Year: 2021
Type: conference-paper
Abstract: We describe the canonical weak distributive law $$\delta :\mathcal S\mathcal P\rightarrow \mathcal P\mathcal S$$ δ : S P → P S of the powerset monad $$\mathcal P$$ P over the S-left-semimodule monad $$\mathcal S$$ S , for a class of semirings S. We show that the composition of $$\mathcal P$$ P with $$\mathcal S$$ S by means of such $$\delta $$ 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 $$\mathcal P$$ P to $$\mathbb {EM}(\mathcal S)$$ EM ( 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 $$\mathcal P_f$$ P f .
Authors: Filippo Bonchi, Alessio Santamaria
Institution: University of Pisa
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $5,392
OpenAlex ID: W3115858079