The points and diameters of quantales

Shaohui Liang

doi:10.1515/math-2018-0057

Article Open Access

The points and diameters of quantales

Shaohui Liang

Published/Copyright: June 21, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we investigate some properties of points on quantales. It is proved that the two sided prime elements are in one to one correspondence with points. By using points of quantales, we give the concepts of p-spatial quantales, and some equivalent characterizations for P-spatial quantales are obtained. It is shown that two sided quantale Q is a spatial quantale if and only if Q is a P-spatial quantale. Based on a quantale Q, we introduce the definition of diameters. We also prove that the induced topology by diameter coincides with the topology of the point spaces.

Keywords: Quantale; Point; P-spatial; Diameter; Topology

MSC 2010: 06F07; 54A05

1 Introduction

Quantale was proposed by Mulvey in 1986 to study the foundations of quantum logic and non-commutative C*-algebras. The term quantale was coined as a combination of quantum logic and locale by Mulvey in [1]. Since quantale theory provides a powerfull tool in studying non-commutative structure, it has a wide range of applications, especially in studying linear logic which supports part of the foundation of theoretical computer science [2, 3, 4, 5]. It is known that quantales are one of the semantics of linear logic, a logic system developed by Girard [6]. A systematic introduction of quantale theory can be found in book [7] written by Rosenthal in 1990. Following Mulvey, quantale theory has had a wide range of applications in computer science, logic, topological, category, C*-algebras, fuzzy theory, roughness theory and so on [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].

A point in topological space is considered to be a point space embedded in a space, which corresponds to a mapping from local to binary chain. The analogous term in commutative C*-algebras is a non-trivial representation onto C, which is a pure state of the algebra. The points of quantales are studied mainly in the context of C*-algebras [28]. A very precise way how to transfer properties of C*-algebras into terms of quantales was introduced by Mulvey and Pelletier. In paper [29, 30, 31] Pultr extended the metric structure to pontless spaces (frames or locales). The classical distance function is being replaced by the notion of diameter satisfying certain properties. Some good properties of diameter in locales are given. In this paper, we discuss the points and diameters of quantales. For the notions and concepts not explained in this paper, refer to [32].

This paper is organized as follows: In Section 2, we review some facts about quantales and category theory, which are needed in the sequel. In Section 3, we discuss some properties of points in quantales. We prove that the set of all completed files is isomorphic to the set of all points of quantales. The definition of P-spatial is given and some equivalent characterizations for P-spatial quantales are obtained. In Section 4, the definition of diameters of quantales is given. We prove that the induced topology by diameter coincides with the topology of the point spaces. In Section 5, finally, we give a summary of the paper.

2 Preliminaries

Definition 2.1

([2]). A quantale is a complete lattice Q with an associative binary operation “&” satisfying: a&(⋁i∈Ibi)=⋁i∈I(a&bi)and(⋁i∈Ibi)&a=⋁i∈Ifor all a, b_i ∈ Q, where I is a set, 0 and 1 denote the smallest element and the greatest element of Q, respectively. An element r ∈ Q is called right-sided if r&1 ≤ r. Similarly, l ∈ Q is called left-sided if 1&l ≤ l. q ∈ Q is called two-sided if q&1 ≤ q, 1&q ≤ q. The sets of right and left-sided elements will be denoted by R(Q), L(Q).

A subset K of Q is called a subquantale if K is closed under ∨ and &. Let Q and P, the function f : Q → P is called a homomorphism of quantales if f preserves ∨ and &.

Definition 2.2

([13, 14, 26]). Let Q be a quantale. A nonempty subset I ⊆ Q is called an ideal of Q if it satisfies the following conditions:

a ∨ b ∈ I for all a, b ∈ I;
for all a, b ∈ Q, if a ∈ I and b ≤ a, then b ∈ I;
for all a ∈ Q and x ∈ I, we have a&x ∈ I and x&a ∈ I.

Let Idl(Q) denote the set of all ideals of Q.

Definition 2.3

([13, 14, 26]). Let Q be a quantale. An ideal I of Q is called:

a prime ideal if I ≠ Q and for all a, b ∈ Q, a&b ∈ I implies a ∈ I or b ∈ I;
a semi-prime ideal if a&a ∈ I implies a ∈ I for all a ∈ Q;
a primary ideal if I ≠ Q, and for all a, b ∈ Q, a&b ∈ I, a ∉ I imply bⁿ ∈ I for some n > 0.

Definition 2.4

([13, 14]). Let Q be a quantale. A nonempty subset F ⊆ Q is called a file of Q if it satisfies the following conditions:

0 ∉ F;
for all a, b ∈ Q, if a ∈ F and a ≤ b, then b ∈ F;
for all a, b ∈ F, we have a&b ∈ F.

Let Fil(Q) denote the set of all files of Q.

Definition 2.5

([7]). Let Q be a quantale. A file F of Q is called a prime file if F ≠ Q and for all a, b ∈ Q, a ∨ b ∈ F implies a ∈ F or b ∈ F.

Definition 2.6

([7]). Let Q be a quantale. An element p ∈ Q, p ≠ 1 is said to be a prime if r&l ≤ p ⇒ r ≤ p or l ≤ p for all r, l ∈ Q. The set of all primes elements is denoted by P_r(Q).

Theorem 2.7

([7]). A quantale is spatial iff every element is a meet of primes.

Theorem 2.8

([30]). If A and B are categories, then a functor F from A to B is a function that assigns to each A-object A and a B-object F(A), to each A-morphism f : A → A′ and a B morphism F(f): F(A) → F(A′), in such a way that:

(i)F(f ∘ g) = F(f) ∘ F(g); (ii) F(id_A) = id_F(A).

3 The points of quantales

Definition 3.1

([3, 7]). Let Q and 2 = ({0, 1}, &) be quantales, a binary multiplication “&” of 2 satisfying: 1&1 = 1, 0&1 = 1&0 = 0&0 = 0. A mappingf : Q → 2 is said to be a point if f is a epimorphism of quantales. Let Pt(Q) denote the set of all points of Q.

Remark 3.2

Let x ∈ Q, Σ_x = {p ∈ Pt(Q)∣ p(a) = 1}.

Σa&b=Σa∩Σb,Σ0=∅,Σ1=Pt(Q),Σ⋁i∈Iai=∪i∈IΣai.
∀ p ∈ Pt(Q), then p⁻¹(1) is a prime filter and p⁻¹(0) is a prime ideal of Q. It is easy to prove that ⋁ p⁻¹(0) ∈ Pr(Q), p⁻¹(0) =↓(⋁ p⁻¹(0)).
Define a mapping f : TPr(Q)⟶Pt(Q),r⟼pr(x)=0,x≤r,1,otherwise,where TPr(Q) denotes the set of all two sided prime elements of Q. It is easy to prove that f is a bijection.
Let x ∈ Q, and x ∈ p⁻¹(1), then x^T = ⋁{y ∈ Q∣ x&y = 0} ∈ p⁻¹(0).

Theorem 3.3

Let Q be a quantale, then:

ψ : Q ⟶ P(Pt(Q)), ∀ x ∈ Q, ψ(x) = Σ_x = {p ∈ Pt(Q)∣ p(x) = 1} is a morphism of quantale.
The set Σ_x ∣ x ∈ Q is a topology of Pt(Q), denoted by Σ_Q.

Proof

It is easy to verify that (P(Pt(Q)), ∩) is a quantale and the function ψ preserves ∨.
Let ∅ ≠ S ⊆ Q, p ∈ Pt(Q), then p ∈ ψ(⋁ S) ⟺ p(⋁ S) = 1 ⟺ ⋁{p(s)∣ s ∈ S} = 1 ⟺ ∃ s₀ ∈ S, p(s₀) = 1 ⟺ p ∈ ∪ {ψ(s)∣ s ∈ S}, we have ψ(∨ S) = ∪ {ψ(s)∣ s ∈ S}.
Let p ∈ Pt(Q), ∀ a, b ∈ Q, then p ∈ ψ(a&b) ⟺ p(a&b) = 1 ⟺ p(a)&p(b) = 1 ⟺ p(a) = 1, p(b) = 1 ⟺ p ∈ψ(a), p ∈ψ(b) ⟺ p ∈ψ(a)∩ψ(b), We have ψ(a&b) = ψ(a)∩ψ(b).
It is easy to prove by definition of topology. □

Theorem 3.4

Let Q be a quantale, then:
φ : Q ⟶ P(TPr(Q)), ∀ x ∈ Q, φ(x) = {r ∈ Pr(Q) ∣ x ≰ r} is a homomorphism of quantale.
The set {φ(x)∣ x ∈ Q}is a topology of TPr(Q), denoted byΩ(TPr(Q)).

Proof

Obviously, (P(Pr(Q)), ∩) is a quantale, the function φ preserves the empty set.

Let ∅ ≠ S ⊆ Q, r ∈ Pr(Q), then r ∈ φ(⋁ S) ⟺ ⋁ S ≰ r ⟺ ∃ s ∈ S, s ≰ r ⟺ r ∈ ∪ {φ(s)∣ s ∈ S}, that is φ(⋁ S) = ∪ {φ(s)∣ s ∈ S}.

Let r ∈ Pr(Q), ∀ a, b ∈ Q, we have r ∈ φ(a&b) ⟺ a&b ≰ r ⟺ a ≰ r, b ≰ r ⟺ r ∈φ(a), r ∈φ(b) ⟺ r ∈φ(a) ∩ φ(b), that is φ(a&b) = φ(a)∩φ(b). Therefore φ is a quantale homorphism.

(2) It is easy to prove by definition of topology. □

Definition 3.5

Let Q be a quantale, the topology space (Pt(Q), Σ_Q) is called points space on Q.

Lemma 3.6

Let h : Q ⟶ K be a quantale epimorphism, the map Σ (h) : (Pt(K), Σ_K)⟶ (Pt(Q), Σ_Q), ∀ f ∈ Pt(K), Σ (h)(f) = f ∘ h, then ∀ x ∈ Q, such that (Σ (h))⁻¹(Σ_x) = Σ_h(x).

Proof

Since (Σ (h))(f) ∈ Σ_x ⇔ (Σ (h))(f)(x) = 1 ⇔ f ∈ Σ_h(x), then f ∘ h(x) = 1. □

Lemma 3.7

Let h : Q ⟶ K be a quantale epimorphism, then Σ (h) : (Pt(K), Σ_K)⟶ (Pt(Q), Σ_Q) is a continuous map.

Let Quant_e denote the category of quantales and epimorphisms, Top^op denote the dual category of topology spaces and continuous maps.

Define a map

Σ:Quante⟶TopopQ⟶(Pt(Q),ΣQ)h:Q⟶K⟶Σ(h):(Pt(K),ΣK)⟶(Pt(Q),ΣQ)f⟼f∘h.

Proof

It is easy to be verified that Σ is well defined by Lemmas 3.6 and 3.7.

∀ f ∈ Pt(Q), ∀ x ∈ Q, we have Σ (id_Q)(f)(x) = f ∘ id_Q(x) = f(x) = id_{Σ_Pt(Q)}(f)(x);
Let s : Q ⟶ P and t : P ⟶ H be quantale epimorphisms, then ∀ f ∈ Σ_H, Σ (t ∘ s)(f) = f ∘ t ∘ s = (Σ (s) ∘ Σ (t))(f) = (Σ(t) ∘^opΣ (s))(f). □

Definition 3.8

Let Q be a quantale, F is a file of Q. The file F is called completed file of Q if it satisfies the following condition: ⋁ A ∈ F ⇒ A ∩ F ≠ ∅. Let CFil(Q) denote the set of all completed files of Q.

Theorem 3.9

Let Q be a two sided quantale. We define a map φ : CFil(Q)⟶ Pt(Q) such that ∀ F ∈ CFil(Q), ∀ x ∈ Q, φ(F)(x)=0,x∉F,1,x∈F,and ψ: Pt(Q) ⟶ CFil(Q) such that ∀ f ∈ Pt(Q), ψ(f) = {x ∈ Q∣ f(x) = 1}, then φ ∘ ψ = id_CFil(Q), ψ ∘ φ = id_Pt(Q).

Proof

∀ x, y ∈ Q, ∀ F ∈ CFil(Q). Obviously, φ(F) is onto map.

φ(F)(x&y)=0,x&y∉F,1,x&y∈F,=0,x∉F,y∉F,1,x∈F,y∈F,=φ(F)(x)&φ(F)(y).
φ(F)(⋁i∈Ixi)=0,⋁i∈Ixi∉F,1,⋁i∈Ixi∈F,=0,∀i∈I,xi∉F,1,∃xi∈F,=⋁i∈Iφ(F)(xi).

Therefore the map φ is well defined.

∀ f ∈ Pt(Q), ψ(f) = {x ∈ Q∣ f(x) = 1}, we have

0 ∉ ψ(f);
∀ x₁, x₂ ∈ ψ(f), then f(x₁&x₂) = f(x₁)&f(x₂) = 1&1 = 1, thus x₁&x₂ ∈ ψ(f);
Let x ∈ ψ(f), y ∈ Q, and x ≤ y, then 1 = f(x) ≤ f(y), that is f(y) = 1;
∀ A ⊆ ψ(f), and ⋁a∈Aa∈ψ(f), then f(⋁a∈Aa)=⋁a∈Af(a)=1, there exists a₀ ∈ A, such that f(a₀) = 1, thus a₀ ∈ ψ(f), that is A ∩ ψ(f) ≠ ∅.

Hence, ψ(f) ∈ CFil(Q), the map ψ(f) is well defined.

In the following, we will prove ψ ∘ φ = id_CFil(Q) and φ ∘ ψ = id_Pt(Q).

∀ F′ ∈ CFil(Q), then ψ ∘ φ(F′) = ψ(φ(F′)) = {x ∈ Q∣ φ(F′)(x) = 1}, ∀ x ∈ {x ∈ Q∣ φ(F′)(x) = 1}, that is φ(F′)(x) = 1, then x ∈ F′, therefore ψ ∘φ(F′) ⊆ F′. ∀ x ∈ F′, then φ(F′)(x) = 1, that is F′ ⊆ ψ ∘φ(F′), therefore ψ ∘ φ = id_CFil(Q).

Conversely, ∀ f′ ∈ Pt(Q), ∀ x ∈ Q, then φ∘ψ(f′)(x)=0,x∉ψ(f′),1,x∈ψ(f′),=0,f′(x)=0,1,f′(x)=1,=f′(x). Therefore φ ∘ ψ = id_Pt(Q).

Theorem 3.10

Let Q be a two sides quantale. ∀ x ∈ Q. DefineΣx′ = {F ∈ CFil(Q)∣ x ∈ F}, then (CFil(Q), ΣQ′ = { Σx′ ∣ x ∈ Q}) is a topological spaces.

Proof

Since Σ0′=∅,Σ1′=CFil(Q), then ∅, CFil(Q) ∈ { Σx′ ∣ x ∈ Q};
Σx&y′={F∈CFil(Q)∣x&y∈F}={F∈CFil(Q)∣x∈F,y∈F}=Σx′∩Σy′;
Σ⋁i∈Ixi′={F∈CFil(Q)∣⋁i∈Ixi∈F}=⋁i∈I⋃{F∈CFil(Q)∣xi∈F}=∪i∈IΣxi′.

Therefore ΣQ′={Σx′∣x∈Q} is a topology on CFil(Q).

Let TQuant denote the category of the two sides quantales and quantale homomorphisms.

Define

Σ′:TQuant⟶TopopQ⟶(CFil(Q),ΣQ′)f:Q⟶K⟶Σ′(f):(CFil(K),ΣK′)⟶(CFil(Q),ΣQ′)F⟼f−1(F).◻

Theorem 3.11

The map Σ′ : Quant ⟶ Top^op is a functor.

Proof

Obviously, the map Σ′ is well defined. In what follows, we will prove that the map Σ′(f) : Σ′(Y) → Σ′(X) is a quantale homomorphism with f is a contionous map.

∀ U₁, U₂ ∈ Ω(Y), then Ω(f)(U₁ ∩ U₂) = f⁻¹(U₁ ∩ U₂) = f⁻¹(U₁)∩ f⁻¹(U₂) = Ω(f)(U₁) ∩ Ω(f)(U₂);
∀ {U_i}_{i ∈ I} ⊆ Ω(Y), then Ω(f)(∪i∈I⁡Ui)=f−1(∪i∈I⁡Ui)=∪i∈I⁡f−1(Ui)=∪i∈IΩ(f)(Ui),

Next, we check that the map Σ′ preserves unit element and composition.

∀ X ∈ ob(Top), then Ω(idX)(U)=idX−1(U)=U=idΩ(X)(U);
For any continous map f : X ⟶ Y and g : Y ⟶ Z, ∀ U ∈ Ω(X), then Ω(g ∘ f)(U) = (g ∘ f)⁻¹(U) = f⁻¹ ∘ g⁻¹(U) = Ω(f) ∘Ω(g)(U) = Ω(g) ∘^opΩ(f)(U).

□

Definition 3.12

Let Q be a quantale. Q is called P-spatial quantale or sufficient points if the quantale epimorphisms ψ : Q ⟶ Σ_Q, x ⟼ Σ_x is a injective function for all x ∈ Q.

Now, we shall give some charaterizations of P-spatial quantales. It is easy to prove that two sided quantale is spatial if and only if Q is p-spatial.

Theorem 3.13

Let Q be a quantale. Then the following statements are equivalent:

Q is a P-spatial quantale;
if a, b ∈ Q, a ≰ b, then there exists an element p ∈ Pt(Q), such that p(a) = 1, p(b) = 0;
if a, b ∈ Q, a ≰ b, then there exists an elemen r ∈ Pr(Q), such that a ≰ r, b ≤ r;
for all a ∈ Q, a = ∧{p ∈ Q∣ a ≤ p, p ∈ Pr(Q)}.

Proof

(1)⇒ (2) Let a, b ∈ Q, a ≰ b. Since Q is a P-spatial quantale, that is the map ψ : Q ⟶ Σ_Q is a one to one map, then a ≰ b implies ψ(a) ⊈ ψ(b), there exists p ∈ Pt(Q) such that p ∈ ψ(a), p ∉ ψ(b). Thus p(a) = 1, p(b) = 0.

(2) ⇒ (3) Let a, b ∈ Q, a ≰ b. By (2), we have that there exists p ∈ Pt(Q), that is epimorphisms p : Q ⟶ 2 such that p(a) = 1, p(b) = 0. By Remark 3.2, we have that ⋁ p⁻¹(0) ∈ Pr(Q), and p⁻¹(0) = ↓ (⋁ p⁻¹(0)), then a ∉ p⁻¹(0), b ∈ p⁻¹(0), therefore a ≰ ⋁ p⁻¹(0), b ≤ ⋁ p⁻¹(0).

(3) ⇒ (1) Let a, b ∈ Q, a ≠ b. By (3), we have that there exists r ∈ Pr(Q), such that a ≰ r, b ≤ r. By Remark 3.2, we have that ⋁ pr−1(0) = r, and pr−1(0) = ↓ (⋁ pr−1(0)) = ↓ r. Since a ≰ r, b ≤ r, then a ∉ pr−1(0), b ∈ pr−1(0), that is p_r(a) = 1, p_r(b) = 0. Therefore p_r ∈ ψ(a), p_r ∉ ψ(b), that is ψ(a) ≠ ψ(b), then ψ : Q ⟶ Σ_Q is a injection map. Thus Q is a P-spatial quantale.

(3) ⇒ (4) Let a ∈ Q, denote a^∗ = ⋀ {r ∈ Pr(Q) ∣ a ≤ r}, then a ≤ a^∗. Suppose a ≠ a^∗, then there exists r^∗ ∈ TPr(Q), such that a^∗ ≰ r^∗, a ≤ r^∗ by (3). Since r^∗ ∈ TPr(Q), and a ≤ r^∗, then a^∗ ≤ r^∗, this contradicts the assumption that a^∗ ≰ r^∗. Thus a = a^∗.

(4) ⇒ (3) Let a, b ∈ Q, a ≠ b, by (4) we have b = ⋀ {r ∈ Pr(Q) ∣ b ≤ r}, then there exists r^∗ ∈ {r ∈ Pr(Q) ∣ b ≤ r}, such that a ≰ r^∗, otherwise ∀ r ∈ {r ∈ Pr(Q) ∣ b ≤ r}, then a ≤ r, thus a ≤{r ∈ Pr(Q) ∣ b ≤ r} = b, this contradicts the assumption that a ≰ b, therefore a ≰ r^∗, b ≤ r^∗. □

Theorem 3.14

LetQbe a quantale, Kis a frame, f : Q ⟶ Kis left adjoint ofg : K ⟶ Q, (f ⊣ g). IfKis a spatial frame (P-spatial quantale) andfis a onto map, the mapfis a quantale homomorphism if and only ifgpreserve the prime elements.

Proof

It is easy to verify that f preserves arbitrary sups. Next we check if f preserves the operation &. Since K is spatial frame, then for all x, y ∈ K, we have

f(x&y)=⋀{r∈Pr(k)∣f(x&y)≤r},=⋀{r∈Pr(k)∣x&y≤g(r)},=⋀({r∈Pr(k)∣x≤g(r)}∪{r∈Pr(k)∣y≤g(r)}),=⋀({r∈Pr(k)∣f(x)≤r}∪{r∈Pr(k)∣f(y)≤r}),=f(x)∧f(y).

Hence f is a quantale homomorphism.

Conversely, let f be a quantale homomorphism, r ∈ Pr(K), then g(r) ≠ 1. Otherwise, g(r) = 1, then r ≥ f ∘ g(r) = f(1) = 1. Therefore r = 1, this is in contradiction with r ∈ Pr(K).

Let x, y ∈ Q, and x & y ≤ g(r). Since f ⊣ g, then f(x)& f(y) = f(x & y) ≤ r. Because K is a prime element of K, then f(x) ≤ r or f(y) ≤ r. Therefore g(r) is a prime element of Q, that is the map g preserves prime elements. □

4 The diameters of quantale

In this section, we introduce the concept of diametric quantale, which extends the metric structure to quantale. The classical distance function is here being replaced by the notion of diameter satisfying certain properties. We give the respective characterizations in the terms of diametric quantales.

Definition 4.1

The setCis called a conver of q quantaleQprovided thatC ⊆ Qand ∨ C = 1.

Definition 4.2

A diameter on a quantaleQis a mapping d : Q ⟶ R⁺ ∪ {0} (whereR⁺ ∪ {0} is the set of nonegative reals) satisfying:

d(0) = 0;
a ≤ b ⇒ d(a) ≤ d(b);
a&b ≠ 0⇒ d(a ∨ b) ≤ d(a) + d(b);
∀ ϵ > 0, the setUϵd = {x ∈ Q ∣ d(x) < ϵ} is a cover ofQ.

Sometimes we will drop condition (iv). In such a case, we will note that d is a prediameter.

Definition 4.3

A diameter is said to be a star diameter if for eachx ∈ Q, and Y ⊆ Qsuch that ∀ y ∈ Y, x&y ≠ 0, thend(x ∨ ⋁ Y) ≤ d(x) + Sup{d(s) + d(t) ∣ s, t ∈ Y, s ≠ t}.
A diameter is said to be a metric diameter if for eachx ∈ Q, ∀ ϵ > 0, there arey, z ∈ Qsuch that y, z ≤ x, andd(y), d(z) < ϵ, d(x) < d(y ∨ z) + ϵ.

It is easy to check that (M) implies (∗). If C is a cover of a quantale Q and x ∈ Q, we denote Cx = ⋁{c ∈ C ∣ c&x ≠ 0}.

Remark 4.4

Let {x_i}_i∈Ibe a family of quantaleQ, Cis a cover of a quantaleQ, thenC⋁i∈I⁡xi=⋁i∈I⁡Cxi.
LetCbe a cover of a quantaleQ. DefinedC : Q ⟶ Qfor ∀ x ∈ Q, C(x) = Cx, thenCpreserves ∨.

Definition 4.5

LetQbe a quantale, dbe a diameter ofQ, μ(d) = {Uϵd∣ϵ>0,ϵ∈R}is a family covers ofQ. Letαϵdbe the right adjoin ofUϵd. If ∀ x ∈ Q, x = ⋁ϵ>0⁡αϵdx. We say that the diameterdis compatible.

If x ≥ ⋁ϵ>0⁡αϵdxforx ∈ Q, we say thatdis week compatile.

Letf : Q ⟶ Pbe a quantale epimorphism, dis prediameter ofQ. We defined(y) = inf{d(x) ∣ y ≤ f(x)}.

Theorem 4.6

Letf : Q ⟶ Pbe a quantale epimorphism, dis a prediameter (diameter, ∗-diameter), thendis a prediameter (diameter, ∗-diameter).

Proof

Firstly, we prove d is a prediameter of P.

By the definition of d, we have that d(0) = inf{d(x) ∣ 0 ≤ f(x)} = 0;
Let a, b ∈ Q, and a ≤ b, then d(a) = inf{d(x) ∣ a ≤ f(x)}, d(b) = inf{d(y) ∣ b ≤ f(y)}. Since a ≤ b ≤ f(y), then d(a) ≤ d(b).
Let a, b ∈ P, and a&b ≠ 0, then d(a ∨ b) = inf{d(x) ∣ a ∨ b ≤ f(x)}, d(a) = inf{d(y) ∣ a ≤ f(y)}, d(b) = inf{d(z) ∣ b ≤ f(z)}.

Since inf{d(y) ∣ a ≤ f(y)} + inf{d(z) ∣ b ≤ f(z)} = inf{d(y) + d(z) ∣ a ≤ f(y), b ≤ f(z)}, d is a prediameter of P, and a ≤ f(y), b ≤ f(z). 0 ≠ a&b, a&b ≤ f(y&z), then f(y&z) ≠ 0, and y&z ≠ 0. We have d(y ∨ z) ≤ d(y) + d(z), a ∨ b ≤ f(y) ∨ f(z) = f(y ∨ z). Thus d(a ∨ b) ≤ d(a) + d(b).

If d is a diameter of Q, we can prove that Uϵd = {x ∈ Q ∣ d(x) < ϵ} for ϵ > 0 is a cover of Q. Since f(1) = 1, then f(⋁Uϵd)=1, and d(y) < ϵ, d(f(y)) < ϵ. Hence 1=f(⋁Uϵd)=⋁f(Uϵd)≤⋁Uϵd¯, that is ⋁Uϵd¯=1.

Secondly, we can show that d is a ∗-diameter when d is a ∗-diameter of P.

Let a ∈ P, B ⊆ P, and ∀ b ∈ B, a&b ≠ 0, then d(a) = inf{d(x) ∣ a ≤ f(x)}, d(b) = inf{d(y) ∣ b ≤ f(y)}. By the definition of infimum, for all ϵ > 0, there exist x_a,x_b ∈ Q, such that d(x_a) < d(a) + ϵ, d(x_b) < d(b) + ϵ, a ≤ f(x_a), b ≤ f(x_b), then 0 ≠ a&b ≤ f(xa∨(⋁b∈B⁡xb)), Therefore d(a ∨ ⋁ B) ≤ d(xa∨(⋁b∈B⁡xb)) ≤ d(x_a)+ Sup{d(x_s) + d(x_t) ∣ s ≠ t, s, t ∈ B} < d(a) + Sup{d(s) + d(t) ∣ s ≠ t, s, t ∈ B} + 3ϵ. Thus d is a ∗-diameter of P. □

Theorem 4.7

LetQbe a quantale, dis a diameter onQ, f : Q ⟶ Pis a epimorphism. For all p ∈ P, ∀ ϵ > 0, we defineΦ_ϵ(p) = {q ∈ Q ∣ q ∈ Uϵd, f(q)&p ≠ 0}, ϕ_ϵ(p) = ⋁ Φ_ϵ(p). We have:

LetQbe a idempotent quantale, and ∀ q ∈ Uϵd, f(q) ≠ 0, thenϕ_ϵ(f(q)) ≥ q;
ϕ_ϵ(A) = ⋁a∈Aϕ_ϵ(a) for allA ⊆ P;
Ifp ≤ f(q), then ϕ_ϵ(p) ≤ Uϵd (q).

Proof

∀ q ∈ Uϵd, f(q) ≠ 0, then ϕ_ϵ(f(q)) = ⋁ Φ_ϵ(f(q)) = ⋁ {x ∈ Uϵd ∣ f(x)&f(q) ≠ 0}. Since Q is a idempotent quantale, and f(q) ≠ 0, then ϕ_ϵ(f(q)) ≥ q.
∀ A ⊆ P, then
ϕϵ(⋁A)=⋁Φϵ(∨A)=⋁{q∈Q∣f(q)&(∨A)≠0}=⋁{q∈Q∣(⋁a∈A⁡(f(q)&a))≠0}=⋁{q∈Q∣∃a∈A,f(q)&a≠0}=⋁⋁a∈A⁡{q∈Q∣f(q)&a≠0}=⋁a∈A⁡⋁{q∈Q∣f(q)&a≠0}=⋁a∈A⁡Φϵ(a).
Since ϕ_ϵ(p) = ⋁ Φ_ϵ(p) = ⋁ {q ∈ Uϵd ∣ f(q)&p ≠ 0}, and Uϵdq = ⋁ {x ∈ Q ∣ d(x) < ϵ, x&q ≠ 0}. let A = {q ∈ Q ∣ f(q)&p ≠ 0}, B = {x ∈ Q ∣ d(x) < ϵ, x&q ≠ 0}. ∀ q ∈ A, and q ∈ Uϵd, f(q)&p ≠ 0, then 0 ≠ f(q)&p ≤ f(q&q). We can see that q&q ≠ 0, thus q ∈ B, that is ϕ_ϵ(p) ≤ Uϵdq.

□

Theorem 4.8

LetQandPbe quantales, f : Q ⟶ Pbe an epimorphism. dbe induced by a compatible diameterd, thendis compatible.

Proof

Let d(u) = inf{d(a) ∣ u ≤ f(a)} < ϵ, u&f(αϵd)(x) ≠ 0, then there exists v ∈ Q, such that d(v) < ϵ, u ≤ f(v). Since 0≠u&f(αϵd)(x)≤f(v)&f(αϵd)(x)=f(v&αϵd(x)), then f(v&αϵd(x))≠0,so thatv&αϵd(x)≠0. Thus v&(⋁{y ∈ Q ∣ Uϵdy ≤ x}) ≠ 0, that is (⋁{v&y ∣ y ∈ Q, Uϵdy ≤ x}) ≠ 0, then ∃ y₀ ∈ Q, Uϵdy₀ ≤ x, v&y₀ ≠ 0, therefore v ≤ x. Thus u ≤ f(v) ≤ f(x), Uεd¯(fαϵd(x)) ≤ f(x), then fαϵd(x)≤αϵd¯(f(x)). We have that ∀ p ∈ P, there exists x ∈ Qsuch that p = f(x), then p = f(x) ≤ f(⋁ϵ>0⁡αϵdx)=⋁ϵ>0⁡f(αϵdx)≤⋁ϵ>0⁡αϵd¯f(x)=⋁ϵ>0⁡αϵd¯p. □

Definition 4.9

Let (Q, d), (P, d′) be prediametric quantales, andf : Q ⟶ Pbe a quantale homomorphism. Ifd′(p) < ϵfor allp ∈ P, there exists an elementq ∈ Q, such thatd(q) < ϵ, p ≤ f(q), the quantale homomorphismfis called contractive.

Theorem 4.10

Let (Q, d), (P, d′), (K, d″) be prediametric quantales andfbe a quantale homomorphism. Consider the following statements:

f : (Q, d) ⟶ (P, d′) is a contractive homomorphism;
∀ ϵ > 0, x ∈ Q, we haveUϵd′∘f(x)≤f(Uϵdx);
∀ ϵ > 0, x ∈ Q, we havef∘αϵd(x)≤αϵd′f(x).

That (1) ⟹ (2) ⟺ (3).

Proof

Firstly, we will prove the implication (1) ⟹ (2).

Let p ∈ Uϵd′ and p&f(x) ≠ 0. Since f is a contractive homomorphism, we have that there exists q′ ∈ Q, such that d(q′) < ϵ, p ≤ f(q′). But p&f(x) ≠ 0, then 0 ≠ p&f(x) ≤ f(q′)&f(x) = f(q′&x), which implies that f(q′&x) ≠ 0, q′&x ≠ 0. Thus

Uϵd′f(x)=⋁{p∈P∣d′(p)<ϵp&f(x)≠0}≤{f(q)∈P∣q∈Q,d(q)<ϵ,q&x≠0}=f(⋁{q∈Q∣d(q)<ϵ,q&x≠0})=f(Uϵd(x)).

(2) ⟹ (3) Let ε > 0, x ∈ Q, then f∘αϵd(x)=f(⋁{q∈Q∣Uϵdq≤x})=⋁{f(q)∣Uϵdq≤x,q∈Q}, and αϵd′f(x)=⋁{p∈P∣Uϵd′p≤f(x)}.

Let p ∈ P, and d′(p) < ϵ, p&f(q) ≠ 0. By(2), we have Uϵd′f(q)≤f(Uϵdq), then p≤f(Uϵdq) = f(⋁{y ∈ Q ∣ d(y) < ϵ, y&q ≠ 0}) ≤ f(x). Thus ⋁{p ∈ P ∣ d′(p) < ϵ, p&f(q) ≠ 0} ≤ f(x), which implies Uϵd′f(q) ≤ f(x), thereforef(αϵd(x))≤αϵd′f(x).

(3) ⟹ (2) Since Uϵd⊣αϵd, then Uϵd′f(x)≤f(Uϵdx)⟺f(x)≤αϵdf(Uϵd′(x)). But f(x) ≤ f(αϵdUϵdx) ≤ αϵd′f(Uϵdx), which implies Uϵd′f(x)≤f(Uϵdx). □

Lemma 4.11

Let (Q, d) be a diametric quantale, anddbe a compatible diameter, β : Q ⟶ 2 is a quantale epimorphism. Leta, b ∈ Q, β(a) = 1, then there is aϵ > 0, such that for allb ∈ Qwithd(b) < ϵeitherβ (b) = 0 orb ≤ a.

Proof

Since 1 = β(a) = β(⋁ϵ>0⁡αϵd(a))=⋁ϵ>0⁡β(αϵd(a)), then there exists ϵ > 0, such that β(αϵd(a))=1. Let d(b) < ϵ, and b ≰ a, then b&αϵd(a) = 0. Otherwise, if b&αϵd(a) ≠ 0, that is b&(⋁{x ∈ Q ∣ Uϵdx ≤ a}) = ⋁{b&x ∈ Q ∣ Uϵd(x) ≤ a} ≠ 0, then there exists x ∈ Q, such that Uϵd(x) ≤ a, b&x ≠ 0, then b ≤ a, but this contradicts the assumption that b ≰ a. Thus b&αϵd(a) = 0, and 0 = β(b&αϵd(a))=β(b)&β(αϵd(a)), then β(b) = 0. Since β(b) ≠ 0, then β(b) = 1, that is β(b)&β(αϵd(a))=β(b&αϵd(a))=1, then b&αϵd(a)≠0, Therefore b ≤ a. □

Let (Q, d) be a diametric quantale, d be a compatible diameter. Pt(Q) denotes the collection of all points of Q.

Define: ρ_d : Pt(Q) × Pt(Q) ⟶ R⁺ ∪ {0}, ρ_d(ξ, η) = inf{d(a) ∣ a ∈ Q, ξ(a) = η(a) = 1}, for all ξ, η ∈ Pt(Q).

Theorem 4.12

Let (Q, d) be a diameteric quantale, and 1&1 ≠ 0, dbe a compatible diameter, thenρ_dis a metric onPt(Q) and the induced topology coincides with the topologyΣ_Q = {Σ_a ∣ a ∈ Q} ofPt(Q), Σ_a = {ξ ∣ ξ(a) = 1}.

Proof

For every element ξ, η ∈ Pt(Q), it is easy to verify that ρ_d(ξ, η) ≥ 0.
Since ⋁ Uϵd = 1 for all ϵ > 0, then ∀ ξ ∈ Pt(Q), ρ_d(ξ,ξ) = inf{d(a) ∣ ξ(a) = 1}, hence ρ_d(ξ,ξ) = 0. For all ξ, η ∈ Pt(Q), ϵ > 0, we have ρ_d(ξ, η) = inf{d(a) ∣ ξ(a) = η(a) = 1} = 0. If ξ(a) = 1, there is an element b ∈ Q, such that d(b) < ϵ, and ξ(b) = η(b) = 1. Thus b ≤ a, therefore η (a) = 1. The symmetry is obvious. If η(a) = 1, then ξ(a) = 1. Thus ξ = η.
For all ξ, η ∈ Pt(Q), ρ_d(ξ, η) = ρ_d(η,ξ) is obvious.
For all ξ, η, δ ∈ Pt(Q), in the following, we will prove that ρ_d(ξ, η) ≤ ρ_d(ξ, σ) + ρ_d(σ, η).

Let A = {a ∈ Q ∣ ξ(a) = η(a) = 1}, B = {b ∈ Q ∣ ξ(b) = δ(b) = 1}, C = {c ∈ Q ∣ δ(c) = η(c) = 1},

∀ b ∈ B, ∀ c ∈ C, ξ(b ∨ c) = ξ(b) ∨ ξ(c) = 1 ∨ ξ(c) = 1, η(b ∨ c) = η(b) ∨ η(c) = 1 ∨ η(b) = 1, then b ∨ c ∈ A. Suppose b&c = 0, then 0 = ξ(b&c) = ξ(b)& ξ(c) = 1&1, but this contradicts the fact 1&1 ≠ 0. From Definition 4.2(iii), we have d(b ∨ c) ≤ d(b)+ d(c), then inf{d(a) ∣ ξ(a) = η(a) = 1} ≤ inf{d(b) ∣ ξ(b) = δ(b) = 1} +inf{d(c) ∣ δ(c) = η(c) = 1} = inf{d(b) + d(c) ∣ ξ(b) = δ(b) = 1, δ(c) = η(c) = 1}, that is ρ_d(ξ, η) ≤ ρ_d(ξ, δ ) + ρ_d( δ, η). Thus ρ_d is a metric on Pt(Q). Therefore (Pt(Q),ρ_d) is a metric space.

In the following, we will prove the induced topology (Pt(Q), τ_{ρ_d}) as ρ_d coincides with the topology Σ_Q = {Σ_a ∣ a ∈ Q} of Pt(Q), Σ_a = {ξ ∣ ξ(a) = 1}.

∀ a ∈ A, ∀ ξ ∈ Σ_a, then ξ(a) = 1. By Lemma 4.11, there exists an ϵ > 0. If ρ_d(ξ, η) < ϵ, we have an element b ∈ Q, such that ξ(b) = η(b) = 1, and d(b) ≤ ϵ, then b ≤ a. Thus η(a) = 1, that is η ∈ Σ_a, ξ ∈ {η ∈ Pt(Q) ∣ ρ_d(ξ, η) < ϵ} ⊆ Σ_a. Hence Σ_Q ⊆ τ_{ρ_d}.

Conversely, for all ξ ∈ Pt(Q), ∀ ϵ > 0, we put a = ∨ {b ∈ Q ∣ d(b) < ϵ, ξ(b) = 1}. For all η ∈ Σ_a, there exists b ∈ Q, such that d(b) < ϵ, and ξ(b) = η(b) = 1. Therefore ρ_d(ξ, η) ≤ d(b) < ϵ. If ρ_d(ξ, η) < ϵ, then there exists an element b ∈ B, such that η(b) = ξ(b) = 1, d(b) < ϵ. Thus b ≤ a, η(a) = 1. Hence Σ_a = {η ∣ ρ_d(ξ, η) < ϵ} Bϵρd(ξ).SinceU=⋃{Bϵρd(u)∣u∈U}=⋁u∈U⁡Σau=Σ⋁u∈U⁡au for all U ∈ τ_{ρ_d}, then U ∈ Σ_Q, therefore τ_{ρ_d} ⊆ Σ_Q. □

5 Conclusion

The term quantale was coined as a combination of quantum logic and locale by Mulvey. Since quantale theory provides a powerful tool in noncommutative structures, it has a wide range of applications. In this paper, we discussed some properties of points on quantales. We proved that the set of all completed files is isomorphic to all points of quantales, and showed that two sided prime elements and points of quantales are in one to one correspondence. Furthermore, a functor from the category of the two sided quantales to the dual category of the topology was constructed. We introduced the definition of P-spatial quantales, and some equivalent characterizations for P-spatial quantales were given. The definition of diameter on frame was generalized to quantales. Finally, we proved that the topology induced by diameter coincides with the topology of the point spaces.

Acknowledgement

The author would like to thank the editors and the reviewers for their valuable comments and helpful suggestions. This work was supported by the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No.17JK0510) and the Engagement Award (2010041) and Dr. Foundation (2010QDJ024) of Xi’an University of Science and Technology, China.

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Received: 2017-12-06

Accepted: 2018-04-05

Published Online: 2018-06-21

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0057

Keywords for this article

Quantale; Point; P-spatial; Diameter; Topology

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