A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoid
- 24 Downloads
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.
Unable to display preview. Download preview PDF.
- [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
- [BPP14]M. Bodirsky, M. Pinsker and A. Pongrácz, Projective clone homomorphisms, Journal od Symbolic Logic, accepted, arXiv:1409.4601, 2014.Google Scholar
- [Las89]D. Lascar, Le demi-groupe des endomorphismes d’une structure N 0-catégorique, in Actes de la Journée Algèbre Ordonnée (Le Mans, 1987), 1989, pp. 33–43.Google Scholar