Combining Semilattices and Semimodules

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 $\mathcal P_f$

Combining probabilistic and non-deterministic choice via weak distributive laws

International audienceCombining probabilistic choice and non-determinism is a long standing problem in denotational semantics. From a category theory perspective, the problem stems from the absence of a distributive law of the powerset monad over the distribution monad. In this paper we prove the existence of a weak distributive law of the powerset monad over the finite distribution monad. As a consequence, we retrieve the well-known convex powerset monad as a weak lifting of the powerset monad to the category of convex algebras. We provide applications to the study of trace semantics and behavioral equivalences of systems with an interplay between probability and non-determinism