MBJ-neutrosophic subalgebras and ﬁlters in BE -algebras

: The concept of a neutrosophic set, which is a generalization of an intuitionistic fuzzy set and a para consistent set etc., was introduced by F. Smarandache. Since then, it has been studied in various applications. In considering a generalization of the neutrosophic set, Mohseni Takallo et al. used the interval valued fuzzy set as the indeterminate membership function because interval valued fuzzy set is a generalization of a fuzzy set, and introduced the notion of MBJ-neutrosophic sets, and then they applied it to BCK / BCI-algebras. The aim of this paper is to apply the concept of MBJ-neutrosophic sets to a BE -algebra, which is a generalization of a BCK-algebra. The notions of MBJ-neutrosophic subalgebras and MBJ-neutrosophic ﬁlters of BE -algebras are introduced and related properties are investigated. The conditions under which the MBJ-neutrosophic set can be a MBJ-neutrosophic subalgebra / ﬁlter are searched. Characterizations of MBJ-neutrosophic subalgebras and MBJ-neutrosophic ﬁlters are considered. The relationship between an MBJ-neutrosophic subalgebra and an MBJ-neutrosophic ﬁlter is established.


Introduction
In 2007, Y. H. Kim and H. S. Kim [4] introduced the notion of a BE-algebra, and investigated its several properties. In [1], Ahn and So introduced the notion of an ideal in BE-algebras. They gave several descriptions of ideals in BE-algebras.
Zadeh [10] introduced the degree of a membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of a fuzzy set, Atanassov [2] introduced the degree of nonmembership/falsehood (f) in 1986, and he defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood). In 2015, neutrosophic set theory was applied to BE-algebra, and the notion of a neutrosophic filter was introduced [5]. As an extension theory of the neutrosophic set, Singh [7] introduced the notion of a type-2 neutrosophic set that could provide a granular representation of features and help model uncertainties with six different memberships. Singh et al. [8] proposed a novel hybrid time series forecasting model using neutrosophic set theory, artificial neural network and gradient descent algorithm. They dealt with three main problems of time series dataset, viz., representation of time series dataset using neutrosophic set, three degrees of memberships of neutrosophic set together, and generation of the forecasting results. In [9], the notion of MBJ-neutrosophic sets was defined as an another generalization of neutrosophic sets to BCK/BCI-algebras. The concept of MBJ-neutrosophic subalgebras in BCK/BCI-algebras was introduced and some related properties were investigated [9].
In this paper, we introduce the notion of an MBJ-neutrosophic subalgebra of a BE-algebra and investigate some related properties of an MBJ-neutrosophic subalgebra. We define the concept of an MBJ-neutrosophic filter of BE-algebras. The relationship between MBJ-neutrosophic subalgebras and MBJ-neutrosophic filters is established. We provide some characterizations of MBJ-neutrosophic filter.

Preliminaries
By a BE-algebra [4] we mean a system (U; * , 1) of type (2, 0) which the following axioms hold: We introduce a relation " ≤ " on U by x ≤ y if and only if x * y = 1.
A BE-algebra (U; * , 1) is said to be transitive if it satisfies that for any x, y, z ∈ U, y * z ≤ (x * y) * (x * z). Note that if (U; * , 1) is a transitive B-algebra, then the relation " ≤ " is a quasi-order on U. A BEalgebra (U; * , 1) is said to be self distributive if it satisfies that for any x, y, z ∈ U, x * (y * z) = (x * y) * (x * z). Note that every self distributive BE-algebra is transitive, but the converse need not be true in general (see [4]).
Every self distributive BE-algebra (U; * , 1) satisfies the following properties: Definition 2.1. Let (U; * , 1) be a BE-algebra and let F be a nonempty subset of U. Then F is called a filter of U [4] if An interval number is defined to be a closed subintervalã = [a − , a + ] of [0, 1], where 0 ≤ a − ≤ a + ≤ 1. Denote by [I] the set of all interval numbers. Let us define what is known as refined minimum (briefly, rmin) and refined maximum (briefly, rmax) of two elements in [I]. We also define the symbols " ", " ", " = " in case of two elements in [I]. Given two interval numbersã 1 and similarly we may haveã 1 ã 2 andã 1 =ã 2 . Letã i ∈ [I], where i ∈ Λ. We define Let U be a nonempty set. Let U be a nonempty set. A neutrosophic set (NC) in U (see [6]) is a structure of the form: Definition 2.2. Let U be a nonempty set. By an MBJ-neutrosophic set in U, we mean a structure of the form: where A M and A J are fuzzy sets in U, which are called a truth membership function and a false membership function, respectively, and AB is an IVF set in U which is called an indeterminate intervalvalued membership function.
For the sake of simplicity, we shall use the symbol A = (A M , AB, A J ) for the MBJ-neutrosophic set In an MBJ-neutrosophic set A = (A M , AB, A J ) in U, if we take

MBJ-neutrosophic subalgebras in BE-algebras
Definition 3.1. Let U be a BE-algebra. An MBJ-neutrosophic set A = (A M , AB, A J ) in U is called an MBJ-neutrosophic subalgebra of U if it satisfies: Example 3.2. Let U := {1, a, b, c} be a BE-algebra [3] with a binary operation " * " which is given in Table 1.  Table 2. It is easy to check that Proof. For any x ∈ U, we have This completes the proof. Proof. It is enough to show that AB satisfies: For any x, y ∈ U, we obtain for all x, y ∈ U. It follows that is not an intuitionistic fuzzy subalgebra of U as seen in Example 3.2. This shows that the converse of Theorem 3.5 is not true.
As a generalization of Proposition 4.8, we get the following results. Proof

2) that
This completes the proof. Let U be a BE-algebra. For two elements a, b ∈ U, we consider an MBJ-neutrosophic set In the following, we know that there exist a, b ∈ U such that A a,b is not an MBJ-neutrosophic filter of U.