A non-commutative approach to uniform structures

Abstract

This paper presents a theory of uniform-type structures in the non-commutative sense. The theory comprises the theory of covering-uniformities for quantales as a generalization of the classical theory of covering-uniformities for frames. Also, by introducing quantale-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L a quantale). The categories of (covering) uniform quantales and L-valued uniform spaces here introduced and the adjunction between them is studied.

Keywords

Covering uniformities uniform frames quantales

1 Introduction

It was Tukey’s approach to uniform spaces via a system of covers that was first studied in the pointfree context of frames. In [8] Isbell introduced (covering) uniformities on frames, as the precise translation into frame terms of Tukey’s notion of a uniformity of a space expressed in terms of open covers, later developed in detail by Pultr [13, 14]. Subsequently, Frith [3] studied uniform-type structures from a more categorical point of view, introducing in frame theory other topological structures such as quasi-uniformities and proximities.

In 1986, the term quantale was proposed by C.J. Mulvey [9] as a “quantization” of the term frame. Its definition provides a non-commutative extension of the concept of frame. The intention was to develop the concept of non-commutative topology, introduced by R. Giles and H. Kummer [4], while providing constructive foundations for quantum mechanics and non-commutative logic. Further details about quantales can be found in [11] and [15].

In [7], U. Höhle proposed a non-commutative extension of the adjunction between frames and topological spaces to an adjunction between the category of quantales and the category of many valued topological spaces. In [16] Rodabaugh, used the concept of semi-quantale, as a generalization of quantale, to generalizing the Höhle-Šostak foundations for fixed-basis lattice-valued topology and the Rodabaugh foundations for variable-basis lattice-valued topology. In [2], El-Saady extended the above Höhle’s adjunction to a more general one between the category of semi-quantales and the category of lattice-valued quasi-topological spaces.

The aim of this paper to introduce and study uniform-type structures in the non-commutative sense. The paper is organized as follows. In Section 2,we will review some useful concepts about quantales, quantic conucleus and L-quasi-topologies. In Section 3, we will present a covering-like theory of uniformities for quantales and study some properties of the resulting (covering)uniform quantales. In Section 4, by introducing quantale-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L a quantale). Finally in Section 5, we introduce and study the adjunction between the category of (covering) uniform quantales and and the category of L-valued uniform spaces.

2 Preliminaries

Throughout this paper, L = (L, ≤ , ⋁ , ⋀ , ⊤ , ⊥) is a complete lattice with the smallest element ⊥ and the largest element ⊤ (⊤ ≠ ⊥).

Definition 2.1. [16]

A semi-quantale L = (L, ≤ , ⊗) is a complete lattice (L, ≤) equipped with a binary operation ⊗ : L × L ⟶ L, with no additional assumptions, called a tensor product.

A unital semi-quantale (L, ≤ , ⊗) is a semi-quantale in which ⊗ has an identity element e ∈ L called the unit. If the unit e of the groupoid (L, ⊗) coincides with the top element ⊤ of L, then a unital semi-quantale is called a strictly two-sided semi-quantale.

A commutative semi-quantate (L, ≤ , ⊗) is a semi-quantale in which, ⊗ that is, a ⊗ b = b ⊗ a for every a, b ∈ L.

A quantale [15] (L, ≤ , ⊗) is a semi-quantale whose multiplication is associative and satisfies $a \otimes (\underset{j \in J}{⋁} b_{j}) = \underset{j \in J}{⋁} (a \otimes b_{j})$ and $(\underset{j \in J}{⋁} b_{j}) \otimes a = \underset{j \in J}{⋁} (b_{j} \otimes a)$ for all a ∈ L, {b} _j∈J ⊆ L.

A frame is a unital quantale whose multiplication and unit are ∧ and ⊤ respectively.

Definition 2.2. [15] Let L be a quantale. A mapping κ : L ⟶ L is said to be a quantic conucleus on L if it satisfies:

κ is isotone, i.e., ∀ a, b ∈ L, if a ≤ b, then κ (a) ≤ κ (b);

κ is restricting, i.e., ∀ a ∈ L, κ (a) ≤ (a);

κ is idempotent, i.e., $\forall a \in L, κ (κ (a)) = κ (a);$

κ (a) ⊗ κ (b) ≤ κ (a ⊗ b) for all a, b ∈ L.

Theorem 2.3.[15] LetLbe a quantale. If the mappingκ : L ⟶ Lis a quantic conucleus onL, then $L_{κ} = {a \in L : κ (a) = a}$ (1) is a subquantale of L. Moreover, if S is any subquantale of L, then S = L_κ for some quantic conucleus κ.

A semi-quantale morphism [16] h from a semi-quantale L = (L, ≤ , ⊗) to an other semi-quantale K = (K, ≤ , ⊗) is a map h : L ⟶ K preserving the tensor product and the arbitrary joins, i.e., the next conditions are satisfied for all a, b ∈ L and {a_j : j ∈ J} ⊆ L:

h (a ⊗ b) = h (a) ⊗ h (b);

$h (\underset{j \in J}{⋁} a_{j}) = \underset{j \in J}{⋁} h (a_{j})$ .

If a semi-quantale morphism h additionally preserves the top element, i.e. h (⊤) = ⊤, then it is said to be strong. Furthermore we say that a semi-quantale morphism h between unital semi-quantales is unital iff h additionally preserves the unit element, i.e. h (e) = e. The category SQuant comprises all semi-quantales together with semi-quantale mprphisms. The non-full subcategory USQuant of SQuant comprises all unial semi-quantales and all unital semi-quantale morphisms [16]. Quant is the full subcategory of of SQuant has as objects all quantales. We define the non-full subcategory StQuant of SQuant to be the category comprising all quantales and all strong semi-quantale morphisms.

Definition 2.4. [10] Let L ∈ |Quant|.

For a, b ∈ L, we say that a is well inside b and we write $a ⊲ b \equiv \exists c \in L with c \otimes a = ⊥, c \lor b = ⊤ .$

An element a ∈ L is said to be regular if $\exists {b_{j}} \subseteq {b \in Q : b ⊲ a}$ such that a = ⋁ _jb_j .

An L ∈ |Quant| is called regular, if every element a ∈ L, is regular.

Definition 2.5. [1] Let L be a semi-quantale. A subset S ⊆ L is a subsemi-quantale of L iff it is closed under the tensor product ⊗ and arbitrary joins. A subsemi-quantale of L is said to be strong iff it contains the top element of L. If L is a unital semi-quantale with the identity e, then a subsemi-quantale S of L is called a unital subsemi-quantale of L iff e belongs to S.

For a non-empty set X and L ∈ |SQuant| let L^X be the set of all L-valued maps μ : X ⟶ L. The smallest element and the largest element in L^X are denoted by $\underline{⊥}$ and $\underline{⊤}$ , respectively. One can extend pointwisely the partial ordering and multiplication from L = (L, ≤ , ⊗) to L^X: $\begin{matrix} μ \leq ν \Leftrightarrow \forall x \in X : μ (x) \leq ν (x) . \\ (μ \otimes ν) (x) = μ (x) \otimes ν (x), x \in X . \end{matrix}$

Then L^X is again a semi-quantale with respect to the multiplication ⊗. If L is a unital semi-quantale with unit e, then L^X becomes a unital semi-quantale with the unit $\underline{e}$ .

For a fixed L ∈ |SQuant| and a set X, a subsemi-quantale τ of L^X = (L^X, ≤ , ⊗) is an L-quasi-topology on X [16], i.e., the following axioms are satisfied:

For all A, B ∈ L^X, A, B ∈ τ ⇒ A ⊗ B ∈ τ.

For all {A_j : j ∈ J} ⊆ L^X, we have ${A_{j} : j \in J} \subseteq τ \Rightarrow ⋁_{j} A_{j} \in τ .$

The pair (X, τ) is defined to be an L-quasi-topological space [16]. By L-QTop, we mean the category of all L-quasi-topological spaces and all L-continuous maps, i.e., a mapping $f : (X, τ_{1}) ⟶ (Y, τ_{2})$ provided that for all G ∈ τ₂, G ∘ f ∈ τ₂ [6, 16].

For a fixed L ∈ |SQuant|(resp., USQuant|) a subsemi-quantale (resp., subunital semi-quantale) τ of L is a semi-quantale Hutton quasi-topology (resp., Hutton topology) on L [1]. The pair (L, τ) is defined to be a semi-quantale Hutton quasi-topological space (resp., Hutton topological space).

By SQH-QTop (resp., SQH-Top) we mean the category of all semi-quantale Hutton quasi-topological spaces (resp., semi-quantale Hutton topological spaces) as objects and h : (L, τ) ⟶ (M, ν) as morphisms, where h : M ⟶ L are SQuant (resp., USQuant)-morphisms such that h maps elements of ν into τ i.e., h|_ν : ν ⟶ τ is SQuant (resp., USQuant)-morphism.

In the case of L ∈ |Quant| (resp., UQuant|), the above category is denoted by QH-QTop (resp., QH-Top).

As a consequence of Theorem 2.3, we have the following corollary:

Corollary 2.6. Let L ∈ |Quant| and κ : L ⟶ L is a quantic conucleus on L, then a subquantale

τ_{κ} = {a \in L : κ (a) = a}

(2)

is a quantale Hutton quasi-topology on L, called the quantale Hutton quasi-topology induced by the quantic conucleus κ.

3 Uniform quantales

Let L ∈ |Quant|. A cover of L is a subset $A$ of L such that $⋁ A$ equals the unit ⊤ of L. In the rest of this section, for a pair of covers $A, B$ of L, we say that $A$ refines $B$ , and write $A ⪯ B$ , if for each $a \in A$ there exists $b \in B$ such that a ≤ b.

The relation ⪯ makes the set of all covers Cov (L) of L, a preordered set. Further one defines $A \otimes B = {a \otimes b : a \in A, b \in B} .$ (3)

Lemma 3.1. For every covers $A, B$ of a quantale L, $A \otimes B$ is also a cover of L.

Proof. Since ⊗ distributes over arbitrary joins,

$⋁ (A \otimes B) = (⋁ A \otimes ⋁ B) = ⊤ \otimes ⊤ = ⊤,$

and it follows that $A \otimes B$ is a cover of L.

Example 3.2. For a quantale L and a non-empty set X, a subset $A \subseteq L^{X}$ is a cover of the quantale L^X (or a L-cover of X) if $⋁ A = \underline{⊤}$ .

For a cover $A$ of L and an element a ∈ L, we define:

$st (a, A) = ⋁ {b \in A : b \otimes a \neq ⊥}$ ; and

$st (A) = {st (a, A) : a \in A}$ .

Lemma 3.3. Let $A, B \subseteq L$ and a, b ∈ L,

If a ≤ b then $st (a, A) \leq st (b, A)$ ;

$A ⪯ B \Rightarrow st (a, A) \leq st (a, B)$ ;

$st (a, (A \otimes B)) = st (a, A) \otimes st (a, B)$ ;

$st (⋁ B, A) = \underset{b \in B}{⋁} st (b, A)$ ;

If $A \in Cov (L)$ then $st (st (a, A), A) \leq st (a, st (A))$ .

Proof. (1) and (2) are obvious.

(3) One have

$\begin{matrix} st (a, (A \otimes B)) \\ = ⋁ {x \in (A \otimes B) : x \otimes a \neq ⊥} \\ = ⋁ {c \otimes d \in (A \otimes B) : x = c \otimes d, x \otimes a \neq ⊥} \\ = ⋁ {c \otimes d : c \in A, d \in B : x = c \otimes d, x \otimes a \neq ⊥} \\ = ⋁ {c \in A : c \otimes a \neq ⊥} \otimes ⋁ {d \in B : d \otimes a \neq ⊥} \\ = st (a, A) \otimes ⋁ st (a, B) . \end{matrix}$ $\begin{matrix} (4) st (⋁ B, A) & = & ⋁ {a \in A : a \otimes (⋁ B) \neq ⊥} \\ = & ⋁ {a \in A : \underset{b \in B}{⋁} (a \otimes b) \neq ⊥} \\ = & \underset{b \in B}{⋁} {⋁ {a \in A : a \otimes b \neq ⊥}} \\ = & \underset{b \in B}{⋁} st (b, A) . \end{matrix}$ (5) Using (4), we have $st (st (a, A), A) = ⋁ {st (b, A) : b \in A, b \otimes a \neq ⊥}$ . Since $st (b, A) \in st (A)$ and $st (b, A) \otimes a \geq b \otimes a \neq ⊥$ , we have

$st (st (a, A), A) = ⋁ {st (b, A) : b \in A, b \otimes a \neq ⊥} \leq ⋁ {st (b, A) \in st (A) : st (b, A) \otimes a \neq ⊥} = st (a, st (A))$ .

Lemma 3.4. Let h : L₁ ⟶ L₂ be a quantale morphism. If $A \subseteq L_{1}$ and b ∈ L₁, then $st (h (b), h (A)) \leq h (st (b, A))$ , where $h (A) = {h (a) : a \in A} \subseteq L_{2}$ .

Proof. Since h preserves both ⋁ and ⊗, we have $\begin{matrix} st (h (b), h (A)) & = & ⋁ {h (a) \in h (A) : h (a) \otimes h (b) \\ = & h (a \otimes b) \neq ⊥} \\ = & h (⋁ {a \in A : h (a \otimes b) \neq ⊥}) \\ \leq & h (⋁ {a \in A : a \otimes b \neq ⊥}) \\ = & h (st (b, A)) . \end{matrix}$

Let $D \subseteq Cov (L)$ be a system of covers on L. The relation $\overset{D}{⊲}$ on L is defined by setting $a \overset{D}{⊲} b \equiv \exists A \in D, st (a, A) \leq b .$ (4)

Proposition 3.5. Let $A \in Cov (L)$ and a, b ∈ L such that $st (a, A) \leq b$ . Then a ⊲ b.

Proof. Put $c = ⋁ {x : x \in A, x \otimes a = ⊥}$ . We have that $c \otimes a = ⊥ and c \lor b \geq c \lor st (a, A) = ⋁ A = ⊤ .$

So by (1) of Definition 2.4, we have that a ⊲ b.

Corollary 3.6. $a \overset{D}{⊲} b \Rightarrow a ⊲ b$ .

Lemma 3.7. Let L be a quantale and $D \subseteq Cov (L)$ such that $A, B \in D \Rightarrow A \otimes B \in D$ . Then

a \overset{D}{⊲} x and b \overset{D}{⊲} y \Rightarrow a \otimes b \overset{D}{⊲} x \otimes y for a, b, x, y \in L

Proof. Let $A, B \in D$ such that $A \otimes B \in D$ and $a \overset{D}{⊲} x$ and $b \overset{D}{⊲} y$ . Then we have $st (a, A) \leq x$ and $st (b, B)$ ≤y. So $\begin{matrix} x \otimes y & \geq & st (a, A) \otimes st (b, B) \\ \geq & ⋁ {c \in A : c \otimes a \neq ⊥} \\ \otimes ⋁ {d \in B : d \otimes b \neq ⊥} \\ \geq & ⋁ {c \otimes d : c \in A, d \in B, \\ c \otimes a \neq ⊥, d \otimes b \neq ⊥} \\ \geq & ⋁ {c \otimes d \in A \otimes B : (c \otimes d) \otimes (a \otimes b) \\ = & (c \otimes a) \otimes (d \otimes b) \neq ⊥} \\ \geq & ⋁ {c \otimes d \in C : C \\ = & A \otimes B, (c \otimes d) \otimes (a \otimes b) \neq ⊥} \\ = & st (a \otimes b, C); \end{matrix}$ which means that $(a \otimes b) \overset{D}{⊲} (x \otimes y)$ .

Theorem 3.8. Let L be a quantale and $D \subseteq Cov (L)$ such that $A, B \in D \Rightarrow A \otimes B \in D$ . Then the family $L_{D} = {a \in L : a = \lor {c \in L : c \overset{D}{⊲} a}}$ (5) is a regular subquantale of L.

Proof. To show that L_𝒟 is a subquantale, we need to show that L_𝒟 is closed under ⊗ and the arbitrary joins.

Let a, b ∈ L_𝒟. By the distributivity and Lemma 3.7, we have $\begin{matrix} a \otimes b & = & ⋁ {c \in L : c \overset{D}{⊲} a}} \otimes ⋁ {d \in L : d \overset{D}{⊲} b} \\ = & ⋁ {c \otimes d : c \overset{D}{⊲} a and d \overset{D}{⊲} b} \\ \leq & ⋁ {c \otimes d : (c \otimes d) \overset{D}{⊲} (a \otimes b)} . \end{matrix}$ which means that a ⊗ b ∈ L_𝒟.

Let $a_{j} \in L_{D}$ , then $\begin{matrix} a_{j} = ⋁ {y \in L : y \overset{D}{⊲} a_{j}} \\ = ⋁ {y \in L : st (y, A) \leq a_{j} \leq ⋁ a_{j} for some A \in D} \\ \leq ⋁ {y \in L : y \overset{D}{⊲} (⋁ a_{j})} \end{matrix}$ for all j and hence $⋁ (a_{j}) \leq ⋁ {y \in L : y \overset{D}{⊲} (⋁ a_{j})} \leq (⋁ a_{j}),$ which means that $⋁ (a_{j}) \in L_{D}$

By Definition 2.4 and Corollary 3.6, we have that the subquantale L_𝒟 is regular.

A system of covers $D$ of L is said to be admissible if L = L_𝒟.

Definition 3.9.Let L be a quantale, $D \subseteq Cov (L)$ a non-empty family of covers. The pair $(L, D)$ is said to be a preuniform quantale if:

$A \in D$ and $A \subseteq B \Rightarrow B \in D$ ;

if $A, B \in D \Rightarrow A \otimes B \in D$ ;

for every $A \in D$ there is a $B \in D$ such that $st (B) \leq A$ .

A (covering) preuniformity $D$ on L that satisfies the condition L = L_𝒟 is called a (covering) uniformity on L an the pair $(L, D)$ is a (covering) uniform quantale. UniQu denotes the category of (covering) uniform quantales and the morphism between two objects $(L_{1}, D_{1})$ and $(L_{2}, D_{2})$ is a quantale morphism h : L₁ → L₂ satisfying $h (A) = {h (a) : a \in A} \in D_{2}$ for each $A \in D_{1}$ .

Definition 3.10. A quantale L is said to be uniformizable if there is a uniformity $D$ on L such that L = L_𝒟.

Proposition 3.11. Let L be a quantale and $D$ a preuniformity on L. Then $κ (a) = ⋁ {b \in L : st (b, A) \leq a for some A \in D}$ is a quanti conucleus on L, the corresponding quantale Hutton quasi-topology on L is called the quasi-topology induced by $D$ and denoted by τ_𝒟.

Proof. (QC₁) and (QC₂) are trivially satisfied.

For (QUC₃), we have

$κ (κ (a)) = ⋁ {b \in L : st (b, A) \leq κ (a) \leq a}$ . Let a, b ∈ L such that b ∈ κ (a), then $\exists A \in D$ such that $st (b, A) \leq a$ . We can find $B \in D$ with $st (B) \leq A$ . From (5) Lemma 3.3, we have $st (st (b, B), B) \leq st (b, st (B)) \leq st (b, A) \leq a$ in particular h (h (b)) ≤ a. Thus $st (b, B) \leq κ (a)$ and therefore $\begin{matrix} κ (a) & = & ⋁ {b \in L : st (b, A) \leq a for some A \in D} \\ \leq & ⋁ {b \in L : st (st (b, B), B) \\ \leq & st (b, st (B)) \leq st (b, A) \leq a for some A \in D} \\ = & ⋁ {b \in L : st (b, B) \leq κ (a) for some B \in D} \\ = & κ (κ (a)) . \end{matrix}$

(QC₄) For a, b ∈ L, we have $\begin{matrix} κ (a) \otimes κ (b) \\ = ⋁ {x \in L : st (x, A) \leq a for some \\ A \in D} \otimes ⋁ {y \in L : st (y, B) \\ \leq b for some B \in D} \\ = ⋁ {x \otimes y : x, y \in L : st (x, A) \leq a and \\ st (y, B) \leq b for some A, B \in D} \\ = ⋁ {x \otimes y : x, y \in L : st (x, A) \otimes st (y, B) \\ \leq a \otimes b for some A, B \in D} \\ = ⋁ {x \otimes y : x, y \in L : st (x \otimes y, A \otimes B) \\ \leq a \otimes b for some A, B \in D} \\ = κ (a \otimes b) . \end{matrix}$

Theorem 3.12. Let $(L, D)$ and $(P, E)$ be two uniform quantales and $f : (L, C) ⟶ (P, E)$ an UniQu-morphism. Then $f : (L, τ_{D}) ⟶ (P, τ_{ℰ})$ is an QH-QTop-morphisms. Therefore $T : UniQu ⟶ QH - QTop$ is a functor.

Proof. Suppose that $f : (L, D) ⟶ (P, E)$ an UniQu-morphism. For all a ∈ τ_ɛ, we have

$a = ⋁ {b \in P : st (b, A) \leq a$ for some $A \in E}$ . We only need to prove that f : P ⟶ L maps elements of τ_ɛ into τ_𝒟 i.e., $\begin{matrix} f (a) & \leq & ⋁ {f (b) : st (f (b), B) \\ \leq & f (a) for some B \in D} . \end{matrix}$

Since f preserves arbitrary joins, we have $\begin{matrix} f (a) & = & f (⋁ {b \in P : st (b, A) \\ \leq & a for some A \in E}) \\ = & ⋁ {f (b) : b \in P, st (f (b), f (A)) \\ \leq & f (st (b, A)) \leq f (a) for some A \in E} . \\ \leq & ⋁ {f (b) \in L : st (f (b), f (A)) \\ \leq & f (a) for some f (A) \in D} . \end{matrix}$

The inequalities follow from Lemma 3.4.

4 Covering quantale-valued uniform spaces

It is known that if L is a semi-quantale (or quantale), then so is the set L^X of all L-valued maps μ : X ⟶ L. So if we use the quantale L^X instead of L in Definition 3.9, we have the notion of L-covering uniformity on X which is nothing else but a covering uniformity on L^X.

As known, a subset $A \subseteq L^{X}$ is said to be an L-cover of X if $⋁ A = \underline{⊤}$ .

Definition 4.1. A pair $(X, U)$ , consisting of a non-empty set X and a non-empty family $U$ of L-covers of X, is called a covering L-uniform space whenever the following conditions are satisfied:

$A \in U$ and $A \subseteq B \Rightarrow B \in U$ ;

if $A, B \in U \Rightarrow A \otimes B \in U$ ;

for every $A \in U$ there is a $B \in U$ such that $st (B) \leq A$ .

A map $f : (X, U) \to (Y, V)$ is uniformly continuous if, $\forall B \in V, f^{- 1} [B] = {f^{\leftarrow} (b) : b \in B} \in U .$ (6)

The resulting category will be denoted by L-UniSp.

For the case ⊗ =∧, this is precisely the category of lattice-valued uniform spaces of [5].

Lemma 4.2.For a quantale L and $(X, U) \in | L$ -UniSp|, the mapping κ : L^X ⟶ L^X defined by $κ (μ) = {ν \in L^{X} : st (ν, A) \leq μ for some A \in U}$ is a quantic conucleus on L^X, the corresponding quasi-topology on X is denoted by τ_𝒰, called the quasi-topology induced by $U$ .

Proof. Similar to the proof of Proposition 3.11.

Proposition 4.3. Suppose that $f : (X, U) ⟶ (Y, V)$ is a uniformly continuous map between covering L-uniform spaces, then $f : (X, τ_{𝒰}) ⟶ (Y, τ_{V})$ is L-continuous.

Proof. Since f^→ : L^X ⟶ L^Y preserves both ⊗ and ⋁, so the proof becomes similar to the proof of Theorem 3.12.

Proposition 4.4. Let L be a quantale and $(X, U) \in | L$ -Unif|, then for each $A \in U$ we have $κ (A) = {κ (a) : a \in A} \in U .$ (7)

Proof. Let $A \in U$ , then by (U₃), there is a $B \in U$ such that $st (B) \leq A$ . Then $B ⪯ κ (A)$ . Now for each $b \in B$ we have $b \leq st (b, B) \leq a$ for some $a \in A$ . Thus b ≤ κ (a) which makes $κ (A)$ a L-cover and an element of $U$ .

For $(X, U) \in | L$ - Unif |, set $κ (U) = {κ (A) : A \in U} .$

Proposition 4.5. Let L be a quantale and $(X, U) \in | L$ -Unif|. Then

$a \in τ_{u} \Leftrightarrow a = \lor {b \in τ_{u} : s t (b, A) \leq a, f o r s o m e A \in k (U)} .$

Proof. The arbitrary join is trivially τ_uopen. Let a ∈ τ_u By Lemma 4.2, we have $\begin{matrix} a & = & ⋁ {b \in L^{X} : st (b, κ (A)) \leq st (b, A) \\ \leq & a for some A \in U} . \\ \leq & ⋁ {b \in τ_{U} : st (b, κ (A)) \\ \leq & st (b, A) \leq a for some A \in U)} \\ \leq & ⋁ {b \in τ_{U} : st (b, A) \\ \leq & a for some A \in U)} . \end{matrix}$

Proposition 4.6. Let $(X, U), (Y, V) \in | L$ -Unif|. A mapping $f : (X, U) ⟶ (Y, V)$ is a uniform homomorphism if and only if $f^{- 1} [B] \in κ (U) for each B \in κ (V)$ (8)

Proof. Let $\forall B \in κ (V), f^{- 1} [B] \in κ (U)$ , then by Proposition 4.4, the map $f : (X, U) ⟶ (Y, V)$ is a uniform homomorphism.

Conversely, let $f : (X, U) ⟶ (Y, V)$ is a uniform homomorphism and $B \in κ (V)$ . Since $κ (V) \in V$ , then $\begin{matrix} f^{- 1} [B] & = & {f^{\leftarrow} (b) : b \in B} \\ \geq & {f^{\leftarrow} (b) : b \in κ (B)} . \end{matrix}$

5 Duality between the categories and UniQu

In this section, we will study the duality between the category L-UniSp of covering L-uninform spaces and the category UniQu of (covering) uniform quantales. For $(X, U) \in | L$ -UniSp| set $Ω_{L} (X, U) = (τ_{U}, κ (U))$ (9) where $κ (U) = {κ (A) : A \in U}$ . By Propositions 4.4 and 4.5, $Ω_{L} (X, U)$ is a uniform quantale. Now if $f : (X, U) ⟶ (Y, V)$ is a uniformly continuous mapping, the map $Ω_{L} (f) : Ω_{L} (Y, V) ⟶ Ω_{L} (X, U),$ defined by $\begin{matrix} Ω_{L} (f : (X, U) ⟶ (Y, V)) \\ = (f_{L}^{\leftarrow}) |_{τ_{V}} : (τ_{V}, κ (V)) ⟶ (τ_{U}, κ (U)) \end{matrix}$ is a UniQu-morphism (Proposition 4.4 again), and we have a contravariant functor $Ω_{L} : L - UniSp ⟶ UniQu .$

For Q, L ∈ |Quant| put $Lpt (Q) = {p : Q ⟶ L : p \in StQuant} .$

These are called the L-points of the quantale Q and comprise all maps from Q to L preserving ⊗, arbitrary ⋁, and ⊤. On the L-powerset of Lpt (Q) we define the map φ_L as follows: $φ_{L} : Q ⟶ L^{Lpt (Q)} by φ_{L} (a) (p) = p (a)$ (10)

Then it can be shown that φ_L preserves ⊗, arbitrary ⋁, and ⊤, where these are inhertied by the codomain of φ_L from L.

Lemma 5.1. For Q, L ∈ |Quant| and $A$ is a cover of Q. Then

The family $φ_{L}^{\to} (A) \subset L^{Lpt (Q)}$ is an L-cover on Lpt (Q);

The family $φ_{L}^{\to} (Q) \subset L^{Lpt (Q)}$ is an L-valued topology on Lpt (Q);

$st (φ_{L} (a), φ_{L}^{\to} (A)) \leq φ_{L}^{\to} (st (a, A))$ .

Proof.

Since φ_L preserves both the arbitrary ⋁, and ⊤, we have $⋁ (φ_{L}^{\to} (A)) = φ_{L}^{\to} (⋁ A) = φ_{L}^{\to} (⊤) = ⊤ .$

From the definition of φ_L, one can easily prove that $φ_{L}^{\to} (Q) \subset L^{Lpt (Q)}$ is a strong subquantale of L^Lpt(Q) and therefore it is L-valued topology.

Follows as a consequence of Lemma 3.4.

Proposition 5.2. For Q, L ∈ |Quant| and a (covering) uniformity $D$ on Q, the family $φ_{L}^{\to} (D) = {φ_{L}^{\to} (A) \subset L^{Lpt (Q)} : A \in D} .$ (11) is a covering L-uniformity on Lpt (Q).

Proof.

take $A \in D$ . Then $⋁ φ_{L}^{\to} (A) = φ_{L}^{\to} (⋁ A) = φ_{L}^{\to} (⊤) = \underline{⊤} ($ the top of L^Lpt(Q)).

Let $A, B \in D$ . Trivially $φ_{L}^{\to} (A), φ_{L}^{\to} (B) \in φ_{L}^{\to} (D)$ and $A \otimes B \in D$ . So $φ_{L}^{\to} (A) \otimes φ_{L}^{\to} (B) = φ_{L}^{\to} (A \otimes B) \in φ_{L}^{\to} (D) .$

Let $A, B \in D$ with $st (B) \leq A$ . It is clear that for each $b \in B$ there exists $a \in A$ such that $st (b, B) \leq a$ , then $\begin{matrix} st (φ (b), φ_{L}^{\to} (B)) & \leq & φ_{L}^{\to} (st (b, B)) \\ \leq & φ (a) . \end{matrix}$

Hence $st (φ_{L}^{\to} (B)) \leq φ_{L}^{\to} (A)$ and this completes the proof.

For $(Q, D) \in | UniQu |$ , we have that $(Q, D) \mapsto (Lpt (Q), φ_{L}^{\to} (D))$ (12) i.e., $LPT (Q, D) = (Lpt (Q), φ_{L}^{\to} (D)) \in L$ - UniSp .

For a uniform homomorphism $h : (Q_{1}, D_{1}) ⟶ (Q_{2}, D_{2})$ define $Lpt (h) : Lpt (Q_{2}) ⟶ Lpt (Q_{1})$ (13) by $Lpt (h) (p) = p \circ h^{op} .$ (14)

Note 5.3. By Lemmas 4.2 and 5.1, we note that there is a quasi-topology $τ_{ϕ_{L}^{\to} (D)}$ and an L-valued topology $φ_{L}^{\to} (Q)$ on Lpt (Q).

Remark 5.4. If ⊗ =∧, then $τ_{ϕ_{L}^{\to} (D)}$ and $φ_{L}^{\to} (Q)$ coincide on Lpt (Q).

Lemma 5.5. For a ∈ Q₁, we have $(Lpt (h))^{\leftarrow} (φ_{L} (a)) = φ_{L} (h (a)) .$

Proof. $\begin{matrix} (Lpt (h))^{\leftarrow} (φ_{L} (a) (p) & = & (φ_{L} (a) \circ Lpt (h)) (p) \\ = & φ (a) (p) \circ Lpt (h) (p) \\ = & (p (a)) \circ (p (h^{op}) \\ = & p (a \circ h^{op}) \\ = & p (h (a)) = φ_{L} (h (a)) . \end{matrix}$

Corollary 5.6. The map $Lpt (h) : (Lpt (Q_{2}), φ_{L}^{\to} (D_{2})) ⟶ (Lpt (Q_{1}), φ_{L}^{\to} (D_{1})) \in L$ -UniSp.

Thus we obtain a contravariant functor $LPT : UniQu ⟶ L - UniSp .$

Now, we turn to study the adjunction between the functors $\begin{matrix} LPT : UniQu ⟶ L - UniSp . \\ and Ω_{L} : L - UniSp ⟶ UniQu . \end{matrix}$

To this aim we give the following definitions

For L, Q ∈ |Quant|, $(X, U) \in | L$ - UniSp |, and $(Q, D) \in | UniQu |$ define the map: $\begin{matrix} η_{(X, U)} : (X, U) \to L P T (Ω_{L} (X, U)) \end{matrix}$ by setting, $η_{(X, 𝒰)} (x) (p) = p (x)$ for every x ∈ X and $p \in τ_{𝒰}$ , and the map: $\begin{matrix} ε_{(Q, D)} : (Q,D) \to Ω_{L} (L P T (Q, D)) \end{matrix}$ by setting $ε_{(Q . 𝒟)} (a) = ϕ_{L} (a)$ .

Lemma 5.7. η _(X,𝒰) is uniform homomorphism and $ε_{(Q, 𝒟)}$ is a uniform quantale homomorphism.

Proof. Continuity of the map $\begin{matrix} η_{(X, U)} : (X, U) \to L P T (Ω_{L} (X, U)) \end{matrix}$ follows from the observation that

$LPT (Ω_{L} (X, U)) = (Lpt (τ_{U}), φ_{L}^{\to} (κ (U)))$ , where $φ_{L}^{\to} (κ (U)) = {φ_{L}^{\to} (κ (A)) : A \in U}$ .

For x ∈ X, we have that $\begin{matrix} η_{(X, U)}^{\leftarrow} (ϕ_{L}^{\to} (k (A)) (x) = ϕ_{L}^{\to} (k (A) (η_{(X, U)}) (x)) \\ = η_{(X, U)}) (x) (k (A) \\ = (k (A) (x) \end{matrix}$

It follows that $η_{X, 𝒰}^{\leftarrow} (ϕ_{L}^{\to} (κ (A)) = κ (A) \in 𝒰$ , which means that $η_{(X, 𝒰)}$ is uniformly continuous.

To show that $ε_{(Q, 𝒟)} : (Q, 𝒟) \to Ω_{L} (L P T (Q, 𝒟))$ is an UniQu-homomorphism. At first, we need to show that $ε_{(Q, 𝒟)}$ is a quantale homomorphism.

To this end let a, b, a_j ∈ Q for j ∈ J, then $\begin{matrix} (i) ε_{(Q, D)} (\underset{j}{\lor} (a_{j})) = ϕ_{L} (\underset{j}{\lor} (a_{j})) \\ = \underset{j}{\lor} (ϕ_{L} (a_{j})) \\ = \underset{j}{\lor} (ε_{(Q, D)} (a_{j})) . \end{matrix}$

$\begin{matrix} (i i) ε_{(Q, D)} (a \otimes b) = ϕ_{L} (a \otimes b) \\ = ϕ_{L} (a) \otimes ϕ_{L} (b) \\ = ε_{(Q,D)} (a) \otimes ε_{(Q,D)} (b) . \end{matrix}$

To prove that the map ε_(Q,D) is an UniQu-morphism, we let $A \in D$ , then $\begin{matrix} ε_{(Q,D)} (A) = ϕ_{L} (A) \\ = {ϕ_{L} (a) : a \in A \\ \leq k (ϕ_{L} (A)) \in k (ϕ_{L}^{\to} (D)) . \end{matrix}$

Lemma 5.8. The systems $\begin{matrix} η = (η_{(X, U)}) a n d ε = (ε_{(Q, D)}) \end{matrix}$ constitute natural transformations $\begin{matrix} η : I d_{L - U n i s p} \to L P T Ω_{L} \end{matrix}$ and $\begin{matrix} ε : I d_{U n i Q u} \to Ω_{L} L P T . \end{matrix}$

Proof. Let $f : (X, U) ⟶ (Y, V)$ is an L - UniSp -homomorphism. Then we have, for any x ∈ X $\begin{matrix} (LPT Ω_{L} (f) \circ (η_{(X, U)}) (x) (p)) \\ = ((η_{(X, U)}) (x) \circ Ω_{L} (f)) (p) \\ = η_{(X, U)}) (x) \circ (f_{L}^{\leftarrow}) |_{τ V} (P) . \\ = η_{(X, U)} (x) \circ (p \circ f) . \\ = p (f (x)) \\ = η_{(Y, V)} (f (x)) (p) . \end{matrix}$

So, we have that the following diagram commutes.

On the other hand, if $h : (Q, D) ⟶ (P, C)$ be an UniQu-homomorphism, then for all a ∈ Q we have $\begin{matrix} (Ω_{L} LPT (h) (ɛ_{(Q, D)} (a))) & = & (LPT (h))^{\leftarrow} (φ_{L} (a)) \\ = & φ_{L} (h (a)) = ɛ_{(P, C)} (h (a)) \end{matrix}$ and this makes the following diagram commutes

By the above two lemmas, we have the following result

Theorem 5.9. The functors $LPT : UniQu ⟶ L - UniSp$ and $Ω_{L} : L - UniSp ⟶ UniQu$ constitute a dual adjunction.

Footnotes

Acknowledgments

The author would like to thank the anonymous referees and the editors for their valuable comments and helpful suggestions.

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