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International Conference on Relational and Algebraic Methods in Computer Science

RAMiCS 2021: Relational and Algebraic Methods in Computer Science pp 292–308Cite as

Polyadic Spaces and Profinite Monoids

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 13027)

Abstract

Hyperdoctrines are an algebraization of first-order logic introduced by Lawvere in [11]. In [9], Joyal defines a polyadic space as the Stone dual of a Boolean hyperdoctrine. He also proposed to recover a polyadic space from a simpler core, its Stirling kernel. We generalize this here in order to adapt polyadic spaces to certain classes of first-order theories. We will see how these ideas can be applied to give a correspondence between some first-order theories with a linear order symbol and equidivisible profinite semigroup with open multiplication. The inspiration comes from the paper [6] of van Gool and Steinberg, where model theory is used to study pro-aperiodic monoids.

Keywords

  • Categorical logic
  • Profinite monoids
  • Logic on words

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 670624).

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Notes

  1. 1.

    More precisely, if there is a binary relational symbol \(\le \) subject to the axioms of partial orders.

  2. 2.

    Trees and forests are considered as special ordered sets for the notion to be first-order.

  3. 3.

    The set \(B \sqcup B \subseteq \mathscr {P}(A^*\times A^*)\) is the Boolean subalgebra generated by sets of the form \(L\times A^*\) and \(A^*\times L\) where \(L \in B\). It is also the coproduct of two copies of B in the category of Boolean algebras.

References

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Acknowledgments

This work has been done under the supervision of Mai Gehrke and André Joyal.

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Authors and Affiliations

  1. Université Côte d’Azur, CNRS, LJAD (UMR 7351), Nice, France

    Jérémie Marquès

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  1. Jérémie Marquès
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Correspondence to Jérémie Marquès .

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Editors and Affiliations

  1. École polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Université Côte d’Azur, Nice, France

    Mai Gehrke

  3. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  4. Brock University, St Catharines, ON, Canada

    Prof. Dr. Michael Winter

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Marquès, J. (2021). Polyadic Spaces and Profinite Monoids. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_18

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  • DOI: https://doi.org/10.1007/978-3-030-88701-8_18

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