Multipolar Fuzzy KU-ideals in KU-algebras

The notion of an m-polar fuzzy KU-ideal is introduced and its properties are investigated. The relationship between m-polar fuzzy KU-subalgebra, m-polar fuzzy KU-ideal, are discussed.


Introduction
In order to deal with possibilistic uncertainty which is connected with perceptions, imprecision of states, and preferences, fuzzy set is useful tool and it is introduced by Zadeh [11]. Since then, fuzzy set theory has become an active research area in various fields, including graph theory, statistics, life and medical sciences, engineering, social sciences, decision making, computer network, robotics and automata theory, artificial intelligence, and pattern recognition, etc. In [8] and [9] constructed a new algebraic structure which is called KU-algebras. Mostafa et al [7] introduced the notion of fuzzy KU-ideals of KU-algebras and then they investigated several basic properties which are related to fuzzy KU-ideals.
Recently, the notion of m-polar fuzzy set theory was applied to graph theory (Akram and Sarwar, 2018[3]), Al-Masarwah and Ahmad in 2019 [4] discussed the notion of m-polar fuzzy sets with an application to BCK/BCI-algebras. We introduce the notions of m-polar fuzzy subalgebras and m-polar fuzzy (closed, commutative) ideals, and then we investigate several properties.
The purpose of this manuscript is to apply the notion of m-polar fuzzy set to fuzzy KU-ideal in KU-algebras. We introduce the notions of an m-polar fuzzy KU-ideal, and investigate their properties. We examine the relationship between m-polar fuzzy KU-subalgebra and m-polar fuzzy KU-ideal. We show the relationship of an m-polar fuzzy KU-ideal and ideal of BCK/BCI-algebras.

Preliminaries
We first recall some elementary aspects which are used to present the paper. Throughout this paper, X always denotes a KU-algebra without any specifications. Definition 2.1. [8] Let X be a nonempty set with a binary operation  and a constant 0, then ( , , 0) is called a KU -algebra, if for all , ,  the following axioms are holds: On a KU-algebra ( , , 0) we can define a binary relation  on X by putting:     0. Then ( , )is a partially ordered set and 0 is its smallest element.
Thus ( , , 0)satisfies the following conditions: for all , ,  . A subset S of a KU-algebra X is called KU-subalgebra of X, if ,  , implies   .
A non-empty subset I of a KU-algebra X is said to be an KU-ideal of X if it satisfies: (K1) 0 I, (K2)  (  )  and  imply   for all , and  .

Definition 2.3: [7]
Let  be a fuzzy set on a KU-algebra X, then  is called a fuzzy KU- Definition 2.4: [7] Let X be a KU-algebra. A fuzzy set  in X is called a fuzzy KU-ideal of X if it satisfies: Lemma 2.5: [7] If fuzzy KU-subalgebra of X, then (0) ≥ ( ), for all ∈ .
Theorem 2.7: [7] A fuzzy KU-ideal of X is a fuzzy KU-subalgebra of X.
By a m-polar fuzzy set a set X (see [5]), we mean a function ̂ ∶ → [0, 1] . The membership value of every element ∈ is denoted by Given an m-polar fuzzy set on a set X, we consider the set which is called a m-polar ̂−level cut set of ̂.
Definition 2.8. [4] An m-polar fuzzy set ̂ of BCK/BCI-algebra X is called an m-polar fuzzy subalgebra if the following assertion is valid:

Conclusion
An m-polar fuzzy model is a generalized form of a bipolar fuzzy model. The m-polar fuzzy models provide more precision, flexibility and compatibility to the system when more than one agreement is to be dealt with. In this article, we have discussed the KU-ideal of KU-algebras based on m-polar fuzzy sets. We have introduced the notions of m-polar fuzzy KU-subalgebras and m-polar fuzzy KU-ideals, and investigated several properties.