# Reconstructing the topology of the elementary self-embedding monoids of countable saturated structures

## TL;DR

This paper investigates conditions under which the topology of elementary self-embedding monoids of countable saturated structures can be reconstructed from their algebraic structure, extending the concept of automatic homeomorphicity.

## Contribution

It proves that if the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then its elementary self-embedding monoid also has this property, and strengthens existing results on isomorphisms.

## Key findings

- Elementary self-embedding monoids inherit automatic homeomorphicity under certain conditions.
- Isomorphisms between monoids of elementary self-embeddings are homeomorphisms for specific structures.
- Conditions on automorphism groups ensure topological reconstruction from algebraic structure.

## Abstract

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever $\mathbf{A}$ is a countable $\aleph_0$-categorical $G$-finite structure whose automorphism group has a trivial center and if $\mathbf{B}$ is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism.