prove.
Theorem 2.10. Lef ω, ϖ and κ be IVF set of a semigroup G. Then the following statements hold: (1) If ω ϖ then ω m ϖ m for all m ∈ Z.

ON INTERVAL VALUED FUZZY ALMOST (m, n)-BI-IDEAL IN SEMIGROUPS
In this section, we give the concept of an interval valued fuzzy almost (m, n)-bi-ideals and investigate properties of an interval valued fuzzy almost (m, n)-bi-ideal in semigroups. Theorem 3.2. Suppose that ω is an IVF almost (m, n)-bi-ideal and ϖ is an IVF subsemigroup of a semigroup G and m, n ∈ Z. Then the following statements hold: (1) If ω ϖ, then ϖ is an IVF almost (m, n)-bi-ideal of G.
an IVF almost (m, n)-bi-ideal of G.
Note that for a subset L of G, define L 0 := G.
Thus χ L is an IVF almost (m, n)-bi-ideal of G.
Conversely, suppose that χ L is an IVF almost (m, n)-bi-ideal of G. Then χ L is an IVF subsemigroup. Thus by Theorem 2.11, L is a subsemigroup of G. Let d ∈ G. Then Thus there exists e ∈ G such that [((χ L ) m • G • (χ L ) n ) χ L ](e) = 0. By Lemma 3.3, Thus e ∈ L m GL n ∩ L. Hence L m GL n ∩ L = / 0.
We conclude that L is an almost (m, n)-bi-ideal of G.
Definition 3.6. An IVF almost bi-ideal ω is called minimal if for all nonzero IVF almost biideals ϖ of a semigroup G such that ϖ ω implies supp(ω) =supp(ϖ).
Definition 3.7. An IVF almost (m, n)-bi-ideal ω is called minimal if for all nonzero IVF almost (m, n)-bi-ideals ϖ of a semigroup G such that ϖ ω implies supp(ω) =supp(ϖ). Proof. Suppose that L is a minimal almost (m, n)-bi-ideal of G. Then by Theorem 3.4, χ L is an IVF almost (m, n)-bi-ideal of G. Let ω be an IVF almost (m, n)-bi-ideal of S such that ω χ L .
Conversely, suppose that χ L is a minimal IVF almost (m, n)-bi-ideal of G and let D be an almost (m, n)-bi-ideal of G such that D ⊆ L. Then χ D is an IVF almost (m, n)-bi-ideal of G such that χ D χ L . Thus D = supp(χ B ) = supp(χ L ) = L. Therefore L is a minimal almost (m, n)-bi-ideal of G. Proof. It follows from Theorem 3.8.