1. Introduction
As a ramification of general algebra,
-algebras first appeared in the mathematics literature in 1966, in work by Imai and Iséki [
1,
2]. These ideas are created from two distinct approaches: propositional calculi and set theory.
-algebras are algebraic patterns of the
-system in combinatory logic. The name of
-algebras arises from the combinatories
in combinatory logic. Various properties of
-algebras are explored within [
3,
4,
5,
6].
Bipolar fuzzy sets [
7]—a generalization of Zadeh’s idea of the fuzzy set [
8] which itself expands the classical set—are sets whose elements have positive and negative membership degrees. Hybrid models of fuzzy sets have been applied in many different sciences [
9,
10,
11,
12]. The first definition of fuzzy ideals in
-algebras was by Xi [
13] in 1991. Bipolar information is applied in many algebraic structures—for instance,
-algebras [
14,
15,
16,
17],
-algebras [
18],
-semihypergroups [
19], and hemirings [
20]. In many real-life issues, information sometimes comes from
m factors
, that is, multi-attribute data arise which cannot be handled using the existing ideals (e.g., fuzzy ideals, bipolar fuzzy ideals, etc.). For the time being, experts trust that the real world is proceeding to multipolarity. Multi-polar vagueness in information performs a crucial role in different areas of the sciences. In neurobiology, multi-polar neurons have numerous dendrites, permitting the integration of a great deal of data from different neurons. In technology, multi-polar technology can be utilized to build and perform large-scale IT structures.
In view of this inspiration, Chen et al. [
21] introduced an
m-polar fuzzy set (
m-pF set, for short) in 2014, which was an extension of the bipolar fuzzy set. In an
m-pF set, the degree of membership of an object ranges over
, which depicts
m distinct characteristics of the object. The theory of
m-pF sets was essentially created to deal with the absence of a mathematical method towards multi-attribute, multi-polar, and multi-index information. Since that time,
m-pF sets have been utilized in mathematical theories such as graph theory [
22,
23,
24] and matroid theory [
25]. Additionally,
m-pF sets have applications in real-life issues such as decision-making problems [
26,
27]. For the first time, Akram et al. [
28] implemented the idea of
m-pF sets into algebraic structures and gave the notion of
m-pF lie subalgebras. In addition, Akram and Farooq [
29] established
m-pF lie ideals of lie subalgebras. Applying the idea of
m-pF sets to group theory, Farooq et al. [
30] initiated the concept of
m-pF subgroups, and investigated some of their properties. Furthermore, Al-Masarwah and Ahmad [
31] applied
m-pF sets to
-algebras. They presented the concepts of
m-pF subalgebras,
m-pF ideals, and
m-pF commutative ideals, and investigated related results.
In 1971, Rosenfeld [
32] used fuzzy sets in the theory of groups and established the notion of fuzzy subgroups. In 1996, Bhakat and Das [
33] generalized the idea of fuzzy subgroups to
-fuzzy subgroups by using the concept of fuzzy points and its “belongingness (∈)” and “quasi-coincidence (
q)” with a fuzzy set. After that, Bhakat [
34,
35] studied this concept in detail. Actually, the notion of an
-fuzzy subgroup is a fundamental and valuable generalization of the fuzzy subgroup. In
-algebras,
-fuzzy subalgebras were created and discussed by Jun [
36,
37], and further studied by Muhiuddin and Al-Roqi in [
38]. Jun [
39] and Zhan et al. [
40] proposed and discussed a generalization of a fuzzy ideal in a
-algebra. As an extension of generalized fuzzy ideals in
-algebras, Ma et al. [
41] considered
-interval-valued fuzzy ideals and Jana et al. [
42] proposed the concept of
-bipolar fuzzy ideals. Recently, Ibrara et al. [
43] proposed the
-bipolar fuzzy generalized bi-ideal in ordered semigroups. In hemirings, Abdullah [
44] defined the concepts of N-dimensional
-fuzzy H-ideals.
Motivated by the previous studies, here we combine the notions of
m-pF sets and
m-pF points to introduce a new notion called
m-polar
-fuzzy ideals in
-algebras. The defined concept is a generalization of fuzzy ideals, bipolar fuzzy ideals,
-fuzzy ideals, and
-bipolar fuzzy ideals. We prove that every
m-polar
-fuzzy ideal is an
m-polar
-fuzzy ideal, and every
m-polar
-fuzzy ideal is an
m-polar
-fuzzy ideal. For the characterizations of
-algebras, we give a fundamental bridge between crisp ideals and
m-polar
-fuzzy ideals, since sometimes it is difficult to comprehend whether a particular ideal is an
m-polar
-fuzzy (resp., crisp) ideal or not. In this case, to provide the required information, we describe the characterization of
m-polar
-fuzzy ideals by level cut subsets. However, this technique has some gaps. One of them is that all outcomes have similarities in crisp ideals. In other words,
m-polar
-fuzzy ideals become a mirror of the crisp case. Moreover, we define
m-polar
-fuzzy commutative ideals and discuss some relevant properties. To show the novelty of this model, some contributions of several authors toward
m-polar
-fuzzy ideals in
-algebras are analyzed in
Table 1.
2. Preliminaries
We recall basic concepts of -algebras, m-pF sets, m-pF ideals, and m-pF commutative ideals. From now on, X stands for a -algebra, unless something else is indicated.
A -algebra is an algebraic structure satisfying the axioms below: for all
- (I)
- (II)
- (III)
- (IV)
and imply
A -algebra X is called a -algebra if for any In any -algebra the following hold: for all
- (1)
- (2)
- (3)
- (4)
- (5)
where means
A non-empty subset J of X is said to be an ideal of X if for all
A
-algebra
X is called commutative if
where
A non-empty subset
D of a
-algebra
X is a commutative ideal of
X (see [
45]) if for all
Definition 1 ([
21])
. An m-pF set on is a function whereis the membership value of every element and is the i-th projection mapping for all . The values and are the smallest and largest values in respectively. Al-Masarwah and Ahmad [
31] proposed
m-pF ideals and
m-pF commutative ideals as follows:
Definition 2 ([
31])
. An m-pF set of X is said to be an m-pF ideal if the assertions below are valid: for all (J1)
(J2)
That is,
(J1)
(J2)
for all
Definition 3 ([
31])
. An m-pF set of a -algebra X is said to be an m-pF commutative ideal of X if it satisfies and for all (J3)
That is, (J3)
for all
For an
m-pF set
of
the set
for all
is called the level cut subset of
An
m-pF set
of
X of the form
is called an
m-pF point, denoted by
, with support
x and value
An m-pF point
Belongs to , denoted by , if , that is, for each
Is quasi-coincident with , denoted by , if , that is, for each
We say that
if does not hold,
if or
if and
3. -Polar -Fuzzy Ideals
In this section, we propose and discuss
m-polar
-fuzzy ideals, where
Theorem 1. For an m-pF set of the set for all is an ideal of X if and only if satisfies the assertions below: for all
- (1)
- (2)
Proof. Let
be an ideal of
Suppose that there exists
such that
Then, and thus But implies a contradiction. Thus, (1) holds. Assume for some Then, and However, since a contradiction. Thus, (2) holds.
Conversely, suppose that (1) and (2) hold. Let
be such that
For any
we get
Thus,
Therefore,
Let
be such that
This implies that
Thus, that is, Hence, is an ideal of □
Definition 4. An m-pF set of X is called an m-polar -fuzzy ideal of X if for all and
- (1)
- (2)
and
Theorem 2. Let be an m-pF subset of X and J be an ideal of X such that
- (1)
for all
- (2)
for all
Then, is an m-polar -fuzzy ideal of
Proof. (a) (For
) Let
and
be such that
Then,
Since
we have
If
then
and we have
If
then
and we have
Thus,
Let
and
be such that
and
Then,
This implies that and so . That is, If then and we have If then and we have Therefore, Hence, is an m-polar -fuzzy ideal of
(b) (For
) Let
and
be such that
Then,
This implies
and so
Thus,
If
then
and we have
If
then
and we have
Hence,
Let
and
be such that
and
Then,
Thus and so That is, If then and we have If then and we have Thus, Hence, is an m-polar -fuzzy ideal of
(c) (For ) It follows from (a) and (b). □
The following example illustrates Theorem 2.
Example 1. Let be a -algebra which is defined in Table 2: Let be a 3-pF set defined as: Then, is an ideal of Therefore, is a 3-polar -fuzzy ideal of
4. -Polar -Fuzzy Ideals
In this section, we define m-polar -fuzzy ideals of X as a special case of m-polar -fuzzy ideals, and discuss several results.
Definition 5. An m-pF set of X is called an m-polar -fuzzy ideal of X if for all and
- (1)
- (2)
and
Example 2. Consider a -algebra which is defined in Table 3: Let be a 4-pF set defined as: Clearly, is a 4-polar -fuzzy ideal of
Lemma 1. For an m-pF set of the following conditions are equivalent for all
- (1)
- (2)
Proof. Let
be an
m-pF set of
X and
Assume
If
then
for some
It follows that
but
Since
, we get
Therefore,
a contradiction to (1). Thus,
If
then
and so
which implies that
or
Hence,
Otherwise,
a contradiction. Therefore,
for all
Let
and
be such that
Then,
Assume that
If
then
This is a contradiction. Therefore,
which implies that
Thus, □
Lemma 2. For an m-pF set of the following conditions are equivalent for all
- (1)
and
- (2)
Proof. Let
be an
m-pF set of
X and
such that
If
then
Choose
such that
This implies that and but and that is, a contradiction. Thus, whenever If then and It follows that by (1), so that or If then which is a contradiction. Therefore, Consequently, for all
For any
Let
be such that
Then,
Suppose that
If
then
a contradiction, so
This implies that
Hence, □
From Lemmas 1 and 2, we deduce that
Theorem 3. An m-pF set of X is an m-polar -fuzzy ideal of X if and only if for all
- (i)
- (ii)
Theorem 4. Any m-polar -fuzzy ideal of X satisfies: for all
- (1)
- (2)
Proof. (1) Suppose that
for all
Then,
We have
(2) Assume that
hold in
Then,
This completes the proof. □
The next theorem gives the bridge between m-polar -fuzzy ideals and crisp ideals.
Theorem 5. An m-pF set of X is an m-polar -fuzzy ideal of X if and only if is an ideal of X for all
Proof. Suppose
is an
m-polar
-fuzzy ideal of
Let
and
Then,
Using Theorem 3 (i) implies that
Thus,
Again, let
Then,
and
Using Theorem 3 (ii), we have
Hence, Therefore, is an ideal of
Conversely, let
be an ideal of
X for all
If there exists
such that
then
for some
It follows that
but
a contradiction. Therefore,
for all
Suppose there exist
v,
such that
Then,
for some
It follows that
and
but
a contradiction. Thus,
for all
Hence,
is an
m-polar
-fuzzy ideal of
X by Theorem 3. □
Theorem 6. An m-pF set of X is an m-polar fuzzy ideal of X if and only if is an m-pF ideal of
Proof. Assume that
is an
m-polar
fuzzy ideal of
Suppose that there exists
such that
Select
such that
Then,
but
a contradiction. Thus,
for all
Assume there exist
such that
Select
such that
Then, and but a contradiction. Thus, for all Hence, is an m-pF ideal of
Conversely, suppose
is an
m-pF ideal of
Let
for
Then,
By hypothesis
that is,
Let
for
Then,
By hypothesis
This implies that Therefore, is an m-polar fuzzy ideal of □
Remark 1. The above theorem shows that m-polar fuzzy ideals are the same as m-pF ideals of
Remark 2. Every m-polar -fuzzy ideal is an m-polar -fuzzy ideal, but the converse may not be true, as shown in the next example.
Example 3. Reconsider the -algebra X given in Example 2. An m-pF set of X defined byis an m-polar -fuzzy ideal of X which is not an m-polar -fuzzy ideal of since We provide a condition for an m-polar -fuzzy ideal to be an m-polar -fuzzy ideal.
Theorem 7. If is an m-polar -fuzzy ideal of X and then is an m-polar -fuzzy ideal of
Proof. Let
be an
m-polar
-fuzzy ideal of
X and
Let
for
Then,
Using Theorem 3 (i), we get
Therefore,
Now, let
for
Then,
and
Using Theorem 3 (ii), we have
Thus, Hence, is an m-polar -fuzzy ideal of □
Next, we discuss the relation between an m-polar -fuzzy ideal and an m-polar -fuzzy ideal.
Theorem 8. Every m-polar -fuzzy ideal of X is an m-polar -fuzzy ideal of
Proof. Let be an m-polar -fuzzy ideal of Let and be such that Then, It follows from Definition 4 (1) that Let and be such that and Then, and It is implied from Definition 4 (2) that Hence, is an m-polar -fuzzy ideal of □
The converse of the above theorem is not true in general.
Example 4. Reconsider the -algebra X given in Example 2. An m-pF set of X defined byis an m-polar -fuzzy ideal of X which is not an m-polar -fuzzy ideal of sincebut Let
be an
m-pF set of
we define the following sets for all
and
The sets and are called q-level cut subset of and -level cut subset of respectively.
Theorem 9. If is an m-polar -fuzzy ideal of then is an ideal of X for all
Proof. Suppose
is an
m-polar
-fuzzy ideal of
Let
and
Then,
Using Theorem 3 (i), we have
i.e.,
Hence,
Again, let
Then,
and
Using Theorem 3 (ii), we have
so that
that is,
Thus,
is an ideal of
□
Theorem 10. An m-pF set of X is an m-polar -fuzzy ideal of X if and only if is an ideal of X for all
Proof. Suppose
is an
m-polar
-fuzzy ideal of
Let
and
Then,
that is,
Using Theorem 3 (i), we have
We consider two cases:
Case (1): Hence, or Therefore, or Thus, , that is,
Case (2): Hence,
or
Therefore,
or
Thus,
, that is,
Hence, in any case, we get
, that is,
Suppose
for
Then,
Thus,
or
and
or
Using Theorem 3 (ii), we have
We consider four cases:
Case (1): and
Hence, or Therefore, or Thus, , that is,
Case (2): and
Hence, or Therefore, or Thus, , that is,
Case (3): and This is similar to Case (2).
Case (4): and
Hence, or Therefore, or Thus, , that is, Therefore, in any case, we get , that is, Thus, is an ideal of
Conversely, suppose
is an
m-pF set of
X and
such that
is an ideal of
If there exists
such that
then
for some
This implies that
but
Additionally,
and so
that is,
Therefore,
a contradiction. Thus,
for all
Suppose that there exist
such that
Then,
for some
This implies that
Since
is an ideal of
Thus,
or
a contradiction. Therefore,
for all
Hence,
is an
m-polar
-fuzzy ideal of
□
5. -Polar -Fuzzy Commutative Ideals
In this section, we propose the notion of m-polar -fuzzy commutative ideals in -algebras and discuss the related properties.
Definition 6. An m-pF set of a -algebra X is called an m-polar -fuzzy commutative ideal of X if for all and
- (1)
- (2)
and
Example 5. Let be a -algebra which is defined in Table 4: Let be a 3-pF set defined as: Clearly, is a 3-polar -fuzzy commutative ideal of
Theorem 11. An m-pF set of a -algebra X is an m-polar -fuzzy commutative ideal of X if and only if for all :
- (1)
- (2)
Proof. Assume
is an
m-polar
-fuzzy commutative ideal of a
-algebra
Let
and suppose that
If
then
for some
It follows that
, but
Since
we get
Therefore,
a contradiction. Thus,
for all
If
then
and so
which implies that
Hence,
Otherwise,
a contradiction. Hence,
for all
Let
Assume that
If not, then
for some
This implies that
and
but
a contradiction. Hence,
whenever
If
then
It follows that
Therefore,
or
If
then
a contradiction. Therefore,
for all
Conversely, suppose that (1) and (2) hold. Let
and
be such that
Then,
Assume
If
then
a contradiction. Therefore,
which implies that
Thus,
Let
and
be such that
Then,
Suppose that
If
then
a contradiction. Hence,
This implies that
So, Hence, is an m-polar -fuzzy commutative ideal of □
Theorem 12. Every m-polar -fuzzy commutative ideal of a -algebra X is an m-polar -fuzzy ideal of
Proof. Let
be an
m-polar
-fuzzy commutative ideal of a
-algebra
Let
and
Then, by taking
in (2) of Definition 6, we have
Since for all
and
so
Hence, satisfies (2) of Definition 5. Combining with (1) of Definition 5 implies that is an m-polar -fuzzy ideal of □
In general, the converse of Theorem 12 is not true.
Example 6. Let be a -algebra which is defined in Table 5: A 3-pF set defined by:is a 3-polar -fuzzy ideal of X which is not a 3-polar -fuzzy commutative ideal of since Theorem 13. If is an m-polar -fuzzy ideal of a -algebra X andfor all then is an m-polar -fuzzy commutative ideal of Proof. Suppose
is an
m-polar
-fuzzy ideal of a
-algebra
Then,
for all
Also, by assumption and (ii) of Theorem 3, we have
Hence, is an m-polar -fuzzy commutative ideal of □
The next theorem provides necessary and sufficient condition for the crisp commutative ideal to be an m-polar -fuzzy commutative ideal.
Theorem 14. An m-pF set of a -algebra X is an m-polar -fuzzy commutative ideal of X if and only if is a commutative ideal of X for all
Proof. Assume that
is an
m-polar
-fuzzy commutative ideal of
Let
and
Then,
Theorem 11 (1) implies that
Thus,
Again, let
Then,
and
Theorem 11 (2) implies that
Therefore, Thus, is a commutative ideal of
Conversely, let
be an
m-pF set of
X be such that
is a commutative ideal of
X for all
If there exists
such that
then
for some
It follows that
but
a contradiction. Therefore,
for all
Assume there exist
such that
Then,
for some
This implies that
and
but
This is impossible. Thus,
for all
Hence,
is an
m-polar
-fuzzy commutative ideal of
X by Theorem 11. □