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

TL;DR

This paper provides a counterexample showing that $$-categorical structures cannot be uniquely reconstructed from their endomorphism monoids, automorphism groups, or polymorphism clones, especially when considering topological properties.

Contribution

It presents the first known example demonstrating the failure of reconstructing $$-categorical structures from their algebraic and topological endomorphism monoids.

Findings

Endomorphism monoids can be isomorphic as abstract monoids but not as topological monoids.

Automorphism groups and polymorphism clones can be isomorphic but not topologically isomorphic.

Certain $$-categorical structures cannot be reconstructed from their algebraic or topological automorphism groups or 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.