Combining Semilattices and Semimodules
This work is foundational for category theory and universal algebra, specifically for researchers interested in monads and their applications in algebraic structures.
This paper describes a canonical weak distributive law between the powerset monad and the S-left-semimodule monad, showing that their composition results in a monad similar to the monad of convex subsets, differing only by the exclusion of the empty set. It also provides an algebraic theory for the resulting composed monad and its restriction to finitely generated convex subsets.
We describe the canonical weak distributive law $δ\colon \mathcal S \mathcal P \to \mathcal P \mathcal S$ of the powerset monad $\mathcal P$ over the $S$-left-semimodule monad $\mathcal S$, for a class of semirings $S$. We show that the composition of $\mathcal P$ with $\mathcal S$ by means of such $δ$ 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 $\mathcal P$ to $\mathbb{EM}(\mathcal S)$ 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 $\mathcal P_f$.