Minimal and intrinsic topologies on monoids of elementary embeddings

TL;DR

This paper studies the minimality of natural topologies on automorphism groups and elementary embeddings of omega-categorical structures, revealing conditions under which these topologies are minimal or coincide with algebraic Zariski topologies.

Contribution

It establishes criteria for minimality of pointwise convergence topologies on automorphism monoids, compares them with Zariski topologies, and characterizes minimal topologies for specific structures like Urysohn spaces.

Findings

$ au_{pw}$ differs from $ au_Z$ when $ ext{Aut}(M)$ has a non-trivial center.

Conditions on algebraic closure imply minimality of $ au_{pw}$ on $ ext{EEmb}(M)$.

On Urysohn spaces, $ au_{mpw}$ is minimal, equals $ au_Z$, and is coarser than $ au_{pw}$.

Abstract

To every $\omega$-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 $\tau_{\mathrm{pw}}$. We investigate the relation of $\tau_{\mathrm{pw}}$ to other topologies on these spaces: in particular, when $\tau_{\mathrm{pw}}$ is minimal, i.e.~does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of $\tau_{\mathrm{pw}}$ on $\mathrm{EEmb}(M)$ is to show that it coincides with the algebraically defined semigroup Zariski topology $\tau_{\mathrm{Z}}$. We show that $\tau_{\mathrm{pw}}$ differs from $\tau_{\mathrm{Z}}$ on $\mathrm{EEmb}(M)$ whenever $\mathrm{Aut}(M)$ has non-trivial centre. We then provide general conditions on the behaviour of algebraic closure on $M$ that imply minimality of $\tau_{\mathrm{pw}}$. These condition cover, for example, countable vector spaces and projective spaces over finite fields. Turning to $\mathrm{Aut}(M)$, we describe the minimal $T_1$ semigroup topologies on the automorphism groups of model-theoretically simple one-based $\omega$-categorical structures with weak elimination of imaginaries. We conclude by proving that the metric pointwise topology $\tau_{\mathrm{mpw}}$ is minimal, equals $\tau_{\mathrm{Z}}$, and is strictly coarser than $\tau_{\mathrm{pw}}$, on $\mathrm{EEmb}(M)$ for the real and the rational Urysohn space and sphere.