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

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.