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Preservation of Equations by Monoidal Monads Parlant, Louis Rot, Jurriaan Silva, Alexandra Westerbaan, Bas monoidal monads algebraic theories preservation of equations If a monad T is monoidal, then operations on a set X can be lifted canonically to operations on TX. In this paper we study structural properties under which T preserves equations between those operations. It has already been shown that any monoidal monad preserves linear equations; affine monads preserve drop equations (where some variable appears only on one side, such as x⋅ y = y) and relevant monads preserve dup equations (where some variable is duplicated, such as x ⋅ x = x). We start the paper by showing a converse: if the monad at hand preserves a drop equation, then it must be affine. From this, we show that the problem whether a given (drop) equation is preserved is undecidable. A converse for relevance turns out to be more subtle: preservation of certain dup equations implies a weaker notion which we call n-relevance. Finally, we identify a subclass of equations such that their preservation is equivalent to relevance. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Louis Parlant and Jurriaan Rot and Alexandra Silva and Bas Westerbaan 2020 Is Part Of LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2020.77 urn:nbn:de:0030-drops-127460 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.77 eng https://creativecommons.org/licenses/by/3.0/legalcode