The powerset monad on the category of sets does not distribute over itself. Nevertheless a weaker form of distributive law of the powerset monad over itself exists and it essentially stems from the canonical Egli-Milner extension of the powerset to the category of relations. On the other hand, any regular category yields a category of relations, and some regular categories also possess a powerset-like monad, as is the Vietoris monad on compact Hausdorff spaces. We derive the Egli-Milner extension in three different frameworks : sets, toposes, and compact Hausdorff spaces. We prove that it corresponds to a monotone weak distributive law in each case by showing that the multiplication extends to relations but the unit does not. We provide an application to coalgebraic determinization of alternating automata.

Combining 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.