Combining Semilattices and Semimodules

TL;DR

This paper introduces a canonical weak distributive law between powerset and semimodule monads over semirings, leading to a new monad for convex subsets with algebraic characterizations.

Contribution

It defines a canonical weak distributive law for powerset and semimodule monads, extending convex subset monads and providing algebraic presentations.

Findings

The composed monad models convex subsets including the empty set.

A characterization of the weak lifting of the powerset monad is provided.

The monad restricted to finitely generated convex sets is algebraically presented.

Abstract

We describe the canonical weak distributive law $\delta \colon \mathcal S \mathcal P \to \mathcal P \mathcal S$ 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 $\delta$ 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 $\mathcal P_f$.