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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# The points and diameters of quantales

Shaohui Liang
• Corresponding author
• Department of Mathematics, Xi’an University of Science and Technology, Xi’an 710054, China
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Published Online: 2018-06-21 | DOI: https://doi.org/10.1515/math-2018-0057

## 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: $\begin{array}{}a\mathrm{&}\left(\underset{i\in I}{\bigvee }{b}_{i}\right)=\underset{i\in I}{\bigvee }\left(a\mathrm{&}{b}_{i}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\underset{i\in I}{\bigvee }{b}_{i}\right)\mathrm{&}a=\underset{i\in I}{\bigvee }\end{array}$ for all a, biQ, where I is a set, 0 and 1 denote the smallest element and the greatest element of Q, respectively. An element rQ is called right-sided if r&1 ≤ r. Similarly, lQ is called left-sided if 1&ll. qQ is called two-sided if q&1 ≤ q, 1&qq. 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 underand &. Let Q and P, the function f : QP is called a homomorphism of quantales if f preservesand &.

#### Definition 2.2

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

1. abI for all a, bI;

2. for all a, bQ, if aI and ba, then bI;

3. for all aQ and xI, we have a&xI and x&aI.

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:

1. a prime ideal if IQ and for all a, bQ, a&bI implies a ∈ I or bI;

2. a semi-prime ideal if a&aI implies aI for all aQ;

3. a primary ideal if IQ, and for all a, bQ, a&bI, aI imply bnI for some n > 0.

#### Definition 2.4

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

1. 0 ∉ F;

2. for all a, bQ, if aF and ab, then bF;

3. for all a, bF, we have a&bF.

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 FQ and for all a, bQ, abF implies aF or bF.

#### Definition 2.6

([7]). Let Q be a quantale. An element pQ, p ≠ 1 is said to be a prime if r&lprp or lp for all r, lQ. The set of all primes elements is denoted by Pr(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 : AAand a B morphism F(f): F(A) → F(A′), in such a way that:

(i)F(fg) = F(f) ∘ F(g); (ii) F(idA) = idF(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 mapping f : 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 xQ, Σx = {pPt(Q)∣ p(a) = 1}.

1. $\begin{array}{}{\mathit{\Sigma }}_{a\mathrm{&}b}={\mathit{\Sigma }}_{a}\cap {\mathit{\Sigma }}_{b},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Sigma }}_{0}=\mathrm{\varnothing },\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Sigma }}_{1}=Pt\left(Q\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Sigma }}_{\underset{i\in I}{\bigvee }{a}_{i}}=\underset{i\in I}{\cup }{\mathit{\Sigma }}_{{a}_{i}}.\end{array}$

2. pPt(Q), then p−1(1) is a prime filter and p−1(0) is a prime ideal of Q. It is easy to prove thatp−1(0) ∈ Pr(Q), p−1(0) =↓(⋁ p−1(0)).

3. Define a mapping f : $\begin{array}{}TPr\left(Q\right)⟶Pt\left(Q\right),r⟼{p}_{r}\left(x\right)=\left\{\begin{array}{ll}0,& x\le r,\\ 1,& otherwise,\end{array}\right\\end{array}$ where TPr(Q) denotes the set of all two sided prime elements of Q. It is easy to prove that f is a bijection.

4. Let xQ, and xp−1(1), then xT = ⋁{yQx&y = 0} ∈ p−1(0).

#### Theorem 3.3

Let Q be a quantale, then:

1. ψ : QP(Pt(Q)), ∀ xQ, ψ(x) = Σx = {pPt(Q)∣ p(x) = 1} is a morphism of quantale.

2. The set ΣxxQ is a topology of Pt(Q), denoted by ΣQ.

#### Proof

1. It is easy to verify that (P(Pt(Q)), ∩) is a quantale and the function ψ preserves ∨.

Let ∅ ≠ SQ, pPt(Q), then pψ(⋁ S) ⟺ p(⋁ S) = 1 ⟺ ⋁{p(s)∣ sS} = 1 ⟺ ∃ s0S, p(s0) = 1 ⟺ p ∈ ∪ {ψ(s)∣ sS}, we have ψ(∨ S) = ∪ {ψ(s)∣ sS}.

Let pPt(Q), ∀ a, bQ, then pψ(a&b) ⟺ p(a&b) = 1 ⟺ p(a)&p(b) = 1 ⟺ p(a) = 1, p(b) = 1 ⟺ pψ(a), pψ(b) ⟺ pψ(a)∩ψ(b), We have ψ(a&b) = ψ(a)∩ψ(b).

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

#### Theorem 3.4

1. Let Q be a quantale, then:

φ : QP(TPr(Q)), ∀ xQ, φ(x) = {rPr(Q) ∣ xr} is a homomorphism of quantale.

2. The set {φ(x)∣ xQ}is a topology of TPr(Q), denoted by Ω(TPr(Q)).

#### Proof

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

Let ∅ ≠ SQ, rPr(Q), then rφ(⋁ S) ⟺ ⋁ Sr ⟺ ∃ sS, srr ∈ ∪ {φ(s)∣ sS}, that is φ(⋁ S) = ∪ {φ(s)∣ sS}.

Let rPr(Q), ∀ a, bQ, we have rφ(a&b) ⟺ a&brar, brrφ(a), rφ(b) ⟺ rφ(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), ΣQ) is called points space on Q.

#### Lemma 3.6

Let h : QK be a quantale epimorphism, the map Σ (h) : (Pt(K), ΣK)⟶ (Pt(Q), ΣQ), ∀ fPt(K), Σ (h)(f) = fh, thenxQ, such that (Σ (h))−1(Σx) = Σh(x).

#### Proof

Since (Σ (h))(f) ∈ Σx ⇔ (Σ (h))(f)(x) = 1 ⇔ fΣh(x), then fh(x) = 1. □

#### Lemma 3.7

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

Let Quante denote the category of quantales and epimorphisms, Topop 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.

1. fPt(Q), ∀ xQ, we have Σ (idQ)(f)(x) = fidQ(x) = f(x) = idΣPt(Q)(f)(x);

2. Let s : QP and t : PH be quantale epimorphisms, then ∀ fΣH, Σ (ts)(f) = fts = (Σ (s) ∘ Σ (t))(f) = (Σ(t) ∘op Σ (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: ⋁ AFAF ≠ ∅. 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 thatFCFil(Q), ∀ xQ, $\begin{array}{}\phi \left(F\right)\left(x\right)=\left\{\begin{array}{l}0,x\notin F,\\ 1,x\in F,\end{array}\right\\end{array}$ and ψ: Pt(Q) ⟶ CFil(Q) such thatfPt(Q), ψ(f) = {xQf(x) = 1}, then φψ = idCFil(Q), ψφ = idPt(Q).

#### Proof

x, yQ, ∀ FCFil(Q). Obviously, φ(F) is onto map.

1. $\begin{array}{}\phi \left(F\right)\left(x\mathrm{&}y\right)=\left\{\begin{array}{l}0,\phantom{\rule{thinmathspace}{0ex}}x\mathrm{&}y\notin F,\\ 1,\phantom{\rule{thinmathspace}{0ex}}x\mathrm{&}y\in F,\end{array}\right\=\left\{\begin{array}{l}0,\phantom{\rule{thinmathspace}{0ex}}x\notin F,y\notin F,\\ 1,\phantom{\rule{thinmathspace}{0ex}}x\in F,\phantom{\rule{thinmathspace}{0ex}}y\in F,\end{array}\right\=\phi \left(F\right)\left(x\right)\mathrm{&}\phi \left(F\right)\left(y\right).\end{array}$

2. $\begin{array}{}\phi \left(F\right)\left(\underset{i\in I}{\bigvee }{x}_{i}\right)=\left\{\begin{array}{l}0,\phantom{\rule{thinmathspace}{0ex}}\underset{i\in I}{\bigvee }{x}_{i}\notin F,\\ 1,\phantom{\rule{thinmathspace}{0ex}}\underset{i\in I}{\bigvee }{x}_{i}\in F,\end{array}\right\=\left\{\begin{array}{l}0,\phantom{\rule{thinmathspace}{0ex}}\mathrm{\forall }\phantom{\rule{thinmathspace}{0ex}}i\in I,{x}_{i}\notin F,\\ 1,\mathrm{\exists }\phantom{\rule{thinmathspace}{0ex}}{x}_{i}\in F,\end{array}\right\=\underset{i\in I}{\bigvee }\phi \left(F\right)\left({x}_{i}\right).\end{array}$

Therefore the map φ is well defined.

fPt(Q), ψ(f) = {xQf(x) = 1}, we have

1. 0 ∉ ψ(f);

2. x1, x2ψ(f), then f(x1&x2) = f(x1)&f(x2) = 1&1 = 1, thus x1&x2ψ(f);

3. Let xψ(f), y ∈ Q, and xy, then 1 = f(x) ≤ f(y), that is f(y) = 1;

4. Aψ(f), and $\begin{array}{}\underset{a\in A}{\bigvee }a\in \psi \left(f\right),\text{\hspace{0.17em}then\hspace{0.17em}}f\left(\underset{a\in A}{\bigvee }a\right)=\underset{a\in A}{\bigvee }f\left(a\right)=1,\end{array}$ there exists a0A, such that f(a0) = 1, thus a0ψ(f), that is Aψ(f) ≠ ∅.

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

In the following, we will prove ψφ = idCFil(Q) and φψ = idPt(Q).

F′ ∈ CFil(Q), then ψφ(F′) = ψ(φ(F′)) = {xQφ(F′)(x) = 1}, ∀ x ∈ {xQφ(F′)(x) = 1}, that is φ(F′)(x) = 1, then xF′, therefore ψφ(F′) ⊆ F′. ∀ xF′, then φ(F′)(x) = 1, that is F′ ⊆ ψφ(F′), therefore ψφ = idCFil(Q).

Conversely, ∀ f′ ∈ Pt(Q), ∀ xQ, then $\begin{array}{}\phi \circ \psi \left({f}^{\prime }\right)\left(x\right)=\left\{\begin{array}{l}0,x\notin \psi \left({f}^{\prime }\right),\\ 1,x\in \psi \left({f}^{\prime }\right),\end{array}\right\=\left\{\begin{array}{l}0,\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime }\left(x\right)=0,\\ 1,\phantom{\rule{thinmathspace}{0ex}}{f}^{\prime }\left(x\right)=1,\end{array}\right\={f}^{\prime }\left(x\right).\end{array}$ Therefore φψ = idPt(Q).

#### Theorem 3.10

Let Q be a two sides quantale. ∀ xQ. Define $\begin{array}{}{\mathit{\Sigma }}_{x}^{\prime }\end{array}$ = {FCFil(Q)∣ xF}, then (CFil(Q), $\begin{array}{}{\mathit{\Sigma }}_{Q}^{\prime }\end{array}$ = { $\begin{array}{}{\mathit{\Sigma }}_{x}^{\prime }\end{array}$xQ}) is a topological spaces.

#### Proof

1. Since $\begin{array}{}{\mathit{\Sigma }}_{0}^{\prime }=\mathrm{\varnothing },\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Sigma }}_{1}^{\prime }=CFil\left(Q\right),\end{array}$ then ∅, CFil(Q) ∈ { $\begin{array}{}{\mathit{\Sigma }}_{x}^{\prime }\end{array}$xQ};

2. $\begin{array}{}{\mathit{\Sigma }}_{x\mathrm{&}y}^{\prime }=\left\{F\in CFil\left(Q\right)\mid x\mathrm{&}y\in F\right\}=\left\{F\in CFil\left(Q\right)\mid x\in F,\phantom{\rule{thinmathspace}{0ex}}y\in F\right\}={\mathit{\Sigma }}_{x}^{\prime }\cap {\mathit{\Sigma }}_{y}^{\prime };\end{array}$

3. $\begin{array}{}{\mathit{\Sigma }}_{\underset{i\in I}{\bigvee }{x}_{i}}^{\prime }=\left\{F\in CFil\left(Q\right)\mid \underset{i\in I}{\bigvee }{x}_{i}\in F\right\}=\underset{i\in I}{\bigvee }\bigcup \left\{F\in CFil\left(Q\right)\mid {x}_{i}\in F\right\}=\underset{i\in I}{\cup }{\mathit{\Sigma }}_{{x}_{i}}^{\prime }.\end{array}$

Therefore $\begin{array}{}{\mathit{\Sigma }}_{Q}^{\prime }=\left\{{\mathit{\Sigma }}_{x}^{\prime }\mid x\in Q\right\}\end{array}$ 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′ : QuantTopop 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.

1. U1, U2Ω(Y), then Ω(f)(U1U2) = f−1(U1U2) = f−1(U1)∩ f−1(U2) = Ω(f)(U1) ∩ Ω(f)(U2);

2. ∀ {Ui}iIΩ(Y), then $\begin{array}{}\mathit{\Omega }\left(f\right)\left(\underset{i\in I}{\cup }{U}_{i}\right)={f}^{-1}\left(\underset{i\in I}{\cup }{U}_{i}\right)=\underset{i\in I}{\cup }{f}^{-1}\left({U}_{i}\right)=\underset{i\in I}{\cup }\mathit{\Omega }\left(f\right)\left({U}_{i}\right),\end{array}$

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

1. Xob(Top), then $\begin{array}{}\mathit{\Omega }\left(i{d}_{X}\right)\left(U\right)=i{d}_{X}^{-1}\left(U\right)=U=i{d}_{\mathit{\Omega }\left(X\right)}\left(U\right);\end{array}$

2. For any continous map f : XY and g : YZ, ∀ UΩ(X), then Ω(gf)(U) = (gf)−1(U) = f−1g−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 : QΣQ, xΣx is a injective function for all xQ.

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:

1. Q is a P-spatial quantale;

2. if a, bQ, ab, then there exists an element pPt(Q), such that p(a) = 1, p(b) = 0;

3. if a, bQ, ab, then there exists an elemen rPr(Q), such that ar, br;

4. for all aQ, a = ∧{pQap, pPr(Q)}.

#### Proof

(1)⇒ (2) Let a, bQ, ab. Since Q is a P-spatial quantale, that is the map ψ : QΣQ is a one to one map, then ab implies ψ(a) ⊈ ψ(b), there exists pPt(Q) such that pψ(a), pψ(b). Thus p(a) = 1, p(b) = 0.

(2) ⇒ (3) Let a, bQ, ab. By (2), we have that there exists pPt(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 ap−1(0), bp−1(0), therefore a ≰ ⋁ p−1(0), b ≤ ⋁ p−1(0).

(3) ⇒ (1) Let a, bQ, ab. By (3), we have that there exists rPr(Q), such that ar, br. By Remark 3.2, we have that ⋁ $\begin{array}{}{p}_{r}^{-1}\end{array}$(0) = r, and $\begin{array}{}{p}_{r}^{-1}\end{array}$(0) = ↓ (⋁ $\begin{array}{}{p}_{r}^{-1}\end{array}$(0)) = ↓ r. Since ar, br, then a$\begin{array}{}{p}_{r}^{-1}\end{array}$(0), b$\begin{array}{}{p}_{r}^{-1}\end{array}$(0), that is pr(a) = 1, pr(b) = 0. Therefore prψ(a), prψ(b), that is ψ(a) ≠ ψ(b), then ψ : QΣQ is a injection map. Thus Q is a P-spatial quantale.

(3) ⇒ (4) Let aQ, denote a = ⋀ {rPr(Q) ∣ ar}, then aa. Suppose aa, then there exists rTPr(Q), such that ar, ar by (3). Since rTPr(Q), and ar, then ar, this contradicts the assumption that ar. Thus a = a.

(4) ⇒ (3) Let a, bQ, ab, by (4) we have b = ⋀ {rPr(Q) ∣ br}, then there exists r ∈ {rPr(Q) ∣ br}, such that ar, otherwise ∀ r ∈ {rPr(Q) ∣ br}, then ar, thus a ≤{rPr(Q) ∣ br} = b, this contradicts the assumption that ab, therefore ar, br. □

#### Theorem 3.14

Let Q be a quantale, K is a frame, f : QK is left adjoint of g : KQ, (fg). If K is a spatial frame (P-spatial quantale) and f is a onto map, the map f is a quantale homomorphism if and only if g preserve 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, yK, 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, rPr(K), then g(r) ≠ 1. Otherwise, g(r) = 1, then rfg(r) = f(1) = 1. Therefore r = 1, this is in contradiction with rPr(K).

Let x, yQ, and x & yg(r). Since fg, 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 set C is called a conver of q quantale Q provided that CQ andC = 1.

#### Definition 4.2

A diameter on a quantale Q is a mapping d : QR+ ∪ {0} (where R+ ∪ {0} is the set of nonegative reals) satisfying:

1. d(0) = 0;

2. abd(a) ≤ d(b);

3. a&b ≠ 0⇒ d(ab) ≤ d(a) + d(b);

4. ϵ > 0, the set $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ = {xQd(x) < ϵ} is a cover of Q.

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 each xQ, and YQ such thatyY, x&y ≠ 0, then d(x ∨ ⋁ Y) ≤ d(x) + Sup{d(s) + d(t) ∣ s, tY, st}.

• (M)

A diameter is said to be a metric diameter if for each xQ, ∀ ϵ > 0, there are y, zQ such that y, zx, and d(y), d(z) < ϵ, d(x) < d(yz) + ϵ.

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

#### Remark 4.4

1. Let {xi}iI be a family of quantale Q, C is a cover of a quantale Q, then $\begin{array}{}{C}_{\underset{i\in I}{\bigvee }{x}_{i}}=\underset{i\in I}{\bigvee }{C}_{{x}_{i}}.\end{array}$

2. Let C be a cover of a quantale Q. Defined C : QQfor ∀ xQ, C(x) = Cx, then C preserves ∨.

#### Definition 4.5

Let Q be a quantale, d be a diameter of Q, μ(d) = $\begin{array}{}\left\{{U}_{ϵ}^{d}\mid ϵ>0,\phantom{\rule{thinmathspace}{0ex}}ϵ\in R\right\}\end{array}$ is a family covers of Q. Let $\begin{array}{}{\alpha }_{ϵ}^{d}\end{array}$ be the right adjoin of $\begin{array}{}{U}_{ϵ}^{d}\end{array}$. IfxQ, x = $\begin{array}{}\underset{ϵ>0}{\bigvee }{\alpha }_{ϵ}^{d}x\end{array}$. We say that the diameter d is compatible.

If x$\begin{array}{}\underset{ϵ>0}{\bigvee }{\alpha }_{ϵ}^{d}x\end{array}$ for xQ, we say that d is week compatile.

Let f : QP be a quantale epimorphism, d is prediameter of Q. We define d(y) = inf{d(x) ∣ yf(x)}.

#### Theorem 4.6

Let f : QP be a quantale epimorphism, d is a prediameter (diameter, ∗-diameter), then d is a prediameter (diameter, ∗-diameter).

#### Proof

Firstly, we prove d is a prediameter of P.

1. By the definition of d, we have that d(0) = inf{d(x) ∣ 0 ≤ f(x)} = 0;

2. Let a, bQ, and ab, then d(a) = inf{d(x) ∣ af(x)}, d(b) = inf{d(y) ∣ bf(y)}. Since abf(y), then d(a) ≤ d(b).

3. Let a, bP, and a&b ≠ 0, then d(ab) = inf{d(x) ∣ abf(x)}, d(a) = inf{d(y) ∣ af(y)}, d(b) = inf{d(z) ∣ bf(z)}.

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

If d is a diameter of Q, we can prove that $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ = {xQd(x) < ϵ} for ϵ > 0 is a cover of Q. Since f(1) = 1, then $\begin{array}{}f\left(\bigvee {U}_{ϵ}^{d}\right)=1,\end{array}$ and d(y) < ϵ, d(f(y)) < ϵ. Hence $\begin{array}{}1=f\left(\bigvee {U}_{ϵ}^{d}\right)=\bigvee f\left({U}_{ϵ}^{d}\right)\le \bigvee {U}_{ϵ}^{\overline{d}},\end{array}$ that is $\begin{array}{}\bigvee {U}_{ϵ}^{\overline{d}}=1.\end{array}$

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

Let aP, BP, and ∀ bB, a&b ≠ 0, then d(a) = inf{d(x) ∣ af(x)}, d(b) = inf{d(y) ∣ bf(y)}. By the definition of infimum, for all ϵ > 0, there exist xa,xbQ, such that d(xa) < d(a) + ϵ, d(xb) < d(b) + ϵ, af(xa), bf(xb), then 0 ≠ a&b$\begin{array}{}f\left({x}_{a}\vee \left(\underset{b\in B}{\bigvee }{x}_{b}\right)\right),\end{array}$ Therefore d(a ∨ ⋁ B) ≤ $\begin{array}{}d\left({x}_{a}\vee \left(\underset{b\in B}{\bigvee }{x}_{b}\right)\right)\end{array}$d(xa)+ Sup{d(xs) + d(xt) ∣ st, s, tB} < d(a) + Sup{d(s) + d(t) ∣ st, s, tB} + 3ϵ. Thus d is a ∗-diameter of P. □

#### Theorem 4.7

Let Q be a quantale, d is a diameter on Q, f : QP is a epimorphism. For all pP, ∀ ϵ > 0, we define Φϵ(p) = {qQq$\begin{array}{}{U}_{ϵ}^{d}\end{array}$, f(q)&p ≠ 0}, ϕϵ(p) = ⋁ Φϵ(p). We have:

1. Let Q be a idempotent quantale, andq$\begin{array}{}{U}_{ϵ}^{d}\end{array}$, f(q) ≠ 0, then ϕϵ(f(q)) ≥ q;

2. ϕϵ(A) = $\begin{array}{}\underset{a\in A}{\bigvee }\end{array}$ ϕϵ(a) for all AP;

3. If pf(q), then ϕϵ(p) ≤ $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ (q).

#### Proof

1. q$\begin{array}{}{U}_{ϵ}^{d}\end{array}$, f(q) ≠ 0, then ϕϵ(f(q)) = ⋁ Φϵ(f(q)) = ⋁ {x$\begin{array}{}{U}_{ϵ}^{d}\end{array}$f(x)&f(q) ≠ 0}. Since Q is a idempotent quantale, and f(q) ≠ 0, then ϕϵ(f(q)) ≥ q.

2. AP, then

$ϕϵ(⋁A)=⋁Φϵ(∨A)=⋁{q∈Q∣f(q)&(∨A)≠0}=⋁{q∈Q∣(⋁a∈A⁡(f(q)&a))≠0}=⋁{q∈Q∣∃a∈A,f(q)&a≠0}=⋁⋁a∈A⁡{q∈Q∣f(q)&a≠0}=⋁a∈A⁡⋁{q∈Q∣f(q)&a≠0}=⋁a∈A⁡Φϵ(a).$

3. Since ϕϵ(p) = ⋁ Φϵ(p) = ⋁ {q$\begin{array}{}{U}_{ϵ}^{d}\end{array}$f(q)&p ≠ 0}, and $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ q = ⋁ {xQd(x) < ϵ, x&q ≠ 0}. let A = {qQf(q)&p ≠ 0}, B = {xQd(x) < ϵ, x&q ≠ 0}. ∀ qA, and q$\begin{array}{}{U}_{ϵ}^{d}\end{array}$, f(q)&p ≠ 0, then 0 ≠ f(q)&pf(q&q). We can see that q&q ≠ 0, thus qB, that is ϕϵ(p) ≤ $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ q.

□

#### Theorem 4.8

Let Q and P be quantales, f : QP be an epimorphism. d be induced by a compatible diameter d, then d is compatible.

#### Proof

Let d(u) = inf{d(a) ∣ uf(a)} < ϵ, $\begin{array}{}u\mathrm{&}f\left({\alpha }_{ϵ}^{d}\right)\left(x\right)\end{array}$ ≠ 0, then there exists vQ, such that d(v) < ϵ, uf(v). Since $\begin{array}{}0\ne u\mathrm{&}f\left({\alpha }_{ϵ}^{d}\right)\left(x\right)\le f\left(v\right)\mathrm{&}f\left({\alpha }_{ϵ}^{d}\right)\left(x\right)=f\left(v\mathrm{&}{\alpha }_{ϵ}^{d}\left(x\right)\right),\end{array}$ then $\begin{array}{}f\left(v\mathrm{&}{\alpha }_{ϵ}^{d}\left(x\right)\right)\ne 0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{so that}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}v\mathrm{&}{\alpha }_{ϵ}^{d}\left(x\right)\ne 0.\end{array}$ Thus v&(⋁{yQ$\begin{array}{}{U}_{ϵ}^{d}\end{array}$ yx}) ≠ 0, that is (⋁{v&yyQ, $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ yx}) ≠ 0, then ∃ y0Q, $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ y0x, v&y0 ≠ 0, therefore vx. Thus uf(v) ≤ f(x), $\begin{array}{}{U}_{\epsilon }^{\overline{d}}\left(f{\alpha }_{ϵ}^{d}\left(x\right)\right)\end{array}$f(x), then $\begin{array}{}f{\alpha }_{ϵ}^{d}\left(x\right)\le {\alpha }_{ϵ}^{\overline{d}}\left(f\left(x\right)\right).\end{array}$ We have that ∀ pP, there exists xQsuch that p = f(x), then p = f(x) ≤ $\begin{array}{}f\left(\underset{ϵ>0}{\bigvee }{\alpha }_{ϵ}^{d}x\right)=\underset{ϵ>0}{\bigvee }f\left({\alpha }_{ϵ}^{d}x\right)\le \underset{ϵ>0}{\bigvee }{\alpha }_{ϵ}^{\overline{d}}f\left(x\right)=\underset{ϵ>0}{\bigvee }{\alpha }_{ϵ}^{\overline{d}}p.\end{array}$ □

#### Definition 4.9

Let (Q, d), (P, d′) be prediametric quantales, and f : QP be a quantale homomorphism. If d′(p) < ϵ for all pP, there exists an element qQ, such that d(q) < ϵ, pf(q), the quantale homomorphism f is called contractive.

#### Theorem 4.10

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

1. f : (Q, d) ⟶ (P, d′) is a contractive homomorphism;

2. ϵ > 0, xQ, we have $\begin{array}{}{U}_{ϵ}^{{d}^{\prime }}\circ f\left(x\right)\le f\left({U}_{ϵ}^{d}x\right);\end{array}$

3. ϵ > 0, xQ, we have $\begin{array}{}f\circ {\alpha }_{ϵ}^{d}\left(x\right)\le {\alpha }_{ϵ}^{{d}^{\prime }}f\left(x\right).\end{array}$

That (1) ⟹ (2) ⟺ (3).

#### Proof

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

Let p$\begin{array}{}{U}_{ϵ}^{{d}^{\prime }}\end{array}$ and p&f(x) ≠ 0. Since f is a contractive homomorphism, we have that there exists q′ ∈ Q, such that d(q′) < ϵ, pf(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, xQ, then $\begin{array}{}f\circ {\alpha }_{ϵ}^{d}\left(x\right)=f\left(\bigvee \left\{q\in Q\mid {U}_{ϵ}^{d}q\le x\right\}\right)=\bigvee \left\{f\left(q\right)\mid {U}_{ϵ}^{d}q\le x,\phantom{\rule{thinmathspace}{0ex}}q\in Q\right\},\end{array}$ and $\begin{array}{}{\alpha }_{ϵ}^{{d}^{\prime }}f\left(x\right)=\bigvee \left\{p\in P\mid {U}_{ϵ}^{{d}^{\prime }}p\le f\left(x\right)\right\}.\end{array}$

Let pP, and d′(p) < ϵ, p&f(q) ≠ 0. By(2), we have $\begin{array}{}{U}_{ϵ}^{{d}^{\prime }}f\left(q\right)\le f\left({U}_{ϵ}^{d}q\right),\end{array}$ then $\begin{array}{}p\le f\left({U}_{ϵ}^{d}q\right)\end{array}$ = f(⋁{yQd(y) < ϵ, y&q ≠ 0}) ≤ f(x). Thus ⋁{pPd′(p) < ϵ, p&f(q) ≠ 0} ≤ f(x), which implies $\begin{array}{}{U}_{ϵ}^{{d}^{\prime }}\end{array}$ f(q) ≤ f(x), thereforef$\begin{array}{}\left({\alpha }_{ϵ}^{d}\left(x\right)\right)\le {\alpha }_{ϵ}^{{d}^{\prime }}f\left(x\right).\end{array}$

(3) ⟹ (2) Since $\begin{array}{}{U}_{ϵ}^{d}⊣{\alpha }_{ϵ}^{d},\end{array}$ then $\begin{array}{}{U}_{ϵ}^{{d}^{\prime }}f\left(x\right)\le f\left({U}_{ϵ}^{d}x\right)⟺f\left(x\right)\le {\alpha }_{ϵ}^{d}f\left({U}_{ϵ}^{{d}^{\prime }}\left(x\right)\right).\end{array}$ But f(x) ≤ $\begin{array}{}f\left({\alpha }_{ϵ}^{d}{U}_{ϵ}^{d}x\right)\end{array}$$\begin{array}{}{\alpha }_{ϵ}^{{d}^{\prime }}f\left({U}_{ϵ}^{d}x\right),\end{array}$ which implies $\begin{array}{}{U}_{ϵ}^{{d}^{\prime }}f\left(x\right)\le f\left({U}_{ϵ}^{d}x\right).\end{array}$ □

#### Lemma 4.11

Let (Q, d) be a diametric quantale, and d be a compatible diameter, β : Q ⟶ 2 is a quantale epimorphism. Let a, bQ, β(a) = 1, then there is a ϵ > 0, such that for all bQ with d(b) < ϵ either β (b) = 0 or ba.

#### Proof

Since 1 = β(a) = $\begin{array}{}\beta \left(\underset{ϵ>0}{\bigvee }{\alpha }_{ϵ}^{d}\left(a\right)\right)=\underset{ϵ>0}{\bigvee }\beta \left({\alpha }_{ϵ}^{d}\left(a\right)\right),\end{array}$ then there exists ϵ > 0, such that $\begin{array}{}\beta \left({\alpha }_{ϵ}^{d}\left(a\right)\right)=1.\end{array}$ Let d(b) < ϵ, and ba, then $\begin{array}{}b\mathrm{&}{\alpha }_{ϵ}^{d}\end{array}$(a) = 0. Otherwise, if $\begin{array}{}b\mathrm{&}{\alpha }_{ϵ}^{d}\end{array}$(a) ≠ 0, that is b&(⋁{xQ$\begin{array}{}{U}_{ϵ}^{d}\end{array}$ xa}) = ⋁{b&xQ$\begin{array}{}{U}_{ϵ}^{d}\end{array}$(x) ≤ a} ≠ 0, then there exists xQ, such that $\begin{array}{}{U}_{ϵ}^{d}\end{array}$(x) ≤ a, b&x ≠ 0, then ba, but this contradicts the assumption that ba. Thus $\begin{array}{}b\mathrm{&}{\alpha }_{ϵ}^{d}\end{array}$(a) = 0, and 0 = $\begin{array}{}\beta \left(b\mathrm{&}{\alpha }_{ϵ}^{d}\left(a\right)\right)=\beta \left(b\right)\mathrm{&}\beta \left({\alpha }_{ϵ}^{d}\left(a\right)\right),\end{array}$ then β(b) = 0. Since β(b) ≠ 0, then β(b) = 1, that is $\begin{array}{}\beta \left(b\right)\mathrm{&}\beta \left({\alpha }_{ϵ}^{d}\left(a\right)\right)=\beta \left(b\mathrm{&}{\alpha }_{ϵ}^{d}\left(a\right)\right)=1,\end{array}$ then $\begin{array}{}b\mathrm{&}{\alpha }_{ϵ}^{d}\left(a\right)\ne 0,\end{array}$ Therefore ba. □

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

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

#### Theorem 4.12

Let (Q, d) be a diameteric quantale, and 1&1 ≠ 0, d be a compatible diameter, then ρd is a metric on Pt(Q) and the induced topology coincides with the topology ΣQ = {ΣaaQ} of Pt(Q), Σa = {ξξ(a) = 1}.

#### Proof

1. For every element ξ, ηPt(Q), it is easy to verify that ρd(ξ, η) ≥ 0.

2. Since ⋁ $\begin{array}{}{U}_{ϵ}^{d}\end{array}$ = 1 for all ϵ > 0, then ∀ ξPt(Q), ρd(ξ,ξ) = inf{d(a) ∣ ξ(a) = 1}, hence ρd(ξ,ξ) = 0. For all ξ, ηPt(Q), ϵ > 0, we have ρd(ξ, η) = inf{d(a) ∣ ξ(a) = η(a) = 1} = 0. If ξ(a) = 1, there is an element bQ, such that d(b) < ϵ, and ξ(b) = η(b) = 1. Thus ba, therefore η (a) = 1. The symmetry is obvious. If η(a) = 1, then ξ(a) = 1. Thus ξ = η.

3. For all ξ, ηPt(Q), ρd(ξ, η) = ρd(η,ξ) is obvious.

4. For all ξ, η, δPt(Q), in the following, we will prove that ρd(ξ, η) ≤ ρd(ξ, σ) + ρd(σ, η).

Let A = {aQξ(a) = η(a) = 1}, B = {b ∈ Qξ(b) = δ(b) = 1}, C = {cQδ(c) = η(c) = 1},

bB, ∀ cC, ξ(bc) = ξ(b) ∨ ξ(c) = 1 ∨ ξ(c) = 1, η(bc) = η(b) ∨ η(c) = 1 ∨ η(b) = 1, then bcA. 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(bc) ≤ 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 ρd(ξ, η) ≤ ρd(ξ, δ ) + ρd( δ, η). Thus ρd is a metric on Pt(Q). Therefore (Pt(Q),ρd) is a metric space.

In the following, we will prove the induced topology (Pt(Q), τρd) as ρd coincides with the topology ΣQ = {ΣaaQ} of Pt(Q), Σa = {ξξ(a) = 1}.

aA, ∀ ξΣa, then ξ(a) = 1. By Lemma 4.11, there exists an ϵ > 0. If ρd(ξ, η) < ϵ, we have an element bQ, such that ξ(b) = η(b) = 1, and d(b) ≤ ϵ, then ba. Thus η(a) = 1, that is ηΣa, ξ ∈ {ηPt(Q) ∣ ρd(ξ, η) < ϵ} ⊆ Σa. Hence ΣQτρd.

Conversely, for all ξPt(Q), ∀ ϵ > 0, we put a = ∨ {bQd(b) < ϵ, ξ(b) = 1}. For all ηΣa, there exists bQ, such that d(b) < ϵ, and ξ(b) = η(b) = 1. Therefore ρd(ξ, η) ≤ d(b) < ϵ. If ρd(ξ, η) < ϵ, then there exists an element bB, such that η(b) = ξ(b) = 1, d(b) < ϵ. Thus ba, η(a) = 1. Hence Σa = {ηρd(ξ, η) < ϵ} $\begin{array}{}{B}_{ϵ}^{{\rho }_{d}}\left(\xi \right).\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{Since}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}U=\bigcup \left\{{B}_{ϵ}^{{\rho }_{d}}\left(u\right)\mid u\in U\right\}=\underset{u\in U}{\bigvee }{\mathrm{\Sigma }}_{{a}_{u}}={\mathrm{\Sigma }}_{\underset{u\in U}{\bigvee }{a}_{u}}\end{array}$ for all Uτρd, then UΣQ, therefore τρdΣQ. □

## 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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Accepted: 2018-04-05

Published Online: 2018-06-21

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 636–645, ISSN (Online) 2391-5455,

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