Paolo Marimon (TU Wien) – On topological reconstruction for monoids of elementary embeddings

The automorphism group $\mathrm{Aut}(A)$ and the monoid of elementary embeddings $\mathrm{EEmb}(A)$ of a first-order structure $A$ are both endowed with a natural topology of pointwise convergence. When $A$ is $\omega$-categorical, these spaces of symmetries (together with their topologies) can be used to reconstruct the original structure up to bi-interpretability. This raises the question of when, given $\mathrm{Aut}(A)$ as a pure group, or $\mathrm{EEmb}(A)$ as a pure monoid, one can reconstruct its topology of pointwise convergence. Whilst the automorphism group version of this problem has been intensively studied over the last 40 years, its analogue for monoids has only recently received attention. In this talk, I will discuss various topological reconstruction problems for monoids of elementary embeddings of $\omega$-categorical structures. We prove that for a countable saturated structure $A$, if $\mathrm{Aut}(A) $ has automatic homeomorphicity with respect to closed subgroups of $S_\omega$ then $\mathrm{EEmb}(A)$ has automatic homeomorphicity with respect to closed submonoids of $\mathbb{N}^{\mathbb{N}}$. This result builds on previous work of Pech and Pech (2018), Behrisch, Truss, and Vargas-García (2017), and Bodirsky, Pinsker, and Pongrácz (2017), who proved special cases of it. We will also discuss when the topology of pointwise convergence ends up being minimal amongst Hausdorff semigroup topologies on $\mathrm{EEmb}(A)$. Interestingly, this seems to happen more easily than for $\mathrm{Aut}(A)$. This talk is based on an upcoming survey paper with Michael Pinsker, and on ongoing work with Javi de la Nuez Gonzalez, Zaniar Ghadernezhad, and Michael Pinsker.