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 l-monoid (an l-semigroup 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 i-semiring. 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 i-semiring 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.

Title	quantale
Canonical name	Quantale
Date of creation	2013-03-22 17:00:08
Last modified on	2013-03-22 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