# Hesitant Anti-Fuzzy Soft Set in BCK-Algebras

Authors: Halimah Alshehri; Noura Alshehri
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

We introduce the notions of hesitant anti-fuzzy soft set (subalgebras and ideals) and provide relation between them. However, we study new types of hesitant anti-fuzzy soft ideals (implicative, positive implicative, and commutative). Also, we stated and proved some theorems which determine the relationship between these notions.


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## Body

## 1. Introduction

In the real world, there are many complicated problems in economic science, engineering, environment, social science, and management science. They are characterized by uncertainty, imprecision, and vagueness. We cannot successfully utilize the classical methods to deal with these problems because there are various types of uncertainties involved in these problems. Moreover, although there are many theories, such as theory of probability, theory of fuzzy sets, theory of interval mathematics, and theory of rough sets, to be considered as mathematical tools to deal with uncertainties, Molodtsov [1] pointed out that all these theories had their own limitations. Also, in order to overcome these difficulties, Molodtsov [1] firstly proposed a new mathematical tool named soft set theory to deal with uncertainty and imprecision. This theory has been demonstrated to be a useful tool in many applications such as decision-making, measurement theory, and game theory.The soft set model can be combined with other mathematical models. Maji et al. [2] firstly presented the concept of fuzzy soft set by combining the theories of fuzzy set and soft set together. The hesitant fuzzy set, as one of the extensions of Zadeh’s [3] (1965) fuzzy set, allows the membership degree of an element to a set presented by several possible values, and it can express the hesitant information more comprehensively than other extensions of fuzzy set. In [4], Torra introduced the concept of hesitant fuzzy set and Babitha and John (2013) [5] defined another important soft set, hesitant fuzzy soft set. They introduced basic operations such as intersection, union, and compliment, and De Morgan’s law was proven. In 2014, Jun et al. [6] applied the notion of hesitant fuzzy soft sets to subalgebras and ideals in BCK/BCI-algebras. In this paper, in Section 3, we introduce the concepts of hesitant anti-fuzzy soft set of subalgebra. In Section 4, we define the hesitant anti-fuzzy soft ideal in BCK-algebras and give some basic relations. In Section 5, we discuss notion of hesitant anti-fuzzy soft implicative ideals and provide some properties. In Section 6, we investigate concept of hesitant anti-fuzzy soft positive implicative ideals and give some relations. In Section 7, we introduce the notion of hesitant anti-fuzzy soft commutative ideals in BCK-algebras and related properties are investigated. Finally, conclusions are presented in the last section.


## 2. Preliminaries

An algebra(X;∗,0) of type (2, 0) is said to be a BCK-algebra if it satisfies the axioms: for all x,y,z∈X,(BCK-1)((x∗y)∗(x∗z))∗(z∗y)=0,(BCK-2)(x∗(x∗y))∗y=0,(BCK-3)x∗x=0,(BCK-4)0∗x=0,(BCK-5)x∗y=0 and y∗x=0 imply that x=y.Define a binary relation≤ on X by letting x≤y if and only if x∗y=0.Then(X;≤) is a partially ordered set with the least element 0. In any BCK-algebra X, the following hold:(1)
x∗y∗z=x∗z∗y.(2)
x∗y≤x.(3)
x∗0=x.(4)
x∗z∗y∗z≤x∗y.(5)
x∗x∗x∗y=x∗y.(6)
x≤y implies that x∗z≤y∗z and z∗y≤z∗x, for all x,y,z∈X.A nonempty subsetI of X is called a subalgebra of X if, for any x,y∈I, x∗y∈I. That is, it is closed under the binary operation ∗ of X.A nonempty subsetI of X is called an ideal of X if (I1) 0∈I;(I2)x∗y∈I and y∈I imply that x∈I.A nonempty subsetI of X is called an implicative ideal if it satisfies (I1) and (I3)x∈I whenever (x∗(y∗x))∗z∈I and z∈I.It is called a commutative ideal if it satisfies(I1) and (I4)x∗(y∗(y∗x))∈I whenever (x∗y)∗z∈I and z∈I; and it is called a positive implicative ideal if it satisfies (I1) and (I5)x∗z∈Iwhenever (x∗y)∗z∈I and y∗z∈I.A BCK-algebraX is said to be implicative if it satisfies ∀x,y∈X:x=x∗(y∗x).A BCK-algebraX is said to be positive implicative if it satisfies ∀x,y,z∈X:(x∗z)∗(y∗z)=(x∗y)∗z.A BCK-algebraX is said to be commutative if it satisfies ∀x,y∈X:x∗(x∗y)=y∗(y∗x).Definition 1 (see [3]).
LetS be a set. A fuzzy set in S is a function μ:S→[0,1]Definition 2 (see [2]).
LetU be an initial universe set and let E be a set of parameters. Let F(U) denote the set of all fuzzy sets in U. Then (F~,A) is called a fuzzy soft set over U, where A⊆E and F~ is a mapping given by F~:A→F(U).Definition 3 (see [4, 7]).
LetE be a reference set. A hesitant fuzzy set on E is defined in terms of a function that when applied to E returns a subset of [0,1] which can be viewed as the following mathematical representation:(1)HE≔e,hEe:e∈E,where hE:E→p([0,1]).Definition 4 (see [7]).
Given a nonempty subsetA of X, a hesitant fuzzy set HX≔{(x;hX(x):x∈X} on X satisfying the condition(2)hXx=ϕ;∀x∉Ais called a hesitant fuzzy set related to A (briefly, A-hesitant fuzzy set) on X and is represented by HA≔{(x,hA(x)):x∈X}; hA is a mapping from X to p([0,1]) with hA(x)=ϕ, for all x∉A.Proposition 5 (see [8]).
LetHA≔{x,hAx∣x∈X} be an A-hesitant anti-fuzzy ideal of X. Then the following hold: for all x,y,z∈A,(a)
ifx≤y, then hA(x)⊆hA(y), which means that hA preserves the order,(b)
ifx∗y≤z, then hA(x)⊆hA(y)∪hA(z).Proposition 6 (see [9]).
In a BCK-algebraX, the following hold: for all x,y,z∈X,(i)
((x∗z)∗z)∗(y∗z)≤(x∗y)∗z,(ii)
(x∗z)∗(x∗(x∗z))=(x∗z)∗z,(iii)
(x∗(y∗(y∗x)))∗(y∗(x∗(y∗(y∗x))))≤x∗y.Definition 7 (see [5, 6]).
Denote byHF(U) the set of all hesitant fuzzy sets. A pair (H~,A) is called a hesitant fuzzy soft set over a reference set U, where H~ is a mapping given by H~:A→HF(U).


## 3. Hesitant Anti-Fuzzy Soft Subalgebras

Definition 8.
Given a nonempty subset (subalgebra as much as possible)A of X, let HA≔{(x,hA(x))∣x∈X} be an A-hesitant fuzzy set on X. Then HA≔{(x,hA(x))∣x∈X} is called a hesitant anti-fuzzy subalgebra of X related to A (briefly, A-hesitant anti-fuzzy subalgebra of X) if it satisfies the following condition:(3)hAx∗y⊆hAx∪hAy∀x,y∈A.
AnA-hesitant anti-fuzzy subalgebra of X with A=X is called a hesitant anti-fuzzy subalgebra of X.Definition 9.
For a subsetA of E, a hesitant fuzzy soft set (H~,A) over X is called a hesitant anti-fuzzy soft subalgebra based on e∈A (briefly, e-hesitant anti-fuzzy soft subalgebra) over X if the hesitant fuzzy set,(4)H~e≔x,hH~ex∣x∈X,on X is a hesitant anti-fuzzy subalgebra of X. If (H~,A) is an e-hesitant anti-fuzzy soft subalgebra over X, for all e∈A, we say that (H~,A) is a hesitant anti-fuzzy soft subalgebra.Example 10.
LetX={0,a,b,c} be a BCK-algebra in Table 1 (Cayley).
Consider a set of parametersE≔{e1,e2,e3,e4}. Let (H~,A) be a hesitant fuzzy soft set over X, where A={e1,e2,e3}, which is given in Table 2. It is routine to verify that H~[e1],H~[e2] and H~[e3] are hesitant anti-fuzzy subalgebra over X based on parameters e1,e2, and e3. Therefore (H~,A) is a hesitant anti-fuzzy soft subalgebra over X.Table 1
∗
0
a
b
c
0
0
0
0
0
a
a
0
a
0
b
b
b
0
0
c
c
b
a
0Table 2
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
0
a
b
c
e
1
(
0.2,0.3
)
[
0.2,0.8
]
[
0.2,0.8
]
[
0.2,0.8
]
e
2
{
0.3
}
{
0.3
}
[
0.1,0
,
2
)
∪
(
0,2
,
0.8
]
[
0.1,0.9
]
e
3
{
0.4
}
[
0.3,0.7
]
[
0.3,0.5
]
[
0.3,0.7
]Proposition 11.
If(H~,A) is a hesitant anti-fuzzy soft subalgebra over X, then(5)hH~e0⊆hH~ex,∀x∈X,where e is any parameter in A.Proof.
For anyx∈X and e∈A, we have(6)hH~e0=hH~ex∗x⊆hH~ex∪hH~ex=hH~ex.This completes the proof.Theorem 12.
Let(H~,A) be a hesitant anti-fuzzy soft subalgebra over X. If B⊆A, then (H~B,B) is a hesitant anti-fuzzy soft subalgebra over X.Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft subalgebra over X. Then H~A is a hesitant anti-fuzzy subalgebra of X, ∀x∈A. Since B⊆A, hB(b) is a hesitant anti-fuzzy subalgebra of X, ∀b∈B. Hence, (H~B,B) is a hesitant anti-fuzzy soft subalgebra over X.The following example shows that there exists a hesitant fuzzy soft set(H~,A) over X such that(i)
(H~,A) is not a hesitant anti-fuzzy soft subalgebra over X,(ii)
there exists a subsetB of A such that (H~B,B) is a hesitant anti-fuzzy soft subalgebra over X.Example 13.
LetX={0,a,b,c} be a BCK-algebra in Table 3 (Cayley).
Consider a set of parametersA={e1,e2,e3}. Let (H~,A) be a hesitant fuzzy soft set over X which is described in Table 4.
ThenH~[e3]≔{(x,hH~[e3](x))∣x∈X} on X is not hesitant anti-fuzzy subalgebra of X, because (7)hH~e3a∗a=hH~e30=0.1⊈0.1,0.2=hH~e3a∪hH~e3a.
Therefore(H~,A) is not hesitant anti-fuzzy soft subalgebra of X. But if we take B={e1,e2}, then (H~B,B) is a hesitant anti-fuzzy soft subalgebra over X.Table 3
∗
0
a
b
c
0
0
0
0
0
a
a
0
a
0
b
b
b
0
0
c
c
c
c
0Table 4
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
0
a
b
c
e
1
(
0.3,0.5
)
(
0.3,0.8
)
[
0.3,0.8
]
[
0.3,0.5
]
e
2
[
0.3,0.6
)
[
0.3,0.9
)
[
0.3,0.8
)
[
0.3,0.6
]
e
3
{
0.1
}
(
0.1,0.2
)
[
0.1,0.3
]
[
0.1,0.4
]Definition 14.
Let(H~,A) be a hesitant fuzzy soft set over X. For each ε∈P([0,1]), the set (H~,A)ε=(H~ϵ,A) is called a hesitant anti-ε-level soft set of (H~,A), where H~ϵ={x∈X:hH~[e](x)⊆ε} for all e∈A.Theorem 15.
Let(H~,A) be a hesitant fuzzy soft set over X.(H~,A) is a hesitant anti-fuzzy soft subalgebra of X if and only if (H~,A)ε is a subalgebra over X for each ε∈P([0,1]).Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft subalgebra of X. For each ε∈P([0,1]),e∈A and x1,x2∈H~ϵ, hH~[e](x1)⊆ε and hH~[e](x2)⊆ε. Thus(8)hH~ex1∗x2⊆hH~ex1∪hH~ex⊆ε.
This implies thatx1∗x2∈H~ϵ. Hence, (H~,A)ε is a subalgebra over X.
Conversely, assume that(H~,A)ε is a subalgebra over X for each ε∈P([0,1]). For each e∈A and x,y∈X, let hH~[e](x)=εx and let hH~[e](y)=εy. Take ε=εx∪εy. Then x,y∈H~ϵ and so x∗y∈H~ϵ. Hence,(9)hH~ex∗y⊆ε=εx∪εy=hH~ex∪hH~ey.
Therefore(H~,A) is a hesitant anti-fuzzy subalgebra over X. Then, by Definition 8, we conclude that(H~,A) is a hesitant anti-fuzzy soft subalgebra of X. This completes the proof.


## 4. Hesitant Anti-Fuzzy Soft Ideals

Definition 16.
Given a nonempty subset (subalgebra as much as possible)A of X, let HA≔{(x,hA(x))∣x∈X} be an A-hesitant fuzzy set on X. Then HA≔{(x,hA(x))∣x∈X} is called a hesitant anti-fuzzy ideal of X related to A (briefly, A-hesitant anti-fuzzy ideal of X) if it satisfies the following conditions:(HAFI1)hA(0)⊆hA(x).(HAFI2)hA(x)⊆hA(x∗y)∪hA(y) for all x,y∈A.
AnA-hesitant anti-fuzzy ideal of X with A=X is called a hesitant anti-fuzzy ideal of X.Definition 17.
Let(H~,A) be a hesitant fuzzy soft set over X, where A is a subset of E. Given e∈A,(H~,A) is called a hesitant anti-fuzzy soft ideal based on e (briefly, e-hesitant anti-fuzzy soft ideal) over X, if the hesitant fuzzy set,(10)H~e≔x,hH~ex∣x∈X,on X is a hesitant anti-fuzzy ideal of X. If (H~,A) is an e-hesitant anti-fuzzy soft ideal over X for all e∈A, we say that (H~,A) is a hesitant anti-fuzzy soft ideal over X.Example 18.
LetX ≔ apple, banana, carrot, peach, radish be a reference set, and consider a soft machine ◊ which produces the following products:(11)apple◊x=appleifx∈carrot,peach,radishcarrotifx∈apple,bananabanana◊y=bananaify∈apple,carrot,peach,radishcarrotify=bananapeach◊z=peachifz∈carrot,radishcarrotifz∈apple,banana,peachradish◊u=radishifu∈carrot,peachcarrotifu∈apple,banana,radishcarrot◊v=carrot∀v∈X.
ThenX is a BCK-algebra under the soft machine ◊. Consider a set of parameters A≔Cat, Cow, Dog, Horse; let (H~,A) be a hesitant fuzzy soft set over X which is described in Table 5.
It is routine to verify that(H~,A) is a hesitant anti-fuzzy soft ideal over X based on parameters “cat,” “cow,” and “dog.” But (H~,A) is not a hesitant anti-fuzzy soft ideal of X based on parameter “horse” because (12)H~horsepeach=0.3,0.9⊈0.3,0.4=H~horsepeach◊banana∪H~horsebanana.Table 5
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
Apple
Banana
Carrot
Peach
Radish
Cat
(
0.2,0.3
)
[
0.2,0.5
]
[
0.2,0.5
]
[
0.2,0.5
]
[
0.2,0.5
]
Cow
[
0.3,0.4
)
[
0.3,0.4
)
[
0.3,0.4
]
[
0.3,0.4
)
[
0.3,0.4
)
Dog
[
0.1,0.3
]
[
0.1,0.7
)
[
0.1,0.8
)
[
0.1,0.7
)
[
0.1,0.7
)
Horse
{
0.3
}
[
0.3,0.4
)
[
0.3,0.4
)
[
0.3,0.9
)
[
0.3,0.6
)Theorem 19.
Let(H~,A) be a hesitant fuzzy soft set over X.(H~,A) is a hesitant anti-fuzzy soft ideal of X if and only if (H~,A)ε is an ideal over X for each ε∈P([0,1]).Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft ideal of X. For each ε∈P([0,1]),e∈A and x,y∈H~ϵ such that x∗y∈H~ϵ and y∈H~ϵ; then hH~[e](x∗y)⊆ε and hH~[e](y)⊆ε. Thus,(13)hH~e0⊆hH~ex⊆hH~ex∗y∪hH~ey⊆ε.
Hence,0∈H~ϵ and x∈H~ϵ, and this implies that (H~,A)ε is an ideal over X.
Conversely, assume that(H~,A)ε is an ideal over X for each ε∈P([0,1]). For each e∈A and x,y∈X, let hH~[e](x)=εx, and then x∈H~ϵ. Since (H~,A)ε is an ideal over X, we have 0∈H~ϵ and so hH~[e](0)⊆εx=hH~[e](x). Let hH~[e](x∗y)=εx∗y and let hH~[e](y)=εy. If we take ε=εx∗y∪εy, then x∗y∈H~ϵ and y∈H~ϵ which imply that x∈H~ϵ. Thus,(14)hH~ex⊆ε=εx∗y∪εy=hH~ex∗y∪hH~ey.
Therefore(H~,A) is a hesitant anti-fuzzy soft ideal of X.Proposition 20.
Every hesitant anti-fuzzy soft ideal(H~,A) over a BCK/BCI-algebra X satisfies the following condition for all e∈A and x,y,z∈X:(a)
Ifx≤y, then hH~[e](x)⊆hH~[e](y).(b)
hH~[e](x∗y)⊆hH~[e](x∗z)∪hH~[e](z∗y).(c)
Ifx∗y≤z, then hH~[e](x)⊆hH~[e](y)∪hH~[e](z).(d)
IfhH~[e](x∗y)=hH~[e](0), then hH~[e](x)⊆hH~[e](y).(e)
hH~[e](0∗(0∗x))⊆hH~[e](x).Proof.
Lete∈A and x,y,z∈X.
(a) Ifx≤y, then x∗y=0. Since (H~,A) is a hesitant anti-fuzzy soft ideal of X,(15)hH~ex⊆hH~ex∗y∪hH~ey=hH~e0∪hH~ey=hH~ey.
(b) Sincex∗y∗x∗z≤z∗y, it follows from (a) that hH~[e]((x∗y)∗(x∗z))⊆hH~[e](z∗y). Hence (16)hH~ex∗y⊆hH~ex∗y∗x∗z∪hH~ex∗z⊆hH~ex∗z∪hH~ez∗y,
(c) Ifx∗y≤z, then x∗y∗z=0. Since (17)hH~ez=hH~e0∪hH~ez=hH~ex∗y∗z∪hH~ez⊇hH~ex∗y,it follows that hH~[e](x)⊆hH~[e](x∗y)∪hH~[e](y)⊆hH~[e](y)∪hH~[e](z).
(d) IfhH~[e](x∗y)=hH~[e](0), then we have (18)hH~ex⊆hH~ex∗y∪hH~ey=hH~e0∪hH~ey=hH~ey.
(e) For allx∈X, (19)hH~e0∗0∗x⊆hH~e0∗0∗x∗x∪hH~ex=hH~e0∪hH~ex=hH~ex.
The proof is complete.Theorem 21.
Let(H~,A) be a hesitant fuzzy soft set over X which satisfies condition (5) and (c) from Proposition 20. Then (H~,A) is a hesitant anti-fuzzy soft ideal over X.Proof.
Lete∈A. Since x∗(x∗y)≤y, for all x,y∈X, it follows from Proposition 20(c) that(20)hH~ex⊆hH~ex∗y∪hH~ey.
Hence,(H~,A) is a hesitant anti-fuzzy soft ideal over X.Theorem 22.
Every hesitant anti-fuzzy soft ideal (based on a parameter) over BCK-algebraX is a hesitant anti-fuzzy soft subalgebra (based on the same parameter) over X.Proof.
For anye∈A, assume that (H~,A) is a hesitant anti-fuzzy soft ideal over X. Then (21)hH~ex∗y⊆hH~ex∗y∗x∪hH~ex=hH~ex∗x∗y∪hH~ex=hH~e0∗y∪hH~ex=hH~e0∪hH~ex⊆hH~ex∪hH~ey,for all x,y∈X, and so (H~,A) is a hesitant anti-fuzzy soft subalgebra over X.The following example shows that the converse of Theorem22 is not true in general.Example 23.
LetX={0,a,b,c} in Example 13 and B=e1,e2, and then (H~,B) is a hesitant anti-fuzzy soft subalgebra over X. But it is not a hesitant anti-fuzzy soft ideal over X based on parameters “e1” and “e2” since (22)hH~e1a=0.3,0.8⊈0.3,0.5=0.3,0.5∪0.3,0.5=hH~e10∪hH~e1c=hH~e1a∗c∪hH~e1c.
Also,(23)hH~e2a=0.3,0.9⊈0.3,0.6=hH~e2a∗c∪hH~e2c.
We provide a condition for a hesitant anti-fuzzy soft subalgebra overX to be a hesitant anti-fuzzy soft ideal over X.Theorem 24.
Let(H~,A) be a hesitant anti-fuzzy soft subalgebra over X. If the inequality x∗y≤z holds in X, then (H~,A) is a hesitant anti-fuzzy soft ideal over X.Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft subalgebra over X. Then, from Proposition 11, we have(24)hH~e0⊆hH~ex,∀x∈X,e∈A.
Assume thatx∗y≤z holds in X. Then, by Proposition 20(c), we get(25)hH~ex⊆hH~ey∪hH~ez∀x,y,z∈X.
Sincex∗(x∗y)≤y,(26)hH~ex⊆hH~ex∗y∪hH~ey∀x,y∈X.
Hence,(H~,A) is a hesitant anti-fuzzy soft ideal over X.


## 5. Hesitant Anti-Fuzzy Soft Implicative Ideals

Definition 25.
A hesitant fuzzy setHX≔{(x,hX(x))∣x∈X} in a BCK-algebra X is called a hesitant anti-fuzzy implicative ideal (briefly, HAFII) of X if it satisfies the following conditions:(HAFII1)hX(0)⊆hX(x), for all x∈X.(HAFII2)hX(x)⊆hX((x∗(y∗x))∗z)∪hX(z), for all x,y,z∈X.Definition 26.
Let(H~,A) be a hesitant fuzzy soft set over X, where A is a subset of E. Given e∈A,(H~,A) is called a hesitant anti-fuzzy soft implicative ideal based on e (briefly, e-hesitant anti-fuzzy soft implicative ideal) over X if the hesitant fuzzy set,(27)H~e≔x,hH~ex∣x∈X,on X is a hesitant anti-fuzzy implicative ideal of X. If (H~,A) is an e-hesitant anti-fuzzy soft implicative ideal over X, for all e∈A, we say that (H~,A) is a hesitant anti-fuzzy soft implicative ideal over X.Proposition 27.
Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebraX is order-preserving.Proof.
Let(H~,A) be a hesitant anti-fuzzy soft implicative ideal over X. Let e∈A and x,y∈X be such that x≤y; then(28)hH~ex⊆hH~ex∗z∗x∗y∪hH~ey=hH~ex∗y∗z∗x∪hH~ey=hH~e0∗z∗x∪hH~ey=hH~e0∪hH~ey=hH~ey.
Hence,hH~[e](x)⊆hH~[e](y), and this completes the proof.Proposition 28.
Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebraX is a hesitant anti-fuzzy soft ideal of X.Proof.
Let(H~,A) be a hesitant anti-fuzzy soft implicative ideal over X. Let e∈A and x,y,z∈X; then(29)hH~ex⊆hH~ex∗y∗x∗z∪hH~ez.
Replacey=x, and using x∗x=0 we get (30)hH~ex⊆hH~ex∗x∗x∗z∪hH~ez=hH~ex∗z∪hH~ez.
For allx,z∈X, (H~,A) is a hesitant anti-fuzzy soft ideal.Combining Proposition28 and Theorem 22 yields the following result.Corollary 29.
Every hesitant anti-fuzzy soft implicative ideal of a BCK-algebraX is a hesitant anti-fuzzy soft subalgebra of X.The following example shows that the converse of Proposition28 may not be true in general.Example 30.
LetX={0,a,b,c} be a BCK-algebra in Table 6 (Cayley).
Consider a set of parametersA={e1,e2}. Let (H~,A) be a hesitant fuzzy soft set over X which is described in Table 7.
Then(H~,A) is a hesitant anti-fuzzy soft ideal (subalgebra) of X. But H~[e1]≔{(x,hH~[e1](x))∣x∈X} and H~[e2]≔{(x,hH~[e2](x))∣x∈X} on X is not hesitant anti-fuzzy soft implicative ideal of X, because (31)hH~e1a=0,12⊈hH~e1a∗b∗a∗0∪hH~e10=hH~e10∪hH~e10=14,12.
Also,(32)hH~e2a=0,12⊈hH~e2a∗b∗a∗0∪hH~e20=hH~e20∪hH~e20=0,14.
Now we give a condition for a hesitant anti-fuzzy soft ideal overX to be a hesitant anti-fuzzy soft implicative ideal over X.Table 6
∗
0
a
b
c
0
0
0
0
0
a
a
0
0
a
b
b
a
0
b
c
c
c
c
0Table 7
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
0
a
b
c
e
1
1
4
,
1
2
0
,
1
2
0
,
1
2
0
,
3
4
e
2
0
,
1
4
0
,
1
2
0
,
1
2
0
,
3
4Theorem 31.
IfX is an implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal over X is a hesitant anti-fuzzy soft implicative ideal over X.Proof.
LetX be an implicative BCK-algebra; it follows that x=x∗(y∗x),∀x,y∈X. And let (H~,A) be hesitant anti-fuzzy soft ideal over X. Then, for any e∈A, we have (33)hH~ex⊆hH~ex∗z∪hH~ez=hH~ex∗y∗x∗z∪hH~ezfor all x,y,z∈X. Hence, it is a hesitant anti-fuzzy implicative ideal of X. That is, (H~,A) is hesitant anti-fuzzy soft implicative ideal of X.Theorem 32.
Let(H~,A) be a hesitant fuzzy soft set in a BCK-algebra X. Then (H~,A) is a hesitant anti-fuzzy soft implicative ideal of X if and only if for each ε∈P([0,1]) the hesitant anti-fuzzy soft ε-level set (H~,A)ε of (H~,A) is empty or an implicative ideal of X.Proof.
Let(H~,A) be a hesitant anti-fuzzy soft implicative ideal of X and assume that H~ε≠⌀ for every ε∈P([0,1]). Note that hH~[e](0)⊆hH~[e](x), for all e∈A and x∈X. In particular, for each x∈H~ε,hH~[e](0)⊆hH~[e](x)⊆ε and so 0∈H~ε. Let (x∗(y∗x))∗z∈H~ε and z∈H~ε, and since (H~,A) is a hesitant anti-fuzzy implicative ideal over X, it follows that(34)hH~ex⊆hH~ex∗y∗x∗z∪hH~ez⊆ε.
Hencex∈H~ε, and H~ε is an implicative ideal of X.
Conversely, letH~ε be an implicative ideal of X; we first show hH~[e](0)⊆hH~[e](x), for all e∈A and x∈X. If not, then there exists x0∈X such that hH~[e](0)⊇hH~[e](x0). Let ε0=1/2{hH~[e](0)+hH~[e](x0)}; then 0≤hH~[e](x0)⊂ε0⊂hH~[e](0)≤1. Thus, x0∈H~ε0 and H~ε0≠⌀. As H~ε0 is an implicative ideal of X,0∈H~ε0, which implies that hH~[e](0)⊆ε0. This is a contradiction and so hH~[e](0)⊆hH~[e](x) for all e∈A and x∈X. Now assume that hH~[e](x)⊆hH~[e]((x∗(y∗x))∗z)∪hH~[e](z) does not hold. Then, hH~[e](x0)⊇hH~[e]((x0∗(y0∗x0))∗z0)∪hH~[e](z0) for some x0,y0,z0∈X. If we take(35)ε0=12hH~ex0+hH~ex0∗y0∗x0∗z0∪hH~ez0,then 0≤hH~[e]((x0∗(y0∗x0))∗z0)∪hH~[e](z0)⊆ε0⊆hH~[e](x0)≤1. It follows that hH~[e]((x0∗(y0∗x0))∗z0)⊆ε0 and hH~[e](z0)⊆ε0, which imply that (x0∗(y0∗x0))∗z0∈H~ε0 and z0∈H~ε0. But H~ε0 is an implicative ideal of X, so x0∈H~ε0 or hH~[e](x0)⊆ε0, and this is a contradiction. Therefore (H~,A) is a hesitant anti-fuzzy soft implicative ideal of X.Theorem 33.
Let(H~,A) be a hesitant anti-fuzzy soft ideal of a BCK-algebra X. Then (H~,A) is a hesitant anti-fuzzy soft implicative ideal of X if and only if it satisfies the condition(36)hH~ex⊆hH~ex∗y∗x
for allx,y∈X and e∈A.Proof.
Assume that(H~,A) is a hesitant anti-fuzzy soft implicative ideal of X. Take z=0 in (37)hH~ex⊆hH~ex∗y∗x∗z∪hH~ez=hH~ex∗y∗x∗0∪hH~e0=hH~ex∗y∗x.
Conversely, suppose that(H~,A) satisfies the condition. As (H~,A) is a hesitant anti-fuzzy soft ideal of X, we have(38)hH~ex⊆hH~ex∗y∗x⊆hH~ex∗y∗x∗z∪hH~ez.
Then(H~,A) is a hesitant anti-fuzzy soft implicative ideal of X and the proof is completed.Now we give characterizations of hesitant anti-fuzzy soft implicative ideals.Theorem 34.
Let(H~,A) be a hesitant anti-fuzzy soft ideal of a BCK-algebra X. Then the following are equivalent:(i)
(H~,A) is a hesitant anti-fuzzy soft implicative ideal of X.(ii)
hH~[e](x)⊆hH~[e](x∗(y∗x)) for all x,y∈X and e∈A.(iii)
hH~[e](x)=hH~[e](x∗(y∗x)) for all x,y∈X and e∈A.Proof.
(
i
)
⇒
(
i
i
) Let (H~,A) be a hesitant anti-fuzzy soft implicative ideal of X. Then by Theorem 33 we have (39)hH~ex⊆hH~ex∗y∗x∗0∪hH~e0=hH~ex∗y∗x∪hH~e0=hH~ex∗y∗x,for all x,y∈X. Hence condition (ii) holds.
(
i
i
)
⇒
(
i
i
i
) Observe that, in BCK-algebra X,x∗(y∗x)≤x. Applying Proposition 20(a), we obtain hH~[e](x∗(y∗x))⊆hH~[e](x). Since (ii) holds, hH~[e](x)=hH~[e](x∗(y∗x)) for all x,y∈X and e∈A. So, condition (iii) holds.
(iii) ⇒ (i) Suppose that condition (iii) holds. Since(H~,A) is a hesitant anti-fuzzy soft ideal of X, by Definition 16, we have(40)hH~ex∗y∗x⊆hH~ex∗y∗x∗z∪hH~ez.
Combining (iii), we obtain(41)hH~ex⊆hH~ex∗y∗x∗z∪hH~ez.
Thus(H~,A) is a hesitant anti-fuzzy soft implicative ideal of X. So, condition (i) holds.Theorem 35.
Let(H~,A) be a hesitant anti-fuzzy soft implicative ideal of a BCK-algebra X; then the set (42)HhH~e≔x∈X∣hH~e0=hH~exis an implicative ideal of X.Proof.
Clearly,0∈HhH~[e]. Let x,y,z∈HhH~[e] be such that (x∗(y∗x))∗z∈HhH~[e] and z∈HhH~[e], for all e∈A; then(43)hH~ex∗y∗x∗z=hH~e0=hH~ez.
Since(H~,A) is a hesitant anti-fuzzy soft implicative ideal of X, it follows that (44)hH~ex⊆hH~ex∗y∗x∗z∪hH~ez=hH~e0∪hH~e0=hH~e0.
Combining Definition25(HAFII1), we get hH~[e](x)=hH~[e](0) and hence x∈HhH~[e]. Therefore HhH~[e] is an implicative ideal of X.Theorem 36.
A hesitant anti-fuzzy soft subalgebrahH~[e] of X is a hesitant anti-fuzzy soft implicative ideal if and only if it satisfies the condition x∗(y∗x)∗z≤u implying that hH~[e](x)⊆hH~[e](z)∪hH~[e](u) for all x,y,z,u∈X and ∀e∈A.Proof.
Assume thathH~[e] is a hesitant anti-fuzzy soft implicative ideal of X and let x,y,z,u∈X be such that x∗(y∗x)∗z≤u. Since hH~[e] is also hesitant anti-fuzzy soft ideal of X, by Proposition 28, it follows from Proposition 20(c) that(45)hH~ex∗y∗x⊆hH~ez∪hH~eu.
Making use of Theorem34, we obtain hH~[e](x)⊆hH~[e](z)∪hH~[e](u).
Conversely, suppose thathH~[e] satisfies the condition in theorem. Obviously, hH~[e] satisfies (5) from Proposition 11. Since (x∗(y∗x))∗((x∗(y∗x))∗z)≤z, it follows from Proposition 20(c) and Theorem 34 that(46)hH~ex⊆hH~ex∗y∗x∗z∪hH~ez,which shows that hH~[e] satisfies conditions of a hesitant anti-fuzzy implicative ideal of X and so hH~[e] is a hesitant anti-fuzzy soft implicative ideal of X. The proof is complete.


## 6. Hesitant Anti-Fuzzy Soft Positive Implicative Ideals

Definition 37.
A hesitant fuzzy setHX≔{(x,hX(x))∣x∈X} in a BCK-algebra X is called a hesitant anti-fuzzy positive implicative ideal (briefly, HAFPII) of X if it satisfies the following conditions:(HAFPII1)hX(0)⊆hX(x), for all x∈X.(HAFPII2)hX(x∗z)⊆hX((x∗y)∗z)∪hX(y∗z), for all x,y,z∈X.Definition 38.
Let(H~,A) be a hesitant fuzzy soft set over X, where A is a subset of E. Given e∈A,(H~,A) is called a hesitant anti-fuzzy soft positive implicative ideal based on e (briefly, e-hesitant anti-fuzzy soft positive implicative ideal) over X if the hesitant fuzzy set,(47)H~e≔x,hH~ex∣x∈X,on X is a hesitant anti-fuzzy positive implicative ideal of X. If (H~,A) is an e-hesitant anti-fuzzy soft positive implicative ideal over X, for all e∈A, we say that (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal over X.Proposition 39.
Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebraX is order-preserving.Proof.
Letx,y∈X and e∈A be such that x≤y. Since (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X,(48)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z=hH~e0∗z∪hH~ey∗z=hH~e0∪hH~ey∗z=hH~ey∗z.
Puttingz=0,(49)hH~ex⊆hH~ey.
The proof is complete.Proposition 40.
Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebraX is a hesitant anti-fuzzy soft ideal of X.Proof.
Let(H~,A) be a hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra X, so for all x,y,z∈X and e∈A we have(50)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z.
Puttingz=0,(51)hH~ex⊆hH~ex∗y∪hH~ey.
Therefore(H~,A) is a hesitant anti-fuzzy soft ideal of X.Combining Theorem21 and Proposition 39 yields the following result.Proposition 41.
Every hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebraX is a hesitant anti-fuzzy soft subalgebra of X.Remark 42.
A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebraX may not be a hesitant anti-fuzzy soft positive implicative ideal of X as shown in the following example.Example 43.
LetX be the BCK-algebra in Example 30. Routine calculations give that (H~,A) is a hesitant anti-fuzzy soft ideal (subalgebra) of X, but it is not a hesitant anti-fuzzy soft positive implicative ideal of X, because (52)hH~e1b∗a=hH~e1a=0,12⊈hH~e1b∗a∗a∪hH~e1a∗a=hH~e10∪hH~e10=14,12.
The converse of Proposition40 is valid if the following condition holds.Proposition 44.
IfX is a positive implicative BCK-algebra, then every hesitant anti-fuzzy soft ideal of X is a hesitant anti-fuzzy soft positive implicative ideal of X.Proof.
Assume that(H~,A) is a hesitant anti-fuzzy soft ideal of a positive implicative BCK-algebra X, for all x,y∈X and e∈A, and then(53)hH~ex⊆hH~ex∗y∪hH~ey.
By replacingx by x∗z and y by y∗z, we get(54)hH~ex∗z⊆hH~ex∗z∗y∗z∪hH~ey∗z.
SinceX is a positive implicative BCK-algebra, (x∗z)∗(y∗z)=(x∗y)∗z for all x,y,z∈X. Hence,(55)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z.
This shows that(H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X.Theorem 45.
Let(H~,A) be a hesitant anti-fuzzy soft ideal over X; then (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X if and only if(56)hH~ex∗z∗y∗z⊆hH~ex∗y∗z,
for alle∈A and x,y,z∈X.Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft ideal over X and(57)hH~ex∗z∗y∗z⊆hH~ex∗y∗z,for all e∈A and x,y,z∈X. (58)hH~ex∗z⊆hH~ex∗z∗y∗z∪hH~ey∗z⊆hH~ex∗y∗z∪hH~ey∗z.
Thus(H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X.
Conversely, assume that(H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X implying that (H~,A) is a hesitant anti-fuzzy soft ideal of X by Proposition 40. Let a=x∗(y∗z) and let b=x∗y; since(59)x∗y∗z∗x∗y≤y∗y∗z,using Proposition 20(a), we have(60)hH~ea∗b∗z=hH~ex∗y∗z∗x∗y∗z⊆hH~ey∗y∗z∗z=hH~e0.
And so(61)hH~ex∗z∗y∗z=hH~ex∗y∗z∗z=hH~ea∗z⊆hH~ea∗b∗z∪hH~eb∗z⊆hH~e0∪hH~eb∗z=hH~eb∗z=hH~ex∗y∗z.
ThereforehH~[e]((x∗z)∗(y∗z))⊆hH~[e]((x∗y)∗z), for all e∈A and x,y,z∈X.Theorem 46.
Let(H~,A) be a hesitant anti-fuzzy soft positive implicative ideal of X. Then, for all x,y,z,a,b∈X and e∈A,(62)x∗y∗z∗a≤b⟹hH~ex∗z∗y∗z⊆hH~ea∪hH~eb.Proof.
Letx,y,z,a,b∈X be such that ((x∗y)∗z)∗a≤b. Since (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X, it follows from Theorem 45 and Proposition 20(c) that(63)hH~ex∗z∗y∗z⊆hH~ex∗y∗z⊆hH~ea∪hH~eb.
This completes the proof.Theorem 47.
Let(H~,A) be a hesitant anti-fuzzy soft positive implicative ideal of a BCK-algebra X, for all e∈A. Then the set(64)HhH~e≔x∈X∣hH~e0=hH~exis a positive implicative ideal of X.Proof.
Lete∈A. Clearly, 0∈HhH~[e]. Let x,y,z∈HhH~[e] be such that (x∗y)∗z∈HhH~[e] and y∗z∈HhH~[e], and then(65)hH~ex∗y∗z=hH~e0=hH~ey∗z.
It follows that(66)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z=hH~e0∪hH~e0=hH~e0,since (H~,A) is a hesitant anti-fuzzy positive implicative ideal of X, and then hH~[e](x∗z)=hH~[e](0) and hence x∗z∈HhH~[e].Therefore HhH~[e]is a positive implicative ideal of X.Theorem 48.
Let(H~,A) be a hesitant anti-fuzzy soft ideal over X; then (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X if and only if it satisfies the inequalities(67)hH~ex∗y⊆hH~ex∗y∗y∀x,y∈X,e∈A,for all e∈A and x,y,z∈X.Proof.
Suppose that the hesitant anti-fuzzy soft ideal(H~,A) of a BCK-algebra X is a hesitant anti-fuzzy soft filter positive implicative ideal of X. So(68)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z∀x,y,z∈X,e∈A.
Substitutingz=y, we have (69)hH~ex∗y⊆hH~ex∗y∗y∪hH~ey∗y=hH~ex∗y∗y∪hH~e0=hH~ex∗y∗y,for all x,y∈X and e∈A.
Conversely, suppose that(H~,A) is a hesitant anti-fuzzy soft ideal over X and satisfies inequality (67).(70)hH~ex∗y⊆hH~ex∗y∗y∀x,y∈X,e∈A,since hH~[e](0)⊆hH~[e](x). Now we can prove that hH~[e](x∗z)⊆hH~[e]((x∗y)∗z)∪hH~[e](y∗z) for all x,y,z∈X and e∈A. In contrast, there exist x`,y`∈X such that (71)hH~ex`∗y`⊇hH~ex`∗y`∗y`∪hH~ey`∗y`=hH~ex`∗y`∗y`∪hH~e0=hH~ex`∗y`∗y`,which is a contradiction. Therefore hH~[e](x∗z)⊆hH~[e]((x∗y)∗z)∪hH~[e](y∗z) for all x,y,z∈X and e∈A. Thus (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X.Now we give the condition that makes equality in Theorem48 hold.Theorem 49.
IfX is positive implicative, then a hesitant anti-fuzzy soft ideal of X is a hesitant anti-fuzzy soft positive implicative ideal of X if and only if it satisfies(72)hH~ex∗y=hH~ex∗y∗y∀x,y∈X,e∈A.Proof.
Suppose that the hesitant anti-fuzzy soft ideal(H~,A) of a BCK-algebra X is a hesitant anti-fuzzy soft positive implicative ideal of X. So by Theorem 48 we have(73)hH~ex∗y⊆hH~ex∗y∗y∀x,y∈X,e∈A.
On the other hand, since(x∗y)∗y≤x∗y, it follows from Proposition 20(a) that hH~[e]((x∗y)∗y)⊆hH~[e](x∗y). Thus we have(74)hH~ex∗y=hH~ex∗y∗y∀x,y∈X,e∈A.
Conversely, assume thathH~[e](x∗y)=hH~[e]((x∗y)∗y) for all x,y∈X and e∈A. We want to prove that a hesitant anti-fuzzy soft ideal (H~,A) of a BCK-algebra X is a hesitant anti-fuzzy soft positive implicative ideal of X. It is clear that hH~[e](0)⊆hH~[e](x) for all x∈X and e∈A. Since (H~,A) is a hesitant anti-fuzzy soft ideal and X is positive implicative,(75)hH~ex∗z=hH~ex∗z∗z⊆hH~ex∗z∗y∗z∪hH~ey∗z=hH~ex∗y∗z∪hH~ey∗z,for all x,y,z∈X and e∈A. Therefore a hesitant anti-fuzzy soft ideal (H~,A) of X is a hesitant anti-fuzzy soft positive implicative ideal of X.Theorem 50.
Let(H~,A) be a hesitant anti-fuzzy soft positive implicative ideal of X if and only if for each ε∈P([0,1]) the hesitant anti-fuzzy soft ε-level set (H~,A)ε of (H~,A) is a positive implicative ideal of X.Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X and for every ε∈P([0,1]) define the sets(76)H~,Aε≔x∈X∣hH~ex⊆ε∀e∈A.Since (H~,A)ε≠⌀, let x∈(H~,A)ε⇒hH~[e](x)⊆ε. By Definition 37, we have hH~[e](0)⊆hH~[e](x), for all e∈A and x∈X, implying that 0∈(H~,A)ε. Let x,y,z∈X be such that (x∗y)∗z∈(H~,A)ε and y∗z∈(H~,A)ε, implying that hH~[e]((x∗y)∗z)⊆ε and hH~[e](y∗z)⊆ε. Since (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal over X, it follows that(77)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z⊆ε.
Hencex∗z∈(H~,A)ε, and (H~,A)ε is a positive implicative ideal of X.
Conversely, suppose that(H~,A)ε is a positive implicative ideal of X for all ε∈P([0,1]). Put hH~[e](x)=ε for any x∈X. Hence 0∈(H~,A)ε⇒hH~[e](0)⊆ε=hH~[e](x), for all e∈A and x∈X. Now we prove that hH~[e](x∗z)⊆hH~[e]((x∗y)∗z)∪hH~[e](y∗z). In contrast, there exists x0,y0,z0∈X such that hH~[e](x0∗z0)⊇hH~[e]((x0∗y0)∗z0)∪hH~[e](y0∗z0). Taking(78)ε0=12hH~ex0∗z0+hH~ex0∗y0∗z0∪hH~ey0∗z0,it follows that hH~[e]((x0∗y0)∗z0)⊆ε0 and hH~[e](y0∗z0)⊆ε0, which imply that (x0∗y0)∗z0∈(H~,A)ε0 and y0∗z0∈(H~,A)ε0. But (H~,A)ε0 is a positive implicative ideal of X, and thus x0∗z0∈(H~,A)ε0 or hH~[e](x0∗z0)⊆ε0, and this is a contradiction. Therefore(79)hH~ex∗z⊆hH~ex∗y∗z∪hH~ey∗z∀x,y,z∈X,e∈A.Hence (H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X.


## 7. Hesitant Anti-Fuzzy Soft Commutative Ideals

Definition 51.
A hesitant fuzzy setHX≔{(x,hX(x))∣x∈X} in a BCK-algebra X is called a hesitant anti-fuzzy commutative ideal (briefly, HAFCI) of X if it satisfies the following conditions:(HAFCI1)hX(0)⊆hX(x), for all x∈X.(HAFCI2)hX(x∗(y∗(y∗x)))⊆hX((x∗y)∗z)∪hX(z), for all x,y,z∈X.Definition 52.
Let(H~,A) be a hesitant fuzzy soft set over X, where A is a subset of E. Given e∈A,(H~,A) is called a hesitant anti-fuzzy soft commutative ideal based on e (briefly, e-hesitant anti-fuzzy soft commutative ideal) over X if the hesitant fuzzy set,(80)H~e≔x,hH~ex∣x∈X,on X is a hesitant anti-fuzzy commutative ideal of X. If (H~,A) is an e-hesitant anti-fuzzy soft commutative ideal over X, for all e∈A, we say that (H~,A) is a hesitant anti-fuzzy soft commutative ideal over X.Proposition 53.
Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebraX is order-preserving.Proof.
Letx,y∈X and e∈A be such that x≤y. Since (H~,A) is a hesitant anti-fuzzy soft commutative ideal of X,(81)hH~ex∗y∗y∗x⊆hH~ex∗y∗z∪hH~ez=hH~e0∗z∪hH~ez=hH~e0∪hH~ez=hH~ez.
Puttingy=0 and replacing z=y, we obtain(82)hH~ex⊆hH~ey.
The proof is complete.Theorem 54.
Any hesitant anti-fuzzy soft commutative ideal of BCK-algebraX is hesitant anti-fuzzy soft ideal of X.Proof.
Letx,y,z∈X and let e∈A. And let (H~,A) be a hesitant anti-fuzzy soft commutative ideal of a BCK-algebra X, so we have (83)hH~ex=hH~ex∗0∗0∗x⊆hH~ex∗0∗z∪hH~ez=hH~ex∗z∪hXz,for all x,z∈X and e∈A. Hence (H~,A) is a hesitant anti-fuzzy soft ideal of X.Combining Theorems22 and 54 yields the following result.Corollary 55.
Every hesitant anti-fuzzy soft commutative ideal of a BCK-algebraX is a hesitant anti-fuzzy soft subalgebra of X.Remark 56.
A hesitant anti-fuzzy soft ideal (subalgebra) of a BCK-algebraX may not be a hesitant anti-fuzzy soft commutative ideal of X as shown in the following example.Example 57.
LetX=0,a,b be the BCK-algebra with in Table 8 (Cayley).
Consider a set of parametersA={e1,e2}. Let (H~,A) be a hesitant fuzzy soft set over X which is described in Table 9.
Then(H~,A) is a hesitant anti-fuzzy soft ideal (subalgebra) of X. Routine calculations give that (H~,A) is a hesitant anti-fuzzy soft ideal (subalgebra) of X, but H~[e1]≔{(x,hH~[e1](x))∣x∈X} and H~[e2]≔{(x,hH~[e2](x))∣x∈X} on X is not hesitant anti-fuzzy soft commutative ideal of X, because (84)hH~e1a∗b∗b∗a=0.2,0.7⊈0.2,0.4=hH~e1a∗b∗0∪hH~e10.
Also,(85)hH~e2a∗b∗b∗a=0,0.5⊈0,0.2=hH~e2a∗b∗0∪hH~e20
The condition in the following theorem makes converse Theorem54 valid.Table 8
∗
0
a
b
0
0
0
0
a
a
0
0
b
b
b
0Table 9
Tabular representation of the hesitant fuzzy soft set(H~,A).
H
~
0
a
b
e
1
(
0.2,0.4
)
[
0.2,0.7
)
[
0,1
)
e
2
(
0,0.2
)
(
0,0.5
)
[
0,0.7
)Theorem 58.
In a commutative BCK-algebraX. Every hesitant anti-fuzzy soft ideal of X is a hesitant anti-fuzzy soft commutative ideal of X.Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft ideal of a BCK-algebra X. It is sufficient to show that (H~,A) satisfies condition (HAFCI2). Let x,y,z∈X. Then(86)x∗y∗y∗x∗x∗y∗z∗z=x∗y∗y∗x∗z∗x∗y∗z≤x∗y∗y∗x∗x∗y=x∗x∗y∗y∗y∗x=0;that is,(87)x∗y∗y∗x∗x∗y∗z≤z.It follows from Proposition 20(c) that(88)hH~ex∗y∗y∗x⊆hH~ex∗y∗z∪hH~ez,for all x,y,z∈X and e∈A. Thus (H~,A) is a hesitant anti-fuzzy soft commutative ideal of a BCK-algebra X, and the proof is complete.Theorem 59.
Let(H~,A) be a hesitant anti-fuzzy soft ideal of a BCK-algebra X. Then (H~,A) is a hesitant anti-fuzzy soft commutative ideal of X if and only if it satisfies the condition(89)hH~ex∗y∗y∗x⊆hH~ex∗y∀x,y∈X,e∈A.Proof.
Letx,y,z∈X and let e∈A. And suppose that (H~,A) is a hesitant anti-fuzzy soft commutative ideal of X. Taking z=0, we have (90)hH~ex∗y∗y∗x⊆hH~ex∗y∗0∪hH~e0=hH~ex∗y.
Conversely, assume that(H~,A) satisfies condition (89) as (H~,A) is a hesitant anti-fuzzy soft ideal of a BCK-algebra X. Hence(91)hH~ex∗y⊆hH~ex∗y∗z∪hH~ez∀x,y,z∈X,e∈A.
Combining (91) and (89), we obtain(92)hH~ex∗y∗y∗x⊆hH~ex∗y∗z∪hH~ez,for all x,y,z∈X and e∈A. Hence (H~,A) is hesitant anti-fuzzy soft commutative ideal of X, which completes the proof.Observingx∗y≤x∗(y∗(y∗x)) and using Proposition 20(a), we have hH~[e](x∗y)⊆hH~[e](x∗(y∗(y∗x))) for all x,y∈X and e∈A. Hence, Theorem 59 can be improved as follows.Theorem 60.
A hesitant anti-fuzzy soft ideal(H~,A) of a BCK-algebra X is a hesitant anti-fuzzy soft commutative ideal of X if and only if it satisfies the identity(93)hH~ex∗y=hH~ex∗y∗y∗x∀x,y∈X,e∈A.Now we describe a relation between hesitant anti-fuzzy soft implicative ideals, hesitant anti-fuzzy soft commutative ideals, and hesitant anti-fuzzy soft positive implicative ideals.Theorem 61.
A hesitant anti-fuzzy soft ideal(H~,A) of a BCK-algebra X is a hesitant anti-fuzzy soft implicative ideal if and only if (H~,A) is both hesitant anti-fuzzy soft commutative ideal and hesitant anti-fuzzy soft positive implicative ideal.Proof.
Assume that(H~,A) is a hesitant anti-fuzzy soft implicative ideal. By Propositions 6(i) and 20(c), for all x,y,z∈X and e∈A, we have (94)hH~ex∗y∗z∪hH~ey∗z⊇hH~ex∗z∗z=hH~ex∗z∗x∗x∗z,byProposition6ii=hH~ex∗z,byTheorem34iii.
Hence(H~,A) is a hesitant anti-fuzzy soft positive implicative ideal of X.
Also, By Propositions20 and 6(iii) and Theorem 34(iii), we have(95)hH~ex∗y∗y∗x=hH~ex∗y∗y∗x∗y∗x∗y∗y∗x⊆hH~ex∗y.
It follows from Theorem59 that (H~,A) is hesitant anti-fuzzy soft commutative ideal of X.
Conversely, suppose that(H~,A) is both hesitant anti-fuzzy soft positive implicative ideal of X and hesitant anti-fuzzy soft commutative ideal of X. Since (y∗(y∗x))∗(y∗x)≤x∗(y∗x), it follows from Proposition 20(a) that(96)hH~ey∗y∗x∗y∗x⊆hH~ex∗y∗x
Using Theorem49, we have hH~[e]((y∗(y∗x))∗(y∗x))=hH~[e](y∗(y∗x)) and so(97)hH~ey∗y∗x⊆hH~ex∗y∗x.
On the other hand,x∗y≤x∗(y∗x) implies that hH~[e](x∗y)⊆hH~[e](x∗(y∗x)). Since (H~,A) is a hesitant anti-fuzzy soft commutative ideal of X, By Theorem 60, we have(98)hH~ex∗y=hH~ex∗y∗y∗x.
Hence,(99)hXx∗y∗y∗x⊆hXx∗y∗x.
Combining (97), we obtain(100)hH~ex∗y∗x⊇hH~ex∗y∗y∗x∪hH~ey∗y∗x⊇hH~ex.
So,(H~,A) is a hesitant anti-fuzzy soft implicative ideal of X by Theorem 34. The proof is complete.Theorem 62.
Let(H~,A) be hesitant anti-fuzzy soft commutative ideal of X if and only if for each ε∈P([0,1]) the hesitant anti-fuzzy soft ε-level set (H~,A)ε of (H~,A) is a commutative ideal of X.Proof.
Suppose that(H~,A) is a hesitant anti-fuzzy soft commutative ideal of X and for every ε∈P([0,1]) define the sets(101)H~,Aε≔x∈X∣hH~ex⊆ε∀e∈A.Since (H~,A)ε≠⌀, let x∈(H~,A)ε⇒hH~[e](x)⊆ε. By definition, we have hH~[e](0)⊆hH~[e](x), for all e∈A and x∈X, implying that 0∈(H~,A)ε. Let x,y,z∈X be such that (x∗y)∗z∈(H~,A)ε and z∈(H~,A)ε, implying that hH~[e]((x∗y)∗z)⊆ε and hH~[e](z)⊆ε. Since (H~,A) is a hesitant anti-fuzzy soft commutative ideal over X, it follows that(102)hH~ex∗y∗y∗x⊆hH~ex∗y∗z∪hH~ez⊆ε.
Namely,x∗(y∗(y∗x))∈(H~,A)ε, and (H~,A)ε is a commutative ideal of X.
Conversely, suppose that(H~,A)ε is a commutative ideal of X for all ε∈P([0,1]). Put hH~[e](x)=ε for any x∈X and e∈A. Hence 0∈(H~,A)ε⇒hH~[e](0)⊆ε=hH~[e](x), for all e∈A and x∈X. Now we prove that hH~[e](x∗(y∗(y∗x)))⊆hH~[e]((x∗y)∗z)∪hH~[e](z). In contrast, there exists x0,y0,z0∈X such that(103)hH~ex0∗y0∗y0∗x0⊇hH~ex0∗y0∗z0∪hH~ez0.Taking(104)ε0=12hH~ex0∗y0∗y0∗x0+hH~ex0∗y0∗z0∪hH~ez0,it follows that hH~[e]((x0∗y0)∗z0)⊆ε0 and hH~[e](z0)⊆ε0, which imply that ((x0∗y0)∗z0)∈(H~,A)ε0 and z0∈(H~,A)ε0. But (H~,A)ε0 is a commutative ideal of X, and thus x0∗(y0∗(y0∗x0))∈(H~,A)ε0 or hH~[e](x0∗(y0∗(y0∗x0)))⊆ε0, and this is a contradiction. Therefore (H~,A) is a hesitant anti-fuzzy soft commutative ideal of X.


## 8. Conclusions

When people make a decision, they are usually hesitant and irresolute for one thing or another which makes it difficult to reach a final agreement. So, we can see that hesitant fuzzy set is a very useful tool to deal with uncertainty. Thus, it is very necessary to develop some concepts about hesitant fuzzy set. Therefore, we introduce the hesitant anti-fuzzy soft set for basic notions in BCK-algebras. In this paper, we introduced the notions of hesitant anti-fuzzy soft subalgebras and hesitant anti-fuzzy soft ideals of BCK-algebras and investigated their relations and properties. However, we defined the notions of hesitant anti-fuzzy soft ideals (implicative, positive implicative, and commutative) in BCK-algebras. Also, we presented the relationship between them and we gave conditions for a hesitant anti-fuzzy soft ideal to be a hesitant anti-fuzzy soft (implicative, positive implicative, and commutative) ideal in BCK-algebras.



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