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Preservation of Equations by Monoidal Monads

Authors Louis Parlant, Jurriaan Rot, Alexandra Silva, Bas Westerbaan

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Author Details

Louis Parlant
  • University College London, UK
Jurriaan Rot
  • Radboud University, Nijmegen, The Netherlands
Alexandra Silva
  • University College London, UK
Bas Westerbaan
  • University College London, UK

Funding

This work was partially supported by the ERC Starting Grant ProFoundNet (grant code 679127), the Marie Curie Fellowship CARBS (grant code 795119), and the EPSRC Standard Grant CLeVer (EP/S028641/1).

Acknowledgements

We are grateful to Dan Marsden, Robin Piedeleu, Edmund Robinson and Maaike Zwart for inspiring discussions and suggestions.

Cite As Get BibTex

Louis Parlant, Jurriaan Rot, Alexandra Silva, and Bas Westerbaan. Preservation of Equations by Monoidal Monads. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.77

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
Keywords
  • monoidal monads
  • algebraic theories
  • preservation of equations

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