# Examples of value quantales

In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of distances with ordinary addition, distance distribution functions with (reversed) pointwise order and suitable triangle function as addition, free locales.

I'm looking for more examples, especially those which are concrete and possibly simple.

I'm especially interested in examples of the following types (or argument that there are no such examples if this is the case!):

$\bullet$ Consider the set $P=\{f:I \rightarrow [0,+\infty]\}$ ($I$ is an interval in $\mathbb{R}$, let's say $[0,1]$ or $(0,1)$) with usual pointwise order and pointwise addition of functions. Now let $Q\subseteq P$. Can $Q$, considered with inherited pointwise order and pointwise addition, be a value quantale (this question is particularly interesting for "nice" subsets $Q$, like $Q$ being all nondecreasing functions, or some continous functions etc.)?

$\bullet$ More generally: let $\{Q_j\}_{j\in J}$ be a family of value quantales. Consider their (set-theoretic) product $\prod_{j\in J}Q_j$ equipped with index-wise order and index-wise addition. When it is a value quantale, or does it posess any subsets (with inherited operations) that are value quantales?

$\bf{Definitions:}$ Let $V$ be a poset. First the well above relation: $x\prec y$ iff for all subsets $S\subseteq V$, if $x\geq\inf S$ then $y\geq s_0$ for some $s_0\in S$. If $V$ is a complete lattice we say that it is completely distributive if $y=\inf\{x\in V\ |\ \ x\succ y\}$ for all $y\in V$; moreover, if $\infty\succ 0$ and $x\wedge y\succ 0\Rightarrow x\succ 0\ \&\ y\succ 0$ we say that $V$ is a value distributive lattice ($0$ is the least element, and $\infty$ is the greatest element in $V$). A quantale is a complete lattice $Q$ equipped with an associative and commutative binary operation $+$ such that $x+0=x$ and $x+\inf S=\inf(x+S)$ for all $x\in Q,\ S\subseteq Q$. Finally, a quantale $V$ is a value quantale iff as a lattice it is a value distributive lattice.

• What's a value quantale? It would be best to give the definition in your question, rather than make the reader try to track down Flagg's paper. (By the way, why do you offer a bounty on a question that may admit a wide variety of possible correct answers? Usually we've used Community Wiki in such a situation.) – Todd Trimble Sep 12 '15 at 22:21
• @Todd: I have added some definitions. Since the second part of my question requires a bit more than just giving an example I think offering a bounty is quite ok. – Godot Sep 17 '15 at 16:11