A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoid

  • Manuel Bodirsky
  • David Evans
  • Michael Kompatscher
  • Michael Pinsker
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Abstract

We present an example of two countable ω-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids— in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable ω-categorical structure in a finite relational language which can neither be reconstructed up to first-order biinterpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.

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References

  1. [AZ86]
    G. Ahlbrandt and M. Ziegler, Quasi-finitely axiomatizable totally categorical theories, Annals of Pure and Applied Logic 30 (1986), 63–82MathSciNetCrossRefMATHGoogle Scholar
  2. [BK96]
    H. Becker and A. Kechris, The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, Vol. 232, Cambridge University Press, Cambridge, 1996.Google Scholar
  3. [Bod05]
    M. Bodirsky, The core of a countably categorical structure, in STACS 2005, Lecture Notes in Computer Science, Vol. 3404, Springer, Berlin, 2005, pp. 110–120.CrossRefMATHGoogle Scholar
  4. [BP14]
    M. Bodirsky and M. Pinsker, Minimal functions on the random graph, Israel Journal of Mathematics 200 (2014), 251–296MathSciNetCrossRefMATHGoogle Scholar
  5. [BP15]
    M. Bodirsky and M. Pinsker, Topological Birkhoff, Transactions of the American Mathematical Society 367 (2015), 2527–2549MathSciNetCrossRefMATHGoogle Scholar
  6. [BP16]
    L. Barto and M. Pinsker, The algebraic dichotomy conjecture for infinite domain constraint satisfaction problems, in Proceedings of the 31st Annual ACM/IEEE Symposium in Logic in Computer Science (New York, NY, 2016), ACM, New York, NY, 2016, pp. 615–622.CrossRefGoogle Scholar
  7. [BPP]
    M. Bodirsky, M. Pinsker and A. Pongrácz, Reconstructing the topology of clones, Transactions of the American Mathematical Society 369 (2017), 3707–3740MathSciNetCrossRefMATHGoogle Scholar
  8. [BPP14]
    M. Bodirsky, M. Pinsker and A. Pongrácz, Projective clone homomorphisms, Journal od Symbolic Logic, accepted, arXiv:1409.4601, 2014.Google Scholar
  9. [EH90]
    D. M. Evans and P. R. Hewitt, Counterexamples to a conjecture on relative categoricity, Annals of Pure and Applied Logic 46 (1990), 201–209MathSciNetCrossRefMATHGoogle Scholar
  10. [Hod97]
    W. Hodges, A shorter Model Theory, Cambridge University Press, Cambridge, 1997.MATHGoogle Scholar
  11. [Kec95]
    A. Kechris, Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics, Vol. 156, Springer, New York, 1995.Google Scholar
  12. [Las82]
    D. Lascar, On the category of models of a complete theory, Journal of Symbolic Logic 47 (1982), 249–266MathSciNetCrossRefMATHGoogle Scholar
  13. [Las89]
    D. Lascar, Le demi-groupe des endomorphismes d’une structure ℋ0-catégorique, in Actes de la Journée Algèbre Ordonnée (Le Mans, 1987), 1989, pp. 33–43.Google Scholar
  14. [Las91]
    D. Lascar, Autour de la propriété du petit indice, Proceedings of the London Mathematical Society 62 (1991), 25–53, 1991.MathSciNetCrossRefMATHGoogle Scholar
  15. [RZ00]
    L. Ribes and P. Zalesskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 40, Springer, Berlin, 2000.Google Scholar
  16. [She84]
    S. Shelah, Can you take Solovay’s inaccessible away?, Israel Journal of Mathematics 48 (1984), 1–47MathSciNetCrossRefMATHGoogle Scholar
  17. [Wit54]
    E. Witt, Über die Kommutatorgruppe kompakter Gruppen, Rendiconti di Matematica e delle sue Applicazioni. Serie V 14 (1954), 125–129MATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Manuel Bodirsky
    • 1
  • David Evans
    • 2
  • Michael Kompatscher
    • 3
  • Michael Pinsker
    • 4
    • 5
  1. 1.Institut für AlgebraTU DresdenDresdenGermany
  2. 2.Department of Mathematics, Huxley BuildingSouth Kensington Campus, Imperial College LondonLondonUK
  3. 3.Institut für ComputersprachenTheory and Logic Group Technische Universität WienWienAustria
  4. 4.Institut für Diskrete Mathematik und Geometrie, FG AlgebraTU WienAustria
  5. 5.Department of AlgebraCharles UniversityPragueCzech Republic

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