A note on some filters in residuated lattices

1 Introduction

It is well known that much of human reasoning and decision making is based on imprecise, uncertain, incomplete information [1]. Then how to simulate and deal with imperfect information becomes a significant problem. For this reason, various kinds of logical algebras as the semantics of corresponding logic systems have been extensively introduced and studied, for example, residuated lattices [2], MTL-algebras [3], BL-algebras [4], IMTL-algebras [3], DRL-monoids [5], etc. Among these algebras, residuated lattices are the most important because the others are all particular classes of residuated lattices.

Filters play a key role in studying the above algebras and the completeness of corresponding logics. From algebraic point of view, filters correspond to congruences in algebras. For example, if we consider a Boolean filter F in residuated lattice L, which is defined by x ∨ ¬ x ∈ F for all x ∈ L, then the quotient algebra L/F induced by Boolean filter F is a Boolean algebra [6]. This implies that Boolean algebras can be characterized by residuated lattices with x ∨ ¬ x = 1. On the other hand, from logical point of view, filters correspond to sets of provable formulae. For example, using prime filters of BL-algebras, Hájek proved the completeness of Basic Logic BL [4]. Hence it is necessary to study the theory of filters.

Up to now, various types of filters have been proposed and some results of them have been obtained. For example, in [7], Víta introduced the notion of t-filters in residuated lattices and indicated that many filters such as Boolean filters, implicative filters, regular filters, positive implicative filters can be regarded as a t-filter for a suitable term t ( x ¯ ). In order to develop a unifying definition for some specific filters, Ma and Hu [8] proposed the notion of I-filters in residuated lattices and obtained some of its characterizations. In [9], Borzooei et al. established EIMTL-filters and associative filters theory in MTL-algebras and proved that EIMTL-filters, IMTL-filters and fantastic filters coincide in BL-algebras, whereas they are different in MTL-algebras. In [10], Kondo presented characterizations of extended filters in residuated lattices. For more details, readers may refer to the relative references given in this paper.

In this paper, we study some properties of EIMTL-filters, associative filters, t-filters and I-filters in residuated lattices. The conditions under which an EIMTL-filter is a t-filter, and which a residuated lattice has a proper associative filter are considered. The existence of a proper EIMTL-filter is studied and it is proved that every residuated lattice has a proper EIMTL-filter. In addition, we discuss the relationship between t-filters and I-filters.

2 Preliminaries

We recollect some definitions and results which will be used in the following and we shall not cite them every time they are used.

Definition 2.1. [4] A residuated lattice is a structure L = (L, ∨ , ∧ , ⊙ , → , 0, 1) of type (2,2,2,2,0,0) satisfying the following axioms:

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

Let L be a residuated lattice. For any x ∈ L, we define ¬x = x → 0.

Let L be a residuated lattice, x, y ∈ L. If x ≤ y or y ≤ x, then x and y are called comparable, otherwise, x and y are called incomparable.

Definition 2.2. [2–5, 8] Let L be a residuated lattice. L is called

a Glivenko residuated lattice, if ¬¬ (¬¬x → x) =1 for any x ∈ L.

Clearly, each linearly ordered residuated lattice is an MTL-algebra, and each BL-algebra is a DRL-monoid.

Lemma 2.3. [4] Let L be a residuated lattice. The following hold:

x ≤ y if and only if x → y = 1.

Definition 2.4. [11] Let L be a residuated lattice,x ∈ L. The order of x is the least positive integer m such that x ^m  = 0, denoted by ord (x). If m does not exist, then the order of x is denoted by ord (x) =∞.

Let L be a residuated lattice. L is locally finite if for every x ∈ L ∖ {1}, ord (x)< ∞.

Definition 2.5. [12] Let X be a set of (distinct) objects called variables. Let H be a type of algebras. The set T (X) of terms of type H over X is the smallest set such that

X ∪ H₀ ⊂ T (X),

Definition 2.6. [4] Let L be a residuated lattice, ∅ ¬ = F ⊆ L. F is called a filter in L, if the following conditions are fulfilled:

If x, y ∈ F, then x ⊙ y ∈ F.

Let F be a filter in L. F is proper if F ¬ = L. A proper filter F is maximal if it is not contained in any other proper filter of L. The intersection of maximal filters is denoted by Rad (L).

For a proper filter F in L, let Rad (F) denote the intersection of all maximal filters that contain F, D _s  (F) = {x ∈ F| ¬ x = 0} denote the set of dense elements of F, and D (F) = {x ∈ L| ¬¬ x ∈ F} denote the set of double complemented elements [13]. Obviously, Rad (L) = Rad ({1}), D _s  (F) ⊆ F ⊆ Rad (F). It is easily checked that D _s  (F) , D (F) and Rad (F) are filters in L [13].

Lemma 2.7. [4] Let L be a residuated lattice. Then F is a filter if and only if the following hold:

1 ∈ F.

For convenience, in the sequel, we will use x ¯ as an abbreviation of a finite sequence x₁, x₂, ….

Definition 2.8. [7, 8] Let L be a residuated lattice, F be a filter in L. F is called

a strong filter, if ¬¬ (¬¬ x → x) ∈ F for allx ∈ L.

Clearly, regular filters and strong filters are t-filter. Each regular filter is a strong filter.

Lemma 2.9. [7] (Extension property) Let F and G be filters in residuated lattice L such that F ⊆ G. If F is a t-filter, then so is G.

Lemma 2.10. [7] (Triple of equivalent characteristics) Let B be a variety of residuated lattices and L ∈ B. The following statements are equivalent:

Every filter in L is a t-filter.

Lemma 2.11. [7] (Quotient characteristics) Let B be a variety of residuated lattices and L ∈ B, and F be a filter in L. Then F is a t-filter, if and only if L/F is in B [t] .

Definition 2.12. [10] Let L be a residuated lattice. For a subset B ⊆ L and a filter F in L, E F ( B ) = { x ∈ L | x ∨ b ∈ F for all b ∈ B } is called an extended filter associated with B.

For convenience, we denote E _F  (x) instead of E _F  ({x}).

Lemma 2.13. [10] Let F be a filter in residuated lattice L, B ⊆ L. The following hold:

E _F  (B) is a filter in L,

In view of Lemmas 2.9 and 2.13, we have the following corollary:

Corollary 2.14. Let L be a residuated lattice, B ⊆ L. If F is a t-filter, then so is E _F  (B).

For a filter F, we define relation ≡ _F on L as follows:x ≡  _F y if and only if x → y ∈ F and y → x ∈ F.

It is easily verified that ≡ _F is a congruence relation. Let [x]  _F denote the congruence class of x, L/F denote the set of the congruence classes of ≡ _F . Then L/F is a residuated lattice [4].

3 Some properties of t-filters

The following theorem points out the relationship between t-filters and I-filters in residuated lattices.

Theorem 3.1. t-filters and I-filters are equivalent in residuated lattices.

Proof. Suppose that F is an I-filter. In view of Definition 2.8, there exist terms u ( x ¯ ) and s ( x ¯ ) such that u ( x ¯ ) ≤ s ( x ¯ ) and s ( x ¯ ) → u ( x ¯ ) ∈ F. Let t ( x ¯ ) = s ( x ¯ ) → u ( x ¯ ), then t ( x ¯ ) ∈ F. By Definition 2.8, we get that F is a t-filter. Conversely, suppose that F is a t-filter, then there is a term t ( x ¯ ) such that t ( x ¯ ) ∈ F. Since t ( x ¯ ) ≤ 1 and 1 → t ( x ¯ ) = t ( x ¯ ) ∈ F, by Definition 2.8, F is an I-filter.

As an application of Theorem 3.1 and Lemmas 2.9, 2.10 and 2.11, we have the following corollaries:

Corollary 3.2. [8] (Extension property) Let F and G be filters in residuated lattice L such that F ⊆ G. If F is an I-filter, then so is G.

Corollary 3.3. [8] (Triple of equivalent characteristics) Let B be a variety of residuated lattices and L ∈ B. The following statements are equivalent:

Every filter in L is an I-filter.

Corollary 3.4. [8] (Quotient characteristics) Let B be a variety of residuated lattices and L ∈ B, and F be a filter in L. Then F is an I-filter, if and only if L/F is in B [t] .

Corollary 3.5. Let L be a residuated lattice, B ⊆ L. If F is an I-filter, then so is E _F  (B).

Theorem 3.6. Let L be a residuated lattice, F be a filter in L, t ( x ¯ ) be a term in L. Then F is a t-filter if and only if E F ( t ( x ¯ ) ) = L.

Proof. Suppose that F is a t-filter, then t ( x ¯ ) ∈ F for all x ¯ ∈ L, so t ( x ¯ ) ∨ y ∈ F for all y ∈ L. Hence E F ( t ( x ¯ ) ) = L. Conversely, suppose E F ( t ( x ¯ ) ) = L. Then for all x ¯ ∈ L, we have t ( x ¯ ) = t ( x ¯ ) ∨ t ( x ¯ ) ∈ F. By Definition 2.8, F is a t-filter.

Corollary 3.7. Let F be a filter in residuated lattice L, t ( x ¯ ) and s ( x ¯ ) are terms in L, t ( x ¯ ) ≤ s ( x ¯ ). Then F is an I-filter if and only if E F ( s ( x ¯ ) → t ( x ¯ ) ) = L.

Corollary 3.8. Let L be a residuated lattice, F be a filter in L. Then F is a strong filter if and only if E _F  (¬¬ (¬¬ x → x)) = L.

The following is a characterization of strong filters.

Theorem 3.9. Let L be a residuated lattice, F be a filter in L. The following are equivalent:

F is a strong filter.

Proof. (1) ⇒ (2) Suppose that F is a strong filter, then ¬¬ (¬¬ x → x) ∈ F for all x ∈ L. From Lemma 2.3 and ¬x ≤ x → y, it follows that ¬¬ x → x ≤ ¬ (x → y) → x, so ¬¬ (¬¬ x → x) ≤ ¬¬ (¬ (x → y) → x). Hence ¬¬ (¬ (x → y) → x) ∈ F.

(2) ⇒ (1) It follows immediately by taking y = 0 in item (2).

(1) ⇒ (3) Suppose that (1) holds. Then ¬¬ (¬¬ x → x) ∈ F for all x ∈ L. By Lemma 2.3, we have ¬¬ x → x ≤ (y ⊙ ¬¬ x) → (y ⊙ x). Hence ¬¬ ((y ⊙ ¬¬ x) → (y ⊙ x)) ∈ F.

(3)⇒ (1) It follows immediately by taking y = 1 in item (3).

4 EIMTL-filters in residuated lattices

The notion of EIMTL-filters was firstly proposed by Borzooei et al. in MTL-algebras. In this section, we study this notion in the scope of residuated lattices.

Definition 4.1 [9]. Let L be a residuated lattice, ∅ ¬ = F ⊆ L. F is called an EIMTL-filter in L, if the following are fulfilled:

1 ∈ F.

Obviously, if L satisfies ¬¬ x = x for all x ∈ L, then the notion of EIMTL-filters coincides with that of filters in L.

The following example shows that there exist EIMTL-filters in residuated lattices.

Example 4.2. Let L = {0, a, b, c, d, 1} be a lattice whose Hasse diagram is given by Fig. 1.

(1) Define ⊙ and → on L as follows:

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice. It is easily checked that {1}, {1, c, d} and L are EIMTL-filters in L.

(2) Define ⊙ and → on L as follows:

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice. It is easily checked that {1} and L are EIMTL-filters in L. Similar to the proofs of [9, Theorem 3.4 and Lemma 3.8], we have the following Lemmas 4.3 and 4.4.

Lemma 4.3. Each EIMTL-filter is a filter in residuated lattice L.

Lemma 4.4. Let L be a residuated lattice, F be a filter in L. Then F is an EIMTL-filter if and only if ¬¬ x ∈ F implies x ∈ F for all x ∈ L.

From Lemma 4.4, we know that EIMTL-filter is not a t-filter. Then under what condition an EIMTL-filter is a t-filter?

Theorem 4.5. Let L be a residuated lattice, F be a filter in L. If F satisfies the condition (*): ¬ ¬ ( x → y ) ∈ F if and only if ¬ ¬ x → ¬ ¬ y ∈ F

Then F is an EIMTL-filter if and only if ¬¬ x →x ∈ F for all x ∈ L.

Proof. Suppose that F is an EIMTL-filter. Since ¬¬ x → ¬¬ x = 1 ∈ F, from (*), we know that ¬¬ (¬¬ x → x) ∈ F. By Lemma 4.4, we have ¬¬ x → x ∈ F. Conversely, suppose that F is a filter and ¬¬ x → x ∈ F for all x ∈ L. If ¬¬ x ∈ F, then x ∈ F. By Lemma 4.4, F is an EIMTL-filter.

Corollary 4.6. Let L be a residuated lattice, F be a filter in L. If F satisfies the condition (*), then F is an EIMTL-filter if and only if F is a regular filter.

It is proved in [5,14, 5,14] that if L is a Glivenko residuated lattice (DRL-monoid), then the identity ¬¬ (x → y) = ¬¬ x → ¬¬ y holds for any x, y ∈ L. In view of Theorem 4.5, we have the following corollary:

Corollary 4.7. Let L be a Glivenko residuated lattice (DRL-monoid), F be a filter. Then F is an EIMTL-filter if and only if ¬¬ x → x ∈ F for any x ∈ L.

Remark 4.8. Corollary 4.7 extends a result from [9, Theorem 3.9].

Remark 4.9. Corollary 4.7 indicates that EIMTL-filters are t-filters in Glivenko residuated lattices and DRL-monoids.

In view of Remark 4.9 and Lemma 2.9, we have the following:

Theorem 4.10. Let L be a Glivenko residuated lattice (DRL-monoid), F and G be filters such that F ⊆ G. If F is an EIMTL-filter, then so is G.

Theorem 4.11. Let L be a residuated lattice, B ⊆ L. If F is an EIMTL-filter, then so is E _F  (B).

Proof. Let ¬¬ x ∈ E _F  (B), by Definition 2.12, for all b ∈ B, we have ¬¬ x ∨ b ∈ F. From b ≤ ¬¬ b, it follows that ¬¬ x ∨ ¬¬ b ∈ F. Since ¬¬ (x ∨ b) = ¬ (¬ x ∧ ¬ b) ≥ ¬¬ x ∨ ¬¬ b, we have ¬¬ (x ∨ b) ∈ F. By Lemma 4.4, x ∨ b ∈ F, i.e., x ∈ E _F  (B). Therefore E _F  (B) is an EIMTL-filter by Lemma 4.4.

Theorem 4.12. Let L be a residuated lattice, F be a filter in L. Then F is an EIMTL-filter if and only if x ∈ E _F  (¬¬ x) implies x ∈ F.

Proof. Suppose that F is an EIMTL-filter and x ∈ E _F  (¬¬ x). Then x ∨ ¬¬ x ∈ F, so ¬¬ x ∈ F. By Lemma 4.4, x ∈ F. Conversely, for all x ∈ L, if ¬¬ x ∈ F, then ¬¬ x ∨ x = ¬¬ x ∈ F, thus x ∈ E _F  (¬¬ x). Hence x ∈ F. From Lemma 4.4, F is an EIMTL-filter.

Lemma 4.13. Let L be a residuated lattice. If F _i  (i ∈ Γ) are EIMTL-filters, then ∩ {F _i  : i ∈ Γ} is an EIMTL-filter.

Proof. The proof is straightforward.

Lemma 4.14. Let L be a residuated lattice. If F is an EIMTL-filter, then so is D _s  (F).

Proof. For any x ∈ L, if ¬¬ x ∈ D _s  (F), then ¬¬ x ∈ F and ¬x = ¬ (¬¬ x) =0. Since F is an EIMTL-filter, we have x ∈ F, and it follows that x ∈ D _s  (F). By Lemma 4.4, D _s  (F) is an EIMTL-filter.

Theorem 4.15. Let F be a filter in residuated lattice L. Then F is an EIMTL-filter if and only if D (F) = F.

Proof. Suppose that F is an EIMTL-filter. Clearly, F ⊆ D (F). If x ∈ D (F), then ¬¬ x ∈ F. Since F is an EIMTL-filter, we have x ∈ F, thus D (F) ⊆ F. This shows that D (F) = F. Conversely, if ¬¬ x ∈ F, then by definition of D (F), x ∈ D (F). From D (F) = F, we have x ∈ F. Hence F is an EIMTL-filter.

The following example shows that there is not a general relationship between EIMTL-filters and strong filters in residuated lattices.

Example 4.16. Let L = {0, a, b, c, d, e, 1} be a lattice whose Hasse diagram is given by Fig. 2.

(1) Define ⊙ and → on L as follows:

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a locally finite residuated lattice. It is easily checked that {1} is an EIMTL-filter, but it is not a strong filter, because ¬¬ (¬¬ a → a) = e ∉ {1}.

(2) Define ⊙ and → on L as follows:

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice. It is easily checked that {1} is a strong filter. However, {1} is not an EIMTL-filter because ¬¬ e ∈ {1}, but e ∉ {1}.

5 The existence of proper EIMTL-filters in residuated lattices

In the preceding section, we know that L is always an EIMTL-filter. A natural question is that whether residuated lattices have proper EIMTL-filters? In this section, we mainly study this question.

Lemma 5.1. Let L be a locally finite residuated lattice, then the following hold:

x = 0 if and only if ¬x = 1.

Proof. (1) If x = 0, then by Lemma 2.3, ¬x = 1. Conversely, if ¬x = 1, then ¬¬ x = 0. From x ≤ ¬¬ x, we have x = 0.

(2) If x = 1, then ¬x = 0. Conversely, if ¬x = 0, then x ¬ =0. Suppose that x ¬ =1, then 0 < x < 1. Since L is locally finite, there exists n > 1 such that x ⁿ  = 0 and x^n-1 ¬ =0. So 1 = ¬ x ⁿ  = x ⁿ  → 0 = x^n-1 → ¬ x. Hence x^n-1 ≤ ¬ x, i.e., x^n-1 = 0. This is a contradiction. Therefore x = 1.

(3) It follows from (1) and (2).

Remark 5.2. From the proof of Lemma 5.1, we know that the results of Lemma 5.1 also hold in any non-commutative locally finite residuated lattices.

Lemma 5.3. Let L be a locally finite residuated lattice, x, y ∈ L ∖ {0, 1}. If x and y are comparable, then x ⊙ y < min {x, y}.

Proof. Let x ≤ y and suppose x ⊙ y = min {x, y}, then x ⊙ y = x. Since L is locally finite, there exists n > 1 such that y ⁿ  = 0. Hence 0 = y ⁿ  ⊙ x = y^n-1 ⊙ (y ⊙ x) = y^n-1 ⊙ x = ⋯ = y ⊙ x = x. This is a contradiction. Therefore x ⊙ y < min {x, y}.

The following example indicates that if x and y are incomparable, then the result of Lemma 5.3 may not be true.

Example 5.4. Let L be the residuated lattice defined in Example 4.2(2). Calculation shows that ord (0) =1, ord (a) = ord (b) = ord (c) =2, ord (d) =3, hence L is locally finite. Obviously, b and c are incomparable, but b ⊙ c = b ∧ c.

Corollary 5.5. If L is a locally finite linearly ordered residuated lattice, then x ⊙ y < min {x, y} for all x, y ∈ L ∖ {0, 1}.

Remark 5.6. Corollary 5.5 extends a result from [15, 16, Lemmas 4.1.1 and 4.1].

Corollary 5.7. Let L be a finite residuated lattice, μ ¬ =1 be an atom of L, 0 < x < 1. If L is locally finite, then μ ⊙ x = 0.

Lemma 5.8. Let L be a finite residuated lattice, μ ¬ =1 be an atom of L. Then ord (μ)< ∞ if and only if μ ⊙ μ = 0.

Proof. If μ ⊙ μ = 0, then ord (μ)< ∞. Conversely, from μ ⊙ μ ≤ μ, we have μ ⊙ μ ∈ {0, μ}. Since ord (μ)< ∞, then μ ⊙ μ < μ, so μ ⊙ μ = 0.

Theorem 5.9. If L is a locally finite residuated lattice, then {1} is an EIMTL-filter.

Proof. For all x ∈ L, if ¬¬ x ∈ {1}, then ¬¬ x = 1, so ¬x = 0. By Lemma 5.1, x = 1, i.e., x ∈ {1}. Hence {1} is an EIMTL-filter.

Theorem 5.10. Every maximal filter is an EIMTL-filter in residuated lattice L.

Proof. Let M be a maximal filter in residuated lattice L. Then L/M is a locally finite residuated lattice [11]. For all x ∈ L, if ¬¬ x ∈ M, then ¬¬ x ∈ [1]  _M . From Theorem 5.9, we know that [1]  _M is an EIMTL-filter, so x ∈ [1]  _M , i.e., x ∈ M. Therefore M is an EIMTL-filter by Lemma 4.4.

Corollary 5.11. Let F be a filter in residuated lattice L. Then Rad (F) is an EIMTL-filter.

Corollary 5.12. Let L be a residuated lattice. Then Rad (L) is an EIMTL-filter.

Corollary 5.13. Every residuated lattice has a proper EIMTL-filter.

6 Associative filters in residuated lattices

Associative filters were firstly introduced by Borzooei et al. in MTL-algebras. In this section, we study this notion in the scope of residuated lattices.

Definition 6.1. [9] Let L be a residuated lattice, ∅ ¬ = F ⊆ L, x ∈ L, x ¬ =0. F is called an associative filter, if the following hold:

1 ∈ F,

Theorem 6.2. Let L be a residuated lattice, ∅ ¬ = F ⊆ L. The following are equivalent:

F is an associative filter.

Proof. (1) ⇒ (2) The proof is similar to that of [9, Theorem 3.18, 3.20].

(2) ⇒ (3) If y → (y → x) ∈ F, by (2), we have (y → y) → x ∈ F, so x ∈ F.

(3) ⇒ (4) From x → (x → x²) =1 ∈ F and (3), we get x² ∈ F.

(4) ⇒ (5) If 1, z, z → (y → (y → x)) ∈ F, then y² → x = y → (y → x) ∈ F. By (4) and y ¬ =0, we have y² ∈ F, so x ∈ F.

(5)⇒ (1) For any x, y ∈ L, if x ∈ F and x ≤ y, then x → (1 → (1 → y)) = x → y = 1 ∈ F. By (5), we have y ∈ F. For any x, y, z ∈ L, x ¬ =0, we have x → (y → z) ≤ (x → y) → (x → (x → z)). If x → (y → z) ∈ F and x → y ∈ F, then (x → y) → (x → (x → z)) ∈ F. In view of (5), we have z ∈ F. Therefore F is an associative filter by Definition 6.1.

Corollary 6.3. Let L be a residuated lattice. If L has a proper associative filter, then ord (x) =∞ for all x ∈ L ∖ {0}.

The following example shows that the converse of Corollary 6.3 may not be true.

Example 6.4. Let L = {0, a, b, 1} be a lattice whose Hasse diagram is given by Fig. 3. Define ⊙ and → on L as follows:

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice [13]. Obviously, {1} , {1, a} , {1, b} and L are filters in L, and ord (a) = ord (b) = ord (1) =∞. But L has no proper associative filters.

Theorem 6.5. Let L be a residuated lattice. L has a proper associative filter if and only if ¬x = 0 for all x ¬ =0.

Proof. Suppose L has a proper associative filter F, and there exists x₀ ¬ =0 such that ¬x₀ > 0. Then from Theorem 6.2, we have x₀, ¬ x₀ ∈ F, so 0 = x₀ ⊙ ¬ x₀ ∈ F. This contradicts with 0 ∉ F. Hence ¬x = 0 for all x ¬ =0. Conversely, suppose ¬x = 0 for all x ¬ =0. Let F = {x ∈ L : x ¬ =0}. Clearly, 0 ∉ F and x² ∈ F for all x ¬ =0. Now we prove F is a filter. If x, y ∈ F, then x ¬ =0, y ¬ =0. Suppose x ⊙ y ∉ F, then x ⊙ y = 0, so y ≤ ¬ x. From ¬x = 0, we have y = 0. This is a contradiction. Hence x ⊙ y ∈ F. If x ≤ y and x ∈ F, then x ¬ =0, thus y ∈ F. Hence F is a filter. By Theorem 6.2, F is a proper associative filter.

Corollary 6.6. Every linearly ordered Heyting algebra has a proper associative filter.

Corollary 6.7. Let L be a residuated lattice. L ∖ {0} is a filter if and only if ¬x = 0 for all x ¬ =0.

Corollary 6.8. Let L be a residuated lattice. L has a proper associative filter if and only if L ∖ {0} is a filter.

Remark 6.9. Corollary 6.8 extends a result from [17, Theorem 2.5].

Acknowledgements

The authors express sincere thanks to the referees for their valuable suggestions and comments. This work is supported by NSFC (Nos. 60875084,61273017), by the Fundamental Research Funds for the Central Universities (JUSRP21118, JUSRP211A24), by Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents and the Project-sponsored by SRF for ROCS, SEM.