Lukasiewicz semantic MV-topology for MV-algebra and its application to Lukasiewicz propositional logic

Abstract

We introduce an MV-topology on the set of all valuations of MV-algebra and then establish Lukasiewicz semantic MV-topological space. We study the topological properties of Lukasiewicz semantic MV-topology, and prove that the Lukasiewicz semantic MV-topological space is a compact zero dimension Hausdorff MV-topological space and a N-compact space. We also establish a classical topology $D$ on the valuations set of MV-algebra, and prove that topology $D$ is finer than the cut topology $C$ generated by Lukasiewicz semantic MV-topology. We prove that a σ-complete lattice is an MV-algebra if and only if it is isomorphic to an MV-clopen lattice of a Stone MV-space. As an application, we use the compactness of topology $D$ to prove the compactness of Lukasiewicz semantic and Lukasiewicz propositional logic system.

Keywords

MV-algebra MV-topological space Lukasiewicz semantic MV-topology compactness Hausdorff MV-space Stone MV-space

1 Introduction

In recent years, algebra, probability, topology and other mathematics methods are widely used in the study of logic. The combined investigation of logic and other mathematical branches, such as algebra, probability, topology and so on, promotes the development of logic and also enriches the content of these mathematical branches [1 –4]. While classical set theory finds its natural algebraic framework in the theory of Boolean algebras and its proper logical setting in classical logic, the family of fuzzy subsets of a given set may have many different algebraic structures such as MV-algebras, R₀-algebras, BL-algebras, MTL-algebras etc, which, on their turn, correspond to as many logics and can also be viewed as a generalization of Boolean algebras [5 –8].

As we all know, a classical topology on a set X is a family of subsets containing X and empty set, which is closed under finite intersection and arbitrary union [9]. A fuzzy set α on a given set X is defined by a membership function α : U → [0, 1], α (x) is the value of α on the element x. The negation α′ of fuzzy set α, the intersection α ∧ β and the union α ∨ β of two fuzzy sets α and β are interpreted in the usual sense, namely α (x) ′ = 1 - α (x), (α ∧ β) (x) = α (x) ∧ β (x) and (α ∨ β) (x) = α (x) ∨ β (x) for all x ∈ X. By using fuzzy sets instead of classical sets and fuzzy sets intersection and union instead of classical sets intersection and union, the classical topology on a set X is extended to the fuzzy topology on a set X which is a family of fuzzy subsets containing X and empty set, and closed under finite union and arbitrary intersection [10 –12].

The topology representation of logic algebra is an important aspect in study of logic theory. Classically, bounded distributive lattices are represented by prime filter spaces which are compact, as shown in [13], and the Stone representation theorem and Loomis-Sikorski representation theorem for Boolean algebras have also been given early in [14]. There are also many scholars began to study fuzzy topology representation of fuzzy logic algebra [15, 16], for example, Wang in [17] and Zhang in [18] have established the fuzzy sets representation of implication lattices and MTL-algebras respectively, however, the results are still far from the classical Stone representation theorem.

On the other hand, we also note that the real unit interval [0,1] can be endowed with some algebraic operations, the previous extension from the classical topologies to the fuzzy topologies focuses only on the partial order structure in [0,1] and it does not take into account that its algebraic operations, for instance, [0,1] with Lukasiewicz operations a → b = min {1, 1 - a + b} and a ⊗ b = max {0, a + b - 1} can be viewed as an MV-algebra. This should be the main reason why Stone representation theorem in Boolean algebras can not be well extended to fuzzy logic algebras [17 –22]. It is undoubtable that, among the various fuzzy logics and corresponding algebraic semantics, Lukasiewicz logic and MV-algebras are the ones that best succeed in both having a rich expressive power and preserving many properties of symmetry that are inborn qualities of classical logic and Boolean algebras. For the above reasons, Russo takes into account both the partial order and the MV-operations on real unit interval [0, 1], which is the range of membership function of fuzzy set, in establishing fuzzy topological space. He introduces a concept of MV-topology which reflects as many properties of classical topology as possible. In an MV-topology, the Boolean algebra of the subsets of the universe is replaced by an MV-algebra of fuzzy subsets, and the algebraic structure of the family of open (fuzzy) subsets has a quantale reduct (Ω, ⋁ , ⊕ , 0), which replaces the classical sup-lattice (Ω, ⋁ , 0), and an idempotent semiring one (Ω, ⋀ , ⊗ , 1) in place of the classical meet-semilattice (Ω, ⋀ , 1) [22].

In this paper, we focus on extending the Stone representation theorem of Boolean algebras to MV-algebras by establishing MV-topology on the set of all valuations of MV-algebras, prove that the Lukasiewicz semantic MV-topological space is a compact zero dimension Hausdorff MV-topological space and a N-compact fuzzy topological space. We introduce a classical topology $D$ on the valuations set of MV-algebra, and prove that topology $D$ is finer than the cut topology $C$ generated by MV-topology. We obtain a conclusion that every Stone MV-space is homeomorphic to one Lukasiewicz semantic MV-topological space, and prove the Stone representation theorem for MV-algebras that a σ-complete lattice is an MV-algebra if and only if M is isomorphic to an MV-clopen set lattice of a Stone MV-space, and then the Stone representation for Boolean algebras is only an example of the representation theorem for MV-algebras. As an application, we prove the compactness of Lukasiewicz semantic and Lukasiewicz propositional logic system by using the compactness of topology on valuations set.

2 Preliminaries

In order to make this paper self-contained, we summarize the basic concepts and recall some useful conclusions of MV-algebras, MV-topologies, etc. More details about MV-algebras and MV-topologies can be found in [6, 9] and [22] respectively.

2.1 MV-algebras

MV-algebras have some equivalent definitions. In this paper, we cite the following definition which is defined by means of concept of adjoint pair.

Let ⊗ and → be two binary operations on a poset M. (⊗ , →) is called an adjoint pair if it satisfies the following conditions: (1) ⊗ : M × M → M is monotone increasing with respect to two variables; (2) → : M × M → M is monotone decreasing with respect to the first variable and monotone increasing with respect to the second one; (3) x ⊗ y ≤ z if and only if x ≤ y → z for all x, y, z ∈ M.

Definition 2.1. [6, 9] An MV-algebra is a structure (M ; ∨ , ∧ , ⊗ , → , 0, 1) such that

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

(M ; ⊗ , 1) is a commutative semigroup with unit element 1;

(⊗ , →) is an adjoint pair in M;

x ∧ y = x ⊗ (x → y);

(x → y) ∨ (y → x) =1;

(x → 0) →0 = x.

If M is also totally ordered then M is called a totally ordered MV-algebra. An MV-algebra M is called complete (σ-complete) if every non-empty subset (countable subset) of M has a supremum in M.

Example 2.2. The real unit interval [0,1] with Lukasiewicz operations $a \to b \overset{def}{=} min {1, 1 - a + b}$ and $a \otimes b \overset{def}{=} max {0, a + b - 1}$ for a, b ∈ [0, 1] is a σ-complete MV-algebra called the MV-unit interval.

Example 2.3. Let X ≠ ∅ , M = [0, 1] ^X, where [0, 1] is the MV-unit interval. The order and the operations ⊗, → on M are defined in point-wise: for A, B ∈ M $\begin{matrix} A \leq B iff x \in X, A (x) \leq B (x), \\ (A \otimes B) (x) \overset{def}{=} A (x) \otimes B (x), x \in X, \\ (A \to B) (x) \overset{def}{=} A (x) \to B (x), x \in X . \end{matrix}$

Then M is a σ-complete MV-algebra called the MV-cube.

On every MV-algebra it is possible to define two further operations as follows: $x^{'} \overset{def}{=} x \to 0$ , $x \oplus y \overset{def}{=} x^{'} \to y$ . It follows immediately from the definitions that operation ^′ : M → M is an order-reversing involution and (M ; ⊕ , 1) is a commutative semigroup with unit element 0. In order to simplify the notation we also define inductively $x^{k} : x^{1} \overset{def}{=} x, x^{k + 1} \overset{def}{=} x^{k} \otimes x, \forall k \in N$ , where N denotes the set of all positive integers.

An element x in an MV-algebra M is called a Boolean element if x ⊕ x = x. Obviously, if x and y are Boolean elements, then x ⊕ y and x′ are Boolean elements as well, moreover we have x ⊕ y = x ∨ y, x ⊗ y = x ∧ y, x ∨ x′ = 1, x ∧ x′ = 0. It is easy to check that the set B (M) of all Boolean elements in an MV-algebra M forms a Boolean algebra.

A nonempty subset F in an MV-algebra M is called a filter of M if (i) 1 ∈ F; (ii) x ∈ F and x ≤ y imply y ∈ F; (iii) x, y ∈ F imply x ⊗ y ∈ F. A filter F of M is said to be a proper filter if F ≠ M, is said to be a prime filter if F is proper and x ∨ y ∈ F implies x ∈ F or y ∈ F, and is said to be a maximal filter if the only filter strictly containing F is the improper filter M.

For a subset A of M, the intersection of all filters containing A is clearly a minimal filter containing A, called it the generated filter of A and denoted it by 〈A〉, and $\begin{matrix} 〈 A 〉 & = & {x \in M | there are n \in N and a_{1}, \dots, a_{n} \in A \\ such that a_{1} \otimes \dots \otimes a_{n} \leq x}, \end{matrix}$ where N denotes the set of all positive integers. If A = {a} then 〈 {a} 〉 is also simplified as 〈a〉, and $〈 a 〉 = {x \in M | there exists n \in N such that a^{n} \leq x} .$

Definition 2.4. Let M be a σ-complete MV-algebra and [0,1] the MV-unit interval. If a mapping v : M → [0, 1] is a type of (¬ , ∨ , →) homomorphism, i.e. v (¬ a) = ¬ v (a) , v (a ∨ b) = v (a) ∨ v (b) , v (a → b) = v (a) → v (b) for any a, b ∈ M, then v is called a Lukasiewicz valuation, for briefly, valuation. If a valuation v is also a σ-homomorphism, i.e. v (⋁ _i∈Na_i) = ⋁ _i∈Nv (a_i) for any sequence {a_n | n ∈ N} ⊆ M, then v is called a σ-valuation. The set of all valuations of M is denoted by Σ. Obviously, a valuation v preserves order and preserves ∧, ⊗ , ⊕ operations, and for every valuation v, v (1) =1, v (0) =0.

Remark 2.5. In the view of Tarski [1], an abstract semantic in a complete lattice L is a nonempty set S if 1 ∉ S, where 1 is the greatest element of L. Clearly, Σ ⊆ [0, 1] ^M, and since ∀v ∈ Σ, v (0) =0, we know 1 ∉ Σ. This means that Σ is an abstract semantic in [0, 1] ^M, we call it the Lukasiewicz semantic.

Example 2.6. Let M = {0, a, b, c, d, 1}. The order and the operations ∨, ∧ on M defined as Fig. 1, the operations ⊗, → on M defined as Tables 1 and 2 respectively. It is easy to know that M is an MV-algebra.

Fig.1

Partially order set M.

Table 1

Two binary operation ⊗

⊗	a	b	c	d	1
0	0	0	0	0	0
a	a	d	0	d	a
b	d	c	c	0	b
c	0	c	c	0	c
d	d	0	0	0	d
1	a	b	c	d	1

Table 2

Two binary operation →

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

Then Σ = {v₁, v₂}, where $v_{1}, v_{2} : M \to [0, 1], v_{1} (1) = 1, v_{1} (a) = 1, v_{1} (b) = \frac{1}{2}, v_{1} (c) = 0, v_{1} (d) = \frac{1}{2}, v_{1} (0) = 0; v_{2} (1) = 1, v_{2} (a) = 0, v_{2} (b) = 1, v_{2} (c) = 1, v_{2} (d) = 0, v_{2} (0) = 0$ .

2.2 MV-topologies

Definition 2.7. Let X be a nonempty set and Ω a subset of MV-cube [0, 1] ^X. If

0, 1 ∈ Ω;

for {α_i} _i∈I ⊆ Ω, ⋁_i∈Iα_i ∈ Ω;

and, for any α₁, α₂ ∈ Ω,

α₁ ⊗ α₂ ∈ Ω;

α₁ → α₂ ∈ Ω;

then Ω is called an MV-topology on X, (X, Ω) is called an MV-topological space (MV-space, for short), and the elements of Ω are called the MV-open sets.

Obviously, for any α₁, α₂ ∈ Ω we have $α_{1}^{'}, α_{1} \land α_{2}, α_{1} \oplus α_{2} \in Ω$ .

In an MV-topology Ω, if we denote Ω′ = {α′ | α ∈ Ω}, then

0, 1 ∈ Ω;

for β₁, β₂ ∈ Ω′, β₁ ∨ β₂ ∈ Ω′, β₁ ⊕ β₂ ∈ Ω′, β₁ ⊗ β₂ ∈ Ω′;

for any {β_i} _i∈I ⊆ Ω′, ⋀_i∈Iβ_i ∈ Ω′.

The elements of Ω′ is called the MV-closed sets.

As in classical topology, given an MV-topological space $T = (X, Ω)$ , a subset Δ of Ω is called a base for $T$ if every MV-open set of $T$ is the union of some elements in Δ.

Let (X, Ω) be an MV-topological space and Y a subset of X. If we denote Ω_Y = {Y ⊗ α | α ∈ Ω} then it is easy to check that (Y, Ω_Y) is also an MV-topological space. For any subset Y of X, we call (Y, Ω_Y) the MV-subspace of (X, Ω), if Y is an MV-closed set, then we call (Y, Ω_Y) the closed MV-subspace.

Example 2.8. (1) Any classical topology is an MV-topological space.

(2) (X, [0, 1] ^X) is an MV-topological space.

(3) Let X be a non-empty set and C (Ω) the set of all constant fuzzy subsets of X, then (X, C (Ω)) is an MV-topological space.

(4) If we denote B (Ω) = Ω ∩ {0, 1} ^X = Ω ∩ B ([0, 1] ^X) for an MV-topological space (X, Ω), then (X, B (Ω)) is a classical topological space, and we call it the shadow space of (X, Ω).

Let Γ be a subset of [0, 1] ^X. If ⋁_α∈Γα = 1, then Γ is called a covering of X. If every element of Γ is MV-open set, then Γ is called an open covering of X. If a finite subset {α₁, α₂, ⋯ , α_n} such that α₁ ⊕ α₂ ⊕ ⋯ ⊕ α_n = 1, then {α₁, α₂, ⋯ , α_n} is called a finite additive covering (⊕-covering, for short).

Proposition 2.9. Let X be a nonempty set of X. Any covering of fuzzy subsets of X which is closed under operations ⊕, ⊗ and ∧ is a base for an MV-topology on X.

Proof. Let Γ ∈ [0, 1] ^X be a covering closed under operations ⊕, ⊗ and ∧, and let Ω = {⋁ G | G ⊆ Γ}. It is easy to check that Ω is an MV-topology and Γ is a base of it.

Proposition 2.10. Let $T = (X, Ω)$ be an MV-topological space. Suppose that Clop $T \overset{def}{=} Ω \cap Ω^{'}$ is the set of all clopen subsets of X, i.e., the fuzzy subsets in Clop $T$ are both open and closed. Then $Clop T$ is a σ-complete MV-algebra.

Proof. Obviously, $0, 1 \in Clop T$ . If $α, β \in Clop T$ , then it is easy to check $α \oplus β, α^{'} \in Clop T$ . This means that $Clop T$ is closed under ⊕, ′. Hence $Clop T$ is a sub-MV-algebra. Since [0, 1] ^X is σ-complete, we know that $Clop T$ is σ-complete.

An MV-topological space (X, Ω) is said to be compact if every open covering of X contains a finite additive covering; it is called strongly compact if every open covering contains a finite covering.

It is obvious that strong compactness implies compactness and, since the operations ⊕ and ∨ coincide on Boolean elements of MV-algebras, in the case of topologies of crisp subsets the two notions collapse to the classical one. For the same reason, it is evident as well that the shadow spaces of both compact and strongly compact MV-spaces are compact. It is easy to check that compactness does not imply strong compactness, i.e., they are not equivalent.

Let $T = (X, Ω)$ be an MV-topological space, x ≠ y ∈ M and x_λ, y_μ be two fuzzy points with support x and y respectively. $T$ is called an Hausdorff (or separated) space if there exist αα₁, αα₂ ∈ Ω such that

x_λ ≤ α₁, y_μ ≤ α₂;

α₁ (y) = α₂ (x) =0;

α₁ ⊗ α₂ = 0.

$T$ is said to be strongly Hausdorff (or strongly separated) if there exist α₁, α₂ ∈ Ω satisfying (1) and

α₁ ∧ α₂ = 0.

It is self-evident that (4) implies both (2) and (3), therefore strong separation implies separation. Furthermore, as for compactness and strong compactness, both separation and strong separation coincide with the classical T₂ property on classical topologies and imply that the corresponding shadow space is Hausdorff in the classical sense.

Proposition 2.11. [22] Every closed MV-subspace (Y, Ω_Y) of a compact space (strongly compact space) (X, Ω) is also a compact space (strongly compact space).

Proposition 2.12. [22] Every subspace (Y, Ω_Y) of an Hausdorff space (strongly Hausdorff space) (X, Ω) is also an Hausdorff space (strongly Hausdorff space).

Proposition 2.13. [22] If (X, Ω) is an Hausdorff space, then all crisp singletons of X are MV-closed sets.

Let X and Y be two sets. Any mapping f : X → Y naturally induce two mapping [11]: $\begin{matrix} f^{\leftarrow} : [0, 1]^{Y} & ⟶ & [0, 1]^{X}, α \mapsto α \circ f, \\ f^{\leftarrow} (α) (x) & = & α (f (x)), \forall x \in X; \\ f^{\to} : [0, 1]^{X} & ⟶ & [0, 1]^{Y}, α \mapsto f^{\to} (α, \\ f^{\to} (α) (y) & = & ⋁_{f (x) = y} α (y), \forall y \in Y . \end{matrix}$

Obviously, f^← (0) =0, and if αα, β ∈ [0, 1] ^Y then for any x ∈ X, $\begin{matrix} f^{\leftarrow} (α \oplus β) (x) & = & (α \oplus β) (f (x)) \\ = & α (f (x)) \oplus β (f (x)) \\ = & f^{\leftarrow} (α) (x) \oplus f^{\leftarrow} (β) (x), \\ f^{\leftarrow} (α^{'}) (x) & = & α^{'} (f (x)) = 1 - α (f (x)) \\ = & 1 - f^{\leftarrow} (α) (x) = (f^{\leftarrow} (α))^{'} (x) . \end{matrix}$

Hence f^← is an MV-algebra homomorphism.

Let (X, Ω_X) and (Y, Ω_Y) be two MV-topological spaces. A mapping f : X → Y is said to be MV-continuous if f^← (β) ∈ Ω_X for all β ∈ Ω_Y, is said to be an MV-homeomorphism if it is bijective and both f and f^-1 are MV-continuous.

It is already prove in [22] that the MV-topological properties, such as compactness, strong compactness, separated and strongly separated etc, remain unchanged under MV-homeomorphism.

3 MV-topologies on the set of all valuations for MV-algebras

In this section, we establish Lukasiewicz semantic MV-topology Ω_Σ on the set Σ of all valuations of an MV-algebra and discuss the topological properties of MV-topological space (Σ, Ω_Σ). Because of the basis set Σ with a certain structure, the MV-topological space (Σ, Ω_Σ) should have some better topological properties.

A valuation v of MV-algebra M is also called a state-morphism in [15]. By means of valuation of MV-algebra, Zhang proves that the order in a σ-complete MV-algebra is determined by the set of allvaluations [23], and then discusses the sizes of elements of M and the pseudo-metric of pairs of elements of M [24].

Theorem 3.1. Let M be a σ-complete MV-algebra. Define a mapping φ : M → [0, 1] ^Σ as follows: $φ (x) (v) \overset{def}{=} v (x), v \in Σ, x \in M .$

Denote δ_M = {φ (x) | x ∈ M}. Then δ generates an MV-topology Ω_Σ, δ_M is a base of Ω_Σ.

Proof. It is easy to know that δ_M is closed under operations ⊕, ⊗ , ∧. Hence by Proposition 2.9, δ_M generates an MV-topology on the set Σ of all valuations, denote it by Ω_Σ, Ω_Σ is called a Lukasiewicz semantic MV-topology and (Σ, Ω_Σ) a Lukasiewicz semantic MV-topological space.

Obviously, if M is a complete MV-algebra then Ω_Σ = δ_M.

Example 3.2. We consider the MV-algebra M as shown in Example 2.6. Then δ_M = {φ (1) , φ (a) , φ (b), φ (c) , φ (d) , φ (0)}, where $φ (1) = \frac{1}{v_{1}} + \frac{1}{v_{2}}$ , $φ (a) = \frac{1}{v_{1}} + \frac{0}{v_{2}}$ , $φ (b) = \frac{0.5}{v_{1}} + \frac{1}{v_{2}}$ , $φ (c) = \frac{0}{v_{1}} + \frac{1}{v_{2}}$ , $φ (d) = \frac{0.5}{v_{1}} + \frac{0}{v_{2}}$ , $φ (0) = \frac{0}{v_{1}} + \frac{0}{v_{2}}$ , generates an MV-topology Ω_Σ and Ω_Σ = δ_M.

Example 3.3. Let M be a complete MV-algebra and F be a filter of M. Denote δ_F = {φ (x) | x ∈ F}, then it is easily to check that (Σ, δ_F ∪ {0}) is an MV-topological space.

Theorem 3.4. The Lukasiewicz semantic MV-topological space (Σ, Ω_Σ) is a compact Hausdorff space.

Proof. Firstly, we prove that (Σ, Ω_Σ) is compact. Let Γ = {φ (x_i) | x_i ∈ M, i ∈ I} be a covering of Σ. If Γ does not contain any finite ⊕-covering, then any finite elements {φ (x₁) , ⋯ , φ (x_n)} in Γ don’t form one ⊕-covering of Σ, hence there is v ∈ Σ such that $({oplus}_{i = 1}^{n} φ (x_{i})) (v) = φ ({oplus}_{i = 1}^{n} x_{i}) (v) = v ({oplus}_{i = 1}^{n} x_{i}) < 1$ . It follows that ${oplus}_{i = 1}^{n} x_{i} < 1$ , and so $⨂_{i = 1}^{n} x_{i}^{'} > 0$ . Let φ^-1 (Γ) = {x | φ (x) ∈ Γ} and φ^* (Γ) = {x′ | x ∈ φ^-1 (Γ)}, then the generated filter [φ^* (Γ)) of φ^* (Γ) is a proper filter of M. Hence there is a prime filter F₀ such that [φ^* (Γ)) contains F₀. Hence the congruence class [x^′]F₀ of every element x′ in φ^* (Γ) with respect to F₀ equals to the congruence class 1_{F
₀} of 1, this also means that [x]_{F
₀} = 0_{F
₀} for every element x in φ^-1 (Γ).

Note that $M ≅_{ψ} M^{'} \subseteq \prod_{F \in P (M)} M_{F}, ψ (x) = {({[x]}_{F})}_{F \in P (M)}$ [6]. Since the quotient algebra M_{F
₀} is a totally ordered MV-algebra, by Proposition 4.3 in [18] we know that there is a unique valuation v_{F
₀} of M_{F
₀} such that v_{F
₀}([x]_{F
₀})=v_{F
₀}(0_{F
₀}) for every element x in φ^-1 (Γ). Then it follows from that v₀=v_{F
₀}∘π_{F
₀} ∘ ψ is a valuation of M, where π_{F
₀} is the project mapping from $\prod_{F \in P (M)} M_{F}$ to M_{F
₀}, and v₀(x)=(v_{F
₀}∘π_{F
₀} ∘ ψ) (x) =0 for every element x in φ^-1 (Γ). Hence φ (x) (v₀) = v₀ (x) =0 for every element x in φ^-1 (Γ). This shows that Γ = {φ (x_i) | x_i ∈ M, i ∈ I} is not a covering of Σ. This is a contradiction, therefore (Σ, Ω_Σ) is a compact space.

Next, we prove that (Σ, Ω_Σ) is Hausdorff. Let v₁ and v₂ be two distinct valuations of M. Then there exists x₀ such that v₁(x₀) ≠ v₂(x₀). Without loss of generality, we may assume v₁ (x₀) < v₂ (x₀). From the proof of Proposition 4.3 in [18] there is $A_{s}^{r} (x_{0})$ such that $v_{1} (A_{s}^{r} (x_{0})) = 0$ and $v_{2} (A_{s}^{r} (x_{0})) = 1$ . It follows that $φ (A_{s}^{r} (x_{0})) (v_{1}) = 0$ , $φ ((A_{s}^{r} (x_{0}))^{'}) (v_{1}) = 1$ and $φ (A_{s}^{r} (x_{0})) (v_{2}) = 1$ , $φ ((A_{s}^{r} (x_{0}))^{'}) (v_{2}) = 0$ . This shows that $φ (A_{s}^{r} (x_{0}))$ and $φ ((A_{s}^{r} (x_{0}))^{'})$ are two MV-open sets satisfying conditions (1) and (2) of definition of Hausdorff space, and it follows from $φ (A_{s}^{r} (x_{0})) \otimes φ ((A_{s}^{r} (x_{0}))^{'}) = 0$ that the above two MV-open sets also satisfy condition (3) of definition of Hausdorff space. Therefore (Σ, Ω_Σ) is Hausdorff.

The following Theorem 3.5 illustrates that the Lukasiewicz semantic MV-topological space (Σ, Ω_Σ) may also possess the other fuzzy topology compactness, such as N-compactness [11].

Theorem 3.5. The Lukasiewicz semantic MV-topological space (Σ, Ω_Σ) is N-compact.

Proof. Suppose that {(v_n) _{λ
_n}, n ∈ D} is a α-net in Ω_Σ, where D is a directed set. Then {(v_n) ₁, n ∈ D} is also a net in [0, 1] ^Σ. Since the MV-unit interval [0, 1] is compact, [0, 1] ^Σ is also compact. Hence there is a subnet {(v_{n
_i}) ₁, i ∈ I} of {(v_n) ₁, n ∈ D} which converges to (F₀) ₁. For any F-neighborhood γ of (v₀) _α, there exists a close F-neighborhood β of (v₀) _α such that γ ≤ β. Since δ_M = {φ (x) | x ∈ M} is an MV-clopen set base of Ω_Σ, we have β′ = φ (x₁) ∨ ⋯ ∨ φ (x_m), that is, $β = φ (x_{1}^{'}) \land \dots \land φ (x_{m}^{'})$ . Hence $(φ (x_{1}^{'}) \land \dots \land φ (x_{m}^{'})) (v_{0}) < α$ . If we take $(φ (x_{1}^{'}) \land \dots \land φ (x_{m}^{'})) (v_{0}) < r < α$ , then $v_{0} \in π_{φ (x_{1}^{'})}^{- 1} ([0, r)) \cup \dots \cup π_{φ (x_{m}^{'})}^{- 1} ([0, r)) \in [0, 1]^{Σ},$ where $π_{φ (x_{k}^{'})}$ is the project mapping from [0, 1] ^Σ to [0, 1]. Since subnet {(v_{n
_i}) ₁, i ∈ I} converges to (v₀) ₁, there exists i₀ such that $v_{n_{i}} \in π_{φ (x_{1}^{'})}^{- 1} ([0, r)) \cup \dots \cup π_{φ (x_{m}^{'})}^{- 1} ([0, r)) \in [0, 1]^{Σ},$ as i₀ ≤ i. Thus when i₀ ≤ i there exists k ∈ {1, ⋯ , m} such that $φ (x_{k}^{'}) (v_{n_{i}}) < r$ , that is, $⋀_{k = 1}^{m} φ (x_{k}^{'}) (v_{n_{i}}) < r$ . Since {(v_n) _{λ
_n}, n ∈ D} is a α-net in Ω_Σ, by the definition of α-net for r < α, there exists n₀ such that λ_n > r as n > n₀. If we take N = max {n_{i
₀}, n₀}, then $γ (v_{n_{i}}) \leq ⋀_{k = 1}^{m} φ (x_{k}^{'}) (v_{n_{i}}) < r < λ_{n}$ as n_i > N. This shows that for any F-neighborhood γ of (v₀) _α, there exists n > n₀ such that (v_n) _{λ
_n} is not in F-neighborhood γ, that is, α-net {(v_n) _{λ
_n}, n ∈ D} frequent absences from γ. Hence α-net {(v_n) _{λ
_n}, n ∈ D} have a cluster point (v₀) _α with height α in Ω_Σ [11]. Therefore (Σ, Ω_Σ) is N-compact.

4 An classical topology on the set of all valuations for MV-algebras

In this section we establish a classical topology on the set of all valuations for MV-algebras and discuss in detail the topological properties of it.

Definition 4.1. Let X ⊆ M. We define a subset ⌈X ⌋ ^* of Σ as follows: $⌈ X ⌋^{*} \overset{def}{=} {v | v \in Σ, \exists x \in X, v (x) \neq 1} .$

Proposition 4.2. Let X, Y ⊆ M and {X_i} _i∈I be some subsets of M. Then

If X ⊆ Y then ⌈X ⌋ ^* ⊆ ⌈ Y ⌋ ^*.

⋃_i∈I ⌈ X_i ⌋ ^* = ⌈ ⋃ _i∈IX_i ⌋ ^*.

⌈X ⌋ ^* = Σ iff 〈X〉 = M.

⌈X⌋ ^* = ∅ iff X =∅ or X = {1}.

⌈X ⌋ ^* = ⌈ 〈X〉 ⌋ ^*.

⌈X ⌋ ^* ∩ ⌈ Y ⌋ ^* = ⌈ 〈X〉 ∩ 〈Y〉 ⌋ ^*.

Proof. The proof of (1) is trivial.

(2) We have from (1) that ⋃_i∈I ⌈ X_i ⌋ ^* ⊆ ⌈ ⋃ _i∈IX_i ⌋ ^*. Let v ∈ ⌈ ⋃ _i∈IX_i ⌋ ^*, then there is x ∈ ⋃ _i∈IX_i such that v (x) ≠1. Hence there exist at least one i₀ ∈ I and one x ∈ X_{i
₀} such that v (x) ≠1. This shows that v ∈ ⌈ X_{i
₀} ⌋ ^*, and so v ∈ ⋃ _i∈I ⌈ X_i ⌋ ^*. Thus ⌈ ⋃ _i∈IX_i ⌋ ^* ⊆ ⋃ _i∈I ⌈ X_i ⌋ ^*. Therefore ⋃_i∈I ⌈ X_i ⌋ ^* = ⌈ ⋃ _i∈IX_i ⌋ ^*.

(3) Suppose that ⌈X ⌋ ^* = Σ. If 〈X〉 ≠ M then 〈X〉 is a proper filter. Hence there exists a prime filter P such that 〈X〉 ⊆ P [6]. By Proposition 4.3 in [18] we know that there is a unique valuation v_P of quotient algebra M_P such that v_P([X]_P)=v_P(1_P)=1 for every element x in P. It follows from that v₀(x)=v_P∘π_P∘ψ(x)=v_P([x]_P)=v_P(1_P) =1 for every element x in X. This means that v ∉ ⌈ X ⌋ ^*. This contradict the fact that ⌈X ⌋ ^* = Σ. Thus 〈X〉 = M.

Conversely, suppose that 〈X〉 = M. Then for any v ∈ Σ, 0 ∈ 〈X〉 and v (0) =0 ≠ 1. This means that for all v ∈ Σ, v ∈ ⌈ X ⌋ ^*. Hence ⌈X ⌋ ^* = Σ.

(4) For any valuation v, we have v (1) =1 and v (0) =0. It follows that ⌈∅ ⌋ ^* = ⌈ {1} ⌋ ^* = ∅. Conversely, suppose that ⌈X⌋ ^* = ∅. If X≠ ∅ and X ≠ {1} then there exists a ≠ 1 such that a ∈ X. It follows there exists a prime filter F such that a ∉ F [6]. By Proposition 4.3 in [18] we know that there is a unique valuation v_F of quotient algebra M_F such that v_F([a]_F)=v_F(0_F)=0 and v_F([X]_F)=v_P(1_P)=1 for every element x in F. It follows from that v₀(a)=v_F∘π_F∘ψ(a)=v_F([a]_F)=v_F(0_F) =0 and v₀(x)=v_F∘π_F∘ψ(x)=v_F([x]_F)=v_F(1_F) =1 for every element x in X. This means that v₀ ∈ ⌈ X ⌋ ^*. This contradict the fact that ⌈X⌋ ^* = ∅. Hence X =∅ or X = {1}.

(5) By (1) we know that ⌈X ⌋ ^* ⊆ ⌈ 〈X〉 ⌋ ^*. Conversely, let v ∈ ⌈ 〈X〉 ⌋ ^*. Then there is x ∈ 〈X〉 such that v (x) ≠1. It follows that there exist x₁, x₂, ⋯ , x_n ∈ X such that x₁ ⊗ x₂ ⊗ ⋯ ⊗ x_n ≤ x, and v (x₁) ⊗ v (x₂) ⊗ ⋯ ⊗ v (x_n) ≤ v (x) ≠1. Hence there is at least x_i such that v (x_i) ≠1. This shows that v ∈ ⌈ X ⌋ ^*, so ⌈〈X〉 ⌋ ^* ⊆ ⌈ X ⌋ ^*. Hence ⌈X ⌋ ^* = ⌈ 〈X〉 ⌋ ^*.

(6) Let v ∈ ⌈ X ⌋ ^* ∩ ⌈ Y ⌋ ^*. Then there exist x ∈ X and y ∈ Y such that v (x) ≠1 and v (y) ≠1. Hence x ∨ y ∈ 〈X〉 ∩ 〈Y〉 and v (x ∨ y) = v (x) ∨ v (y) ≠1. It follows that v ∈ ⌈ 〈X〉 ∩ 〈Y〉 ⌋ ^*. This shows that ⌈X ⌋ ^* ∩ ⌈ Y ⌋ ^* ⊆ ⌈ 〈X〉 ∩ 〈Y〉 ⌋ ^*.

Conversely, by (1) and (5) we have that ⌈〈X〉 ∩ 〈Y〉 ⌋ ^* ⊆ ⌈ 〈X〉 ⌋ ^* = ⌈ X ⌋ ^* and ⌈〈X〉 ∩ 〈Y〉 ⌋ ^* ⊆ ⌈ 〈Y〉 ⌋ ^* = ⌈ Y ⌋ ^*, so ⌈〈X〉 ∩ 〈Y〉 ⌋ ^* ⊆ ⌈ X ⌋ ^* ∩ ⌈ Y ⌋ ^*. Therefore ⌈X ⌋ ^* ∩ ⌈ Y ⌋ ^* = ⌈ 〈X〉 ∩ 〈Y〉 ⌋ ^*.

If A = {a} then we denote ⌈ {a} ⌋ ^* by ⌈a ⌋ ^*, namely ⌈a ⌋ ^* = {v | v ∈ Σ, v (a) ≠1}. We have the following proposition by Proposition 4.2.

Proposition 4.3. Let a, b ∈ M. Then

If a ≤ b then ⌈b ⌋ ^* ⊆ ⌈ a ⌋ ^*.

⌈a ⌋ ^* ∩ ⌈ b ⌋ ^* = ⌈ a ∨ b ⌋ ^*.

⌈a ⌋ ^* ∪ ⌈ b ⌋ ^* = ⌈ a ⊗ b ⌋ ^* = ⌈ a ∧ b ⌋ ^*.

Definition 4.4. We define a subset $T$ of the power set of Σ as follows: $D \overset{def}{=} {⌈ X ⌋^{*} | X \subseteq M} .$

Then by Proposition 4.2 we have the following assertions:

$\emptyset, Σ \in D$ .

If $⌈ X ⌋^{*}, ⌈ Y ⌋^{*} \in D$ then $⌈ X ⌋^{*} \cap ⌈ Y ⌋^{*} \in D$ .

If ${⌈ X_{i} ⌋^{*} | i \in I} \subseteq D$ then $⋃_{i \in I} ⌈ X_{i} ⌋^{*} \in D$ .

Hence $D$ is a topology on Σ, $(Σ, D)$ is called a valuation topological space and ⌈X ⌋ ^* is called an open set.

Proposition 4.5. Set family {⌈ a ⌋ ^* | a ∈ M} is a base of topology $D$ .

Proof. Let X ⊆ L. Then ⌈X ⌋ ^* is an open set in $(Σ, D)$ . It follows from Proposition 4.2 that $⌈ X ⌋^{*} = ⌈ ⋃_{x \in X} {x} ⌋^{*} = ⋃_{x \in X} ⌈ x ⌋^{*} .$

This means that every open set in $T$ is always the union of some elements in {⌈ a ⌋ ^* | a ∈ M}. Hence set family {⌈ a ⌋ ^* | a ∈ M} is a base of topology $D$ .

Example 4.6. We consider the MV-algebra M as shown in Example 2.6. Then ⌈1⌋ ^* = ∅, ⌈a ⌋ ^* = {v₂}, ⌈b ⌋ ^* = ⌈ c ⌋ ^* = {v₁}, ⌈d ⌋ ^* = ⌈0 ⌋ ^* = {v₁, v₂}. $D = {\emptyset, {v_{1}}, {v_{2}}, {v_{1}, v_{2}}}$ .

Proposition 4.7. Let a ∈ M. Then ⌈a ⌋ ^* is a compact set in $(Σ, D)$ .

Proof. Suppose that ⌈a ⌋ ^* = ⋃ _i∈I ⌈ a_i ⌋ ^* = ⌈ ⋃ _i∈I {a_i} ⌋ ^*. It follows from Proposition 4.2 that 〈a〉 = 〈 ⋃ _i∈I {a_i} 〉. Hence a ∈ 〈 ⋃ _i∈I {a_i} 〉, this means that there exist finite elements a_{i
₁}, a_{i
₂}, ⋯ , a_{i
_n} ∈ ⋃ _i∈I {a_i} such that a_{i
₁} ⊗ a_{i
₂} ⊗ ⋯ ⊗ a_{i
_n} ≤ a. By Proposition 4.3 we have $\begin{matrix} ⌈ a ⌋^{*} & \subseteq & ⌈ a_{i_{1}} \otimes a_{i_{2}} \otimes \dots \otimes a_{i_{n}} ⌋^{*} \\ = & ⌈ a_{i_{1}} ⌋^{*} \cup ⌈ a_{i_{2}} ⌋^{*} \cup \dots \cup ⌈ a_{i_{n}} ⌋^{*} \subseteq ⌈ a ⌋^{*} . \end{matrix}$

Hence $⌈ a ⌋^{*} = ⌈ a_{i_{1}} ⌋^{*} \cup ⌈ a_{i_{2}} ⌋^{*} \cup \dots \cup ⌈ a_{i_{n}} ⌋^{*} .$

This shows that ⌈a ⌋ ^* is a compact set in $(Σ, D)$ .

Theorem 4.8. $(Σ, D)$ is a compact Hausdorff topological space.

Proof. Since Σ = ⌈0 ⌋ ^*, we know that $(Σ, D)$ is compact by Proposition 4.7. In the following we prove that $(Σ, D)$ is also a Hausdorff space. For v₁, v₂ ∈ Σ, v₁ ≠ v₂, then there exists x ∈ M such that v₁ (x) ≠ v₂ (x). Without loss of generality, we may assume that v₁ (x) < v₂ (x). From the proof of Proposition 4.3 in [18] there is $A_{s}^{r} (x)$ such that $v_{1} (A_{s}^{r} (x)) = 0$ and $v_{2} (A_{s}^{r} (x)) = 1$ . Analogously, from v₂ (x′) < v₁ (x′) we know that there is $A_{l}^{k} (x^{'})$ such that $v_{2} (A_{l}^{k} (x^{'})) = 0$ and $v_{1} (A_{l}^{k} (x^{'})) = 1$ . Taking $a = A_{l}^{k} (x^{'}) \to A_{s}^{r} (x)$ and $b = A_{s}^{r} (x) \to A_{l}^{k} (x^{'})$ , then $v_{1} (a) = v_{1} (A_{l}^{k} (x^{'})) \to v_{1} (A_{s}^{r} (x)) = 0$ and $v_{2} (b) = v_{2} (A_{s}^{r} (x)) \to v_{2} (A_{l}^{k} (x^{'})) = 0$ . This means that v₁ ∈ ⌈ a ⌋ ^* and v₂ ∈ ⌈ b ⌋ ^*. Observe that $a \lor b = (A_{l}^{k} (x^{'}) \to A_{s}^{r} (x)) \lor (A_{s}^{r} (x) \to A_{l}^{k} (x^{'})) = 1,$ and then $⌈ a ⌋^{*} \cap ⌈ b ⌋^{*} = ⌈ a \lor b ⌋^{*} = ⌈ 1 ⌋^{*} = \emptyset .$

This proves that $(Σ, D)$ is a Hausdorff space.

The valuations MV-topology Ω_Σ induces a cut topology $C$ which is a classical topology. The cut topology $C$ of Ω_Σ is generated by subbase $C (Ω_{Σ}) \overset{def}{=} {f_{λ} | f \in Ω_{Σ}, λ \in [0, 1]},$ where f_λ = {v ∈ Σ | f (v) > λ} is the λ-cut set of f. Since δ_M is a base for Ω_Σ, for any f ∈ Ω_Σ, there exists a subset {φ (x_i) | i ∈ I} of δ_M such that f = ⋁ {φ (x_i) | i ∈ I}. It is easy to prove that f (v) > λ if and only if there exists i ∈ I such that φ (x_i) (v) > λ. Hence $f_{λ} = ⋃ {(φ (x_{i}))_{λ} | i \in I} .$

It follows that $C (δ_{M}) = {(φ (x))_{λ} | x \in M, λ \in [0, 1]}$ is also a subbase for $C$ .

Example 4.9. We consider the MV-topological space (Σ, Ω_Σ) as shown in Example 3.2. Then the cut topology $C = {\emptyset, {v_{1}}, {v_{2}}, {v_{1}, v_{2}}}$ .

We say that the topology $C$ is finer than $D$ if $D \subseteq C$ as subsets of the power set of Σ. Then we have the following conclusion.

Theorem 4.10. Topology $C$ is finer than $D$ .

Proof. By Proposition 4.5, we know that set family {⌈ x ⌋ ^* | x ∈ M} is a base of topology $D$ , where ⌈x ⌋ ^* = {v | v ∈ Σ, v (x) ≠1}. In the following we prove that ⌈x ⌋ ^* = (φ (x′)) ₀ for every ⌈x ⌋ ^* ∈ {⌈ x ⌋ ^* | x ∈ M}. In fact, for any u ∈ ⌈ x ⌋ ^* = {v | v ∈ Σ, v (x) ≠1}, then u (x) <1. This means that φ (x′) (u) = u (x′) =1 - u (x) >0. Hence u ∈ (φ (x′)) ₀ = {v | v ∈ Σ, φ (x′) (v) >0}. This shows that ⌈x ⌋ ^* ⊆ (φ (x′)) ₀. Conversely, suppose that v ∈ (φ (x′)) ₀, then φ (x′) (v) >0. Hence φ (x) (v) =1 - φ (x′) (v) =1 - v (x′) = v (x) <1. This means that v ∈ ⌈ x ⌋ ^*. This shows that (φ (x′)) ₀ ⊆ ⌈ x ⌋ ^*. Therefore we prove that every element of {⌈ x ⌋ ^* | x ∈ M} is also an element of $C (δ_{M})$ . It follows that $D \subseteq C$ , and hence topology $C$ is finer than $D$ .

5 Stone representation theorem for MV-algebras

In this section, we extend the Stone representation theorem of Boolean algebras to MV-algebras. Similar to the classical cases that a zero dimension space is a topological space with a base consisting of open closed sets and a Stone space is a compact zero dimension Hausdorff topological space, an MV-topological space with a base consisting of MV-open closed set is called a zero dimension MV-topological space and a compact zero dimension Hausdorff MV-topological space is called a Stone MV-space.

Obviously, there is a base consisting of MV-open closed set in Lukasiewicz semantic MV-topological space (Σ, Ω_Σ), and hence it is a zero dimension space.

Theorem 5.1. (Representation theorem for MV-algebras) Let M be a σ-complete lattice. Then the following are equivalent:

M is an MV-algebra.

M is isomorphic to a MV-clopen set lattice of a Stone MV-space.

Proof. Suppose that M is a σ-complete MV-algebra. Let (Σ, Ω_Σ) be the Lukasiewicz semantic MV-topological space on M. By Proposition 2.8 and Theorem 3.4, we know that $T = (Σ, Ω_{Σ})$ is a compact zero dimension Hausdorff MV-topological space, and δ_M = {φ (x) | x ∈ M} is an MV-clopen set lattice of $T$ . In the following we prove the mapping $φ : M \to [0, 1]^{Σ}, φ (x) (v) = v (x), v \in Σ, x \in M,$ is an isomorphic mapping from M to δ_M. Since φ (x ⊕ y) = v (x ⊕ y) = v (x) ⊕ v (y) = φ (x) ⊕ φ (y), we need only prove that φ (x) is an injection. In fact, suppose that φ (x) = φ (y). Then φ (x) (v) = φ (y) (v), i.e. v (x) = v (y) for any v ∈ Σ. By Theorem 3.12 in [23] we know x = y. This shows that φ is an injection, and hence φ is an isomorphic mapping from M to δ_M.

Conversely, suppose that M is isomorphic to an MV-clopen set lattice of a Stone MV-space. Since a compact zero dimension Hausdorff MV-topological space is a σ-complete MV-algebra, we know that M is an MV-algebra.

We noticed an obvious fact that an MV-topological space degenerate to a classical topological space as the MV-cube [0, 1] ^X degenerate to the classical power set {0, 1} ^X. Hence by using the representation theorem for MV-algebras, we can easily give another proof of Stone representation theorem for Boolean algebras.

Corollary 5.2. (Representation theorem for Boolean algebras) A poset M is a Boolean algebra if and only if M is isomorphic to a clopen set lattice of a Stone space.

Proof. In the proof of Theorems 3.4 and 5.1, if M is a Boolean algebra, since the elements in a Boolean algebra satisfying conditions: x ∨ x′ = 1, x ∧ x′ = 0, we know that valuation v takes only value 0 and 1. This means that the MV-topological space in Theorem 5.1 is virtually a classical topological space. In this case the compactness, the Hausdorff separability of MV-topological space and a compact zero dimension Hausdorff MV-topological space degenerate to the classical one. Therefore the conclusion holds by Theorem 5.1.

The following theorem illustrate that any Stone MV-space is homeomorphic to a Lukasiewicz semantic MV-topological space.

Theorem 5.3. Suppose that (X, Ω) is a StoneMV-space. Then there exists a Lukasiewicz semantic MV-topological space (Σ, Ω_Σ) such that (X, Ω) is MV-homeomorphic to (Σ, Ω_Σ).

Proof. Suppose that $T = (X, Ω)$ is a Stone MV-space and Clop $T$ is the set of all MV-clopen sets in $T$ . Then Clop $T$ is a σ-complete MV-algebra by Proposition 2.10. In the following, we firstly introduce a valuation set Σ on the MV-algebra $Clop T$ and then prove that (X, Ω) is MV-homeomorphic to (Σ, Ω_Σ).

At first, for any x ∈ X, we define a corresponding mapping $v_{x} : Clop T \to [0, 1], v_{x} (α) \overset{def}{=} α (x)$ . Obviously, for $0, 1, α, β \in Clop T$ , v_x (0) =0 (x) =0, v_x (1) =1 (x) =1, v_x (α′) = α′ (x) =1 - α (x) =1 - v_x (α) = (v_x (α)) ′, v_x (α → β) = (α → β) (x) = α (x) → β (x) = v_x (α) → v_x (β). This shows v_x is a valuation on the MV-algebra $Clop T$ . The set of all such valuations is also denoted by Σ. We define a mapping $φ : Clop T \to [0, 1]^{Σ}$ as follows: $φ (α) (v_{x}) \overset{def}{=} v_{x} (α) = α (x), α \in Clop T, v_{x} \in Σ .$

Then the set family $δ_{Clop T} = {φ (α) | α \in Clop T}$ generates an MV-topology denoted it by Ω_Σ. We then prove that (X, Ω) is MV-homeomorphic to (Σ, Ω_Σ), the MV-homeomorphic mapping is defined by $ψ : X \to Σ, ψ (x) \overset{def}{=} v_{x} .$

Obviously, ψ is a surjective. We now prove that ψ is an injective. Let x ≠ y ∈ X. By the Hausdorff separability of $T$ , there exist α, β ∈ Ω such that (1) x₁ ≤ α, y₁ ≤ β; (2) α (y) = β (x) =0; (3) α ⊗ β = 0. Hence v_x (α) = α (x) =1, v_x (β) =0 and v_y (α) =0, v_y (β) = β (y) =1. This means that v_x ≠ v_y, and so ψ (x) ≠ ψ (y). This shows that ψ is an injective.

It remains to prove that ψ and ψ^-1 are MV-continuous. For any MV-open set ξ in Ω_Σ, there exist some set families {φ (α_i)} _i∈I such that ξ = ⋁ _i∈Iφ (α_i). Hence $\begin{matrix} ψ^{\leftarrow} (ξ) (x) & = & ξ (ψ (x)) = (⋁_{i \in I} φ (α_{i})) (ψ (x)) \\ = & ⋁_{i \in I} φ (α_{i}) (v_{x}) = ⋁_{i \in I} α_{i} (x) \\ = & (⋁_{i \in I} α_{i}) (x) . \end{matrix}$

Obviously ⋁_i∈Iα_i ∈ Ω, this means that the inverse image of every MV-open set in Ω_Σ is an MV-open set in Ω. This shows that ψ is MV-continuous.

Let η ∈ Ω. Since (X, Ω) is a Stone MV-space, there exist some ${α_{1}, \dots, α_{n}} \in Clop T$ such that η = α₁ ⊕ ⋯ ⊕ α_n. $\begin{matrix} (ψ^{- 1})^{\leftarrow} (η) (v_{x}) = η (ψ^{- 1} (v_{x})) = η (x) \\ = (α_{1} \oplus \dots \oplus α_{n}) (x) = α_{1} (x) \oplus \dots \oplus α_{n} (x) \\ = φ (α_{1}) (v_{x}) \oplus \dots \oplus φ (α_{1}) (v_{x}) \\ = (φ (α_{1}) \oplus \dots \oplus φ (α_{n})) (v_{x}) . \end{matrix}$

Obviously φ (α₁) ⊕ ⋯ ⊕ φ (α_n) ∈ Ω_Σ, this means that the inverse image of every MV-open set in Ω is an MV-open set in Ω_Σ. This proves that ψ^-1 is also MV-continuous.

6 An application to Lukasiewicz propositional logic system

In this section we prove the compactness of Lukasiewicz semantics and Lukasiewicz propositional logic system by using the compactness of topology $D$ .

Definition 6.1. Let Γ be a subset of MV-algebra M and Σ the Lukasiewicz semantic of M. We say that Γ is satisfiable if there exists a valuation v such that v (x) >0 for all x ∈ Γ. The set Γ is said to be finitely satisfiable if each finite subset of Γ is satisfiable. We say that Lukasiewicz semantics Σ of MV-algebra M is compact if its satisfiability is equivalent to finite satisfiability.

Theorem 6.2. Lukasiewicz semantic Σ is campact.

Proof. Let Γ be a subset of M. For an element x ∈ M, it is easy to check that $\begin{matrix} {v | v \in Σ, v (x) > 0} = {v | v \in Σ, v (x^{'}) < 1} \\ = Σ - {v | v \in Σ, v (x^{'}) = 1} = Σ - ⌈ x^{'} ⌋^{*} . \end{matrix}$

By Proposition 4.5 we know that Σ - ⌈ x′ ⌋ ^* is a closed set. Hence the collection {Σ - ⌈ x′ ⌋ ^* | x ∈ Γ} is a centered closed sets family (i.e. each finite subset has a nonempty intersection). Since $(Σ, D)$ is a compact Hausdorff topological space, we have that the intersection {Σ - ⌈ x′ ⌋ ^* | x ∈ Γ} is nonempty. It follows that there is at least one v ∈ ⋂ _x∈Γ (Σ - ⌈ x′ ⌋ ^*). This shows that there exists v ∈ Σ such that v (x′) <1 for any x ∈ Γ, and then Γ is satisfiable.

Suppose that S = {p₁, p₂, ⋯} and F (S) is the free algebra of type (¬ , ∨ , →) generated by S, where ¬ is an unary operator, ∨ and → are binary operators, then elements of S are atomic propositions (atomic formulas), elements of F (S) are propositions (formulas, or, wffs). Let [0, 1] be an MV-unit interval. A valuation v of F (S) is a homomorphism v : F (S) → [0, 1] of type (¬ , ∨ , →), v (A) is the valuation of A with respect to v. The set consisting of all valuations of F (S) is denoted by Σ.

Let A, B ∈ F (S). If v (A) = v (B) for all v ∈ Ω then we say that A and B are logically equivalent, denote it by A ≈ B. It is easy to verify that ≈ is a congruence relation with respect to (¬ , ∨ , →) and the quotient algebraF (S)/≈ is an MV-algebra, call it Lindenbaum MV-algebra and denote it by M_L [7]. For any A ∈ F (S), if we employ [A] to denote the congruence class of A, then v has constant value on [A]. Hence we can define a valuation v^* : M_L → [0, 1] as follows: $v^{*} ([A]) \overset{def}{=} v (A), [A] \in M_{L} .$

Then v^* is a homomorphism of type (¬ , ∨ , →), for briefly, v^* is called a valuation of M_L and is also denoted by v. The set consisting of all valuations of M_L is denoted by Σ. According to the discussion in Section 3, we can get an MV-topological space (Σ, Ω), and call it a Lukasiewicz logic space.

Definition 6.3. Let A ∈ F (S), Γ ⊆ F (S) and v ∈ Σ. If v (A) =1 then v is called a model of A. If v (A) =1 for every A ∈ Γ then v is called a model of Γ.

Theorem 6.4. Lukasiewicz logic is compact, i.e., let Γ ⊆ F (S), if there exist some models for every finite subset of Γ then there exists a model for Γ.

Proof. Suppose that there exist some models for every finite subset of Γ. Then for every finite formulas set {A₁, A₂, ⋯ , A_n} ⊆ Γ, $\begin{matrix} {v | v \in Ω, v ([A_{1}]) \\ = v ([A_{2}]) = \dots = v ([A_{n}]) = 1} \\ = ⋂_{i = 1}^{n} (Σ - ⌈ [A_{i}] ⌋^{*}) \neq \emptyset . \end{matrix}$

By Proposition 4.5 we know that Σ - ⌈ [A_i] ⌋ ^* is a closed set. Hence the collection {Σ - ⌈ [A] ⌋ ^* | A ∈ Γ} is a centered closed sets family (i.e. each finite subset has a nonempty intersection). Since $(Σ, D)$ is compact Hausdorff topological space, we have that the intersection {Σ - ⌈ [A] ⌋ ^* | A ∈ Γ} is nonempty. It follows that there is v ∈ ⋂ _A∈Γ (Σ - ⌈ [A] ⌋ ^*). This shows that v (A) =1 for any A ∈ Γ, and v is a model for Γ.

7 Conclusion

In this paper, we focus on establishing Lukasiewicz semantic MV-topological space (Σ, Ω_Σ) and investigate in detail its topological properties, such as compactness, separability etc. We prove that the Lukasiewicz semantic MV-topological space (Σ, Ω_Σ) is a compact zero dimension Hausdorff MV-topological space and is also a N-compact fuzzy topological space in Theorems 3.4 and 3.5 respectively. We also establish a classical topological space $(Σ, D)$ on the valuations set Σ, discuss the topological properties of it and prove that the topology $D$ is finer than the cut topology $C$ which is generated by Lukasiewicz semantic MV-topology Ω_Σ in Theorem 4.10. We prove the representation theorem for MV-algebras that a σ-complete lattice is an MV-algebra if and only if M is isomorphic to an MV-clopen set lattice of a Stone MV-space, and then the Stone representation for Boolean algebras is only an example of the representation theorem for MV-algebras. We give the characterization of Stone MV-space, which any Stone MV-space is homeomorphic to a Lukasiewicz semantic MV-topological space in Theorem 5.3. As an application, we use the compactness of topology $D$ to prove the compactness of Lukasiewicz semantics and Lukasiewicz propositional logic system.

Footnotes

Acknowledgments

The work of this paper has been supported by the Key Science Research Project of Hunan Province Education Department (No. 2014A135), Natural Science Fund Project in Hunan Province (No. 16JJ6138), Social Sciences Fund Project in Hunan Province (No. 16YBA329), Construction Program of the Key Discipline in Hunan Province.

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