A COUNTEREXAMPLE TO THE RECONSTRUCTION OF omega-CATEGORICAL STRUCTURES FROM THEIR ENDOMORPHISM MONOID
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 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.