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

Manuel Bodirsky; David Evans; Michael Kompatscher; Michael Pinsker

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arXiv · 1510.00356

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

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 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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Manuel Bodirsky, David Evans, Michael Kompatscher, Michael Pinsker. 2016-07-01. A counterexample to the reconstruction of $ω$-categorical structures from their endomorphism monoids. https://arxiv.org/abs/1510.00356

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