Localization and Flatness in Quantale Theory

Abstract

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc.

1. Introduction

Quantales were introduced by Mulvey in [1] as algebraic structures to model the logic of quantum mechanics and the spectra of non-commutative $C^{\ast }$-algebras. A quantale is a complete lattice endowed with a new unary operation · which is distributive with respect to the suprema. One of the most important examples of a quantale is the complete multiplicative lattice $Id\left(R\right)$ of ideals in a (unital) commutative ring R. A lot of research in quantale theory has aimed to abstract some notions and theorems from ring theory. There are at least two motivations for such an approach. Firstly, abstraction is also a process of generalization and unification, through which a better grounded architecture of theory is reached. Secondly, in addition to the interest in itself of such an approach, such abstract results can be particularized to other classes of algebraic structures (commutative $C^{\ast }$-algebras, semirings, various algebras of logic, etc.).

Now we mention four important research themes in the commutative algebra: localization theory, flat ring morphisms, Krull dimension, and Going-down and Going-up properties (see [2,3]). An abstract approach to localization was made by Anderson in [4], as well as by other authors (see [5,6], etc.). A treatment of Krull dimension in the setting of frame theory can be found in [7]. The paper [8] studied the Going-down and Going-up properties in the frame theory. As can be seen, the development of the abstract theory of flatness is missing and presents itself as a natural problem. It also seems necessary to study the connections between the abstract versions of the four themes.

In this paper, the flat quantale morphisms are defined as an abstraction of flat ring morphisms. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat rings morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional quantales are obtained, formulated in terms of Going-down, Going-up, and localization.

In Section 2, we recall some fundamental notions and elementary facts on quantale theory: the residuation and the annihilator operation, m-prime and maximal elements of a quantale, radical elements, etc.

Section 3 focuses on quantale morphisms and their right adjoints. The few lemmas of the section present some basic properties of coherent quantale morphisms.

Section 4 concerns the localization theory in a coherent quantale A. We find a short form of the localization $a_p$ of an element $a\in A$ at an m-prime element p of A. Using this short form for $a_p$, in this section are presented (possibly new) proofs of some results from [4,5,6]. Also, the quantale versions of some results from [8] on the localization in frame theory are proved.

Section 5 defines the notion of a flat quantale morphism. The definition is inspired by a characterization of the flat ring morphisms by means of residuation (see [3,9]). We generalize some classical results concerning the flat ring morphisms. The paper [8] studied the Going-down and Going-up properties in a coherent frame. The results of Dube abstract for frames some theorems of [10] regarding the relationship between the Krull dimension of commutative rings and the Going-down and Going-up properties. The formulation of Going-down and Going-up properties in a quantale follows word for word the definitions in [8]. The main contributions in this section consist of characterizations of zero-dimensional coherent quantales and of coherent quantales of dimension at most one by means of flatness, localization, the Going-down property, and the Going-up property. These characterization theorems constitute quantale versions of the main results of [8,10].

2. Preliminaries

This section contains some background in quantale theory (cf. [11,12,13]) and frame theory (cf. [14,15]).

In this paper, by a ring we mean a unital commutative ring. Let $Id\left(R\right)$ be the set of ideals of a ring R. Then, $Id\left(R\right)$ is a multiplicative complete lattice: the join of a family of ideals is their sum, and the multiplication is the ideal product. Then, $Id\left(R\right)$ is a prototype of the abstract notion of quantales and a source of inspiration for developing the quantale theory.

A quantale is a structure $(A,\bigvee ,\bigwedge ,·,0,1)$, whenever $(A,\bigvee ,\bigwedge )$ is a complete lattice, $(A,·)$ is a semigroup and, for all $T\subseteq A$ and $x\in A$, the following infinite distributive law holds: $x·(\bigvee T)=\bigvee \{x·t\mid t\in T\}$ and $(\bigvee T)·x=\bigvee \{t·x\mid t\in T\}$. We shall write $xy$ instead of $x·y$, and the quantale $(A,\bigvee ,\bigwedge ,·,0,1)$ is denoted by A.

The quantale A is integral, if $(A,·,1)$ is a monoid and commutative, if · is a commutative operation. A frame is a particular case of quantales: the operations · and ∧ coincide (see [14,15]).

In this paper, by the word “quantale”, we shall mean an integral and commutative quantale.

Let A be a quantale. Then, $d\in A$ compact if for any $T\subseteq A$ such that $d\le \bigvee T$, there exists a finite subset $T_0$ of T such that $d\le \bigvee T_0$. We denote by $K\left(A\right)$ the set of all compact elements of A. Recall from [13] that for each ring R, $K\left(Id\right(R\left)\right)$ is the set of finitely generated ideals in R.

A quantale in which any element is a join of compact elements is said to be algebraic. Then, for each element x of an algebraic quantale A, we have $x=\bigvee \{d\in K(A)\mid d\le x\}$. An algebraic quantale A is coherent if $1\in K\left(A\right)$ and the set $K\left(A\right)$ is closed under the operation ·. If R is a ring, then $Id\left(R\right)$ is a coherent quantale.

We note that in several papers, the coherent quantales are named C-lattices (see e.g., [5,6]).

Lemma 1 

([16]). If A is a quantale and $x,y,z\in A$, then the following hold:

(1) 

If $x\vee y=1$, then $xy=x\wedge y$;

(2) 

If $x\vee y=1$, then $x^n\vee y^n=1$, for each integer $n\ge 1$;

(3) 

If $x\vee y=1$ and $x\vee z=1$, then $x\vee \left(yz\right)=x\vee (y\wedge z)=1$.

According to [13], on any quantale A, the following operations are defined:

• the implication →: for all $x,y\in A$,

  $$x\to y=\bigvee \{a\in A\mid xa\le y\};$$
• the annihilator operation: for any $x\in A$,

  $$x^{\perp }=x^{{\perp }_A}=x\to 0=\bigvee \{a\in A\mid xa=0\}.$$

According to [13], the following residuation rule holds: for all $x,y,z\in A$, $x\le y\to z$ if and only if $xy\le z$. Thus, $(A,\vee ,\wedge ,·,\to ,0,1)$ has a structure of a (commutative) residuated lattice. In what follows, some basic properties of residuated lattices are used without mention (cf. [17]).

Following [13], we say that an element $p<1$ of an arbitrary quantale A is m-prime if for all $x,y\in A$, $xy\le p$ implies $x\le p$ or $y\le p$. In an algebraic quantale A, the following characterization of the m-prime elements holds: an element $p<1$ is m-prime if and only if for all $e,f\in K\left(A\right)$, $ef\le p$ implies $e\le p$ or $f\le p$. We say that an element $m<1$ is maximal if for each $a\in A$ such that $m\le a<1$ we have $m=a$. Following the standard notations [13], $Spec\left(A\right)$ is the set of m-prime elements of A, and $Max\left(A\right)$ is the set of maximal elements of A. Keeping the denominations in ring theory [2], $Spec\left(A\right)$ is called the m-prime spectrum of A, and $Max\left(A\right)$ is called the maximal spectrum of A. If $1\in K\left(A\right)$, then $Max\left(A\right)\subseteq Spec\left(A\right)$, and for each $x<1$ there exists $m\in Max\left(A\right)$ such that $x\le m$.

Let x be an arbitrary element of a quantale A and $q\in Spec\left(A\right)$. We say that q is minimal over x if for any $r\in Spec\left(A\right)$, $x\le r\le q$ implies $q=r$. By using Zorn’s axiom, it follows that for any $x\in A$, there exists $q\in Spec\left(A\right)$ which is minimal over x. If $q\in Spec\left(A\right)$ is minimal over 0, then we say that q is a minimal m-prime element of A. The minimal m-prime spectrum of A is defined as the set $Min\left(A\right)$ of minimal m-prime elements of A.

Recall from ring theory [2] that the Jacobson radical of a ring R is the ideal $J\left(A\right)=\bigcap Max\left(A\right)$ (cf. [2]). We abstract this notion into quantale theory: the Jacobson radical of a quantale A is $r\left(A\right)=\bigwedge Max\left(A\right)$ (cf. [18]). Keeping a terminology from ring theory, the quantale A is local if it has a unique maximal element $m_A$. We remark that for any local quantale A, we have $r\left(A\right)=m_A$.

Let us denote by $B\left(A\right)$ the set of complemented elements of a quantale A. It is well known that $B\left(A\right)$ is a Boolean algebra (see [17]). According to [18], a quantale A is hyperarchimedean if for each $c\in K\left(A\right)$ there exists an integer $n\ge 1$ such that $c^n\in B\left(A\right)$.

Let x be an element of a quantale A. In accordance with [13], the radical $\rho \left(x\right)$ of x is defined by

$$\rho \left(x\right)={\rho }_A\left(x\right)=\bigwedge \{q\in Spec\left(A\right)|x\le q\}$$

If $x=\rho \left(x\right)$, then we say that x is a radical element of A. We shall denote by $R\left(A\right)$ the set of radical elements of A. If $\rho \left(0\right)=0$, then we say that A is a semiprime quantale.

Recall from [13] that for all $x,y\in A$, the following properties hold:

$\left(2.1\right)$$x\le \rho \left(x\right)$;

$\left(2.2\right)$$\rho \left(xy\right)=\rho \left(x\right)\wedge \rho \left(y\right)$;

$\left(2.3\right)$$\rho \left(\rho \right(x\left)\right)=\rho \left(x\right)$;

$\left(2.4\right)$$\rho (x\vee y)=\rho \left(\rho \right(x)\vee \rho (y\left)\right)$;

$\left(2.5\right)$$\rho \left(x\right)=1$ if and only if $x=1$;

$\left(2.6\right)$$\rho \left(x\right)\vee \rho \left(y\right)=1$ if and only if $x\vee y=1$;

$\left(2.7\right)$ for any integer $n\ge 1$, $\rho \left(x^n\right)=\rho \left(x\right)$.

For any $T\subseteq A$, we have the following:

$\left(2.8\right)$$\rho (\bigvee T)=\rho (\bigvee \{\rho \left(t\right)|t\in T\})$.

Since the set $R\left(A\right)$ of radical elements of A is closed under arbitrary meets, $R\left(A\right)$ is a complete lattice. For any $T\subseteq R\left(A\right)$, we set $\dot{\bigvee }T=\rho (\bigvee T)$. Using the previous properties of the map $\rho :A\to A$, it is easy to prove that $(R\left(A\right),\dot{\bigvee },\wedge ,0,1)$ is a frame (see [13]). By virtue of Lemma 8 of [18], for any coherent quantale A, we have $K\left(R\right(A\left)\right)=\rho \left(K\right(A\left)\right)$, and $R\left(A\right)$ is a coherent frame.

Lemma 2 

([19]). If A is a coherent quantale and $a\in A$, then $a\to \rho \left(0\right)=\rho \left(a\right)\to \rho \left(0\right)$.

Lemma 3 

([4,20]). Let A be a coherent quantale. For any $x\in A$, the following properties hold:

(1) 

$\rho \left(x\right)=\bigvee \{d\in K\left(A\right)\mid d^n\le x$ for some integer $n\ge 1\}$;

(2) 

For any compact element d of A, $d\le \rho \left(x\right)$ if and only if $d^n\le x$ for some integer $n\ge 1$;

(3) 

A is semiprime if and only if for all $d\in K\left(A\right)$ and $n\ge 1$, $d^n=0$ implies $d=0$.

3. Quantale Morphisms

Suppose that A and B are two quantales. According to [13], a map $u:A\to B$ is said to be a quantale morphism if it preserves the arbitrary joins and the multiplicative operation ·. Therefore, any quantale morphism preserves the bottom element 0. A quantale morphism $u:A\to B$ is integral if it preserves the top element 1. If $A,B$ are two coherent quantales, then an integral quantale morphism $u:A\to B$ is called coherent if $u\left(c\right)\in K\left(B\right)$, for any $c\in K\left(A\right)$.

Assume that $u:A\to B$ is a quantale morphism. Consider the map $u_{\star }:B\to A$, defined by $u_{\star }\left(y\right)=\bigvee \{x\in A\mid u\left(x\right)\le y\}$, for any $y\in B$. The pair $(u,u_{\star })$ verifies the following adjointness property: for all $x\in A$ and $y\in B$, $u\left(x\right)\le y$ if and only if $x\le u_{\ast }\left(y\right)$. Thus, $u_{\ast }$ is a right adjoint of u.

Lemma 4. 

Let $u:A\to B$ be a quantale morphism. Then, the following hold:

(1) 

$a\le u_{\ast }\left(u\left(a\right)\right)$, for any $a\in A$;

(2) 

$u\left(u_{\ast }\left(b\right)\right)\le b$, for any $b\in B$.

Lemma 5. 

Let $A,B$ be two algebraic quantales and $u:A\to B$ a surjective quantale morphism. If $u\left(K\right(A\left)\right)\subseteq K\left(B\right)$, then $u\left(K\right(A\left)\right)=K\left(B\right)$.

Proof. 

The inclusion $u\left(K\right(A\left)\right)\subseteq K\left(B\right)$ is ensured by the hypothesis that u preserves the compact elements. Assume that $d\in K\left(B\right)$, so $d=u\left(x\right)$ for some $x\in A$. But $x=\bigvee S$ for some $S\subseteq K\left(A\right)$, so $d=u(\bigvee S)=\bigvee \left\{u\right(s\left)\right|s\in S\}$ and $u\left(s\right)\in K\left(B\right)$, for any $s\in S$. Then, there exists a finite subset T of S such that $d=\bigvee \left\{u\right(t\left)\right|t\in T\}$. We set $c=\bigvee T$, so $c\in K\left(A\right)$ and $d=u\left(c\right)$. We prove that $K\left(B\right)\subseteq u\left(K\right(A\left)\right)$, so $u\left(K\right(A\left)\right)=K\left(B\right)$. □

Lemma 6. 

Let $u:A\to B$ be a coherent quantale morphism. If $q\in Spec\left(B\right)$, then $u_{\ast }\left(q\right)\in Spec\left(A\right)$.

Proof. 

Assume that $q\in Spec\left(B\right)$. Let $c,d$ be two compact elements of A such that $cd\le u_{\ast }\left(q\right)$. By the adjointness property, we obtain $u\left(c\right)u\left(d\right)=u\left(cd\right)\le q$, so $u\left(c\right)\le q$ or $u\left(d\right)\le q$; hence, $c\le u_{\ast }\left(q\right)$ or $d\le u_{\ast }\left(q\right)$. Then, $u_{\ast }\left(q\right)$ is an m-prime element of B. □

Lemma 7. 

Let $A,B$ be two coherent quantales and $v:K\left(A\right)\to K\left(B\right)$ a map that preserves multiplication, finite joins, and the top element. Then, there exists a unique coherent quantale morphism $v^{\ast }:A\to B$ such that $v^{\ast }\left(a\right)=v\left(a\right)$ for any $a\in A$.

Proof. 

Firstly, we shall prove that for all $S,T\subseteq K\left(A\right)$, the following implication holds:

$(\ast )$$\bigvee S=\bigvee T\Rightarrow \bigvee \left\{v\right(c\left)\right|c\in S\}=\bigvee \{v\left(d\right)|d\in T\}$.

In order to prove $(\ast )$, assume that $\bigvee S=\bigvee T$. Let c be an element of S; hence, $c\le \bigvee S$. Then, $c\le \bigvee T$, so $c\le \bigvee T_0$ for some finite subset $T_0$ of T. Since v preserves finite joins, we have $v\left(c\right)\le \bigvee \left\{v\left(d\right)\right|d\in T_0\}\le \bigvee \left\{v\left(d\right)\right|d\in T\}$. Then, $\bigvee \left\{v\right(c\left)\right|c\in S\}\le \bigvee \{v\left(d\right)|d\in T\}$, and the converse inequality follows in a similar manner.

Let a be an element of A, so $a=\bigvee S$ for some subset S of $K\left(A\right)$. We set $v^{\ast }\left(a\right)=\bigvee \left\{v\left(s\right)\right|s\in S\}$. By $(\ast )$, we obtain a map $v^{\ast }:A\to B$ such that $v^{\ast }\left(c\right)=v\left(c\right)$ for any $c\in K\left(A\right)$. The rest of the proof is straightforward. □

Let A be a coherent quantale and $a\in A$. An easy computation shows that the set ${\left[a\right)}_A=\{x\in A\mid a\le x\}$ is closed under arbitrary joins of A. Given two elements $x,y\in {\left[a\right)}_A$, we set $x{·}_{a}y=xy\vee a$. Therefore, ${\left[a\right)}_A$ is closed under the operation ${·}_a$. It is straightforward to verify that $({\left[a\right)}_A,\bigvee ,\wedge ,{·}_a,a,1)$ is a quantale.

Consider the map $u_a^A:A\to {\left[a\right)}_A$ defined by $u_a^A\left(x\right)=a\vee x$, for each $x\in A$.

Lemma 8 

([18]). For any element a of a coherent quantale A, the following properties are fulfilled:

(1) 

${\left[a\right)}_A$ is a coherent quantale;

(2) 

$u_a^A:A\to {\left[a\right)}_A$ is a coherent frame morphism;

(3) 

$K\left({\left[a\right)}_A\right)=\{a\vee c|c\in K\left(A\right)\}$;

(4) 

${\rho }_{{\left[a\right)}_A}\left(x\right)={\rho }_A\left(x\right)$, for any $x\in {\left[a\right)}_A$;

(5) 

$Spec\left({\left[a\right)}_A\right)=\{p\in Spec\left(A\right)|a\le p\}$.

Lemma 9. 

Let A be a coherent quantale and $a\in A$. Then, ${\left(u_a^A\right)}_{\ast }\left(x\right)=x$, for any $x\in {\left[a\right)}_A$.

Proof. 

Assume that $x\in {\left[a\right)}_A$. According to Lemma 8, any compact element of the quantale ${\left[a\right)}_A$ has the form $a\vee c$, where $c\in K\left(A\right)$. Therefore, using the adjointness property, it follows that for any $c\in K\left(A\right)$, the following equivalences hold:

$$c\le {\left(u_a^A\right)}_{\ast }\left(x\right)\iff a\vee c\le {\left(u_a^A\right)}_{\ast }\left(x\right)\iff u_a^A(a\vee c)\le x\iff a\vee c\le x\iff c\le x$$

From these equivalences, we obtain that ${\left(u_a^A\right)}_{\ast }\left(x\right)=x$. □

4. Localization in Coherent Quantales

The main notions and results in the localization of multiplicative lattices were due to Anderson in [4]. Subsequently, the definitions and theorems of [4] were completed and put in a slightly modified context (see [5,6]). A frame version of the localization was proposed by Dube in [8].

In this section, a short form of the definition of the localization of an element in a coherent quantale is found. A fundamental lemma is then proved, on the basis of which proofs of a few propositions from [4,5,6] are obtained. The section also contains new results, in particular, quantale generalizations of some theorems in [8] on localization in algebraic frames.

Recall that the coherent quantales coincide with the C-lattices (in the sense of [5,6], etc.).

Let us fix a coherent quantale A.

Following [4,5,6], for any $p\in Spec\left(A\right)$ and $x\in A$, we denote $x_p=\bigvee \{c\in K\left(A\right)|cs\le x$ for some $s\in K\left(A\right)$ such that $s\nleqq p\}$.

The element $x_p$ is called the localization of x at the m-prime element p.

Remark 1. 

The definition of $x_p$ can be written as

$$x_p=\bigvee \{c\in K\left(A\right)|\exists s\in K\left(A\right)[(s\le c\to x)\&(s\nleqq p)]\}$$

Therefore, we obtain the following short expression of $x_p$:

$$x_p=\bigvee \{c\in K\left(A\right)|c\to x\nleqq p\}$$

Lemma 10. 

For any $d\in K\left(A\right)$, $d\le x_p$ if and only if $d\to x\nleqq p$.

Proof. 

Assume that $d\le x_p$, so $d\le \bigvee \{c\in K(A\left)\right|c\to x\nleqq p\}$ (according to the previous remark). By the compactness of d, there exists an integer $n\ge 1$ and $c_1,\dots ,c_n\in K\left(A\right)$ such that $d\le {\bigvee }_{i=1}^n{c}_i$ and $c_i\to x\nleqq p$, for all $i=1,\dots ,n$. Denote $c={\bigvee }_{i=1}^n{c}_i$. Then, $c\in K\left(A\right)$ and $d\le c$. We observe that

$$c\to x=(\underset{i=1}{\overset{n}{\bigvee }}{c}_i)\to x=\underset{i=1}{\overset{n}{\bigwedge }}(c_i\to x)$$

and hence $c\to x\nleqq p$ (because p is m-prime). But $d\le c$ implies $c\to p\le d\to p$, so $d\to x\nleqq p$. The converse implication follows using Remark 1. □

Using Remark 1 and Lemma 10, we shall obtain some proofs of certain results of [4] (see also [5,6]). We shall use the terminology and the formulations existing in [5,6].

Lemma 11 

([4,5,6]). For all elements $x,y\in A$ and $p\in Spec\left(A\right)$, the following items hold:

(1) 

If $x\le y$, then $x_p\le y_p$;

(2) 

$x\le x_p$;

(3) 

${\left(x_p\right)}_p=x_p$;

(4) 

${(x\wedge y)}_p=x_p\wedge x_p$;

(5) 

$x_p=1$ if and only if $x\nleqq p$;

(6) 

$x\le y_p$ if and only if $x_p\le y_p$.

Proof. 

$\left(1\right)$ Obviously.

$\left(2\right)$ Let c be a compact element of A such that $c\le x$. Since $c\to x=1\nleqq p$, using Lemma 10, we obtain $c\le x_p$. It follows that $x\le x_p$.

$\left(3\right)$ Let c be a compact element of A such that $c\le {\left(x_p\right)}_p$; hence, $c\to x_p\nleqq p$ (by Lemma 10). We remark that $x\le x_p$ implies $c\to x_p\ge c\to x$, so $c\to x\nleqq p$. A new application of Lemma 2 gives $c\le x_p$, so we obtain ${\left(x_p\right)}_p\le x_p$. The converse inequality $x_p\le {\left(x_p\right)}_p$ follows by $\left(2\right)$.

$\left(4\right)$ Let c be a compact element of A such that $c\le x_p\wedge y_p$. By Lemma 10, we obtain $c\to x\nleqq p$ and $c\to y\nleqq p$. Since p is m-prime, it follows that $c\to (x\wedge y)=(c\to x)\wedge (c\to y)\nleqq p$. Hence, $c\le {(x\wedge y)}_p$, so we obtain $x_p\wedge x_p\le {(x\wedge y)}_p$. The converse inequality ${(x\wedge y)}_p\le x_p\wedge x_p$ follows using $\left(2\right)$.

$\left(5\right)$ Assume that $x\nleqq p$. We want to prove that $x_p=1$. Let c be an arbitrary compact element of A. We remark that $c\to x\nleqq p$ (because $c\to x\le p$ and $x\le c\to x$ contradicts $x\nleqq p$). According to Lemma 10, we obtain $c\le x_p$ for any $c\in K\left(A\right)$, so $x_p=1$.

Assume now $x_p=1$, and hence $1=\bigvee \{c\in K(A\left)\right|c\to x\nleqq p\}$ (cf. Remark 1). Since $1\in K\left(A\right)$, there exist an integer $n\ge 1$ and the compact elements $c_1,\dots ,c_n$ such that $1={\bigvee }_{i=1}^n{c}_i$ and $c_i\to x\nleqq p$, for all $i=1,\dots ,n$. By residuation theory, we have

$$x=1\to x=(\underset{i=1}{\overset{n}{\bigvee }}{c}_i)\to x=\underset{i=1}{\overset{n}{\bigwedge }}(c_i\to x)$$

Therefore, $x\nleqq p$ (because p is m-prime).

$\left(6\right)$ If $x\le y_p$, then $x_p\le {\left(y_p\right)}_p=y_p$ (by $\left(1\right)$ and $\left(3\right)$). The converse implication follows using $\left(1\right)$. □

Definition 1 

([13,21]). A function $j:A\to A$ is said to be a nucleus of the quantale A if the following conditions are fulfilled:

(1) 

j is a closure operator on A;

(2) 

$j\left(a\right)j\left(b\right)\le j\left(ab\right)$, for all $a,b\in A$.

For any nucleus j of A, we denote $A_j=\{a\in A|j\left(a\right)=a\}$.

Lemma 12 

([13,21]). If $j:A\to A$ is a nucleus, then for all $a,b\in A$, the following hold:

(1) 

$j\left(ab\right)=j\left(aj\right(b\left)\right)=j\left(j\right(a\left)b\right)=j\left(j\right(a\left)j\right(b\left)\right)$;

(2) 

$j\left(a\right)\to j\left(b\right)=a\to j\left(b\right)$;

(3) 

$A_j$ is closed under ∧ and →.

For all $a,b\in A$ and $S\subseteq A$, we set $a{·}_{j}b=j\left(ab\right)$ and ${\bigvee }_{j}S=j(\bigvee S)$.

Lemma 13 

([13,21]). $(A_j,{\bigvee }_j,\wedge ,{·}_j,0,1)$ is a quantale.

For any $p\in Spec\left(A\right)$, consider the map $j_p:A\to A$ defined by $j_p\left(x\right)=x_p$, for any $x\in A$.

Proposition 1. 

$j_p:A\to A$ is a nucleus on A.

Proof. 

The fact that $j_p$ is a closure operator follows from Lemma 11. We need to verify condition $\left(2\right)$ of Definition 1. Assume that $a,b\in A$. We have to show that $j_p\left(a\right)j_p\left(b\right)\le j_p\left(ab\right)$. According to Remark 1, we have

$$j_p\left(a\right)=\bigvee \{c\in K\left(A\right)|c\to a\nleqq p\};j_p\left(b\right)=\bigvee \{d\in K\left(A\right)|d\to b\nleqq p\}$$

and hence, using the distributive property, we obtain

$$j_p\left(a\right)j_p\left(b\right)=\bigvee \left\{cd\right|c,d\in K\left(A\right),c\to a\nleqq p,d\to b\nleqq p\}$$

Assume that $c,d\in K\left(A\right)$, $c\to a\nleqq p$ and $d\to b\nleqq p$; hence, using that p is m-prime, we obtain $(c\to a)(d\to b)\nleqq p$. On the other hand, the inequalities $c(c\to a)\le a$ and $d(d\to b)\le b$ imply $cd(c\to a)(d\to b)\le ab$; therefore, we obtain $(c\to a)(d\to b)\le cd\to ab$ (by residuation theory). It follows that $cd\to ab\nleqq p$. Thus, the following inequality is obtained:

$$\bigvee \left\{cd\right|c,d\in K\left(A\right),c\to a\nleqq p,d\to b\nleqq p\}\le \bigvee \{e\in K\left(A\right)|e\to ab\nleqq p\}$$

Therefore, we conclude that $j_p\left(a\right)j_p\left(b\right)\le j_p\left(ab\right)$. □

Applying Lemma 13, we obtain the following result regarding the quantale structure of $L_p=\left\{a_p\right|a\in A\}$.

Corollary 1. 

$L_p$ has a quantale structure with respect to the following operations:

(1) 

${\bigvee }_{j_p}\left\{{\left(a_i\right)}_p\right|i\in I\}={\left({\bigvee }_{i\in I}{a}_i\right)}_p$;

(2) 

$a_p\wedge b_p={(a\wedge b)}_p$;

(3) 

$a_p{·}_{j_p}{b}_p={\left(ab\right)}_p$.

In the rest of this paper, we shall denote ${\bigvee }_p\left\{{\left(a_i\right)}_p\right|i\in I\}={\bigvee }_{j_p}\left\{{\left(a_i\right)}_p\right|i\in I\}$ and $a_p{·}_p{b}_p=a_p{·}_{j_p}{b}_p$.

Lemma 14. 

If $c,e\in K\left(A\right)$, $a\in A$ and $p\in Spec\left(A\right)$, then the following hold:

(1) 

$e\le {(c\to a)}_p\iff e\to (c\to a)\nleqq p$;

(2) 

$e\le c_p\to a_p\iff ed\to a\nleqq p$, for any $d\in K\left(A\right)$ such that $d\to c\nleqq p$.

Proof. 

$\left(1\right)$ By Lemma 10.

$\left(2\right)$ Using Lemma 10 and residuation theory, the following properties are equivalent:

• $e\le c_p\to a_p$;

• $e\le \bigvee \{d\in K\left(A\right)|d\to c\nleqq p\}\to a_p$;

• $e\le \bigwedge \{d\to a_p|d\in K\left(A\right),d\to c\nleqq p\}$;

• $e\le d\to a_p$, for any $d\in K\left(A\right)$ such that $d\to c\nleqq p$;

• $ed\le a_p$, for any $d\in K\left(A\right)$ such that $d\to c\nleqq p$;

• $ed\to a\nleqq p$, for any $d\in K\left(A\right)$ such that $d\to c\nleqq p$; □

Proposition 2 

([4,5,6]). For all $c\in K\left(A\right)$ and $a\in A$ the following equality holds:

$${(c\to a)}_p=c_p\to a_p$$

Proof. 

$Claim\left(1\right)$${(c\to a)}_p\le c_p\to a_p$

Let e be a compact element of A such that $e\le {(c\to a)}_p$. By Lemma 14(1), we have $e\to (c\to a)\nleqq p$. In order to show that $e\le c_p\to a_p$, consider a compact element d such that $d\to c\nleqq p$. By Lemma 14(2), it suffices to prove that

$\left(i\right)$ $e\to (c\to a)\nleqq p,d\to c\nleqq p\Rightarrow ed\to c\nleqq p$.

We remark that $e(e\to (c\to a\left)\right)\le c\to a$ and $d(d\to c)\le c$, and therefore we obtain $ed(e\to (c\to a\left)\right)(d\to c)\le c(c\to a)\le a$.

Thus, we obtain the following inequality:

$\left(ii\right)$ $(e\to (c\to a\left)\right)(d\to c)\le ed\to a$.

Since p is m-prime, from $e\to (c\to a)\nleqq p$ and $d\to c\nleqq p$, we obtain that $(e\to (c\to a\left)\right)(d\to c)\nleqq p$. Therefore, by virtue of $\left(ii\right)$, it follows that $ed\to a\nleqq p$. Then, $\left(i\right)$ is verified.

$Claim\left(2\right)$$c_p\to a_p\le {(c\to a)}_p$

Let e be a compact element of A such that $e\le c_p\to a_p$. By using Lemma 14(2), it follows that for any $d\in K\left(A\right)$ such that $d\to c\nleqq p$, we have $ed\to a\nleqq p$. Particularly, for $d=c$, we have $c\to c=1\nleqq p$, and hence $ec\to a\nleqq p$. But the inequality $ec(ec\to a)\le a$ implies $ec\to a\le e\to (c\to a)$, so $e\to (c\to a)\nleqq p.$ According to Lemma 14(1), we obtain the inequality $c_p\to a_p\le {(c\to a)}_p$.

We conclude that ${(c\to a)}_p=c_p\to a_p$. □

Lemma 15 

([4]). If $c\in K\left(A\right)$, then $c_p\in K\left(A_p\right)$.

Proof. 

Assume that $c_p\le {\bigvee }_p\left\{s_p\right|s\in S\}$, where S is a subset of A. Then, $c\le {(\bigvee S)}_p$; hence, $c\to \bigvee S\nleqq p$ (cf. Lemma 10). Since the quantale A is coherent, we can find $d\in K\left(A\right)$ such that $d\le c\to \bigvee S$ and $d\nleqq p$. Then, $dc\le \bigvee S$ and $dc\in K\left(A\right)$ so there exists a finite subset T of S such that $dc\le \bigvee T$. Therefore, $d\le c\to \bigvee T$ and $d\nleqq p$; hence, $c\to \bigvee T\nleqq p$. According to Lemma 10, we obtain $c\le {(\bigvee T)}_p={\bigvee }_p\left\{s_p\right|p\in T\}$, so $c_p\le {\bigvee }_p\left\{s_p\right|p\in T\}$. Therefore, we conclude that $c_p$ is a compact element of $A_p$. □

Corollary 2 

([4]). $K\left(A_p\right)=\left\{c_p\right|c\in K\left(A\right)\}$.

Proof. 

By Lemma 15, the quantale morphism $j_p:A\to A_p$ preserves the compact elements. By virtue of Lemma 5, it follows that $K\left(A_p\right)=j_p\left(K\left(A\right)\right)=\left\{c_p\right|c\in K\left(A\right)\}$. □

Corollary 3. 

$A_p$ is a coherent quantale.

Proof. 

Assume that $c,d$ are two compact elements of A. Then, $c_p{·}_p{d}_p={\left(cd\right)}_p$ and $cd\in K\left(A\right)$, so $K\left(A_p\right)$ is closed under multiplication ${·}_p$ (cf. Corollary 2). □

Corollary 4. 

$j_p$ is a coherent quantale morphism.

Proposition 3 

([5,6,8]). $Spec\left(A_p\right)=\{q\in Spec\left(A\right)|q\le p\}$.

Proof. 

Assume that $q\in Spec\left(A\right)$ and $q\le p$. Firstly, we shall prove that $q_p\le q$. Let c be a compact element of A such that $c\le q_p$; hence, $c\to q\nleqq p$ (cf. Lemma 10). Then, there exists $d\in K\left(A\right)$ such that $d\le c\to q$ and $d\nleqq q$. But p is m-prime, so $dc\le q$ and $d\nleqq q$ imply $c\le q$. It follows that $q_p\le q$; hence, $q_p=q$ (by Lemma 11(2)).

Now we shall prove that $q_p\in Spec\left(A_p\right)$. It is clear that $q_p\ne 1$ because $q\le p$. Consider two compact elements $x,y$ of $A_p$ such that $x{·}_{p}y\le q_p$. By Corollary 2, there exist $c,d\in K\left(A\right)$ such that $x=c_p,y=d_p$, so ${\left(cd\right)}_p=c_p{·}_p{d}_p\le q_p$. Thus, $cd\le q_p=q$; hence, $c\le q$ or $d\le q$. It follows that $c_p\le q_p$ or $d_p\le q_p$, i.e., $x\le q_p$ or $y\le q_p$. We prove the inclusion $\{q\in Spec\left(A\right)|q\le p\}\subseteq Spec\left(A_p\right)$.

In order to show that $Spec\left(A_p\right)\subseteq \{q\in Spec\left(A\right)|q\le p\}$, assume that $q\in Spec\left(A_p\right)$. Thus, $q=q_p$ and $q_p\ne 1$; hence, $q\le p$ (by Lemma 11(5)). It remains to be proved that $q\in Spec\left(A\right)$. Let $c,d$ be two compact elements of A such that $cd\le q$, so $c_p{·}_p{d}_p={\left(cd\right)}_p\le q_p=q$. Since $q\in Spec\left(A_p\right)$, it follows that $c_p\le q$ or $d_p\le q$, so $c\le q$ or $d\le q$ (because $c\le c_p$ and $d\le d_p$). □

In accordance with the previous proposition, for any $p\in Spec\left(A\right)$, $A_p$ is a local quantale and $Max\left(A_p\right)=\left\{p\right\}$.

The following lemma shows that the right adjoint of quantale morphism $j_p:A\to A_p$ is the inclusion map $A_p\to A$.

Lemma 16. 

${\left(j_p\right)}_{\ast }\left(x\right)=x$, for any $x\in A_p$.

Proof. 

Let $x=a_p$ be an arbitrary element of $A_p$. Using the adjointness property for the pair $(j_p,{\left(j_p\right)}_{\ast })$ and Lemma 11(6), for any $c\in K\left(A\right)$, we have the following equivalence:

$$c\le {\left(j_p\right)}_{\ast }\left(a_p\right)\iff j_p\left(c\right)\le a_p\iff c_p\le a_p\iff c\le a_p$$

It follows that ${\left(j_p\right)}_{\ast }\left(a_p\right)=a_p$. □

For any $x\in A_p$, we shall denote ${\rho }_p\left(x\right)={\rho }_{A_p}\left(x\right)$.

Proposition 4. 

For any $a\in A$ we have ${\rho }_p\left(a_p\right)={\left(\rho \left(a\right)\right)}_p$.

Proof. 

By Corollary 2, any compact element of $A_p$ has the form $c_p$, where $c\in K\left(A\right)$. Therefore, in order to prove the equality ${\rho }_p\left(a_p\right)={\left(\rho \left(a\right)\right)}_p$, we need to verify the following equivalence:

$$c_p\le {\rho }_p\left(a_p\right)\iff c_p\le {\left(\rho \left(a\right)\right)}_p$$

Assume that $c_p\le {\rho }_p\left(a_p\right)$; hence, by Lemma 3(3) and Corollary 1(3), we obtain ${\left(c^n\right)}_p\le a_p$ for some integer $n\ge 1$. Then, $c^n\le a_p$ and $c^n\in K\left(A\right)$, so, using Lemma 10, it follows that $c^n\to a\nleqq p$. Then, one can find $d\in K\left(A\right)$ such that $d\le c^n\to a$ and $d\nleqq p$. We remark that $d\le c^n\to a$ implies $d^n{c}^n\le d{c}^n\le a$, so $dc\le \rho \left(a\right)$ (by Lemma 3(3)). Therefore, $d\le c\to \rho \left(a\right)$ and $d\nleqq p$, so $c\to \rho \left(a\right)\nleqq p$. A new application of Lemma 10 gives $c\le {\left(\rho \left(a\right)\right)}_p$, so $c_p\le {\left(\rho \left(a\right)\right)}_p$.

Conversely, assume that $c_p\le {\left(\rho \left(a\right)\right)}_p$; hence, $c\le {\left(\rho \left(a\right)\right)}_p$. By a straightforward argument, one can find an integer $m\ge 1$ and a compact element e such that $e^m{c}^m\le a$ and $e\nleqq p$. We set $d=e^m$; hence, $d\in K\left(A\right)$ and $d{c}^m\le a$. Thus, $d\le c^m\to a$ and $d\nleqq p$, so $c^m\le a_p$ (by Lemma 10). The last inequality implies ${\left(c^m\right)}_p\le a_p$. By virtue of Lemma 3(3) and Corollary 1(3), we obtain $c_p\le {\rho }_p\left(a_p\right)$. □

Recall that $Spec\left(A\right)=Spec\left(R\right(A\left)\right)$ and $Spec\left(A_p\right)=Spec\left(R\left(A_p\right)\right)$.

Corollary 5. 

$R\left(A_p\right)={\left(R\left(A\right)\right)}_p$.

Proof. 

According to Proposition 4, the following equalities hold:

$$R\left(A_p\right)=\left\{{\rho }_{A_p}\left(a_p\right)\right|a\in A\}=\left\{{\left({\rho }_A\left(a\right)\right)}_p\right|a\in A\}=\left\{x_p\right|x\in R\left(A\right)\}={\left(R\left(A\right)\right)}_p$$

□

Now we fix a coherent quantale morphism $u:A\to B$ and $q\in Spec\left(B\right)$. Then, $p=u_{\ast }\left(q\right)$ is an m-prime element of A (cf. Lemma 6).

Lemma 17. 

For all $c,a\in A$, we have $u(c\to a)\le u\left(c\right)\to u\left(a\right)$.

Proof. 

Assume that $c\in K\left(A\right)$ and $a\in A$. Since $c\to a\le c\to a$, by using the residuation theory, we have $c(c\to a)\le a$; hence, $u\left(c\right)u(c\to a)=u\left(c\right(c\to a\left)\right)\le u\left(a\right)$. Then, we obtain the inequality $u(c\to a)\le u\left(c\right)\to u\left(a\right)$. □

Lemma 18. 

For all $c,d\in K\left(A\right)$, the following hold:

(1) 

$c_p\le d_p$ implies ${\left(u\left(c\right)\right)}_q\le {\left(u\left(d\right)\right)}_q$;

(2) 

$c_p=d_p$ implies ${\left(u\left(c\right)\right)}_q={\left(u\left(d\right)\right)}_q$.

Proof. 

$\left(1\right)$ Assume that $c_p\le d_p$. By Lemma 11(2), we have $c\le c_p$, so $c\le d_p$. Therefore, applying Lemma 10, we obtain $c\to d\nleqq p$. Recall that $p=u_{\ast }\left(q\right)$. Thus, $c\to d\nleqq u_{\ast }\left(q\right)$; hence, $u(c\to d)\nleqq q$ (by the adjointness property). By Lemma 17, we have $u(c\to d)\le u\left(c\right)\to u\left(d\right)$; hence, $u\left(c\right)\to u\left(d\right)\nleqq q$. Since the quantale morphism u is coherent, $u\left(c\right)$ is a compact element of B. Then, a new application of Lemma 10 gives $u\left(c\right)\le {\left(u\left(d\right)\right)}_q$, so ${\left(u\left(c\right)\right)}_q\le {\left(u\left(d\right)\right)}_q$.

$\left(2\right)$ By virtue of $\left(1\right)$. □

Recall from Corollary 2 that $K\left(A_p\right)=\left\{c_p\right|c\in K\left(A\right)\}$ and $K\left(B_q\right)=\left\{d_q\right|d\in K\left(B\right)\}$. In accordance with Lemma 18 one can consider the function $v:K\left(A_p\right)\to K\left(B_q\right)$, defined by $v\left(c_p\right)={\left(u\left(c\right)\right)}_q$, for any $c\in K\left(A\right)$.

Lemma 19. 

v preserves multiplication, finite joins and the top element.

Proof. 

In accordance with Corollary 1(3), for all $c,d\in K\left(A\right)$ the following equalities hold:

$$v\left(c_p{·}_p{d}_p\right)=v\left({\left(cd\right)}_p\right)={\left(u\left(cd\right)\right)}_q={\left(u\left(c\right)u\left(d\right)\right)}_q={\left(u\left(c\right)\right)}_q{·}_q{\left(u\left(d\right)\right)}_q=v\left(c_p\right){·}_{q}v\left(d_p\right)$$

so v preserves multiplication. The equalities $v\left(c_p{\vee }_p{d}_p\right)=v\left(c_p\right){\vee }_{p}v\left(d_p\right)$ and $v\left(1\right)=1$ follow in a similar way. □

The following result is a quantale generalization of Theorem 3.12 of [8].

Theorem 1. 

If $u:A\to B$ is a coherent quantale morphism, then for any m-prime element q of B there exists a unique coherent quantale morphism

$$u_q:A_{u_{\ast }\left(q\right)}\to B_q$$

such that the following diagram is commutative:

Proof. 

We apply Lemmas 7 and 19. □

Lemma 20. 

Let $u:A\to B$ be a coherent quantale morphism, $q\in Spec\left(B\right)$ and $p=u_{\ast }\left(q\right)$. Then, ${\left(u_q\right)}_{\ast }\left(b_q\right)=u_{\ast }\left(b_q\right)$, for any $b\in B$.

Proof. 

According to Theorem 1, we have $u_q\circ j_p=j_q\circ u$. We know from Corollary 2 that a compact element of $A_p$ has the form $c_p$, for some $c\in K\left(A\right)$. Using the adjointness property and Lemma 11(6), we obtain the following equivalences: $c_p\le {\left(u_q\right)}_{\ast }\left(b_q\right)$ iff $u_q\left(c_p\right)\le b_q$ iff $j_q\left(u\left(c\right)\right)\le b_q$ iff ${\left(u\left(c\right)\right)}_q\le b_q$ iff $u\left(c\right)\le b_q$ iff $c\le u_{\ast }\left(b_q\right)$ iff $c_p\le u_{\ast }\left(b_q\right)$. Then, we obtain the equality ${\left(u_q\right)}_{\ast }\left(b_q\right)=u_{\ast }\left(b_q\right)$. □

Recall from [18,22] that a quantale A is normal if for each $p\in Spec\left(A\right)$ there exists a unique $m\in Max\left(A\right)$ such that $p\le m$. We note that a similar concept of normality was also introduced for other classes of multiplicative lattices (see [23]).

Proposition 5. 

If A is a coherent quantale, then the following are equivalent:

(1) 

A is a normal quantale;

(2) 

For all distinct $m,n\in Max\left(A\right)$ there exist $c,d\in K\left(A\right)$ such that $c\nleqq m$, $d\nleqq n$ and $cd=0$;

(3) 

For all distinct $m,n\in Max\left(A\right)$ there exists $c\in K\left(A\right)$ such that $c_m=1$ and $c_n=0_n$.

Proof. 

$\left(1\right)\iff \left(2\right)$ By Proposition 7.7 of [24].

$\left(2\right)\Rightarrow \left(3\right)$ Assume that m and n are two distinct maximal elements of A. Then, there exist $c,d\in K\left(A\right)$ such that $c\nleqq m$, $d\nleqq n$ and $cd=0$. In accordance with Lemma 11(5), we have $c_m=1$ and $d_n=1$. Using Corollary 1(3), from $cd=0$ we infer $c_n{·}_p{d}_n={\left(cd\right)}_n=0_n$; hence, $c_n=0_n$.

$\left(3\right)\Rightarrow \left(2\right)$ Assume that m and n are two distinct maximal elements of A. Then, there exists $c\in K\left(A\right)$ such that $c_m=1$ and $c_n=0_n$. By Lemma 11(5), we obtain $c\nleqq m$. But $c\le c_n$, so we have $c\le 0_n$. Therefore, applying Lemma 10, we obtain $c^{\perp }=c\to 0\nleqq n$. Thus, one can find a compact element d such that $d\le c^{\perp }$ and $d\nleqq n$. We prove that $c,d\in K\left(A\right)$, $c\nleqq m$, $d\nleqq n$ and $cd=0$. □

Let $u:A\to B$ be a coherent quantale morphism. According to Lemma 6.2 of [25], there exists a unique coherent frame morphism $u^{\rho }:R\left(A\right)\to R\left(B\right)$ such that the following diagram is commutative:

Recall that $Spec\left(R\right(A\left)\right)=Spec\left(A\right)$ and $Spec\left(R\right(B\left)\right)=Spec\left(B\right)$. Let q be an m-prime element of B and $p={\left(u^{\rho }\right)}_{\ast }\left(q\right)$. By Proposition 6.4 of [25], we have $p=u_{\ast }\left(q\right)$. Applying Theorem 1 to the coherent frame morphism $u^{\rho }:R\left(A\right)\to R\left(B\right)$, there exists a unique frame morphism

$${\left(u^{\rho }\right)}_q:{\left(R\left(A\right)\right)}_p\to {\left(R\left(B\right)\right)}_q$$

such that the following diagram is commutative:

Using the previous two commutative diagrams, we obtain the following lemma:

Lemma 21. 

For any $a\in A$ we have ${\left(u^{\rho }\right)}_q{\left({\rho }_A\left(a\right)\right)}_p)={\left({\rho }_B\left(u\left(a\right)\right)\right)}_q$.

On the other hand, by Theorem 1, we obtain the following commutative diagram:

Applying Lemma 6.2 of [25] to the coherent quantale morphism $u_q:A_p\to B_q$. there exists a unique frame morphism

$${\left(u_q\right)}^{\rho }:R\left(A_p\right)\to R\left(B_q\right)$$

such that the following diagram is commutative:

Using the previous two commutative diagrams, we obtain the following lemma:

Lemma 22. 

For any $a\in A$ we have ${\left(u_q\right)}^{\rho }\left({\rho }_{A_p}\left(a_p\right)\right)={\rho }_{B_q}\left({\left(u\left(a\right)\right)}_q\right)$.

Recall from Corollary 5 that $R\left(A_p\right)={\left(R\left(A\right)\right)}_p$ and $R\left(B_q\right)={\left(R\left(B\right)\right)}_q$.

Proposition 6. 

${\left(u^{\rho }\right)}_q={\left(u_q\right)}^{\rho }$.

Proof. 

Let x be an arbitrary element of $R\left(A_p\right)={\left(R\left(A\right)\right)}_p$. Thus, there exists $a\in A$ such that $x={\rho }_{A_p}\left(a_p\right)={\left({\rho }_A\left(a\right)\right)}_p$ (cf. Proposition 4). According to Proposition 4, we have ${\rho }_{B_q}\left({\left(u\left(a\right)\right)}_q\right)={\left({\rho }_B\left(u\left(a\right)\right)\right)}_q$. Therefore, using Lemmas 21 and 22 the following equalities hold:

$${\left(u_q\right)}^{\rho }\left(x\right)={\left(u_q\right)}^{\rho }\left({\rho }_{A_p}\left(a_p\right)\right)={\rho }_{B_q}\left({\left(u\left(a\right)\right)}_q\right)$$

$$={\left({\rho }_B\left(u\left(a\right)\right)\right)}_q={\left(u^{\rho }\right)}_q{\left({\rho }_A\left(a\right)\right)}_p)={\left({\rho }_A\left(a\right)\right)}_p\left(x\right)$$

□

5. Flat Quantale Morphisms

The notion of a flat ring morphism is usually defined using the tensor product (see [2], p. 29). A characterization of the flat ring morphisms in terms of the ideal residuation theory can be found in Exercise 8 of [3], p. 65 or [9], p. 46: a ring morphism $f:R\to S$ is flat if and only if for each ideal I of R and for each finitely generated ideal J of R, we have $(I:J)S=(IS:JS)$. This observation leads us to the following notion of a flat quantale morphism.

Definition 2. 

A quantale morphism $u:A\to B$ is flat if for all $c\in K\left(A\right)$ and $a\in A$ we have $u(c\to a)=u\left(c\right)\to u\left(a\right)$.

To each ring morphism $f:R\to S$, one can assign the quantale morphism $f^{•}:Id\left(R\right)\to Id\left(Q\right)$ defined by $f^{•}\left(I\right)=IS$, for any $I\in Id\left(R\right)$. Then, f is a flat ring morphism if and only if $f^{•}$ is a flat quantale morphism.

We note that the flat quantale morphisms also abstract the flat morphisms of bounded distributive lattices [26] and the flat morphisms of residuated lattices [27].

It is straightforward to prove that the composition of two flat quantale morphisms is a flat quantale morphism.

Example 1. 

Let p be an m-prime element of a coherent quantale A, $A_p$ the localization of A at p, and $j_p:A\to A_p$ the associated quantale morphism. Due to Proposition 2, $j_p$ is a flat quantale morphism.

Lemma 23. 

If $u:A\to B$ is a quantale morphism, then the following are equivalent:

(1) 

u is flat;

(2) 

For all $c\in K\left(A\right)$ and $a\in A$, $u\left(c\right)\to u\left(a\right)\le u(c\to a)$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Obviously.

$\left(2\right)\Rightarrow \left(1\right)$ By Lemma 17. □

Proposition 7. 

For any element a of a quantale A, the following are equivalent:

(1) 

$u_a^A:A\to {\left[a\right)}_A$ is flat;

(2) 

For all $c\in K\left(A\right)$ and $x\in A$, $c\to (x\vee a)\le a\vee (c\to x)$.

Proof. 

Assume that $c\in K\left(A\right)$ and $x\in A$. In accordance with the definition of $u_a^A$, we have:

$$u_a^A\left(c\right)\to u_a^A\left(x\right)=(c\vee a)\to (x\vee a)=(c\to (x\vee a))\wedge (a\to (x\vee a))=c\to (x\vee a)$$

$$u_a^A(c\to x)=a\vee (c\to x)$$

By Lemma 23, the following properties are equivalent:

• $u_a^A:A\to {\left[a\right)}_A$ is flat;
• For all $c\in K\left(A\right)$ and $x\in A$, $u_a^A\left(c\right)\to u_a^A\left(x\right)\le u_a^A(c\to x)$;
• For all $c\in K\left(A\right)$ and $x\in A$, $c\to (x\vee a)\le a\vee (c\to x)$.

□

Recall from [22,23] that an element a of a quantale A is pure if

$$\forall c\in K\left(A\right)[c\le a\Rightarrow a\vee c^{\perp }=1]$$

Proposition 8. 

Let A be a coherent quantale. For any $a\in A$, the following are equivalent:

(1) 

$u_a^A:A\to {\left[a\right)}_A$ is flat;

(2) 

a is a pure element of A.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Assume that the quantale morphism $u_a^A$ is flat. Let c be a compact element of A such that $c\le a$. By hypothesis, the following equalities hold:

$$a\vee c^{\perp }=a\vee (c\to 0)=u_a^A(c\to 0)=u_a^A\left(c\right)\to u_a^A\left(0\right)=(c\vee a)\to a=1$$

Therefore, a is a pure element of A.

$\left(2\right)\Rightarrow \left(1\right)$ Assume that a is a pure element of A. According to Proposition 7, in order to show that $u_a^A$ is flat, we need to check that for all $c\in K\left(A\right)$ and $x\in A$, we have $c\to (x\vee a)\le a\vee (c\to x)$. Let d be a compact element of A such that $d\le c\to (x\vee a)$, so $dc\le x\vee a$. But $cd\in K\left(A\right)$, so there exist $e,f\in K\left(A\right)$ such that $e\le x$, $f\le a$ and $cd\le e\vee f$.

Since a is pure, $f\le a$ implies $a\vee f^{\perp }=1$; hence, there exist $s,t\in K\left(A\right)$ such that $s\le a$, $t\le f^{\perp }$ and $s\vee t=1$ (because $1\in K\left(A\right)$). From $cd\le e\vee f$ and $tf=0$, we obtain $cdt\le (e\vee f)t\le et\vee ft=et\le e\le x$, so $dt\le c\to x$. Further, we remark that $d=d1=d(s\vee t)=ds\vee dt\le s\vee (c\to x)\le a\vee (c\to x)$. We prove that

$$\forall d\in K\left(A\right)[d\le c\to (x\vee a)\Rightarrow d\le a\vee (c\to x\left)\right]$$

It follows that $c\to (x\vee a)\le a\vee (c\to x)$, so $u_a^A$ is flat. □

Extending Definition 4.4 of [8], a coherent quantale morphism $u:A\to B$ is said to be slightly open if $u\left(c^{{\perp }_A}\right)={\left(u\left(c\right)\right)}^{{\perp }_B}$, for any $c\in K\left(A\right)$. We remark that any flat quantale morphism $u:A\to B$ is slightly open: for any compact element c of A, we have $u(c\to 0)=u\left(c\right)\to u\left(0\right)=u\left(c\right)\to 0$.

Theorem 2. 

Let A be a coherent quantale. For any $p\in Spec\left(A\right)$, the following are equivalent:

(1) 

$u_p^A:A\to {\left[p\right)}_A$ is flat;

(2) 

p is a pure element of A;

(3) 

$u_p^A:A\to {\left[p\right)}_A$ is slightly open.

Proof. 

$\left(1\right)\iff \left(2\right)$ By Proposition 8.

$\left(1\right)\Rightarrow \left(3\right)$ Obviously.

$\left(3\right)\Rightarrow \left(2\right)$ Assume that $u_p^A:A\to {\left[p\right)}_A$ is slightly open; therefore,, for any $c\in K\left(A\right)$, the following equality holds: $u_p^A\left(c^{{\perp }_A}\right)={\left(u_p^A\left(c\right)\right)}^{{\perp }_{{\left[p\right)}_A}}$.

Therefore, taking into account that

$$u_p^A\left(c^{{\perp }_A}\right)=p\vee c^{{\perp }_A}$$

$${\left(u_p^A\left(c\right)\right)}^{{\perp }_{{\left[p\right)}_A}}=(p\vee c)\to p=(p\to p)\wedge (c\to p)=c\to p$$

we obtain the equality $p\vee c^{{\perp }_A}=c\to p$. Suppose that $c\le p$. Then, we have $p\vee c^{{\perp }_A}=c\to p=1$. In conclusion, p is a pure element of A. □

Lemma 24 

(see [4], Theorem 2.8). Let A be a coherent quantale and $a,b\in A$. Then, the following are equivalent:

(1) 

$a=b$;

(2) 

$a_p=b_p$, for all $p\in Spec\left(A\right)$;

(3) 

$a_m=b_m$, for all $m\in Max\left(A\right)$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)\Rightarrow \left(3\right)$ Obviously.

$\left(3\right)\Rightarrow \left(1\right)$ Assume by absurdum that $a\ne b$, so $a\nleqq b$ or $b\nleqq a$. Consider the case $a\nleqq b$, so there exists $c\in K\left(A\right)$ such that $c\le a$ and $c\nleqq b$. Thus, $c\to b\ne 1$, so $c\to b\le m$ for some $m\in Max\left(A\right)$. By hypothesis, we have $a_m=b_m$. According to Lemma 11(5), $c\to b\le m$ implies ${(c\to b)}_m\ne 1$. By Proposition 2, we have ${(c\to b)}_m=c_m\to b_m$, so $c_m\to b_m\ne 1$; hence, we obtain $c_m\nleqq b_m$. On the other hand, $c\le a$ implies $c_m\le a_m$, contradicting that $a_m=b_m$. The case $b\nleqq a$ can be treated in a similar way, so $a=b$. □

Lemma 25. 

Assume that $A,B$ are two coherent quantales, $p\in Spec\left(A\right)$, and $v:A_p\to B$ is a coherent quantale morphism. Then, the following are equivalent:

(1) 

v is a flat quantale morphism;

(2) 

$v\circ j_p$ is a flat quantale morphism.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ This implication follows by Example 1 and the fact that the composition of two flat quantale morphisms is a flat quantale morphism.

$\left(2\right)\Rightarrow \left(1\right)$ In order to show that v is a flat quantale morphism, assume that $x\in K\left(A_p\right)$ and $y\in A_p$. Then, $x=c_p$ and $y=a_p$ for some $c\in K\left(A\right)$ and $a\in A$ (see Corollary 2). By Example 1 and the hypothesis, we know that $j_p$ and $v\circ j_p$ are flat quantale morphisms, so the following equalities hold:

$$v(x\to y)=v(j_p\left(c\right)\to j_p\left(a\right))=v\left(j_p(c\to a)\right)=v\left(j_p\left(c\right)\right)\to v\left(j_p\left(a\right)\right)=v\left(x\right)\to v\left(y\right)$$

Therefore, v is a flat quantale morphism. □

Theorem 3. 

Let $A\to$ be a coherent quantale morphism. Then, the following are equivalent:

(1) 

u is a flat quantale morphism;

(2) 

For any $q\in Spec\left(A\right)$, $u_q:A_{u_{\ast }\left(q\right)}\to B_q$ is a flat quantale morphism.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Let q be an m-prime element of B. In accordance with Theorem 1, we have $u_q\circ j_{u_{\ast }\left(q\right)}=j_q\circ u$. Recall from Example 1 that $j_q$ and $j_{u_{\ast }\left(q\right)}$ are flat quantale morphisms. Since $j_q\circ u$ is a flat quantale morphism (as a composition of flat quantale morphisms), it follows that $u_q\circ j_{u_{\ast }\left(q\right)}$ is flat. Applying Lemma 25, it follows that $u_q$ is flat.

$\left(2\right)\Rightarrow \left(1\right)$ Assume that $c\in K\left(A\right)$ and $a\in A$. We have to show that $u(c\to a)=u\left(c\right)\to u\left(a\right)$. According to Lemma 24, it suffices to check that ${\left(u(c\to a)\right)}_q={(u\left(c\right)\to u\left(a\right))}_q$, for any $q\in Spec\left(B\right)$. By virtue of Example 1 and the hypothesis, $j_q,j_{u_{\ast }\left(q\right)}$ and $u_q$ are flat quantale morphisms. Therefore, using Theorem 1, we obtain the following equalities:

$${\left(u(c\to a)\right)}_q=j_q\left(u(c\to a)\right)=u_q\left(j_{u_{\ast }\left(q\right)}(c\to a)\right))$$

$$=u_q\left(j_{u_{\ast }\left(q\right)}\left(c\right)\right)\to u_q\left(j_{u_{\ast }\left(q\right)}\left(a\right)\right)=j_q\left(u\left(c\right)\right)\to j_q\left(u\left(a\right)\right)$$

$$=j_q(u\left(c\right)\to u\left(a\right))={(u\left(c\right)\to u\left(a\right))}_q$$

Therefore, u is a flat quantale morphism. □

Proposition 9. 

Let A be a local coherent quantale and q a non-maximal m-prime element of A. Then, the following are equivalent:

(1) 

$u_q^A$ is a flat quantale morphism;

(2) 

$q=0$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Let $m_A$ be the unique maximal element of the local coherent quantale A; hence, $q\le m_A$. Assume by absurdum that $q\ne 0$, so $0<c\le q$ for some $c\in K\left(A\right)$. From $c<0$, we obtain $c^{\perp }\ne 1$ (because $c^{\perp }=1$ implies $c\le c^{\perp \perp }=0$); hence, $c^{\perp }\le m_A$. On the other hand, q is a pure element of A (by Proposition 8), so $c\le q$ implies $q\vee c^{\perp }=1$. Therefore, $1=q\vee c^{\perp }\le m_A$, contradicting that $m_A\in Max\left(A\right)$. We conclude that $q=0$.

$\left(2\right)\Rightarrow \left(1\right)$ Obviously. □

6. Dimension Versus Going-Down and Going-Up

Let A be an algebraic quantale such that $1\in K\left(A\right)$. Recall from Section 2 that $p\in Spec\left(A\right)$ is minimal over $a\in A$ if $a\le p$ and for any $q\in Spec\left(A\right)$, $a\le q\le p$ implies $q=p$. Then, $p\in Spec\left(A\right)$ is minimal over the bottom element 0 if and only if p is a minimal m-prime element of A.

Lemma 26 

(see [24], Proposition 8.3). Let A be a coherent quantale and $p\in Spec\left(A\right)$. Then, the following are equivalent:

(1) 

p is a minimal m-prime element;

(2) 

For any $c\in K\left(A\right)$, $c\le p$ implies $c\to \rho \left(0\right)\nleqq p$.

Corollary 6. 

Let A be a coherent quantale, $a\in A$ and $p\in Spec\left(A\right)$. If $a\le p$, then the following are equivalent:

(1) 

p is minimal over a;

(2) 

For any $c\in K\left(A\right)$, $c\le p$ implies $c\to \rho \left(a\right)\nleqq p$.

Proof. 

According to Lemma 8(4) we have ${\rho }_{{\left[a\right)}_A}\left(a\right)=\rho \left(a\right)$. We know that $Spec\left({\left[a\right)}_A\right)=\{p\in Spec\left(A\right)|a\le p\}$ and $K\left({\left[a\right)}_A\right)=\{a\vee c|c\in K\left(A\right)\}$. Hence, using Lemma 26, it follows that the following properties are equivalent:

• p is minimal over a;
• $p\in Min\left({\left[a\right)}_A\right)$;
• For any $x\in K\left({\left[a\right)}_A\right)$, $x\le p$ implies $c\to {\rho }_{{\left[a\right)}_A}\left(a\right)\nleqq p$;
• For any $c\in K\left(A\right)$, $c\le p$ implies $c\to \rho \left(a\right)\nleqq p$.

□

The Going-down property in frame theory was introduced in Definition 3.1 of [8]. In a very similar way, we shall define the Going-down property for the quantale morphisms.

Definition 3. 

Let $u:A\to B$ be a quantale morphism and $p\in Spec\left(A\right)$. We say that

(1) 

u goes to p if for each $q\in Spec\left(B\right)$ such that $p\le u_{\ast }\left(q\right)$ there exists $r\in Spec\left(B\right)$ such that $r\le q$ and $p=u_{\ast }\left(r\right)$;

(2) 

u satisfies the Going-down property (= $GD$-property) if u goes to each m-prime element of A.

Proposition 10. 

Let $A\to B$ be a coherent quantale morphism and $p\in Spec\left(A\right)$. Then, the following are equivalent:

(1) 

u goes down to p;

(2) 

If $q\in Spec\left(B\right)$ is minimal over $u(p$), then $u_{\ast }\left(q\right)=p$.

Proof. 

$\left(1\right)\iff \left(2\right)$ This equivalence follows by imitating in every detail the proof of $\left(1\right)\iff \left(2\right)$ in Theorem 3.6 of [8]. □

Corollary 7. 

Let $A\to B$ be a coherent quantale morphism. Then, the following are equivalent:

(1) 

u satisfies the $GD$-property;

(2) 

For all $p\in Spec\left(A\right)$ and $q\in Spec\left(B\right)$ such that q is minimal over $u(p$), we have $u_{\ast }\left(q\right)=p$.

Corollary 8. 

Let $A\to B$ be a coherent quantale morphism. Then, the following are equivalent:

(1) 

u satisfies the $GD$-property;

(2) 

For any $q\in Spec\left(B\right)$, $u_q:A_{u_{\star }\left(q\right)}\to B_q$ satisfies the $GD$-property;

(3) 

For any $q\in Spec\left(B\right)$, the map $Spec\left(u_q\right):Spec\left(B_q\right)\to Spec\left(A_{u_{\star }\left(q\right)}\right)$ is surjective.

Proof. 

The equivalence of the three statements follows in a very similar way to the proof of Corollary 3.13 of [8]. As an illustration, we shall prove the implication $\left(1\right)\Rightarrow \left(2\right)$. Assume that u satisfies the $GD$-property. Let q be an m-prime element of $B_q$. We have to show that $u_q:A_{u_{\star }\left(q\right)}\to B_q$ satisfies the $GD$-property.

Assume that $k\in Spec\left(A_{u_{\star }\left(q\right)}\right)$, $l\in Spec\left(B_q\right)$ and l is minimal over $u_q\left(k\right)$. Recall from Proposition 3 that

$$Spec\left(A_{u_{\star }\left(q\right)}\right)=\{r\in Spec\left(A\right)|r\le u_{\ast }\left(q\right)\};Spec\left(B_q\right)=\{s\in Spec\left(B\right)|s\le q\}$$

Then, $k\in Spec\left(A\right)$, $l\in Spec\left(B\right)$ and l is minimal over $u\left(k\right)$. By hypothesis, u satisfies the $GD$-property; hence, using Corollary 7, there exists $t\in Spec\left(B\right)$ such that $t\le l$ and $u_{\star }\left(t\right)=k$. By Lemma 20, we have ${\left(u_q\right)}_{\ast }\left(t\right)=u_{\star }\left(t\right)$, so ${\left(u_q\right)}_{\ast }\left(t\right)=k$. By virtue of Corollary 7, $u_q:A_{u_{\star }\left(q\right)}\to B_q$ satisfies the $GD$-property. □

The Going-up property in frame theory was introduced in [28], p. 586. We shall define the Going-up property for a quantale morphism by using the form of the $GU$-property given in [8], p. 1805.

Definition 4. 

A quantale morphism $u:A\to B$ satisfies the Going-up property (=$GU$-property) if for all $p\in Spec\left(A\right)$ and $q\in Spec\left(B\right)$ such that $p\ge u_{\ast }\left(q\right)$, there exists $r\in Spec\left(B\right)$ such that $r\ge q$ and $p=u_{\ast }\left(r\right)$.

Lemma 27. 

For any element a of a coherent quantale A, $u_a^A:A\to {\left[a\right)}_A$ satisfies the $GU$-property.

Proof. 

Assume $p\in Spec\left(A\right)$, $q\in Spec\left({\left[a\right)}_A\right)$ and $p\ge {\left(u_a^A\right)}_{\ast }\left(q\right)$. By Lemma 8(5), we have $Spec\left({\left[a\right)}_A\right)=\{r\in Spec\left(A\right)|a\le r\}$. On the other hand, using Lemma 9, we obtain ${\left(u_a^A\right)}_{\ast }\left(q\right)=q$. Thus, $p\ge q\ge a$, so $p\in Spec\left({\left[a\right)}_A\right)$. But ${\left(u_a^A\right)}_{\ast }\left(p\right)=p$, so $u_a^A$ satisfies the $GU$-property. □

If $p_0<p_1<\dots <p_n$ is a chain of m-prime elements of a quantale A, then the integer n is the chain length. The dimension of A, denoted by $dim\left(A\right)$, is the supremum of the lengths of m-prime elements of A. A is said to be zero-dimensional if $dim\left(A\right)=0$. It is clear that A is a zero-dimensional quantale iff $Spec\left(A\right)=Max\left(A\right)$ iff $Spec\left(A\right)=Min\left(A\right)$ iff $Max\left(A\right)=Min\left(A\right)$.

The following result is a quantale generalization of Proposition 2.1 of [10], as well as of some parts of Propositions 4.4 and 4.6 of [8].

Theorem 4. 

For any coherent quantale A, the following are equivalent:

(1) 

$dim\left(A\right)=0$;

(2) 

Any coherent quantale morphism $u:A\to B$ satisfies the $GD$-property;

(3) 

For any $p\in Spec\left(A\right)$, $u_p^A:A\to {\left[p\right)}_A$ satisfies the $GD$-property;

(4) 

For any $p\in Spec\left(A\right)-Min\left(A\right)$, $u_p^A:A\to {\left[p\right)}_A$ satisfies the $GD$-property;

(5) 

Any coherent quantale morphism $u:A\to B$ satisfies the $GU$-property;

(6) 

For any $p\in Spec\left(A\right)$, $u_p^A:A\to A_p$ satisfies the $GU$-property;

(7) 

For any $p\in Spec\left(A\right)-Max\left(A\right)$, $u_p^A:A\to A_p$ satisfies the $GU$-property;

(8) 

For any $p\in Spec\left(A\right)-Min\left(A\right)$, $u_p^A:A\to {\left[p\right)}_A$ is flat;

(9) 

A is hyperarchimedean.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ and $\left(1\right)\Rightarrow \left(5\right)$ follows by $Spec\left(A\right)=Max\left(A\right)=Min\left(A\right)$.

$\left(2\right)\Rightarrow \left(3\right)\Rightarrow \left(4\right)$ and $\left(5\right)\Rightarrow \left(6\right)\Rightarrow \left(7\right)$ Obviously.

$\left(1\right)\Rightarrow \left(8\right)$ Because $Spec\left(A\right)-Min\left(A\right)=\varnothing$.

$\left(4\right)\Rightarrow \left(1\right)$ Assume by absurdum that $dim\left(A\right)\ne 0$; therefore, there exist $p,q\in Spec\left(A\right)$ such that $p<q$. Then, $q\in Spec\left(A\right)-Min\left(A\right)$, so the quantale morphism $u_q^A:A\to {\left[q\right)}_A$ satisfies the $GD$-property. Thus, $u_q^A$ goes down to p. We know that $Spec\left({\left[q\right)}_A\right)=\{r\in Spec\left(A\right)|q\le r\}$, so $q\in Spec\left({\left[q\right)}_A\right)$ and q is minimal over q. On the other hand, $u_q^A\left(p\right)=q\vee p=q$, so q is minimal over $u_q^A\left(p\right)$. Applying Proposition 10, we obtain $u_q^A\left(q\right)=p$, contradicting $u_q^A\left(q\right)=q$ and $p<q$. Then, the implication $\left(4\right)\Rightarrow \left(1\right)$ is proven.

$\left(8\right)\Rightarrow \left(1\right)$ Assume by absurdum that $dim\left(A\right)\ne 0$; therefore, there exist $p,q\in Spec\left(A\right)$ such that $p>q$. Then, $p\in Spec\left(A\right)-Min\left(A\right)$; hence, $u_p^A:A\to {\left[p\right)}_A$ is a flat quantale morphism. By Proposition 8, p is a pure element of A. The strict inequality $p>q$ implies that there exists $c\in K\left(A\right)$ such that $c\le p$ and $c\nleqq q$. Since p is pure, $c\le p$ implies $p\vee c^{\perp }=1$. If $c^{\perp }\le q$, then $c^{\perp }\le p$, so $p=p\vee c^{\perp }=1$, contradicting $p\in Spec\left(A\right)$. Therefore, $c^{\perp }\nleqq q$, so we obtain $c\le q$ (because $q\in Spec\left(A\right)$). We obtain a contradiction, so $dim\left(A\right)=0$.

$\left(7\right)\Rightarrow \left(1\right)$ Assume by absurdum that $dim\left(A\right)\ne 0$; therefore, there exist $p,q\in Spec\left(A\right)$ such that $p<q$, so $p\in Spec\left(A\right)-Max\left(A\right)$. By hypothesis, $j_p:A\to A_p$ satisfies the $GU$-property.

Consider the coherent quantale morphisms

$$j_p:A\to A_p,u_p^{A_p}:A_p\to {\left[p\right)}_{A_p}$$

and the map $u:A\to {\left[p\right)}_{A_p}$ defined by $u=u_p^{A_p}\circ j_p$. Then, u is a coherent quantale morphism as a composition of two coherent quantale morphisms. We know from Lemma 27 that $u_p^{A_p}$ satisfies the $GD$-property. Since the two quantale morphisms $j_p$ and $u_p^{A_p}$ verify the $GU$-property, it follows that u satisfies the $GU$-property.

We observe that $u\left(x\right)=u_p^{A_p}\left(j_p\left(x\right)\right)=p{\vee }_p{x}_p={(p\vee x)}_p$, for any $x\in A$. Therefore, using the adjointness property and Lemma 11(6), it follows that for any $c\in K\left(A\right)$, the following equivalences hold:

$$c\le u_{\ast }\left(p\right)\iff u\left(c\right)\le p\iff {(p\vee c)}_p\le p\iff p\vee c\le p\iff c\le p$$

Then, we obtain the equality $u_{\ast }\left(p\right)=p$, so $q>u_{\ast }\left(p\right)$. Since the quantale morphism u satisfies the $GU$-property, from $q>u_{\ast }\left(p\right)$, it follows that there exists $r\in Spec\left({\left[p\right)}_{A_p}\right)$ such that $r\ge p$ and $u_{\star }\left(r\right)=q$. Using Lemma 8(5) and Proposition 3 we have

$$Spec\left({\left[p\right)}_{A_p}\right)=\{s\in Spec\left(A_p\right)|p\le s\}=\left\{p\right\}.$$

Thence $r=p$, so $p=u_{\star }\left(p\right)=u_{\star }\left(r\right)=q$, contradicting $q>p$. Therefore, we conclude that $dim\left(A\right)=0$.

$\left(1\right)\iff \left(9\right)$ See Proposition 6.2 of [18]. □

Corollary 9. 

Let A be a coherent quantale. Then, the following are equivalent:

(1) 

A is a semiprime zero-dimensional quantale;

(2) 

For any $p\in Spec\left(A\right)$, $u_p^A$ is a flat quantale morphism.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Applying Corollary 5.5 of [19], if A is a semiprime zero-dimensional quantale, then any $p\in Spec\left(A\right)$ is pure. According to Proposition 8, $u_p^A$ is a flat quantale morphism.

$\left(2\right)\Rightarrow \left(1\right)$ By the implication $\left(8\right)\Rightarrow \left(1\right)$ of Theorem 4, A is a zero-dimensional quantale. It remains to show that A is semiprime. Assume by absurdum that A is not semiprime; hence, in accordance with Lemma 3(3), there exists a non-zero compact element c such that $c^2=0$.

Let us denote $x=\bigvee \{d\in K\left(A\right)|c\le d^{\perp }\}$. Assume that $x=1$. Using the compactness of 1, one can find an integer $n\ge 1$ and $d_1,\dots ,d_n\in K\left(A\right)$ such that ${\bigvee }_{i=1}^n=1$ and $c\le d_i^{\perp }$, for any $i=1,\dots ,n$. Using the residuation theory, we obtain $c\le {\bigwedge }_{i=1}^n{d}_i^{\perp }={\left({\bigvee }_{i=1}^n\right)}^{\perp }=1^{\perp }=0$, contradicting the assumption that $c\ne 0$.

It follows that $x<1$; hence, $x\le p$ for some $p\in Spec\left(A\right)$. We remark that $c\le c^{\perp }$, so $c\le x\le p$. Since $u_p^A$ is a flat, it follows that p is pure (cf. Proposition 8), so $c\le p$ implies that $p\vee c^{\perp }=1$. On the other hand, in accordance with the definition of x, for each $d\in K\left(A\right)$, the following implications hold: $d\le c^{\perp }\Rightarrow c\le d^{\perp }\Rightarrow d\le x\le p$. Then, $c^{\perp }\le p$; hence, $p=p\vee c^{\perp }=1$, contradicting that p is m-prime. In conclusion, A is a semiprime quantale. □

Recall from [19] the following notions in a quantale A:

• An element $a\in A$ is N-pure if for any $a\in K\left(A\right)$ such that $c\le a$ there exists an integer $n\ge 1$ such that ${\left(c^n\right)}^{\perp }\vee a=1$;
• An element $a\in A$ is w-pure if for any $a\in K\left(A\right)$ such that $c\le a$ we have $(c\to \rho (0\left)\right)\vee a=1$;
• A is an $NJ$-quantale if ${\rho }_A\left(0\right)=r\left(A\right)$.

We note that the N-pure elements in a quantale are abstractions of the N-pure ideals in a commutative ring [29].

Lemma 28 

([19]). An element a of a coherent quantale A is N-pure if and only it is w-pure.

Theorem 5. 

For any coherent quantale A, the following are equivalent:

(1) 

$dim\left(A\right)=0$;

(2) 

Any $a\in A$ is N-pure;

(3) 

Any $c\in K\left(A\right)$ is N-pure;

(4) 

Any $m\in Max\left(A\right)$ is N-pure;

(5) 

For any $p\in Spec\left(A\right)$, $A_p$ is an $NJ$-quantale;

(6) 

For any $m\in Max\left(A\right)$, $A_m$ is an $NJ$-quantale.

Proof. 

$\left(1\right)\iff \left(2\right)\iff \left(3\right)\iff \left(4\right)$ See Theorem 5.4 of [19].

$\left(2\right)\Rightarrow \left(5\right)$ Assume that any $a\in A$ is N-pure. Let p be an m-prime element of A. We have to prove that ${\rho }_{A_p}\left(0_p\right)=r\left(A_p\right)$. By Proposition 3 we have $Max\left(A_p\right)=\left\{p\right\}$, so $r\left(A_p\right)=p$. On the other hand, ${\rho }_{A_p}\left(0_p\right)={\left(\rho \left(0\right)\right)}_p$ (cf. Proposition 4). Hence, we need to verify the equality $p={\left(\rho \left(0\right)\right)}_p$.

Since $\rho \left(0\right)\le p$, we obtain ${\left(\rho \left(0\right)\right)}_p\le p$. Recall from Corollary 2 that any compact element of $A_p$ has the form $c_p$, where $c\in K\left(A\right)$. Therefore, in order to prove the converse inequality $p\le {\left(\rho \left(0\right)\right)}_p$, consider a compact element c of A such that $c_p\le p$. Then, $c\le p$, so $p\vee (c\to \rho (0\left)\right)=1$ (according to Lemma 28, the hypothesis and that p is w-pure). It follows that $c\to \rho \left(0\right)\nleqq p$; hence, $c\le {\left(\rho \left(0\right)\right)}_p$ (by Lemma 10). Then, $c_p\le {\left(\rho \left(0\right)\right)}_p$ for any $c\in K\left(A\right)$; hence, $p\le {\left(\rho \left(0\right)\right)}_p$. Thus, $p={\left(\rho \left(0\right)\right)}_p$, so $A_p$ is an $NJ$-quantale.

$\left(5\right)\Rightarrow \left(6\right)$ Obviously.

$\left(6\right)\Rightarrow \left(4\right)$ Assume that $m\in Max\left(A\right)$. We have to show that m is N-pure. Let c be a compact element of A such that $c\le m$. Using hypothesis $\left(6\right)$ together with Propositions 3 and 4, we have $m=r\left(A_m\right)={\rho }_{A_m}\left(0_m\right)={\left(\rho \left(0\right)\right)}_m$; hence, $c\le {\left(\rho \left(0\right)\right)}_m$. By Lemma 10, we obtain $c\to \rho \left(0\right)\nleqq m$, so $m\vee (c\to \rho (0\left)\right)=1$ (because $m\in Max\left(A\right)$). Therefore, m is an N-pure element of A. □

Following [24], a quantale A is an $mp$-quantale if for any $p\in Spec\left(A\right)$, there exists a unique $q\in Min\left(A\right)$ such that $q\le p$. We note that the $mp$-quantales abstractize the notion of $mp$-rings [29].

Lemma 29 

([24]). Let A be a coherent quantale. Then, the following are equivalent:

(1) 

A is an $mp$-quantale;

(2) 

For all distinct $p,q\in Min\left(A\right)$, we have $p\vee q=1$.

Theorem 6. 

Let A be a coherent quantale. Then, the following are equivalent:

(1) 

For any non-maximal $p\in Spec\left(A\right)$, $u_p^A:A\to {\left[p\right)}_A$ satisfies the $GD$-property;

(2) 

$dim\left(A\right)\le 1$ and A is an $mp$-quantale.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Assume by absurdum that $dim\left(A\right)\ge 2$, so there exist three m-prime elements $p,q,r$ of A such that $q<p<r$. Then, $p\notin Max\left(A\right)$; hence, $u_p^A$ satisfies the $GD$-property. We know from Lemma 8(5) that $Spec\left({\left[p\right)}_A\right)=\{s\in Spec\left(A\right)|p\le s\}$, so $r\in Spec\left({\left[p\right)}_A\right))$. By Lemma 9, we have ${\left(u_p^A\right)}_{\ast }\left(r\right)=r$, so $q<{\left(u_p^A\right)}_{\ast }\left(r\right)$. Thus, there exists $s\in Spec\left({\left[p\right)}_A\right)$ such that ${\left(u_p^A\right)}_{\ast }\left(s\right)=q$. A new application of Lemma 9 gives $s=q$, contradicting $q<p\le s$. It follows that $dim\left(A\right)\le 1$.

In order to show that A is an $mp$-quantale, consider two distinct minimal m-prime elements p and q. Let us assume that $p\nleqq q$. We have to prove that $p\vee q=1$. Suppose that $p\vee q\ne 1$, so $p\vee q\le m$ for some $m\in Max\left(A\right)$.

We remark that $dim\left(A\right)=1$ (because $dim\left(A\right)=0$ implies $p=m=q$) and $p<m,q<m$. Thus, $p\notin Max\left(A\right)$, so, by hypothesis $\left(1\right)$, it follows that the quantale morphism $u_p^A:A\to {\left[p\right)}_A$ satisfies the $GD$-property. Therefore, $m\in Spec\left({\left[p\right)}_A\right)$ and ${\left(u_p^A\right)}_{\ast }\left(q\right)=q<m$ imply that there exists $r\in Spec\left({\left[p\right)}_A\right)$ such that $r\le m$ and ${\left(u_p^A\right)}_{\ast }\left(r\right)=q$, so $r=q$ (by Lemma 9). Then, $q\in Spec\left({\left[p\right)}_A\right)$; hence, $p\le q$, contradicting the hypothesis $p\nleqq q$. It follows that $p\vee q=1$; therefore, A is an $mp$-quantale (cf. Lemma 29).

$\left(2\right)\Rightarrow \left(1\right)$ Assume that $p\in Spec\left(A\right)$ is non-maximal. We have to prove that $u_p^A$ satisfies the $GD$-property. Since $dim\left(A\right)\le 1$, we consider two cases:

$\left(a\right)$ $dim\left(A\right)=0$. By virtue of Theorem 4, $u_p^A$ satisfies the $GD$-property.

$\left(b\right)$ $dim\left(A\right)=1$. We can find a maximal element m of A such that $p<m$. Since A is an $mp$-quantale, p is the unique minimal m-prime element below m. We also remark that $Spec\left({\left[p\right)}_A\right)=\{p,m\}$. In order to prove that $u_p^A$ satisfies the $GD$-property, we must check the following condition:

$(\star )$ for all $r\in Spec\left(A\right)$ and $q\in Spec\left({\left[p\right)}_A\right)$ such that $r\le {\left(u_p^A\right)}_{\ast }\left(q\right)$ there exists $t\in Spec\left({\left[p\right)}_A\right)$ such that $t\le q$ and ${\left(u_p^A\right)}_{\ast }\left(t\right)=r$.

Since $Spec\left({\left[p\right)}_A\right)=\{p,m\}$, we need to verify condition $(\star )$ in the following two situations:

$\left(i\right)$ $r\in Spec\left(A\right),q=p$ and $r\le {\left(u_p^A\right)}_{\ast }\left(p\right)$.

According to Lemma 9, we obtain $r\le p$; therefore, $r=p$ (because p is a minimal m-prime element). Hence, condition $(\star )$ follows by taking $t=p$.

$\left(ii\right)$ $r\in Spec\left(A\right),q=m$ and $r\le {\left(u_p^A\right)}_{\ast }\left(m\right)$.

But ${\left(u_p^A\right)}_{\ast }\left(m\right)=m$, so $r\le m$; therefore, we have two possibilities: either $r=p$ or $r=m$. To verify $(\star )$ in these two cases is routine. □

Theorem 7. 

Let A be a coherent quantale. Then, the following are equivalent:

(1) 

For any $p\in Spec\left(A\right)-Min\left(A\right)$, $j_p:A\to A_p$ satisfies the $GU$-property;

(2) 

$dim\left(A\right)\le 1$ and A is a normal quantale.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ Assume that $dim\left(A\right)\ge 2$, so there exist the m-prime elements $p,q,r$ of A such that $q<p<r$. Thus, $p\notin Min\left(A\right)$, so $j_p:A\to A_p$ satisfies the $GU$-property. By Lemma 20, we have ${\left(j_p\right)}_{\ast }\left(q\right)=q$; hence, $r>{\left(j_p\right)}_{\ast }\left(q\right)$. Then, there exists $t\in Spec\left(A_p\right)$ such that $t\ge r$ and ${\left(j_p\right)}_{\ast }\left(t\right)=r$. A new application of Lemma 20 gives $t={\left(j_p\right)}_{\ast }\left(t\right)=r$. According to Proposition 3, $t\in Spec\left(A_p\right)$ implies that $t\in Spec\left(A\right)$ and $r=t\le p$, contradicting the hypothesis that $p<r$. It follows that $dim\left(A\right)\le 1$.

In order to show that A is a normal quantale, assume that $p\in Spec\left(A\right)$, $m,n\in Max\left(A\right)$ and $p\le m\wedge n$. Since $m\notin Min\left(A\right)$, $j_m:A\to A_m$ satisfies the $GU$-property. By Lemma 20, we have ${\left(j_m\right)}_{\ast }\left(p\right)=p$; hence, $n\ge {\left(j_m\right)}_{\ast }\left(p\right)$. Thus, there exists $r\in Spec\left(A_m\right)$ such that $r\ge n$ and $r={\left(j_m\right)}_{\ast }\left(r\right)=p$. From $r\in Spec\left(A_m\right)$, we obtain $r\le m$ (by Proposition 3); hence, $n\le m$. It follows that $m=n$ (because m and n are maximal elements), so A is normal.

$\left(2\right)\Rightarrow \left(1\right)$ Assume that $p\in Spec\left(A\right)-Min\left(A\right)$. We want to prove that $j_p:A\to A_p$ satisfies the $GU$-property. Since $dim\left(A\right)\le 1$, we will treat two cases:

$\left(a\right)$ $dim\left(A\right)=0$. According to Theorem 4, $j_p$ satisfies the $GU$-property.

$\left(b\right)$ $dim\left(A\right)=1$. Then, p is a maximal element of A (because $p\notin Min\left(A\right)$). Assume that $r\in Spec\left(A\right)$, $q\in Spec\left(A_p\right)$ and $r\ge {\left(j_p\right)}_{\ast }\left(q\right)$. By Proposition 3, $q\in Spec\left(A_p\right)$ implies $q\le p$. On the other hand, using Lemma 20, we have ${\left(j_p\right)}_{\ast }\left(q\right)=q$; therefore, $q\le r$. Let m be a maximal element of A such that $r\le m$. Then, $q\le m$, so we obtain $m=p$ (because A is normal and $q\le p$, $q\le m$). It follows that $r\le p$; hence, $r\in Spec\left(A_p\right)$ (cf. Proposition 3). On the other hand, ${\left(j_p\right)}_{\ast }\left(r\right)=r$ and $r\ge r$; therefore, $j_p$ satisfies the $GU$-property. □

7. Final Remarks

The present paper contains three types of contributions:

$\left(I\right)$ The proof of some basic properties of flat quantale morphisms in close connection with Anderson localization theory [4];

$\left(II\right)$ Establishing some relationships between the flat quantale morphisms and the quantale versions of the Going-down and Going-up properties;

$\left(III\right)$ Proving some theorems for characterizing a coherent quantale of dimension 0 and of dimension at most 1.

Here are some suggestions for future research:

$\left(1\right)$ Flat morphisms in universal algebra.

Let $\mathcal{V}$ be a congruence-modular variety [16], A an algebra of $\mathcal{V}$, and $Con\left(A\right)$ the complete lattice of congruences of A. Then, $Con\left(A\right)$, endowed with the commutator operation $[·,·]$, becomes a commutative multiplicative lattice (see [30]). Using the commutator operation, one can define a residuation → on $Con\left(A\right)$ in a canonical way. Denote by $K\left(Con\right(A\left)\right)$ the set of compact congruences of A. Let $f:M\to N$ be a morphism of $\mathcal{V}$. Consider the map $f^{•}:Con\left(A\right)\to Con\left(B\right)$ defined by $f^{•}\left(\theta \right)=C{g}_N\left(f\left(\theta \right)\right)$, i.e., the congruence of N generated by the subset $f\left(\theta \right)$ of $N^2$. We can see that the map $f^{\ast }$ preserves the arbitrary joins but not the commutator operation.

Definition 5. 

We will say that f is a flat morphism of the variety $\mathcal{V}$ if for all $\alpha \in K\left(Con\right(A\left)\right)$ and $\theta \in Con\left(A\right)$, we have $f^{•}(\alpha \to \theta )=f^{•}\left(\alpha \right)\to f^{•}\left(\theta \right)$.

We note that the notion of the flat morphism in a congruence-modular variety $\mathcal{V}$ generalizes the following three particular cases: the flat morphisms of commutative rings [2,3], the flat morphisms of bounded distributive lattices [26], and the flat morphisms of commutative residuated lattices [27]. It would be interesting to investigate how much of the theory of flat ring morphisms can be generalized to congruence-modular varieties.

$\left(2\right)$ Flat morphisms in other types of multiplicative lattices.

The theory of flat quantale morphisms developed in this paper has two important limitations: we focus on commutative and coherent quantales.

In obtaining the most important results of the work, Lemma 3 is used, and the hypothesis of coherence is essential in the proof of this lemma. We do not know of a characterization of quantales in which the statement of this lemma is valid.

Building a flatness theory for non-commutative quantales seems to be a more complicated goal. Recall that in any quantale A, we have two implications: ${\to }_l$ and ${\to }_r$ (cf. [13]). This would require the following definition:

Definition 6. 

Let $A,B$ be two arbitrary quantales and $u:A\to B$ a quantale morphism. We will say that f is a flat quantale morphism if for all $c\in K\left(A\right)$ and $a\in A$, we have $u\left(c{\to }_{l}a\right)=u\left(c\right){\to }_{l}u\left(a\right)$ and $u\left(c{\to }_{l}a\right)=u\left(c\right){\to }_{l}u\left(a\right)$.

It remains to be seen what conditions we must put on the quantales A and B for this definition to lead to notable results, which would apply to non-commutative algebraic structures.

A more general problem is the construction of a flatness theory that abstracts the notion of the flat morphism in a congruence-modular variety $\mathcal{V}$. We point out two difficulties that arise in studying this problem:

• In general, the commutator operation is non-associative.
• If $f:M\to N$ is a morphism of $\mathcal{V}$, then $f^{•}:Con\left(M\right)\to Con\left(N\right)$ does not preserve the commutator operation, so it is not a morphism of multiplicative lattices.