math.LO

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# Title: A counterexample to the reconstruction of $ω$-categorical structures from their endomorphism monoids

Abstract: We present an example of two countable $\omega$-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.
 Comments: 17 pages Subjects: Logic (math.LO) Cite as: arXiv:1510.00356 [math.LO] (or arXiv:1510.00356v2 [math.LO] for this version)

## Submission history

From: Michael Pinsker [view email]
[v1] Thu, 1 Oct 2015 18:20:20 GMT (23kb)
[v2] Fri, 1 Jul 2016 13:04:19 GMT (24kb)
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