Abstract
As a generalization of a neutrosophic set, the notion of MBJ-neutrosophic sets is introduced by Mohseni Takallo, Borzooei and Jun, and it is applied to BCK/BCI-algebras. In this article, MBJ-neutrosophic set is used to study commutative ideal in BCI-algebras. The concept of closed MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal is introduced and their properties and relationships are studied. The conditions for an MBJ-neutrosophic ideal to be a commutative MBJ-neutrosophic ideal are given. The conditions for an MBJ-neutrosophic ideal to be a closed MBJ-neutrosophic ideal are provided. Characterization of a commutative MBJ-neutrosophic ideal is established. Finally, the extension property for a commutative MBJ-neutrosophic ideal is founded.
Highlights
One of the extended concepts of the fuzzy set, the intuitionistic fuzzy set was introduced by Atanassov in 1983, and it has been applied in several fields
The concept of neutrosophic set has been introduced by Smarandache [3–5] and it is a generalization of classic set, intuitionistic fuzzy set, interval valued fuzzy set, picture fuzzy set, ternary fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, spherical fuzzy set, and n-hyperspherical fuzzy set
We can see that neutrosophic set is widely applied to a variety of areas
Summary
The fuzzy set introduced by L.A. Zade [1] in 1965 is a useful tool for dealing with uncertainties in many of these real-world applications. In [6], the notion of MBJ-neutrosophic sets has been introduced as a little extended concept of neutrosophic set and it has been applied to BCK/BCI-algebras. Jun et al [7] and Hur et al [8] applied the concept of MBJ-neutrosophic sets to ideals and positive implicative ideals in BCK/BCI-algebras, respectively. The purpose of this paper is to study commutative ideal in BCI-algebra using the MBJ-neutrosophic set. We introduce the notion of closed MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal, and investigate their properties. Presenting a condition for an MBJ-neutrosophic set to be a closed MBJ-neutrosophic ideal. Commutative ideal in BCI-algebra using MBJneutrosophic set will be studied in the third section
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