Minimal and intrinsic topologies on monoids of elementary embeddings

Abstract

To every $ω$-categorical structure $M$ one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group $\mathrm{Aut}(M)$ of its automorphisms and the topological monoid $\mathrm{EEmb}(M)$ of its elementary embeddings, both equipped with the topology of pointwise convergence $τ_{\mathrm{pw}}$. We investigate the relation of $τ_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $τ_{\mathrm{pw}}$ is minimal, i.e. does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of $τ_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $τ_{\mathrm{Z}}$. We show that $τ_{\mathrm{pw}}$ differs from $τ_{\mathrm{Z}}$ on $\mathrm{EEmb}(M)$ whenever $\mathrm{Aut}(M)$ has a non-trivial centre. In spite of this, we then prove that whenever algebraic closure on $M$ is modular, then $τ_{\mathrm{pw}}$ is minimal on $\mathrm{EEmb}(M)$. This covers, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{Aut}(M)$, we describe the semigroup topologies coarser than $τ_{\mathrm{pw}}$ on the automorphism groups of structures for which algebraic independence satisfies independent 3-amalgamation. We conclude by proving that for the real and the rational Urysohn space and sphere, the metric pointwise topology $τ_{\mathrm{mp}}$ is minimal on $\mathrm{EEmb}(M)$, equals $τ_{\mathrm{Z}}$, and is strictly coarser than $τ_{\mathrm{pw}}$.