Convexity via Weak Distributive Laws

TL;DR

This paper characterizes a weak distributive law between the powerset and semimodule monads for certain semirings, leading to a monad of convex subsets in free semimodules, advancing the theoretical understanding of convexity in algebraic structures.

Contribution

It introduces a novel characterization of the weak distributive law as a convex closure in free semimodules, expanding the algebraic framework for convexity via monad composition.

Findings

Characterization of the weak distributive law as convex closure

Construction of a monad of convex subsets in free semimodules

Application to semirings including positive semifields

Abstract

We study the canonical weak distributive law $\delta$ of the powerset monad over the semimodule monad for a certain class of semirings containing, in particular, positive semifields. For this subclass we characterise $\delta$ as a convex closure in the free semimodule of a set. Using the abstract theory of weak distributive laws, we compose the powerset and the semimodule monads via $\delta$, obtaining the monad of convex subsets of the free semimodule.