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Minimal and intrinsic topologies on monoids of elementary embeddings

This paper investigates the minimality and relationships between the pointwise, Zariski, and metric pointwise topologies on the monoids of elementary embeddings and automorphism groups of $\omega$ -categorical structures, establishing conditions for minimality, demonstrating the distinction between topologies when automorphism groups have non-trivial centers, and analyzing specific cases like vector spaces and Urysohn spaces.

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

Published 2026-03-31

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite puzzle. This puzzle represents a mathematical structure (like a network of points, a vector space, or a geometric shape). The "symmetries" of this puzzle are the ways you can shuffle the pieces around without breaking the rules of how the puzzle is built.

In mathematics, we study these shuffles in two main ways:

The Group of Perfect Shuffles (Automorphisms): You can move pieces around, but you must be able to move them back to where they started. It's like a reversible dance.
The Monoid of Partial Shuffles (Elementary Embeddings): You can move pieces around, but you don't necessarily have to be able to reverse the move. You might stretch the puzzle or map it into a larger version of itself. It's like a one-way street.

Mathematicians usually study these shuffles using a specific "ruler" called the Pointwise Convergence Topology ( $\tau_{pw}$ ). Think of this ruler as a magnifying glass that checks if two shuffles are "close" by seeing if they move the same specific pieces to the same specific spots.

The Big Question:
Is this magnifying glass the only way to measure how close two shuffles are? Or is there a "coarser" (blurrier) ruler that still works? If the magnifying glass is the only possible ruler that makes sense, we say the system is minimal. If there are blurrier rulers, the system is "loose."

This paper investigates when these systems are "tight" (minimal) and when they are "loose."

The Main Characters and Their Tools

1. The Algebraic Ruler (The Zariski Topology)

Imagine you have a set of algebraic equations (like $x^2 + y = 5$ ). The Zariski Topology ( $\tau_Z$ ) is a ruler defined purely by these equations. It doesn't care about the specific points; it only cares about the rules of the game.

The Standard Trick: Usually, if the "Algebraic Ruler" and the "Magnifying Glass" give the exact same results, we know the system is minimal. It's like finding that your map and your GPS agree perfectly.

2. The Center of the Puzzle (The Centre)

Some puzzles have a "center" that doesn't move no matter how you shuffle the rest.

The Discovery: The authors found that if a puzzle has a non-trivial center (a part that stays put while everything else moves), the "Algebraic Ruler" breaks. It becomes too blurry to distinguish between different shuffles.
The Analogy: Imagine a spinning top. If the top has a heavy, unmoving core, you can't tell the difference between two spins just by looking at the algebraic equations of the spin. The "Algebraic Ruler" fails to be precise. In these cases, the standard trick to prove minimality doesn't work.

The New Strategy: "Sinks" and "Universally Embedded Models"

Since the standard trick failed for some puzzles, the authors invented a new way to prove minimality. They looked for "Sinks."

The Analogy of the Sink: Imagine a drain in a bathtub. No matter how you swirl the water, eventually, everything gets pulled toward the drain.
The Mathematical Sink: They found that in many complex structures (like infinite vector spaces), there is a special "sub-puzzle" (a sink) that acts like a drain. Any shuffle that tries to move things around eventually gets "sucked" into this sub-puzzle in a very specific way.
The Result: If a puzzle has this "Sink" property, the "Magnifying Glass" ( $\tau_{pw}$ ) is indeed the only ruler that works. The system is minimal. This covers many important cases, like infinite vector spaces over finite fields, even though the "Algebraic Ruler" trick failed for them.

The Twist: The Urysohn Spaces (The Metric Puzzle)

The paper ends with a fascinating twist involving Urysohn Spaces. These are special geometric puzzles where the distance between points matters (like a city map where you care about miles, not just connections).

The Metric Topology ( $\tau_{mp}$ ): In these spaces, there is a new ruler based on distance. It asks: "Are these two shuffles close if they move points to nearby locations?"
The Surprise: For these metric puzzles, the "Magnifying Glass" ( $\tau_{pw}$ ) is NOT the tightest ruler. There is a "blurrier" ruler (the Metric Topology) that still works perfectly.
The Connection: The authors proved that for these spaces, the "Algebraic Ruler" ( $\tau_Z$ ) and the "Metric Ruler" ( $\tau_{mp}$ ) are actually the same thing.
The Takeaway: In the world of metric spaces, the geometry (distance) dictates the rules. The "Magnifying Glass" is too strict; the "Metric Ruler" is the natural, minimal way to measure these shuffles.

Summary in Plain English

The Goal: We want to know if the standard way of measuring "similarity" between mathematical shuffles is the only way to do it.
The Problem: Sometimes, the standard way of checking (using algebra) fails because the puzzle has a "fixed center."
The Solution (Part 1): For many puzzles (like vector spaces), the authors found a new "drain" mechanism (Sinks) that proves the standard measurement is still the only one that works, even if the algebraic check fails.
The Solution (Part 2): For geometric puzzles (Urysohn spaces), the standard measurement is actually too strict. There is a "distance-based" measurement that is the true minimal ruler, and it turns out to be identical to the algebraic ruler.

In a nutshell: The paper maps out the landscape of mathematical symmetries. It tells us when the "standard ruler" is the only one that fits, when it breaks down, and when a "distance-based ruler" takes over. It's like discovering that for some cities, you must check every single street corner to know if two maps are the same, while for others, just checking the main highways is enough.

1. Problem Statement and Context

The paper investigates the topological structure of spaces of symmetries associated with countable first-order structures, specifically focusing on the monoid of elementary embeddings ( $\text{EEmb}(M)$ ) and the automorphism group ( $\text{Aut}(M)$ ).

The Core Object: For a countable structure $M$ , $\text{EEmb}(M)$ is the set of injective maps preserving all first-order formulas. It is equipped with the topology of pointwise convergence ( $\tau_{pw}$ ), which makes it a Polish topological semigroup.
The Central Question: Is $\tau_{pw}$ a minimal topology? A topology is minimal if it admits no strictly coarser Hausdorff semigroup topology.
The Gap in Knowledge:
- For groups ( $\text{Aut}(M)$ ), minimality of $\tau_{pw}$ is well-studied. It often fails due to "localization" (non-transitivity), "intervals" (order structures), or "metrics" (metric structures).
- For monoids ( $\text{EEmb}(M)$ ), the behavior is less understood. A standard method to prove minimality is showing that $\tau_{pw}$ coincides with the Zariski topology ( $\tau_Z$ ), which is the coarsest $T_1$ semigroup topology defined algebraically. However, it was known that $\tau_{pw} = \tau_Z$ fails in some cases (e.g., when $\text{Aut}(M)$ has a non-trivial center), but general conditions for minimality in the monoid case were lacking.

The authors aim to:

Characterize when $\tau_{pw} \neq \tau_Z$ for monoids.
Establish general conditions under which $\tau_{pw}$ is minimal for monoids of elementary embeddings, even when $\tau_{pw} \neq \tau_Z$ .
Analyze the case of Urysohn spaces (metric structures), where minimality is expected to fail for groups, and determine the behavior for their embedding monoids.

2. Methodology

The authors employ a blend of model theory, topological algebra, and combinatorial group theory.

Algebraic vs. Topological Analysis: They compare the pointwise topology $\tau_{pw}$ with the Zariski topology $\tau_Z$ (defined by solution sets of semigroup equations). They utilize the fact that $\tau_Z$ is contained in every Hausdorff semigroup topology.
Independence Relations: A significant portion of the work relies on model-theoretic notions of independence (generalizing linear independence). They use axioms such as base monotonicity, stationarity, and independent 3-amalgamation to control the behavior of algebraic closures.
Rigidity and Dispersivity: They introduce concepts of $\tau$ -rigidity (elements fixing points within algebraic closures) and $\tau$ -dispersivity (elements that can move points arbitrarily far from finite sets). They prove that if a topology is strictly coarser than $\tau_{pw}$ , it must exhibit specific "dispersive" behaviors that lead to contradictions with the Hausdorff property or the structure of the monoid.
Universally Embedded Models: They develop a method using "universally embedded models" (related to lovely pairs in model theory) to construct "sinks"—subsets of the domain that force any coarser topology to be non-Hausdorff.
Metric Techniques: For Urysohn spaces, they adapt techniques to the metric setting, defining pinching and spreading properties for isometric embeddings to relate the metric pointwise topology ( $\tau_{mp}$ ) to the Zariski topology.

3. Key Contributions and Results

The paper presents four main theorems (A, B, C, D) that significantly advance the understanding of minimal topologies on monoids.

Theorem A: Failure of $\tau_{pw} = \tau_Z$

Result: If a closed permutation group $G$ has a non-trivial center and locally finite algebraic closure, then the Zariski topology $\tau_Z$ on $G$ (and its closure $\bar{G}$ ) is not Hausdorff.
Significance: This provides the first general condition where the standard method of proving minimality (showing $\tau_{pw} = \tau_Z$ ) fails. It applies to structures like countable vector spaces over finite fields and disjoint unions of complete graphs.

Theorem B: Minimality via Base Monotonicity

Result: Let $G$ be the automorphism group of a countable saturated structure with locally finite algebraic closure. If algebraic independence satisfies base monotonicity, then $\tau_{pw}$ is minimal among Hausdorff semigroup topologies on the monoid $\bar{G}$ .
Significance: This is a major breakthrough. It proves minimality even in cases where $\tau_{pw} \neq \tau_Z$ (e.g., infinite-dimensional vector spaces). The condition of "base monotonicity" is satisfied by many "tame" structures, including simple one-based structures and smoothly approximable structures.
Scope: Covers countable vector spaces, projective spaces over finite fields, and various geometric structures.

Theorem C: Minimal Topologies on Groups (Simple Structures)

Result: For the automorphism group $G$ of a simple one-based countable saturated structure with weak elimination of imaginaries, every $T_1$ semigroup topology coarser than $\tau_{pw}$ is a generalized pointwise topology. If $G$ is transitive, $\tau_{pw}$ is minimal.
Significance: This improves upon previous results (e.g., from [33]) by removing the assumption of algebraic closedness of points in non-transitive settings and characterizing all coarser $T_1$ topologies, not just Hausdorff group topologies.

Theorem D: Urysohn Spaces and Metric Topologies

Result: For the monoids of isometric embeddings of Urysohn spaces (rational/real Urysohn space and sphere, and ultrametric variants), the metric pointwise topology ( $\tau_{mp}$ $τ_{m p}$ ) is:
1. Minimal among Hausdorff semigroup topologies.
2. Equal to the Zariski topology ( $\tau_{mp} = \tau_Z$ ).
3. Strictly coarser than the standard pointwise topology ( $\tau_{mp} \subsetneq \tau_{pw}$ ).
Significance: This resolves the tension for metric structures. While $\tau_{pw}$ is not minimal for the group of isometries of Urysohn spaces (due to the metric), the monoid of embeddings admits a natural minimal topology ( $\tau_{mp}$ ) that is distinct from $\tau_{pw}$ . This demonstrates that the presence of a metric does not necessarily destroy minimality for monoids, provided one considers the correct intrinsic topology.

4. Detailed Technical Mechanisms

Sinks and Universally Embedded Models: To prove Theorem B, the authors construct "sinks" (specific subsets $\Omega_1 \subset \Omega$ ) using universally embedded models. If a structure admits such a model, any Hausdorff topology strictly coarser than $\tau_{pw}$ leads to a contradiction because the "sink" property forces the topology to fail the Hausdorff separation axiom. This relies heavily on the coheir property and base monotonicity.
Pinching and Spreading: In Section 8, to prove Theorem D, the authors define a monoid $S$ as pinching if it can separate points within a small metric ball while agreeing outside, and spreading if it can map images into specific balls. They prove that for Urysohn spaces, the monoid of isometric embeddings is both pinching and spreading. This allows them to show that any Zariski-open set can approximate any metric-open set, proving $\tau_{mp} = \tau_Z$ .
Independence and Chains: The proofs for Theorems B and C utilize "chains" of algebraically closed sets and independence relations. They show that if a topology is coarser, it must allow for "deeply dispersive" elements, which, combined with the independence axioms (like $k$ -narrowness), forces the topology to be the pointwise topology.

5. Significance and Impact

Unification of Group and Monoid Theory: The paper bridges the gap between the study of automorphism groups and elementary embedding monoids. It shows that monoids often behave "better" (more likely to be minimal) than groups, even in the presence of algebraic closure.
New Criteria for Minimality: It moves beyond the "Zariski coincidence" method. Theorem B establishes that base monotonicity is a sufficient condition for minimality, even when the Zariski topology is not Hausdorff. This expands the class of structures known to have minimal symmetry topologies.
Resolution of Metric Cases: The analysis of Urysohn spaces provides a nuanced view of how metrics affect topology. It identifies $\tau_{mp}$ as the intrinsic minimal topology for these structures, distinguishing it from the standard pointwise topology.
Model-Theoretic Applications: The results have implications for the reconstruction of structures from their symmetry monoids. If $\tau_{pw}$ is minimal, the algebraic structure of the monoid uniquely determines the topology, and thus the structure itself (up to bi-interpretability).

In summary, this paper provides a comprehensive framework for understanding the intrinsic topologies of symmetry monoids, introducing powerful new tools (universally embedded models, pinching/spreading) and establishing minimality for a wide range of model-theoretically significant structures, including those with non-trivial centers and metric structures.