(∈ ,∈ ∨ q )-fuzzy t-filters on residuated lattices 1

Abstract

On residuated lattices, the notion of (∈ , ∈ ∨ q)-fuzzy t-filters is introduced and relative properties (e.g. Extension property, Triple of equivalent characteristics and Quotient characteristics) are uniformly covered by this simple general framework. The relations among (∈ , ∈ ∨ q)-fuzzy t-filters are investigated.

Keywords

Residuated lattice t-filter fuzzy t-filter

1 Introduction

Non-classical logic has become a considerable formal tool for computer science and artificial intelligence to deal with fuzzy information and uncertain information. Many-valued logic, a great extension and development of classical logic [3] has always been a crucial direction in non-classical logic. In the meantime, Many logical algebras are introduced as the algebraic semantics of so-called fuzzy logics, such as MV-algebras, BL-algebras, MTL-algebras, Gödel algebras, NM-algebras and R0-algebras, etc. These algebras are special residuated lattices [8,29, 8,29]. Besides their logical interest, residuated lattices have lots of interesting properties. In [10], Idziak proved that the varieties of residuated lattices is equational.

Filters play a vital role when the algebraic semantics were investigated for different logics. From logical point of views, filters correspond to sets of provable formulae. In [24] and [28], Ma and Víta found the common features of filters on residuated lattices. They respectively proposed the notion of τ-filters and t-filters on residuated lattices. In [28], Víta studied some basic properties of t-filters and gave the simple general framework of special types of filters.

The theory of fuzzy sets was introduced by Zadeh [31] and has been applied to many branches. The fuzzification of the filters was originated in 1995 [30]. Whereafter, lots of papers about special types of fuzzy filters was published in many journals on different logical algebras [11–14,17–19,25,27,34,35 , 11–14,17–19,25,27,34,35]. In [27], Víta noted the common characteristics of fuzzy filters on residuated lattices. He proposed the notion of fuzzy t-filters and discussed their properties.

In [32], Zadeh introduced the concept of interval valued fuzzy subset. The interval valued fuzzy subgroups were first defined and studied by Biswas [1] which are the subgroups of the same nature of the fuzzy subgroups defined by Rosenfeld. The (∈, ∈ ∨q)-fuzzy subgroups was introduced in an earlier paper of Bhakat and Das [2] by using the combined notions of "belongingness" and "quasi-coincidence" of fuzzy points and fuzzy sets, which was introduced by Pu and Liu [26]. The (∈, ∈ ∨q)-fuzzy subgroup is the most viable generalization of Rosenfeld’s fuzzy subgroup. In [6], Davvaz applied this theory to near-rings.

Recently, Ma [21 –23], Jun [15] applied this theory to some logical algebras and obtained some useful results. As a generalization of those papers, the concepts of (∈ , ∈ ∨ q)-fuzzy filters are introduced on residuated lattices. However, the paper do not want to increase the amount of papers about particular types of (∈ , ∈ ∨ q)-fuzzy filters. Rather, the unified form of(∈ , ∈ ∨ q)-fuzzy filters is given. Hence, the notion of (∈ , ∈ ∨ q)-fuzzy t-filters is proposed. And the relative properties about special types of (∈ , ∈ ∨ q)-fuzzy filters (e.g. Extension property, Triple of equivalent characteristics and Quotient characteristics) are unified by this simple general framework.

The paper is organized as follows: In Section 2, the basic definitions and results are recalled. In Section 3, the notion of (∈ , ∈ ∨ q)-fuzzy t-filters is proposed. The relations between (∈ , ∈ ∨ q)-fuzzy t-filters and fuzzy t-filters are discussed. The general properties of (∈ , ∈ ∨ q)-fuzzy t-filters (e.g. Extension property, Triple of equivalent characteristics and Quotient characteristics) are obtained. In Section 4, the relations among (∈ , ∈ ∨ q)-fuzzy t-filters are investigated.

2 Preliminaries

Definition 2.1. [8, 29] A residuated lattice is an algebra structure (L, ∧, ∨ , ⊗ , → , 0, 1) such that for all x, y, z∈ L,

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

(L, ⊗, 1) is a commutative monoid,

(⊗, →) forms an adjoint pair, i.e. x ⊗ y ≤ z if and only if x ≤ y → z.

Definition 2.2. [4, 5, 8, 35] Let L be a residuated lattice. Then L is called

an MTL-algebra if (x → y) ∨ (y → x) =1 for all x, y∈ L (prelinear axiom).

an Rl-monoid if x ∧ y = x ⊗ (x → y) for all x, y∈ L (divisible axiom).

a Heyting algebra if x ⊗ y = x ∧ y for all x, y∈ L, which is equivalent to an idempotent residuated lattice, that is, x = x ⊗ x = x ² for all x∈ L.

a regular residuated lattice if it satisfies double negation i.e. x ^∗∗ = x for all x∈ L, where x ^∗ = x → 0.

a BL-algebra if it satisfies both prelinear and divisible axioms.

an MV-algebra if it is a regular Rl-monoid.

a Boolean algebra if it is an idempotent MV-algebra.

In what follow, L is a residuated lattice unless otherwise specified.

Definition 2.3. [8] A non-empty subset F of L is called a filter on L if

for all x∈ F, x ≤ y implies y∈ F,

for all x, y∈ F, x⊗ y ∈ F.

Definition 2.4. [4, 5, 16, 24, 28, 34, 35] Let F be a filter on L. Then F is called

an implicative filter if x→ x ² ∈ F for all x∈ L.

a regular filter if x ^∗∗→ x ∈ F for all x∈ L.

a prelinear filter if (x→ y) ∨ (y → x) ∈ F for all x, y∈ L.

a divisible filter if (x∧ y) → (x ⊗ (x → y)) ∈ F for all x, y∈ L.

an n-contractive filter if x ⁿ→ x ⁿ⁺¹ ∈ F for all x∈ L, where x ⁿ⁺¹ = x ⁿ ⊗ x, n ≥ 1.

a Boolean filter if x∨ x ^∗ ∈ F for all x∈ L.

a fantastic filter if (y→ x) → (((x → y) → y) → x) ∈ F for all x, y∈ L.

Remark 2.5. On residuated lattices, x → (y → z) = y → (x → z) holds (see [8,16, 8,16]). Using this property, we have F is a fantastic filter if and only if ((x→ y) → y) → ((y → x) → x) ∈ F.

Now, some fuzzy concepts are given. A fuzzy set on L is a function μ: L⟶[0, 1]. For any α∈ [0,1] and an arbitrary fuzzy set μ, the set {x∈ L |μ (x) ≥ α} (i.e. the α-cut) is denoted by the symbol μ _α. A fuzzy set μ of L having the form

$μ (y) = {\begin{matrix} s (\neq 0), & y = x, \\ 0, & y \neq x . \end{matrix}$

is said to be a fuzzy point with support x and value s and is denoted by U (x ; s). A fuzzy point U (x ; s) is said to belong to (resp. be quasi-coincident with) a fuzzy set μ, written as U (x ; s) ∈ μ (resp. U (x ; s) q μ)if μ (x) ≥ s (resp. μ (x) + s > 1). The symbol U (x ; s) ∈ ∨ qμ means that U (x ; s) ∈ μ or U (x ; s) q μ.

Definition 2.6. [34, 35] A fuzzy set μ of L is a fuzzy filter on L if and only if it satisfies the following two conditions for all x, y∈ L:

μ (x ⊗ y) ≥ min {μ (x) , μ (y)},

if x ≤ y, then μ (x) ≤ μ (y).

Lemma 2.7. [34, 35] A fuzzy set μ on L is a fuzzy filter if it satisfies:

μ (1) ≥ μ (x) for all x∈ L,

μ (y) ≥ min {μ (x) , μ (x → y)} for all x, y∈ L.

Definition 2.8. A fuzzy set μ of L is called an (∈ , ∈ ∨ q)-fuzzy filter on L if for all s, r ∈ (0, 1] and x, y∈ L,

U (x ; r) ∈ μ implies U (y ; r) ∈ ∨ qμ with x ≤ y,

U (x ; s) ∈ μ and U (y ; r) ∈ μ imply U (x⊗ y ;min {s, r}) ∈ ∨ qμ.

Lemma 2.9. A fuzzy set μ on L is an (∈ , ∈ ∨ q)-fuzzy filters if and only if it satisfies the following two conditions for all x, y∈ L:

μ (x ⊗ y) ≥ min {μ (x) , μ (y) , 0.5},

if x ≤ y, then μ (y) ≥ min {μ (x) , 0.5}.

Proof. The proof is similar to that of [22, Theorem 3.3].

Lemma 2.10. A fuzzy set μ on L is an (∈ , ∈ ∨ q)-fuzzy filters if and only if it satisfies the following two conditions for all x, y∈ L:

μ (1) ≥ min {μ (x) , 0.5},

μ (y) ≥ min {μ (x) , μ (x → y) , 0.5}.

Proof. The proof is similar to that of [22, Theorem3.6].

3 Fuzzy t-filters and (∈ , ∈ ∨ q)-fuzzy t-filters

In the following, the symbol $\bar{x}$ denotes the abbreviation of x, y, …, i.e. $\bar{x}$ is a formal listing of variables used in a given content. By the term t, it is always meant a term in the language of residuated lattices. Another useful convention is also used: given a variety B of residuated lattices, its subvariety given by the equation t=1 is denoted by the symbol B[t], this algebra is called t-algebra.

Definition 3.1. [28] Let t be an arbitrary term on the language of residuated lattices. A filter F on L is a t-filter if t $(\bar{x}) \in$ F for all $\bar{x} \in$ L.

Definition 3.2. [27] A fuzzy filter μ of L is called a fuzzy t-filter on L, if for all $\bar{x} \in$ L it satisfies $μ (t (\bar{x})) = μ (1)$ .

Lemma 3.3. A fuzzy filter μ on L is a fuzzy t-filter, if and only if for all $\bar{x} \in$ L it satisfies $μ (t (\bar{x})) \geq μ (1)$ .

Proof. Since μ is a fuzzy filter, then by Lemma 2.7, $μ (1) \geq μ (t (\bar{x}))$ . Thus $μ (t (\bar{x})) = μ (1) \Leftrightarrow μ (t (\bar{x})) \geq μ (1)$ .

Definition 3.4. An (∈ , ∈ ∨ q)-fuzzy filter μ of L is called an (∈ , ∈ ∨ q)-fuzzy t-filter on L, if for all $\bar{x} \in$ L, it satisfies $μ (t (\bar{x})) \geq \min {μ (1), 0.5}$ .

Remark 3.5. The above definition incorporates many different types of (∈ , ∈ ∨ q)-fuzzy filters on residuated lattices. For example, (∈ , ∈ ∨ q)-fuzzy implicative filters are (∈ , ∈ ∨ q)-fuzzy t-filters for t equal to x → x ². (∈ , ∈ ∨ q)-fuzzy Boolean filters are (∈ , ∈ ∨ q)-fuzzy t-filters for t equal to x ∨ x ^*. (∈ , ∈ ∨ q)-fuzzy regular filters are (∈ , ∈ ∨ q)-fuzzy t-filters for t equal to x ^** → x.

Proposition 3.6. Every fuzzy filter on L is an (∈ , ∈ ∨ q)-fuzzy filter.

Proof. By Lemma 2.7 and Lemma 2.10, it is clear.

Proposition 3.7. If μ is an (∈ , ∈ ∨ q)-fuzzy filter on L with μ (1) <0.5, then μ is indeed a fuzzy filter.

Proof. Let μ be an (∈ , ∈ ∨ q)-fuzzy filter on L with μ (1) <0.5. Then by Lemma 2.10(1), μ (x) <0.5 and μ (1)≥ μ (x) , ∀ x ∈ L. Also by Lemma 2.10(2), μ (y) ≥ min {μ (x) , μ (x → y) , 0.5} = min {μ (x) , μ (x → y)}. This shows that μ is a fuzzy filter.

Theorem 3.8. Every fuzzy t-filter on L is an (∈ , ∈ ∨ q)-fuzzy t-filter.

Proof. Let μ be a fuzzy t-filter. Then μ is a fuzzy filter. By Proposition3.6, μ is an (∈ , ∈ ∨ q)-fuzzy filter. Since μ is a fuzzy t-filter, then $μ (t (\bar{x})) \geq μ (1)$ . Thus $μ (t (\bar{x})) \geq \min {μ (1), 0.5}$ . This shows that μ is an (∈ , ∈ ∨ q)-fuzzy t-filter.

Theorem 3.9. If μ is an (∈ , ∈ ∨ q)-fuzzy t-filter on L with μ (1) <0.5, then μ is indeed a fuzzy t-filter.

Proof. Suppose μ is an (∈ , ∈ ∨ q)-fuzzy t-filter on L with μ (1) <0.5, then by Proposition 3.7, μ is a fuzzy filter. Also $μ (t (\bar{x})) \geq \min {μ (1), 0.5} = μ (1)$ , by Lemma 3.3, μ is a fuzzy t-filter.

Theorem 3.10. A fuzzy set μ on L is an (∈ , ∈ ∨ q)-fuzzy filter if and only if μ _α (≠ ∅) is a filter for all α ∈ [0, 0.5].

Proof. The proof is similar to that of [22, Theorem 3.1].

Corollary 3.11. F is a filter if and only χ_F is an (∈ , ∈ ∨ q)-fuzzy filter of L, where $χ_{F} = {\begin{matrix} 1, & x \in F, \\ 0, & x \notin F . \end{matrix}$

Proof. Suppose F is a filter, ∀α ∈ (0, 0.5], the α-cut of χ _F is equal to F. When α = 0, the α-cut of χ _F is L. By Theorem 3.10, χ _F is an (∈ , ∈ ∨ q)-fuzzy filter. Conversely, Let χ _F be an (∈ , ∈ ∨ q)-fuzzy filter. By Theorem 3.10, the 0.5-cut of (∈ , ∈ ∨ q) is a filter. Also the 0.5-cut of χ _F is equal to F. Thus F is a filter.

Corollary 3.12. χ_{1} is an (∈ , ∈ ∨ q)-fuzzy filter.

Theorem 3.13. A fuzzy set μ on L is an (∈ , ∈ ∨ q)-fuzzy t-filter if and only if μ _α (≠ ∅) is a t-filter for all α ∈ [0, 0.5].

Proof. There exist two cases:

μ (1) ≥0.5,

μ (1) <0.5.

Case a) Let μ be an (∈ , ∈ ∨ q)-fuzzy t-filter on L and α ∈ [0, 0.5]. Then, by Theorem 3.10, μ _α is a filter on L. Also,

μ (t (\bar{x})) \geq \min {μ (1), 0.5} = 0.5 \geq α

. Thus

t (\bar{x}) \in μ_{α}

. Hence, μ _α is a t-filter. Conversely, let μ be a fuzzy set on L such that μ _α is a t-filter on L for all 0 ≤ α ≤ 0.5. Then by Theorem 3.10, μ is an (∈ , ∈ ∨ q)-fuzzy filter on L. Since μ (1) ≥0.5, then μ _0.5≠ ∅ is a t-filter on L. Thus, for all x∈ L,

t (\bar{x}) \in μ_{0.5}

. That is,

μ (t (\bar{x})) \geq 0.5 = \min {μ (1), 0.5}

. Therefore, μ is an (∈ , ∈ ∨ q)-fuzzy t-filter.

Case b) By Theorem 3.9, when μ (1) <0.5, every (∈ , ∈ ∨ q)-fuzzy t-filter becomes a fuzzy t-filter. The similar result can refer to [27, Theorem18]. It only need to note the α-cut is empty when α ≥ 0.5.

Theorem 3.14. F is a t-filter if and only if χ_F is an (∈ , ∈ ∨ q)-fuzzy t-filter of L.

Proof. The proof is similar to that of Corollary 3.11.

Theorem 3.15. (Extension property) Let μ, γ be (∈ , ∈ ∨ q)-fuzzy filters on L, μ (x) ≤ γ (x) for all x∈ L, and moreover μ (1) ≥0.5. If μ is an (∈ , ∈ ∨ q)-fuzzy t-filter, then γ is an (∈ , ∈ ∨ q)-fuzzy t-filter.

Proof. Let μ be an (∈ , ∈ ∨ q)-fuzzy t-filter and μ (1) ≥0.5. Thus for all $\bar{x} \in$ L, $μ (t (\bar{x})) \geq \min {μ (1),$ 0.5} =0.5. Since μ ≤ γ(pointwise), then γ (1) ≥0.5 and $(γ (t (\bar{x}))) \geq μ (t (\bar{x})) \geq 0.5$ . That is, $γ (t (\bar{x})) \geq \min {γ (1), 0.5}$ . Hence γ is an (∈ , ∈ ∨ q)-fuzzy t-filter.

Theorem 3.16. (Triple of equivalent characteristics) Let B be a variety of residuated lattices and B ∈B. Then the following statements are equivalent:

Every (∈ , ∈ ∨ q)-fuzzy filter on B is an (∈ , ∈ ∨ q)-fuzzy t-filter.

χ _{1} is an (∈ , ∈ ∨ q)-fuzzy t-filter.

B ∈B[t].

Proof. (1) ⇒ (2)

Since χ _{1} is an (∈ , ∈ ∨ q)-fuzzy filter, by assumption, χ _{1} is an (∈ , ∈ ∨ q)-fuzzy t-filter.

(2) ⇒ (3)

Suppose that χ _{1} is an (∈ , ∈ ∨ q)-fuzzy t-filter. Since the 0.5-cut of χ _{1} is equal to {1}, then {1} is a t-filter. Thus t = 1. Hence B ∈B[t].

(3) ⇒ (1)

Let μ be an arbitrary (∈ , ∈ ∨ q)-fuzzy filter on B and B ∈B[t]. Thus $t (\bar{x}) = 1$ . for all $\bar{x} \in$ B. Hence $t (\bar{x}) \geq 1$ . Therefore $μ (t (\bar{x})) \geq \min {μ (1), 0.5}$ . This shows that μ is an (∈ , ∈ ∨ q)-fuzzy t-filter.

The next part of this paper concerns fuzzy quotient constructions. Some results and constructions concerning fuzzy quotients residuated lattices are given.

Theorem 3.17. Let μ be an (∈ , ∈ ∨ q)-fuzzy filter on L and x, y∈ L. For any z∈ L, we define μ ^x : L → [0,1], μ ^x (z) = min {μ (x → z) , μ (z → x) , 0.5}. Then μ ^x = μ ^y if and only if μ (x → y) ≥ min {μ (1) , 0.5} and μ (y → x) ≥ min {μ (1) , 0.5}.

Proof. The proof is similar to that of [20, Theorem 5.4], it only need to put k = 0.

Theorem 3.18. Let μ be an (∈ , ∈ ∨ q)-fuzzy filter on L and L/μ : = {μ ^x|x∈ L}. For any μ ^x, μ ^y∈ L/μ, we define

μ ^x ∧ μ ^y = μ ^x∧y,

μ ^x ∨ μ ^y = μ ^x∨y,

μ ^x ⊗ μ ^y = μ ^x⊗y,

μ ^x → μ ^y = μ ^x→y,

Then L/μ = (L/μ, ∧ , ∨ , ⊗ , → , μ ⁰, μ ¹) is a residuated lattice called the fuzzy quotient residuated lattice.

Proof. The proof is similar to that of [20, Theorem 5.6], it only need to put k = 0.

Theorem 3.19. (Quotient characteristics) Let B be a variety of residuated lattices and B ∈B. Let μ be an (∈ , ∈ ∨ q)-fuzzy filter on B. Then the fuzzy quotient L/μ belongs to B[t] if and only if μ is an (∈ , ∈ ∨ q)-fuzzy t-filter on B.

Proof. Let μ be an (∈ , ∈ ∨ q)-fuzzy t-filter on B.Then μ (t (x, y, . . .)) ≥ min {μ (1) , 0.5} for all x, y, . . .∈ L. Also t (x, y, . . .) =1 → t (x, y, . . .) and t (x, y, . . .)→1 =1. Thus μ (1 → t (x, y, . . .)) = μ (t (x, y, . . .)) ≥ min {μ (1) , 0.5} . And μ (t (x, y, . . .) →1) = μ (1) ≥ min {μ (1) , 0.5}. Hence, by Theorem 3.17, for all x, y, . . .∈ L, μ ^t(x,y,...) = μ ¹ holds. According to the definition of L/μ, t (μ ^x, μ ^y, . . .) = μ ¹. Starting with arbitrary μ ^x, μ ^y, . . .∈ L/μ, t (μ ^x, μ ^y, . . .) = μ ¹ holds. Thus L/μ belongs to B[t]. Since μ ¹ is the greatest element in B[t], then t = 1.

Conversely, let L/μ ∈ B[t]. Then t = 1, so for all μ ^x, μ ^y, . . .∈ L/μ (i.e. for all x, y, . . .∈ L):t (μ ^x, μ ^y, . . .) = μ ¹. Thus μ ^t(x,y,...) = μ ¹. Thus, by Theorem3.17, μ (t (x, y, . . .)) = μ (1 → t (x, y, . . .)) ≥ min {μ (1) , 0.5}. Therefore μ is an (∈ , ∈ ∨ q)-fuzzy t-filter.

4 The relations among (∈ , ∈ ∨ q)-fuzzy t-filters

Theorem 4.1. Suppose that there are (∈ , ∈ ∨ q)-fuzzy t₁-filter and (∈ , ∈ ∨ q)-fuzzy t₂-filter on L and B[t₁] ⊆ B[t₂]. If μ is an (∈ , ∈ ∨ q)-fuzzy t₁-filter, then μ is an (∈ , ∈ ∨ q)-fuzzy t₂-filter.

Proof. μ is an (∈ , ∈ ∨ q)-fuzzy t₁-filter ⇒ L/μ ∈ B[t₁] ⇒ L/μ ∈ B[t₂] ⇒ μ is an (∈ , ∈ ∨ q)-fuzzy t₂-filter.

Theorem 4.2. Suppose that there are (∈ , ∈ ∨ q)-fuzzy t₁-filter and (∈ , ∈ ∨ q)-fuzzy t₂-filter on L. If B[t₁] = B[t₂]. Then μ is an (∈ , ∈ ∨ q)- fuzzy t₁-filter if and only if μ is an (∈ , ∈ ∨ q)- fuzzy t₂-filter.

Proof. μ is an (∈ , ∈ ∨ q)-fuzzy t₁-filter ⇔ L/μ ∈ B[t₁] ⇔ L/μ ∈ B[t₂] ⇔ μ is an (∈ , ∈ ∨ q)-fuzzy t₂-filter.

Theorem 4.3. Let L be a residuated lattice. If μ is an (∈ , ∈ ∨ q)- fuzzy implicative filter, then μ is an (∈ , ∈ ∨ q)-fuzzy n-contractive filter.

Proof. It is obvious that B [x → x ²] ⊆ B [x ⁿ → x ⁿ⁺¹]. By Theorem 4.1, the result is clear.

Lemma 4.4. [5] Let L be a residuated lattice. If L is a Heyting algebra, then L is an Rl-monoid.

Lemma 4.5. [35] Let L be a residuated lattice. Then the following are equivalent:

L is an MV-algebra.

(x→ y) → y = (y → x) → x, ∀ x, y ∈ L.

Lemma 4.6. [8] Let L be a residuated lattice. Then L is an MV-algebra if and only if L is a regular BL-algebra.

Lemma 4.7. Let L be a residuated lattice. Then the following are equivalent:

L is a Boolean algebra.

x∨ x ^∗ = 1, ∀ x ∈ L.

L is regular and idempotent.

Proof. (1)⇔(2) [35, Proposition 2.10]

(1)⇒(3)

If L is a Boolean algebra, then L is an idempotent MV-algebra. Thus L is regular and idempotent.

(3)⇒(1)

If L is regular and idempotent, then L is a regular Rl-monoid. Thus L is an MV-algebra. Also L is idempotent. Hence L is a Boolean algebra.

Proposition 4.8. Let L be a residuated lattice. t₁, t₂ are arbitrary terms on L. Then B[t₁⊗ t₂] = B[t₁] ∩ B[t₂].

Proof. L ∈B[t₁⊗ t₂]⇔ L ∈B and t₁⊗ t₂ = 1⇔ L ∈B, t₁ = 1 and t₂ = 1⇔ L ∈B[t₁] ∩ B[t₂].

Theorem 4.9. Let L be a residuated lattice. Then μ is an (∈ , ∈ ∨ q)- fuzzy Boolean filter if and only if μ is both an (∈ , ∈ ∨ q)- fuzzy regular and (∈ , ∈ ∨ q)-fuzzy implicative filter.

Proof. μ is an (∈ , ∈ ∨ q)-fuzzy Boolean filter ⇔ L /μ∈ B [x ∨ x ^∗] ⇔ L/μ∈ B [(x ^∗∗ → x) ⊗ (x → x ²)] ⇔ L/μ ∈B [x ^∗∗→ x] ∩ B [x → x ²] ⇔ L/μ ∈ B [x ^∗∗ → x] and L/μ ∈ B [x → x ²] ⇔ μ is both an (∈ , ∈ ∨ q)- fuzzy regular and (∈ , ∈ ∨ q)-fuzzy implicative filter.

Similarly, the following results are true.

Theorem 4.10. Let L be a residuated lattice. If μ is an (∈ , ∈ ∨ q)-fuzzy Boolean filter if and only if μ is an (∈ , ∈ ∨ q)-fuzzy fantastic and (∈ , ∈ ∨ q)-fuzzy implicative filter.

Theorem 4.11. Let L be a residuated lattice. If μ is an (∈ , ∈ ∨ q)-fuzzy implicative filter, then μ is an (∈ , ∈ ∨ q)-fuzzy divisible fliter.

Theorem 4.12. Let L be a residuated lattice. If μ is an (∈ , ∈ ∨ q)-fuzzy fantastic filter if and only if μ is both an (∈ , ∈ ∨ q)-fuzzy regular and (∈ , ∈ ∨ q)-fuzzy divisible filter.

5 Conclusions and future research

On residuated lattices, the notion of (∈ , ∈ ∨ q)-fuzzy t-filters are proposed and relative properties are discussed. The relations among (∈ , ∈ ∨ q)-fuzzy t-filter are studied. The typical results about special types of (∈ , ∈ ∨ q)-fuzzy filter can be covered by a simple uniform theory. In fact, the results in this paper are applicable not only on residuated lattices, but also on all their subvarieties. And the same method can be used to unify special types of (∈ , ∈ ∨ q _k)-fuzzy filters [20].

In addition, Zhu [36] proposed the $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy (implicative, positive implicative, fantastic and regular) filters on residuated lattices. In the future work, these special types of $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy filter will be unified and and the notion of $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy t-filters will be introduced. The properties and the relations among various special $(\bar{\in}, \bar{\in} \lor \bar{q})$ -fuzzy t-filters will be investigated. For more details, they will be given out in the future paper.

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