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A guide to topological reconstruction on endomorphism monoids and polymorphism clones

Paolo Marimon, Michael Pinsker

TL;DR

This survey maps the landscape of topological reconstruction for spaces of symmetries beyond automorphism groups, focusing on endomorphism monoids and polymorphism clones of countable, ω-categorical structures. It clarifies how pw topology, oligomorphicity, and various interpretation notions interact to determine when algebraic structure enforces topology (AH, AC, UPP) and provides both lifting techniques from automorphism groups and direct semigroup/clone methods (Property X, Zariski topology). The authors present a central transfer result showing that AH of Aut(A) implies AH of EEmb(A) for countable saturated A, while also highlighting counterexamples where End(A) fails AH and where clone reconstruction is delicate. Across this dual focus, the paper surveys techniques (gate coverings, SAPHG, AEP) and outlines open questions driving progress in reconstruction for monoids and clones, with implications for bi-interpretability and model theory.

Abstract

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the topology of pointwise convergence. In some cases, the algebraic structure of such space alone is sufficiently rich to determine its topology (under some requirements on the topology). For automorphism groups, the problem of when this happens has been actively pursued over the last 40 years. With the exception of some early work of Lascar, the analogue of this problem for endomorphism monoids and polymorphism clones has only received attention in the past 15 years. In this guide, we survey the current state of affairs in this relatively young line of research. We moreover use this opportunity to polish several existing results and to extend them beyond what was hitherto known.

A guide to topological reconstruction on endomorphism monoids and polymorphism clones

TL;DR

This survey maps the landscape of topological reconstruction for spaces of symmetries beyond automorphism groups, focusing on endomorphism monoids and polymorphism clones of countable, ω-categorical structures. It clarifies how pw topology, oligomorphicity, and various interpretation notions interact to determine when algebraic structure enforces topology (AH, AC, UPP) and provides both lifting techniques from automorphism groups and direct semigroup/clone methods (Property X, Zariski topology). The authors present a central transfer result showing that AH of Aut(A) implies AH of EEmb(A) for countable saturated A, while also highlighting counterexamples where End(A) fails AH and where clone reconstruction is delicate. Across this dual focus, the paper surveys techniques (gate coverings, SAPHG, AEP) and outlines open questions driving progress in reconstruction for monoids and clones, with implications for bi-interpretability and model theory.

Abstract

Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. For example, the automorphism group of a structure is, on top of being a group, a topological group when equipped with the topology of pointwise convergence. In some cases, the algebraic structure of such space alone is sufficiently rich to determine its topology (under some requirements on the topology). For automorphism groups, the problem of when this happens has been actively pursued over the last 40 years. With the exception of some early work of Lascar, the analogue of this problem for endomorphism monoids and polymorphism clones has only received attention in the past 15 years. In this guide, we survey the current state of affairs in this relatively young line of research. We moreover use this opportunity to polish several existing results and to extend them beyond what was hitherto known.

Paper Structure

This paper contains 17 sections, 34 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces of symmetries and their topologies
  4. omega-categorical structures and related notions
  5. Interpretations
  6. Notions of reconstruction
  7. Reconstruction
  8. Automatic homeomorphicity
  9. Automatic homeomorphicity for monoids of elementary embeddings
  10. Automatic homeomorphicity for endomorphism monoids
  11. On the number of Polish topologies
  12. Multiple Polish topologies on monoids of elementary embeddings
  13. Minimality of the topology of pointwise convergence and the Zariski topology
  14. Maximality of the topology of pointwise convergence, Property X, and automatic continuity
  15. More on automatic continuity and open mappings
  16. ...and 2 more sections

Key Result

Theorem 2.8

Let $\mathbb{A}$ and $\mathbb{B}$ be $\omega$-categorical structures of size $>1$. Then:

Figures (1)

  • Figure 1: Implications amongst automatic continuity (AC) and automatic homeomorphicity (AH) notions with respect to different classes, with the relevant citations. Above, $G$ indicates the automorphism group of an $\omega$-categorical structure, $\overline{G}$ its monoid of elementary embeddings, and $\mathcal{S}$ its endomorphism monoid.

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Lemma 2.9
  • proof
  • ...and 55 more