Hesitant fuzzy dot subalgebra and dot ideals of B-algebra

Abstract

In this paper, the notions of hesitant fuzzy dot subalgebras, hesitant fuzzy normal dot subalgebras, and hesitant fuzzy dot ideals of B-algebras are presented, and some of their features are examined, in this study. The homomorphic image and inverse image of hesitant fuzzy dot subalgebras, as well as the hesitant fuzzy dot ideal, are investigated. We also explore some related characteristics of hesitant fuzzy relations on the family of hesitant fuzzy dot subalgebras and hesitant fuzzy dot ideal of B-algebras.

Keywords

B-algebra hesitant fuzzy dot subalgebra hesitant fuzzy normal dot subalgebra hesitant fuzzy dot ideal

1 Introduction

Fuzzy sets were introduced by Zadeh [1] in 1965. That marks the birth of fuzzy mathematics. After continuous development, fuzzy theory has also made great progress. Since Zadeh put forward the concept of fuzzy sets in his paper, it has attracted the attention of a large number of scientists. In 1971, Rosenfeld introduced fuzzy subgroups [2], which mark the beginning of the study of fuzzy algebra. Fuzzy sets and its extensions have provided successful results in dealing with uncertainty provided in different problems. Worldwide, there has been a rapid growth interest in applications of fuzzy sets and some generalization of this is discussed by authors, such as intuitionistic fuzzy set [3], bipolar valued fuzzy set [4], these fuzzification ideas have also been applied to other algebraic structures, and a series of conclusions have emerged one after another. For example, C. H. Liu et al. [5] studied the bipolar fuzzy ideals in negative non-involutive residual lattices and discussed their properties, H. R. Zhang [6] introduced the intuitionistic fuzzy filter theory in algebraic structures, more conclusions can be found in reference [7 –11].

In order to study the algebraic system composed of truth values in Fuzziness theory and solve some problems in logic proof, literature [12, 13] introduced the concept of B-algebra and discusses its properties. Literature [14] introduced the concept of fuzzy subalgebra of B-algebra, the concepts of fuzzy B-algebra and fuzzy normal B-algebra are introduced in literature [15], the properties of fuzzy B-algebra are obtained. The fuzzy closed ideals of B-algebra and the fuzzy closed ideals under t-norm are studied in document [16, 17]. The concept of interval valued fuzzy sets is applied to B-algebra in document [18], the interval valued fuzzy subalgebras of B-algebra and their related properties are given. Using the concept of interval valued intuitionistic fuzzy sets, the concept of interval valued intuitionistic fuzzy subalgebras of B-algebra is introduced in literature [19]. The relevant literature on the fuzzy structure of B-algebra gives the fuzzy characteristics of B-algebra from different angles, which enriches the fuzzy theory of B-algebra.

In 2010, Torra [20] introduced hesitant fuzzy set theory, hesitant fuzzy set is a useful tool to express people’s indecision in real life, which solves the problem of uncertainty. Hesitant fuzzy numbers are more comprehensive than traditional fuzzy elements, they contain different groups and have a certain degree of hesitation, which applied in many mathematical models [21 –23]. Hesitant fuzzy sets can help with group decision making when experts are undecided about which of multiple viable memberships for a set element. In comparison to the fuzzy set and its many classical variants, the hesitant fuzzy set can more effectively reflect people’s uncertainty in expressing their preferences for objects.

These prospective memberships may be not only crisp numbers in [0, 1], but also interval values during the assessing procedure in practice. The hesitant fuzzy set can handle cases where determining the membership degree is difficult due to our hesitation among a few different values, rather than a margin of error (as in intuitionistic or interval-valued fuzzy sets) or a specified possibility distribution of the possible values (as in type-2 fuzzy sets) [26].

New distances for dual hesitant fuzzy sets and their application in clustering algorithm were proposed in [27]. Dual hesitant fuzzy sets can model situations in the actual world more completely than hesitant fuzzy sets and intuitionistic fuzzy sets. Dual hesitant fuzzy sets clearly display a set of membership degrees and a set of non-membership degrees, implying a set of crucial data: hesitant degrees. The standard definition of distance between dual hesitant fuzzy sets only takes into account membership and non-membership degrees, but hesitant degree should be considered as well. Hesitant fuzzy set has also several extensions, e.g., complex hesitant fuzzy set; an extension of hesitant fuzzy set which includes both (time/phase and periodicity) such type of potential characters of phase & periodicity can make information more accurate [28].

The ideal of hesitant fuzzy sets is closer to general thinking, ideal as a reasoning criterion in the study of B-algebra, it is of a great significance to study their properties. At present, there are few conclusions about the algebraic structure of hesitant fuzzy sets, therefore, in this paper, we further study B-algebras, several main properties are proved and we obtained the related concepts and properties of hesitant fuzzy dot subalgebra (normal dot subalgebra/dot ideal) on B-algebras, we discussed the notion of hesitant fuzzy relations on the family of hesitant fuzzy dot subalgebras and hesitant fuzzy dot ideal of B-algebras.

The remainder of this article is organized as follows. In Section 2, we introduce basic definitions and knowledge. Section 3 give the definition of hesitant fuzzy dot subalgebra and discuss their properties in details. Section 4, studies the properties of hesitant fuzzy normal dot subalgebra. In Section 5, (strongest) hesitant fuzzy ρ-product relation is defined and present some of its properties. Section 6, introduces the definition of hesitant fuzzy dot ideal and discuss their properties in detail. Section 7, concludes this paper. numbers; if necessary, one may refer to sections.

2 Preliminaries

In this section, we recollect some basic definitions and knowledge which will be used in the following.

Definition 1. [13] A non-empty set X with a constant 0 and a binary operation * is said to be B-algebra if it satisfies the following axioms: for all x, y, z ∈ X,

x * x = 0,

x * 0 = x,

(x * y) * z = x * (z * (0 * y)).

Lemma 1. [12] Let X be a B-algebra, then for all x ∈ X, 0 * (0 * x) = x.

Lemma 2. [13] Let X be a B-algebra, then for all x, y ∈ X, (x * y) * (0 * y) = x.

A non-empty subset A of a B-algebra X is called a subalgebra [24] of X, if x * y ∈ A for any x, y ∈ A. A non-empty subset N of a B-algebra X is called normal [24] if (x * a) * (y * b) ∈ N whenever x * y ∈ N, a * b ∈ N. A non-empty subset N of a B-algebra X is called a normal subalgebra of X if it is both a subalgebra and a normal subset of X. A partial ordering “⩽” on X can be defined by x ⩽ y if and only if x * y = 0. A mapping f : X → Y of B-algebra is called a homomorphism [24] if f (x * y) = f (x) * f (y), for all x, y ∈ X. Note that if f : X → Y is a B-homomorphism [24], then f (0) =0.

Definition 2. [15] A fuzzy set A in X is called a fuzzy subalgebra if it satisfies A (x * y) ⩾ min {A (x) , A (y)} for all x, y ∈ X.

Definition 3. [16] A fuzzy set A in X is called a fuzzy ideal of X if it satisfies the following conditions:

A (0) ⩾ A (x),

A (x) ⩾ min {A (x * y) , A (y)} for all x, y ∈ X.

In the following, we review the concept of a hesitant fuzzy set defined by Torra [20]:

Definition 4. [20] Let X be a reference set. Then a hesitant fuzzy set (briefly HF) A is represented mathematically as: $A = {< x, h_{A} (x) > : h_{A} (x) \in P ([0, 1]), x \in X}$

Where h_A (x) is called the membership value of x in A and P ([0, 1]) is the power set of [0, 1].

The complement of A is denoted by A^c and is given by A^c = {< x, h_{A
^c} (x) > : x ∈ X}, where h_{A
^c} (x) =1 - h_A (x).

For any two hesitant fuzzy sets A and B in X, the following operations are defined $A \subseteq B \Leftrightarrow h_{A} (x) \subseteq h_{B} (x), and$ $A \cap B = h_{A} (x) \cap h_{B} (x) .$

Definition 5. [20] Let A be a hesitant fuzzy set in X and γ ∈ P ([0, 1]), then the set $X (A, γ) : = {x \in X | γ \subseteq h_{A} (x)}$

is called a level subset of A, and the set $X (A, γ) : = {x \in X | γ \subset h_{A} (x)}$

is called a strong level subset of A.

Definition 6. [25] Let X and Y be two non-empty sets and f : X → Y be a mapping. Let A and B be HFS of X and Y respectively. Then the image of A under the map f is denoted by f (A) and is defined as f (A) (y) =< y, h_f(A) (y) >, where $h_{f (A)} (y) = {\begin{matrix} \underset{f (x) = y}{\cup} h_{A} (x) : x \in f^{- 1} (y) \\ 0, otherwise \end{matrix}$

Also, pre-image of B under f is denoted by f^-1 (B) and is defined by $h_{f^{- 1} (B)} (x) = h_{B} (f (x)), x \in X .$

3 Hesitant fuzzy dot subalgebras of B-algebras

In this section, hesitant fuzzy dot subalgebras of B-algebras are defined and some important properties are presented. In what follows, let (X, * , 0) or simply X denoted a B-algebra unless otherwise specified.

Definition 7. Let A be a HF of X. Then A is called a hesitant fuzzy subalgebra (briefly HFS) of X if for all x, y ∈ X, $h_{A} (x * y) \supseteq h_{A} (x) \cap h_{A} (y) .$

Definition 8. Let A be a HF of X, then A is called a hesitant fuzzy dot subalgebra (briefly HFDS) of X if for all x, y ∈ X, h_A (x * y) ⊇ h_A (x) · h_A (y), where * denotes the ordinary multiplication of x and y.

Example 1. Let X = {0, 1, 2, 3, 4, 5} be a set with the following Cylay Table:

Table 1
Table1

* 0 1 2 3 4 5

0 0 2 1 3 4 5

1 1 0 2 4 5 3

2 2 1 0 5 3 4

3 3 4 5 0 2 1

4 4 5 3 1 0 2

5 5 3 4 2 1 0

*	0	1	2	3	4	5
0	0	2	1	3	4	5
1	1	0	2	4	5	3
2	2	1	0	5	3	4
3	3	4	5	0	2	1
4	4	5	3	1	0	2
5	5	3	4	2	1	0

Obviously, (X, * , 0) is a B-algebra. Define a hesitant fuzzy set in X by h_A (1) = h_A (2) =0.7 and h_A (x) =0.5 for all x ∈ X ∖ {1, 2}. Then A is a HFDS of X.

Note that if A is HFS in X, then we have A is a HFDS of X, but the converse is not true, in general.

In fact, the hesitant fuzzy dot subalgebra in Example 1, is not a hesitant fuzzy subalgebra, since $\begin{matrix} h_{A} (1 * 1) = h_{A} (0) = 0.5 < 0.7 = h_{A} (1) \\ = h_{A} (1) \cap h_{A} (1) . \end{matrix}$

Proposition 1. Every HFDSA of X satisfies the inequality h_A (0) ⊇ (h_A (x)) ² for all x ∈ X.

Proof. For all x ∈ X, we have x * x = 0. Then $h_{A} (0) = h_{A} (x * x) \supseteq h_{A} (x) \cdot h_{A} (x) = (h_{A} (x))^{2} .$

Theorem 1. Let A be a HFDS of X. If there exists a sequence {x_n} in X such that $lim_{n \to \infty} (h_{A} (x_{n}))^{2} = 1$ , then h_A (0) =1.

Proof. By Proposition 1, we have h_A (0) ⊇ (h_A (x)) ² for all x ∈ X. Therefore, h_A (0) ⊇ (h_A (x_n)) ² for every positive integer n.

Consider, $1 \supseteq h_{A} (0) \supseteq lim_{n \to \infty} (h_{A} (x_{n}))^{2} = 1$ .

Hence, h_A (0) =1.

Proposition 2. Let A be a HFDS of X, then A^m is a HFDS of X, where m is any positive integer.

Proof. For all x ∈ X, A^m is a HFS of X, defined by $h_{A^{m}} (x) = h_{A}^{m} (x)$ , where m is any positive integer. Let A be a HFDS of X, then

h_A (x * y) ⊇ h_A (x) · h_A (y) for all x, y ∈ X. We have $\begin{matrix} h_{A}^{m} (x * y) = [h_{A} (x * y)]^{m} \\ = h_{A}^{m} (x) \cdot h_{A}^{m} (y) \\ \supseteq [h_{A} (x) \cdot h_{A} (y)]^{m} \\ = [h_{A} (x)]^{m} \cdot [h_{A} (y)]^{m} \end{matrix}$

Therefore, A^m is a HFDS of X^.

Theorem 2. Let A_i (i = 1, 2) be two HFDS of X, then A₁ ∩ A₂ is also a HFDS of X.

Proof. Let x, y ∈ A₁ ∩ A₂. Then x, y ∈ A₁ and x, y ∈ A₂. Now $\begin{matrix} h_{A_{1} \cap A_{2}} (x * y) \\ = h_{A_{1}} (x * y) \cap h_{A_{2}} (x * y), \\ \supseteq (h_{A_{1}} (x) \cdot h_{A_{1}} (y)) \cap (h_{A_{2}} (x) \cdot h_{A_{2}} (y)) \\ = (h_{A_{1}} (x) \cap h_{A_{2}} (x)) \cdot (h_{A_{1}} (y) \cap h_{A_{2}} (y)) \\ = h_{A_{1} \cap A_{2}} (x) \cdot h_{A_{1} \cap A_{2}} (y) \end{matrix}$

Hence, A₁ ∩ A₂ is a HFDS of X.

Theorem 3. Let A be a HF of X^. Define a hesitant fuzzy set X_{h
_A} of X as follows $X_{h_{A}} : X \to P ([0, 1]), X_{h_{A}} (x) = {\begin{matrix} 1, x \in A \\ 0, otherwise \end{matrix}$

Then A is a subalgebra of X if and only if X_{h
_A} is a HFDS of X.

Proof. Let A be a subalgebra of X and x, y ∈ A. Then x * y ∈ A, and we have $X_{h_{A}} (x * y) = 1 \supseteq X_{h_{A}} (x) \cdot X_{h_{A}} (y) .$

If x ∈ A and y ∉ A (or x ∉ A and y ∈ A), then we have X_{h
_A} (x) =1 or X_{h
_A} (y) =0.

Therefore X_{h
_A} (x * y) ⊇ X_{h
_A} (x) · X_{h
_A} (y) =1 · 0 =0.

Conversely, assume that X_{h
_A} is a HFDS of X and let x, y ∈ A. Then we have $X_{h_{A}} (x * y) \supseteq X_{h_{A}} (x) \cdot X_{h_{A}} (y) = 1 \cdot 1 = 1$

Therefore x * y ∈ A.

Theorem 4. Let f : X → Y be a homomorphism of B-algebras. If B is a HFDS of Y, then the pre-image f^-1 (B) of B under f is a HFDS of X.

Proof. Suppose that B is a HFDS of Y and let x, y ∈ X. Then $\begin{matrix} h_{f^{- 1} (B)} (x * y) = h_{B} (f (x * y)) \\ = h_{B} (f (x) * f (y)) \\ \supseteq h_{B} (f (x)) \cdot h_{B} (f (y)) \\ = h_{f^{- 1} (B)} (x) \cdot h_{f^{- 1} (B)} (y) \end{matrix}$

Therefore, f^-1 (B) is a HFDS of X.

Theorem 5. Let f : X → Y be a homomorphism from a B-algebras X onto a B-algebras Y. If A is a HFDS of X, then the image f (A) of A under f is a HFDS of Y.

Proof. Let A be a HFDS of X and let y₁, y₂ ∈ Y.

Noticing that {x₁ * x₂ : x₁ ∈ f^-1 (y₁) , x₂ ∈ f^-1 (y₂)} $\subseteq {x \in X : x \in f^{- 1} (y_{1} * y_{2})}$

Then we have $\begin{matrix} h_{f (A)} (y_{1} * y_{2}) = \underset{x \in f^{- 1} (y_{1} * y_{2})}{\cup} h_{μ} (x) \\ \supseteq \underset{x_{1} \in f^{- 1} (y_{1}), x_{2} \in f^{- 1} (y_{2})}{\cup} h_{μ} (x_{1} * x_{2}) \\ \supseteq \underset{x_{1} \in f^{- 1} (y_{1}), x_{2} \in f^{- 1} (y_{2})}{\cup} (h_{μ} (x_{1}) \cdot h_{μ} (x_{2})) \\ = (\underset{x_{1} \in f^{- 1} (y_{1})}{\cup} h_{μ} (x_{1})) \cdot (\underset{x_{2} \in f^{- 1} (y_{2})}{\cup} h_{μ} (x_{2})) \\ = h_{μ} (y_{1}) \cdot h_{μ} (y_{2}) \end{matrix}$

Therefore, f (A) is a HFDS of Y.

Theorem 6. Let A be a HF of X, A is a HFS of X if and only if for every γ ∈ P ([0, 1]), a non-empty level subset $R (A, γ) : = {x \in A | γ \subseteq h_{A} (x)}$

is a subalgebra of X.

Proof. Let x, y ∈ X be such that x, y ∈ R (A, γ), for any γ ∈ P ([0, 1]), then h_A (x) ⊇ γ, h_A (y) ⊇ γ. Since A is a HFS of X, so h_A (x* y) ⊇ h_A (x) ∩ h_A (y) ⊇ γ ∩ γ = γ. We have x * y ∈ R (A, γ).

Conversely, let R (A, γ) be a subalgebra of X, for any x, y ∈ X, γ ∈ P ([0, 1]) with R (A, γ)≠ ø.

Let h_A (x) = α, h_A (y) = β, put γ = α ∩ β, then x, y ∈ R (A, γ). Since R (A, γ) is a subalgebra of X, hence x * y ∈ R (A, γ) and so h_A (x * y) ⊇ γ, h_A (x* y) ⊇ γ = α ∩ β = h_A (x) ∩ h_A (y).

Therefore, A is a HFS of X.

Remark 1. If A is a HFDS of X, then R (A, γ) need not be a subalgebra of X. In Example 1, A is a HFDS of X, but R (A, 0.7) : = { x ∈ A|h_A (x) ⊇0.7 } = {1, 2} is not a subalgebra of X since 1 * 1 =0 ∉ R (A, 0.7).

Theorem 7. Let A be a HFDS of X, then R (A, 1) : ={ x ∈ A|h_A (x) =1 } is either empty or is a subalgebra of X.

Proof. Assume that R (A, 1)≠ ø.

Obviously, 0 ∈ R (A, 1).

If x, y ∈ R (A, 1), then h_A (x* y) ⊇ h_A (x) · h_A (y) =1.

Hence h_A (x * y) =1, which implies that x * y ∈ R (A, 1).

Consequently, R (A, 1) is a subalgebra of X.

Definition 9. Let A and B be two HF in X. The Cartesian product A × B : X × X → P ([0, 1]) is defined by $h_{A \times B} (x, y) = h_{A} (x) \cdot h_{B} (y) for all x, y \in X .$

Proposition 3. Let A and B be two HFDS of X, then A × B is a HFDS of X × X.

Proof. Let (x₁, y₁) , (x₂, y₂) ∈ X × X. Then $\begin{matrix} h_{A \times B} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \\ = h_{A \times B} (x_{1} * x_{2}, y_{1} * y_{2}) \\ = h_{A} (x_{1} * x_{2}) \cdot h_{B} (y_{1} * y_{2}) \\ \supseteq (h_{A} (x_{1}) \cdot h_{A} (x_{2})) \cdot (h_{B} (y_{1}) \cdot h_{B} (y_{2})) \\ = (h_{A} (x_{1}) \cdot h_{B} (y_{1})) \cdot (h_{A} (x_{2}) \cdot h_{B} (y_{2})) \\ = h_{A \times B} (x_{1}, y_{1}) \cdot h_{A \times B} (x_{2}, y_{2}) . \end{matrix}$

Hence, A × B is a HFDS of X × X.

4 Hesitant fuzzy normal dot subalgebras of B-algebras

In this section, the hesitant fuzzy normal dot subalgebras of B-algebras are defined and discussed the relationship between hesitant fuzzy normal dot subalgebra, hesitant fuzzy normal subalgebras and hesitant fuzzy dot subalgebra of B-algebras.

Definition 10. Let A be a HF of X. Then A is called a hesitant fuzzy normal subalgebras (briefly HFNS) of X if h_A ((x * a) * (x * b)) ⊇ h_A (x * y) ∩ h_A (a * b) for all x, y ∈ X.

Definition 11. Let A be a HF of X. Then A is called a hesitant fuzzy normal dot subalgebras (briefly HFNDS) of X if

h_A ((x * a) * (x * b)) ⊇ h_A (x * y) · h_A (a * b) for all x, y ∈ X.

Example 2. Let A be a HF of X, defined by h_A (0) =0.8, h_A (1) = h_A (2) =0.7 and h_A (3) = h_A (4) = h_A (5) =0.3 for all x ∈ X in Example 1. Then A is a HFNDS of X.

Note that every HFNDS of X is a HFNS of X, but the converse is not true.

In fact, if we define a hesitant fuzzy set A in the Example 1, by h_A (0) =0.8, h_A (1) =0.5, h_A (2) =0.6 and h_A (3) = h_A (4) = h_A (5) =0.7 for all x ∈ X, then A is a HFNDS of X but not a HFNS of X, Since h_A (1) = h_A (0 *2) = h_A ((0* 0) * (0 * 1)) =0.5 ⊂ h_A (0 *0) ∩ h_A (0 *1) =h_A (0) ∩ h_A (2) =0.6.

Theorem 8. If A is a HFNDS of X, then A is a HFDS of X.

Proof. Let A be a HFNDS of X and x, y ∈ X. Then $\begin{matrix} h_{A} (x * y) = h_{A} ((x * y) * (0 * 0)) \\ \supseteq h_{A} (x * 0) \cdot h_{A} (y * 0) \\ = h_{A} (x) \cdot h_{A} (y) . \end{matrix}$

Consequently, A is a HFDS of X.

Remark 2. The converse of the above theorem is not true in general. If we define a set A in the Example 1 such that h_A (2) =0.46, h_A (0) = h_A (3) =0.7 and h_A (3) = h_A (4) = h_A (5) =0.6 for all x ∈ X, then A is a HFDS of X, but not a HFNDS of X, since h_A ((2 * 5) * (4 * 1)) = h_A (2) =0.46 < 0 . 49 = h_A (2*4) · h_A (5 *1).

Definition 12. A HFS A in X is called a hesitant fuzzy normal dot B-algebra if it is a hesitant fuzzy dot B-algebra which is a hesitant fuzzy normal.

Proposition 4. If a HFSA in X is a hesitant fuzzy normal dot B-algebra with h_A (0) =1, then h_A (x * y) = h_A (y * x) for all x, y ∈ X.

Proof. Let x, y ∈ X. Then by (B1) and (B2), $\begin{matrix} h_{A} (x * y) = h_{A} ((x * y) * (x * x)) \\ \supseteq h_{A} (x * x) \cdot h_{A} (y * x) \\ = h_{A} (0) \cdot h_{A} (y * x) \\ = h_{A} (y * x) . \end{matrix}$

If we exchange x and y, we can get h_A (y* x) ⊇ h_A (x * y).

Therefore, h_A (x * y) = h_A (y * x).

5 Hesitant fuzzy ρ-product relation of B-algebras

In this section, hesitant fuzzy relation and strongest hesitant fuzzy ρ- product relation of B-algebra are defined and presented some of its properties.

Definition 13. Let ρ be a HF of X. The strongest hesitant fuzzy ρ- relation on X is the hesitant fuzzy subset μ_ρ of X × X given by h_{μ_ρ} (x, y) = h_ρ (x) · h_ρ (y) for all x, y ∈ X.

Theorem 9. Let μ_ρ be the strongest hesitant fuzzy ρ- relation on X, where ρ is a subset of X. If ρ is a HFDS of X, then μ_ρ is a HFDS of X × X.

Proof. Assume that ρ is a HFDS of X. For any (x₁, y₁) and (x₂, y₂) ∈ X × X,we have $\begin{matrix} h_{μ_{ρ}} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \\ = h_{μ_{ρ}} (x_{1} * x_{2}, y_{1} * y_{2}) \\ = h_{ρ} (x_{1} * x_{2}) \cdot h_{ρ} (y_{1} * y_{2}) \\ \supseteq (h_{ρ} (x_{1}) \cdot h_{ρ} (x_{2})) \cdot (h_{ρ} (y_{1}) \cdot h_{ρ} (y_{2})) \\ = (h_{ρ} (x_{1}) \cdot h_{ρ} (y_{1})) \cdot (h_{ρ} (x_{2}) \cdot h_{ρ} (y_{2})) \\ = h_{μ_{ρ}} (x_{1}, y_{1}) \cdot h_{μ_{ρ}} (x_{2}, y_{2}) . \end{matrix}$

Therefore, μ_ρ is a HFDS of X × X.

Definition 14. Let ρ be a HF of X. A hesitant fuzzy relation μon X is called a hesitant fuzzy ρ-product relation h_μ (x, y) ⊇ h_ρ (x) · h_ρ (y) for all x, y ∈ X.

Definition 15. Let ρ be a HF of X. A hesitant fuzzy relation μon X is called a left hesitant fuzzy relation h_μ (x, y) = h_ρ (x) for all x, y ∈ X.

Similarly, we can define a right hesitant fuzzy relation on ρ. Note that a left (respectively, right) hesitant fuzzy relation on ρ is a ρ- product relation.

Theorem 10. Let μ be a left hesitant fuzzy relation on a hesitant fuzzy subset ρ of X. If μ is a HFDS of X × X, then ρ is a HFDS of X.

Proof. Assume that a left hesitant fuzzy relation μ on ρ is a hesitant fuzzy dot subalgebra of X × X. Then for all x₁, x₂, y₁, y₂ ∈ X. $\begin{matrix} h_{ρ} (x_{1} * x_{2}) = h_{μ} (x_{1} * x_{2}, y_{1} * y_{2}) \\ = h_{μ} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \\ \supseteq h_{μ} (x_{1}, y_{1}) \cdot h_{μ} (x_{2}, y_{2}) \\ = h_{ρ} (x_{1}) \cdot h_{ρ} (x_{2}) . \end{matrix}$

Hence, ρ is a HFDS of X.

Theorem 11. Let μ be a hesitant fuzzy relation on satisfying the inequality h_μ (x, y) ⊆ h_μ (x, 0) for all x, y ∈ X. Given s ∈ X, let ρ_s be a HF of X defined by h_{ρ
_s} (x) = h_μ (x, s) for all x ∈ X. If μ is a HFDS of X × X, then ρ_s is a HFDS of X for all s ∈ X.

Proof. Let x, y, s ∈ X. Then $\begin{matrix} h_{ρ_{s}} (x * y) = h_{μ} (x * y, s) \\ = h_{μ} (x * y, s * 0) \\ = h_{μ} ((x, s) * (y, 0)) \\ \supseteq h_{μ} (x, s) \cdot h_{μ} (y, 0) \\ \supseteq h_{μ} (x, s) \cdot h_{μ} (y, s) \\ = h_{ρ_{s}} (x) \cdot h_{ρ_{s}} (y) . \end{matrix}$

Hence, ρ_s is a HFDS of X.

Theorem 12. Let μ be a hesitant fuzzy relation on X and let ρ_μ be a HF of X given by $h_{ρ_{μ}} (x) = \underset{y \in X}{\cap} h_{μ} (x, y) \cdot h_{μ} (y, x)$ for all x ∈ X. If μ is a HFDS of X × X satisfying the equality h_μ (x, 0) =1 =h_μ (0, x) for all x ∈ X, then ρ_μ is a HFDS of X.

Proof. Let x, y, z ∈ X, we have $\begin{matrix} h_{μ} (x * y, z) = h_{μ} (x * y, z * 0) \\ = h_{μ} ((x, z) * (y, 0)) \\ \supseteq h_{μ} (x, z) \cdot h_{μ} (y, 0) = h_{μ} (x, z) \\ h_{μ} (z, x * y) = h_{μ} (z * 0, x * y) \\ = h_{μ} ((z, x) * (0, y)) \\ \supseteq h_{μ} (z, x) \cdot h_{μ} (0, y) = h_{μ} (z, x) . \end{matrix}$

It follows that $\begin{matrix} h_{μ} (x * y, z) \cdot h_{μ} (z, x * y) \\ \supseteq h_{μ} (x, z) \cdot h_{μ} (z, x) \\ \supseteq (h_{μ} (x, z) \cdot h_{μ} (z, x)) \cdot (h_{μ} (y, z) \cdot h_{μ} (z, y)) \end{matrix}$

Hence, $\begin{matrix} h_{ρ_{μ}} (x * y) \\ = \underset{z \in X}{\cap} h_{μ} (x * y, z) \cdot h_{μ} (z, x * y) \\ \supseteq (\underset{z \in X}{\cap} h_{μ} (x, z) \cdot h_{μ} (z, x)) \cdot (\underset{z \in X}{\cap} h_{μ} (y, z) \cdot h_{μ} (z, y)) \\ = h_{ρ_{μ}} (x) \cdot h_{ρ_{μ}} (y) . \end{matrix}$

Therefore, ρ_μ is a HFDS of X.

6 Hesitant fuzzy dot ideals of B-algebras

In this section, hesitant fuzzy dot ideals (HFDI) of B-algebras are defined and studied some related results. It is proved that every HFI is a HFDI but the converse is not true. The relationship between a HFI and the non-empty level set is discussed. Furthermore, the homomorphic image and inverse image of hesitant fuzzy dot subalgebras, as well as the hesitant fuzzy dot ideal, are investigated.

Definition 16. Let A be a HF of X. Then A is called a hesitant fuzzy ideal (briefly HFI) if it satisfies:

h_A (0) ⊇ h_A (x),

h_A (x) ⊇ h_A (x * y) ∩ h_A (y) for all x, y ∈ X.

Definition 17. Let A be HFS of X. Then A is called a hesitant fuzzy dot ideals of X (briefly HFDI) if it satisfies (B4) and

(B6) h_A (x) ⊇ h_A (x * y) · h_A (y) for all x, y ∈ X.

Example 3. Let X = {0, 1, 2, 3} be a B-algebra with the following table:

Table 2
Table2

* 0 1 2 3

0 0 1 2 3

1 1 0 3 2

2 2 3 0 1

3 3 2 1 0

*	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

Obviously, (X, * , 0) is a B-algebra. Define a hesitant fuzzy set in X by h_A (0) = h_A (1) =0.8, h_A (2) =0.6 and h_A (3) =0.5 for all x ∈ X. Then A is a HFDI of X.

Note that if A is a HFI of X, then A is a HFDI of X, but the converse is not true. In fact, A is a HFDI of X in Example 3, but A is not a HFDI of X, because $0.5 = h_{A} (3) \subset 0.6 \cap 0.8 = h_{A} (3 * 1) \cap h_{A} (1) .$

Theorem 13. If A is a HFI of X such that h_A (0 * x) ⊇ h_A (x) for all x ∈ X.

Then A is a HFDI of X.

Proof. Let x, y ∈ X. Then (x * y) * (0 * y) = x by lemma 2, we have $\begin{matrix} h_{A} (x) = h_{A} ((x * y) * (0 * y)) \\ \supseteq h_{A} (x * y) \cdot h_{A} (0 * y) \\ \supseteq h_{A} (x * y) \cdot h_{A} (y) . \end{matrix}$

Therefore, A is a HFDI of X.

Proposition 5. Let A be a HFDI of X and for all x ∈ X, if x ⩽ y in X, then we have h_A (x)⊇ h_A (0) · h_A (y).

Proof. Let x, y ∈ X such that x ⩽ y. Then x * y = 0 and thus h_A (x) ⊇ h_A (x * y) · h_A (y) = h_A (0) · h_A (y).

Proposition 6. Let A be a HFDI of X. If x * y ⩽ z in X for all x, y, z ∈ X, then we have h_A (x) ⊇ h_A (0) · h_A (y) · h_A (z).

Proof. Let x, y, z ∈ X such that x * y ⩽ z. Then (x * y) * z = 0, and thus $\begin{matrix} h_{A} (x) \supseteq h_{A} (x * y) \cdot h_{A} (y) \\ \supseteq h_{A} ((x * y) * z) \cdot h_{A} (z) \cdot h_{A} (y) \\ = h_{A} (0) \cdot h_{A} (z) \cdot h_{A} (y) \\ = h_{A} (0) \cdot h_{A} (y) \cdot h_{A} (z) . \end{matrix}$

Proposition 7. Let A be a HFDI of X, then A^m ∈ HFDI (X), and m is every positive integer.

Proof. For all x ∈ X, A^m ∈ HF (X) and defined by $h_{A^{m}} (x) = h_{A}^{m} (x)$ , where m is any positive integer. Let A ∈ HFDI (X), then we get h_A (0) ⊇ h_A (x) and h_A (x) ⊇ h_A (x * y) · h_A (y) for all x, y ∈ X.

Hence, we have $h_{A}^{m} (0) = [h_{A} (0)]^{m} \supseteq [h_{A} (x)]^{m} = h_{A}^{m} (x)$ and $h_{A}^{m} (x) = [h_{A} (x)]^{m} \supseteq [h_{A} (x * y) \cdot h_{A} (y)]^{m} =$ $[h_{A} (x * y)]^{m} \cdot [h_{A} (y)]^{m} = h_{A}^{m} (x * y) \cdot h_{A}^{m} (y)$ .

Therefore, A^m ∈ HFDI (X).

Proposition 8. If A and A^c are both HFDI of X, then A is a constant function.

Proof. Let A and A^c be both HFDI of X. Then h_A (0) ⊇ h_A (x) and h_{A
^c} (0) ⊇ h_{A
^c} (x),

Therefore, 1 - h_A (0)⊇ 1 - h_A (x). Hence h_A (0) ⊆ h_A (x).

Thus, we have that A is a constant function.

Theorem 14. Let A₁ and A₂ be two HFDI of X, then A₁ ∩ A₂ is also a HFDI of X.

Proof. Let A₁ and A₂ be two HFDI of X. Then for any x ∈ X,

h_{A
₁} (0) ⊇ h_{A
₁} (x) and h_{A
₂} (0) ⊇ h_{A
₂} (x). $\begin{matrix} Now, h_{A_{1} \cap A_{2}} (0) = h_{A_{1}} (0) \cap h_{A_{2}} (0) \\ \supseteq h_{A_{1}} (x) \cap h_{A_{2}} (x) \\ = h_{A_{1} \cap A_{2}} (x) . \end{matrix}$

Again, for any x, y ∈ X, we have $\begin{matrix} h_{A_{1} \cap A_{2}} (x) = h_{A_{1}} (x) \cap h_{A_{2}} (x) \\ \supseteq (h_{A_{1}} (x * y) \cdot h_{A_{1}} (y)) \cap (h_{A_{2}} (x * y) \cdot h_{A_{2}} (y) \\ = (h_{A_{1}} (x * y) \cap h_{A_{2}} (x * y)) \cdot (h_{A_{1}} (y) \cap h_{A_{2}} (y)) \\ = h_{A_{1} \cap A_{2}} (x * y) \cdot h_{A_{1} \cap A_{2}} (y) . \end{matrix}$

Hence, A₁ ∩ A₂ is a HFDI of X.

Theorem 15. Let A and B be two HFDI of X. Then A × B is a HFDI of X × X.

Proof. Let (x₁, y₁) and (x₂, y₂) ∈ X × X.

Then we have $\begin{matrix} h_{A \times B} (x_{1}, y_{1}) \\ = h_{A} (x_{1}) \cdot h_{B} (y_{1}) \\ \supseteq (h_{A} (x_{1} * x_{2}) \cdot h_{A} (x_{2})) \cdot (h_{B} (y_{1} * y_{2}) \cdot h_{B} (y_{2})) \\ = (h_{A} (x_{1} * x_{2}) \cdot h_{B} (y_{1} * y_{2})) \cdot (h_{A} (x_{2}) \cdot h_{B} (y_{2})) \\ = h_{A \times B} (x_{1} * x_{2}, y_{1} * y_{2}) \cdot h_{A \times B} (x_{2}, y_{2}) \\ = h_{A \times B} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \cdot h_{A \times B} (x_{2}, y_{2}) . \end{matrix}$

Hence, A × B is a HFDI of X × X.

Theorem 16. Let A be a HFS of X, A is a HFI of X if and only if for any γ ∈ P ([0, 1]), a non-empty level subset R (A, γ) : ={ x ∈ A|γ ⊆ h_A (x) } is an ideal of X.

Proof. Let x ∈ X, for any γ ∈ P ([0, 1]) with R (A, γ)≠ ø, there exists x ∈ X such as h_A (x) ⊇ γ, Since A is a HFI of X, we have h_A (0) ⊇ h_A (x) ⊇ γ, therefore 0 ∈ R (A, γ).

Let x, y ∈ X be such that y, x * y ∈ R (A, γ), for any γ ∈ P ([0, 1]), then h_A (y) ⊇ γ, h_A (x * y) ⊇ γ.

Therefore h_A (x) ⊇ h_A (x * y) ∩ h_A (y) ⊇ γ ∩ γ = γ.

Then we have x ∈ R (A, γ).

Conversely, let R (A, γ) be an ideal of X, for any γ ∈ P ([0, 1]) with R (A, γ)≠ ø. For any x ∈ X, Put h_A (x) = γ, then we have x ∈ R (A, γ). Since R (A, γ) is an ideal of X, hence 0 ∈ R (A, γ) and so h_A (0) ⊇ γ ⊇ h_A (x).

For any x, y ∈ X, Put h_A (x * y) = α, h_A (y) = β, and let γ = α ∩ β. Then we have y, x * y ∈ R (A, γ).

Since R (A, γ) is an ideal of X, Hence x ∈ R (A, γ).

Therefore, h_A (x) ⊇ γ = α ∩ β = h_A (x * y) ∩ h_A (y).

Remark 3. Let A be a HFDI of X, then R (A, γ) need not be an ideal of X. In Example 3, A is a HFDI of X but R (A, 0.6) : ={ x ∈ A|h_A (x) ⊇0.6 } = {0, 1, 2} is not an ideal of X since 1 * 2 =3 ∉ R (A, 0.6).

Theorem 17. Let A be a HFDI of X, then R (A, 1) : ={ x ∈ A|h_A (x) =1 } is either empty or an ideal of X.

Proof. Assume that R (A, 1)≠ ø. Obviously, 0 ∈ R (A, 1).

Let x, y ∈ X such that x * y and y ∈ R (A, 1). Then h_A (x * y) =1 = h_A (y).

It follows that h_A (x)⊇ h_A (x * y) · h_A (y) =1.

So, h_A (x) =1 i.e., x ∈ R (A, 1).

Therefore, R (A, 1) is an ideal of X.

Theorem 18. Let f : X → Y be a homomorphism of B-algebras. If B is a HFDI of Y, then f^-1 (B) of B under f is a HFDI of X.

Proof. Suppose that B is a HFDI of Y.

For all x ∈ X, h_f^-1(B) (x) = h_B (f (x))⊆ h_B (0) = h_B (f (0)) = h_f^-1(B) (0).

Again let x, y ∈ X. Then $\begin{matrix} h_{f^{- 1} (B)} (x) = h_{B} (f (x)) \\ \supseteq h_{B} (f (x) * f (y)) \cdot h_{B} (f (y)) \\ \supseteq h_{B} (f (x * y)) \cdot h_{B} (f (y)) \\ = h_{f^{- 1} (B)} (x * y) \cdot h_{f^{- 1} (B)} (y) . \end{matrix}$

Hence, f^-1 (B) is a HFDI of X.

Theorem 19. Let f : X → Y be an epimorphism of B-algebras. If f^-1 (B) of B under f in X is a HFDI of X, then B is a HFDI of Y.

Proof. For any x ∈ Y, there exists a ∈ X such that f (a) = x. Then $\begin{matrix} h_{B} (x) = h_{B} (f (a)) \\ = h_{f^{- 1} (B)} (a) \subseteq h_{f^{- 1} (B)} (0) \\ = h_{B} (f (0)) = h_{B} (0) . \end{matrix}$

Let x, y ∈ Y. Then f (a) = x and f (b) = y for some a, b ∈ X. Thus $\begin{matrix} h_{B} (x) = h_{B} (f (a)) \\ = h_{f^{- 1} (B)} (a) \\ \supseteq h_{f^{- 1} (B)} (a * b) \cdot h_{f^{- 1} (B)} (b) \\ = h_{B} (f (a * b)) \cdot h_{B} (f (b)) \\ = h_{B} (f (a) * f (b)) \cdot h_{B} (f (b)) \\ = h_{B} (x * y) \cdot h_{B} (y) . \end{matrix}$

Therefore, B is a HFDI of Y.

Theorem 20. Let μ_ρ be the strongest hesitant fuzzy ρ- relation on X, where ρ is a subset of X.

Then ρ is a HFDI of X if and only if μ_ρ is a HFDI of X × X.

Proof. Assume that ρ is a HFDI of X. For any x, y ∈ X, we have

h_{μ_ρ} (0, 0) = h_ρ (0) · h_ρ (0) ⊇ h_ρ (x) · h_ρ (y) = h_{μ_ρ} (x, y).

Let (x₁, y₁) and (x₂, y₂) ∈ X × X. Then we have $\begin{matrix} h_{μ_{ρ}} (x_{1}, y_{1}) \\ = h_{ρ} (x_{1}) \cdot h_{ρ} (y_{1}) \\ = (h_{ρ} (x_{1} * x_{2}) \cdot h_{ρ} (x_{2})) \cdot (h_{ρ} (y_{1} * y_{2}) \cdot h_{ρ} (y_{2})) \\ = (h_{ρ} (x_{1} * x_{2}) \cdot h_{ρ} (y_{1} * y_{2})) \cdot (h_{ρ} (x_{2}) \cdot h_{ρ} (y_{2})) \\ = (h_{μ_{ρ}} (x_{1} * x_{2}, y_{1} * y_{2}) \cdot h_{μ_{ρ}} (x_{2}, y_{2}) \\ = h_{μ_{ρ}} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \cdot h_{μ_{ρ}} (x_{2}, y_{2}) . \end{matrix}$

Hence μ_ρ is a HFDI of X × X.

Conversely, assume that μ_ρ is a HFDI of X × X. By applying (B4), we get (h_ρ (0)) ² = h_{μ_ρ} (0, 0) ⊇ h_{μ_ρ} (x, x) = (h_ρ (x)) ² and so h_ρ (0) ⊇ h_ρ (x) for all x ∈ X. Next, we have $\begin{matrix} (h_{ρ} (x))^{2} = h_{μ_{ρ}} (x, x) \\ \supseteq h_{μ_{ρ}} ((x, x) * (y, y)) \cdot h_{μ_{ρ}} (y, y) \\ = h_{μ_{ρ}} (x * y, x * y) \cdot h_{μ_{ρ}} (y, y) \\ = (h_{ρ} (x * y) \cdot h_{ρ} (y))^{2} . \end{matrix}$

Which implies that h_ρ (x) = h_ρ (x * y) · h_ρ (y) for all x, y ∈ X. Therefore, ρ is a HFDI of X.

7 Conclusion

In this paper, we drew into the concept of hesitant fuzzy dot subalgebras (hesitant fuzzy normal dot subalgebras / hesitant fuzzy dot ideals) of B-algebra, the equivalent characterizations and properties of them are studied. In our opinion, results obtained in this paper can be generalized to extend other algebraic systems including BF-algebras and MV-algebras. We hope that this paper will establish a foundation for further studies in the theory of other logical-algebras.

Footnotes

Acknowledgments

This work was Supported by the Shaanxi Natural Science Foundation (NO. 2022JM-053). The authors are very grateful to the referees for their valuable comments and suggestions for improving this paper.

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Hesitant fuzzy dot subalgebra and dot ideals of B-algebra

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Hesitant fuzzy dot subalgebras of B-algebras

Table 1 Table1 * 0 1 2 3 4 5 0 0 2 1 3 4 5 1 1 0 2 4 5 3 2 2 1 0 5 3 4 3 3 4 5 0 2 1 4 4 5 3 1 0 2 5 5 3 4 2 1 0

5 Hesitant fuzzy ρ-product relation of B-algebras

6 Hesitant fuzzy dot ideals of B-algebras

Table 2 Table2 * 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0

Footnotes

Acknowledgments

References

Table 1
Table1

* 0 1 2 3 4 5

0 0 2 1 3 4 5

1 1 0 2 4 5 3

2 2 1 0 5 3 4

3 3 4 5 0 2 1

4 4 5 3 1 0 2

5 5 3 4 2 1 0

Table 2
Table2

* 0 1 2 3

0 0 1 2 3

1 1 0 3 2

2 2 3 0 1

3 3 2 1 0