Distance functions and filter topological residuated lattices 1

Abstract

In this paper, we firstly extend C. C. chang’s distance functions from MV-algebras into residuated lattices. But in general, the functions may not be a distance function on residuated lattices. We introduce weak involutory residuated lattices, in which Chang’s function is a pseduo distance function. Moreover we prove that the functions become distance functions on involutory residuated lattices. Secondly by use of the function and a lattice filter, we define F-ball on residuated lattices, and we prove that the set of all F-balls forms a base of a topology τ_LF on an involutory residuated lattice. Moreover we show that the topology τ_LF on an involutory residuated lattice L makes L to be a topological residuated lattice. At last, we use filters instead of lattice filters to set up filter topologies on an involutory residuated lattice, and study the properties of the filter topologies.

Keywords

(Weak) involutory residuated lattice distance function (lattice) filter topology

1 Introduction

In order to have more than two values for evaluation of formulas, a lot of work has been done to generalize the classical algebra for logic. On this way, a special structure, namely residuated lattices, is a very basic and important algebraic structure. Many kinds of non-classical logical algebras, such as MTL-algebra, IMTL-algebra, BL-algebra, Heyting algebra and MV-algebra can be constructed based on residuated lattices. In [6], P. Hájek introduced the notion of BL-algebra as a residuated lattice with two more conditions, namely divisibility and prelinearity.

In [18], the notion of co-annihilators was introduced by Turunen in BL-algebras and some of its properties were studied. In [7], with the help of co-annihilators, M. Haveshki proposed the concept of α-filters in BL-algebras which was inspired by the dual notion of α-ideal from the lattice theory [4].

In 2017, Luan and Yang [9] applied proper lattice filters of an MV-algebra to construct a topology and called it filter topology. They showed that, each filter topology on an MV-algebra making it a topological MV-algebra. Also, using some new types of sequences in MV-algebras, they observed that a 2-divisible MV-algebra is semisimple if and only if it is Hausdorff with respect to any 2-divisible lattice filter. In addtion, there is another way to induce topologies. By two novel norms, topology is induced on neutrosophic soft set, and some topological properties are discussed in [17].

In 2019, Di Nola etc. [5] studied topological spaces of monadic MV-algebras and constructed a covariant functor γ from the category of monadic MV-algebras into the category of Q-distributive lattices, i.e., distributive lattices with quantifier introduced by R. Cignoli. For every monadic MV-algebra, they constructed a dual object named QM-space, and these objects formed a special subcategory of spectral spaces and of Q-spaces developed by R. Cignoli for Q-distributive lattices.

In 2019, Asadzadeh etc. [3] studied some basic facts and some notable examples, including the standard MV-algebra. Then authors showed that every filter topology on a finite MV-chain and also finite simple MV-algebra is discrete. Moreover, they completely described the shape of open sets in the filter topology which generated by a proper filter. This description applies to present basic topological properties of a large class of filter topologies. As an application, they characterized an algebraic property of MV-algebras: an MV-algebra is bipartite if and only if it has a proper filter which generates a filter topology with exactly four open sets.

A (semi)topological residuated lattice is introduced. By means of the inverse limit of an inverse system, completion of a residuated lattice is characterized in [14]. By a lattice ideal and a distance function on an involutive residuated lattice, Wang and Zhao constructed a semitopological residuated lattice in [19]. However in our paper, we use lattice filter and distance function to construct a toplogical residuated lattice, which is more strong than the above result. There is also a close relationship between topology and rough sets. By using a more general topology, authors [1, 2] designed new rough set models with better measurements, and applied the models to disease diagnosis, achieving good diagnostic results.

2 Preliminaries

In this section, we give some notions and results in residuated lattices, which will be used in the following sections of this paper.

Definition 2.1. [6, 20] An algebra structure (L, ∧ , ∨ , ⊙ , → , 0, 1) of type (2, 2, 2, 2, 0, 0) is called a residuated lattice if it satisfies the following conditions:

(1) (L, ∧ , ∨ , 0, 1) is a bounded lattice relative to the order ≤;

(2) (L, ⊙ , 1) is a commutative monoid;

(3) x ⊙ y ≤ z if and only if x ≤ y → z, for all x, y, z ∈ L.

In the following, we denote a residuated lattice (L, ∧ , ∨ , ⊙ , → , 0, 1) by L unless specified.

For a ∈ L and a natural number n, we define aⁿ = a^n-1 ⊙ a for n ≥ 1 and ¬a = a → 0.

Lemma 2.2. [6, 18] In any residuated lattice L, the following properties hold: for any x, y, z ∈ L,

1 → x = x, x → 1 =1,

x ≤ y if and only if x → y = 1,

x ⊙ ¬ x = 0, and x ⊙ y = 0 if and only if x ≤ ¬ y,

(x ∨ y) → z = (x → z) ∧ (y → z),

x ⊙ (x → y) ≤ y,

x ⊙ y ≤ x ∧ y, x ≤ y → x,

x → (y → z) = (x ⊙ y) → z = y → (x → z),

x ≤ y implies ¬y ≤ ¬ x,

¬0 =1, ¬1 =0, x ≤ ¬¬ x and ¬x = ¬¬ ¬ x,

x ∨ (y ⊙ z) ≥ (x ∨ y) ⊙ (x ∨ z),

x ⊙ (y ∨ z) = (x ⊙ y) ∨ (x ⊙ z),

x ≤ y implies y → z ≤ x → z and z → x ≤ z → y,

x ≤ (x → y) → y and y ≤ (x → y) → y,

x ≤ y implies x ⊙ z ≤ y ⊙ z.

Definition 2.3. Let L be a residuated lattice.

If for all x ∈ L, ¬¬ x = x, then L is called an involutory residuated lattice.

Definition 2.4. [6, 10] Let L be a residuated lattice. Then L is called:

(1) a Rl-monoid if x ∧ y = x ⊙ (x → y) for any x, y ∈ L,

(2) a Heyting algebra if x ⊙ y = x ∧ y = x ⊙ (x → y) for any x, y ∈ L.

Definition 2.5. [13] A nonempty set F of a residuated lattice L is called a filter of L if it satisfies the following conditions:

(1) x, y ∈ F implies x ⊙ y ∈ F,

(2) x ∈ F, y ∈ L and x ≤ y implies y ∈ F.

For any nonempty subset X of L, the least filter containing X is called the filter generated by X, which is denoted by 〈X〉. From [13], we get 〈X〉 = {x ∈ L|x ≥ x₁ ⊙ x₂ ⊙ ·· · ⊙ x_n, x_i ∈ X, i = 1, 2 ·· · , n}. For any a ∈ L, we call 〈 {a} 〉 the principal filter of L, denoted by 〈a〉. Then 〈a〉 = {x ∈ L|x ≥ aⁿ}.

Definition 2.6. [13 , 21] Let F be a filter of a residuated lattice L. Then we have the following

(a) F is said to be proper if F ≠ L.

(b) A proper filter is said to be prime if for all x, y ∈ L, x ∨ y ∈ F implies x ∈ F or y ∈ F.

(d) A prime filter P is called minimal if it is the minimal element in the class of all prime filters.

(e) A filter F is called a Boolean filter, if for all x ∈ L, x ∨ x^∗ ∈ F.

(f) A filter F is called a positive implicative filter if (i) 1 ∈ F; (ii) for all x, y, z ∈ L, x → (y → z) ∈ F and x → y ∈ F imply x → z ∈ F.

(g) A filter F is called implicative if 1 ∈ F and x → ((y → z) → y) ∈ F and x ∈ F imply that y ∈ F.

(h) A proper filter F is called obstinate if for all x, y ∉ F, x → y ∈ F and y → x ∈ F.

Theorem 2.7. [18, 21] Let F be a subset of a residuated lattice L. Then F is an implicative filter of L if and only if F is a Boolean filter of L if and only if F is a positive implicative filter of L.

Theorem 2.8. [15] Let F be a filter of a residuated lattice L. Then the following conditions are equivalent:

(i) F is a maximal and Boolean filter(positive implicative filter,implicative filter),

(ii) F is a prime and Boolean filter (positive implicative filter,implicative filter),

(iii) F is an obstinate filter.

3 (Weak) involutory residuated lattices and distance functions

In this section, weak involutory residuated lattices are introduced. Some properties of weak involutory residuated lattices are given. Especially we introduce a metric function on weak involutory residuated lattices.

In any residuated lattice L, we define binary operations ⊕ and ⊖ by x ⊕ y = ¬ (¬ x ⊙ ¬ y) and x ⊖ y = x ⊙ ¬ y for any x, y ∈ L, respectively.

Lemma 3.1. Let L be a residuated lattice. Then we have the following: for all x, y, z ∈ L,

(1) x ⊕ y ≥ x;

(2) x ⊖ y ≤ x;

(3) x ⊕ 0 = ¬¬ x. If L is involutory, then x ⊕ 0 = x;

(4) x ⊖ 0 = x;

(5) x ⊖ z ≤ (x ⊖ y) ⊕ (y ⊖ z);

(6) (x ⊕ y) ⊖ y ≤ ¬¬ x. If L is involutory, then (x ⊕ y) ⊖ y ≤ x;

(7) x ∗ y ≤ x ⊕ y, for ∗ ∈ {⊙ , ∨ , ∧}.

Proof. (1) Note that x ⊕ y = ¬ (¬ x ⊙ ¬ y) ≥ ¬¬ x ≥ x.

(2)-(4) Straightforward.

(5) Since y ⊙ ¬ z ⊙ ¬ (y ⊙ ¬ z) =0, then we have ¬z ⊙ ¬ (y ⊙ ¬ z) ≤ ¬ y. Therefore (x ⊖ z) ⊙ (¬ (x ⊖ y) ⊙ ¬ (y ⊖ z)) = (x ⊙ ¬ z) ⊙ (¬ (x ⊙ ¬ y) ⊙ ¬ (y ⊙ ¬ z)) = x ⊙ ¬ (x ⊙ ¬ y) ⊙ ¬ z ⊙ ¬ (y ⊙ ¬ z) ≤ x ⊙ ¬ (x ⊙ ¬ y) ⊙ ¬ y = 0. It follows that x ⊖ z ≤ (¬ (x ⊖ y) ⊙ ¬ (y ⊖ z)) →0 = ¬ (¬ (x ⊖ y) ⊙ ¬ (y ⊖ z)) = (x ⊖ y) ⊕ (y ⊖ z).

(6) Note that ((x ⊕ y) ⊖ y) ⊙ ¬ x = ¬ (¬ x ⊙ ¬ y) ⊙ ¬ y ⊙ ¬ x = 0, we have (x ⊕ y) ⊖ y ≤ ¬¬ x.

(7) Straightforward. □

The following lemma shows the interplay between algebra and order.

Lemma 3.2. Let L be a residuated lattice. If x ≤ y, then for every z ∈ L:

(1) x ⊕ z ≤ y ⊕ z;

(2) x ⊖ z ≤ y ⊖ z;

(3) z ⊖ y ≤ z ⊖ x.

Proof. (1) Let x ≤ y. Then ¬y ≤ ¬ x and hence ¬y ⊙ ¬ z ≤ ¬ x ⊙ ¬ z. Thus x ⊕ z = ¬ (¬ x ⊙ ¬ z) ≤ ¬ (¬ y ⊙ ¬ z) = y ⊕ z.

(2) Let x ≤ y. Then x ⊖ z = x ⊙ ¬ z ≤ y ⊙ ¬ z = y ⊖ z. Similarly we can prove z ⊖ y ≤ z ⊖ x.

(3) Similarly to (2). □

Definition 3.3. Let L be a residuated lattice. If L satisfies ¬x ⊙ ¬ y ∈ ¬ L for all x, y ∈ L, where ¬L = {¬ x ∣ x ∈ L}, then it is said to be a weak involutory residuated lattice (for short, a W-I-residuated lattices).

Proposition 3.4. Every involutory residuated lattice is clearly a W-I-residuated lattice, but the converse is not always true by the following example.

Example 3.5. Let L = {0, a, b, 1} be a chain and operations ⊙ and → be defined

⊙	a	b	1
0	0	0	0
a	a	a	a
b	a	a	b
1	a	b	1

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

We can easily find that L is a W-I-residuated lattice, but it is not an involutory residuated lattice since ¬¬ a = 1 ≠ a.

Now we give a example to show that a residuated lattice may not be a W-I-residuated lattice.

Example 3.6. Let L = {0, a, b, c, d, 1} be a chain. The operations ⊙ and → are defined as

⊙	a	b	c	d	1
0	0	0	0	0	0
a	0	0	0	a	a
b	0	0	0	a	b
c	0	0	a	c	c
d	a	a	c	d	d
1	a	b	c	d	1

→	0	a	b	c	d	1
0	1	1	1	1	1	1
a	c	1	1	1	1	1
b	c	d	1	1	1	1
c	b	c	c	1	1	1
d	0	b	b	c	1	1
1	0	a	b	c	d	1

respectively. Then, L is a residuated lattice, which is not a W-I-residuated lattice, since ¬a ⊙ ¬ b = c ⊙ c = a ∉ ¬ L.

Proposition 3.7. Let L be a W-I-residuated lattice. Then we have the following: for all x, y, z ∈ L

(1) ¬¬ (¬ x ⊙ ¬ y) = ¬ x ⊙ ¬ y;

(2) ¬x ⊙ ¬ y = ¬ (x ⊕ y);

(3) x ≤ (x ⊖ y) ⊕ y;

(4) (x ⊕ y) ⊕ z = x ⊕ (y ⊕ z);

(5) x ⊖ (y ⊕ z) = (x ⊖ y) ⊖ z.

Proof. (1) Since ¬x ⊙ ¬ y = ¬ z for some z ∈ L, we have ¬¬ (¬ x ⊙ ¬ y) = ¬¬ ¬ z = ¬ z = ¬ x ⊙ ¬ y.

(2) By (1), we have ¬ (x ⊕ y) = ¬¬ (¬ x ⊙ ¬ y) = ¬ x ⊙ ¬ y.

(3) Since x ⊙ ¬ y ⊙ ¬ (x ⊙ ¬ y) =0, we have x ≤ (¬ y ⊙ ¬ (x ⊙ ¬ y)) →0, and hence x ≤ ¬ (¬ y ⊙ ¬ (x ⊙ ¬ y)) = ¬ (¬ y ⊙ ¬ (x ⊖ y)) = (x ⊖ y) ⊕ y.

(4) By (1), (x ⊕ y) ⊕ z = ¬ (¬¬ (¬ x ⊙ ¬ y) ⊙ ¬ z) = ¬ ((¬ x ⊙ ¬ y) ⊙ ¬ z) = ¬ (¬ x ⊙ (¬ y ⊙ ¬ z)) = x ⊕ (y ⊕ z).

(5) By (2), we have (x ⊖ y) ⊖ z = (x ⊙ ¬ y) ⊙ ¬ z = x ⊙ (¬ y ⊙ ¬ z) = x ⊙ ¬ (y ⊕ z) = x ⊖ (y ⊕ z). □

Proposition 3.8. In any W-I-residuated lattice L, the following conditions are equivalent:

(1) ¬y ≤ ¬ x;

(2) ¬¬ x ≤ ¬¬ y;

(3) x ⊖ y = 0;

(4) ¬ (¬ x ⊕ y) =0.

Proof. (1) ⇒ (2) It is from (8) in Lemma 2.2.

(2) ⇒ (1) Let ¬¬ x ≤ ¬¬ y. By (8) and (9) in Lemma 2.2, we have ¬y ≤ ¬ x.

(2) ⇒ (3) Let ¬¬ x ≤ ¬¬ y. Then ¬y = ¬¬ ¬ y ≤ ¬¬ ¬ x = ¬ x. Moreover x ⊖ y = x ⊙ ¬ y ≤ x ⊙ ¬ x = 0 by Lemma 2.2(3), and hence x ⊖ y = 0. We get (3).

(3) ⇒ (2) Let x ⊖ y = 0. Then x ⊙ ¬ y = 0 and hence x ≤ ¬ y → 0 = ¬¬ y. This shows that ¬y = ¬¬ ¬ y ≤ ¬ x, and hence ¬¬ x ≤ ¬¬ y.

(2)⇒(4) Let ¬¬ x ≤ ¬¬ y. Then ¬y ≤ ¬ x and hence ¬ (¬ x ⊕ y) = ¬¬ x ⊙ ¬ y ≤ ¬¬ x ⊙ ¬ x = 0 by Lemma 2.2(3) and Proposition 3.7(1). So we get ¬ (¬ x ⊕ y) =0.

(4)⇒(2) Let ¬ (¬ x ⊕ y) =0. By Proposition 3.7(1), we have ¬¬ x ⊙ ¬ y = 0 and thus ¬¬ x ≤ ¬¬ y. □

Corollary 3.9. Let L be a W-I-residuated lattice. If x ≤ y, then x ⊖ y = 0 and ¬ (¬ x ⊕ y) =0.

In the following we extend the Chang’s distance function to residuated lattices. Let L be a residuated lattice. Define a binary operation d : L × L → L on L by d (x, y) = (x ⊖ y) ⊕ (y ⊖ x) for all x, y ∈ L.

The following example to show that in general, d may not be a distance function on residuated lattice.

Example 3.10. Consider the residuated lattice L given in Example 3.6. We can see that d (a, b) = (a ⊖ b) ⊕ (b ⊖ a) = (a ⊙ ¬ b) ⊕ (b ⊖ a) = (a ⊙ c) ⊕ (b ⊙ c) =0 ⊕ 0 =0, which shows that d is not a distance function.

Proposition 3.11. Let L be a residuated lattice. Then we have the following:

(1) d (x, y) ≤ d (¬ x, ¬ y). If L is involutory, then d (x, y) = d (¬ x, ¬ y);

(2) d (x, y) ≤ d (¬¬ x, ¬¬ y). If L is involutory, then d (x, y) = d (¬¬ x, ¬¬ y);

(3) d (x, x) =0;

(4) d (x, 0) = ¬¬ x. If L is involutory, then d (x, 0) = x;

(5) d (x ⊕ t, t ⊕ y) ≤ d (¬ x, ¬ y). If L is involutory, then d (x ⊕ t, t ⊕ y) ≤ d (x, y).

Proof. (1) Note that d (¬ x, ¬ y) = (¬ x ⊖ ¬ y) ⊕ (¬ y ⊖ ¬ x) = ¬ (¬ (¬ x ⊙ ¬¬ y) ⊙ ¬ (¬ y ⊙ ¬¬ x)) ≥ ¬ (¬ (¬ x ⊙ y) ⊙ ¬ (¬ y ⊙ x)) = ¬ (¬ (y ⊖ x) ⊙ ¬ (x ⊖ y)) = (y ⊖ x) ⊕ (x ⊖ y) = d (x, y). Moreover let L be involutory, then d (x, y) ≤ d (¬ x, ¬ y) ≤ d (¬¬ x, ¬¬ y) = d (x, y), and hence d (x, y) = d (¬ x, ¬ y).

(2) It follows from (1).

(3) Since x ⊖ x = x ⊙ ¬ x = 0, we have d (x, x) = (x ⊖ x) ⊕ (x ⊖ x) =0 ⊕ 0 = ¬ (¬0 ⊙ ¬0) = ¬ (1 ⊙ 1) =0.

(4) By Lemma 3.1(3), we have d (x, 0) = (x ⊖ 0) ⊕ (0 ⊖ x) = (x ⊙ 1) ⊕0 = ¬¬ x.

(5) It follows from Lemma 3.1(6) and Proposition 3.7(5) that d (x ⊕ t, t ⊕ y) = ((x ⊕ t) ⊖ (t ⊕ y)) ⊕ ((t ⊕ y) ⊖ (x ⊕ t))

= (((x ⊕ t) ⊖ t) ⊖ y) ⊕ (((t ⊕ y) ⊖ t) ⊖ x)

≤ (¬¬ x ⊖ y) ⊕ (¬¬ y ⊖ x)

= (¬¬ x ⊙ ¬ y) ⊕ (¬¬ y ⊙ ¬ x)

= (¬ y ⊖ ¬ x) ⊕ (¬ x ⊖ ¬ y)

= d (¬ x, ¬ y).

If L is involutory, we have d (x ⊕ t, t ⊕ y) ≤ d (x, y) by Proposition 3.11(1). □

Proposition 3.12. Let L be a W-I-residuated lattice. Then d (x, y) satisfies the following: for all x, y, z ∈ L,

(1) d (x, y) ≥0, and d (x, y) =0 if and only if ¬¬ x = ¬¬ y if and only if ¬x = ¬ y;

(2) d (x, y) = d (y, x);

(3) d (x, z) ≤ d (x, y) ⊕ d (y, z).

(4) d (x ⊕ s, y ⊕ t) ≤ d (¬ x, ¬ y) ⊕ d (¬ s, ¬ t). If L is involutory, then d (x ⊕ s, y ⊕ t) ≤ d (x, y) ⊕ d (s, t).

Proof. (1) It is clear that d (x, y) ≥0. Let ¬¬ x = ¬¬ y. Then by Lemma 2.2(3), we have d (¬¬ x, ¬¬ x) = (¬¬ x ⊙ ¬¬ ¬ x) ⊕ (¬¬ x ⊙ ¬¬ ¬ x) =0. It follows from Proposition 3.11(1) that d (x, y) ≤ d (¬¬ x, ¬¬ y) = d (¬¬ x, ¬¬ x) =0, and hence d (x, y) =0. Conversely, let d (x, y) =0. Then (x ⊖ y) =0 and (y ⊖ x) =0. Hence ¬¬ x ≤ ¬¬ y and ¬¬ y ≤ ¬¬ x by Proposition 3.8. It follows that ¬¬ x = ¬¬ y. By Proposition 3.8, we have that ¬¬ x = ¬¬ y if and only if ¬x = ¬ y.

(2) It is clear.

(3) By use of Lemma 3.1(5) and Proposition 3.7(4), we have d (x, y) ⊕ d (y, z) = (x ⊖ y) ⊕ (y ⊖ x) ⊕ (y ⊖ z) ⊕ (z ⊖ y) = (x ⊖ y) ⊕ (y ⊖ z) ⊕ (y ⊖ x) ⊕ (z ⊖ y) ≥ (x ⊖ z) ⊕ (z ⊖ x) = d (x, z).

(4) By Propositions 3.11(5) and 3.12(3), we have

d (x ⊕ s, y ⊕ t)

≤d (x ⊕ s, s ⊕ y) ⊕ d (s ⊕ y, y ⊕ t)

≤d (¬ x, ¬ y) ⊕ d (¬ s, ¬ t).

If L is involutory, it follows from Proposition 3.11(1) that d (x ⊕ s, y ⊕ t) ≤ d (x, y) ⊕ d (s, t). □

Corollary 3.13. Let L be an involutory residuated lattice. Then d (x, y) satisfies the following: for all x, y, z ∈ L,

(1) d (x, y) ≥0, and d (x, y) =0 if and only if x = y;

(2) d (x, y) = d (y, x);

(3) d (x, z) ≤ d (x, y) ⊕ d (y, z).

Therefore d is a distance function on L.

A non-empty subset F of a poset (P, ≤) is called an order filter if and only if x ∈ F and x ≤ y imply y ∈ F, also we call F is upwards closed with respect to ≤.

Lemma 3.14. Let F be an order filter of a residuated lattice L. Then for any x, y ∈ L, the following properties hold:

(1) If x ⊖ y ∈ F, then x, ¬ y ∈ F;

(2) If x ⊙ y ∈ F, then x, y ∈ F.

Proof. (1) Let x ⊖ y ∈ F. Since x ⊖ y ≤ x, ¬ y, then x, ¬ y ∈ F.

(2) Let x ⊙ y ∈ F. Since x ⊙ y ≤ x, y, we have x, y ∈ F. □

4 Lattice filter topologies on residuated lattices

In this section, by use of lattice filter F and distance function, we define the F-balls which induce a topology τ_LF on residuated lattice L. Moreover we study properties of the F-balls and the topology τ_LF. Finally we obtain a sufficient and necessary condition for (L, τ_LF) being a topological residuated lattice.

Definition 4.1. An order filter F of a residuated lattice L is called a lattice filter of L, if F is closed under ∧ operation. A lattice filter F of L is proper if F ≠ L.

We denote by F (L) and L (L) respectively the set of all proper filters of L and the set of all proper lattice filters of L. Let us note that, from the inequality x ⊙ y ≤ x ∧ y for all x, y ∈ L, we obtain F (L) ⊆ L (L), and the inclusion is proper as the following example shows.

Example 4.2. Let L = {0, a, b, c, d, 1} with 0 < a, b < c < d < 1 and a, b incomparable, ⊙ and → be defined as follows.

⊙	a	b	c	d	1
0	0	0	0	0	0
a	a	0	a	a	a
b	0	b	b	b	b
c	a	b	c	c	c
d	a	b	c	c	d
1	a	b	c	d	1

→	0	a	b	c	d	1
0	1	1	1	1	1	1
a	b	1	b	1	1	1
b	a	a	1	1	1	1
c	0	a	b	1	1	1
d	0	a	b	d	1	1
1	0	a	b	c	d	1

Then we can see that L is a residuated lattice. We can easily get that F = {d, 1} is a lattice filter but it is not a filter since d ⊙ d = c ∉ F.

Definition 4.3. Let L be a residuated lattice and let F be a lattice filter of L. The open F-ball of radius r ∈ F, with the center a ∈ L, is denoted by ∪_a.r = {x ∈ L ∣ r ⊖ d (x, a) ∈ F}.

Proposition 4.4. Let L be a W-I-residuated lattice and let F be a lattice filter of L. Then open F-balls constitute a base of a topology on L, we call this topology a lattice filter topology, denoted by τ_LF.

Proof. By Lemma 3.14(4), we have r ⊖ 0 = r for all r ∈ F, and hence by Proposition 3.11(3), for each a ∈ L, r ⊖ d (a, a) = r ⊖ 0 = r ∈ F. It follows that a ∈ ∪ _a,r. Thus, the open F-balls form an open cover of the space L. Let z₀ ∈ ∪ _a,r ∩ ∪ _b,R be arbitrary. By the definition of F-balls, we can assume that r ⊖ d (z₀, a) = c₁, R ⊖ d (z₀, b) = c₂ for some c₁, c₂ ∈ F. Taking ɛ = c₁ ∧ c₂, we have ɛ ∈ F. Let x ∈ ∪ _z₀,ɛ. Then we have ɛ ⊖ d (x, z₀) = c₃ ∈ F. By Proposition 3.12(3), Proposition 3.7(5) and Lemma 3.2(3), we get, r ⊖ d (x, a) ≥ r ⊖ (d (z₀, a) ⊕ d (z₀, x)) = (r ⊖ d (z₀, a)) ⊖ d (z₀, x) = c₁ ⊖ d (z₀, x) ≥ ɛ ⊖ d (z₀, x) = c₃ ∈ F.

It follows that r ⊖ d (x, a) ∈ F and thus x ∈ ∪ _a,r. Similar arguments imply that x ∈ ∪ _b,R. This shows that ∪_z₀,ɛ ⊆ ∪ _a,r ∩ ∪ _b,R. Hence the open F-balls form a base of a topology on L. □

Example 4.5. Consider the W-I-residuated lattice L given in Example 3.5. We can see that F = {b, 1} is a lattice filter of L and ∪_0,b = ∪ _0,1 = {0}, ∪_a,b = ∪ _b,b = ∪ _1,b = ∪ _a,1 = ∪ _b,1 = ∪ _1,1 = {a, b, 1}. Therefore τ_LF = {∅ , {0} , {a, b, 1} , L}.

Recall that a topological residuated lattice is a residuated lattice and a topological space such that the operations ∧, ∨, ⊙ and → are continuous.

Proposition 4.6. Let L be an involutory residuated lattice with a ⊙ (b ∧ c) = (a ⊙ b) ∧ (a ⊙ c) for a, b, c ∈ L, F be a lattice filter of L and τ_LF be a lattice filter topology given in Proposition 4.4. Then we have the following:

(1) The operation ⊕ is continuous on lattice filter topology τ_LF, iff for each r ∈ F there exists s ∈ F such that for any x ∈ L, s ⊖ x ∈ F implies r ⊖ (x ⊕ x) ∈ F.

(2) The operation ¬ is continuous on lattice filter topology τ_LF.

(3) The operation ∨ is continuous on lattice filter topology τ_LF if for each r ∈ F there exists s ∈ F such that for any x ∈ L, s ⊖ x ∈ F implies r ⊖ (x ⊕ x) ∈ F.

Proof. (1) We prove that the mapping ⊕: L × L → L, (a, b) ↦ a ⊕ b is continuous. For any a, b ∈ L, and r ∈ F, In the following, we will prove ∪_a,s ⊕ ∪ _b,s ⊆ ∪ _a⊕b,r, which means that ⊕ is continuous. For any x ∈ ∪ _a,s and y ∈ ∪ _b,s, we have s ⊖ d (x, a) ∈ F and s ⊖ d (y, b) ∈ F. It follows that (s ⊖ d (x, a)) ∧ (s ⊖ d (y, b)) = s ⊙ (¬ d (x, a) ∧ ¬ d (y, b)) = s ⊙ ¬ (d (x, a) ∨ d (y, b)) = s ⊖ (d (x, a) ∨ d (y, b)) ∈ F. And thus r ⊖ (d (x, a) ∨ d (y, b)) ⊕ (d (x, a) ∨ d (y, b)) ∈ F. It is obvious that d (x, a) ∨ d (y, b)) ⊕ (d (x, a) ∨ d (y, b) ≥ d (x, a) ⊕ d (y, b). And then r ⊖ (d (x, a) ⊕ d (y, b)) ∈ F. From ¬ (d (x, a) ⊕ d (y, b)) ≤ ¬ d (x ⊕ y, a ⊕ b), we have r ⊖ d (x ⊕ y, a ⊕ b) ∈ F. Hence x ⊕ y ∈ ∪ _a⊕b,r. Conversely suppose ⊕ is continuous, so it is also continuous at (0, 0). So for any r ∈ F, there exist r₁, r₂ ∈ F such that ∪_0,r₁ ⊕ ∪ _0,r₂ ⊆ ∪ _0,r. First, denote s = r₁ ∧ r₂, then s ∈ F and ∪_0,r₁ ∩ ∪ _0,r₂ = ∪ _0,s. Now we have ∪_0,s ⊕ ∪ _0,s ⊆ ∪ _0,r. For any x ∈ ∪ _0,s, we have x ⊕ x ∈ ∪ _0,r, which means that s ⊖ d (x, 0) = s ⊖ x ∈ F implies r ⊖ d (x ⊕ x, 0) = r ⊖ (x ⊕ x) ∈ F.

(2) We prove that the operation ¬ is continuous. Let ∪₀ be an open F-ball of radius ɛ around ¬x₀, where x₀ ∈ L and ɛ ∈ F, and let V₀ be an open F-ball of radius ɛ around x₀. For any x ∈ V₀, we have ɛ ⊖ d (x, x₀) ∈ F, and thus, by Proposition 3.11(1), we get ɛ ⊖ d (x, x₀) = ɛ ⊖ d (¬ x, ¬ x₀). Therefore ɛ ⊖ d (¬ x, ¬ x₀) ∈ F, and hence ¬x ∈ ∪ ₀. It follows that ¬ (V₀) ⊆ ∪ ₀. Hence we prove that ¬ is continuous.

(3) We prove that the operation ∨ is continuous. Now we prove d (x ∨ y, a ∨ b) ≤ d (x, a) ⊕ d (y, b) for any x, y, a, b ∈ L. By Lemma 2.2(11) and Lemma 3.1(7), we have

d (x ∨ y, a ∨ b)

= ((x ∨ y) ⊖ (a ∨ b)) ⊕ ((a ∨ b) ⊖ (x ∨ y))

= ((x ∨ y) ⊙ ¬ (a ∨ b)) ⊕ ((a ∨ b) ⊙ ¬ (x ∨ y))

= ((x ⊙ ¬ (a ∨ b)) ∨ (y ⊙ ¬ (a ∨ b))) ⊕ ((a ⊙ ¬ (x ∨ y)) ∨ (b ⊙ ¬ (x ∨ y)))

= ((x ⊖ (a ∨ b)) ∨ (y ⊖ (a ∨ b))) ⊕ ((a ⊖ (x ∨ y)) ∨ (b ⊖ (x ∨ y)))

≤ ((x ⊖ a) ∨ (y ⊖ b)) ⊕ ((a ⊖ x) ∨ (b ⊖ y)))

≤ ((x ⊖ a) ⊕ (y ⊖ b)) ⊕ ((a ⊖ x) ⊕ (b ⊖ y)))

= (x ⊖ a) ⊕ (a ⊖ x) ⊕ (y ⊖ b) ⊕ (b ⊖ y)

= d (x, a) ⊕ d (y, b).

The remaining proof is similar to Proposition 4.6(1). □

Proposition 4.7. Let L be an involutory residuated lattice. Then we have the following: for all x, y ∈ L

(1) x ⊙ y = ¬ (¬ x ⊕ ¬ y),

(2) x → y = ¬ x ⊕ y = ¬ (x ⊙ ¬ y),

(3) x ∧ y = ¬ (¬ x ∨ ¬ y).

Proof. (1) It follows from the definition of ⊕ and involutory property of L.

(2) Note that ¬x ⊕ y = ¬ (¬¬ x ⊙ ¬ y) = ¬ (x ⊙ y). Then (¬ x ⊕ y) ⊙ x ⊙ ¬ y = ¬ (x ⊙ ¬ y) ⊙ x ⊙ ¬ y = 0 and hence ¬ (x ⊙ ¬ y) ⊙ x ≤ ¬ y → 0 = ¬¬ y = y by Definition 2.1(3). Using Definition 2.1(3) again, we get ¬x ⊕ y = ¬ (x ⊙ ¬ y) ≤ x → y. Conversely, note that (x → y) ⊙ (x ⊙ ¬ y) = x ⊙ (x → y) ⊙ ¬ y = (x ∧ y) ⊙ ¬ y ≤ y ⊙ ¬ y = 0, and thus x → y ≤ (x ⊙ ¬ y) →0 = ¬ (x ⊙ ¬ y) = ¬ x ⊕ y. Combining the above arguments we get x → y = ¬ (x ⊙ ¬ y) = ¬ x ⊕ y.

(3) Note that ¬ (¬ x ∨ ¬ y) ≤ ¬¬ x = x and ¬ (¬ x ∨ ¬ y) ≤ ¬¬ y = y. Hence ¬ (¬ x ∨ ¬ y) is a lower bound of x and y. On the other hand, let z ≤ x and z ≤ y. Then ¬x ≤ ¬ z and ¬y ≤ ¬ z, and thus ¬ (¬ x ∨ ¬ y) ≥ ¬ (¬ z ∨ ¬ z) = ¬¬ z = z. It follows that ¬ (¬ x ∨ ¬ y) is the largest lower bound, that is ¬ (¬ x ∨ ¬ y) = x ∧ y. □

Theorem 4.8. Let L be an involutory residuated lattice with a ⊙ (b ∧ c) = (a ⊙ b) ∧ (a ⊙ c) for a, b, c ∈ L. (L, τ_LF) is a topological residuated lattice iff for each r ∈ F there exists s ∈ F such that for any x ∈ L, s ⊖ x ∈ F implies r ⊖ (x ⊕ x) ∈ F.

Proof. By Proposition 4.6, operations ⊕, ¬ and ∨ are continuous on lattice filter topology τ_LF. By Proposition 4.7, each operation ∗ ∈ {⊙ , → , ∧} is a function of ⊕, ¬ and ∨, and hence it is also continuous. □

The above theorem generalize the result on MV-algebras. The following example shows that it is a proper generalization, that is, an involutory residuated lattice with a ⊙ (b ∧ c) = (a ⊙ b) ∧ (a ⊙ c) may not be an MV-algebra.

Example 4.9. Let L = {0, a, b, 1} be a set with Cayley tables as follows.

⊙	a	b	1
0	0	0	0
a	0	0	a
b	0	b	b
1	a	b	1

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

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is an involutory residuated lattice with x ⊙ (y ∧ z) = (x ⊙ y) ∧ (x ⊙ z) for any x, y, z ∈ L. But it is not an MV-algebra since b ⊙ (b → a) ≠ b ∧ a.

Proposition 4.10. Let L be an involutory residuated lattice and let F be a proper lattice filter of L. Then we have the following:

(1) F = ∪ _1,1,

(2) ∪_¬a,m = ¬ ∪ _a,m,

(3) If m ≤ r, then ∪_a,m ⊆ ∪ _a,r,

(4) If y ∈ ∪ _a,m, then y → a, a → y ∈ F.

Proof. (1) Since d (a, 1) =1 ⊖ a = ¬ a for any a ∈ L, then we have ∪_1,1 = {x : 1 ⊖ d (x, 1) ∈ F} = {x : 1 ⊖ ¬ x ∈ F} = {x : ¬¬ x ∈ F} = {x : x ∈ F} = F.

(2) Note that the following:

z ∈ ∪ _¬a,m

⇔m ⊖ d (z, ¬ a) ∈ F

⇔m ⊖ d (¬ z, a) ∈ F

⇔ ¬ z ∈ ∪ _a,m

⇔z ∈ ¬ ∪ _a,m.

Therefore ∪_¬a,m = ¬ ∪ _a,m.

(3) Let m ≤ r and y ∈ ∪ _a,m. Then m ⊖ d (y, a) ≤ r ⊖ d (y, a) by Lemma 3.2(2), and hence y ∈ ∪ _a,r.

(4) Let y ∈ ∪ _a,m. Then m ⊖ d (y, a) ∈ F. By Lemma 3.14(1) and Proposition 4.7(2), ¬d (y, a) = ¬ ((y ⊖ a) ⊕ (a ⊖ y)) = ¬ ((y ⊙ ¬ a) ⊕ (a ⊙ ¬ y)) = (¬ y ⊕ a) ⊙ (¬ a ⊕ y) = (y → a) ⊙ (a → y) ∈ F. By Lemma 3.14(2), we get y → a, a → y ∈ F. □

Corollary 4.11. Let L be an involutory residuated lattice and let F be a proper lattice filter of L. Then |τ_FL|≥4.

Proof. By (1)and (2) of Proposition 4.10, we have that F and ¬F are open sets. Since F is proper, then F ≠ L, and thus F ≠ ¬ F. Otherwise let F = ¬ F, then taking x ∈ F, we get ¬x ∈ F. It follows that 0 = x ∧ ¬ x ∈ F, and thus F = L, a contradiction. Therefore τ_LF is not be trivial and |τ_FL|≥4. □

Example 4.12. (1) Consider I = [0, 1] with Lukasiewicz operations a → b : = min {1, 1 - a + b} and a ⊙ b : = max {0, a + b - 1} for a, b ∈ I. Define a ∧ b = min {a, b} and a ∨ b = max {a, b}. Then (I, ⊙ , → , ∨ , ∧ , 1) is a residuated lattice. Note that ¬a = a → 0 = min {1, 1 - a + 0} =1 - a and hence ¬¬ a = (a → 0) →0 = 1 - (1 - a) = a, then we have that I is an involutory residuated lattice. Firstly we note that d (x₀, x) = (x₀ ⊖ x) ⊕ (x ⊖ x₀) = |x - x₀| for x, x₀ ∈ L, that is, Chang’s distance function on I is the usual Euclidean distance. Note that L (I) is of two forms: ↑a = [a, 1] with a ≠ 0 and (a, 1] with a ≠ 1. If F ∈ L (I) and F = ↑ a with a ≠ 0, we claim that τ_↑a is discrete. In fact, note that

y ∈ ∪ _x₀,a

⇔a ⊖ d (y, x₀) ∈ F

⇔a ⊖ |y - x₀| ≥ a

⇔|y - x₀|≤0

⇔|y - x₀|=0

⇔y = x₀.

Therefore ∪_x₀,a = {x₀}. For any x₀ ∈ I and r ∈ ↑ a, we have ∪_x₀,a ⊆ ∪ _x₀,r by Proposition 4.10(3). Therefore {∪ _x₀,a = {x₀} : x₀ ∈ I} forms a base of topology τ_LF, where F = ↑ a, and hence τ_LF is discrete. Now we consider the lattice filter F with the form F = (a, 1] for any a ∈ I with a ≠ 1. For x₀ ∈ L and m ∈ F, if d (x, x₀) ≥ m, then m ⊖ d (x, x₀) =0, and thus x ∉ ∪ _x₀,m. Therefore we have the following:

x ∈ ∪ _x₀,m

⇔d (x₀, x) < m, m ⊖ d (x, x₀) ∈ F

⇔m - d (x, x₀) > a

⇔m - |x - x₀| > a

⇔x₀ - (m - a) < x < x₀ + (m - a).

It follows that ∪_x₀,m = (x₀ - (m - a) , x₀ + (m - a)), and hence τ_LF coincides with the usual topology on I.

(2) Consider $L_{n + 1} : = {0, \frac{1}{n}, \dots, \frac{n - 1}{n}, 1}$ for n ∈ N and n ≥ 2. Define a → b : = min {1, 1 - a + b}, a ⊙ b : = max {0, a + b - 1}, a ∧ b = min {a, b} and a ∨ b = max {a, b} for all a . b ∈∈ L_n+1. Then (L_n+1, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice. Taking F = ↑ a for a ≠ 0. By similar arguments of Example 4.12(1), τ_LF is discrete.

In [9], authors propose an open question: How do homomorphisms of MV-algebras behave with respect to the lattice filter topologies? Now we answer the question on the general case and study the interaction between involutory residuated lattices and lattice filter topologies in the following theorem.

Theorem 4.13. Let h : A → B be a homomorphism between involutory residuated lattices A and B.

(1) If h is onto and G ∈ L (B), then h : (A, τ_h^-1(G)) → (B, τ_G) is continuous and open,

(2) If h is a residuted lattice isomorphism from A to B, and F ∈ L (A), then (A, τ_F) ≅ (B, τ_h(F)).

Proof. (1) Let G ∈ L (B). Since h is order preserving, then h^-1 (G) ∈ L (A). In order to prove the continuityof h, we consider ∪_a,m ∈ τ_G. Then there exist (m_∗, a_∗) ∈ h^-1 (G) × A such that h (m_∗) = m and h (a_∗) = a. Therefore we have the following:

z ∈ h^-1 (∪ _a,m)

⇔m ⊖ d (a, h (z)) ∈ G

⇔h (m_∗) ⊖ d (h (a_∗) , h (z)) ∈ G

⇔h (m_∗ ⊖ d (a_∗, z)) ∈ G

⇔m_∗ ⊖ d (a_∗, z) ∈ h_-1 (G)

⇔z ∈ ∪ _{a_∗,m_∗}.

This shows that h^-1 (∪ _a,m) = ∪ _{a_∗,m_∗} and hence h is continuous. Next we prove that h is open. Let z ∈ h (∪ _a,m) for (m, a) ∈ h^-1 (G) × A. Then z = h (t) for some t ∈ ∪ _a,m. Hence m ⊖ d (a, t) ∈ h^-1 (G), which implies h (t) ∈ ∪ _h(a),h(m). This shows that h (∪ _a,m) ⊆ ∪ _h(a),h(m). On the other hand, if z ∈ ∪ _h(a),h(m), then h (m) ⊖ d (h (a) , z) ∈ G. Moreover h (m) ⊖ d (h (a) , h (z_∗)) ∈ G for some z_∗ ∈ A, and thus h (m ⊖ d (a, z_∗)) ∈ G. It follows that m ⊖ d (a, z_∗) ∈ h^-1 (G), and hence z_∗ ∈ ∪ _a,m. This means z = h (z_∗) ∈ h (∪ _a,m), and thus ∪_h(a),h(m) ⊆ h (∪ _a,m). Combining the above arguments we can get ∪_h(a),h(m) = h (∪ _a,m), and hence h is open.

(2) It follows from Theorem 4.13(1). □

5 Topologies induced by filters on residuated lattices

Let L be a residuated lattice and F ∈ F (L). Now we discuss the relation between F-ball ∪_a,r and equivalent class [a], where a ∈ L, r ∈ F, and [a] = {x ∈ L : x → a, a → x ∈ F}.

Proposition 5.1. Let L be an involutory residuated lattice and F ∈ F (L). Then ∪_a,r = [a], that is, ∪_a,r is independent with r.

Proof. Taking (a, r) ∈ L × F, then by Proposition 4.7(2), we have that x ∈ ∪ _a,r iff r ⊖ d (a, x) ∈ F iff r ⊙ ¬ d (a, x) ∈ F iff r, ¬ d (a, x) ∈ F iff ¬ (a ⊖ x) ⊙ ¬ (x ⊖ a) ∈ F iff ¬ (a ⊖ x) ⊙ (x ⊖ a) ∈ F iff ¬ (a ⊙ ¬ x) ⊙ (x ⊙ ¬ a) ∈ F iff (¬ a ⊕ x) ⊙ (¬ x ⊕ a) ∈ F iff (a → x) ⊙ (x → a) ∈ F iff x → a, a → x ∈ F iff x ∈ [a]. Hence ∪_a,r = [a]. □

By Propositions 4.10 and 5.1, we can get the following corollaries.

Corollary 5.2. Let L be an involutory residuated lattice and F ∈ F (L). Then ∪_1,r = F and ∪_0,r = ¬ F for any r ∈ F.

Corollary 5.3. Let L be an involutory residuated lattice and F be a proper filter of L. Then {∅} ∪ {[x] : [x] ∈ L/F} is a base of τ_F.

Remark 5.4. In the case given in Corollary 5.3, τ_F is called the partition topology generated by L/F (see [16]).

Theorem 5.5. Let L be an involutory distributive residuated lattice. The the following are equivalent:

(1) L is a Boolean algebra,

(2) L satisfies the distributive law: x ⊙ (y ⊕ z) = (x ⊙ y) ⊕ (x ⊙ z),

(3) F (L) = L (L).

Proof. (1)⇒(2) Let L be a Boolean algebra. By (8) and (10) in Lemma 2.2 and Proposition 3.7(2), we have the following:

(x ⊙ (y ⊕ z)) ∨ ¬ ((x ⊙ y) ⊕ (x ⊙ z))

= (x ⊙ (y ⊕ z)) ∨ (¬ (x ⊙ y) ⊙ ¬ (x ⊙ z))

≥ (x ∨ ¬ (x ⊙ y)) ⊙ (x ∨ ¬ (x ⊙ z)) ⊙ ((y ⊕ z) ∨ ¬ (x ⊙ y)) ⊙ ((y ⊕ z) ∨ ¬ (x ⊙ z))

≥ (x ∨ ¬ x) ⊙ (x ∨ ¬ x) ⊙ (y ∨ ¬ y) ⊙ (z ∨ ¬ z) =1.

Therefore x ⊙ (y ⊕ z) = (x ⊙ y) ⊕ (x ⊙ z).

(2)⇒(1) Let (2) hold. Then for all x, y ∈ L, ¬ (¬ x ⊕ y) ⊕ y = ¬ ((¬ x ⊕ y) ⊙ ¬ y) = ¬ ((¬ x ⊙ ¬ y) ⊕ (y ⊙ ¬ y)) = ¬ ((¬ x ⊙ ¬ y) ⊕0) = ¬ (¬ x ⊙ ¬ y). Similarly we can prove ¬ (¬ y ⊕ x) ⊕ x = ¬ (¬ x ⊙ ¬ y) and thus ¬ (¬ x ⊕ y) ⊕ y = ¬ (¬ y ⊕ x) ⊕ x. This shows that L is an MV-algebra. By Proposition 4.7(3), we have x ∨ y = ¬ (¬ x ∧ ¬ y) and hence x ∨ ¬ x = ¬ (¬ x ∧ ¬¬ x) = ¬ (¬ x ∧ x) = ¬0 = 1, and thus L becomes a Boolean algebra.

(1)⇒(3) Note that y ≤ x → y for all x, y ∈ L. Let F ∈ L (L) and x, y ∈ F. Then x ∧ y ∈ F. Since L is a Boolean algebra, then x ∧ y = x ⊙ (x → y) ≤ x ⊙ y. Since x ∧ y ∈ F, then x ⊙ y ∈ F. Therefore F ∈ F (L). It follows that F (L) = L (L).

(3)⇒(2) Let F (L) = L (L). For a, b ∈ L, consider F = ↑ (a ∧ b) and then F is a Lattice filter of L. Thus F ∈ F (L). Clearly a, b ∈ F and hence a ⊙ b ∈ F since F ∈ F (L). It follows that a ⊙ b ≥ a ∧ b, and thus a ⊙ b = a ∧ b. In order to prove that a ∨ b = a ⊕ b, we consider F = ↑ (a ⊕ b). Note that

a ⊕ b ∈ F

⇒ ¬ (¬ a ⊙ ¬ b) ∈ F

⇒ ¬ (¬ a ∧ ¬ b) ∈ F

⇒a ∨ b ∈ F,

and so a ⊕ b ≤ a ∨ b. It follows that a ⊕ b = a ∨ b. Since L is distributive, we have x ⊙ (y ⊕ z) = x ∧ (x ∨ z) = (x ∧ y) ∨ (x ∧ z) = (x ⊙ y) ⊕ (x ⊙ z). That is, (2) holds. □

By the above theorem we have the following corollary.

Corollary 5.6. All lattice filter topologies are characterized by Corollary 5.3 on Boolean algebras.

Now we give a notion of involutory filters.

Definition 5.7. A filter F of a residuated lattice L is called involutory if it satisfies the following condition: ¬¬ x → x ∈ F for all x ∈ L.

Example 5.8. Consider the residuated lattice given in Example 4.2. Then we can check that F = {c, d, 1} is an involutory filter. Note that F₁ = {1} is a filter, but it is not involutory.

Proposition 5.9. Let L be a residuated lattice and F a filter of L. Then the following are equivalent:

(1) L/F is an involutory residuated lattice,

(2) F is an involutory filter.

Proof. (1) ⇒ (2) Let L/F be an involutory residuated lattice. Then [¬¬ x] = ¬¬ [x] = [x] for all x ∈ L. This means that ¬¬ x → x ∈ F for all x ∈ L, and hence F is an involutory filter.

(2) ⇒ (1) Let F be an involutory filter. Then x → ¬¬ x = 1 ∈ F and ¬¬ x → x ∈ F for all x ∈ L. Hence [x] = [¬¬ x] = ¬¬ [x] for all x ∈ L, which shows that L/F is an involutory residuated lattice. □

Theorem 5.10. Let (L, τ) be a topological residuated lattice and let F be an involutory filter of L. Then L/F with the quotient topology is a topological residuated lattice.

Proof. By Proposition 5.9, we have that L/F is an involutory residuated lattice. By Theorem 4.8, we get L/F with the quotient topology is a topological residuated lattice. □

Corollary 5.11. Let F be an filter of an involutory residuated lattice L and let (L, τ_F) be a topological residuated lattice. Then L/F with the quotient topology is a discrete topological residuated lattice.

Proof. By Theorem 5.10, we get L/F with the quotient topology is a topological residuated lattice. It follows from Corollary 5.3 that [x] ∈ τ_F for each x ∈ L. Thus each single point set {[x]} in L/F is an open set with respect to quotient topology. Hence L/F with the quotient topology is discrete. □

Theorem 5.12. Let τ be a topology on an involutory residuated lattice L. Then the following are equivalent:

(1) (L, τ) is a topological residuated lattice,

(2) the function F : L × L ↪ L defined by F (x, y) = ¬ x ⊙ y is continuous and ∨ is continuous.

Proof. (1)⇒(2) Suppose that (L, τ) is a topological residuated lattice. Since the mapping I (x) = x and ¬ (x) = ¬ x of L into L are continuous, the mapping f (x, y) = (¬ x, y) = (¬ (x) , I (y)) is a continuous map of L × L into L × L. Hence F = ⊙ ∘ f is continuous. (2)⇒(1) Let F and ∨ be continuous. Then K = F ∣ _L×0 is continuous. We define h : L ↪ L × L by h (x) = (x, 0). Clearly, h is continuous. Therefore, ¬ = K ∘ h is continuous. Now we show that ⊙ is continuous. Let f (x, y) = (¬ x, y) be a map of L × L ⇌ L × L. Since the mapping I (x) = x and ¬ are continuous, f is also continuous. The following diagram shows that ⊙ = F ∘ f is a continuous map, (x, y) ↪ ^f (¬ x, y) ↪ ^F ¬¬ x ⊙ y = x ⊙ y.

Moreover, by Proposition 4.7(2), we have x → y = ¬ (x ⊙ ¬ y). Since the mapping ¬ (x) = ¬ x of L into L is continuous, the mapping g (x, y) = (¬ x, ¬ y) = (¬ (x) , ¬ (y)) is a continuous map of L × L into L × L. Hence → = ¬ ∘ F ∘ g is continuous. Finally, by Proposition 4.7(3), we have x ∧ y = ¬ (¬ x ∨ ¬ y), and hence ∧ is continuous. □

Theorem 5.13. Let (L, τ) be a topological residuated lattice and let F be a filter of L. If the natural homomorphism π : L → L/F is open and continuous, then L/F with the quotient topology is a topological residuated lattice.

Proof. Note that π × π is an open surjection, and therefore a quotient map. The function h : L × L → L/F defined by: h (x, y) = π ∘ ⊙ ∘ (¬ × Id) (x, y) is constant on each set x/F × y/F, (x/F, y/F) ∈ L/F × L/F. In fact, h (x/F × y/F) = ¬ x ⊙ y/F. Hence by universal property of quotient topology, there is a continuous map θ : L/F × L/F → L/F which commute the following diagram, that is θ ∘ (π × π) = h.

Consequently, we get that θ (x/F, y/F) = (¬ x ⊙ y)/F. Moreover, we can see that $\bar{\lor} (x / F, y / F) = (x / F) \bar{\lor} (y / F) = (x \lor y) / F$ and thus $\bar{\lor} = π \circ \lor$ is continuous. By Theorem 5.12, we have L/F with the quotient topology is a topological residuated lattice. □

Theorem 5.14. Let F ∈ F (L). The following conditions are equivalent:

(1) F = {1},

(2) τ_F is discrete,

(3) τ_F is Hausdorff.

Proof. (1)⇒(2) Note that y ∈ U_a,1 iff d (y, a) =0, and hence U_a,1 = {a}. It follows that τ_F is discrete.

(2)⇒(3) It is straightforward.

(3)⇒(1) Now let τ_F be Hausdorff and 1 ≠ x ∈ F. Then there exist a, b ∈ F such that (x, 1) ∈ a/F × b/F and a/F∩ b/F = ∅. Hence, b/F = 1/F and x ∈ b/F, a contradiction. Therefore F = {1}. □

6 Conclusion

C. C. Chang’s distance function is an important function, which makes MV algebras form a distance space. But they are not distance function on residuated lattices. In this paper, we introduce weak involutory residuated lattices, in which Chang’s function is a pseduo distance function. Moreover we prove that the functions become distance functions on involutory residuated lattices. Note that involutory residuated lattices are not MV-algenras, and so, it is a substantial promotion to extend C. C. chang’s distance functions from MV-algebras into residuated lattices. By use of the function and a lattice filter, we define F-ball on residuated lattices, and we prove that the set of all F-balls forms a base of a topology τ_LF on an involutory residuated lattice. Moreover we show that the topology τ_LF on an involutory residuated lattice L makes L to be a lattice filter topological residuated lattice. At last, we instead of lattice filters by filters to set up filter topologies on an involutory residuated lattice, and study the properties of the filter topologies. In the future, we study properties of topologies induced by lattice filters and diatance functions, such as separation and connectivity. In addition, we will construct topological residuated lattices by other methods instead of ones in our paper.

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