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Minimal and intrinsic topologies on monoids of elementary embeddings

J. de la Nuez Gonzalez, Zaniar Ghadernezhad, Paolo Marimon, Michael Pinsker

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 non-trivial centre. We then provide general conditions on the behaviour of algebraic closure on $M$ that imply minimality of $τ_{\mathrm{pw}}$. 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 $ω$-categorical structures with weak elimination of imaginaries. We conclude by proving that the metric pointwise topology $τ_{\mathrm{mpw}}$ is minimal, equals $τ_{\mathrm{Z}}$, and is strictly coarser than $τ_{\mathrm{pw}}$, on $\mathrm{EEmb}(M)$ for the real and the rational Urysohn space and sphere.

Minimal and intrinsic topologies on monoids of elementary embeddings

Abstract

To every -categorical structure one can associate two spaces of symmetries which determine the structure up to first-order bi-interpretability: the topological group of its automorphisms and the topological monoid of its elementary embeddings, both equipped with the topology of pointwise convergence . We investigate the relation of to other topologies on these spaces: in particular, when is minimal, i.e.~does not admit any strictly coarser Hausdorff semigroup topology. A common method to prove minimality of on is to show that it coincides with the algebraically defined semigroup Zariski topology . We show that differs from on whenever has non-trivial centre. We then provide general conditions on the behaviour of algebraic closure on that imply minimality of . These condition cover, for example, countable vector spaces and projective spaces over finite fields. Turning to , we describe the minimal semigroup topologies on the automorphism groups of model-theoretically simple one-based -categorical structures with weak elimination of imaginaries. We conclude by proving that the metric pointwise topology is minimal, equals , and is strictly coarser than , on for the real and the rational Urysohn space and sphere.

Paper Structure

This paper contains 33 sections, 43 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Spaces of symmetries and their topologies
  3. The case of groups
  4. The case of monoids
  5. The Zariski topology
  6. Results
  7. Preliminaries
  8. Spaces of symmetries
  9. Infinite permutation groups
  10. The topology of pointwise convergence
  11. Model-theoretic background
  12. Imaginaries
  13. Independence relations
  14. More topologies
  15. Zariski topology
  16. ...and 18 more sections

Key Result

Theorem A

Let $G\curvearrowright \Omega$ be a closed permutation group with locally finite algebraic closure. Suppose that $G$ has non-trivial centre. Let $S$ be either $G$ or $\overline{G}$. Then, the semigroup Zariski topology $\tau_{\mathrm{Z}}$ on $S$ is not Hausdorff. In particular, it is properly contai

Figures (3)

  • Figure 1: An illustration of the definition of a universally embedded model. In the figure, we have that $(\overline{a}_1; \overline{a}_2; \overline{b})$ is an absorbing configuration for $\Omega'$. In particular, $\overline{b}\mathop{\hbox{$\mid$} \hbox{$\smile$}}^{a}_{\overline{b}\cap\Omega'} \overline{a}_1$ and $\overline{b}\cap\Omega'\mathop{\hbox{$\mid$} \hbox{$\smile$}}^{a}_{\overline{a}_1} \overline{a}_2$. This allows us to "absorb" $\overline{a}_1$ into $\Omega'$: that is, to find $\overline{a}_1'\overline{a}_2'\equiv_{\overline{b}} \overline{a}_1 \overline{a}_2$ such that $\Omega'\cap \overline{a}_2'=\overline{a}_1'$.
  • Figure 2: Illustration of two isometries $\phi$ and $\psi$ witnessing pinching at $a$. Note that $\phi$ and $\psi$ disagree everywhere in the ball $B_\epsilon(a)$, but agree everywhere else.
  • Figure 3: Illustration of two isometries $\sigma$ and $\theta$ witnessing spreading at $a$. Note that $a$ is in the images of both $\sigma$ and $\theta$ and that the ball around the image of $\theta$ of radius $\epsilon$ only contains points in the image of $\sigma$ inside of the ball $B_\epsilon(a)$.

Theorems & Definitions (135)

  • Theorem A: \ref{['thm:zar']}
  • Theorem B: \ref{['t:minimality endomorphism monoids']}
  • Theorem C: \ref{['thm:onebased']}
  • Theorem D: \ref{['thm:urysohnmain']}
  • proof
  • Definition 3.3
  • Definition 3.4
  • Example 3.7
  • Remark 3.8
  • Remark 3.9
  • ...and 125 more