Lattice Isomorphism and L -homeomorphism

Abstract

This paper deals with lattice isomorphic L-topological spaces. We are concerned with a question: Under what conditions will a lattice isomorphic L-topological spaces be L-homeomorphic. We give contributions to this question in three different ways.

Keywords

lattice isomorphism

1 Introduction

Fuzzy set theory was first introduced by L.A. Zadeh [21] in 1965, and it provided a mathematical description of fuzziness. A fuzzy set on a set X is simply a mapping from X to [0, 1]. The concept of fuzzy set offers a new framework of set theory, and this new framework generalize many of the concepts of general topology which form the content of fuzzy topology. Later an extensive study of fuzzy topological space has been carried out by several mathematicians and they developed a theory of fuzzy topological spaces. Later mathematicians replaced [0, 1] by a complete lattice and introduced L-subset. Goguen in [8] proposed the natural generalization of fuzzy topology by substituting L-subsets for fuzzy sets.

Topology and Lattice Theory are two interrelated branches of mathematics that influence one another. In a topological space the collection of all open sets forms a lattice. So in addition to geometrical inputs, topology inextricably incorporates lattice theoretic material. The application of lattice theoretic ideas to research on topological spaces began with Marshall Stone’s work on the topological representation of Boolean algebras [14–16] and distributive lattices. Later many researchers continued this study of topological space using lattice theoretic ideas. Two topological spaces are lattice isomorphic if their corresponding lattices of open set are lattice isomorphic. One famous problem that connect topology and lattice theory is that “under what conditions two lattice isomorphic spaces are homemorphic?”. It is worth noting that, over the years several researchers like Thron [17], Finch [6] dealt with this problem in topology using different aspects.

Coming to L-topology, similar to topology L-topological space have a lattice structure. Let (X, δ) be an L-topological space, then the collection of all L-open sets and the collection of all L-closed sets forms lattices, we denote it by Γ (X) and Ω (X) respectively. So it is natural to study the relation between L-topological spaces and these lattice theoretical ideas. S.E. Rodabaugh’s paper [13] is undoubtedly the first systematic and important work on this respect. In Luo’s research [12] on lattices, the relation between L-topological spaces and lattices was also investigated. The Stone Representation Theorem for Boolean algebras, and its generalized form, the Stone Representation Theorem for distributive lattices, generated important influence in modern mathematics. Rodabaugh’s work [13] and the work [11] of Zhang and Liu established the L-topological representation of distributive lattices from different ways.

The concept of L-homeomorphism has been proved to be of fundamental importance in the field of L-topology. And as an area of mathematics, L-topology is concerned with all questions directly or indirectly related to L-homeomorphism. Main objective of this paper is to study L-homeomorphism in the basis of lattice theoretical ideas. Two L-topological spaces (X, δ) and (Y, η) are said to be lattice isomorphic, if there is a bijective map from Γ (X) to Γ (Y) which together with its inverse is order-preserving. So naturally there is a question, what additional conditions have to be imposed on lattice isomorphic spaces so that they are L-homeomorphic. This is one of the celebrated questions in topological space. Answers to this question may be regarded as a bridge between Lattice Theory and Topology. While this question of determining the conditions remains unsettled in fuzzy topology and L-topology, this paper is devoted to shed some light on this problem.

Let (X, δ) and (Y, η) be two L-topological spaces. Then it is clear that for any L-continuous function q : X → Y, the induced map Γ (q) : Γ (Y) → Γ (X), defined by Γ (q) (F) = q^-1 (F) is a lattice homomorphism. In this study, we also investigate the conditions under which this induced lattice-homomorphism is a lattice-isomorphism for an L-continuous function q : X → Y.

2 Preliminaries

In this section, we include certain definition and known results needed for the subsequent development of the study. Throughout this paper, X be a non empty ordinary set and (L,′) be a F-lattice, that is a complete, completely distributive lattice L equipped with an order-reversing involution ′.

Definition 2.1. [10] Let X be a non-empty ordinary set, L a complete lattice. An L-subsets on X is a mapping from X to L. The family of all L-subsets on X is called L-space and it is denoted by L^X. From the partial order on L, there is a partial order on L^X given by A ≤ B ⇔ A (x) ≤ B (x) , ∀ x ∈ X. Under this partial order L^X is also a complete lattice.

Definition 2.2. [10] Let X be a non-empty ordinary set, L a complete lattice and A ∈ L^X. Then support of A, denoted by supp (A), is defined as supp (A) = {x ∈ X : A (x) >0}.

Definition 2.3. [10] An L-point on X with support x and value λ is an L-subset x_λ ∈ L^X defined for every y ∈ X by $x_{λ} (y) = {\begin{matrix} λ & if y = x \\ 0 & otherwise . \end{matrix}$

The collection of all L-points on L^X is denoted by Pt (L^X). An L-point x_λ belongs to an L-subset u in X if only if λ ≤ u (x). In this case, we shall use the notation x_λ ∈ u and otherwise x_λ ∉ u

Definition 2.4. [10] Let L^X, L^Y be two L-spaces, f : X → Y an ordinary mapping. Then for any L-subset A of X, f (A) is an L-subset in Y defined by

$f (A) (y) = \sup {A (x) : x \in X, f (x) = y}, \forall A \in L^{X}, \forall y \in Y$

For an L-subset B in Y, f^-1 (B) is an L-subset of X defined by $f (B) (x) = B (f (x)), \forall B \in L^{Y}, \forall x \in X$

Definition 2.5. [10] Let X be a non-empty ordinary set and (L,′) be a F-lattice and δ ⊆ L^X. Then δ is called an L-topology on X and (X, δ) is called an L-topological space, if δ satisfies the following three conditions:

$\underline{0}, \underline{1} \in δ$

If $A \subseteq δ$ then $\lor A \in δ$

If U, V ∈ δ then U ∧ V ∈ δ.

Every element in δ is called as an L-open subset, complement of an L-open set is called as an L-closed subset. The smallest closed L-subset containing A is called the closure of A denoted by $\bar{A}$ .

Definition 2.6. [10] Let (X, δ) be an L-topological space, x_α ∈ L^X and A, B ∈ L^X. Then we say:

x_α quasi-coincides with A, denoted by $x_{α} \hat{q} A$ , if αnleqA ′ (x);

A quasi-coincides with B at x if A (x) nleqB ′ (x);

A quasi-coincides with B if A is quasi-coincident with B at some point x ∈ X.

Definition 2.7. [7] Let L be a lattice and a, b, c ∈ L then,

(i) c ≠ 1 is called a meetirreducibleelement if whenever c = a ∧ b then either c = a or c = b.

(ii) c ≠ 1 is called a primeelement, if whenever c ≥ a ∧ b then either c ≥ a or c ≥ b.

The dual notions co-prime, join-irreducible are dually defined. A join irreducible element of L is also called a molecule in L. The collection of molecules in L and the collection of prime elements in L are denoted by M (L) and pr (L) respectively.

Definition 2.8. [10] Let (X, δ) be an L-topological space. Then

(X, δ) is quasi-T₀, if for every distinguished mole-cules x_λ and x_μ in L^X with same support point x, U ∈ Q (x_λ) such that x_μ ≤ U ′ or there exist V ∈ Q (x_μ) such that x_λ ≤ V ′.

(X, δ) is sub-T₀ if for every distinguished ordinary points x, y in X, there exist λ ∈ M (L) such that U ∈ Q (x_λ) exists satisfying y_λ ≤ U ′.

(X, δ) is T₀ if for every distinguished molecules x_λ and y_μ in L^X, there exist U ∈ Q (x_λ) such that y_μ ≤ U ′ or there exist V ∈ Q (y_μ) such that x_λ ≤ V ′.

(X, δ) is T₁ space, if for every two distinguished molecules x_λ and y_μ in L^X such that x_λ≰y_μ, there exist U ∈ Q (x_λ) such that y_μ ≤ U ′.

(X, δ) is called T₂ if for every molecule x_λ and y_μ in L^X with distinguished supporting points x ≠ y, there exist U ∈ Q (x_λ) and V ∈ Q (y_μ) such that $U \land V = \underline{0}$ .

Theorem 2.1. [10] (X, δ) is T₁L-topological space if and only if every molecules in X is L-closed.

Definition 2.9. [10] Let (X, δ) , (Y, η) be L-topological spaces

f : X → Y is called an L-continuous mapping from X to Y if ∀U ∈ η, f^-1 (U) ∈ δ.

f : X → Y is called an L-homeomorphism if f is bijective such that f and f^-1 are L-continuous.

Theorem 2.2. [10] The molecules in L^X i.e. M (L^X) = {x_λ : x ∈ X and λ ∈ M (L)}.

In an L-topological space the collection of all L-open sets forms a lattice, so motivated by the definition of prime element in a lattice, Tomy and Johnson introduced pL-open sets in L-topological space.

Definition 2.10. [18] Let (X, δ) be an L-topological space. An L-open set G in X is called a prime L-open set if H ∧ K ≤ G implies H ≤ G or K ≤ G, where H, K are L-open sets. Prime L-open sets are denoted by pL-open sets. The collection of all pL-open set in (X, δ) is denoted by δ_p.

Pseudo-Complements of pL-open sets are called prime L-closed sets, denoted by pL-closed sets.

Proposition 2.1. [18] Let (X, δ) be an L-topological space and G ∈ δ. Then G is not pL-open in X if and only if there exist two L-open sets H and K such that G < H, G < K and G = H ∧ K

Corollary 2.1. [18] Let (X, δ) be an L-topological space and F be an L-closed set. Then F is not pL-closed in (X, δ) if and only if there exist two L-closed sets M and N such that F > M, F > N and F = M ∨ N

Proposition 2.2. [18] Let (X, δ) be an L-topological space, x_λ ∈ M (L^X). Then ${\bar{x_{λ}}}^{'}$ is pL-open and $\bar{x_{λ}}$ is pL-closed.

Theorem 2.3. [18] Let (X, δ) be an L-topological space. Then every L-open set is the meet of all pL-open sets containing it and every L-closed set is the join of all pL-closed sets contained in it.

3 pL-homeomorphism and lattice isomorphism

Recall that an L-topology (X, δ) is sober if for every L-open set G ∈ pr (δ) there exists a unique x_λ ∈ M (L^X) such that $G = {\bar{x_{λ}}}^{'}$ [10]. So we can say that an L-topology (X, δ) is called sober if for every pL-open set $G \neq \underline{1}$ there exists a unique x_λ ∈ M (L^X) such that $G = {\bar{x_{λ}}}^{'}$ .

Theorem 3.1. An L-topological space which is both T₂ and T₁ is a sober space.

Proof. To show that every L-topological spaces that are both T₂ and T₁ are sober, it is enough to show that for each pL-closed set $F \neq \underline{0}$ , there exist unique x_λ ∈ M (L^X) such that $F = \bar{x_{λ}}$ . Let (X, δ) be an L-topological space which is both T₂ and T₁ and $F \neq \underline{0}$ be a pL-closed set. Suppose that F = x_t where x_t ∈ Pt (L^X). Since (X, δ) is T₁, for F to be a pL-closed set, t must be in M (L). Consider the other case F ≠ x_t for any x_t ∈ Pt (L^X), then we can find x_λ and y_μ ∈ M (L^X) such that x ≠ y and x_λ, y_μ ≤ F. Since (X, δ) is T₂, we can find two L-open sets U and V such that U ∈ Q (x_λ), V ∈ Q (y_μ) and $U \land V = \underline{0}$ . So λnleqU ′ (x), μnleqV ′ (y) and $U \land V = \underline{0}$ implies $U^{'} \lor V^{'} = \underline{1}$ . Consider F ∧ U ′ and F ∧ V ′, then

$(F \land U^{'}) \lor (F \land V^{'}) = F \land (U^{'} \lor V^{'}) = F .$

If F ∧ U ′ = F then F (x) ≤ U ′ (x) for all x ∈ X. Hence x_λ ≤ F ≤ U ′, which is a contradiction. Therefore, F ∧ U ′ ≠ F, similarly F ∧ V ′ ≠ F. Now we get two L-closed sets F ∧ U ′ and F ∧ V ′ such that (F ∧ U ′) ∨ (F ∧ V ′) = F. Then by Corollary, 2.1 F is not pL-closed set which is a contradiction. So F = x_λ for some x_λ ∈ M (L^X). Therefore in all cases $F = \bar{x_{λ}}$ for some x_λ ∈ M (L^X). The uniqueness of x_λ follows since (X, δ) is T₁. Thus (X, δ) is sober. □

Definition 3.1. Let (X, δ) and (Y, η) be two L-topological spaces,

f : X → Y is called a pL-continuous map if f^-1 (U) is pL-open in X for any pL-open set U in Y.

f : X → Y is called a pL-homeomorphism if f is bijective such that both f and f^-1 are pL-continuous.

Remark 3.1. Note that L-continuity and pL-continuity are independent of each other. Consider the following examples.

Example 3.1. Let X = Y = N and L = [0, 1]. Consider discrete L-topology on X and define L-topology η on Y such that $η = \cup {U \in L^{X} : U (1) = 0} \cup \underline{1}$ . Now consider the identity map, then the identity map is a L-continuous map but not pL-continuous.

Example 3.2. Let L be any F-lattice and $X = Y = ℝ$ . Define different L-topologies on X and Y that are both T₂ and T₁. Since all spaces which are both T₂ and T₁ are sober, the pL-open sets on X and Y are full set and complement of closure of molecules. Therefore the identity map is a pL-continuos map but not an L-continuos map.

Theorem 3.2. Let (X, δ) and (Y, η) be two L-topological spaces and f : X → Y be an L-homeomorphism, then f is a pL-homeomorphism.

Proof. Let G be a pL-open set in Y. Since f is L-continuous, f^-1 (G) is L-open in X. We want to show that f^-1 (G) is pL-open. On contradiction assume that f^-1 (G) is not pL-open. Then by Proposition 2.1, we can find two L-open sets H and K in X such that f^-1 (G) ≤ H, f^-1 (G) ≤ K and f^-1 (G) = H ∧ K . Then f (H) and f (K) are L-open sets in Y and we have G ≤ f (H) , G ≤ f (K) and G = f (H) ∧ f (K). Hence by Proposition 2.1, we get that G is not pL-open, which is a contradiction. Therefore f^-1 (G) is pL-open in X and thus f is a pL-continuous map. Similarly we can show that f^-1 is also pL-continuous. Hence f is a pL-homeomorphism. □

Remark 3.2. The converse of above theorem is not true even in ordinary topology. Note that $ℝ$ with discrete topology and Sorgenfrey line are p-homeomorphic but they are not homeomorphic [19].

Lemma 3.1. Let (X, δ) and (Y, η) be two L-topological spaces, f : X → Y be a pL-continuous map and x_λ ∈ M (L^X), then $f (\bar{x_{λ}}) \leq \bar{f (x_{λ})}$ .

Proof. By Proposition 2.2, $\bar{{f (x)}_{λ}} = \bar{f (x_{λ})}$ is a pL-closed set in Y. Since f is pL-continuous, $f^{- 1} (\bar{f (x_{λ})})$ is a pL-closed set in X. We have $x_{λ} \leq f^{- 1} f (x_{λ}) \leq f^{- 1} (\bar{f (x_{λ})})$ . Therefore $\bar{x_{λ}} \leq f^{- 1} (\bar{f (x_{λ})})$ , which implies that $f (\bar{x_{λ}}) \leq \bar{f (x_{λ})}$ .

□

Theorem 3.3. Let (X, δ) and (Y, η) be two L-topological spaces. A map f : X → Y is an L-homeomorphism if and only if f is a pL-homeomorphism and there exists a lattice isomorphism φ : Γ (X) → Γ (Y) satisfying $φ (\bar{x_{λ}}) = f (\bar{x_{λ}}), \forall x_{λ} \in M (L^{X})$ and $φ^{- 1} (\bar{y_{λ}}) = f^{- 1} (\bar{y_{λ}}), \forall y_{λ} \in M (L^{Y})$

Proof. We know that every L-homeomorphism is also a pL-homeomorphism and if f is an L-homeomorphism then the mapping φ : Γ (X) → Γ (Y) given by φ (C) = f (C) is a lattice isomorphism which clearly satisfies our requirements.

For sufficiency part, suppose that there exists a pL-homeomorphism f : X → Y and a lattice isomorphism φ : Γ (X) → Γ (Y) with $φ (\bar{x_{λ}}) = f (\bar{x_{λ}}), \forall x_{λ} \in M (L^{X})$ and $φ^{- 1} (\bar{y_{λ}}) = f^{- 1} (\bar{y_{λ}}), \forall y_{λ} \in M (L^{Y})$ . To prove that f is a homeomorphism, it is enough to prove that φ^-1 (F) = f^-1 (F) , ∀ F ∈ Γ (Y) and f (E) = φ (E) , ∀ E ∈ Γ (X).

Let F be an L-closed set in Y and x_λ ≤ φ^-1 (F), where x_λ ∈ M (L^X). Then φ^-1 (F) is an L-closed set and so $\bar{x_{λ}} \leq φ^{- 1} (F)$ . Thus $φ (\bar{x_{λ}}) \leq F$ and so by our assumption $f (\bar{x_{λ}}) \leq F \Rightarrow f (x_{λ}) \leq F \Rightarrow f^{- 1} f (x_{λ}) \leq f^{- 1} (F)$ . It follows that x_λ ≤ f^-1 (F). Thus φ^-1 (F) ≤ f^-1 (F).

Suppose that x_λ ≤ f^-1 (F) so that f (x_λ) ≤ F. Hence $\bar{f (x_{λ})} \leq F$ . Since f is pL-continuous, by Lemma 3.1 $f (\bar{x_{λ}}) \leq \bar{f (x_{λ})}$ . Therefore, $f (\bar{x_{λ}}) \leq F$ and by our assumption we have $φ (\bar{x_{λ}}) \leq F$ . It follows that $\bar{x_{λ}} \leq φ^{- 1} (F)$ and x_λ ≤ φ^-1 (F), and thus f^-1 (F) ≤ φ^-1 (F).

Hence we have f^-1 (F) = φ^-1 (F) for each F ∈ Γ (Y). Similarly we can show that f (E) = φ (E) for each E ∈ Γ (X). Hence f is a pL-homeomorphism.

□ Remark 3.3. Theorem 3.3 is a solution for the question that we raised, as well as to the question, “under what condition pL-homeomorphism implies L-homeomorphism?”

Recall that an L-topological space (X, δ) is Alexandroff if δ is closed under arbitrary meet [3].

Corollary 3.1. Let (X, δ) and (Y, η) be two alexandroff L-topological spaces. Then f is an L-homeomorphism if and only if f is a pL-homeomorphism.

Proof. Let (X, δ) and (Y, η) be two alexandroff L-topological space and f : X → Y be a pL-homeomorphism. Define a map φ on Γ (X) by φ (A) = f (A). Consider F ∈ Γ (X), then by Theorem 2.3 F = ∨ {G ≤ F : G isa pL - closedset}. Since f is pL-homeomorphic, image of any pL-closed set is pL-closed and hence f (G) is pL-closed set for any pL-closed set G ≤ F. Since Y is alexandroff, $\lor {f (G) : G \leq F, G is a pL - closed set}$ is an L-closed set. Thus φ is a mapping from Γ (X) to Γ (Y). Let E ∈ Γ (Y) then by Theorem 2.3, E = ∨ {H ≤ E : H isa pL - closedset}. Since X is alexandroff and f is a pL-homeomorphism, there exist A ∈ Γ (X) such that φ (A) = E. Thus φ is surjective. Also φ is 1-1 since f is 1-1, and this makes φ bijective. Clearly φ and its inverse φ^-1 = f^-1 are order preserving. Therefore φ is a lattice isomorphism from Γ (X) to Γ (Y). From the construction of φ, it is clear that $φ (\bar{x_{λ}}) = f (\bar{x_{λ}})$ and $φ^{- 1} (\bar{y_{λ}}) = f^{- 1} (\bar{y_{λ}}), \forall λ \in M (L), \forall x \in X$ and ∀y ∈ Y. Thus by Theorem 3.3 we get that f is an L-homeomorphism.

□ The following examples shows that all the sufficiency condition in Theorem 3.3 cannot be deleted.

Example 3.3. Let L = [0, 1]. Consider $ℝ$ with co-finite L-topology and co-countable topology. Clearly both topologies are T₂ and T₁, and hence sober. By Theorem 3.1, in both topology pL-closed sets are full set and $\bar{x_{λ}}$ for all x_λ ∈ M (L^R). Therefore the identity map is a pL-homeomorphism. But we know that they are not L-homeomorphic.

Example 3.4. Let X, Y be two sets with distinct cardinalities and L be any F-lattice. We equip X and Y with the indiscrete L-topology. Then it is easily seen that the map φ : Γ (X) → Γ (Y) defined by $φ ({\underline{1}}_{X}) = {\underline{1}}_{Y}$ and $φ ({\underline{0}}_{X}) = {\underline{0}}_{Y}$ is a lattice isomorphism. But they are not L-homeomorphic since the cardinalities are different.

Example 3.5. Let L = [0, 1], consider $X = {(x, 0) : x \in ℝ}$ with co-finite L-topology and $Y = {(y, 1) : y \in ℝ}$ with co-countable L-topology.

Let W ⊂ Z, then for every A ∈ L^W we define A^* ∈ L^Z as follows: $A^{*} (x) = {\begin{matrix} A (x), & x \in W \\ 0, & x \notin W \end{matrix}$

Let α, β ∉ X ∪ Y such that α ≠ β. Set X′ = X ∪ {α} with topology whose L-closed sets are ${\underline{1}}_{X^{'}}$ and F^* for all L-closed set F ∈ X. Similarly set Y′ = Y ∪ {β} with topology whose L-closed sets are ${\underline{1}}_{Y^{'}}$ and F^* for all L-closed set F in Y.

Note that the free union E + F of disjoint spaces E and F is the set E ∪ F with L-topology: U is L-open if and only if U|_E is L-open in E and U|_F is L-open in F.

Let Λ = X′ + Y and Δ = X + Y′. Consider the map φ : Γ (Λ) → Γ (Δ) defined by $φ (F) = {\begin{matrix} G, & if supp (F) \cap Y \neq Y \\ H, & if supp (F) \cap Y = Y \end{matrix}$ where G and H are defined as $G = {\begin{matrix} F (x), & if x \in X \cup Y \\ 0, & if o . w \end{matrix}$ $H = {\begin{matrix} F (x), & if x \in X \cup Y \\ 1, & if x = β \end{matrix}$ Then we can see that φ is a lattice isomorphism.

The pL-closed sets in X and Y are full set and closure of all molecules. Therefore the map h : X → Y given by (x, 0) → (x, 1) is a pL-homeomorphism. Similarly in case of X′ and Y′, pL-closed sets are full set and closure of all molecules. Therefore the map g : X′ → Y′ given by (x, 0) → (x, 1), and α → β is a pL-homeomorphism. Next define f : Λ → Δ as follows $f (a) = {\begin{matrix} (x, 1), & if a = (x, 0) \\ (x, 0), & if a = (x, 1), \\ β, & if a = α . \end{matrix}$ Then f restricted to X′ is g and f restricted to Y is h^-1. Also, the pL-closed sets in Λ are closure of all molecules, χ_X′, χ_Y and the full set, and the pL-closed sets in Δ are closure of all molecules, χ_X, χ_Y′ and the full set. Combining all this, we can show that f is a pL-homeomorphism. But f is not a L-homeomorphism.

4 Molecule closure lattice isomorphism and L-homeomorphism

This section is based on the work by Finch [6].

Theorem 4.1. Two L-topological spaces (X, δ) and (Y, η) are L-homeomorphic if and only if there exists a lattice isomorphism φ : Γ (X) → Γ (Y) such that:

For each x ∈ X, there exists y ∈ Y such that for any λ ∈ M (L) $φ (\bar{x_{λ}}) = \bar{y_{λ}} .$

For each y ∈ Y, there exists x ∈ X such that for any λ ∈ M (L) $φ^{- 1} (\bar{y_{λ}}) = \bar{x_{λ}} .$

Let x ∈ X and y ∈ Y. Set $X_{x} = {t \in X : \bar{t_{λ}} = \bar{x_{λ}}, \forall λ \in M (L)}$ and $Y_{y} = {t \in Y : \bar{t_{λ}} = \bar{y_{λ}}, \forall λ \in M (L)}$ ,

then $φ (\bar{x_{λ}}) = \bar{y_{λ}}, \forall λ \in M (L)$ implies |X_x| = |Y_y|.

Proof. ⇒ Let f be an L-homeomorphism from X to Y. Then f is an order preserving map from Γ (X) to Γ (Y). Thus the map φ : Γ (X) → Γ (Y) given by φ (C) = f (C) is a lattice isomorphism which has all three stated properties. Since in the first place (i) holds, for if y = f (x) then for all $λ \in M (L), \bar{y_{λ}} = \bar{f (x)_{λ}} = \bar{f (x_{λ})} = f (\bar{x_{λ}}) = φ (\bar{x_{λ}})$ . Similarly (ii) holds.

For (iii), suppose that $\bar{y_{λ}} = φ (\bar{x_{λ}}), \forall λ \in M (L)$ and t = Y_y. Then $\bar{f^{- 1} (t_{λ})} = f^{- 1} (\bar{t_{λ}}) = f^{- 1} (\bar{y_{λ}}) = φ^{- 1} (\bar{y_{λ}}) = \bar{x_{λ}}$ . This is true for all λ ∈ M (L), thus we get that f^-1 (t) ∈ X_x. Hence Y_y ⊆ {f (x) : x ∈ X_x}. Conversely if t ∈ X_x, then $\bar{f (t_{λ})} = f (\bar{t_{λ}}) = f (\bar{x_{λ}}) = φ (\bar{x_{λ}}) = \bar{y_{λ}}$ . Therefore f (t) ∈ Y_y which implies {f (x) : x ∈ X} ⊆ Y_y. Thus Y_y = {f (x) ; x ∈ X_x} and since f is 1-1, it follows that |Y_y| = |X_x|

⇐ The relation $\bar{x_{λ}} = \bar{t_{λ}}, \forall λ \in M (L)$ is an equivalence relation on X and so the sets X_x for x in X forms a partition of X. Let this partition be denoted by $A = {X_{α} : α \in A}$ , where A is an indexing set. Similarly let $B = {Y_{β} : β \in B}$ denote the partition of which is formed by the sets Y_y for y in Y.

Next we will show that |A| = |B|. Consider the map ψ : X_α → ψX_α, where $ψ X_{α} = {y : \exists x \in X_{α}, \forall λ \in M (L), φ (\bar{x_{λ}}) = \bar{y_{λ}}}$ . Clearly by (i), no ψX_α is empty. Suppose that y₁, y₂ ∈ ψX_α, then there are elements x₁, x₂ in X_α such that $\bar{{y_{1}}_{λ}} = φ (\bar{{x_{1}}_{λ}}) = φ (\bar{{x_{2}}_{λ}}) = \bar{{y_{2}}_{λ}}$ . This is true for all λ ∈ M (L). Thus ψX_α is contained in some Y_β. Observe that if y ∈ ψX_α and t ∈ Y_y, then there is an x ∈ X_α such that for each λ ∈ M (L), $φ (\bar{x_{λ}}) = \bar{y_{λ}} = \bar{t_{λ}}$ . This establishes that t ∈ ψX_α and therefore ψX_α = Y_β for some Y_β. Thus X_α → ψX_α is a map from $A$ into $B$ .

Suppose ψX_α = ψX_γ then there is y ∈ Y such that $φ (\bar{{x_{α}}_{λ}}) = \bar{y_{λ}} = φ (\bar{{x_{γ}}_{λ}}), \forall λ \in M (L)$ where x_α ∈ X_α and x_γ ∈ X_γ. Since φ is 1-1, we get that $\bar{{x_{α}}_{λ}} = \bar{{x_{γ}}_{λ}}, \forall λ \in M (L)$ . Thus X_α = X_γ and ψ is 1-1. To prove that the mapping ψ is onto, we argue as follows, let $Y_{β} \in B$ and y ∈ Y_β. Then by (ii) there is at least one element x ∈ X such that $φ^{- 1} (\bar{y_{λ}}) = \bar{x_{λ}}, \forall λ \in M (L)$ . But there is a unique X_α in $A$ which contains this x and since $\bar{y_{λ}} = φ (\bar{x_{λ}}), \forall λ \in M (L)$ , it follows that Y_β = ψX_α.

In light of the result just established there is no loss of generality in supposing that A = B. That is, $A$ and $B$ have the same indexing set. So $Y_{α} = {y : \exists x \in X_{α}, \forall λ \in M (L), φ (\bar{x_{λ}}) = \bar{y_{λ}}} .$ $X_{α} = {x : \exists y \in Y_{α}, \forall λ \in M (L), φ (\bar{y_{λ}}) = \bar{x_{λ}}} .$ From (iii) it follows that |X_α| = |Y_α| for each α in A.

Therefore there exists a bijective map f_α from X_α into Y_α. Define f : X → Y by f|_{X
_α} = f_α. Then f is a bijective map since $A$ is partition of X, $B$ is a partition of Y and since each f_α is bijective.

Then for each x ∈ X there is a unique X_α in $A$ which contains x so that, $\bar{f (x_{λ})} = \bar{f (x)_{λ}} = \bar{y_{λ}} = φ (\bar{x_{λ}}), \forall λ \in M (L) .$ Therefore, $\bar{f (x_{λ})} = φ (\bar{x_{λ}}), \forall λ \in M (L) .$ Similarly $\bar{f^{- 1} (y_{λ})} = φ^{- 1} (\bar{y_{λ}}), \forall λ \in M (L) .$ Next we will prove that f is an L-homeomorphism. It is enough to prove that f (C) = φ (C) , ∀ C ∈ Γ (X) and f^-1 (D) = φ^-1 (D) , ∀ D ∈ Γ (Y). Let C be an L-closed set in X and y_λ ∈ L^Y such that y_λ ≤ f (C). Then f (x_λ) ≤ f (C) which implies $x_{λ} \leq C \Rightarrow \bar{x_{λ}} \leq C$ . Then $\bar{y_{λ}} = \bar{f (x)_{λ}} = φ (\bar{x_{λ}}) \leq φ (C) .$ This show that y_λ ≤ φ (C) and hence f (C) ≤ φ (C).

Conversely suppose that y_λ ≤ φ (C), then $\bar{y_{λ}} \leq φ (C)$ which implies that $\bar{f^{- 1} (y_{λ})} = φ^{- 1} (\bar{y_{λ}}) \leq C .$ It follows that f^-1 (y_λ) ≤ C and y_λ ≤ f (C) . Therefore, $φ (C) \leq f (C) .$ Thus φ (C) = f (C), ∀C ∈ Γ (X). The proof for f^-1 (D) = φ^-1 (D) for each D in Γ (Y) is similar. This concludes the proof. □

Remark 4.1. In a sub-T₀ space, the fact that $\bar{x_{λ}} = \bar{y_{λ}}, \forall λ \in M (L)$ implies that x = y [10]. So in such a space |X_x|=1 for each x ∈ X. It follows that in the statement of Theorem 4.1, the condition (iii) is superfluous when each of the space is a sub-T₀ space.

Corollary 4.1. In order that two sub-T₀ spaces X and Y be L-homeomorphic it is necessary and sufficient that there exists a lattice isomorphism φ : Γ (X) → Γ (Y) with the properties (i) and (ii) above.

Consider the following example

Example 4.1. Let X = Y = {x} and L = [0, 1] with discrete L-topology. Then we know that X and Y are L-homeomorphic and any increasing bijective map f : [0, 1] → [0, 1] corresponds a lattice isomorphism φ from Γ (X) to Γ (Y) given by φ (x_a) = x_f(a). Out of all this, one lattice isomorphism corresponds to the L-homeomorphism between X and Y which has the property that for each x_λ ∈ M (L^X) there exist y_λ ∈ M (L^Y) such that $φ (\bar{x_{λ}}) = \bar{y_{λ}}$ , and for each y_λ ∈ M (L^Y) there exist x_λ ∈ M (L^X) such that $φ^{- 1} (\bar{y_{λ}}) = \bar{x_{λ}}$ .

By this observation, next we define molecule-closure lattice isomorphism in such a way that;

Definition 4.1. Let X, Y be two L-topological spaces and φ : Γ (X) → Γ (Y) be a lattice isomorphism. We say that φ is molecule-closure lattice isomorphism if the following properties are satisfied,

For each x_λ ∈ M (L^X), there exist y_λ ∈ M (L^Y) such that $φ (\bar{x_{λ}}) = \bar{y_{λ}}$ .

For each y_λ ∈ M (L^Y), there exist x_λ ∈ M (L^X) such that $φ^{- 1} (\bar{y_{λ}}) = \bar{x_{λ}}$ .

Theorem 4.2. Let (X, δ) and (Y, η) be two T₁L-topological spaces. If 1 ∈ M (L) and φ : Γ (X) → Γ (Y) is a molecule closure lattice isomorphism then X and Y are L-homeomorphic.

Proof. Let (X, δ) and (Y, η) be T₁ L-topological spaces, 1 ∈ M (L) and x ∈ X. Since 1 ∈ M (L), x₁ ∈ M (L^X). Given that X is T₁, so we have $\bar{x_{1}} = x_{1}$ . Suppose that φ is a molecule closure from Γ (X) to Γ (Y), so there exist y₁ ∈ M (L^X) such that φ (x₁) = y₁. Since (Y, η) is also T₁, $\bar{y_{1}} = y_{1}$ . That is, φ (x₁) = y₁. Let x_λ ∈ M (L^X) then x_λ ≤ x₁ so $φ (\bar{x_{λ}}) \leq φ (\bar{x_{1}}) \Rightarrow φ (x_{λ}) \leq φ (x_{1}) = y_{1}$ . Thus, φ (x_λ) = y_λ since φ is molecule closure. Similarly we can show that ∀y ∈ Y, ∃ x ∈ X such that $φ^{- 1} (\bar{y_{λ}}) = \bar{x_{λ}}, \forall λ \in M (L)$ . So by Corollary 4.1 we get that X and Y are L-homeomorphic. □

Remark 4.2. We can’t remove the condition 1 ∈ M (L) in above theorem. Consider the following example.

Example 4.2. Let $X = Y = ℕ$ and L = [0, 1] ^[0,1]. Equip X with L-topology whose L-closed sets are generated by {A ∈ L^X : supp (A) < ω and ∀ x ∈ supp (A) , supp (A (x)) < ω} ∪ {B, C, D, E} where supp (A) < ω means support of A is finite, and B, C, D and E are given by $B (x) = {\begin{matrix} 0; & x \neq 1 \\ b; & x = 1 \end{matrix}, b (x) = {\begin{matrix} 1; & x \in [0, 1 / 4) \\ 0; & o . w \end{matrix}$

$C (x) = {\begin{matrix} 0; & x \neq 1 \\ c; & x = 1 \end{matrix}, c (x) = {\begin{matrix} 1; & x \in [1 / 4, 1 / 2) \\ 0; & o . w \end{matrix}$

$D (x) = {\begin{matrix} 0; & x \neq 2 \\ b; & x = 2 \end{matrix}, d (x) = {\begin{matrix} 1; & x \in [1 / 2, 3 / 4) \\ 0; & o . w \end{matrix}$

$E (x) = {\begin{matrix} 0; & x \neq 2 \\ d; & x = 2 \end{matrix}, e (x) = {\begin{matrix} 1; & x \in [3 / 4, 1) \\ 0; & o . w \end{matrix} .$

And equip Y with L-topology whose L-closed sets are generated by {A ∈ L^X : supp (A) < ω and ∀ x ∈ supp (A) , supp (A (x)) < ω} ∪ {J, K, L, M} where J, K, L and M are given by

$J (x) = {\begin{matrix} 0; & x \neq 1 \\ b; & x = 1 \end{matrix}, j (x) = {\begin{matrix} 1; & x \in [0, 1 / 4) \\ 0; & o . w \end{matrix}$ $K (x) = {\begin{matrix} 0; & x \neq 1 \\ b; & x = 1 \end{matrix}, k (x) = {\begin{matrix} 1; & x \in [3 / 4, 1) \\ 0; & o . w \end{matrix}$ $L (x) = {\begin{matrix} 0; & x \neq 2 \\ b; & x = 2 \end{matrix}, l (x) = {\begin{matrix} 1; & x \in [1 / 4, 1 / 2) \\ 0; & o . w \end{matrix}$ $M (x) = {\begin{matrix} 0; & x \neq 2 \\ m; & x = 2 \end{matrix}, m (x) = {\begin{matrix} 1; & x \in [1 / 2, 3 / 4) \\ 0; & o . w \end{matrix}$

Then Γ (X) and Γ (Y) are lattice isomorphic with a lattice isomorphism φ : Γ (X) → Γ (Y) which satisfies the following properties.

$φ (x_{a}) = {\begin{matrix} 2_{a}; & x = 1 and a \in [1 / 4, 1 / 2) \\ 1_{a}; & x = 2 and a \in [3 / 4, 1) \\ x_{a}; & o . w \end{matrix}$ Here the conditions in Theorem 4.2 are all met by X, Y and φ except that 1 ∈ M (L), yet X and Y are not L-homeomorphic.

5 quasi L-homeomorphism

In this section we’ll look into the requirements that an L-continuous function q : X → Y must have in order for the induced lattice-homomorphism Γ (q) to be a lattice-isomorphism. Yip Kai Wing [9] used quasi homeomorphism to answer this question in the case of topological spaces. Accordingly, using quasi homeomorphism in topological spaces, we define quasi L-homeomorphism in L-topological space as follows:

Definition 5.1. Let (X, δ) and (Y, η) be two L-topological spaces. An L-continuous map q : X → Y is said to be a quasi L-homeomorphism if

For any L-closed set E in X, $q^{- 1} [\bar{q (E)}] = E$ .

For any L-closed set F in Y, $\bar{q [q^{- 1} (F)]} = F$ .

The next result provides the necessary and sufficient condition for an L-continuous function q : X → Y so that the induced lattice-homomorphism Γ (q) is a lattice-isomorphism.

Theorem 5.1. Let (X, δ) and (Y, η) be two L-topological spaces and let q : X → Y be an L-continuous map. Then q is a quasi L-homeomorphism if and only q : X → Y induces a lattice-isomorphism Γ (q) : Γ (Y) → Γ (X) given by F → q^-1 (F).

Proof. ⇐ : Let Γ (q) : Γ (Y) → Γ (X) given by F → q^-1 (F) be a lattice isomorphism. Let E be an L-closed set in X. Then since Γ (q) is onto there exists an L-closed set C in Y such that q^-1 (C) = E. Then q (E) = q [q^-1 (C)] ≤ C which implies $\bar{q (E)} \leq C$ . Thus we have the containments, $E \leq q^{- 1} [q (E)] \leq q^{- 1} [\bar{q (E)}] \leq q^{- 1} (C) = E .$ So that $q^{- 1} [\bar{q (E)}] = E$ .

Let F be any L-closed set in Y. Then by the condition $q^{- 1} [\bar{q (E)}] = E$ , we also have

$q^{- 1} (F) = q^{- 1} (q [q^{- 1} (F)]) \leq q^{- 1} \bar{(q [q^{- 1} (F)])} = q^{- 1} (F) .$

Hence $q^{- 1} (F) = q^{- 1} \bar{(q [q^{- 1} (F)])}$ . Then since Γ (q) is 1-1, we have $\bar{q [q^{- 1} (F)]} = F$

⇒ : Let q : X → Y be a quasi L-homeomorphism and consider Γ (q) : Γ (Y) → Γ (X) given by F → q^-1 (F). Suppose that F is an L-closed sets in X. Then by condition (i) of quasi L-homeomorphism, $q^{- 1} [\bar{q (F)}] = F$ which implies that Γ (q) is onto. Suppose that there exist two L-closed sets E and F such that q^-1 (E) = q^-1 (F). Hence by condition (ii) of quasi L-homeomorphism, $E = \bar{q [q^{- 1} (E)]} = \bar{q [q^{- 1} (F)]} = F$ . Hence we get that Γ (q) is 1-1 and therefore φ is a lattice isomorphism. It follows that q is a quasi L-homeomorphism. □

Remark 5.1. From the above observation we can also define quasi L-homeomorphism in the following ways.

An L-continuous map q : X → Y is said to be a quasi L-homeomorphism if F → q^-1 (F) defines a bijection from Γ (Y) to Γ (X).

An L-continuous map q : X → Y is said to be a quasi L-homeomorphism if U → q^-1 (U) defines a bijection from Ω (Y) to Ω (X).

In this section we shall study some properties of quasi L-homeomorphism.

Theorem 5.2. Let (X, δ) and (Y, η) be two L-topological spaces and let q : X → Y be a quasi L-homeomorphism. Then we have the following,

If for two L-closed sets F₁ and F₂ in Y, f^-1 (F₁) = f^-1 (F₂) then F₁ = F₂.

If for two L-closed sets C₁ and C₂ in X, $\bar{f (C_{1})} = \bar{f (C_{2})}$ then C₁ = C₂.

$f ({\underline{1}}_{X})$ is dense in Y.

Proof. (i) and (ii) follow from Theorem 5.1. To prove (iii), it is clear that $f^{- 1} [\bar{f ({\underline{1}}_{X})}] = {\underline{1}}_{X} = f^{- 1} ({\underline{1}}_{Y})$ By (i), $\bar{f ({\underline{1}}_{X})} = {\underline{1}}_{Y}$ and hence $f ({\underline{1}}_{X})$ is dense in Y.

□ Remark 5.2. The example that follows illustrates how the quasi-L-homeomorphism is not a symmetric concept. That is, the existence of a L-homeomorphism between Y and X does not follow from the existence of a quasi L-homeomorphism from X to Y.

Example 5.1. Let X be an infinite set and L be a F-lattice. Consider X with co-finite L-topology δ. Put Y = X ∪ {∞}, where ∞ is an object not in X with topology $η = {V \in L^{Y} : V (\infty) = 1, V |_{X} \in δ} \cup \underline{0}$ . It is easy to verify that q : X → Y defined by q (x) = x is a quasi L-homeomorphism.

Now we shall show that there is no quasi homeomorphism from Y to X. Suppose g : Y → X is a quasi L-homeomorphism from Y to X. From the definition of quasi L-homeomorphism we know that, for all L-closed set C in X, $\bar{g [g^{- 1} (C)]} = C$ . Let x = g (∞), we know that x₁ is an L-closed set, therefore $\bar{g [g^{- 1} (x_{1})]} = x_{1}$ . But g^-1 (x₁) is an L-closed set in Y and $\underset{1}{\infty} \leq g^{- 1} (x_{1})$ . Hence $g^{- 1} (x_{1}) = {\underline{1}}_{Y}$ . By (iii) of Theorem 5.2, we have $\bar{g [g^{- 1} (x_{1})]} = \bar{g ({\underline{1}}_{Y})} = {\underline{1}}_{X} .$ This contradiction shows the non-existence of quasi L-homeomorphism from Y to X.

Proposition 5.1. Let (X, δ) and (Y, η) be two L-topological spaces and let q : X → Y be a quasi L-homeomorphism. If X is a sub-T₀ space then q is injective.

Proof. Suppose that there exist x, y ∈ X such that q (x) = q (y). As X is sub-T₀, there exists a λ ∈ M (L) such that $\bar{x_{λ}} \neq \bar{y_{λ}}$ . Since q is a quasi L-homeomorphism there exists a F ∈ Γ (X) such that $q^{- 1} (F) = \bar{x_{λ}}$ . Now q (x) = q (y) implies that F (q (x)) = F (q (y)). Therefore $y_{λ} \leq \bar{x_{λ}}$ . Similarly we can show that $x_{λ} \leq \bar{y_{λ}}$ . Thus $\bar{x_{λ}} = \bar{y_{λ}}$ which is a contradiction. Therefore q is injective. □ To conclude this note, we prove the following:

Theorem 5.3. Let (X, δ) and (Y, η) be two L-topological spaces and let q : X → Y be a quasi L-homeomorphism then the following are equivalent;

q is a surjective mapping.

q is an L-closed mapping.

q is an L-open mapping.

Proof. (ii) → (i) and (iii) → (i)

Let q be L-closed(res. L-open) map. Then $q ({\underline{1}}_{X})$ is an L-closed(res. L-open) subset of Y. Now by (iii) of Theorem 5.2 we have $q^{- 1} (q ({\underline{1}}_{X})) = q^{- 1} ({\underline{1}}_{Y})$ . Then by the Remark 5.1, we get that $q ({\underline{1}}_{X}) = {\underline{1}}_{Y}$ , which implies that q (X) = Y. Thus q is onto.

(i) ⇒ (ii) and (i) ⇒ (ii)

Let C be an L-closed(res. L-open) subset of X. By the definition, there is an L-closed(res. L-open) subset D of Y such that C = q^-1 (D). But since q is onto, we get that q (C) = D, proving that q (C) is L-closed(res. L-open). □ Above theorem proves that surjective quasi L-homeomorphisms are both L-closed and L-open. So we have the following result.

Corollary 5.1. Every bijective quasi L-homeomorphisms are L-homeomorphic.

Theorem 5.4. Let (X, δ) and (Y, η) be two L-topological spaces and let q : X → Y be an L-continuous surjective map. Then the following are equivalent;

q is a quasi L-homeomorphism,

q is L-closed and for each L-closed subset C of X, q^-1 [q (C)] = C,

q is L-open and for each L-open subset U of X, q^-1 [q (U)] = U.

Proof. (i) ⇒ (ii): Let q : X → Y be a quasi L-homeomorphism. By Theorem 5.3 we get that q is an L-closed map. Let C be an L-closed set in X. From Remark 5.1, it is clear that there exists an L-closed set D in Y such that q^-1 (D) = C. Then $q^{- 1} [q (C)] = q^{- 1} (q [q^{- 1} (D)]) = q^{- 1} (D) = C .$ It follows that q^-1 [q (C)] = C.

(ii) → (i): Let q be L-closed and for each L-closed subset C of X, q^-1 [q (C)] = C. Let φ : Γ (Y) → Γ (X) be given by F → q^-1 (F). Since q is L-closed, q (C) is L-closed. Therefore q^-1 [q (C)] = C implies that the map φ is onto. Since q is surjective, we get that φ is 1-1. Hence φ is bijective and therefore φ is lattice isomorphism. So by Remark 5.1, it follows that q is a lattice isomorphism.

Similarly (i) ⇒ (ii) and (iii) ⇒ (ii) since q^-1 (B′) ≥ (q^-1 (B)) ′, ∀B ∈ L^Y. □ Now let’s move to the key theorem in this section. To prove the next result, we need the following lemma;

Lemma 5.1. Let g : Y → X be a quasi L-homeomorphism and f : Y → Z be L-continuous. Suppose that Z is a sub-T₀ space, then there exists at most one L-continuous map F : X → Z such that F ∘ g = f .

Proof. Assume that there exist two distinct L-continuous map F, G : X → Z such that F ∘ g = G ∘ g = f. Let x ∈ X with F (x) ≠ G (X). Then since Z is sub-T₀ space, there exists λ ∈ M (L) such that an L-closed C exists with F (x_λ) ≤ C and G (x_λ) nleqC or G (x_λ) ≤ C and F (x_λ) nleqC. Consider the two L-closed set D = F^-1 (C) and E = G^-1 (C), then D ≠ E. Thus we get $g^{- 1} (D) = g^{- 1} [F^{- 1} (C)] = g^{- 1} [G^{- 1} (C)] = g^{- 1} (E) .$ Since g is a quasi L-homeomorphism, we get D = E, a contradiction. Thus F (x) = G (x) , ∀ x ∈ X. □

Theorem 5.5. [The L-continuous Extension Theorem] Let g : Y → X be a surjective quasi L-homeomorphism, f : Y → Z be an L-continuous map. Let Z be a sub-T₀ space, then there exists one and only one L-continuous map F : X → Z such that F ∘ g = f

Proof. Let x ∈ X, since g is surjective there exists y ∈ Y such that x = g (y). Suppose that there exists z ∈ Y such that x = g (y) = g (z). Take any molecule λ ∈ M (L). Then $\bar{g (y_{λ})} = \bar{g (y)_{λ}} = \bar{x_{λ}} = \bar{g (z)_{λ}} = \bar{g (z_{λ})}$ . Since g is L-continuous, we get $g (\bar{y_{λ}}) \leq \bar{g (y_{λ})}$ and $g (\bar{z_{λ}}) \leq \bar{g (z_{λ})}$ . From Theorem 5.4, we get g is L-closed, therefore $\bar{g (y_{λ})} \leq g (\bar{y_{λ}})$ and $\bar{g (z_{λ})} \leq g (\bar{z_{λ}})$ . Thus we get $g (\bar{y_{λ}}) = \bar{g (y_{λ})}$ and $g (\bar{y_{λ}}) = \bar{g (y_{λ})}$ , therefore $g (\bar{y_{λ}}) = g (\bar{z_{λ}})$ . Again using Theorem 5.4 $\bar{y_{λ}} = g^{- 1} [g (\bar{y_{λ}})] = g^{- 1} [g (\bar{z_{λ}})] = \bar{z_{λ}} .$

Given that f is an L-continuous map, therefore $\bar{f (\bar{y_{λ}})} \leq \bar{f (y_{λ})}$ which implies $\bar{f (\bar{y_{λ}})} = \bar{f (y_{λ})}$ . Similarly we can show that $\bar{f (\bar{y_{λ}})} = \bar{f (y_{λ})}$ . Thus we have $\bar{f (y_{λ})} = \bar{f (\bar{y_{λ}})} = \bar{f (\bar{z_{λ}})} = \bar{f (z_{λ})} .$ Therefore for all λ ∈ M (L), $\bar{f (y)_{λ}} = \bar{f (z)_{λ}}$ . Since Z is sub-T₀, this implies that f (y) = f (z). This provides a map F : X → Z, x = g (y) ↦ f (y) with F ∘ g = f.

It remains to prove that F is L-continuous. To do so, let C be an L-closed subset of Z. Then f^-1 (C) = g^-1 [F^-1 (C)]. Hence g [f^-1 (C)] = g (g^-1 [F^-1 (C)]) = F^-1 (C) so that since g is L-closed, F^-1 (C) is an L-closed subset of X. We have thus proved that F is L-continuous. The uniqueness of the map is clear from Lemma 5.1. □

6 quasi L-homeomorphism and L-homeomorphism

We proved that every bijective quasi L-homeomorphism are L-homeomorphic. Next, we are looking for other instances where quasi L-homeomorphism implies L-homeomorphism. This problem is addressed in the following theorems.

Theorem 6.1. Let (X, δ) be a sober and (Y, η) be a T₀ space and let q : X → Y be a quasi L-homeomorphism. Then q is a L-homeomorphism.

Proof. Let y ∈ Y and λ ∈ M (L). It is obviously seen that $q^{- 1} (\bar{y_{λ}})$ is a pL-closed set in X. Hence $q^{- 1} (\bar{y_{λ}})$ has a generic molecule x_μ. Then $q (x_{μ}) \leq q (\bar{x_{μ}}) \leq q [q^{- 1} (\bar{y_{λ}})] \leq \bar{y_{λ}}$ which implies $\bar{q (x_{μ})} \leq \bar{y_{λ}}$ . Thus we have the containments, $\bar{x_{μ}} \leq q^{- 1} [\bar{q (x_{μ})}] \leq q^{- 1} (\bar{y_{μ}}) = \bar{x_{μ}} .$ So that $q^{- 1} [\bar{q (x_{μ})}] = q^{- 1} (\bar{y_{λ}})$ . It follows, from the fact that q is a quasi L-homeomorphism, $\bar{q (x_{μ})} = \bar{y_{λ}}$ . Since Y is T₀, we get q (x_μ) = y_λ. This proves that q (x) = y and thus q is surjective. By Proposition 5.1, we get q is injective and hence q is bijective. We know that every bijective quasi L-homeomorphism is an L-homeomorphism. Therefore we get that q is an L-homeomorphism. □

The following result establishes some connections between lattice isomorphism induced by an L-homeomorphism and those induced by a quasi L-homeomorphism. For the proof of the next theorem, we need a lemma.

Lemma 6.1. Let (X, δ) and (Y, η) be two L-topological spaces such that (Y, η) is a sub-T₀L-topological space. If f, g : X → Y are two L-continuous maps such that Γ (f) = Γ (g) then f = g.

Proof. Let x ∈ Xand λ ∈ M (L). Then

$f^{- 1} (\bar{f (x_{λ})}) = g^{- 1} (\bar{f (x_{λ})}) and g^{- 1} (\bar{g (x_{λ})}) = f^{- 1} (\bar{g (x_{λ})}) .$

This yields $g (x_{λ}) \leq \bar{f (x_{λ})}$ and $f (x_{λ}) \leq \bar{g (x_{λ})}$ . Thus $\bar{f (x_{λ})} = \bar{g (x_{λ})}$ , so that f (x) = g (x), since Y is sub-T₀. □

Theorem 6.2. Let (X, δ) and (Y, η) be two sub-T₀L-topological spaces and φ : Γ (X) → Γ (Y) be a lattice isomorphism of L-topological spaces. Then the following statements are equivalent;

φ is induced by an L-homeomorphism,

There are two quasi L-homeomorphisms q : Y → X and p : X → Y such that φ = Γ (q) and φ^-1 = Γ (p).

Proof. (i) ⇒ (ii): Straightforward.

(ii) ⇒ (i): Let q : Y → X and p : X → Y be two quasi L-homeomorphisms such that φ = Γ (q) and φ^-1 = Γ (p). We have φφ^-1 = I_Γ(Y) = Γ (I_Y) and φ^-1φ = I_Γ(X) = Γ (I_X). Hence Γ (pq) = Γ (I_Y) and Γ (qp) = Γ (I_X). Thus, by Lemma 6.1, we get qp = I_X and pq = I_Y. Therefore, φ is induced by the L-homeomorphism p. □ The following result establishes the relation between quasi L-homeomorphism and pL-continuity. For the proof of the next theorem, we need a lemma;

Lemma 6.2. Let L and M be two lattices and φ be a lattice isomorphism from L to M. Then a ∈ L is a meet irreducible element (resp. join irreducible, meet prime, join prime) if and only if φ (a) is a meet irreducible (resp. join irreducible, meet prime, join prime) element in M.

Proof. Let a be a meet irreducible element in L and b, c be two elements in M such that φ (a) = b ∧ c. Then a = φ^-1 (b ∧ c) = φ^-1 (b) ∧ φ^-1 (c). Now, since a is meet irreducible in L, a = φ^-1 (b) or a = φ^-1 (c). This yields φ (a) = b or φ (a) = c. This proves that φ (a) is a meet irreducible element. Similarly since φ is a lattice isomorphism, we can also show the converse. The proofs for join irreducible, join prime, meet prime element are same as above. □

Lemma 6.3. Let (X, δ) and (Y, η) be two L-topological spaces. Then f : X → Y is a quasi L-homeomorphism if and only if f is pL-continuous and there exists a lattice isomorphism φ : Γ (Y) → Γ (X) such that φ (F) = f^-1 (F) for every pL-closed set F in Y.

Proof. Suppose that there exists a quasi L-homeomorphism f : X → Y. Then the mapping φ : Γ (Y) → Γ (X), defined by φ (C) = f^-1 (C) for C ∈ Γ (Y) is a lattice isomorphism. By Lemma 6.2, φ maps pL-open set to pL-open set. Thus f is a pL-continuous map. This concludes the necessary part.

Conversely, suppose that there exists a pL-continuous map f : X → Y and a lattice isomorphism φ : Γ (Y) → Γ (X) such that φ (F) = f^-1 (F) for every pL-closed set F in Y. Let E be an L-closed set in Y. By Theorem 2.3, there exists a collection of pL-closed sets {E_α} such that E = ∪ {E_α}. Then f^-1 (E) = f^-1 (∪ E_α) = ∪ (f^-1 (E_α)). By our assumption f^-1 (E_α) = φ (E_α), so we have f^-1 (E) = ∪ (φ (E_α)) = φ (∪ (E_α)) = φ (E). This proves that f is an L-continuous map and so f is a quasi L-homeomorphism. □

Theorem 6.3. Let (X, δ), (Y, η) be two sub-T₀L-topological spaces and φ : Γ (X) → Γ (Y) be a lattice isomorphism of L-topological spaces. Then the following are equivalent;

φ is induced by an L-homeomorphism,

There are two pL-continuous maps q : Y → X and p : X → Y such that φ (F) = q^-1 (F) and φ^-1 (E) = p^-1 (E) for every pL-closed sets F in Y and for every pL-closed sets E in X.

Proof. The result follows from Lemma 6.3 and Theorem 6.2. □

7 T_D axiom and L-homeomorphism

In 2009 Othman Echi and Sami Lazaar [4] proved that every quasihomeomorphism between two T_D-spaces is a homeomorphism. In this section we try to find the analogues result in L-topological space.

Recall that a topological space X is called T_D if for every x ∈ X, derived set of {x} is closed. But in case of L-topological space T_D axiom is not defined, since in some lattices a molecule can be an accumulation point of itself, for example when L = [0, 1]. In such lattices the derived set is L-closed for any L-subset and for any L-topological space. In this section, we first introduce accumulation point in such a way that any molecule is not an accumulation point of itself. Then, we give the definition of T_D so that when L = {0, 1} these separation axiom becomes T_D axiom which was introduced by Aull and Thron [17].

Definition 7.1. Let (X, δ) be an L-topological space, A, B ∈ L^X. The L-difference of A and B, denoted by A ⊢ B is defined as $\begin{matrix} \begin{matrix} A ⊢ B = & \lor {x_{λ} \in M (A) : B (x) = 0} ⋁ \\ \lor {x_{λ} \in M (A) : B (x) ≰ A (x)} . \end{matrix} \end{matrix}$

As a direct consequence of the definition of L-difference and the result that, for all A ∈ L^X, A = ∨ {x_λ ∈ M (L^X) : x_λ ≤ A}, we have the following:

Proposition 7.1. Let (X, δ) be an L-topological space, A, B and C ∈ L^X, and x_a ∈ Pt (L^X). Then the following conclusions hold:

A ⊢ B ≤ A.

$A ⊢ \underline{0} = A$ .

If supp (A) = supp (B) and B ≤ A then $A ⊢ B = \underline{0}$ .

If ∀x ∈ supp (B) , B (x) nleqA (x) , then A ⊢ B = A.

If $A \land B = \underline{0}$ then A ⊢ B = A.

If x_a≰A then A ⊢ x_a = A.

If supp (B) = supp (C) , B ≤ C then A ⊢ B ≤ A ⊢ C

Definition 7.2. Let (X, δ) be an L-topological space, A ∈ L^X and x_λ ∈ M (L^X).

x_λ is called an adherent point of A if for every U ∈ Q (x_λ), U quasi-coincides with A, i.e. $U \hat{q} A$ .

x_λ is called an accumulation point of A if x_λ is an adherent point of A ⊢ x_λ; that is for every U ∈ Q (x_λ), U is quasi-coincides with A ⊢ x_λ, i.e. $U \hat{q} (A ⊢ x_{λ})$ .

The derived set of A, denoted by A^d is defined as A^d = ∨ {x_λ ∈ M (L^X) : x_λ isanaccumulationpointofA}.

Theorem 7.1. Let (X, δ) be an L-topological space and A ∈ L^X. Then $\bar{A} = A \lor A^{d}$ .

Proof. Let x_λ be an accumulation point A. That is, x_λ is an adherent point of A ⊢ x_λ. Since A ⊢ x_λ ≤ A we get that x_λ is an adherent point of A. Hence $A \lor A^{d} \leq \bar{A}$ .

Suppose $x_{λ} \in M (\bar{A})$ and x_λ≰A. Then x_λ is an adherent point of A and x_λ≰A. By Proposition 7.1 (iv), A ⊢ x_λ = A. So x_λ is an adherent point A ⊢ x_λ, which implies x_λ ≤ A^d. Hence $\bar{A} \leq A \lor A^{d}$ . □

Remark 7.1. This new way of defining accumulation point is meaningful because we still have $\bar{A} = A \lor A^{d}$ .

When L = {0, 1}, A ⊢ B and A ⊢ x_λ becomes A \ B and A \ {x} respectively. Therefore, derived set of A becomes ordinary derived set in ordinary topology.

Theorem 7.2. Let (X, δ) be an L-topological space and x_a ∈ Pt (L^X). Then any molecule x_λ ≤ x_a is not an accumulation point of x_a.

Proof. Suppose that there exists an x_λ ∈ M (L^X) with x_λ ≤ x_a and x_λ is an accumulation point of x_a. Then any U ∈ Q (x_λ) is quasi-coincident with x_a ⊢ x_λ. By Proposition 7.1 (iii), $x_{a} ⊢ x_{λ} = \underline{0}$ , which implies that for any U ∈ Q (x_λ), $U ≰ {\underline{0}}^{'} = \underline{1}$ , which is a contradiction. So any molecule x_λ ≤ x_a is not an accumulation point of x_a. □

Remark 7.2. From this theorem, we get that every molecule is not an accumulation point of any molecule which has same support point and greater height. In some lattices the derived set of molecule is always L-closed irrespective of L-topology, and this problem can be resolved by the revised definition of accumulation point. So now we can define T_D separation axiom in L-topological space as follows:

Definition 7.3. An L-topological space is called T_D, if for any x_λ ∈ M (L^X), x_λ^d is an L-closed set.

Proposition 7.2. Let (X, δ) be an L-topological space. Then (X, δ) is T₁ if and only if ${x_{λ}}^{d} = \underline{0}$ , for every x_λ ∈ M (L^X).

Proof. For necessity part, suppose that (X, δ) is T₁. Let x_λ ∈ M (L^X). Then $\bar{x_{λ}} = x_{λ}$ which implies, all adherent point of x_λ is of the form x_μ where μ ≤ λ. But by Theorem 7.2, x_μ is not an accumulation point of x_λ. Hence ${x_{λ}}^{d} = \underline{0}$ .

For sufficiency part, suppose that ${x_{λ}}^{d} = \underline{0}$ for every x_λ ∈ M (L^X). By Theorem 7.1, $\bar{x_{λ}} = x_{λ} \lor {x_{λ}}^{d} = x_{λ} \lor \underline{0} = x_{λ}$ . Hence (X, δ) is T₁. □

Theorem 7.3. Let (X, δ) be an L-topological space then T₁ ⇒ T_D ⇒ sub-T₀.

Proof. By Proposition 7.2, we get that in T₁ space for each x_λ ∈ M (L^X), ${x_{λ}}^{d} = \underline{0}$ which is an L-closed set. Hence T₁ implies T_D.

Let (X, δ) be a T_D space and x, y ∈ X. Let α ∈ M (L) be a maximal element. That is, there does not exists any λ ∈ M (L) with α < λ. Suppose that $\bar{x_{α}} = \bar{y_{α}}$ , then y_α is an accumulation point of x_α. By T_D, x_α^d is an L-closed set. Since there does not exists any λ ∈ M (L^X) with α < λ, x_α≰x_α^d. So x_α≰x_α^d and y_α ≤ x_α^d, which is a contradiction. Hence $\bar{x_{α}} \neq \bar{y_{α}}$ and therefore (X, δ) is sub-T₀. □

Theorem 7.4. If an L-topological space (X, δ) is T_D then for each x_λ ∈ M (L^X), either x_λ ≤ x_λ^d or there exists U ∈ Q (x_λ) such that $U \land x_{λ}^{'}$ is L-open.

Proof. Suppose that (X, δ) is a T_D space and there exists a x_λ ∈ M (L^X) such that x_λ≰x_λ^d. By T_D axiom we have x_λ^d is L-closed. Then x_λ≰x_λ^d implies that x_λ is not an accumulation point of x_λ^d. Thus, we can find U ∈ Q (x_λ) such that U is not quasi-coincident with x_λ^d ⊢ x_λ. By Proposition 7.1 (vi), x_λ^d ⊢ x_λ = x_λ^d which implies $U \leq ({x_{λ}}^{d} ⊢ x_{λ})^{'} = ({x_{λ}}^{d})^{'}$

Consider the L-open set $U \land {\bar{x_{λ}}}^{'}$ , we get that

$U \land {\bar{x_{λ}}}^{'} = U \land ({x_{λ}}^{d} \lor x_{λ})^{'} = U \land ({x_{λ}}^{d})^{'} \land {x_{λ}}^{'} = U \land {x_{λ}}^{'} .$

Hence $U \land x_{λ}^{'}$ is L-open. □

Corollary 7.1. Let (X, δ) be a T_DL-topological space. Then (X, δ) is quasi-T₀ if and only if for each x_λ ∈ M (L^X) there exist U ∈ Q (x_λ) such that $U \land x_{λ}^{'}$ is L-open.

Proof. Suppose that (X, δ) is T_D and quasi-T₀ L-topological space and x_λ ∈ M (L^X). Then x_λ^d is L-closed and hence by the definition of quasi-T₀ we have x_λ≰x_λ^d. So by Theorem 7.4 there exist U ∈ Q (x_λ) such that $U \land x_{λ}^{'}$ is L-open.

Conversely suppose that (X, δ) is an L-topological space with for each x_λ ∈ M (L^X) there exist U ∈ Q (x_λ) such that $U \land x_{λ}^{'}$ is L-open. Let x_λ and x_μ be two distinguished molecules with same support point x. Then by our assumption there exists U ∈ Q (x_λ) such that $U \land x_{λ}^{'}$ is L-open. If U ∉ Q (x_μ) then Q (x_λ) ≠ Q (x_μ). Otherwise $U \land x_{λ}^{'} \in Q (x_{μ})$ and $U \land x_{λ}^{'} \notin Q (x_{λ})$ which implies Q (x_λ) ≠ Q (x_μ). Hence (X, δ) is quasi-T₀. □ Now let’s move to the main theorem in this section.

Theorem 7.5. Every quasi-L-homeomorphism between two T_D spaces is an L-homeomorphism.

Proof. Let (X, δ) and (Y, η) be two T_D L-topological spaces and let q : X → Y be a quasi-L-homeomorphism. Since every T_D space is sub-T₀, by Proposition 5.1, we get that q is injective. Let λ be a maximal element in M (L) and y ∈ Y. Then since Y is T_D and λ is a maximal element we have y_λ^d is an L-closed sets and $\bar{y_{λ}} \neq {y_{λ}}^{d}$ . Since q is a quasi-L-homeomorphism, $q^{- 1} (\bar{y_{λ}}) \neq q^{- 1} ({y_{λ}}^{d})$ . We have $\bar{y_{λ}} (z) = {y_{λ}}^{d} (z)$ for all z ≠ y. Hence we get that $\bar{y_{λ}} (q (x)) = {y_{λ}}^{d} (q (x))$ for all q (x) ≠ y. Therefore for $q^{- 1} (\bar{y_{λ}}) \neq q^{- 1} ({y_{λ}}^{d})$ , there should be an x ∈ X such that q (x) = y. Thus q is onto and bijective. By Corollary 5.1, we get that q is L-homeomorphism. □

Footnotes

Acknowledgment

The first author is supported by Senior Research Fellowship scheme of University Grants Commission (IN).

References

Aull

C.E.

and Thron

W.J.

, Separation axioms between T0 and T1 Indag, Math 24 (1962), 26–37.

Chang

C.L.

, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications 24(1) (1968), 182–190.

Chen

and Zhang

, Alexandroff L-co-topological spaces, Fuzzy Sets and Systems 161(18) (2010), 2505–2514.

Echi

and Lazaar

, Quasihomeomorphisms and lattice equivalent topological spaces, Applied General Topology 10(2) (2009), 227–237.

Ehresmann

, Gattungen von lokalen Strukturen, Jahresbericht der Deutschen Mathematiker-Vereinigung 60 (1958), 49–77.

Finch

P.D.

, On the lattice equivalence of topological spaces, Journal of the Australian Mathematical Society 6(4) (1966), 495–511.

Gierz

, Hofmann

K.H.

, Keimel

, Lawson

J.D.

, Mislove

and Scott

D.S.

, A compendium of continuous lattices, Springer Science and Business Media, 2012.

Goguen

J.A.

, L-fuzzy sets, Journal of Mathematical Analysis and Applications 18(1) (1967), 145–174.

Kai-Wing

, Quasi-homeomorphisms and lattice equivalences of topological spaces, Journal of the Australian Mathematical Society, Cambridge University Press 14(1) (1972), 41–44.

10.

Liu

Y.M.

and Luo

M.K.

, Fuzzy topology (Vol. 9). World Scientific, 1998.

11.

Liu

and Zhang

, Stone’s representation theorem in fuzzy topology, Science in China Series A: Mathematics 46 (2003), 775–788.

12.

Luo

M.K.

, Pointed dispostion of topology on lattices, Diss. PhD thesis, Sichuan University, 1992.

13.

Rodabaugh

S.E.

, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40(2) (1991), 297–345.

14.

Stone

M.H.

, Boolean algebras and their application to topology, Proceedings of the National Academy of Sciences 20(3) (1934), 197–202.

15.

Stone

M.H.

, The theory of representation for Boolean algebras, Transactions of the American Mathematical Society 40(1) (1936), 37–111.

16.

Stone

M.H.

, Topological representations of distributive lattices and Brouwerian logics, Časopis pro pěstovani matematiky a fysiky 67(1) (1938), 1–25.

17.

Thron

W.J.

, Lattice-equivalence of topological spaces, (1962), 671–679.

18.

Tomy

and Johnson

T.P.

, Some separation axioms in Ltopological spaces, Bull. Cal. Math. Soc. 114(5) (2022), 809–820.

19.

Vinitha

and Johnson

T.P.

, p-Compactness and C-p. Compactness, Global Journal of Pure and Applied Mathematics 13(9) (2017), 5539–5550.

20.

Warner

M.W.

, Frame-fuzzy points and membership, Fuzzy Sets and Systems 42(3) (1991), 335–344.

21.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Lattice Isomorphism and L -homeomorphism

Abstract

Keywords

1 Introduction

2 Preliminaries

3 pL-homeomorphism and lattice isomorphism

4 Molecule closure lattice isomorphism and L-homeomorphism

5 quasi L-homeomorphism

6 quasi L-homeomorphism and L-homeomorphism

7 T D axiom and L-homeomorphism

Footnotes

Acknowledgment

References

7 T_D axiom and L-homeomorphism