1. Introduction
Various types of uncertainty arise in many complex systems and/or real-world situations such as behavior, biology, chemistry, etc. 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. One of the extended concepts of the fuzzy set, the intuitionistic fuzzy set was introduced by Atanassov in 1983 (see [
2]), and it has been applied in several fields. Intuitionistic fuzzy set is very useful in providing a flexible model to elaborate uncertainty and vagueness involved in decision making, and it is a tool in modelling real life problems like sale analysis, new product marketing, financial services, negotiation process, psychological investigations etc. The concept of neutrosophic set has been introduced by Smarandache [
3,
4,
5] and it is a generalization of classic set, (inconsistent) intuitionistic fuzzy set, interval valued (intuitionistic) fuzzy set, picture fuzzy set, ternary fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, spherical fuzzy set, and n-hyperspherical fuzzy set. Neutrosophic set is able to handle inconsistency, indeterminacy, and uncertainty for reasoning and computing. Therefore, we can see that neutrosophic set is widely applied to a variety of areas. It can be said that the generalization of the theory shows that the scope of application will be greatly expanded. 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. We discuss the next items.
Using commutative ideal to set up commutative MBJ-neutrosophic ideal
Investigating the relationship between MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal.
Presenting the conditions under which a commutative MBJ-neutrosophic ideal can be made from an MBJ-neutrosophic ideal.
Presenting a condition for an MBJ-neutrosophic set to be a closed MBJ-neutrosophic ideal.
Establishing characterization of a commutative MBJ-neutrosophic ideal (using the MBJ-level sets of an MBJ-neutrosophic set).
Constructing the extension property for a commutative MBJ-neutrosophic ideal.
In the second section, we list the well-known foundations for BCK-algebra and MBJ-neutrosophic set required in this paper. Commutative ideal in BCI-algebra using MBJ-neutrosophic set will be studied in the third section.
3. Commutative MBJ-Neutrosophic Ideals of BCI-Algebras
In this section, let X denote a BCI-algebra unless otherwise specified.
Definition 1. An MBJ-neutrosophic set in X is called a commutative MBJ-neutrosophic ideal (briefly, cMBJ-neutrosophic ideal) of X if it satisfies (17) andfor all . Example 1. Consider a BCI-algebra with the binary operation * which is given in Table 1 (see [11]). Let be an MBJ-neutrosophic set in X defined by Table 2. It is routine to verify that is a cMBJ-neutrosophic ideal of X.
Proposition 1. Every cMBJ-neutrosophic ideal of X satisfies:for all . Proof. The result (
22) is obtained using (
1) and (
17) after choosing
in (
20) and (
21). □
Using a commutative ideal, we establish a cMBJ-neutrosophic ideal.
Theorem 1. Given a commutative ideal I of X, consider an MBJ-neutrosophic set in X as follows:where , in and in . Then is a cMBJ-neutrosophic ideal of X. Proof. It is clear that
for all
. Let
. If
and
, then
since
I is a commutative ideal of
X. Hence
that is,
, and
Assume that or . Then or .
It follows that
and
Therefore is a cMBJ-neutrosophic ideal of X. □
We discuss the relationship between a cMBJ-neutrosophic ideal and an MBJ-neutrosophic ideal.
Theorem 2. Every cMBJ-neutrosophic ideal is an MBJ-neutrosophic ideal.
Proof. Let
be a cMBJ-neutrosophic ideal of
X. If we take
in (
20) and (
21) and use (
1), then
and
for all
. Therefore
is an MBJ-neutrosophic ideal of
X. □
The next example shows that the converse of Theorem 2 is not true.
Example 2. Consider a BCI-algebra with the binary operation * which is given in Table 3 (see [12]). Let be an MBJ-neutrosophic set in X defined by Table 4. It is routine to verify that is an MBJ-neutrosophic ideal of X. We can observe that Hence is not a cMBJ-neutrosophic ideal of X by Proposition 1.
We find and present the conditions under which a cMBJ-neutrosophic ideal can be made from an MBJ-neutrosophic ideal.
Theorem 3. Given an MBJ-neutrosophic set in X, the next assertions are equivalent.
- (i)
is a cMBJ-neutrosophic ideal of X.
- (ii)
is an MBJ-neutrosophic ideal of X that satisfies (22).
Proof. The necessity is evident by Proposition 1 and Theorem 2. Let
be an MBJ-neutrosophic ideal of
X that satisfies (
22). Then
and
It follows from (
22) that
and
Therefore is a cMBJ-neutrosophic ideal of X. □
Given an MBJ-neutrosophic set
in
X, we consider the next assertion.
In the following example, we know that there exists an MBJ-neutrosophic ideal
of
X which does not satisfy the condition (
23).
Example 3. Consider the BCI-algebra where is the set of integers and “−” is the minus operation in . Let be an MBJ-neutrosophic set in defined bywhere is the set of natural numbers, and α is the proper superset of β in . Then is an MBJ-neutrosophic ideal of , but it does not satisfy the condition (23) since . Definition 2 ([
7]).
An MBJ-neutrosophic ideal of X is said to be closed if it satisfies (23). Example 4. Let be an MBJ-neutrosophic set in X defined bywhere , and α is the proper superset of β in . It is routine to verify that is a closed MBJ-neutrosophic ideal of X. We provide a condition for an MBJ-neutrosophic set to be a closed MBJ-neutrosophic ideal.
Theorem 4. Given an element , let be an MBJ-neutrosophic set in X defined bywhere and α is the proper superset of β in . Then is a closed MBJ-neutrosophic ideal of X. Proof. Since
, it is clear that
for all
. For any
, if
, then
and so
. If
, then
and thus
Hence
for all
, and therefore
satisfies the condition (
23). Let
. Assume that
and
. Then
by
, (
1), (
3), (
6) and (
24). On the other hand, we have
by
and (
3). Hence
, that is,
. Therefore
and
If
or
, then
or
. Therefore
and
As a result, is a closed MBJ-neutrosophic ideal of X. □
Lemma 1 ([
7]).
Every MBJ-neutrosophic ideal of X satisfies: Theorem 5. A closed MBJ-neutrosophic ideal of X is commutative if and only if it satisfies: Proof. Assume that a closed MBJ-neutrosophic ideal
of
X is commutative, and let
. Note that
It follows from Proposition 1 and Lemma 1 that
and
Since
is closed, the combination these with (
23) induce
Conversely, suppose that a closed MBJ-neutrosophic ideal
of
X satisfies the condition (
26). For every
, we have
It follows from Lemma 1 and (
26) that
and
Since
is closed, the combination these with (
23) induce
Therefore is commutative by Theorem 3. □
Lemma 2 ([
9]).
A BCI-algebra X is commutative if and only if it satisfies: Theorem 6. In a commutative BCI-algebra, every closed MBJ-neutrosophic ideal is commutative.
Proof. Let
be a closed MBJ-neutrosophic ideal of
X. For every
, we have
by
,
, (
3), (
5) and Lemma 2. It follows from Lemma 1 and (
23) that
Consequently,
is a cMBJ-neutrosophic ideal of
X by Theorem 5. □
We form the characterization of a cMBJ-neutrosophic ideal using the MBJ-level sets of an MBJ-neutrosophic set in X.
Lemma 3 ([
7]).
An MBJ-neutrosophic set in X is an MBJ-neutrosophic ideal of X if and only if the non-empty MBJ-level sets of are ideals of X. Lemma 4 ([
10]).
A subset I of X is a commutative ideal of X if and only if it is an ideal of X that satisfies:for all . Theorem 7. An MBJ-neutrosophic set in X is a cMBJ-neutrosophic ideal of X if and only if the non-empty MBJ-level sets of are commutative ideals of X.
Proof. Assume that
is a cMBJ-neutrosophic ideal of
X. Then it is an MBJ-neutrosophic ideal of
X. Let
and
be such that
and
are non-empty. Then
and
are ideals of
X for all
and
by Lemma 3. Let
be such that
,
and
. Then
,
and
. It follows from Proposition 1 that
and
It follows that
and
Therefore
and
are commutative ideals of
X by Lemma 4.
Conversely, suppose that the non-empty MBJ-level sets
and
are commutative ideals of
X for all
and
. Then they are ideals of
X, and so
is an MBJ-neutrosophic ideal of
X by Lemma 3. Assume that
for some
. Then
or
If
then
and
for
. This is a contradiction. If
then
and
for
, which is impossible. If
then
and
for some
. This is also a contradiction. As a result, we know that
for all
. Therefore
is a cMBJ-neutrosophic ideal of
X by Theorem 3. □
Note that any MBJ-neutrosophic ideal might not be a cMBJ-neutrosophic ideal (see Example 2). But we have the following extension property for a cMBJ-neutrosophic ideal.
Theorem 8. Let and be MBJ-neutrosophic ideals of X such that
- (i)
, and .
- (ii)
, and for all .
If is a cMBJ-neutrosophic ideal and is a closed MBJ-neutrosophic ideal of X, then is a cMBJ-neutrosophic ideal of X.
Proof. Assume that
is a cMBJ-neutrosophic ideal and
is a closed MBJ-neutrosophic ideal of
X. Then
by (
23). Using
, (
1), (
3), Proposition 1 and the given conditions, we have
and
It follows from (
17)–(
19) that
and
Since
we have
and
by Lemma 1. This shows that
. Consequently,
is a cMBJ-neutrosophic ideal of
X by Theorem 5. □
4. Conclusions
Neutrosophic set, which is introduced by Smarandache, is a generalization of (inconsistent) intuitionistic fuzzy set, picture fuzzy set, ternary fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, spherical fuzzy set, and n-hyperspherical fuzzy set. Neutrosophic set is able to handle inconsistency, indeterminacy, and uncertainty for reasoning and computing. Therefore, we can see that neutrosophic set is widely applied to a variety of areas. The generalization of the theory shows that the scope of application will be greatly expanded. From this point of view, Mohseni Takallo et al. tried to introduce the notion of MBJ-neutrosophic sets as a little extended concept of neutrosophic set. The aim of this manuscript was to conduct a study that applied the MBJ-neutrosophic set to commutative ideal in BCI-algebra. We introduced the notion of closed MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal, and investigated their related properties. We used commutative ideal to set up commutative MBJ-neutrosophic ideal, and discussed the relationship between MBJ-neutrosophic ideal and commutative MBJ-neutrosophic ideal. We presented the conditions under which a commutative MBJ-neutrosophic ideal can be made from an MBJ-neutrosophic ideal, and presented a condition for an MBJ-neutrosophic set to be a closed MBJ-neutrosophic ideal. We established characterizations of a commutative MBJ-neutrosophic ideal by using the MBJ-level sets of an MBJ-neutrosophic set, and constructed the extension property for a commutative MBJ-neutrosophic ideal. The ideas and results of this paper are expected to be applicable in related algebraic structures in the future, such as MV-algebra, BL-algebra, EQ-algebra, hoop, equality algebra, etc., so we hope that many mathematicians will proceed with the study and achieve good results.