Semialgebras and Weak Distributive Laws

Daniela Petrişan
(IRIF)
Ralph Sarkis
(ENS de Lyon)

Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition.

In Ana Sokolova: Proceedings 37th Conference on Mathematical Foundations of Programming Semantics (MFPS 2021), Hybrid: Salzburg, Austria and Online, 30th August - 2nd September, 2021, Electronic Proceedings in Theoretical Computer Science 351, pp. 218–241.
Published: 29th December 2021.

ArXived at: http://dx.doi.org/10.4204/EPTCS.351.14 bibtex PDF
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