A positively ordered rig $A$ is a rig together with a (pre-)order $\leq$ such that:
- $x\leq y \Rightarrow x + z \leq y +z$
- $x\leq y \Rightarrow x\cdot z \leq y\cdot z\,\&\, z\cdot x \leq z\cdot y$
- $0\leq x$
It is complete if abitrary infima $\bigwedge$ and suprema $\bigvee$ exist w.r.t. to $\leq$ (Obviously $\bigwedge A = 0$ and we would let $\infty := \bigvee A$) (Note, that one of the conditions is actually redundant)
We can define "abitrary sums" via:
$$\sum_I (x_i)_{x\in I} = \bigvee \{\sum_{i\in A} x_i : A\subseteq I, \# A < \infty\}$$
Motivating examples are (among many others I presume):
- $[0\,..\infty]$, with the usual structure, where $0\cdot \infty = 0$
- $[0\,..\infty]^X$, the set of all functions $X \to [0\,..\infty]$ with the usual structure
- the set of all nonnegative measurable functions from some measurable space
- every complete lattice
A morphism of "cpor's" is a function preserving all this structure. As an example take:
$$\operatorname{eval} :[0\,..\infty]^X \times X \to [0\,..\infty], (f,x) \mapsto fx$$
Then $\operatorname{eval}(\_,x)$ is a morphism of cpor's pretty much by definition for all $x\in X$.
Another example is the obvious inclusion $[0\,..\infty] \to [0\,..\infty]^X$.
What references are there on complete positively ordered rigs and (possibly) their morphisms?