Several types of hesitant fuzzy filters on residuated lattices

Abstract

The present paper investigates the hesitant fuzzy filters on residuated lattices. A one-to-one correspondence between the set of all hesitant fuzzy filters and the set of all hesitant fuzzy congruences is established and a quotient residuated lattice with respect to a hesitant fuzzy filter is induced. Furthermore, several special types of hesitant fuzzy filters such as hesitant fuzzy implicative, regular and Boolean filters are introduced, and some alternative definitions of them are obtained, then some typical logical algebras are characterized by these identity forms.

Keywords

Hesitant fuzzy sets hesitant fuzzy implicative filters hesitant fuzzy regular filters hesitant fuzzy Boolean filters residuated lattices

1 Introduction

Since fuzzy sets were firstly introduced by Zadeh [1], it has been extended to intuitionistic fuzzy sets [2], interval-valued fuzzy sets [3], type-2 fuzzy sets [4] and type-n fuzzy sets [5]. Recently, the concept of hesitant fuzzy sets, a new generalization of fuzzy sets, permitting an element to have, not just one, but a set of several possible membership values was proposed by Torra [6, 7], and the relationships among hesitant fuzzy sets and other existing generalizations of fuzzy sets were discussed. It may be mentioned that hesitant fuzzy sets can reflect the human’s hesitancy more objectively than the other extensions of fuzzy sets. The algebraic structure of various set theories dealing with uncertainties is also an interesting topic. Zhu [8] investigated the filter theory in residuated lattices. Jun [9] proposed the fuzzy positive implicative and fuzzy associative filters in lattice implication algebras. Jun [10] and Liu [11] gave the notion of the fuzzy filters in MTL-algebras and BL-algebras, respectively and Liu [12] provided an answer to the Jun-Shim-Lele’s open problem on the fuzzy filters. Rachunek [13] developed the fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo-BL-algebras. Zhang [14] defined the fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras. Borzooei et al. [15] investigated the fuzzy positive implicative filters of Hoops based on fuzzy points. Ghorbani [16] and Hedayati [17] investigated the intuitionstic fuzzy filters in R₀-algebras and residuated lattices, respectively. Jun [18] proposed the hesitant fuzzy prefilters and filters of EQ-algebras. Muhiuddin [19] defined hesitant fuzzy filters and hesitant fuzzy G-filters in residuated lattices. Shao [20] gave the neutrosophic hesitant fuzzy subalgebras and filters in pseudo-BCI algebras.

Inspired by the work of Jun [18] and Muhiuddin [19], in this paper, we continue to deal with the algebraic structure of residuated lattices by applying hesitant fuzzy set theory. Firstly, a one-to-one correspondence between the set of all hesitant fuzzy filters and the set of all hesitant fuzzy congruences is determined. Then, the notions of hesitant fuzzy implicative, regular and Boolean filters are introduced and some properties of them are obtained. The relationships among the three types of fuzzy hesitant fuzzy filters are discussed and some new characterizations of these filters with identity forms are given, and then their corresponding logical algebras are alternatively described by these identities.

2 Preliminaries

In order for this paper to be as self-contained as possible, we include in this section a brief review of the main concepts needed throughout it. The concepts of residuated lattices, hesitant fuzzy sets and hesitant fuzzy filters are reviewed.

2.1 Residuated lattices

Definition 2.1. [21 –23] An algebra $L = (L, \lor, \land, \otimes, \to, 0, 1)$ is called a residuated lattice if it holds that

(L, ∨ , ∧ , 0, 1) is a bounded lattice;

(L, ⊗ , 1) is a commutative monoid;

(⊗ , →) forms an adjoint pair, that is, x ⊗ y ≤ z iff x ≤ y → z.

Some basic properties of residuated lattices are collected in the following lemma, see, e.g. [21 –23]. In the sequel, we denote x → 0 as ¬x and (x → y) ∧ (y → x) as x ↔ y.

Lemma 2.2. [21 –23] Let L be a residuated lattice. Then the following properties hold for all x, y, z, w ∈ L

x ⊗ y ≤ x ⊗ (x → y) ≤ x ∧ y ≤ x ∧ (x → y);

x → (y → z) = y → (x → z) = (x ⊗ y) → z;

x ∨ y ≤ (x → y) → y ≤ ¬ x → y;

x ≤ y implies z → x ≤ z → y and y → z ≤ x → z;

x ≤ ¬¬ x;

(y → x) ⊗ (z → w) ≤ (x → z) → (y → w)

(x → y) ⊗ (y → z) ≤ x → z;

¬x ∨ y ≤ x → y;

¬¬ x → (x ⊗ y) ≤ x → y;

x → y ≤ (x ⊗ z) → (y ⊗ z);

x → y ≤ ¬ y → ¬ x;

y ≤ x → (x ⊗ y);

(x ↔ y) ⊗ (z ↔ w) ≤ (x ◊ z) ↔ (y ◊ w), where ◊∈ { ⊗ , ∨ , ∧ , → };

(x ↔ y) ⊗ (y ↔ z) ≤ x ↔ z.

Definition 2.3. [21, 22] Let L be a residuated lattice. Then L is said to be

regular if ¬¬ x = x for all x ∈ L;

a Heyting algebra if x ∧ y = x ⊗ y for all x, y ∈ L;

a Boolean algbera if ¬x ∨ x = 1 for all x ∈ L.

Definition 2.4. [22] Let L be a residuated lattice. Then a nonempty subset F of L is called a filter of L if for all x, y ∈ F it holds that

x ∈ F, x ≤ y imply y ∈ F;

x, y ∈ F implies x ⊗ y ∈ F.

However, a filter can also be alternatively described as for all x, y ∈ L it holds that 1 ∈ F and x, x → y ∈ F implies y ∈ F [23]. With any filter F, we can associate a congruence θ_F on L by defining: (x, y) ∈ θ_F iff x → y, y → x ∈ F iff (y → x) ∧ (x → y) ∈ F iff (y → x) ⊗ (x → y) ∈ F.

Conversely, for a congruence θ, the subset of L defined by x ∈ F_θ iff (x, 1) ∈ θ

is a filter of L. Moreover, there is a one-to-one correspondence between the set of all filters of L and the set of all congruences on L. Let L/F denote the set of the congruence classes of θ_F, i.e., $L / F = {[x]_{F} | x \in L},$ where [x] _F : = {y ∈ L| (x, y) ∈ θ_F}. Define operations on L/F as follows: $\begin{matrix} [x]_{F} ⊙ [y]_{F} = & [x \otimes y]_{F}, \\ [x]_{F} ⇀ [y]_{F} = & [x \to y]_{F}, \\ [x]_{F} ⊓ [y]_{F} = & [x \land y]_{F}, \\ [x]_{F} ⊔ [y]_{F} = & [x \lor y]_{F} . \end{matrix}$

Lemma 2.5. [24] Let F be a filter of L. Then (L/F, ⊔ , ⊓ , ⊙ , ⇀ , [0] _F, [1] _F) is a residuated lattice called the quotient residuated lattice w.r.t. F.

2.2 Hesitant fuzzy sets and hesitant fuzzy filters

Definition 2.6. [6, 7]) Let X be a reference set. Then a hesitant fuzzy set (HFS) A in X is mathematically represented as follows: $A = {〈 x, h_{A} (x) 〉 | h_{A} (x) \in P ([0, 1]), x \in X},$ where $P ([0, 1])$ is the power set of [0, 1].

The following additional definitions will be useful for defining union and intersection of hesitant fuzzy sets. For a given hesitant fuzzy set h_A we define

α-upper bound: $(h_{A}^{+})_{α} (x) = {h \in h_{A} (x) | h \geq α}$ ;

α-lower bound: $(h_{A}^{-})_{α} (x) = {h \in h_{A} (x) | h \leq α}$ .

Definition 2.7. [6, 7]) Let A, B be two hesitant fuzzy set. Their complement, union and intersection are represented by A^c, A ∪ B and A ∩ B, respectively, as follows:

$h_{A}^{c} (x) =$ $⋃_{γ \in h_{A} (x)} {1 - γ}$ ;

$(h_{A} \cup h_{B}) (x) = {h \in h_{A} (x) \cup h_{B} (x) | h \geq max (h_{A}^{-}, h_{B}^{-})}$ ;

$(h_{A} \cap h_{B}) (x) = {h \in h_{A} (x) \cup h_{B} (x) | h \leq min (h_{A}^{+}, h_{B}^{+})}$ .

For convenience, Xia [25] named the set h = h_A (x) as a hesitant fuzzy element (HFE). We will represent the family of all hesitant fuzzy elements defined on X by HFE (X). The above operations can be alternatively defined as follows:

Definition 2.8. [26] Let h, h₁, h₂ ∈ HFE (X). Then the operations complement, union and intersection are defined as follows:

h^c = $⋃_{γ \in h} {1 - γ}$ ;

$h_{1} \cup h_{2} = ⋃_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {γ_{1} \lor γ_{2}}$ ;

$h_{1} \cap h_{2} = ⋃_{γ_{1} \in h_{1}, γ_{2} \in h_{2}} {γ_{1} \land γ_{2}}$ .

We donote h₁ ⊆ h₂ iff h₁ ∩ h₂ = h₁. It is trivial to verify that ⊆ satisfies the properties as follows:

h₁ ⊆ h₁;

h₁ ⊆ h₂ and h₂ ⊆ h₁ imply h₁ = h₂;

h₁ ⊆ h₂ and h₂ ⊆ h₃ imply h₁ ⊆ h₃.

That is, ⊆ is a partial order on HFE (X).

Definition 2.9. [19] Let A be a hesitant fuzzy set of L. Then A is called a hesitant fuzzy filter if

x ≤ y implies h_A (x) ⊆ h_A (y) for all x, y ∈ L;

h_A (x) ∩ h_A (y) ⊆ h_A (x ⊗ y) for all x, y ∈ L.

Theorem 2.10. [19] Let A be a hesitant fuzzy set of L. Then A is a hesitant fuzzy filter if and only if the following assertions holds:

h_A (x) ⊆ h_A (1);

h_A (x) ∩ h_A (x → y) ⊆ h_A (y) for all x, y ∈ L.

The items in Definition 2.9 and Theorem 2.10 will be frequently used, so we don’t cite them every time.

3 Relationships between the hesitant fuzzy filters and the hesitant fuzzy congruences

In this section, we introduce the concept of hesitant fuzzy congruences and establish a correspondence between the hesitant fuzzy filters and the hesitant fuzzy congruences.

Proposition 3.1. Let A be a hesitant fuzzy filter of L. Then

h_A (x → y) = h_A (1) implies h_A (x) ⊆ h_A (y);

h_A (x ⊗ y) = h_A (x ∧ y) = h_A (x) ∩ h_A (y) for all x, y ∈ L.

Proof. (1) It follows immediately from Theorem 2.10.

(2) It is obvious that h_A (x ⊗ y) ⊆ h_A (x ∧ y) ⊆ h_A (x) ∩ h_A (y) ⊆ h_A (x ⊗ y). Thus the identity holds.□

Definition 3.2. L be a residuated lattice. A hesitant fuzzy set C of L × L is called a hesitant fuzzy congruence of L if the following assertions hold:

h_C (x, x) = h_C (1, 1);

h_C (x, y) = h_C (y, x);

h_C (x, y) ∩ h_C (y, z) ⊆ h_C (x, z);

h_C (x, y) ∩ h_C (z, w) ⊆ h_C (x ◊ z, y ◊ w), where ◊∈ { ∨ , ∧ , ⊗ , → }.

For a given hesitant fuzzy filter A, we define $C_{A} : L \times L ⟶ P ([0, 1])$ as follows: $h_{C_{A}} (x, y) = h_{A} (x \to y) \cap h_{A} (y \to x) .$ It follows from Proposition 3.1 that h_{C
_A} (x, y) = h_A (x ↔ y).

Proposition 3.3. Let A be a hesitant fuzzy filter of L. Then C_A is a hesitant fuzzy congruence.

Proof. We only prove the item (4) in Definition 3.2. By Lemma 2.2(12) we have $\begin{matrix} h_{C_{A}} (x, y) \cap h_{C_{A}} (z, w) \\ = & h_{A} (x \leftrightarrow y) \cap h_{A} (z \leftrightarrow w) \\ \subseteq & h_{A} ((x \leftrightarrow y) \otimes (z \leftrightarrow w)) \\ \subseteq & h_{A} ((x ◊ z) \leftrightarrow (y ◊ w)) \\ = & h_{C_{A}} (x ◊ z, y ◊ w) . \end{matrix}$ Thus h_{C
_A} (x, y) ∩ h_{C
_A} (z, w) ⊆ h_{C
_A} (x ◊ z, y ◊ w).□ Let C be a hesitant fuzzy congruence of L and x ∈ L. Define a hesitant fuzzy set C^x of L as $h_{C^{x}} (y) = h_{C} (x, y)$ for all y ∈ L which is called a hesitant fuzzy congruence class of x by C. The set $L / C = {C^{x} | x \in L}$ is called a hesitant fuzzy quotient set by C.

Proposition 3.4. Let C be a hesitant fuzzy congruence of L. Then C¹ is a hesitant fuzzy filter.

Proof. It is routine, so we omit it.□

Lemma 3.5. Let C be a hesitant fuzzy congruence of L. Then for all x, y ∈ L it holds that $h_{C} (x \leftrightarrow y, 1) = h_{C} (x, y) .$

Proof. It is obvious that $\begin{matrix} h_{C} (x \leftrightarrow y, 1) \\ = & h_{C} (x \leftrightarrow y, y \leftrightarrow y) \supseteq h_{C} (x, y) . \end{matrix}$ On the other hand, using Lemma 2.2(3) and (13), we have $\begin{matrix} h_{C} (x \leftrightarrow y, 1) = & h_{C} (x \leftrightarrow y, 1) \cap h_{C} x, x) \\ \subseteq & h_{C} ((x \leftrightarrow y) \otimes x, 1 \otimes x) \\ = & h_{C} ((x \leftrightarrow y) \otimes x, x) \cap h_{C} (y, y) \\ \subseteq & h_{C} (((x \leftrightarrow y) \otimes x) \lor y, x \lor y) \\ = & h_{C} (y, x \lor y), \end{matrix}$ and hence h_C (x ↔ y, 1) ⊆ h_C (y, x ∨ y). Similarly, we have h_C (x ↔ y, 1) ⊆ h_C (x, x ∨ y).

Thus h_C (x ↔ y, 1) ⊆ h_C (y, x ∨ y) ∩ h_C (x, x ∨ y ⊆ h_C (x, y). Then the identity holds.□

Proposition 3.6. Let C be a hesitant fuzzy congruence of L. Then C_{C
¹} = C.

Proof. Using Lemma 3.5, it holds that h_{C
_{C
¹}} (x, y) = h_{C
¹} (x ↔ y) = h_C (1, x ↔ y) = h_C (x, y). Thus C_{C
¹} = C.□

Proposition 3.7. Let A be a hesitant fuzzy filter of L. Then $C_{A}^{1} = A$ .

Proof. It is obvious, so we omit it.□

Theorem 3.8. There is a one-to-one correspondence between the set of all hesitant fuzzy filters and the set of all hesitant fuzzy congruences.

Proof. It follows immediately from Propositions 3.3 and 3.4.□

Lemma 3.9. Let A be a hesitant fuzzy filter of L. Then $C_{A}^{x} = C_{A}^{y}$ if and only if $h_{A} (x \leftrightarrow y) = h_{A} (1),$ for all x, y ∈ L.

Proof. Assume that $C_{A}^{x} = C_{A}^{y}$ . That is, $h_{C_{A}^{x}} (z) = h_{C_{A}^{y}} (z)$ for all z ∈ L. Thus h_A (x ↔ z) = h_A (y ↔ z). In particular, taking y = z, then h_A (x ↔ y) = h_A (1).

Conversely, it follows from Lemma 2.2(13) that $\begin{matrix} h_{C_{A}^{x}} (z) = & h_{C_{A}^{x}} (z) \cap h_{A} (1) \\ = & h_{A} (x \leftrightarrow z) \cap h_{A} (x \leftrightarrow y) \\ \subseteq & h_{A} ((x \leftrightarrow z) \otimes (x \leftrightarrow y)) \\ \subseteq & h_{A} (z \leftrightarrow y) \\ = & h_{C_{A}^{y}} (z), \end{matrix}$ and hence $h_{C_{A}^{x}} (z) \subseteq h_{C_{A}^{y}} (z)$ . Similarly, we have $h_{C_{A}^{y}} (z) \subseteq h_{C_{A}^{x}} (z)$ . Thus $h_{C_{A}^{x}} (z) = h_{C_{A}^{y}} (z)$ , that is, $C_{A}^{x} = C_{A}^{y}$ .□ Let A be a hesitant fuzzy filter of L. For any $C_{A}^{x}, C_{A}^{y} \in L / C_{A}$ , we define $\begin{matrix} C_{A}^{x} \lor C_{A}^{y} = & C_{A}^{x \lor y}, \\ C_{A}^{x} \land C_{A}^{y} = & C_{A}^{x \land y}, \\ C_{A}^{x} \otimes C_{A}^{y} = & C_{A}^{x \otimes y}, \\ C_{A}^{x} \to C_{A}^{y} = & C_{A}^{x \to y} . \end{matrix}$

Theorem 3.10. Let A be a hesitant fuzzy filter of L. Then $L / C_{A} = {L / C_{A}, \land, \lor, \otimes, \to, C_{A}^{0}, C_{A}^{1}}$ is a residuated lattice.

Proof. It is routine.□

4 Some types of hesitant fuzzy filters

In this section, we introduce the hesitant fuzzy implicative filters, the hesitant fuzzy regular filters and the hesitant fuzzy Boolean filters.

4.1 Hesitant fuzzy implicative filters

Definition 4.1. Let A be a hesitant fuzzy filter of L. Then A is called a hesitant fuzzy implicative filer if $h_{A} (x \to x^{2}) = h_{A} (1),$ for all x ∈ L.

Example 4.2. Let L be a residuated lattice with the universe {0, a, b, 1} such that 0 < a, b < 1 and a, b are incomparable. The operations ⊗ and → are given by the tables below:

⊗	a	b	1	→	0	a	b	1
0	0	0	0	0	1	1	1	1
a	a	0	a	a	b	1	b	1
b	0	b	b	b	a	a	1	1
1	a	b	1	1	0	a	b	1

Let t₁ = {0.3, 0.4}, t₂ = {0.3, 0.4, 0.5, 0.6}, t₃ = {0.3, 0.4, 0.5, 0.6, 0.7, 0.8}. Define A as h_A (0) = h_A (a) = t₁, h_A (b) = t₂ and h_A (1) = t₃. Routine calculations show that A is a hesitant fuzzy implicative filter.

Theorem 4.3. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy implicative filter if and only if $h_{A} (x \to y) = h_{A} (x \to (x \otimes y)),$ for all x, y ∈ L.

Proof. Assume that A is a hesitant fuzzy implicative filter. Using Lemma 2.2 (6) with z = w = x ⊗ y and y = x² and (10) it holds that $\begin{matrix} h_{A} (1) = & h_{A} (x \to x^{2}) \\ \subseteq & h_{A} ((x^{2} \to (x \otimes y)) \to (x \to (x \otimes y))) \\ \subseteq & h_{A} ((x \to y) \to (x \to (x \otimes y))), \end{matrix}$ and hence h_A (x → y) ⊆ h_A (x → (x ⊗ y)). It follows from Lemma 2.2(4) that h_A (x → (x ⊗ y)) ⊆ h_A (x → y). Thus h_A (x → (x ⊗ y)) = h_A (x → y).

Conversely, it is obvious by taking x = y.□

Theorem 4.4. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy implicative filter if and only if L/C_A is a Heyting algebra.

Proof. Assume that A is a hesitant fuzzy implicative filter. By Theorem 3.10, it holds that L/C_A is a residuated lattice and h_A (x → x²) = h_A (1). Using Lemma 3.9, we have $C_{A}^{x \to x^{2}} = C_{A}^{1}$ , i.e., $C_{A}^{x} \to (C_{A}^{x})^{2} = C_{A}^{1}$ . This implies that L/C_A is a Heyting algebra.

Conversely, assume that L/C_A is a Heyting algebra. Then $C_{A}^{x} \to (C_{A}^{x})^{2} = C_{A}^{1}$ . Using Lemma 3.9, we obtain h_A (x → x²) = h_A (1). Thus A is a hesitant fuzzy implicative filter.□

4.2 Hesitant fuzzy regular filters

Definition 4.5. Let A be a hesitant fuzzy filter of L. Then A is called a hesitant fuzzy regular filer if $h_{A} (\neg \neg x \to x) = h_{A} (1),$ for all x ∈ L.

Example 4.6. The hesitant fuzzy filter A in Example 4.2 is not a hesitant fuzzy regular filter, because h_A (¬¬ a → a) = t₁ ≠ t₃ = h_A (1).

Example 4.7. Let L be a residuated lattice with the universe {0, a, b, 1} such that 0 < a < b < 1. The operations ⊗ and → are given by the tables below:

⊗	a	b	1	→	0	a	b	1
0	0	0	0	0	1	1	1	1
a	0	0	a	a	b	1	1	1
b	0	a	b	b	a	b	1	1
1	a	b	1	1	0	a	b	1

Theorem 4.8. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy regular filter if and only if $h_{A} (x \to y) = h_{A} (\neg \neg x \to y),$ for all x, y ∈ L.

Proof. Assume that A is a hesitant fuzzy regular filter. Considering the following special case of Lemma 2.2(6): ¬¬ x → x ≤ (x → y) → (¬¬ x → y), it holds that h_A ((x → y) → (¬¬ x → y)) = h_A (1). Using Proposition 3.1, we have h_A (x → y) ⊆ h_A (¬¬ x → y). By Lemma 2.2(4) and (5), it holds that h_A (¬¬ x → y) ⊆ h_A (x → y). Thus h_A (¬¬ x → y) = h_A (x → y).

Conversely, it is obvious by taking x = y.□

Theorem 4.9. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy regular filter if and only if L/C_A is a regular residuated lattice.

Proof. The proof is similar to that of Theorem 4.4.□

4.3 Hesitant fuzzy boolean filters

Definition 4.10. Let A be a hesitant fuzzy filter of L. Then A is called a hesitant fuzzy Boolean filer if $h_{A} (x \lor \neg x) = h_{A} (1),$ for all x ∈ L.

Example 4.11. It is routine to verify that the hesitant fuzzy filter A in Example 4.2 is a hesitant fuzzy Boolean filter.

Theorem 4.12. Let A be a hesitant fuzzy filter of L. Then the following assertions are equivalent:

A is a hesitant fuzzy Boolean filter;

h_A (¬ x → x) = h_A (x) for all x ∈ L;

h_A (¬ x → y) = h_A (x ∨ y) for all x, y ∈ L.

Proof. (1)⇒(2) Replacing y with ¬x in Lemma 2.2(3), it holds that x ∨ ¬ x ≤ (¬ x → x) → x. Thus h_A ((¬ x → x) → x) = h_A (1). By Proposition 3.1(1), we get h_A (¬ x → x) ⊆ h_A (x). The inequality x ≤ ¬ x → x leads that h_A (x) ⊆ h_A (¬ x → x). Thus h_A (x) = h_A (¬ x → x).

(2)⇒(3) Replacing x with x ∨ y, then h_A (x ∨ y) = h_A (¬ (x ∨ y) → (x ∨ y)). It follows from Lemma 2.2(4) that ¬x → y ≤ ¬ (x ∨ y) → (x ∨ y), and hence h_A (¬ x → y) ⊆ h_A (x ∨ y). Lemma 2.2(3) leads that h_A (x ∨ y) ⊆ h_A (¬ x → y). Thus h_A (x ∨ y) = h_A (¬ x → y).

(3)⇒(1) It is trivial.□

Theorem 4.13. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy Boolean filter if and only if $h_{A} (x \to y) = h_{A} (\neg x \lor y),$ for all x, y ∈ L.

Proof. Assume that A is a hesitant fuzzy Boolean filter. Replacing x, y with ¬x, x, respectively, we get h_A (¬¬ x → x) = h_A (¬ x ∨ x). Thus it holds that h_A (¬¬ x → x) = h_A (1). Replacing x with ¬x, we have h_A (¬¬ x → y) = h_A (¬ x ∨ y). Using Lemma 2.2(4) and (7), it holds that $\begin{matrix} h_{A} (\neg \neg x \to x) \cap h_{A} (x \to y) \\ \subseteq & h_{A} (\neg \neg x \to x \otimes x \to y) \subseteq h_{A} (\neg \neg x \to y) . \end{matrix}$ Thus h_A (x → y) ⊆ h_A (¬ x ∨ y). Lemma 2.2(8) leads that h_A (¬ x ∨ y) ⊆ h_A (x → y). Thus we have h_A (¬ x ∨ y) = h_A (x → y).

Conversely, it is obvious by taking x = y.□

Theorem 4.14. Let A be a hesitant fuzzy filter. Then A is a hesitant fuzzy Boolean filter if and only if A is both a hesitant fuzzy implicative filter and a hesitant fuzzy regular filter.

Proof. Assume that A is a hesitant fuzzy Boolean filter. Taking x = ¬ y in Theorem 4.12(3), it holds that h_A (¬¬ y → y) = h_A (1), that is, A is a hesitant fuzzy regular filter. By Lemma 2.2(3) and (2) we have h_A (x ∨ ¬ x) ⊆ h_A ((x → ¬ x) → ¬ x) = h_A (x → ¬¬ x²), and hence h_A (x → ¬¬ x²) = h_A (1). Since A is regular, h_A (¬¬ x² → x²) = h_A (1). Thus $\begin{matrix} h_{A} (1) = & h_{A} (x \to \neg \neg x^{2}) \cap h_{A} (\neg \neg x^{2} \to x^{2}) \\ \subseteq & h_{A} ((x \to \neg \neg x^{2}) \otimes (\neg \neg x^{2} \to x^{2})) \\ \subseteq & h_{A} (x \to x^{2}), \end{matrix}$ and hence h_A (x → x²) = h_A (1), that is, A is a hesitant fuzzy implicative filter.

Conversely, since A is a hesitant fuzzy implicative filter, it holds that h_A (1) = h_A (x → x²). Replacing x with x ∧ y, it follows from Lemma 2.2(4) and (1) that $\begin{matrix} h_{A} (1) = & h_{A} ((x \land y) \to (x \land y)^{2}) \\ \subseteq & h_{A} ((x \land y) \to (x \otimes y)) \\ \subseteq & h_{A} ((x \land y) \to (x \otimes (x \to y))), \end{matrix}$ and hence h_A ((x ∧ y) → (x ⊗ (x → y))) = h_A (1). Replacing x, y with ¬x, ¬ y, respectively, we have h_A ((¬ x ∧ ¬ y) → (¬ x ⊗ (¬ x → ¬ y))) = h_A (1). Let u = x ∨ y. Then ¬u = ¬ x ∧ ¬ y. Thus h_A (¬ u → (¬ x ⊗ (¬ x → ¬ y))) = h_A (1).

By the regularity of A, Lemma 3(4), (8), (10) and (11), we have $\begin{matrix} h_{A} (1) \\ = & h_{A} (\neg \neg x \to x) \\ \subseteq & h_{A} ((y \to \neg \neg x) \to (y \to x)) \\ \subseteq & h_{A} ((\neg x \otimes (y \to \neg \neg x)) \to (\neg x \otimes (y \to x))) \\ \subseteq & h_{A} ((\neg u \to (\neg x \otimes (y \to \neg \neg x))) \\ \to (\neg u \to (\neg x \otimes (y \to x)))) \\ \subseteq & h_{A} ((\neg u \to (\neg x \otimes (y \to \neg \neg x))) \\ \to (\neg (\neg x \otimes (y \to x)) \to \neg \neg u)) \\ = & h_{A} ((\neg u \to (\neg x \otimes (y \to \neg \neg x))) \\ \to (((y \to x) \to \neg \neg x) \to \neg \neg u)), \end{matrix}$ and hence h_A ((¬ u → (¬ x ⊗ (y → ¬¬ x))) → (((y → x) → ¬¬ x) → ¬¬ u)) = h_A (1). It follows from Proposition 3.1(1) that h_A (¬ u → (¬ x ⊗ (y → ¬¬ x))) ⊆ h_A (((y → x) → ¬¬ x) → ¬¬ u). Thus h_A (((y → x) → ¬¬ x) → ¬¬ u) = h_A (1). Using the regularity of A, Lemma 2.2(4), (5) and (7), we have $\begin{matrix} h_{A} (1) \\ = & h_{A} (((y \to x) \to \neg \neg x) \to \neg \neg u) \cap h_{A} (\neg \neg x \to x) \\ \subseteq & h_{A} (((y \to x) \to \neg \neg x) \to u) \\ \subseteq & h_{A} (((y \to x) \to x) \to u), \end{matrix}$ and hence h_A (((y → x) → x) → u) = h_A (1). It follows from Proposition 3.1(1) that h_A ((y → x) → x) ⊆ h_A (u). Lemma 2.2(3) leads that h_A (x ∨ y) ⊆ h_A ((y → x) → x). Thus h_A (x ∨ y) = h_A ((y → x) → x). In particular, h_A (¬ y ∨ y) = h_A ((y → ¬ y) → ¬ y). Using Lemma 2.2(2) and (6), it yields that h_A ((y → ¬ y) → ¬ y) = h_A (¬ y² → ¬ y) ⊇ h_A (y → y²). Since A is implicative, it holds that h_A (¬ y ∨ y) = h_A (1), that is, A is a hesitant fuzzy Boolean filter.□

Example 4.15. Let L be a residuated lattice with the universe {0, a, b, 1} such that 0 < a < b < 1. The operations ⊗ and → are given by the tables below:

⊗	a	b	1	→	0	a	b	1
0	0	0	0	0	1	1	1	1
a	a	a	a	a	0	1	1	1
b	a	b	b	b	0	a	1	1
1	a	b	1	1	0	a	b	1

Let t₁ = {0.4}, t₂ = {0.4, 0.5, 0.6}, t₃ = {0.4, 0.5, 0.6, 0.7, 0.8}. Define A as h_A (0) = h_A (a) = t₁, h_A (b) = t₂ and h_A (1) = t₃. Routine calculations show that A is a hesitant fuzzy implicative filter, but not a hesitant fuzzy Boolean filter, because h_A (¬ a ∨ a) = t₁ ≠ t₃ = h_A (1).

Example 4.16. It is routine to verify that the hesitant fuzzy filter A in Example 4.7 is a hesitant fuzzy regular filter, but not a hesitant fuzzy Boolean filter, because h_A (¬ a ∨ a) = t₂ ≠ t₃ = h_A (1).

Corollary 4.17. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy Boolean filter if and only if $h_{A} (\neg \neg x \to x^{2}) = h_{A} (1),$ for all x ∈ L.

Theorem 4.18. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy Boolean filter if and only if $h_{A} (x \to y) = h_{A} (\neg \neg x \to (x \otimes y)),$ for all x, y ∈ L.

Proof. Using Corollary 4.17, Lemma 2.2(10), (1) and (2), we have $\begin{matrix} h_{A} (1) \\ = & h_{A} (\neg \neg x \to x^{2}) \\ \subseteq & h_{A} ((\neg \neg x \otimes (x \to y)) \to (x^{2} \otimes (x \to y))) \\ \subseteq & h_{A} ((\neg \neg x \otimes (x \to y)) \to (x \otimes y)) \\ = & h_{A} ((x \to y) \to (\neg \neg x \to (x \otimes y))), \end{matrix}$ and hence h_A ((x → y) → (¬¬ x → (x ⊗ y))) = h_A (1). It follows from Proposition 3.1 that h_A (x → y) ⊆ h_A (¬¬ x → (x ⊗ y)). Lemma 2.2(9) leads that h_A (¬¬ x → (x ⊗ y)) ⊆ h_A (x → y). Thus the identity holds.

Conversely, it is obvious by taking x = y.□

Theorem 4.19. Let A be a hesitant fuzzy filter of L. Then A is a hesitant fuzzy Boolean filter if and only if L/C_A is a Boolean algebra.

Proof. The proof is similar to that of Theorem 4.4.

□

5 Conclusions

In this paper, we have investigated the relationship between the hesitant fuzzy filters and the hesitant fuzzy congruences and established a one-to-one correspondence between the set of all hesitant fuzzy filters and the set of all hesitant fuzzy congruences; we have also proposed the notions of hesitant fuzzy implicative, regular and Boolean filters and investigated their properties.

Future research will focus on applying the idea/result in this paper to other related algebraic structures.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their insightful and constructive comments and suggestions that have led to this improved version of the paper. This research was supported by the NSF of Shandong Province (No. ZR2017MG027).

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