Really good stuff

[timelets]

...the most basic questions about a resource theory: When are two resources equivalent under the free operations, in the sense that one can convert one to the other and vice-versa? What are the necessary and sufficient conditions on two resources such that the first can be converted to the second (possibly irreversibly) under the free operations? How can we find measures of the quality of a resource, that is, functions from the resources to the reals which are nonincreasing under the transformations allowed by the resource theory? ... Can a given resource conversion be made possible by the presence of a catalyst, i.e. a resource which must be present but is not consumed in the process?

We refer to this minimal framework as a theory of resource convertibility. The requisite mathematical formalism turns out to be that of a commutative (equivalently, Abelian or symmetric) preordered monoid.

First, consider the preorder. This is the order over resources that captures whether one resource can be converted to another or not. Recall that the first of the questions above called us to characterize the equivalence classes of resources under the free operations, and the second called us to find the partial order over these equivalence classes that is induced by the free operations. But this is nothing more than a call to find the preorder over the resources induced by the free relation of convertibility. Once the preorder is identified, one can define measures of the resource, or “resource monotones”, as any map from the resources to the reals which respects the preorder.

BFS. https://doi.org/10.1016/j.ic.2016.02.008

Just a few days ago I wanted to write that I can simulate any system of interest with a preorder and a monoid :)