1. Introduction
As is well known, the fuzzy set, which was first introduced by Zadeh [
1], dealt with the membership degree that is represented by only one function, the so-called truth function. As a generalization of the fuzzy set, the notion of the intuitionistic fuzzy set is introduced by Atanassove [
2]. In 2014, Chen et al. [
3] introduced an
m-polar fuzzy set which is an extension of the bipolar fuzzy set, and then this notion was applied to graph theory, algebraic structure, the decision making problem, etc. For BCK/BCI-algebra, see [
4,
5,
6]; for graph theory, see [
7,
8,
9,
10]; and see [
11,
12,
13,
14] for the decision making problem. In [
15], Kang et al. introduced the concept of a multipolar intuitionistic fuzzy set of finite degree as a generalization of an intuitionistic fuzzy set, and they applied it to BCK/BCI-algebras. The hyperstructure theory was introduced by Marty [
16] in 1934 at the 8th Congress of Scandinavian Mathematicians. Jun et al. [
17,
18] applied the hyperstructure theory to BCK-algebras, and they introduced a hyper BCK-algebra which is a generalization of a BCK-algebra. They studied hyper ideal theory in hyper BCK-algebras. Jun and Xin discussed the fuzzy set theory of hyper BCK-ideals in hyper BCK-algebras (see [
19]), and Bakhshi et al. [
20] studied fuzzy (positive, weak) implicative hyper BCK-ideals. In 2004, Borzooei and Jun [
21] considered the intuitionistic fuzzy set theory of hyper BCK-ideals in hyper BCK-algebras.
In this paper, using the concept of a multipolar intuitionistic fuzzy set of finite degree, we introduce the notions of the k-polar intuitionistic fuzzy hyper BCK-ideal (briefly, k-pIF hBCK-ideal), the k-polar intuitionistic fuzzy weak hyper BCK-ideal (briefly, k-pIF weak hBCK-ideal), the k-polar intuitionistic fuzzy s-weak hyper BCK-ideal (briefly, k-pIF s-weak hBCK-ideal), the k-polar intuitionistic fuzzy strong hyper BCK-ideal (briefly, k-pIF strong hBCK-ideal) and the k-polar intuitionistic fuzzy reflexive hyper BCK-ideal (briefly, k-pIF reflexive hBCK-ideal). We investigate several properties and their relations. We discuss k-pIF (weak, s-weak, strong, reflexive) the hBCK-ideal in relation to k-polar upper and lower level sets.
2. Preliminaries
Let be a nonempty set endowed with a hyperoperation “∘”. For two subsets D and C of , denote by the set . We shall use instead of , , or .
By a
hyper BCK-algebra (briefly, hBCK-algebra), we mean a nonempty set
endowed with a hyperoperation “∘” and a constant 0 satisfying the following axioms (see [
17]):
- (HK1)
;
- (HK2)
;
- (HK3)
;
- (HK4)
and imply ,
for all , where is defined by and for every is defined by , such that . In such case, we call “≪” the hyperorder in .
Note that the condition (HK3) is equivalent to the condition:
A subset D of a hBCK-algebra is called
A
hyper BCK-ideal (briefly, hBCK-ideal) of
(see [
17]) if
A
weak hyper BCK-ideal (briefly, weak hBCK-ideal) of
(see [
17]) if it satisfies (
2) and
A
strong hyper BCK-ideal (briefly, strong hBCK-ideal) of
(see [
18]) if it satisfies (
2) and
A
reflexive hyper BCK-ideal (briefly, reflexive hBCK-ideal) of
(see [
18]) if it is a hBCK-ideal of
which satisfies:
Every hBCK-algebra
satisfies the following assertions.
For any subsets
C,
A and
B of a hBCK-algebra
, the following assertions are valid.
For any family
of real numbers, we define
If , we will also use and instead of and , respectively.
A
multipolar intuitionistic fuzzy set of finite degree k (briefly,
k-pIF set) over a universe
is a mapping
where
and
are
k-pF sets over a universe
such that
for all
, that is,
for all
and
.
Given a
k-pIF set
over a universe
, we consider the sets
where
and
with
, that is,
and
which is called a
k-polar upper (resp.,
lower) level set of
. It is clear that
and
where
3. k-Polar Intuitionistic Fuzzy Hyper BCK-Ideals
Unless otherwise stated, shall represent a hyper BCK-algebra.
Definition 1. A k-pIF set on is called a k-polar intuitionistic fuzzy hyper BCK-ideal (briefly, k-pIF hBCK-ideal) of if it satisfiesthat is, and for all with , and for all and .
Example 1. Let be a set with the hyperoperation “∘”,
which is given by Table 1. Then is a hBCK-algebra (see [17]). Let be a 4-polar intuitionistic fuzzy set over given bywhere m and n are natural numbers. It is routine to verify that is a 4-polar intuitionistic fuzzy hBCK-ideal of . Proposition 1. Let be a k-pIF hBCK-ideal of . Then
- (i)
, that is, and for all and ,
- (ii)
If satisfies the condition that is, for every there exist such that for .
Proof. (i) Since
for all
, it follows from (
18) that
and
for all
.
(ii) Assume that
satisfies the condition (
21). For any
, there exists
such that
and
. It follows from (
20) that
and
for
. which proves (ii). □
Theorem 1. Let be a k-pIF set over . If is a k-pIF hBCK-ideal of , then the k-polar upper level set and the k-polar lower level set are hBCK-ideals of for all with .
Proof. Let
with
. Assume that
is a
k-pIF hBCK-ideal of
. It is clear that
and
by Proposition 1(i). Let
be such that
,
,
and
. Then
,
,
and
for all
. It follows that
and
which imply that
and
for all
and
. Hence
and
, and so
for all
. Thus
and
. Therefore
and
are hBCK-ideals of
.
We need the following lemma for considering the converse of Theorem 1.
Lemma 1 ([
22]).
Let D be a subset of . If K is a hBCK-ideal of such that , then D is contained in K. Theorem 2. Let be a k-pIF set over in which the k-polar upper level set and the k-polar lower level set are hBCK-ideals of for all with . Then is a k-pIF hBCK-ideal of .
Proof. Assume that the
k-polar upper level set
and the
k-polar lower level set
are hBCK-ideals of
for all
with
. Let
be such that
,
,
and
. Then
and
, and so
and
. It follows from Lemma 1 that
and
, i.e.,
and
. Hence
and
. For any
, let
and
. Then
,
and
for all
and
, i.e.,
and
. Thus
and
which imply from (
12) that
and
. Since
and
are hBCK-ideal of
, we have
and
which imply that
and
. Therefore
is a
k-pIF hBCK-ideal of
. □
Definition 2. A k-pIF set over is called a
k-polar intuitionistic fuzzy weak hBCK-ideal (briefly, k-pIF weak hBCK-ideal) of if it satisfies Proposition 1(i) and (19). k-polar intuitionistic fuzzy s-weak hBCK-ideal (briefly, k-pIF s-weak hBCK-ideal) of if it satisfies Proposition 1(i) and (22). k-polar intuitionistic fuzzy strong hBCK-ideal (briefly, k-pIF strong hBCK-ideal) of if it satisfies for all and .
Example 2. Let be a set with the hyperoperation “∘”
which is given by Table 2. Then is a hBCK-algebra (see [17]). Let be a 4-polar intuitionistic fuzzy set over given bywhere n is a natural number and . It is routine to check that is a 4-polar intuitionistic fuzzy strong hBCK-ideal of . We describe the relation between k-pIF weak hBCK-ideal and k-pIF s-weak hBCK-ideal in the following theorem.
Theorem 3. In a hBCK-algebra, every k-pIF s-weak hBCK-ideal is a k-pIF weak hBCK-ideal.
Proof. Let
be a
k-pIF
s-weak hBCK-ideal of
and let
. Then there exist
and
such that
and
by (
22). Since
and
, it follows that
Therefore is a k-pIF weak hBCK-ideal of . □
We consider a condition for a k-pIF weak hBCK-ideal to be a k-pIF s-weak hBCK-ideal.
Theorem 4. Let be a k-pIF weak hBCK-ideal of which satisfies the condition (21). Then is a k-pIF s-weak hBCK-ideal of . Proof. For any
, there exist
and
such that
and
; that is,
and
by (
21). It follows that
Therefore is a k-pIF s-weak hBCK-ideal of . □
Proposition 2. Every k-pIF strong hBCK-ideal of satisfies the following assertions.
- (i)
; that is, and for all and ,
- (ii)
; that is, and for all with and .
- (iii)
.
Proof. (i) Since
for all
, we get
for all
.
(ii) Let
be such that
. Then
and thus
and
for
. It follows from (
23) and (i) that
and
for
, that is,
and
for all
with
.
(iii) Let
be such that
. Then
and
for
. Hence
and
for all
with
. □
Corollary 1. If is a k-pIF strong hBCK-ideal of , thenfor all . Corollary 2. Every k-pIF strong hBCK-ideal is a k-pIF hBCK-ideal and a k-pIF s-weak hBCK-ideal (and hence a k-pIF weak hBCK-ideal).
In general, a
k-pIF (weak) hBCK-ideal may not be a
k-pIF strong hBCK-ideal. In fact, the 4-polar intuitionistic fuzzy hBCK-ideal
of
in Example 1 is not a 4-polar intuitionistic fuzzy strong hBCK-ideal of
since
and/or
It is clear that every k-pIF hBCK-ideal of is a k-pIF weak hBCK-ideal of . However, the converse is not true in general, as seen in the following example.
Example 3. Let be a hBCK-algebra as in Example 1. Let be a 3-polar intuitionistic fuzzy set over given bywhere n is a natural number. Then is a 3-polar, intuitionistic, fuzzy weak hBCK-ideal of . Note that , and/or ; that is, and/or . Hence is not a 3-polar intuitionistic fuzzy hBCK-ideal of . We have a characterization of a k-pIF weak hBCK-ideal in the similar way to the proofs of Theorems 1 and 2.
Theorem 5. Given a k-pIF set over , the following are equivalent.
- (i)
is a k-pIF weak hBCK-ideal of .
- (ii)
The k-polar upper level set and the k-polar lower level set are weak hBCK-ideals of for all with .
Theorem 6. Let be a k-pIF set over . If is a k-pIF strong hBCK-ideal of , then the k-polar upper level set and the k-polar lower level set are strong hBCK-ideals of for all with .
Proof. Assume that
is a
k-pIF strong hBCK-ideal of
and let
with
be such that
and
are nonempty. Then there exists
and
, and so
and
; that is,
and
for all
. It is clear that
and
by Proposition 2(1). Let
be such that
,
,
and
. Then there exist
and and
. Hence
and
, i.e.,
and
, for
. It follows that
and
for all
. Hence
and
. Therefore
and
are strong hBCK-ideals of
. □
Theorem 7. Let be a k-pIF set over which satisfies the condition If the k-polar upper level set and the k-polar lower level set are strong hBCK-ideals of for all with , then is a k-pIF strong hBCK-ideal of .
Proof. Assume that the
k-polar upper level set
and the
k-polar lower level set
are strong hBCK-ideals of
for all
with
. Then
and
for some
, and so
and
. Using Lemma 1, we get
and
. Hence for every
and
, we get
and
. Thus
and
for all
. It follows that
for
. For any
, put
and
, that is,
and
for
. Then
and
are strong hBCK-ideals of
by hypothesis. The condition (
24) implies that there exists
and
such that
and
, i.e.,
and
for
. Hence
and
for
, which imply that
and
. Hence
and
, and thus
and
. It follows that
and
for
. Therefore
is a
k-pIF strong hBCK-ideal of
. □
Definition 3. A k-pIF set over is called a k-pIF reflexive hBCK-ideal of if it satisfies:that is, , , andfor all and . Theorem 8. Every k-pIF reflexive hBCK-ideal is a k-pIF strong hBCK-ideal.
Proof. Straightforward. □
Theorem 9. If is a k-pIF reflexive hBCK-ideal of , then the k-polar upper level set and the k-polar lower level set are reflexive hBCK-ideals of for all with .
Proof. Assume that
is a
k-pIF reflexive hBCK-ideal of
. Then
is a
k-pIF strong hBCK-ideal of
by Theorem 8, and so
is a
k-pIF hBCK-ideal of
. It follows from Theorem 1 that the
k-polar upper level set
and the
k-polar lower level set
are hBCK-ideals of
for all
with
. Let
with
be such that
and
are nonempty. Then
and
for some
. For any
, let
and
. The condition (
25) implies that
and
, that is,
and
. Thus
and
for all
, and therefore
and
are reflexive hBCK-ideals of
. □
Lemma 2 ([
18]).
Every reflexive hBCK-ideal is a strong hBCK-ideal. We need additional conditions to induce the converse of Theorem 9.
Theorem 10. Let be a k-pIF set over which satisfies the condition (24). If the k-polar upper level set and the k-polar lower level set are reflexive hBCK-ideals of for all with , then is a k-pIF reflexive hBCK-ideal of . Proof. Assume that the
k-polar upper level set
and the
k-polar lower level set
are reflexive hBCK-ideals of
for all
with
. Then
and
are strong hBCK-ideals of
by Lemma 2. Using Theorem 7, we know that
is a
k-pIF strong hBCK-ideal of
and so (
26) is valid. For any
, let
and
for
. Since
and
are reflexive hBCK-ideals of
, we get
and
. Hence
for all
and
for all
. Thus
and
which imply that
for all
. Therefore
is a
k-pIF reflexive hBCK-ideal of
. □
Theorem 11. Let be a k-pIF strong hBCK-ideal of which satisfies the condition (24). Then is a k-pF reflexive hBCK-ideal of if and only if and for all and . Proof. Assume that
is a
k-pIF strong hBCK-ideal of
which satisfies the condition (
24). The necessity is clear. Assume that
and
for all
and
. Since
is a
k-pF hBCK-ideal of
by Corollary 2, we have
and
for all
and
. It follows that
and
for all
and
. For any
and
, let
and
. The condition (
24) implies that there exists
such that
and so
, i.e.,
, and there exists
such that
and so
, i.e.,
. Hence
and
. Since
and
are strong hBCK-ideals of
by Theorem 6, it follows that
and
. Hence
and
. Therefore
is a
k-pIF reflexive hBCK-ideal of
. □
4. Conclusions
In 2020, Kang (together with Song and Jun) introduced the concept of a k-polar intuitionistic fuzzy set in a universe, and applied it to BCK/BCI-algebras. In this article, we have applied the k-polar intuitionistic fuzzy set to hyper BCK-algebra. We have introduced the notions of the k-pIF hBCK-ideal, the k-pIF weak hBCK-ideal, the k-pIF s-weak hBCK-ideal, the k-pIF strong hBCK-ideal and the k-pIF reflexive hBCK-ideal, and have investigated related properties and their relations. We have discussed k-pIF (weak, s-weak, strong, reflexive) hBCK-ideals in relation to k-polar upper and lower level sets. In the future work, we will use the idea and results in this paper to study other hyper algebraic structures, for example, hyper hoop, hyper BCI-algebra, hyper equality algebra and hyper MV-algebra.