On the Zariski topology on endomorphism monoids of omega-categorical structures

This paper establishes systematic model-theoretic conditions under which the pointwise convergence and Zariski topologies coincide on the endomorphism monoids of $\omega$-categorical structures, while also providing a counterexample where they differ, thereby resolving a question posed by Elliott et al.

The Gist

Imagine you have a giant, infinite library of books (this is our mathematical "structure"). Inside this library, there are many librarians (these are the "endomorphisms"). Each librarian has a specific job: they can take any book and move it to a different shelf, or leave it where it is, as long as they follow the library's rules.

The paper by Michael Pinsker and Clemens Schindler is about how we can measure the "closeness" or "similarity" between these librarians. They ask: Do two different ways of measuring similarity actually give us the same result?

Here is the breakdown of their discovery using simple analogies.

1. The Two Ways to Measure "Closeness"

The authors compare two different rulers for measuring how similar two librarians are.

Ruler A: The "Pointwise" Topology (The "Spot Check" Method)
Imagine you want to know if two librarians, Alice and Bob, are doing the same job. You pick a specific book (let's call it Book X) and ask: "Where did you put Book X?"

• If Alice and Bob both put Book X on the same shelf, they are "close" regarding that book.
• If you check a few more books (Book Y, Book Z) and they match there too, they are even closer.
• The Rule: Two librarians are considered "identical" in this system only if they move every single book in the library to the exact same shelf.
• Analogy: This is like a teacher checking a student's homework question-by-question. If the answers match on every single question, the students are identical.

Ruler B: The "Zariski" Topology (The "Algebraic Puzzle" Method)
This ruler doesn't look at the books directly. Instead, it looks at the rules the librarians follow.

• Imagine you give the librarians a complex puzzle: "If you take a book, move it, then move it again, then move it a third time, does it end up in the same place as if you just moved it once?"
• The Zariski ruler checks if the librarians solve these algebraic puzzles differently.
• If two librarians solve any of these puzzles differently, they are considered "far apart." If they solve all puzzles the same way, they are "close."
• Analogy: This is like judging two chefs not by tasting their food, but by checking if they follow the exact same recipe steps. If their recipe steps differ, they are different chefs, even if the final dish tastes the same.

2. The Big Question

For many years, mathematicians studied specific types of libraries (called "$\omega$-categorical structures"). In every single case they looked at, Ruler A and Ruler B gave the exact same answer.

If two librarians were close according to the "Spot Check" (Ruler A), they were also close according to the "Algebraic Puzzle" (Ruler B). This led to a big question:

"Is this always true? Is there any library where the 'Spot Check' says two librarians are different, but the 'Algebraic Puzzle' says they are the same?"

3. The Authors' Discovery

The authors found the answer in two parts:

Part 1: When do the rulers agree? (The "Good" Libraries)

They discovered that the rulers agree if the library has a specific "core" structure that is either:

1. Small and Finite: The library's essential rules are simple and limited (like a small town).
2. Infinite but "Fluid": The library is huge, but it has no rigid "anchors." You can move things around freely without getting stuck.

The "Mobile Core" Metaphor:
Imagine the library has a "heart" (the core).

• If the heart is Mobile, it means you can take any book in the library and, by following the rules, move it into the heart.
• If the heart is Small OR Fluid (no algebraicity), then the "Spot Check" and the "Algebraic Puzzle" will always agree. The structure of the library forces the librarians to be so constrained that checking a few books is the same as checking the whole recipe.

Part 2: The Counterexample (The "Bad" Library)

The authors then built a very specific, tricky library to answer "Yes" to the big question. They created a library where the rulers disagree.

The Construction:

• Imagine a library where the shelves are arranged in a giant circle (a complete graph).
• But, every single shelf is actually a tiny, complex machine (a bipartite graph) with two sides: Left and Right.
• The librarians can swap books between Left and Right sides.

The Disagreement:

• The Spot Check (Ruler A): Can tell the difference between a librarian who swaps Left/Right and one who doesn't. If you check one book, you see the difference immediately.
• The Algebraic Puzzle (Ruler B): Is "blind" to this swap. Because the library is so complex and symmetric, the algebraic equations used in the Zariski ruler cannot "see" the difference between swapping and not swapping. It thinks both librarians are following the same rules.

The Result:
In this specific library, the "Spot Check" says: "These two librarians are different!"
But the "Algebraic Puzzle" says: "No, they are the same."

4. Why Does This Matter?

This paper is a landmark because:

1. It explains the "Why": It tells us why the rulers usually agree (it's about the "core" of the structure being simple or fluid).
2. It breaks the pattern: It proves that the agreement isn't a universal law of mathematics. There are complex, well-behaved libraries where the internal rules (Zariski) fail to capture the external reality (Pointwise).
3. The "Heart" of the matter: It shows that the secret to understanding these mathematical structures lies in analyzing their "model-complete core"—essentially, finding the smallest, most essential version of the library that still holds all the rules.

In summary: The authors found a way to predict when two ways of measuring mathematical similarity will match, and they built a specific, weird example where they don't, proving that the "heart" of the structure is the deciding factor.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the relationship between two distinct topologies defined on the endomorphism monoid $\text{End}(\mathcal{A})$ of a model-theoretic structure $\mathcal{A}$:

1. The Pointwise Topology ($\mathcal{T}_{pw}$): An "external" topology induced by the action of endomorphisms on the domain of $\mathcal{A}$. A sequence converges if it converges pointwise. For countable $\omega$-categorical structures, this is a Polish (separable, completely metrizable) semigroup topology.
2. The Zariski Topology ($\mathcal{T}_{Zariski}$): An "internal" topology defined purely algebraically. Its closed sets are solution sets to equations (identities) in the language of semigroups. It is generated by sets of the form $\{s \in S : \phi(s) \neq \psi(s)\}$, where $\phi$ and $\psi$ are semigroup terms.

The Core Question:
For many known $\omega$-categorical structures, $\mathcal{T}_{pw}$ and $\mathcal{T}_{Zariski}$ coincide. When they coincide, it implies that $\mathcal{T}_{pw}$ is the coarsest Hausdorff semigroup topology on $\text{End}(\mathcal{A})$. Previous proofs of this coincidence were ad hoc and specific to individual structures. The authors ask:

• Are there systematic, structural reasons (based on the model-theoretic properties of $\mathcal{A}$) that guarantee $\mathcal{T}_{pw} = \mathcal{T}_{Zariski}$?
• Does there exist an $\omega$-categorical structure where these topologies differ (i.e., $\mathcal{T}_{pw}$ is strictly finer than $\mathcal{T}_{Zariski}$)?

2. Methodology and Key Concepts

The authors employ tools from model theory, universal algebra, and topological semigroups.

A. Model-Complete Cores

A central tool is the model-complete core of a structure. Every $\omega$-categorical structure $\mathcal{A}$ is homomorphically equivalent to a unique (up to isomorphism) model-complete core $\mathcal{C}$.

• $\mathcal{C}$ is model-complete if $\text{End}(\mathcal{C})$ coincides with the pointwise closure of $\text{Aut}(\mathcal{C})$.
• The authors analyze the properties of $\mathcal{C}$ (specifically its cardinality and whether it possesses algebraicity) to determine the topological behavior of $\text{End}(\mathcal{A})$.

B. Mobile Cores

The authors introduce a new concept: structures with a mobile core.

• $\mathcal{A}$ has a mobile core if for every element $a \in \mathcal{A}$, there exists an endomorphism $g$ mapping $\mathcal{A}$ into a copy of the model-complete core $\mathcal{C}$ such that $a$ is in the image of $g$.
• This is a weakening of transitivity. It allows the authors to "move" any element of the domain into the core structure via an endomorphism.

C. Algebraicity

• A structure has no algebraicity if the orbit of any element $a$ under the stabilizer of a finite set $Y$ (where $a \notin Y$) is infinite.
• The presence or absence of algebraicity in the core is the critical differentiator in the authors' results.

3. Key Contributions and Results

The paper provides two main sets of results: sufficient conditions for equality and a counterexample for inequality.

Result 1: Sufficient Conditions for Equality

The authors prove that if $\mathcal{A}$ is an $\omega$-categorical structure without algebraicity and has a mobile core, then $\mathcal{T}_{pw} = \mathcal{T}_{Zariski}$ if either of the following holds:

1. Finite Core: The model-complete core $\mathcal{C}$ is finite.
2. Infinite Core without Algebraicity: The model-complete core $\mathcal{C}$ is infinite but possesses no algebraicity itself.

Implication: In these cases, $\mathcal{T}_{pw}$ is the coarsest Polish semigroup topology on $\text{End}(\mathcal{A})$. This generalizes previous results for structures like the random graph, the rational order $(\mathbb{Q}, <)$, and various equivalence relations.

Proof Sketch:

• Finite Case: The authors show that the basic open sets of $\mathcal{T}_{pw}$ (fixing a point) can be constructed as Zariski-open sets by utilizing the finiteness of the core and the mobility to restrict the image of endomorphisms.
• Infinite Non-Algebraic Case: They utilize a technical lemma (Lemma 3.5) requiring the existence of specific endomorphisms that agree outside a point but differ at that point, and whose images can be made disjoint except at that point. The lack of algebraicity in the core allows the construction of such "almost identical" embeddings (Lemma 3.6).

Result 2: The Counterexample (Strict Inequality)

The authors answer Question 1.1 affirmatively by constructing an $\omega$-categorical, transitive, homogeneous structure $\mathcal{G}$ (in a finite language) without algebraicity where $\mathcal{T}_{pw} \neq \mathcal{T}_{Zariski}$.

The Structure $\mathcal{G}$:

• Construction: $\mathcal{G}$ consists of countably many copies of the complete bipartite graph $K_{2, \omega}$ (with parts $A^+$ and $A^-$).
• Relations:
  • $E_1$: Connects any two points in different copies of $K_{2, \omega}$ (forming a complete graph between copies).
  • $E_2$: Connects points within the same copy according to the bipartite structure of $K_{2, \omega}$.
• Properties: $\mathcal{G}$ is $\omega$-categorical, homogeneous, transitive, and has no algebraicity.
• The Core: The model-complete core of $\mathcal{G}$ is an infinite structure where each point is replaced by a single edge (a copy of $K_2$). Crucially, this core has algebraicity (stabilizing one vertex in a copy forces the stabilization of the other).

The Proof of Inequality:
The authors demonstrate that $\mathcal{T}_{Zariski}$ cannot distinguish between endomorphisms that "flip" the parts of the bipartite graph ($A^+ \leftrightarrow A^-$) and those that do not, whereas $\mathcal{T}_{pw}$ can.

• They define a sign function $\text{sgn}(s) \in \{+1, -1\}$ for endomorphisms of the bipartite component.
• They prove that any Zariski-open set containing the identity (or a specific "positive" endomorphism) must also contain "negative" endomorphisms (those with sign $-1$) that differ only on a "large" subset of the domain.
• Consequently, the pointwise open set $\{s : s(0, a^+) = (0, a^+)\}$ cannot be Zariski-open, proving $\mathcal{T}_{Zariski} \subsetneq \mathcal{T}_{pw}$.

4. Significance and Conclusion

• Systematic Characterization: The paper moves beyond ad hoc proofs to establish that the structure of the model-complete core is the decisive factor. Specifically, the coincidence of topologies holds if the core is finite or infinite without algebraicity.
• Necessity of Algebraicity in the Core: The counterexample shows that if the model-complete core is infinite and possesses algebraicity, the Zariski topology may fail to be Hausdorff (or at least fail to coincide with the pointwise topology). This suggests that algebraicity in the core is the "obstruction" to the topologies coinciding.
• Refinement of Open Questions: The work refines the original question by suggesting that even for transitive, homogeneous structures in finite languages, the topologies can differ. It raises a new question (Question 4.1): Is there a structure where a Hausdorff semigroup topology exists that is not finer than the pointwise topology? (The counterexample suggests the Zariski topology itself might not even be Hausdorff in such cases).

In summary, the paper provides a comprehensive framework for understanding when the algebraic (Zariski) and topological (pointwise) structures of endomorphism monoids align, identifying the model-complete core's algebraicity as the pivotal property.