Combining Semilattices and Semimodules

12/29/2020
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by   Filippo Bonchi, et al.
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We describe the canonical weak distributive law ฮด๐’ฎ๐’ซโ†’๐’ซ๐’ฎ of the powerset monad ๐’ซ over the S-left-semimodule monad ๐’ฎ, for a class of semirings S. We show that the composition of ๐’ซ with ๐’ฎ by means of such ฮด 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 ๐’ซ to ๐”ผ๐•„(๐’ฎ) 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 ๐’ซ_f.

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