A representation theorem for infinite fuzzy distributive lattices

Abstract

In this paper, we show that the category of infinite fuzzy Priestley spaces is equivalent to the dual of the category of infinite fuzzy distributive lattices. A representation theorem for the infinite fuzzy distributive lattices is also given.

Keywords

Fuzzy ordered relation fuzzy ordered set fuzzy lattice fuzzy Priestley space homomorphism of fuzzy lattices homomorphism of fuzzy Priestley spaces

1 Introduction

The representation theorems appeared in the thirties of the last century; M. Stone [12] proved that every Boolean algebra is isomorphic to a set of {I_a : a∈ A } (where I_a denotes the set of prime ideals of A not containing a). The representation theorem for distributive lattices proved by Birkhoff [2]; asserts that any finite distributive lattice L is isomorphic to the lattice of the ideals of the partial order of the join-irreducible elements of L.

H. Priestly developed another kind of duality for bounded distributive lattices see [9, 10]. Such representation theorems enable a deep and a concrete comprehension of the lattices as well their structures.

The duality is central in making the link between syntactical and semantic approaches to logic, also in theoretical computer science this link is central as the two sides correspond to specification languages and the space of computational states. This ability to translate faithfully between algebraic specification and spatial dynamics has often proved itself to be a powerful theoretical tool as well as a handle for making practical problems decidable.

Topological duality for Boolean algebras [11] and distributive lattices [12] is a useful tool for studying relational semantics for propositional logics. Canonical extensions [4 –7], provide a way of looking at these semantics algebraically.

Priestley’s duality for bounded distributive lattices has enjoyed growing attention and has been variously applied in the international literature since its inception in 1970.

After the introduction of the notion of fuzzy relations by Zadeh [14], many concepts and results from the theory of ordered sets have been extended to the fuzzy sets. Venugopalan [13] introduced a definition of a fuzzy ordered set (foset), then extended it to obtain a fuzzy lattice, on which a fuzzy relation is a generalization of an equivalence relation.

Another approach was proposed by Chon [3]. Hence, a fuzzy lattice is defined to be just a fuzzy set equipped with a fuzzy ordering relation, albeit the simple definition, many interesting properties of these lattices were deduced [3].

In [1], Amroune and Davvaz gave a representation theorem for finite fuzzy distributive lattices.

The aim of this paper is to extend some results in [1 , 10]; more precisely, a representation theorem of fuzzy distributive lattices in the infinite case is presented. The next section is devoted to some basic notions and definitions on which the remaining text rests on. In the third section, we give and prove the main result using a definition of fuzzy ordering admitting the minimum t-norm. At the end of this section, several examples are given. The paper is ended by some open questions.

2 Preliminaries

In this section, we recall some definitions and concepts needed in the sequel.

Let X be a non-empty set. A mapping R : X × X → [0, 1] is called a fuzzy binary relation on X.

A fuzzy binary relation R on X is called

Reflexive, if R (x, x) =1, for all x ∈ X.

Antisymmetric, if R (x, y) ∧ R (y, x) =0 whenever x ≠ y, for all x, y ∈ X.

Transitive, if R (x, y) ∧ R (y, z) ≤ R (x, z), for all x, y, z ∈ X.

A reflexive, antisymmetric and transitive fuzzy relation is called a fuzzy partial ordering relation. A set equipped with a fuzzy order relation is called a fuzzy ordered set (foset). The height of R, denoted by h (R), is defined by h (R) = ∨ _{{(x,y)∈X²:x≠y}}R (x, y) .

2.1 Fuzzy lattices

The terminology used in this paper is the same as in [3, 13], where all the necessary definitions and results may be found.

Definition 1. [3] Let (X, R) be a fuzzy poset and let A be a subset of X . An element u ∈ X is said to be an upper bound of A if and only if R (a, u) >0 for all a ∈ A. An upper bound u₀ of A is the least upper bound of A if and only if R (u₀, u) >0 for every upper bound u of A. An element l ∈ X is said to be a lower bound of A if and only if R (l, a) >0 for all a ∈ A. A lower bound l₀ of A is the greatest lower bound of A if and only if R (l, l₀) >0 for every lower bound l of A.

The least upper bound and the greatest lower bound of the set {x, y} are denoted by x ∨ y and x ∧ y respectively.

Proposition 2. [3] Let (X, R) be a fuzzy lattice. For any x, y, z ∈ X, it follows that,

R (x, x ∨ y) >0, R (y, x ∨ y) >0, R (x ∧ y, x) >0 and R (x ∧ y, y) >0.

R (x, z) >0 and R (y, z) >0 implies R (x ∨ y, z) >0.

R (z, x) >0 and R (z, y) >0 implies R (z, x ∧ y) >0.

R (x, y) >0 if and only if x ∨ y = y.

R (x, y) >0 if and only if x ∧ y = x.

If R (y, z) >0, then R (x ∧ y, x ∧ z) >0 and R (x ∨ y, x ∨ z) >0.

Proposition 3. [3] Let (X, R) be a fuzzy lattice. For any x, y, z ∈ X, it follows that,

x ∨ x = x, x ∧ x = x.

x ∨ y = y ∨ x, x ∧ y = y ∧ x.

(x ∨ y) ∨ z = x ∨ (y ∨ z), (x ∧ y) ∧ z = x ∧ (y ∧ z).

(x ∨ y) ∧ x = x, (x ∧ y) ∨ x = x.

The following definition give a characterizations of fuzzy distributive lattices.

Definition 4. [3] Let (X, R) be a fuzzy lattice. (X, R) is distributive if and only if x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) and (x ∨ y) ∧ (x ∨ z) = x ∨ (y ∧ z).

2.1.1 Fuzzy lattices isomorphism

We recall the following definition, see [1] and [13].

Definition 5. [13] Let (L, r, ∧ , ∨), and (M, R, ∧ , ∨) be two fuzzy lattices.

A function f : L → M is called monotone if for all x, y ∈ L, r (x, y) ≤ R (f (x) , f (y)).

Let f : L → M be monotone function between fuzzy lattices. Then f is called a lattice homomorphism if for any x, y ∈ L, f (x ∧ y) = f (x) ∧ f (y), and f (x ∨ y) = f (x) ∨ f (y). If f is a bijection, then f is said to be fuzzy lattices isomorphism.

Let (X, R) be a foset. A subset E of X is called increasing, if for all x belongs to E and R (x, y) >0 (y is an upper bound of x), then y belongs to E. A decreasing set is defined dually. A fuzzy ordered space is a triplet (X, τ, R), where X is a nonempty set, τ is a topology on X and R is a fuzzy order on X.

A fuzzy ordered space (X, τ, R) is called totally order disconnected if for x, y ∈ X and R (x, y) =0, there exists an increasing τ-clopen U and a decreasing τ-clopen V, such that U ∩ V = φ, with x ∈ U and y ∈ V. We recall that a clopen set in a topological space is a set which is both open and closed. A fuzzy ordered space (X, τ, R) is called a fuzzy Priestley space, if it is compact and totally order disconnected.

3 Priestley duality for fuzzy distributive lattices

Throughout this section, all fuzzy lattices are closed distributive lattices (fuzzy closed distributive lattices), and homomorphisms preserve the smallest element (denoted 0) and the greatest element (denoted 1). If (A, ∧ , ∨ , R) is a fuzzy distributive lattice, then its dual space is defined by T (A) = (X, τ, R₁), where X is the set of 0 - 1 homomorphisms from A onto {0, 1}, τ be the topology induced on X by the product topology on {0, 1 } ^A and R₁ is a fuzzy order adequately chosen on X.

If δ = (X, τ, r) is a fuzzy Priestley space, then its dual is defined by: (L (δ) , ∨ , ∧ , r₁), where L (δ) ={ Y ⊆ X/Y isincreasingand τ - clopen } and r₁ is a fuzzy order adequately chosen.

Lemma 6. If (A, ∨ , ∧ , R) is a fuzzy distributive lattice, then there exist two fuzzy orders R₁, R₂ such that

T (A) = (X, τ, R₁) is a fuzzy Priestley space,

(L (T (A)) , ∨ _R
₂, ∧ _R
₂, R₂) is a fuzzy distributive lattice.

Proof. Let R₁ be the relation defined by

{\begin{array}{l} 1 f = g, \\ \lor a \in g^{- 1} \underset{a \neq b}{(1), b \in} f^{- 1} (1) R (a, b) f^{- 1} (1) \subset g^{- 1} (1) \\ 0 otherwise . \end{array},

We show that R₁ is a fuzzy order. Since R₁ (f, f) =1 for all f ∈ X, it follows that R₁ is reflexive. For all f, g ∈ X such that f ≠ g, we have R₁ (f, g) ∧ R₁ (g, f) =0 (it suffices to apply the definition of R₁). Hence, R₁ is antisymmetric. Now, for all f, g, h ∈ X we show that R₁ (f, g) ∧ R₁ (g, h) ≤ R₁ (f, h). Indeed, there are eight possible relations between f^-1 (1) , g^-1 (1) , h^-1 (1) in term of inclusion. Let P be the proposition: R₁ (f, g) ∧ R₁ (g, h) ≤ R₁ (f, h). Let A = (f^-1 (1) ⊆ g^-1 (1)), B = (g^-1 (1) ⊆ h^-1 (1)) and C = (f^-1 (1) ⊆ h^-1 (1)) .

The only case that requires an investigation is f^-1 (1) ⊆ g^-1 (1) and g^-1 (1) ⊆ h^-1 (1). Since $\lor_{\underset{a \neq b}{a \in g^{- 1} (1), b \in f^{- 1} (1)}} R (a, b) \leq \lor_{\underset{c \neq b}{c \in h^{- 1} (1), b \in f^{- 1} (1)}} R (c, b),$ where g^-1 (1) ⊆ h^-1 (1), then it follows that $\lor_{\underset{a \neq b}{a \in g^{- 1} (1), b \in f^{- 1} (1)}} R (a, b) \land \lor_{\underset{c \neq a}{c \in h^{- 1} (1), a \in g^{- 1} (1)}} R (a, c) \leq \lor_{\underset{c \neq b}{c \in h^{- 1} (1), b \in f^{- 1} (1)}} R (c, b) .$ Hence for all f, g, h ∈ X, the inequality R₁ (f, g) ∧ R₁ (g, h) ≤ R₁ (f, h) holds, and R₁ is transitive. It follows that R₁ is a fuzzy order and according to [9, 10], T (A) = (X, τ, R₁) is a fuzzy Priestley space. This completes the proof of (1). For the second case (2), let M₀ =∧ _x∈A ∧ _y∈A { R (x, y)/x ≠ y and R (x, y) >0 } and let R₂ the relation defined by $R_{2} (H, D) = {\begin{matrix} 1 & if H = D, \\ Max (M_{0}, δ) & if H \subset D and H \neq φ, \\ M_{0} & if H = φ, \\ 0 & otherwise . \end{matrix}$ where $δ = \lor_{\underset{(a \neq b)}{a \in \cup_{f \in H} f^{- 1} (1), b \in \cup_{g \in D} g^{- 1} (1)}} R (a, b) .$ Note that both ∪_f∈Hf^-1 (1) and ∨_(a≠b)R (a, b) exist. First, we show that R₂ is a fuzzy order. For the reflexivity, we have R₂ (H, H) =1. To prove the antisymmetric property of R₂, let H, D ∈ L (T (A)). Clearly, R₂ (H, D) ∧ R₂ (D, H)) =0 whenever H = D, i.e., R₂ is antisymmetric. In order to verify the transitivity, we will use the following truth table, where P is the proposition (R₂ (H, D) ∧ R₂ (D, E)) ≤ R₂ (H, E)). First, if one of the three elements H, D, E is empty, then the transitivity holds trivially.

If H ≠φ and D ≠ φ and E ≠ φ, the only case that needs investigation is when H ⊂ D and D ⊂ E. This yields to Max (M₀, α) ∧ Max (M₀, β) = Max (M₀, α ∧ β) ≤ Max (M₀, δ) .

where, $α = \lor_{\underset{(a \neq b)}{a \in \cup_{f \in H} f^{- 1} (1), b \in \cup_{g \in D} g^{- 1} (1)}} R (a, b), β = \lor_{\underset{(b \neq c)}{b \in \cup_{g \in D} g^{- 1} (1), c \in \cup_{h \in E} h^{- 1} (1),}} R (b, c)$ and $δ = \lor_{\underset{(a \neq c)}{a \in \cup_{f \in H} f^{- 1} (1), c \in \cup_{h \in E} h^{- 1} (1)}} R (a, c) .$ This inequality comes from $\lor_{\underset{(a \neq b)}{a \in \cup_{f \in H} f^{- 1} (1), b \in \cup_{g \in D} g^{- 1} (1)}} R (a, b) \leq \lor_{\underset{(a \neq c)}{a \in \cup_{f \in H} f^{- 1} (1), c \in \cup_{h \in E} h^{- 1} (1)}} R (a, c),$ where ∪_g∈Dg^-1 (1) ⊂ ∪ _h∈Eh^-1 (1) . Hence R₂ is transitive. Finally, the least upper and greatest lower bounds of H and D (with respect to the relation R₂) are denoted by H ∨ _{R
₂}D and H ∧ _{R
₂}D, respectively. We prove that H ∨ _{R
₂}D = H ∪ D and H ∧ _{R
₂}D = H ∩ D. It is not difficult to see that H ∪ D is an upper bound of {H, D} since R₂ (H, H ∪ D) >0 and R₂ (D, H ∪ D) >0. If C is the least upper bound of {H, D}, then we have four cases:

if H = φ and D = φ, then R₂ (C, H ∪ D) = R₂ (C, φ), which is different from 0 if and only if C = φ . Hence C = H ∪ D.

if H = φ, D ≠ φ and D ⊂ C. Since R₂ (D, C) >0, we have R₂ (C, H ∪ D) = R₂ (C, D), which is different from 0 if and only if C ⊂ D, hence C = D = H ∪ D .

if H ≠ φ, D = φ and H ⊂ C. Since R (H, C) >0, it follows that R₂ (C, H ∪ D) = R (C, H), which is equivalent to C ⊂ H. Hence C = H = H ∪ D .

if H ≠ φ and D ≠ φ, then we have H ⊂ C and D ⊂ C; this implies that H ∪ D ⊂ C, it follows that R₂ (C, H ∪ D) is different from 0 if and only if C ⊂ H ∪ D. Hence C = H ∪ D.

Similarly, we proof that H ∧ _{R
₂}D = H ∩ D. It is known that H ∩ D and H ∪ D are increasing and τ-clopens. This shows that (L (T (A)) , ∨ _{R
₂}, ∧ _{R
₂}, R₂) is a fuzzy distributive lattice. ■

Lemma 7. If δ = (X, τ, r) is a fuzzy Priestley space, then there exist two fuzzy orders r₁ and r₂ such that

(L (δ) , ∨ _r
₁, ∧ _r
₁, r₁) is a fuzzy distributive lattice.

(T (L (δ)) , τ, r₂) is a fuzzy Priestley space.

Proof. If h (r) =0, then X is an antichain and we can write r₁ as follows: $r_{1} (A, B) = {\begin{matrix} 1 & if A = B, \\ α (α \in] 0, 1 [) & if A \subset B, \\ 0 & otherwise . \end{matrix}$

It is not difficult to see that r₁ is a fuzzy order, A ∨ _{r
₁}B = A ∪ B and A ∧ _{r
₁}B = A ∩ B exist for every A and B in L (δ). Therefore A ∪ B and A ∩ B are increasing and τ-clopen sets of L(δ), where (L (δ) , ∨ _{r
₁}, ∧ _{r
₁}, r₁) is a fuzzy distributive lattice.

If h (r) ≠0, then X is not an antichain and we can choose

m₀ = ∧ _x∈X ∧ _y∈X {r (x, y)/x ≠ y and r (x, y) > 0} , clearly m₀ ≠ 0, we can set $\begin{array}{l} r_{1} (A, B) \\ = {\begin{array}{l} 1 & if A = B, \\ M a x (m_{0}, \lor (a \neq b) a \in A, b \in B r (a, b)) & if A \subset B and A \neq ϕ, \\ m_{0} & if A \neq ϕ, \\ 0 & otherwise . \end{array} \end{array}$

Similarly to Lemma 6, r₁ is a fuzzy order and we can assume that A ∨ _{r
₁}B = A ∪ B and A ∧ _{r
₁}B = A ∩ B for every A and B in L (δ), where (L (δ) , ∨ _{r
₁}, ∧ _{r
₁}, r₁) is a fuzzy distributive lattice. This completes the proof of (1).

To prove the second assertion

$α = \underset{\underset{a \neq b}{a \in \cap_{A \in f^{- 1} (1)} A, b \in \cap_{B \in g^{- 1} (1)} B}}{\lor} r (a, b)$ for all f and g in T (L (δ)) let $\begin{matrix} r_{2} (f, g) \\ = {\begin{matrix} 1 & if f = g, \\ α & if f^{- 1} (1) \subset g^{- 1} (1) and \cap_{B \in g^{- 1} (1)} B \neq φ, \\ m_{0} & if \cap_{B \in g^{- 1} (1)} B = φ, \\ 0 & otherwise . \end{matrix} \end{matrix}$

We will show that the choice of $\lor_{\underset{a \neq b}{a \in \cap_{A \in f^{- 1} (1)} A, b \in \cap_{B \in g^{- 1} (1)} B}} r (a, b)$ is possible.

Firstly, it is not difficult to see that r₂ is well defined since the closed set ∩_A∈f^-1(1)A exists in τ.

Secondly,

∩_A∈f^-1(1)A ≠ φ, since f^-1 (1) ⊂ g^-1 (1) which implies that ∩_B∈g^-1(1)B ⊂ ∩ _A∈f^-1(1)A

with ∩_B∈g^-1(1)B ≠ φ, i.e. ∩_A∈f^-1(1)A ≠ φ.

We show that ∩_A∈f^-1(1)A ≠ ∩ _B∈g^-1(1)B .

Suppose that ∩_A∈f^-1(1)A = ∩ _B∈g^-1(1)B . If f^-1 (1) ={ Y_i : i ∈ I } and g^-1 (1) ={ Y_i : i ∈ I } ∪ { Zi : j ∈ J }, where Z_j ≠ Y_i for all i ∈ I and j ∈ J. Then ∩_i∈IY_i = ∩ _i∈IY_i ∩∩ _j∈JZ_j, which implies that ∩_i∈IY_i ⊂ Z_j, for all j ∈ J. Let j₀ be a fixed element of J. Then ∩_i∈IY_i ∩ Z_{j
₀} = ∩ _i∈IY_i. On the other hand we have ∩_i∈IY_i ⊂ Z_{j
₀}, where Z_{j
₀} is increasing and τ-clopen, which implies that $Z_{j_{0}}^{c} \subset \cup_{i \in I} Y_{i}^{c}$ . Since $Z_{j_{0}}^{c}$ is τ-clopen, it is compact (see [8, page 76]). Therefore $Z_{j_{0}}^{c} \subset \overset{n}{\underset{i = 1}{\cup}} Y_{i}^{c}$ , this implies that $\overset{n}{\underset{i = 1}{\cap}} Y_{i} \subset Z_{j_{0}}$ , hence $f (Z_{j_{0}}) \geq f (\overset{n}{\underset{i = 1}{\cap}} Y_{i}) = \underset{i = 1}{\overset{n}{\land}} f (Y_{i}) = 1$ , which yields Z_{j
₀} ∈ f^-1 (1). This is impossible. Hence, ∩_A∈f^-1(1)A ≠ ∩ _B∈g^-1(1)B .

To show that r₂ is a fuzzy order, we have r₂ (f, f) =1 for all f ∈ X, then r₂ is reflexive. For all f, g ∈ T (L (δ)) , clearly r₂ (f, g) ∧ r₂ (g, f) =0, whenever f = g, then r₂ is antisymmetric.

To verify the transitivity, we use the following truth table, where the proposition P is (r₂ (f, g) ∧ r₂ (g, h) ≤ r₂ (f, h)), for all f, g, h ∈ T (L (δ)); also ∗ means can not occur and $φ_{1}^{- 1} \subset ψ_{1}^{- 1} is φ^{- 1} (1) \subset ψ^{- 1} (1) .$

As shown above, the only case who merits to be checked is f^-1 (1) ⊂ g^-1 (1) and g^-1 (1) ⊂ h^-1 (1).

If ∩_A∈f^-1(1)A = φ or ∩_B∈g^-1(1)B = φ or ∩_C∈h^-1(1)C = φ, the transitivity is trivial.

If ∩_A∈f^-1(1)A ≠ φ and ∩_B∈g^-1(1)B ≠ φ and ∩_C∈h^-1(1)C ≠ φ,

Let $α = \lor_{\underset{(a \neq b)}{a \in \cap_{A \in f^{- 1} (1)} A, b \in \cap_{B \in g^{- 1} (1)} B}} r (a, b), β = \lor_{\underset{(b \neq c)}{b \in \cap_{B \in g^{- 1} (1)} B, c \in \cap_{C \in h^{- 1} (1)} C}} r (b, c)$ and $γ = \lor_{\underset{(a \neq c)}{a \in \cap_{A \in f^{- 1} (1)} A, c \in \cap_{C \in h^{- 1} (1)} C}} r (a, c),$ we have α ∧ β ≤ γ, since β ≤ γ, where ∩_B∈g^-1(1)B ⊂ ∩ _A∈f^-1(1)A . Hence, r₂ is transitive. Furthermore, (T (L (δ)) , τ, r₂) is a fuzzy Priestley space. ■

In this section we extend the result obtained in [1] to the case of infinite fuzzy bounded distributive lattices.

Theorem 8.

Let A be a fuzzy distributive lattice. The map F_A : A → L (T (A)) defined by F_A (a) ={ f ∈ X : f (a) =1 } is a fuzzy lattice isomorphism.

If δ = (X, τ, r) is a Priestley space, then the map

G : δ → T (L (δ)) defined by $G (x) (Y) = {\begin{array}{l} 1 & i f x \in Y, \\ 0 & i f x \notin Y . \end{array}$ for all Y ∈ L (δ), is an isomorphism of fuzzy Priestley space, i.e., a bijection, continuous and increasing map.

If f : A₁ → A₂ is a fuzzy lattice homomorphism, then the map T (f) : T (A₁) → T (A₂) defined by T (f) (g) = g ∘ f, is a homomorphism of fuzzy Priestley space, i.e., a continuous and increasing map.

If h : δ₁ → δ₂ is a homomorphism of fuzzy Priestley space, then the map L (h) : L (δ₂) → L (δ₁) defined by L (h) (y) = h^-1 (y) for every y ∈ L (δ₂), is a fuzzy lattice homomorphism.

If f and h are as in (3) and (4), then the following diagrams

and

are commutative.

Proof. (1) To show that the map F_A (a) ={ f ∈ X : f (a) =1 } is a fuzzy lattice isomorphism, it suffices to prove that R (x, y) ≤ R₂ (F_A (x) , F_A (y)) for all x, y ∈ A. For this, setting $β = M a x (M_{0}, \lor_{\underset{(a \neq b)}{a \in \cup_{f \in F_{A} (x)} f^{- 1} (1), b \in \cup_{g \in F_{A} (y)} g^{- 1} (1)}} R (a, b))$

and $R_{2} (F_{A} (x), F_{A} (y)) = {\begin{matrix} 1 & if F_{A} (x) = F_{A} (y), \\ β & if F_{A} (x) \subset F_{A} (y), \\ 0 & otherwise . \end{matrix}$

Note that F_A (x) ≠ φ for all x ∈ A. If x = y, then R (x, y) = R₂ (F_A (x) , F_A (y)). If x ≠ y, we consider two cases

If R (x, y) =0, then we have R (x, y) ≤ R₂ (F_A (x) , F_A (y)).

If R (x, y) >0, then F_A (x) ⊂ F_A (y), which implies that R (x, y) ≤ R₂ (F_A (x) , F_A (y)).

(2) According to [9] and [10] it suffices to show that r (x, y) ≤ r₂ (G_δ (x) , G_δ (y)) for all x, y ∈ δ

r_{2} (G_{δ} (x), G_{δ} (y)) = {\begin{matrix} 1 & if G_{δ} (x) = G_{δ} (y), \\ θ & if G_{δ}^{- 1} (x) \subset G_{δ}^{- 1} (y), \\ 0 & otherwise . \end{matrix}

where

θ = \lor_{\underset{a \neq b}{a \in \cap_{A \in G_{δ}^{- 1} (x) (1)} A, b \in \cap_{B \in G_{δ}^{- 1} (y) (1)} B}} r (a, b)

If x = y, then r₂ (G_δ (x) , G_δ (y)) = r (x, y) = 1 . Otherwise, we consider the two following cases

If r (x, y) =0, then we have r (x, y) ≤ r₂ (G_δ (x) , G_δ (y)).

If r (x, y) >0, y belongs to each τ-clopen which contains x. Hence, $G_{δ}^{- 1} (x) \subset G_{δ}^{- 1} (y)$ , which implies that $r_{2} (G_{δ} (x), G_{δ} (y)) = \lor_{\underset{a \neq b}{a \in \cap_{A \in G_{δ}^{- 1} (x) (1)} A, b \in \cap_{B \in G_{δ}^{- 1} (y) (1)} B}} r (a, b) \geq$ r (x, y) .

The remaining assertions are obtained by the same reasoning. ■

Example 9. Let (A, ∨ , ∧ , R) be a fuzzy distributive lattice, where A ={ a, b, c, d, e, f } and R be a fuzzy relation defined by.

Then, its dual is The set of 0 - 1 homomorphisms from A onto {0, 1}={f₁, f₂, f₃, f₄} where f₁, f₂, f₃, f₄ are given by

and R₁ is given by

which have the bidual as follows

L (T (A)) = {φ, { f₃ } , { f₁, f₃ } , { f₂, f₃ } , {f₁, f₂, f₃} , X} where R₂ is given by

Finally, F_A : A → L (T (A)) is given by

Example 10. Let (X, τ, r) be a Priestley space, where X ={ x, y, z } and r is given by

Then L (X) = {φ, { x } , { y } , { z } , { x, y } , { x, z } , { y, z } , X} and r₁ is given by $r_{1} (A, B) = {\begin{matrix} 1 & if A = B; \\ α (α \in] 0, 1 [) & if A \subset B; \\ 0 & otherwise . \end{matrix}$

Then r₁ is given by

The set T (L (X)) is equal to {f₁, f₂, f₃ } .

r₂ is given by

The isomorphism G_X : X → T (L (X)) is defined as follows

Example 11. Let (X, τ, r) be a Priestley space, where X ={ x, y, z, t, s } and r is given by

and $L (X) = {\begin{matrix} φ, {s}, {t, s}, {x, t, s}, {y, t, s}, {z, t, s}, \\ {x, y, t, s}, {x, z, t, s}, {y, z, t, s}, X \end{matrix}},$ where r₁ is given by and T (L (X)) ={ f₁, f₂, f₃, f₄, f₅ } such that

The isomorphism G_X : X → T (L (X)) is defined as follows

4 Conclusion and open problems

In this paper, we have proposed a way to represent fuzzy distributive lattices. This, by constructing adequate fuzzy orders. In this context a theory of representation of fuzzy distributive lattices in the infinite case is presented. The main theorem (Theorem 8) shows that the category of infinite fuzzy Priestley spaces is equivalent to the dual of the category of infinite fuzzy bounded distributive lattices. The work in this paper suggests the following question: Is it possible to obtain such representation if we change the definition of fuzzy set? In other words, what happens if we replace the unite interval [0,1] by any closed distributive lattice L?

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