A topological characterisation of endomorphism monoids of countable structures

TL;DR

This paper provides a topological characterization of endomorphism monoids and polymorphism clones of countable structures, focusing on properties like separability, ultrametrics, and non-expansiveness, with special cases for omega-categorical structures.

Contribution

It introduces new topological criteria that precisely identify when a monoid or clone is isomorphic to those arising from countable structures, including omega-categorical ones.

Findings

Characterization of endomorphism monoids via separability and ultrametrics

Topological criteria for polymorphism clones of countable structures

Special case characterization for omega-categorical structures

Abstract

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable omega-categorical structures. Finally we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable omega-categorical structures.