Combining Semilattices and Semimodules

We describe the canonical weak distributive law [Formula: see text] of the powerset monad [Formula: see text] over the S-left-semimodule monad [Formula: see text] , for a class of semirings S. We show that the composition of [Formula: see text] with [Formula: see text] by means of such [Formula: see text] yields almost the monad of convex subsets previously introduced by Jacobs: the only difference consists in the absence in Jacobs’s monad of the empty convex set. We provide a handy characterisation of the canonical weak lifting of [Formula: see text] to [Formula: see text] as well as an algebraic theory for the resulting composed monad. Finally, we restrict the composed monad to finitely generated convex subsets and we show that it is presented by an algebraic theory combining semimodules and semilattices with bottom, which are the algebras for the finite powerset monad [Formula: see text] .