Interval-valued hesitant fuzzy filters in BE-algebras

Abstract

In this paper, we develop fuzzy filter theory on a BE-algebra and apply the concept of interval-valued hesitant fuzzy filter to BE-algebras. Moreover, some types of interval-valued hesitant fuzzy filters such as interval-valued hesitant fuzzy implicative filter and interval-valued hesitant fuzzy fantastic filter are introduced and some of properties are investigated.

Keywords

(distributive) BE-algebra hesitant fuzzy (implicative,fantastic) filter interval-valued hesitant fuzzy (implicative,fantastic) filter

1 Introduction

The concept of fuzzy set was introduced by Zadeh [20]. Fuzzy sets and its extensions have provided successful results dealing with uncertainty in different problems. Worldwide, there has been a rapid growth in interest in applications of fuzzy sets and some generalization of this is discussed by authors such as intuitionistic fuzzy sets, interval-valued fuzzy sets, type-n fuzzy sets and fuzzy multi sets. Intuitionistic fuzzy sets (IFSs, for short) were introduced in 1983 by Atanassov [2]. Interval-valued fuzzy sets (IVFSs, for short), apparently first studied by Sambuc [15] who called them φ-flou functions, serve to capture a feature of uncertainty w.r.t. the assignment of membership degrees. Also, another generalization of this theory was proposed by Torra and Narukawa [17] and Torra [16] as hesitant fuzzy set (HFS), which permitting a set of several possible membership values. Then some researchers who have defined divers concepts, extensions, aggregation operators, decision making and measures to handle with hesitant information [1 , 19]. Z. Pei et al. investigated some properties of operators and algebraic structure of hesitant fuzzy sets with confidence levels and based on closed interval-valued hesitant fuzzy sets, an equivalence relation on hesitant fuzzy sets was defined, then lattices and distributive lattices on hesitant fuzzy sets were constructed [9].

H.S. Kim and Y.H. Kim introduced the notion of a BE-algebra as a generalization of a dual BCK-algebra [7]. A. Borumand Saeid et al. defined some types of filters in BE-algebras and showed the relationship between them [4]. Recently, A. Rezaei et al. introduced the notion of hesitant fuzzy (implicative) filters and got some useful properties [13].

In this paper, we introduce the notion of interval-valued hesitant fuzzy (implicative, fantastic) filters and get some useful properties. In fact, we show that in self distributive BE-algebras two concepts of interval-valued hesitant fuzzy implicative filter and interval-valued hesitant fuzzy filter are equivalent. Translation of interval-valued hesitant fuzzy filters is studied.

2 Preliminaries

In this section, we cite the fundamental definitions that will be used in the sequel:

Definition 2.1. [7] By a BE-algebra we shall mean an algebra $X = (X; *, 1)$ of type (2, 0) satisfying the following axioms:

x ∗ x = 1,

x ∗ 1 =1,

1 ∗ x = x,

x ∗ (y ∗ z) = y ∗ (x ∗ z), for all x, y, z ∈ X .

From now on $X$ is a BE-algebra, unless otherwise is stated. We introduce a relation “≤” on $X$ by x ≤ y if and only if x ∗ y = 1. A BE-algebra $X$ is said to be a self distributive if x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z), for all x, y, z ∈ X.

Proposition 2.2. [7] In a BE-algebra $X$ , the following hold:

x ∗ (y ∗ x) =1,

y ∗ ((y ∗ x) ∗ x) =1, for all x, y ∈ X .

Proposition 2.3. [11] Let $X$ be a self distributive BE-algebra. The following hold:

if x ≤ y, then z ∗ x ≤ z ∗ y,

y ∗ z ≤ (x ∗ y) ∗ (x ∗ z), for all x, y, z ∈ X .

From now on, $X$ is a BE-algebra unless otherwise specified.

A subset F of X is called a filter of $X$ if it satisfies: (F1) 1 ∈ F, (F2) x ∈ F and x ∗ y ∈ F imply y ∈ F. Define $A (x, y) = {z \in X : x * (y * z) = 1},$ which is called an upper set of x and y. It is easy to see that 1, x, y ∈ A (x, y), for any x, y ∈ X . Every upper set A (x, y) need not be a filter of $X$ in general.

Definition 2.4. [4] A non-empty subset F of X is called an implicative filter of $X$ if satisfies the following conditions:

1 ∈ F,

x ∗ (y ∗ z) ∈ F and x ∗ y ∈ F imply that x ∗ z ∈ F, for all x, y, z ∈ X .

Definition 2.5. [12] A fuzzy set μ of X is called a fuzzy filter of $X$ if satisfies the following conditions:

μ (1) ≥ μ (x),

μ (y) ≥ min {μ (x ∗ y), μ (x)}, for allx, y ∈ X.

Definition 2.6. [10] A fuzzy set μ of X is called a fuzzy implicative filter of $X$ if satisfies the following conditions:

μ (1) ≥ μ (x),

μ (x ∗ z) ≥ min {μ (x ∗ (y ∗ z)), μ (x ∗ y)}, for all x, y, z ∈ X.

Definition 2.7. [16] Let X be a reference set. A hesitant fuzzy set (HFS, for short) A in $X$ is represented mathematical as: $A = {< x, h_{A} (x) > : h_{A} (x) \in ρ ([0, 1]), x \in X},$ where ρ ([0, 1]) is the power set of [0, 1].

So, we can define a set of fuzzy sets an HFS by union of their membership functions.

Definition 2.8. [18] Let $h, h_{1}, h_{2} \in HFE (X)$ and λ ∈ (0, 1]. The operations, complement, union and intersection are defined as follows:

h^c = {1 - γ : γ ∈ h},

h₁ ⊔ h₂ = {max(γ₁, γ₂) : γ₁ ∈ h₁, γ₂ ∈ h₂},

h₁ ⊓ h₂ = {min(γ₁, γ₂) : γ₁ ∈ h₁, γ₂ ∈ h₂},

h₁ ⊕ h₂ = {γ₁ + γ₂ - γ₁γ₂ : γ₁ ∈ h₁, γ₂ ∈ h₂},

h₁ ⊗ h₂ = {γ₁γ₂ : γ₁ ∈ h₁, γ₂ ∈ h₂},

λh = {1 - (1 - γ) ^λ : γ ∈ h}.

Definition 2.9. [13] Hesitant fuzzy set A of $X$ is called a hesitant fuzzy filter if satisfies the following conditions:

h_A (x) ⊑ h_A (1),

h_A (x) ⊓ h_A (x ∗ y) ⊑ h_A (y), for allx, y ∈ X .

Denote the set of all hesitant fuzzy filters of $X$ by HFF ( $X$ ).

Theorem 2.10. [13] Let $h, h_{1}, h_{2} \in HFF (X)$ and λ ∈ [0, 1]. Then

$h_{1} ⊔ h_{2} \in HFF (X)$ ,

$h_{1} ⊓ h_{2} \in HFF (X)$ ,

$h_{1} \oplus h_{2} \in HFF (X)$ ,

$h_{1} \otimes h_{2} \in HFF (X)$ ,

$λ h \in HFF (X)$ .

Theorem 2.11. [13] Algebra $(HFF (X); ⊔, ⊓,^{c}, \tilde{0}, \tilde{1})$ of type (2, 2, 1, 0, 0) is a complete lattice, but it is not a Boolean algebra.

By an interval $\tilde{a}$ we mean an interval [a^-, a⁺] where a^-, a⁺ are real numbers and 0 ≤ a^- ≤ a⁺ ≤ 1 .The set of all intervals is denoted by D [0, 1]. The interval [a, a] is identified with the real number a.

For intervals ${\tilde{a}}_{i} = [a_{i}^{-}, a_{i}^{+}]$ , ${\tilde{b}}_{i} = [b_{i}^{-}, b_{i}^{+}]$ , where i ∈ I, I is an index set, we define $r max {{\tilde{a}}_{i}, {\tilde{b}}_{i}} = [max {a_{i}^{-}, b_{i}^{-}}, max {a_{i}^{+}, b_{i}^{+}}]$ , $r min {{\tilde{a}}_{i}, {\tilde{b}}_{i}} = [min {a_{i}^{-}, b_{i}^{-}}, min {a_{i}^{+}, b_{i}^{+}}]$ , $\land_{i \in I} {\tilde{a}}_{i} = [\land_{i \in I} a_{i}^{-}, \land_{i \in I} a_{i}^{+}]$ , $\lor_{i \in I} {\tilde{a}}_{i} = [\lor_{i \in I} a_{i}^{-}, \lor_{i \in I} a_{i}^{+}]$ .

Furthermore, we have

${\tilde{a}}_{i} \leq {\tilde{b}}_{i}$ if and only if $a_{i}^{-} \leq b_{i}^{-}$ and $a_{i}^{+} \leq b_{i}^{+}$ ,

${\tilde{a}}_{i} = {\tilde{b}}_{i}$ if and only if $a_{i}^{-} = b_{i}^{-}$ and $a_{i}^{+} = b_{i}^{+}$ ,

$k \tilde{a} = [{ka}_{i}^{-}, {ka}_{i}^{+}]$ , where 0 ≤ k ≤ 1.

Therefore it can be shown that (D [0, 1], ≤, ∧, ∨) is a complete lattice, $\tilde{0} = [0, 0]$ as its least element and $\tilde{1} = [1, 1]$ as its greatest element.

By an interval-valued fuzzy set F on $X$ , we mean that set $F = {< x, [μ_{F}^{-} (x), μ_{F}^{+} (x)] > | x \in X},$ where $μ_{F}^{-}$ and $μ_{F}^{+}$ are two fuzzy set of $X$ such that $μ_{F}^{-} (x) \leq μ_{F}^{+} (x)$ , for any x ∈ X. Putting ${\tilde{μ}}_{F} (x) = [μ_{F}^{-} (x), μ_{F}^{+} (x)]$ , we see that $F = {(x, {\tilde{μ}}_{F} (x)) | x \in X}$ , where ${\tilde{μ}}_{F} : X \to D [0, 1]$ .

For any $\tilde{t} \in D [0, 1]$ , the set $U (F; \tilde{t}) = {x \in X | {\tilde{μ}}_{F} (x) \geq \tilde{t}}$ is called the interval-valued level subset of $X$ [5].

Definition 2.12. [6] Let X be a reference set. An interval-valued hesitant fuzzy set (IVHFS) on $X$ is $\tilde{F} = {< x, h_{\tilde{F}} (x) > | x \in X},$ where $h_{\tilde{F}} (x)$ denotes all possible interval-valued membership degrees of the element x ∈ X to the set $\tilde{F}$ . For convenience, we call $h_{\tilde{F}} (x)$ an interval-valued hesitant fuzzy element (IVHFE), which reads $h_{\tilde{F}} (x) = {\tilde{r} | \tilde{r} \in h_{\tilde{F}} (x)} .$ Here $\tilde{r}$ is an interval number. An IVHFE is the basic unit of an IVHFS, and it can be considered as a special case of the IVHFS. The relationship between IVHFE and IVHFS is similar to that between interval-valued fuzzy number and interval-valued fuzzy set.

It can easily seen that both hesitant fuzzy sets and interval-valued fuzzy sets are all particular cases of interval-valued hesitant fuzzy sets.

Definition 2.13. [6] Let X be a reference set and $\tilde{F}$ be an interval-valued hesitant fuzzy set on $X$ . For any x, y ∈ X,

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (y)$ if and only if for any interval $\tilde{r} \in h_{\tilde{F}} (x)$ , there exists an interval $\tilde{s} \in h_{\tilde{F}} (y)$ such that $\tilde{r} \leq \tilde{s} .$

$h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) = {r min {\tilde{r}, \tilde{s}} | \tilde{r} \in h_{\tilde{F}} (x), \tilde{s} \in h_{\tilde{F}} (y)} .$

3 Interval-valued hesitant fuzzy filters

In this section, we will apply the interval-valued fuzzy set to generalize the notion of hesitant fuzzy filters in BE-algebras.

Definition 3.1. Interval-valued hesitant fuzzy set $\tilde{F}$ of $X$ is called an interval-valued hesitant fuzzy filter if satisfies the following conditions:

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1),$

$h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} (y)$ , for all x, y ∈ X .

Denote the set of all interval-valued hesitant fuzzy filters of $X$ by IVHFF ( $X$ ).

Note. It can be seen that both hesitant fuzzy filters and interval-valued fuzzy filters are all particular cases of interval-valued hesitant fuzzy filters.

Example 3.2. Let X : = {1, a, b, c} be a set with the following table:

∗	1	a	b	c
1	1	a	b	c
a	1	1	b	c
b	1	a	1	c
c	1	1	b	1

Then $X = (X; *, 1)$ is a BE-algebra. For anyx ∈ X, we define $h_{\tilde{F}} (x)$ as follows:

$h_{\tilde{F}} (1) = {[\frac{5}{6}, \frac{8}{10}]}$ , $h_{\tilde{F}} (a) = {[\frac{1}{5}, \frac{1}{3}], [\frac{2}{3}, \frac{3}{4}]}$ and $h_{\tilde{F}} (b) = h_{\tilde{F}} (c) = {[\frac{2}{5}, \frac{3}{5}], [\frac{2}{3}, \frac{4}{5}]}$ .

Then $\tilde{F} = {< 1, {[\frac{5}{6}, \frac{8}{10}]} >, < a, {[\frac{1}{5}, \frac{1}{3}], [\frac{2}{3}, \frac{3}{4}]} >, < b, {[\frac{2}{5}, \frac{3}{5}], [\frac{2}{3}, \frac{4}{5}]} >, < c, {[\frac{2}{5}, \frac{3}{5}]} >}$ is an interval-valued hesitant fuzzy filter of $X$ .

Proposition 3.3. Let $\tilde{F} \in IVHFF (X)$ and x, y, z, a, b, a_i ∈ X for i = 1, …, n. Then

if x ≤ y, then $h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (y),$

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (y * x)$ ,

$h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} (x * y)$ ,

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} ((x * y) * y)$ ,

$h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} ((x * (y * z)) * z)$ ,

if $h_{\tilde{F}} (y) ⊓ h_{\tilde{F}} ((x * y) * z) ⊑ h_{\tilde{F}} (z * x)$ , then $h_{\tilde{F}}$ is an order reversing (i.e., if x ≤ y, then $h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} (x)$ ),

if z ∈ A (x, y), then $h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} (z)$ ,

if $\prod_{i = 1}^{n} a_{i} * x = 1$ , then $⊓_{i = 1}^{n} h_{\tilde{F}} (a_{i}) ⊑ h_{\tilde{F}} (x)$ , where $\prod_{i = 1}^{n} a_{i} * x = a_{n} * (a_{n - 1} * (\dots (a_{1} * x) \dots)) .$

Proof (i). Let x ≤ y. Then x ∗ y = 1 and so $\begin{matrix} h_{\tilde{F}} (x) & = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (1) \\ = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (x * y) \\ ⊑ & h_{\tilde{F}} (y) . \end{matrix}$

(ii). Since x ≤ y ∗ x, by using (i) we have $h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (y * x)$ .

(iii). By using (ii) we have $h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} (x * y) .$

(iv). It follows from Definition 3.1 that $\begin{matrix} h_{\tilde{F}} (x) & = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (1) \\ = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} ((x * y) * (x * y)) \\ = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (x * ((x * y) * x)) \\ ⊑ & h_{\tilde{F}} ((x * y) * y) . \end{matrix}$

(v). From (iv) we have $\begin{matrix} h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) & ⊑ & h_{\tilde{F}} ((x * (y * x)) * (y * x)) ⊓ h_{\tilde{F}} (y) \\ ⊑ & h_{\tilde{F}} ((x * (y * z)) * z) . \end{matrix}$

(vi). Let x ≤ y, that is, x ∗ y = 1. Then $\begin{matrix} h_{\tilde{F}} (y) & = & h_{\tilde{F}} (y) ⊓ h_{\tilde{F}} (1 * 1) \\ = & h_{\tilde{F}} (y) ⊓ h_{A} ((x * y) * 1) \\ ⊑ & h_{\tilde{F}} (1 * x) \\ = & h_{\tilde{F}} (x) . \end{matrix}$

(vii). Let z ∈ A (x, y) . Then x ∗ (y ∗ z) =1. Hence $\begin{matrix} h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) & = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊓ h_{\tilde{F}} (1) \\ = & h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊓ h_{\tilde{F}} (x * (y * z)) \\ ⊑ & h_{\tilde{F}} (y) ⊓ h_{\tilde{F}} (y * z) \\ ⊑ & h_{\tilde{F}} (z) . \end{matrix}$

(viii). The proof is by induction on n. By (vii) it is true for n = 1, 2. Assume that it satisfies for n = k, that is, $\prod_{i = 1}^{k} a_{i} * x = 1 \Rightarrow ⊓_{i = 1}^{k} h_{\tilde{F}} (a_{i}) ⊑ h_{\tilde{F}} (x),$ for all a₁, …, a_k, x ∈ X . Suppose that $\prod_{i = 1}^{k + 1} a_{i} * x = 1$ , for all a₁, …, a_k, a_k+1, x ∈ X . Then $⊓_{i = 2}^{k + 1} h_{\tilde{F}} (a_{i}) ⊑ h_{\tilde{F}} (a_{1} * x) .$

Since $\tilde{F}$ is a hesitant fuzzy filter of $X$ , we have $\begin{matrix} ⊓_{i = 1}^{k + 1} h_{\tilde{F}} (a_{i}) & = & (⊓_{i = 2}^{k + 1} h_{\tilde{F}} (a_{i})) ⊓ h_{\tilde{F}} (a_{1}) \\ ⊑ & h_{\tilde{F}} (a_{1} * x) ⊓ h_{\tilde{F}} (a_{1}) \\ ⊑ & h_{\tilde{F}} (x) . □ \end{matrix}$

Proposition 3.4. Let $X$ be a self distributive BE-algebra and $\tilde{F} \in IVHFF (X)$ . If x ≤ y, then $h_{\tilde{F}} (z * x) ⊑ h_{\tilde{F}} (z * y) .$

Proof. Let $\tilde{F} \in IVHFF (X)$ . Since $X$ is self distributive and x ≤ y, we have z ∗ x ≤ y ∗ z by Proposition 2.3(i). It follows from Proposition 3.3(i) that $h_{\tilde{F}} (z * x) ⊑ h_{\tilde{F}} (z * y) .$ □

Proposition 3.5. Let ${{\tilde{F}}_{α} | α \in Δ}$ be an indexed family of interval-valued hesitant fuzzy filter of $X$ . Then $⊓_{α \in Δ} {\tilde{F}}_{α}$ , is also an interval-valued hesitant fuzzy filter of $X$ .

Theorem 3.6. Let $X$ be self distributive BE-algebraand $\tilde{F} \in IVHFS (X)$ . Then $\tilde{F} \in IVHFF (X)$ if and only if $\tilde{F}$ satisfies following two conditions:

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1)$ ,

$h_{\tilde{F}} (x * y) ⊓ h_{\tilde{F}} (y * z) ⊑ h_{\tilde{F}} (x * z)$ , for all x, y, z ∈ X .

Proof. Assume that $\tilde{F} \in IVHFF (X)$ . Since $X$ is self distributive BE-algebra, we have x ∗ y ≤ (y ∗ z) ∗ (x ∗ z), it follows from Proposition 3.3(i) that $h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} ((y * z) * (x * z)) .$

Also, since $\tilde{F} \in IVHFF (X)$ , we have $h_{\tilde{F}} (y * z) ⊓ h_{\tilde{F}} ((y * z) * (x * z)) ⊑ h_{\tilde{F}} (x * z) .$

Therefore, $\begin{matrix} h_{\tilde{F}} (x * y) ⊓ h_{\tilde{F}} (y * z) \\ ⊑ h_{\tilde{F}} ((y * z) * (x * z)) ⊓ h_{\tilde{F}} (y * z) \\ ⊑ h_{\tilde{F}} (x * z) . \end{matrix}$

Conversely, if put x = 1 in (ii), by (BE3), we have $\begin{matrix} h_{\tilde{F}} (y) ⊓ h_{\tilde{F}} (y * z) & = & h_{\tilde{F}} (1 * y) ⊓ h_{\tilde{F}} (y * z) \\ ⊑ & h_{\tilde{F}} (1 * z) \\ = & h_{\tilde{F}} (z) . \end{matrix}$

Therefore, $\tilde{F} \in IVHFF (X)$ . □

Theorem 3.7. Let $\tilde{F} \in IVHFS (X)$ . Then $\tilde{F} \in IVHFF (X)$ if and only if $\tilde{F}$ satisfies the following conditions:

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (y * x)$ ,

$h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (y) ⊑ h_{\tilde{F}} ((x * (y * z)) * z)$ , for all x, y, z ∈ X .

Proof. Assume that $\tilde{F} \in IVHFF (X)$ . The proof is similar to the proof of Proposition 3.3(ii) and (v).

Conversely, let $\tilde{F} \in IVHFS (X)$ . If we put y = x in (i), by (BE1), we have $h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (x * x) = h_{\tilde{F}} (1) .$

Also, by using (ii), (BE3) and (BE1), we have $\begin{matrix} h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (x * y) & ⊑ & h_{\tilde{F}} ((x * ((x * y) * y)) * y) \\ = & h_{\tilde{F}} (((x * y) * (x * y)) * y) \\ = & h_{\tilde{F}} (1 * y) \\ = & h_{\tilde{F}} (y) . \end{matrix}$

Therefore, $\tilde{F} \in IVHFF (X)$ . □

Theorem 3.8. Let $\tilde{F} \in IVHFS (X)$ . Then $\tilde{F} \in IVHFF (X)$ if and only if the set ${\tilde{F}}_{\tilde{O}} : = {x \in X | \tilde{O} ⊑ h_{\tilde{F}} (x)}$ is a filter of $X$ , for all $\tilde{O} \in ρ (D [0, 1])$ with ${\tilde{F}}_{\tilde{O}} \neq \emptyset .$

Proof. Let $\tilde{F} \in IVHFF (X)$ and $\tilde{O} \in ρ (D [0, 1])$ be such that $x, x * y \in {\tilde{F}}_{\tilde{O}} .$ Then $\tilde{O} ⊑ h_{\tilde{F}} (x)$ and $\tilde{O} ⊑ h_{\tilde{F}} (x * y) .$ It follows from (IVHFF1) that $\tilde{O} ⊑ h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1)$ and so $1 \in {\tilde{F}}_{\tilde{O}}$ . On the other hand, we have $\tilde{O} = \tilde{O} ⊓ \tilde{O} ⊑ h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} (y) .$

Hence $y \in {\tilde{F}}_{\tilde{O}} .$ Therefore, ${\tilde{F}}_{\tilde{O}} \in F (X) .$

Conversely, assume that ${\tilde{F}}_{\tilde{O}} \in F (X),$ for all $\tilde{O} \in ρ (D [0, 1])$ with ${\tilde{F}}_{\tilde{O}} \neq \emptyset .$ Let x ∈ X. Put $h_{\tilde{F}} (x) = \tilde{α}$ . Then $x \in {\tilde{F}}_{\tilde{α}}$ . Now, since ${\tilde{F}}_{\tilde{α}}$ is a filter of $X$ , then $1 \in {\tilde{F}}_{\tilde{α}}$ . Hence $\tilde{α} = h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1) .$

Also, for any x, y ∈ X, let $h_{\tilde{F}} (x) = \tilde{α}$ and $h_{\tilde{F}} (x * y) = \tilde{β}$ . Hence $x \in {\tilde{F}}_{\tilde{α} ⊓ \tilde{β}}$ and $x * y \in {\tilde{F}}_{\tilde{α} ⊓ \tilde{β}}$ , which imply that $y \in {\tilde{F}}_{\tilde{α} ⊓ \tilde{β}}$ and so $\tilde{α} ⊓ \tilde{β} ⊑ h_{\tilde{F}} (y) .$ Then $h_{\tilde{F}} (x) ⊓ h_{\tilde{F}} (x * y) = \tilde{α} ⊓ \tilde{β} ⊑ h_{\tilde{F}} (y) .$

Therefore, $\tilde{F} \in IVHFF (X)$ . □

Corollary 3.9. Let $\tilde{F} \in IVHFS (X)$ and a ∈ X be such that $h_{\tilde{F}} (a) \neq \emptyset$ . Then the set ${x \in X | h_{\tilde{F}} (a) ⊑ h_{\tilde{F}} (x)}$ is a filter of $X$ .

4 Interval-valued hesitant fuzzy implicative filters

Definition 4.1. Interval-valued hesitant fuzzy set $\tilde{F}$ of $X$ is called an interval-valued hesitant fuzzy implicative filter if satisfies the following conditions:

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1),$

$h_{\tilde{F}} (x * (y * z)) ⊓ h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} (x * z)$ , for all x, y, z ∈ X .

Denote the set of all interval-valued hesitant fuzzy implicative filters on $X$ by IVHFIF( $X$ ).

Taking x = 1 in (IVHFIF2), we can see that every interval-valued hesitant fuzzy implicative filter is an interval-valued hesitant fuzzy filter.

Theorem 4.2. Let $X$ be a self distributive BE-algebra. Then every interval-valued hesitant fuzzy filter is an interval-valued hesitant fuzzy implicative filter.

Proof. Let $\tilde{F} \in IVHFF (X)$ . Obvious that $h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1)$ , for all x ∈ X . By using self distributivity and (IVHFF2), we have $h_{\tilde{F}} (x * (y * z)) ⊓ h_{\tilde{F}} (x * y) = h_{\tilde{F}} ((x * y) * (x * z)) ⊓ h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} (x * z) .$

Therefore, $\tilde{F} \in IVHFIF (X)$ . □

In the following example shows that the condition, self distributivity of Theorem 4.2, is necessary.

Example 4.3. [10] Let X = {1, a, b, c, d} with the following table:

→	1	a	b	c	d
1	1	a	b	c	d
a	1	1	b	c	d
b	1	a	1	b	a
c	1	a	1	1	a
d	1	1	1	b	1

Then $X = (X; *, 1)$ is a BE-algebra, but it is not self distributive, because $\begin{matrix} a * (b * d) & = & a * a = 1 \neq (a * b) * (a * d) \\ = & b * d = a . \end{matrix}$

For any x ∈ X, we define $h_{\tilde{F}} (x)$ to a set $\tilde{F}$ as follows: $h_{\tilde{F}} (1) = h_{\tilde{F}} (a) = {[\frac{9}{16}, \frac{15}{16}]}$ , $h_{\tilde{F}} (b) = h_{\tilde{F}} (b) = h_{\tilde{F}} (d) = {[\frac{11}{16}, \frac{12}{16}], [\frac{14}{16}, \frac{15}{16}]}$ . Then $\tilde{F} = {< 1, {[\frac{9}{16}, \frac{15}{16}]} >, < a, {[\frac{11}{16}, \frac{12}{16}], [\frac{14}{16}, \frac{15}{16}]} >, < b, {[\frac{11}{16}, \frac{12}{16}], [\frac{14}{16}, \frac{15}{16} [} >, < c, {[\frac{11}{16}, \frac{12}{16}], [\frac{14}{16}, \frac{15}{16}]} >, < d, {[\frac{11}{16}, \frac{12}{16}], [\frac{14}{16}, \frac{15}{16}]} >}$ is an interval-valued hesitant fuzzy filter, but it is not an interval-valued hesitant fuzzy implicative filter because, $\begin{matrix} h_{\tilde{F}} (b) & = & h_{\tilde{F}} (b * c) \\ ⊏ & h_{\tilde{F}} (b * (d * c)) ⊓ h_{\tilde{F}} (b * d) \\ = & h_{\tilde{F}} (1) ⊓ h_{\tilde{F}} (a) \\ = & h_{\tilde{F}} (a) . \end{matrix}$

Theorem 4.4. Let $h, h_{1}, h_{2} \in IVHFIF (X)$ and λ ∈ [0, 1]. Then

$h_{1} ⊔ h_{2} \in IVHFIF (X)$ ,

$h_{1} ⊓ h_{2} \in IVHFIF (X)$ ,

$h_{1} \oplus h_{2} \in IVHFIF (X)$ ,

$h_{1} \otimes h_{2} \in IVHFIF (X)$ ,

$λ h \in IVHFIF (X)$ .

Theorem 4.5. Algebra $(IVHFIF (X); ⊑, ⊔, ⊓, \tilde{0}, \tilde{1})$ of type (2, 2, 2, 0, 0) is a complete lattice.

Theorem 4.6. Let $X$ be a self distributive BE-algebra and $F \in HFF (X)$ . Then the following conditions are equivalent:

$\tilde{F} \in IVHFIF (X)$ ,

$h_{\tilde{F}} (y * (y * x)) ⊑ h_{\tilde{F}} (y * x)$ ,

$h_{\tilde{F}} ((z * (y * (y * x)))) ⊓ h_{\tilde{F}} (z) ⊑ h_{\tilde{F}} (y * x),$ for all x, y, z ∈ X .

Proof. (i) ⇒ (ii). Let $\tilde{F} \in IVHFIF (X)$ . By (IVHFIF1) and (BE1) we have $\begin{matrix} h_{\tilde{F}} (y * (y * x) & = & h_{\tilde{F}} (y * (y * x)) ⊓ h_{\tilde{F}} (1) \\ = & h_{\tilde{F}} (y * (y * x)) ⊓ h_{\tilde{F}} (y * y) \\ ⊑ & h_{\tilde{F}} (y * x) . \end{matrix}$

(ii) ⇒ (iii). Let $\tilde{F}$ be a hesitant fuzzy filter of $X$ satisfying the condition (ii). By using (IVHFIF2) and (ii) we have $\begin{matrix} h_{\tilde{F}} (z * (y * (y * x))) & ⊓ & h_{\tilde{F}} (z) ⊑ h_{\tilde{F}} (y * (y * x)) \\ ⊑ & h_{\tilde{F}} (y * x) . \end{matrix}$

(iii) ⇒ (i). Since $x * (y * z) = y * (x * z) \leq (x * y) * (x * (x * z)) .$

Hence $h_{\tilde{F}} (x * (y * z)) ⊑ h_{\tilde{F}} ((x * y) * (x * (x * z))),$ by Proposition 3.3 (i). Thus $h_{\tilde{F}} (x * (y * z)) ⊓ h_{\tilde{F}} (x * y) ⊑$ $h_{\tilde{F}} (((x * y) * (x * (x * z))) ⊓ h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} (x * z) .$

Therefore, $\tilde{F} \in IVHFIF (X)$ . □

Let f : X → Y be a homomorphism of BE-algebras and $\tilde{F} \in HFS (Y)$ , where $Y = (Y; *, 1)$ . Define a mapping $h_{{\tilde{F}}^{f}} : X \to ρ ([0, 1])$ such that $h_{{\tilde{F}}^{f}} (x) = h_{\tilde{F}} (f (x))$ , for all x ∈ X . Then $h_{{\tilde{F}}^{f}} (x)$ is well defined and ${\tilde{F}}^{f} \in HFS (X)$ , in which ${\tilde{F}}^{f} = {x \in X : f (x) \in \tilde{F}} .$

Theorem 4.7. Let f : X → Y be an onto homomorphism of BE-algebras and $F \in HFS (Y)$ . Then $\tilde{F} \in IVHFF (Y)$ (resp., $\tilde{F} \in IVHFIF (Y)$ ) if and only if ${\tilde{F}}^{f} \in IVHFF (X)$ (resp., ${\tilde{F}}^{f} \in IVHFIF (X)$ ).

Proof. Assume that $\tilde{F} \in IVHFF (Y)$ . For any x ∈ X, we have $\begin{matrix} h_{{\tilde{F}}^{f}} (x) & = & h_{\tilde{F}} (f (x)) \\ ⊑ & h_{\tilde{F}} (1_{Y}) \\ = & h_{\tilde{F}} (f (1_{X})) \\ = & h_{{\tilde{F}}^{f}} (1_{X}) . \end{matrix}$

Hence (IVHFF1) is valid. Now, let x, y ∈ X $\begin{matrix} h_{{\tilde{F}}^{f}} (x * y) ⊓ h_{{\tilde{F}}^{f}} (x) & = & h_{\tilde{F}} (f (x * y)) ⊓ h_{\tilde{F}} (f (x)) \\ = & h_{\tilde{F}} (f (x) * f (y)) ⊓ h_{\tilde{F}} (f (x)) \\ ⊑ & h_{\tilde{F}} (f (y)) \\ = & h_{{\tilde{F}}^{f}} (y) \end{matrix}$

Therefore, ${\tilde{F}}^{f} \in IVHFF (X)$ .

Conversely, assume that ${\tilde{F}}^{f} \in IVHFF (X)$ . Lety ∈ Y. Since f is onto, there exists x ∈ X such that f (x) = y. Hence $\begin{matrix} h_{\tilde{F}} (y) & = & h_{\tilde{F}} (f (x)) \\ = & h_{{\tilde{F}}^{f}} (x) \\ ⊑ & h_{{\tilde{F}}^{f}} (1_{X}) \\ = & h_{\tilde{F}} (f (1_{X})) \\ = & h_{\tilde{F}} (1_{Y}) . \end{matrix}$

Now, let x, y ∈ Y. Then there exist a, b ∈ X such that f (a) = x and f (b) = y. Hence we have $\begin{matrix} h_{\tilde{F}} (x * y) ⊓ h_{\tilde{F}} (x) & = & h_{\tilde{F}} (f (a) * f (b)) ⊓ h_{\tilde{F}} (f (a)) \\ = & h_{\tilde{F}} (f (a * b)) ⊓ h_{\tilde{F}} (f (a)) \\ = & h_{{\tilde{F}}^{f}} (a * b) ⊓ h_{{\tilde{F}}^{f}} (a) \\ ⊑ & h_{{\tilde{F}}^{f}} (b) \\ = & h_{\tilde{F}} (f (b)) \\ = & h_{\tilde{F}} (y) . \end{matrix}$

Therefore, $\tilde{F} \in IVHFF (Y)$ . □

5 Interval-valued hesitant fuzzy fantastic filters

Definition 5.1. Interval-valued hesitant fuzzy set $\tilde{F}$ of $X$ is called an interval valued hesitant fuzzy fantastic filter if satisfies the following conditions:

$h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1),$

$h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z) ⊑ h_{\tilde{F}} (((y * x) * x) * y)$ , for all x, y, z ∈ X .

Denote the set of all hesitant fuzzy fantastic filters on $X$ by $IVHFFF (X)$ . It can be considered every interval-valued hesitant fuzzy fantastic filter as a hesitant fantastic filter.

Note. If $| h_{\tilde{F}} (x) | = 1$ , for all x ∈ X, then $h_{\tilde{F}}$ is a fuzzy fantastic filter.

Theorem 5.2. Let $\tilde{F} \in IVHFF (X)$ . Then $\tilde{F} \in IVHFFF (X)$ if and only if $h_{\tilde{F}} (x * y) ⊑ h_{\tilde{F}} (((y * x) * x) * y) .$

Proof. Let $\tilde{F}$ be an interval-valued hesitant fuzzy fantastic filter of $X$ . Using (BE3) we have x ∗ y = 1 ∗ (x ∗ y). Hence $h_{\tilde{F}} (x * y) = h_{\tilde{F}} (1 * (x * y)$ and so by Definition 5.1, we have $\begin{matrix} h_{\tilde{F}} (x * y) & = & h_{\tilde{F}} (1 * (x * y)) \\ = & h_{\tilde{F}} (1 * (x * y)) ⊓ h_{\tilde{F}} (1) \\ ⊑ & h_{\tilde{F}} (((y * x) * x) * y) . \end{matrix}$

Conversely, since $\tilde{F}$ is an interval valued hesitant fuzzy filter and using hypothesis, we have $\begin{matrix} h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z) ⊑ h_{\tilde{F}} (x * y) \\ ⊑ h_{\tilde{F}} (((y * x) * x) * y) . \end{matrix}$

Therefore, $\tilde{F} \in IVHFFF (X)$ . □

Theorem 5.3. Let $h, h_{1}, h_{2} \in IVHFFF (X)$ and λ ∈ [0, 1]. Then

$h_{1} ⊔ h_{2} \in IVHFFF (X)$ ,

$h_{1} ⊓ h_{2} \in IVHFFF (X)$ ,

$h_{1} \oplus h_{2} \in IVHFFF (X)$ ,

$h_{1} \otimes h_{2} \in IVHFFF (X)$ ,

$λ h \in IVHFFF (X)$ .

Theorem 5.4. Algebra $(IVHFFF (X); ⊑, ⊔, ⊓, \tilde{0}, \tilde{1})$ of type (2, 2, 2, 0, 0) is a complete lattice.

Let f : X → Y be a homomorphism of BE-algebras and $\tilde{F} \in IVHFS (Y)$ . Define a mapping $h_{{\tilde{F}}^{f}} : X \to ρ ([0, 1])$ such that $h_{{\tilde{F}}^{f}} (x) = h_{\tilde{F}} (f (x))$ , for all x ∈ X . Then $h_{{\tilde{F}}^{f}} (x)$ is well defined and ${\tilde{F}}^{f} \in HFS (X)$ , where ${\tilde{F}}^{f} = {x \in X : f (x) \in \tilde{F}}$ .

Theorem 5.5. Let f : X → Y be an onto homomorphism of BE-algebras and $\tilde{F} \in HFS (Y)$ . Then $\tilde{F} \in IVHFFF (Y)$ if and only if ${\tilde{F}}^{f} \in IVHFFF (X)$ .

Hence (IVHFFF1) is valid. Now, let x, y, z ∈ X $h_{{\tilde{F}}^{f}} (z * (x * y)) ⊓ h_{{\tilde{F}}^{f}} (z) = h_{\tilde{F}} (f (z * (x * y))) ⊓ h_{\tilde{F}} (f (z)) = h_{\tilde{F}} (f (z) * (f (x) * f (y))) ⊓ h_{\tilde{F}} (f (z)) ⊑ h_{\tilde{F}} (((f (y) * f (x)) * f (x)) * f (y)) = h_{\tilde{F}} ((f (y * x) * f (x)) * f (y)) = h_{\tilde{F}} (f ((y * x) * x) * y) = h_{{\tilde{F}}^{f}} (((y *) * x) * y)$ .

Therefore, ${\tilde{F}}^{f} \in IVHFFF (X)$ .

Conversely, assume that ${\tilde{F}}^{f} \in IVHFFF (X)$ . Let y ∈ Y. Since f is onto, there exist x ∈ X such that f (x) = y. Then $\begin{matrix} h_{\tilde{F}} (y) & = & h_{\tilde{F}} (f (x)) \\ = & h_{{\tilde{F}}^{f}} (x) \\ ⊑ & h_{{\tilde{F}}^{f}} (1_{X}) \\ = & h_{\tilde{F}} (f (1_{X})) \\ = & h_{\tilde{F}} (1_{Y}) . \end{matrix}$

Now, let x, y, z ∈ Y. Then there exist a, b, c ∈ X such that f (a) = x, f (b) = y and f (c) = z. Hence we have $h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z) = h_{\tilde{F}} (f (c) * (f (a) * f (b))) ⊓ h_{\tilde{F}} (f (c)) = h_{\tilde{F}} (f (c * (a * b))) ⊓ h_{\tilde{F}} (f (c)) = h_{{\tilde{F}}^{f}} (c * (a * b)) ⊓ h_{{\tilde{F}}^{f}} (c) ⊑ h_{{\tilde{F}}^{f}} (((b * a) * a) * b) = h_{\tilde{F}} (f (((b * a) * a) * b)) = h_{\tilde{F}} (((f (b) * f (a)) * f (a)) * f (b))) = h_{\tilde{F}} ((((y * x) * x) * y))$

Therefore, $\tilde{F} \in IVHFFF (Y)$ . □

Let $\tilde{F} \in HFS (X)$ . Denote $⊤ : = 1 - sup {sup {\tilde{r} | \tilde{r} \in h_{\tilde{F}} (x)} | x \in X} .$

For any β ∈ [0, ⊤], we define $h_{{\tilde{F}}_{β}} (x) : = h_{\tilde{F}} (x) + β = {\tilde{r} + β | \tilde{r} \in h_{\tilde{F}} (x)}$ , for all x ∈ X, where $\tilde{r} + β = {r + β | r \in \tilde{r}}$ . Obviously, $h_{{\tilde{F}}_{β}}$ is a mapping from X to [0, 1], that is, ${\tilde{F}}_{β} \in HFS (X)$ . $h_{{\tilde{F}}_{β}} (x)$ is well-defined. Assume that β ∈ [0, ⊤] and x₁ = x₂. Then $h_{\tilde{F}} (x_{1}) = h_{\tilde{F}} (x_{2})$ and so $h_{{\tilde{F}}_{β}} (x_{1}) = h_{\tilde{F}} (x_{1}) + β = h_{\tilde{F}} (x_{2}) + β = h_{{\tilde{F}}_{β}} (x_{2})$ . Hence $h_{{\tilde{F}}_{β}}$ is well-defined.

Theorem 5.6. Let β ∈ [0, ⊤]. If $\tilde{F} \in IVHFFF (X)$ , then $h_{{\tilde{F}}_{β}} \in IVHFFF (X)$ .

Proof. Let x, y, z ∈ X . Then $h_{{\tilde{F}}_{β}} (z * (x * y)) ⊓ h_{{\tilde{F}}_{β}} (z) = (h_{\tilde{F}} (z * (x * y)) + β) ⊓ (h_{\tilde{F}} (z) + β) = (h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z)) + β ⊑ (h_{\tilde{F}} (((y * x) * x) * y)) + β = h_{{\tilde{F}}_{β}} (((y * x) * x) * y) .$

Also, $h_{{\tilde{F}}_{β}} (x) = h_{\tilde{F}} (x) + β \subseteq h_{\tilde{F}} (1) + β = h_{{\tilde{F}}_{β}} (1) .$ Therefore, $h_{{\tilde{F}}_{β}} \in IVHFFF (X)$ . □

Theorem 5.7. If there exists β ∈ [0, ⊤] such that $h_{{\tilde{F}}_{β}} \in IVHFFF (X)$ , then $h_{\tilde{F}} \in IVHFFF (X)$ .

Proof. Assume that $h_{{\tilde{F}}_{β}} \in IVHFFF (X)$ , for some β ∈ [0, ⊤] . Let x, y ∈ X . Since $h_{{\tilde{F}}_{β}} (z * (x * y)) ⊓ h_{{\tilde{F}}_{β}} (z) ⊑ h_{{\tilde{F}}_{β}} (((y * x) * x) * y)$ , we can see that $h_{{\tilde{F}}_{β}} (z * (x * y)) ⊓ h_{{\tilde{F}}_{β}} (z) = (h_{\tilde{F}} (z * (x * y)) + β) ⊓ (h_{\tilde{F}} (z) + β) = (h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z)) + β ⊑ h_{\tilde{F}} ((((y * x) * x) * y)) + β$ .

Now, by canceling β we have $h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z) ⊑ h_{\tilde{F}} (((y * x) * x) * y)$ . Also, by a similar argument we have $h_{\tilde{F}} (x) ⊑ h_{\tilde{F}} (1)$ . Therefore, $h_{\tilde{F}} \in IVHFFF (X)$ . □

Let $\tilde{F} \in IVHS (X)$ and γ ∈ [0, 1]. Define ${\tilde{F}}_{γ}^{m} = γ . \tilde{F} = {< x, γ . h_{\tilde{F}} (x) > | x \in X},$ where $γ . h_{\tilde{F}} (x) = {γ . A | A \in h_{\tilde{F}} (x)}$ and $γ . A = {γ . a | a \in A} .$

We note that if γ : =1, then ${\tilde{F}}_{γ}^{m} = \tilde{F}$ .

Theorem 5.8. Let γ ∈ [0, 1]. If $\tilde{F} \in IVHFFF (X)$ , then ${\tilde{F}}_{γ}^{m} \in IVHFFF (X)$ .

Proof. Let x, y, z ∈ X . Then $h_{{\tilde{F}}_{γ}^{m}} (z * (x * y)) ⊓ h_{{\tilde{F}}_{γ}} (z) = h_{γ . \tilde{F}} (z * (x * y)) ⊓ h_{γ . \tilde{F}} (z) = γ . (h_{\tilde{F}} (z * (x * y))) ⊓ γ . (h_{\tilde{F}} (z)) = γ . (h_{\tilde{F}} (z * (x * y)) ⊓ h_{\tilde{F}} (z)) ⊑ γ . (h_{\tilde{F}} (((y * x) * x) * y)) = h_{{\tilde{F}}_{γ}^{m}} (((y * x) * x) * y) .$

Also, $h_{{\tilde{F}}_{γ}^{m}} (x) = γ . h_{\tilde{F}} (x) \subseteq γ . h_{\tilde{F}} (1) = h_{{\tilde{F}}_{γ}^{m}} (1) .$

Therefore, $h_{{\tilde{F}}_{γ}^{m}} \in IVHFFF (X)$ . □

Theorem 5.9. Let γ ∈ (0, 1]. If ${\tilde{F}}_{γ}^{m} \in IVHFFF (X)$ , then $\tilde{F} \in IVHFFF (X)$ .

Proof. See the proof of Theorem 5.8, by canceling γ, since γ ≠ 0. □

We note that if γ = 0, then ${\tilde{F}}_{γ}^{m} = 0$ , for all $\tilde{F} \in IVHS (X)$ . So Theorem 5.9 is not valid in this case.

Corollary 5.10. Let γ ∈ (0, 1]. Then ${\tilde{F}}_{γ}^{m} \in IVHFFF (X)$ if and only if $\tilde{F} \in IVHFFF (X)$ .

6 Conclusions and future works

From a logical point of view, various filters, correspond to various sets of provable formulas, plays an important role in several algebraic structures. In this paper, we applied the theory of hesitant fuzzy sets to BE-algebras and developed the interval-valued hesitant fuzzy (implicative, fantastic) filters in BE-algebras. The relation between interval-valued hesitant fuzzy (implicative, fantastic) filters and interval-valued hesitant fuzzy filters is discussed and some properties is studied. This result can be used in other algebraic structures. In our future research, we will define different types of interval-valued hesitant fuzzy filters and discuss on relation between them and construct quotient algebras via these filters. Congruence relations on a BE-algebra with respect to these notions will be studied.

Compliance with ethical standards

The author declare that there is no conflict of interest. This article does not contain any studies with human participants or animals performed by the author.

Footnotes

Acknowledgments

The author would like to express their thanks to the reviewers for their comments and suggestions which improved the paper.

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