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Preprint Number 936
936. Manuel Bodirsky, David Evans, Michael Kompatscher, Michael Pinsker
A counterexample to the reconstruction of ω-categorical structures from their endomorphism monoids
Submission date: 1st October 2015.
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 $\omega$-categorical structure in a finite relational language which can neither be reconstructed up to first-order bi-interpretations 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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