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Polyadic Spaces and Profinite Monoids

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


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.


  • 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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  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.


  1. Almeida, J., et al.: The linear nature of pseudowords. In: Publicacions Matemàtiques, vol. 63, pp. 361–422, July 2019. ISSN: 0214-1493.

  2. Borceux, F.: Handbook of Categorical Algebra 1 - Basic Category Theory. Cambridge University Press, Cambridge (1994).

  3. Doner, J.E., Mostowski, A., Tarski, A.: The elementary theory of well-odering - a metamathematical study. In: Logic Colloquium 1977. Studies in Logic and the Foundations of Mathematics, vol. 96, pp. 1–54. Elsevier (1978).

  4. Gehrke, M.: Stone duality, topological algebra, and recognition. J. Pure Appl. Algebra 220(7), 2711–2747 (2016)

    CrossRef  MathSciNet  Google Scholar 

  5. Gehrke, M., Grigorieff, S., Pin, J.É.: A topological approach to recognition. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 151–162. Springer, Heidelberg (2010).

    CrossRef  Google Scholar 

  6. van Gool, S.J., Steinberg, B.: Pro-aperiodic monoids via saturated models. Israel J. Math. 234(1), 451–498 (2019).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997). ISBN: 0521587131

    Google Scholar 

  8. Johnstone, P.: Stone Spaces, vol. xxi, p. 370. Cambridge University Press, Cambridge (1982). ISBN: 0521238935

    Google Scholar 

  9. Joyal, A.: Polyadic spaces and elementary theories. In: Notices of the American Mathematical Society, April 1971

    Google Scholar 

  10. Lambek, J., Scott, P.J.: Introduction to Higher Order Categorical Logic. Cambridge University Press, Cambridge (1986). ISBN: 0521246652

    Google Scholar 

  11. Lawvere, F.W.: Adjointness in foundations. In: Dialectica (1969).

  12. Marquis, J.-P., Reyes, G.E.: The history of categorical logic: 1963–1977. In: Sets and Extensions in the Twentieth Century. Handbook of the History of Logic, pp. 689–800. Elsevier (2012).

  13. Pin, J.-E.: Logic on words. In: Current trends in theoretical computer science, pp. 254–273. World Scientific Publishing (2001)

    Google Scholar 

  14. Rasiowa, H., Sikorski, R.: Proof of the completeness theorem of Gödel. In: Fundamenta Mathematicae 37(1), 193–200 (1950).

  15. Rosenstein, J.G.: Linear Orderings, vol. xvii, p. 487. Academic Press, New York (1981). ISBN: 0125976801

    Google Scholar 

  16. Seely, R.A.G.: Hyperdoctrines, natural deduction and the beck condition. Math. Logic Q. 29(10), 505–542 (1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

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

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Correspondence to Jérémie Marquès .

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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.

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