Combining Semilattices and Semimodules

DOI10.1007/978-3-030-71995-1_6arXiv2012.14778MaRDI QIDQ6357167

Alessio Santamaria, Filippo Bonchi

Publication date: 29 December 2020

Abstract: We describe the canonical weak distributive law $deltacolonmathcalSmathcalPomathcalPmathcalS$ of the powerset monad $mathcalP$ over the $S$-left-semimodule monad $mathcalS$, for a class of semirings $S$. We show that the composition of $mathcalP$ with $mathcalS$ 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 $mathcalP$ to $mathbbEM(mathcalS)$ 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 $mathcal{P}_f$.

Mathematics Subject Classification ID

Theory of computing (68Qxx) Theory of software (68Nxx)

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