Hesitant fuzzy translations and extensions of subalgebras and ideals in BCK / BCI -algebras

Abstract

The notions of hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set on BCK/BCI-algebras are introduced, and related properties are investigated. We prove that every hesitant fuzzy translation of a hesitant fuzzy subalgebra (ideal) is a hesitant fuzzy subalgebra (ideal). Conditions for a hesitant fuzzy set to be a hesitant fuzzy subalgebra (ideal) are provided. We show that if a hesitant fuzzy set is a hesitant fuzzy subalgebra (ideal), then its support is a subalgebra (ideal), and also prove that if the support of a hesitant fuzzy set is a subalgebra (ideal), then its hesitant fuzzy translation is a hesitant fuzzy subalgebra (ideal).

Keywords

Hesitant fuzzy subalgebra hesitant fuzzy ideal hesitant fuzzy translation hesitant fuzzy extension support

1 Introduction

As a useful generalization of the fuzzy set, Torra [7] introduced the hesitant fuzzy set that is designed for situations in which it is difficult to determine the membership of an element to a set owing to ambiguity between a few different values. The hesitant fuzzy set permits the membership degree of an element to a set to be represented by a set of possible values between 0 and 1 (see [7] and [8]). The hesitant fuzzy set therefore provides a more accurate representation of hesitancy in stating their preferences over objects than the fuzzy set or its classical extensions. Hesitant fuzzy set theory has been applied to several practical problems, primarily in the area of decision making (see [6 , 8–13]). Jun and Song applied the notion of hesitant fuzzy sets to MTL-algebras and EQ-algebras (see [3] and [4]).

In this paper, we introduce the notions of hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set on BCK/BCI-algebras, and investigate their related properties. We prove that every hesitant fuzzy translation of a hesitant fuzzy subalgebra (ideal) is a hesitant fuzzy subalgebra (ideal), and provide conditions for a hesitant fuzzy set to be a hesitant fuzzy subalgebra (ideal). We show that if a hesitant fuzzy set is a hesitant fuzzy subalgebra (ideal), then its support is a subalgebra (ideal). We also prove that if the support of a hesitant fuzzy set is a subalgebra (ideal), then its hesitant fuzzy translation is a hesitant fuzzy subalgebra (ideal).

2 Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki, and it was extensively investigated by several researchers.

An algebra 𝒳 : = (X ; * , 0) of type (2, 0) is called a BCI-algebra if it satisfies the following conditions:

(∀ x, y, z ∈ X) (((x * y) * (x * z)) * (z * y) =0) ,

(∀ x, y ∈ X) ((x * (x * y)) * y = 0) ,

(∀ x ∈ X) (x * x = 0) ,

(∀ x, y ∈ X) (x * y = 0, y * x = 0 ⇒x = y) .

If a BCI-algebra 𝒳 satisfies the following identity:

(∀ x ∈ X) (0 * x = 0) ,

then 𝒳 is called a BCK-algebra. Any BCK/BCI-algebra 𝒳 satisfies the following axioms:

(∀ x ∈ X) (x * 0 = x) ,

(∀ x, y, z ∈ X) (x ≤ y ⇒ x * z ≤ y * z, z * y ≤ z * x) ,

(∀ x, y, z ∈ X) ((x * y) * z = (x * z) * y) ,

(∀ x, y, z ∈ X) ((x * z) * (y * z) ≤ x * y)

where x ≤ y if and only if x * y = 0 . A nonempty subset S of a BCK/BCI-algebra 𝒳 is called a subalgebra of 𝒳 if x * y ∈ S for all x, y ∈ S . A subset A of a BCK/BCI-algebra 𝒳 is called an ideal of 𝒳 if it satisfies:

0 ∈ A .

(∀ x ∈ X) (∀ y ∈ A) (x * y ∈ A ⇒ x ∈ A) .

We refer to the books [1, 5] for further information regarding on BCK/BCI-algebras.

Torra [7] defined hesitant fuzzy sets in terms of a function that returns a set of membership values for each element in the domain. This will be formally defined in the following definition.

Definition 2.1. [7] Let X be a reference set. Then we define a hesitant fuzzy set on X in terms of a function ℋ that when applied to X returns a subset of [0, 1] .

For a hesitant fuzzy set ℋ on 𝒳 and x, y, z ∈ X, we use the notations _ℋx : = ℋ (x), $ℋ_{x}^{y} : = ℋ (x) \cap ℋ (y)$ and $ℋ_{x}^{y} [z] : = ℋ (x) \cap ℋ (y) \cap ℋ (z)$ . Also, _ℋx (λ) : = ℋ (x) ∪ λ and $ℋ_{x}^{y} (λ) : = (ℋ (x) \cap ℋ (y)) \cup λ$ where λ ∈ 𝒫 ([0, 1]). It is clear that $ℋ_{x}^{y} = ℋ_{y}^{x}$ , $ℋ_{x} = ℋ_{y} \Leftrightarrow ℋ_{x} \subseteq ℋ_{y}, ℋ_{y} \subseteq ℋ_{x},$ and $ℋ_{x}^{y} (λ) = ℋ_{x} (λ) \cap ℋ_{y} (λ) = ℋ_{y}^{x} (λ)$ for allx, y ∈ X.

3 Hesitant fuzzy translations

In what follows, let 𝒳 denote a BCK/BCI-algebra unless otherwise specified, and we take X as a reference set.

Definition 3.1. [2] A hesitant fuzzy set ℋ on 𝒳 is called a hesitant fuzzy subalgebra of 𝒳 if it satisfies the following condition: $(\forall x, y \in X) (ℋ_{x}^{y} \subseteq ℋ_{x * y}) .$ (1)

Definition 3.2. [2] A hesitant fuzzy set ℋ on 𝒳 is called a hesitant fuzzy ideal of 𝒳 if it satisfies the following condition: $(\forall x \in X) (ℋ_{x} \subseteq ℋ_{0}),$ (2) $(\forall x, y \in X) (ℋ_{x * y}^{y} \subseteq ℋ_{x}) .$ (3)

For any hesitant fuzzy set ℋ on 𝒳, we consider the set $Σ_{ℋ} : = [0, 1] ∖ \underset{x \in X}{\cup} ℋ_{x} .$ (4)

It is clear that Σ_ℋ and _ℋx are disjoint for all x ∈ X.

Definition 3.3. For a hesitant fuzzy set ℋ on 𝒳 and λ ⊆ Σ_ℋ, a hesitant fuzzy set $ℋ_{λ}^{t}$ on 𝒳 which is given as follows: $ℋ_{λ}^{t} : X \to 𝒫 ([0, 1]), x \mapsto ℋ_{x} (λ)$ is called the hesitant fuzzy translation of ℋ with respect to λ (briefly, λ-hesitant fuzzy translation of ℋ).

Lemma 3.4. If ℋ is a hesitant fuzzy subalgebra of 𝒳, then $ℋ_{x * y} (λ) \supseteq ℋ_{x}^{y} (λ) and ℋ_{x}^{x} (λ) = ℋ_{x} (λ)$ for all x, y ∈ X and λ ∈ 𝒫 ([0, 1]).

Proof. Straightforward. □

Lemma 3.5. If ℋ is a hesitant fuzzy ideal of 𝒳, then $ℋ_{0}^{x} (λ) = ℋ_{x} (λ)$ for all x ∈ X and λ ∈ 𝒫 ([0, 1]).

Proof. Straightforward. □

Proposition 3.6. For any λ ⊆ Σ_ℋ, the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of a hesitant fuzzy subalgebra ℋ of 𝒳 satisfies: $(\forall x \in L) (ℋ_{λ}^{t} (0) \supseteq ℋ_{λ}^{t} (x)) .$ (5)

If ℋ is a hesitant fuzzy ideal of 𝒳, then the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ satisfies: ∀x, y, z ∈ X, $(x \leq y \Rightarrow ℋ_{λ}^{t} (x) \supseteq ℋ_{λ}^{t} (y)),$ (6) $(x * y \leq z \Rightarrow ℋ_{λ}^{t} (x) \supseteq ℋ_{λ}^{t} (y) \cap ℋ_{λ}^{t} (z)) .$ (7)

Proof. Let ℋ be a hesitant fuzzy subalgebra of 𝒳. For any x ∈ X, we have $\begin{matrix} ℋ_{λ}^{t} (0) = ℋ_{0} (λ) = ℋ_{x * x} (λ) \supseteq ℋ_{x}^{x} (λ) \\ = ℋ_{x} (λ) = ℋ_{λ}^{t} (x) \end{matrix}$

Assume that ℋ is a hesitant fuzzy ideal of X over U. Let x and y be elements of X such that x ≤ y. Then x * y = 0, and so $\begin{matrix} ℋ_{λ}^{t} (y) = ℋ_{y} (λ) = ℋ_{0}^{y} (λ) = ℋ_{x * y}^{y} (λ) \\ \subseteq ℋ_{x} (λ) = ℋ_{λ}^{t} (x) . \end{matrix}$

Now let x, y and z be elements of X such that x * y ≤ z. Then $ℋ_{λ}^{t} (x * y) \supseteq ℋ_{λ}^{t} (z)$ by (3.6). It follows from (3.3) that $\begin{matrix} ℋ_{λ}^{t} (x) & = & ℋ_{x} (λ) \supseteq ℋ_{x * y}^{y} (λ) = ℋ_{x * y} (λ) \cap ℋ_{y} (λ) \\ = & ℋ_{λ}^{t} (x * y) \cap ℋ_{λ}^{t} (y) \supseteq ℋ_{λ}^{t} (y) \cap ℋ_{λ}^{t} (z) . \end{matrix}$

This completes the proof. □

Theorem 3.7. If ℋ is a hesitant fuzzy subalgebra of 𝒳, then the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra of 𝒳 for all λ ⊆ Σ_ℋ.

Proof. Using (3.1), we have $\begin{matrix} ℋ_{λ}^{t} (x * y) & = & ℋ_{x * y} (λ) \supseteq ℋ_{x}^{y} (λ) \\ = & ℋ_{x} (λ) \cap ℋ_{y} (λ) = ℋ_{λ}^{t} (x) \cap ℋ_{λ}^{t} (y) \end{matrix}$ for all x, y ∈ X and λ ⊆ Σ_ℋ. Therefore $ℋ_{λ}^{t}$ is a hesitant fuzzy subalgebra of 𝒳. □

Theorem 3.8. If ℋ is a hesitant fuzzy ideal of 𝒳, then so is the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ for all λ ⊆ Σ_ℋ.

Proof. Using (3.2) and (3.3), we have $ℋ_{λ}^{t} (x) = ℋ_{x} (λ) \subseteq ℋ_{0} (λ) = ℋ_{λ}^{t} (0)$ and $\begin{matrix} ℋ_{λ}^{t} (x) & = & ℋ_{x} (λ) \supseteq ℋ_{x * y}^{y} (λ) = ℋ_{x * y} (λ) \cap ℋ_{y} (λ) \\ = & ℋ_{λ}^{t} (x * y) \cap ℋ_{λ}^{t} (y) \end{matrix}$ for all x, y ∈ X and λ ⊆ Σ_ℋ. Therefore $ℋ_{λ}^{t}$ is a hesitant fuzzy ideal of 𝒳. □

We consider the converse of Theorems 3.7 and 3.8.

Theorem 3.9. Let ℋ be a hesitant fuzzy set on 𝒳. If there exists a subset λ of Σ_ℋ such that the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra of 𝒳, then ℋ is a hesitant fuzzy subalgebra of 𝒳.

Proof. Assume that the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra of 𝒳 for some λ ⊆ Σ_ℋ. Then $\begin{matrix} ℋ_{x * y} (λ) & = & ℋ_{λ}^{t} (x * y) \supseteq ℋ_{λ}^{t} (x) \cap ℋ_{λ}^{t} (y) \\ = & ℋ_{x} (λ) \cap ℋ_{y} (λ) = ℋ_{x}^{y} (λ) \end{matrix}$ for all x, y ∈ X. Now, if $z \in ℋ_{x}^{y}$ , then $z \in ℋ_{x}^{y} (λ) \subseteq ℋ_{x * y} (λ)$ and z ∉ λ since _ℋx and λ are disjoint for all x ∈ X . Hence z ∈ _ℋx *y, and so $ℋ_{x}^{y} \subseteq ℋ_{x * y}$ . Therefore ℋ is a hesitant fuzzy subalgebra of 𝒳. □

Theorem 3.10. Let ℋ be a hesitant fuzzy set on 𝒳. If there exists a subset λ of Σ_ℋ such that the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy ideal of 𝒳, then ℋ is a hesitant fuzzy ideal of 𝒳.

Proof. Assume that the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy ideal of 𝒳 for some λ ⊆ Σ_ℋ. Then $ℋ_{x} (λ) = ℋ_{λ}^{t} (x) \subseteq ℋ_{λ}^{t} (0) = ℋ_{0} (λ)$ and $\begin{matrix} ℋ_{x} (λ) & = & ℋ_{λ}^{t} (x) \supseteq ℋ_{λ}^{t} (x * y) \cap ℋ_{λ}^{t} (y) \\ = & ℋ_{x * y} (λ) \cap ℋ_{y} (λ) = ℋ_{x * y}^{y} (λ) \end{matrix}$ for all x, y ∈ X. Since _ℋx and λ are disjoint for all x ∈ X, it follows that _ℋ0 ⊇ _ℋx and $ℋ_{x} \supseteq ℋ_{x * y}^{y}$ for all x, y ∈ X. Therefore ℋ is a hesitant fuzzy ideal of 𝒳. □

For any hesitant fuzzy set ℋ on 𝒳, consider a set $𝒳_{δ}^{λ} : = {x \in X ∣ δ \ λ \subseteq ℋ_{x}}$ where λ ⊆ Σ_ℋ and δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ. We say that $𝒳_{δ}^{λ}$ is the (δ, λ)-support of ℋ. Note that $(\forall x \in X) (δ \ λ \subseteq ℋ_{x} \Leftrightarrow δ \subseteq ℋ_{x} (λ) = ℋ_{λ}^{t} (x))$

Hence $𝒳_{δ}^{λ} : = {x \in X ∣ δ \subseteq ℋ_{λ}^{t} (x)}$ .

Theorem 3.11. For any λ ⊆ Σ_ℋ, if a hesitant fuzzy set ℋ on 𝒳 is a hesitant fuzzy subalgebra (resp. ideal) of 𝒳, then the (δ, λ)-support of ℋ is a subalgebra (resp. ideal) of 𝒳 for all δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ.

Proof. Assume that ℋ is a hesitant fuzzy subalgebra (resp. ideal) of 𝒳. Let x, y ∈ X. If $x, y \in 𝒳_{δ}^{λ}$ , then δ \ λ ⊆ _ℋx and δ \ λ ⊆ _ℋy. Hence $ℋ_{x * y} \supseteq ℋ_{x}^{y} \supseteq δ \ λ$ and _ℋ0 ⊇ _ℋx ⊇ δ \ λ by (3.1) and (3.2), and so $x * y \in 𝒳_{δ}^{λ}$ and $0 \in 𝒳_{δ}^{λ}$ . Suppose that $x * y \in 𝒳_{δ}^{λ}$ and $y \in 𝒳_{δ}^{λ}$ . Then δ \ λ ⊆ _ℋx *y and δ \ λ ⊆ _ℋy. Using (3.3), we have $ℋ_{x} \supseteq ℋ_{x * y}^{y} \supseteq δ \ λ$ and thus $x \in 𝒳_{δ}^{λ}$ . Therefore the (δ, λ)-support of ℋ is a subalgebra (resp. ideal) of 𝒳. □

Using Theorems 3.9, 3.10 and 3.11, we have the following corollary.

Corollary 3.12. For a hesitant fuzzy set ℋ on 𝒳, if there exists a subset λ of Σ_ℋ such that the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra (resp. hesitant fuzzy ideal) of 𝒳, then the (δ, λ)-support of ℋ is a subalgebra (resp. ideal) of 𝒳 for all δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ.

Given λ ⊆ Σ_ℋ and any δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ, let ℋ be a hesitant fuzzy set on 𝒳 such that the (δ, λ)-support of ℋ is a subalgebra of X. Let x and y be elements of X such that $ℋ_{λ}^{t} (x) = δ_{x}$ and $ℋ_{λ}^{t} (y) = δ_{y}$ . If we take δ = _δx ∩ _δy, then $ℋ_{λ}^{t} (x) = δ_{x} \supseteq δ$ and $ℋ_{λ}^{t} (y) = δ_{y} \supseteq δ$ . Thus $x, y \in 𝒳_{δ}^{λ}$ , and so $x * y \in 𝒳_{δ}^{λ}$ . It follows that $ℋ_{λ}^{t} (x * y) \supseteq δ = δ_{x} \cap δ_{y} = ℋ_{λ}^{t} (x) \cap ℋ_{λ}^{t} (y) .$

Therefore we have the following theorem.

Theorem 3.13. Given λ ⊆ Σ_ℋ and any δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ, if the (δ, λ)-support of a hesitant fuzzy set ℋ on 𝒳 is a subalgebra of 𝒳, then the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra of 𝒳.

Corollary 3.14. Given λ ⊆ Σ_ℋ and any δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ, if the (δ, λ)-support of a hesitant fuzzy set ℋ on 𝒳 is a subalgebra of 𝒳, then ℋ is a hesitant fuzzy subalgebra of 𝒳.

Theorem 3.15. Let 𝒳 be a BCK-algebra. Given λ ⊆ Σ_ℋ and any δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ, if the (δ, λ)-support of a hesitant fuzzy set ℋ on 𝒳 is an ideal of 𝒳, then the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy ideal of 𝒳.

Proof. Suppose that the (δ, λ)-support $𝒳_{δ}^{λ}$ of a hesitant fuzzy set ℋ on a BCK-algebra 𝒳 is an ideal of 𝒳 for any λ ⊆ Σ_ℋ and δ ∈ 𝒫 ([0, 1]) with λ ⊆ δ. Then $𝒳_{δ}^{λ}$ is a subalgebra of X because every ideal is a subalgebra in a BCK-algebra 𝒳. It follows from Theorem 3.13 that the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra of 𝒳 and from Theorem 3.9 that ℋ is a hesitant fuzzy subalgebra of 𝒳. Using (3.5) in Proposition 3.6, we have $ℋ_{λ}^{t} (0) \supseteq ℋ_{λ}^{t} (x)$ for all x ∈ X. Let x and y be elements of X such that $ℋ_{λ}^{t} (x * y) = δ_{x * y}$ and $ℋ_{λ}^{t} (y) = δ_{y}$ . If we take δ : = _δx *y ∩ _δy, then $ℋ_{λ}^{t} (x * y) = δ_{x * y} \supseteq δ$ and $ℋ_{λ}^{t} (y) = δ_{y} \supseteq δ$ , that is, $x * y \in 𝒳_{δ}^{λ}$ and $y \in 𝒳_{δ}^{λ}$ . Since $𝒳_{δ}^{λ}$ is an ideal of 𝒳, we have $x \in 𝒳_{δ}^{λ}$ by (b2). Thus $ℋ_{λ}^{t} (x) \supseteq δ = δ_{x * y} \cap δ_{y} = ℋ_{λ}^{t} (x * y) \cap ℋ_{λ}^{t} (y)$ . Therefore the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy ideal of 𝒳. □

4 Hesitant fuzzy extensions

Definition 4.1. Let ℋ and 𝒢 be hesitant fuzzy sets on 𝒳. We say that 𝒢 is a hesitant fuzzy extension of ℋ if it satisfies:

(∀ x ∈ X) (_ℋx ⊆ _𝒢x),

If ℋ is a hesitant fuzzy subalgebra of 𝒳, then so is 𝒢.

Example 4.2. If ℋ is a hesitant fuzzy subalgebra of 𝒳, then the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy extension of ℋ for all λ ⊆ Σ_ℋ.

The following example shows that there exist hesitant fuzzy sets ℋ and 𝒢 on 𝒳 such that

ℋ is a hesitant fuzzy subalgebra of 𝒳.

𝒢 is a hesitant fuzzy extension of ℋ.

𝒢 is not the λ-hesitant fuzzy translation of ℋ for any λ ⊆ Σ_𝒢.

Example 4.3. Let X = {0, 1, 2, 3, 4} be a set with the following Cayley table which is appeared in Table 1.

Then 𝒳 : = (X, * , 0) is a BCK-algebra (see [5]). Let ℋ be a hesitant fuzzy set on 𝒳 which is defined by $\begin{matrix} ℋ : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.2, 0.3, 0.5} & if x=0, \\ {0.1, 0.3} & if x=1, \\ {0.1, 0.5} & if x=2, \\ {0.3, 0.5} & if x=3, \\ {0.1} & if x=4 . \end{matrix} \end{matrix}$

Then ℋ is a hesitant fuzzy subalgebra of 𝒳.

(1) Let 𝒢 be a hesitant fuzzy set on 𝒳 which is defined by $\begin{matrix} 𝒢 : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.2, 0.3, 0.5} & if x=0, \\ {0.1, 0.3, 0.5} & if x=1, \\ {0.1, 0.2, 0.5} & if x=2, \\ {0.2, 0.3, 0.5} & if x=3, \\ {0.1, 0.2} & if x=4 . \end{matrix} \end{matrix}$

Then _ℋx ⊆ _𝒢x for all x ∈ X, and 𝒢 is a hesitant fuzzy subalgebra of 𝒳. Therefore 𝒢 is a hesitant fuzzy extension of ℋ. Note that Σ_ℋ = {0.4, 0.6} and there is no λ ⊆ Σ_𝒢 such that 𝒢 is the λ-hesitant fuzzy translation of ℋ.

(2) Let ℱ be a hesitant fuzzy set on 𝒳 which is defined by $\begin{matrix} ℱ : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.2, 0.3, 0.5, 0.6} & if x=0, \\ {0.1, 0.3, 0.6} & if x=1, \\ {0.1, 0.5, 0.6} & if x=2, \\ {0.3, 0.5, 0.6} & if x=3, \\ {0.1, 0.6} & if x=4 . \end{matrix} \end{matrix}$

Then $ℱ = ℋ_{λ}^{t}$ where λ = {0.6} ⊆ Σ_ℋ, that is, ℱ is the λ-hesitant fuzzy translation of ℋ. Thus it is a hesitant fuzzy extension of ℋ.

Let ℋ be a hesitant fuzzy subalgebra of 𝒳. For every λ ⊆ Σ_ℋ, the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ is a hesitant fuzzy subalgebra of 𝒳. If 𝒢 is a hesitant fuzzy extension of $ℋ_{λ}^{t}$ , then there exists δ ⊆ Σ_ℋ such that λ ⊆ δ and $ℋ_{δ}^{t} (x) \subseteq 𝒢 (x)$ for all x ∈ X. Also the _λ1-hesitant fuzzy translation and the _λ2-hesitant fuzzy translation of ℋ are hesitant fuzzy subalgebras of 𝒳 for any subsets _λ1 and _λ2 of Σ_ℋ by Theorem 3.7. If _λ1 ⊆ _λ2, then $ℋ_{λ_{1}}^{t} (x) = ℋ_{x} (λ_{1}) \subseteq ℋ_{x} (λ_{2}) = ℋ_{λ_{2}}^{t} (x)$ for all x ∈ X. Hence we have the following theorem.

Theorem 4.4. Let ℋ be a hesitant fuzzy subalgebra of 𝒳 and λ ⊆ Σ_ℋ. For every hesitant fuzzy extension 𝒢 of the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ, there exists δ ⊆ Σ_ℋ such that λ ⊆ δ and 𝒢 is a hesitant fuzzy extension of the δ-hesitant fuzzy translation $ℋ_{δ}^{t}$ of ℋ. Also for any subsets _λ1 and _λ2 of Σ_ℋ with _λ1 ⊆ _λ2, the _λ2-hesitant fuzzy translation $ℋ_{λ_{2}}^{t}$ of ℋ is a hesitant fuzzy extension of the _λ1-hesitant fuzzy translation $ℋ_{λ_{1}}^{t}$ of ℋ.

The following example illustrates the first part of Theorem 4.4.

Example 4.5. Let X = {0, 1, 2, 3, 4} be a set with the following Cayley table which is appeared in Table 2.

Then 𝒳 : = (X, * , 0) is a BCK-algebra (see [5]). Let ℋ be a hesitant fuzzy set on 𝒳 which is defined by $\begin{matrix} ℋ : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.3, 0.5, 0.7} & if x=0, \\ {0.1, 0.5, 0.7} & if x=1, \\ {0.1, 0.3, 0.5} & if x=2, \\ {0.1, 0.3} & if x=3, \\ {0.1} & if x=4 . \end{matrix} \end{matrix}$

Then ℋ is a hesitant fuzzy subalgebra of 𝒳 and Σ_ℋ = {0.2, 0.4, 0.6}. If we take λ = {0.6}, then the λ-hesitant fuzzy translation $(ℋ_{λ}^{t}, X)$ of ℋ is givenby $\begin{matrix} ℋ_{λ}^{t} : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.3, 0.5, 0.6, 0.7} & if x=0, \\ {0.1, 0.5, 0.6, 0.7} & if x=1, \\ {0.1, 0.3, 0.5, 0.6} & if x=2, \\ {0.1, 0.3, 0.6} & if x=3, \\ {0.1, 0.6} & if x=4, \end{matrix} \end{matrix}$ and it is a hesitant fuzzy subalgebra of 𝒳 by Theorem 3.7. Let 𝒢 be a hesitant fuzzy set on 𝒳 which is defined by $\begin{matrix} 𝒢 : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7} & if x=0, \\ {0.1, 0.2, 0.4, 0.5, 0.6, 0.7} & if x=1, \\ {0.1, 0.2, 0.3, 0.5, 0.6} & if x=2, \\ {0.1, 0.2, 0.3, 0.6} & if x=3, \\ {0.1, 0.2, 0.6} & if x=4 . \end{matrix} \end{matrix}$

Then 𝒢 is a hesitant fuzzy subalgebra of 𝒳 and $ℋ_{λ}^{t} (x) \subseteq 𝒢 (x)$ for all x ∈ X. Thus 𝒢 is a hesitant fuzzy extension of the λ-hesitant fuzzy translation $ℋ_{λ}^{t}$ of ℋ. But 𝒢 is not the λ-hesitant fuzzy translation of ℋ for all λ ⊆ Σ_ℋ. If we take δ = {0.2, 0.6}, then λ ⊆ δ and the δ-hesitant fuzzy translation $ℋ_{δ}^{t}$ of ℋ is given as follows: $\begin{matrix} ℋ_{δ}^{t} : X \to 𝒫 ([0, 1]), \\ x \mapsto {\begin{matrix} {0.1, 0.2, 0.3, 0.5, 0.6, 0.7} & if x=0, \\ {0.1, 0.2, 0.5, 0.6, 0.7} & if x=1, \\ {0.1, 0.2, 0.3, 0.5, 0.6} & if x=2, \\ {0.1, 0.2, 0.3, 0.6} & if x=3, \\ {0.1, 0.2, 0.6} & if x=4 . \end{matrix} \end{matrix}$

Then $𝒢 (x) \supseteq ℋ_{δ}^{t} (x)$ for all x ∈ X and $ℋ_{δ}^{t}$ is a hesitant fuzzy subalgebra of 𝒳. Therefore 𝒢 is a hesitant fuzzy extension of the δ-hesitant fuzzy translation of ℋ.

5 Conclusions

Given a hesitant fuzzy set on a BCK/BCI-algebra, we have introduced the notions of hesitant fuzzy translations and hesitant fuzzy extensions, and investigated related properties. We have shown that every hesitant fuzzy translation of a hesitant fuzzy subalgebra (ideal) is a hesitant fuzzy subalgebra (ideal). We have provided conditions for a hesitant fuzzy set to be a hesitant fuzzy subalgebra (ideal). We have shown that if a hesitant fuzzy set is a hesitant fuzzy subalgebra (ideal), then its support is a subalgebra (ideal). We have also shown that if the support of a hesitant fuzzy set is subalgebra (ideal), then its hesitant fuzzy translation is a hesitant fuzzy subalgebra (ideal).

Footnotes

Acknowledgments

The first author would like to acknowledge financial support for this work, from the Deanship of Scientific Research (DRS), University of Tabuk, Tabuk, Saudi Arabia, under grant no. S/0123/1436.

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