Israel Journal of Mathematics

, Volume 224, Issue 1, pp 57–82 | Cite as

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

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


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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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 Algebra, TU DresdenDresdenGermany
  2. 2.Department of MathematicsHuxley Building, South 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 UniversityCharlesCzech Republic

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