  It came to the attention of myself and the coauthors of (S., Rozowski, Silva,
Rot, 2022) that a number of process calculi can be obtained by algebraically
presenting the branching structure of the transition systems they specify.
Labelled transition systems, for example, branch into sets of transitions,
terms in the free semilattice generated by the transitions. Interpreting
equational theories in the category of sets has undesirable limitations, and we
would like to have more examples of presentations in other categories. In this
brief article, I discuss monad presentations in the category of partially
ordered sets and monotone maps. I focus on quantitative monads, namely free
modules over ordered semirings, and give sufficient conditions for one of these
to lift a monad on the category of sets. I also give a description of ordered
semirings that are useful for specifying unguarded recursive calls. Examples
include ordered probability theory and ordered semilattices.
