quantale
A quantale $Q$ is a set with three binary operations^{} on it: $\wedge ,\vee $, and $\cdot $, such that

1.
$(Q,\wedge ,\vee )$ is a complete lattice^{} (with $0$ as the bottom and $1$ as the top), and

2.
$(Q,\cdot )$ is a monoid (with ${1}^{\prime}$ as the identity^{} with respect to $\cdot $), such that

3.
$\cdot $ distributes over arbitrary joins; that is, for any $a\in Q$ and any subset $S\subseteq Q$,
$$a\cdot \left(\bigvee S\right)=\bigvee \{a\cdot s\mid s\in S\}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\left(\bigvee S\right)\cdot a=\bigvee \{s\cdot a\mid s\in S\}.$$
It is sometimes convenient to drop the multiplication symbol, when there is no confusion. So instead of writing $a\cdot b$, we write $ab$.
The most obvious example of a quantale comes from ring theory. Let $R$ be a commutative ring with $1$. Then $L(R)$, the lattice of ideals of $R$, is a quantale.
Proof.
In addition to being a (complete^{}) lattice^{}, $L(R)$ has an inherent multiplication operation^{} induced by the multiplication on $R$, namely,
$$IJ:=\{\sum _{i=1}^{n}{r}_{i}{s}_{i}\mid {r}_{i}\in I\text{and}{s}_{i}\in J\text{,}n\in \mathbb{N}\},$$ 
making it into a semigroup^{} under the multiplication.
Now, let $S=\{{I}_{i}\mid i\in N\}$ be a set of ideals of $R$ and let $I=\bigvee S$. If $J$ is any ideal of $R$, we want to show that $IJ=\bigvee \{{I}_{i}J\mid i\in N\}$ and, since $R$ is commutative^{}, we would have the other equality $JI=\bigvee \{J{I}_{i}\mid i\in N\}$. To see this, let $a\in IJ$. Then $a=\sum {r}_{i}{s}_{i}$ with ${r}_{i}\in I$ and ${s}_{i}\in J$. Since each ${r}_{i}$ is a finite sum of elements of $\bigcup S$, ${r}_{i}{s}_{i}$ is a finite sum of elements of $\bigcup \{{I}_{i}J\mid i\in N\}$, so $a\in \bigvee \{{I}_{i}J\mid i\in N\}$. This shows $IJ\subseteq \bigvee \{{I}_{i}J\mid i\in N\}$. Conversely, if $a\in \bigvee \{{I}_{i}J\mid i\in N\}$, then $a$ can be written as a finite sum of elements of $\bigcup \{{I}_{i}J\mid i\in N\}$. In turn, each of these additive components is a finite sum of products^{} of the form ${r}_{k}{s}_{k}$, where ${r}_{k}\in {I}_{i}$ for some $i$, and ${s}_{k}\in J$. As a result, $a$ is a finite sum of elements of the form ${r}_{k}{s}_{k}$, so $a\in IJ$ and we have the other inclusion $\bigvee \{{I}_{i}J\mid i\in N\}\subseteq IJ$.
Finally, we observe that $R$ is the multiplicative identity^{} in $L(R)$, as $IR=RI=I$ for all $I\in L(R)$. This completes the proof. ∎
Remark. In the above example, notice that $IJ\le I$ and $IJ\le J$, and we actually have $IJ\le I\wedge J$. In particular, ${I}^{2}\le I$. With an added condition, this fact can be characterized in an arbitrary quantale (see below).
Properties. Let $Q$ be a quantale.

1.
Multiplication is monotone in each argument. This means that if $a,b\in Q$, then $a\le b$ implies that $ac\le bc$ and $ca\le cb$ for all $c\in Q$. This is easily verified. For example, if $a\le b$, then $ac\vee bc=(a\vee b)c=bc$, so $ac\le bc$. So a quantale is a partially ordered semigroup, and in fact, an lmonoid (an lsemigroup and a monoid at the same time).

2.
If $1={1}^{\prime}$, then $ab\le a\wedge b$: since $a\le 1$, then $ab\le a1=a{1}^{\prime}=a$; similarly, $b\le ab$. In particular, the bottom $0$ is also the multiplicative zero: $a0\le a\wedge 0=0$, and $0a=0$ similarly.

3.
Actually, $a0=0a=0$ is true even without $1={1}^{\prime}$: since $a\mathrm{\varnothing}=\{ab\mid b\in \mathrm{\varnothing}\}=\mathrm{\varnothing}$ and $0:=\bigvee \mathrm{\varnothing}$, we have $a0=a\bigvee \mathrm{\varnothing}=\bigvee a\mathrm{\varnothing}=\bigvee \mathrm{\varnothing}=0$. Similarly $0a=0$. So a quantale is a semiring^{}, if $\vee $ is identified as $+$ (with $0$ as the additive identity), and $\cdot $ is again $\cdot $ (with ${1}^{\prime}$ the multiplicative identity).

4.
Viewing quantale $Q$ now as a semiring, we see in fact that $Q$ is an idempotent semiring, since $a+a=a\vee a=a$.

5.
Now, view $Q$ as an isemiring. For each $a\in Q$, let $S=\{{1}^{\prime},a,{a}^{2},\mathrm{\dots}\}$ and define ${a}^{*}=\bigvee S$. We observe some basic properties

–
${1}^{\prime}+a{a}^{*}={a}^{*}$: since ${1}^{\prime}\vee (a\bigvee S)={1}^{\prime}\vee (\bigvee \{a{1}^{\prime},aa,a{a}^{2},\mathrm{\dots}\})=\bigvee \{{1}^{\prime},a,{a}^{2},\mathrm{\dots}\}=\bigvee S={a}^{*}$

–
${1}^{\prime}+{a}^{*}a={a}^{*}$ as well

–
if $ab\le b$, then ${a}^{*}b\le b$: by induction^{} on $n$, we have ${a}^{n}b\le b$ whenever $a\le b$, so that ${a}^{*}b=\bigvee \{{a}^{n}b\mid n\in \mathbb{N}\cup \{0\}\}\le b$.

–
similarly, if $ba\le b$, then $b{a}^{*}\le b$
All of the above properties satisfy the conditions for an isemiring to be a Kleene algebra. For this reason, a quantale is sometimes called a standard Kleene algebra.

–

6.
Call the multiplication idempotent^{} if each element is an idempotent with respect to the multiplication: $aa=a$ for any $a\in Q$. If $\cdot $ is idempotent and $1={1}^{\prime}$, then $\cdot =\wedge $. In other words, $ab=a\wedge b$.
Proof.
As we have seen, $ab\le a\wedge b$ in the 2 above. Now, suppose $c\le a\wedge b$. Then $c\le a$ and $c\le b$, so $c={c}^{2}\le cb\le ab$. So $ab$ is the greatest lower bound^{} of $a$ and $b$, i.e., $ab=a\wedge b$. This also means that $ba=b\wedge a=a\wedge b=ab$. ∎

7.
In fact, a locale is a quantale if we define $\cdot :=\wedge $. Conversely, a quantale where $\cdot $ is idempotent and $1={1}^{\prime}$ is a locale.
Proof.
If $Q$ is a locale with $\cdot =\wedge $, then $aa=a\wedge a=a$ and $a1=a\wedge 1=a=1\wedge a=1a$, implying $1={1}^{\prime}$. The infinite^{} distributivity of $\cdot $ over $\vee $ is just a restatement of the infinite distributivity of $\wedge $ over $\vee $ in a locale. Conversely, if $\cdot $ is idempotent and $1={1}^{\prime}$, then $\cdot =\wedge $ as shown previously, so $a\wedge (\bigvee S)=a(\bigvee S)=\bigvee \{as\mid s\in S\}=\bigvee \{a\wedge s\mid s\in S\}$. Similarly $(\bigvee S)\wedge a=\bigvee \{s\wedge a\mid s\in S\}$. Therefore, $Q$ is a locale. ∎
Remark. A quantale homomorphism between two quantales is a complete lattice homomorphism and a monoid homomorphism at the same time.
References
 1 S. Vickers, Topology via Logic, Cambridge University Press, Cambridge (1989).
Title  quantale 

Canonical name  Quantale 
Date of creation  20130322 17:00:08 
Last modified on  20130322 17:00:08 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  12 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06F07 
Synonym  standard Kleene algebra 
Defines  quantale homomorphism 