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A guide to topological reconstruction on endomorphism monoids and polymorphism clones

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Pith's one-line read This paper proves that for countable saturated structures, automatic homeomorphicity of the automorphism group automatically transfers to the monoid of elementary embeddings, so algebraic isomorphisms between such monoids are automatically

desk verdict Theorem 4.7 is the kind of result worth wanting, but the proof of Lemma 4.5 has a load-bearing error, and the survey's claim that countable saturated structures have EEmb=Aut is simply false. read the letter →

arxiv 2512.01086 v2 pith:EM3PLAOH submitted 2025-11-30 math.LO math.GRmath.RA

Paolo Marimon , Michael Pinsker This is my paper · ORCID
classification math.LOmath.GRmath.RA MSC 03C3503C5008A4020B2720M20
keywords topologicalreconstructionautomatichomeomorphicityelementaryembeddingmonoidendomorphismpolymorphismcloneω-categoricitysaturatedstructuresbi-interpretability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is a guide to topological reconstruction for spaces of symmetries, and its main new result is a transfer theorem. For any countable saturated structure, if its automorphism group is so rigid that every algebraic isomorphism to a closed permutation group is automatically a homeomorphism, then the same rigidity holds for its monoid of elementary embeddings: every algebraic isomorphism to a closed transformation monoid is automatically a homeomorphism. The proof isolates the monoid of elementary embeddings as an intermediate object and shows that the only endomorphism of it fixing the automorphism group pointwise is the identity. The paper also polishes several earlier results, including a very tame counterexample showing that full endomorphism monoids need not inherit this rigidity even when the automorphism group is maximally well-behaved. A sympathetic reader should care because automatic homeomorphicity is the precise sense in which purely algebraic information, the monoid structure, determines the topology of these symmetry spaces, and the theorem extends that guarantee to a naturally larger class.

What carries the argument

Key object: the monoid of elementary self-embeddings EEmb(A), with its automorphism group G as the invertible elements. The load-bearing tool is the class of 'free Galois-closed' elementary embeddings: images whose tuple orbits under the setwise stabiliser are just the restrictions of G-orbits, and whose pointwise stabiliser fixes nothing outside. A prior factorization lemma writes any f as g=h∘f with g,h free Galois-closed, so any endomorphism Φ fixing G pointwise has the form Φ(f)=φ(f)∘f for a homomorphism φ: EEmb(A)→centre(G) vanishing on G. A saturation 'commuting-square' argument forces φ trivial, hence Φ is the identity; Lemma 4.1 turns this into automatic homeomorphicity of EEmb(A).

What would settle it

A single concrete counterexample would settle the central claim: a countable saturated structure A whose automorphism group has automatic homeomorphicity with respect to closed permutation groups, together with an algebraic isomorphism from EEmb(A) onto a closed transformation monoid that is not a homeomorphism. Internally, the weakest link to test is the factorization lemma: for instance, try f(x)=2x on the countable dense linear order (Q,<), and check whether f can be written as g=h∘f with free Galois-closed g,h; a failure there breaks Proposition 4.4 and with it the proof of Theorem 4.7.

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Extended reading notes

Core claim

The central claim (Theorem 4.7): for a countable saturated structure A, if Aut(A) has automatic homeomorphicity with respect to closed permutation groups, then the monoid EEmb(A) of elementary self-embeddings has automatic homeomorphicity with respect to all closed transformation monoids. Moreover, any injective monoid homomorphism from EEmb(A) into the full transformation monoid with image closed in the symmetric group is automatically a homeomorphism onto its image. The proof shows that the only endomorphism of EEmb(A) fixing Aut(A) pointwise is the identity, using a factorization through 'free Galois-closed' maps and a saturation argument forcing centre-valued homomorphisms to be trivial.

Load-bearing premise

The load-bearing premise, imported from the literature without proof, is that every elementary embedding f of a countable saturated structure factors as g = h∘f with g and h free Galois-closed; if that fails, the proof that the only automorphism-group-fixing endomorphism of the embedding monoid is the identity, and hence Theorem 4.7, collapses.

Editorial extensions

If this is right

  • For every countable saturated structure, automatic homeomorphicity of Aut(A) among closed permutation groups implies automatic homeomorphicity of EEmb(A) among all closed transformation monoids.
  • Any injective monoid homomorphism from EEmb(A) whose image is a closed subgroup of the full symmetric group is automatically a homeomorphism onto its image; the image itself is a closed permutation group.
  • For countably categorical structures satisfying the G-finite condition, a mild finite-intersection hypothesis on open subgroups, the elementary-embedding monoid has automatic homeomorphicity with respect to closed transformation monoids, extending earlier results that required a trivial-centre assumption.
  • The full endomorphism monoid does not inherit this: there is a countably categorical, ω-stable structure, definable in a finitely homogeneous structure, whose automorphism group has automatic continuity with respect to all second-countable groups, yet whose endomorphism monoid has a discontinuous automorphism.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The transfer theorem suggests a division of labour: to prove automatic homeomorphicity for elementary-embedding monoids, it is now enough to prove it for the underlying automorphism group, where tools such as the small index property and ample generics are already available; this reverses the usual direction of effort.
  • Coupled with the folklore correspondence between topological isomorphism of elementary-embedding monoids and first-order bi-interpretability, the theorem implies that under its hypotheses an algebraic isomorphism between two such monoids already witnesses bi-interpretability, giving a purely algebraic route to a model-theoretic conclusion.
  • The counterexample for endomorphism monoids suggests that the elementary-embedding monoid, not the full endomorphism monoid, is the right intermediate object for general transfer: hypotheses strong enough to force automatic homeomorphicity of Aut(A) must be supplemented by extra structural information before they say anything about End(A).
  • A natural testable extension is whether the two internal ingredients, the factorization lemma and the commuting-square construction, hold outside the countable saturated setting, for example for saturated structures in uncountable languages or for non-saturated structures with quantifier elimination.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper is a survey of topological reconstruction for automorphism groups, monoids of elementary embeddings, endomorphism monoids, and polymorphism clones of countable first-order structures, with emphasis on the omega-categorical case. It organizes the notions of automatic continuity, automatic homeomorphicity, the unique Polish property, the Zariski topology, and automatic action compatibility, and it provides a diagram of known implications. The main new result is Theorem 4.7: for a countable saturated structure whose automorphism group has automatic homeomorphicity with respect to closed permutation groups, the monoid of elementary embeddings has automatic homeomorphicity with respect to closed transformation monoids. A secondary contribution is Theorem 4.9, an omega-categorical, omega-stable structure whose automorphism group has strong automatic continuity but whose endomorphism monoid has a discontinuous automorphism. The remaining sections survey existing techniques and open problems.

Significance. If Theorem 4.7 is correct, it is a genuine advance: it removes the trivial-centre/superhomogeneity assumptions that were present in earlier work of Pech and Pech and provides a clean transfer from automorphism groups to monoids of elementary embeddings for all countable saturated structures. The paper also performs a useful service by collecting and polishing results scattered across the literature, making the landscape of topological reconstruction more accessible. The proof of Theorem 4.7 is not circular and the external dependencies are clearly cited. However, the proof rests on at least two points that are not fully justified in the manuscript: the exact scope of a cited factorization lemma from [47], and the claim that central automorphisms are central in the monoid of elementary embeddings. These need clarification before the paper can be accepted.

major comments (3)
  1. [Section 4.1, Proposition 4.4] The proof of Proposition 4.4 invokes [47, Proposition 5.1] to factor every f in Gbar as g = h ∘ f with free Galois-closed g,h in Gbar. This is the single load-bearing external step of Theorem 4.7: if that factorization carries hidden hypotheses (e.g., superhomogeneity in the stronger sense of [47], or applicability only to the canonical homogeneous companion A' rather than to the arbitrary saturated structure A), then the construction of the homomorphism φ has no basis and Corollary 4.6 cannot be applied. Since Definition 4.2 explicitly translates [47]'s notion of superhomogeneity, the authors should state [47, Proposition 5.1] in full, with its hypotheses, and explain why it applies to every f ∈ EEmb(A) for an arbitrary G ∈ C^sat_G. As written, this is an unverified chokepoint, not an internal inconsistency.
  2. [Section 4.1, Proposition 4.4, displayed equation] After cancelling the injective map h, the proof uses the assertion that z_h, z_g belong to Z(Gbar) 'being in Z(G)'. For arbitrary saturated structures it is not immediate that an automorphism central in Aut(A) commutes with every elementary embedding of A. The manuscript gives no proof or reference for this commutation. If Z(G) is not contained in Z(Gbar), then the step from f ∘ z_g = z_h ∘ Φ(f) to Φ(f) = z_h^{-1} z_g ∘ f is unjustified and the whole argument collapses. The claim is plausible and can be proved by an orbit argument for canonical structures, but it needs to be stated and proved in the generality used here.
  3. [Lemma 4.5] The model-theoretic chain construction is too abbreviated. The sentence 'Consider the type p(z;A0):=tp(A1/A0) ... Let A2 be a realisation of p(x;α0(A0))' does not, as written, ensure that the resulting isomorphism α1 extends α0; one needs to take the pushforward of tp(A1/A0) under α0 and then use saturation or compactness to obtain an extension. In addition, the assertion that the union of a countable elementary chain of countable saturated models is again saturated is used without proof or citation. Since Corollary 4.6 and hence Theorem 4.7 depend on this lemma, the construction should be written out in detail.
minor comments (4)
  1. [Theorem 4.7] In the statement of the 'Moreover' part, the notation is inconsistent: θ is said to be an injective homomorphism from G to Ω^Ω, but in the preceding sentence G denotes the monoid of elementary embeddings; the overline on G appears to be missing. Please fix the notation.
  2. [Section 2.1] Typo: 'Polishtopological group' should be 'Polish topological group'.
  3. [Definition 2.12] The notation C^sat_G, C^sat_G, C^sat_T, C^sat_C is dense and the subscript/superscript conventions are easy to confuse. A small table with the classes and their inclusions would help the reader, especially in the diagram in Figure 1.
  4. [Theorem 4.9] The proof states that α commutes with all elements of Aut(A'). This is true for the flip of all two-element E-classes, but it would be helpful to spell out the centralizer argument, since the rest of the verification that η is an automorphism depends on it.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular dependence: Theorem 4.7 is a genuine transfer result whose main external inputs are cited lemmas from Pech–Pech and an independent published lemma from Bodirsky–Pinsker–Pongrácz.

full rationale

The paper's central new result, Theorem 4.7, does not assume its own conclusion. It assumes automatic homeomorphicity of the automorphism group G and proves that any endomorphism of the elementary-embedding monoid Ḡ fixing G pointwise must be the identity (Proposition 4.4 + Corollary 4.6). That rigidity statement is then fed into Lemma 4.1, an independent transfer lemma from [18]. Although [18] shares an author with the present paper (Pinsker), the lemma is a previously published, standalone result with its own proof; it is not a restatement of Theorem 4.7 and does not smuggle in the desired conclusion. The genuinely load-bearing external ingredient is [47, Proposition 5.1], which supplies the factorization of arbitrary f ∈ Ḡ as g = h∘f with free Galois-closed g,h, together with [47, Corollary 3.12/Proposition 3.13] used in Proposition 4.3. These results are by Pech and Pech, not by the present authors, and are cited explicitly rather than re-derived. A failure of that factorization would indeed threaten Theorem 4.7, but that is an external mathematical dependency, not circularity: the paper does not define free Galois-closedness in a way that makes the factorization true by construction, and it does not fit any parameter to the target conclusion. The survey portions and open questions are also presented honestly, without renaming known results as new predictions or appealing to an author-specific uniqueness theorem to force the choice. Overall, there is no circular step; at most there is a minor self-citation in the derivation chain, which is not circular and does not raise the score beyond 1.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numeric parameters are fitted, and no new entities (particles, forces, dimensions) are introduced. The central theorem rests on cited external lemmas ([47]) and the standard closed-monoid/reconstruction correspondence. The only 'free parameters' in the statistical sense are none; the only assumptions are the cited model-theoretic facts.

assumptions (5)
  • standard math ZFC and basic model theory (e.g., Tent–Ziegler [53])
    The paper operates in ordinary mathematics and assumes model-theoretic background as stated in Section 2.
  • standard math Closed submonoids of Ω^Ω and closed subclones of O correspond to endomorphism monoids and polymorphism clones of relational structures
    Used throughout to identify spaces of symmetries with structures; stated in Section 2.1 as folklore.
  • domain assumption [47, Proposition 5.1]: for every f∈G there are free Galois-closed g,h∈G with g=h∘f
    Load-bearing external lemma in the proof of Proposition 4.4; not proved in this paper.
  • domain assumption [47, Corollary 3.12 & Proposition 3.13]: if Φ∈End(G) fixes G pointwise, then for free Galois-closed h, Φ(h)=h∘g_h for g_h∈Z(G)
    Used in Proposition 4.3 to obtain the centralizer action.
  • domain assumption Countable saturated structures have homogeneous expansions in a countable language
    Cited from [47] and used in Lemma 4.5 to enable the elementary-chain argument.

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Pith. "Pith review of A guide to topological reconstruction on endomorphism monoids and polymorphism clones." pith.science (2026). https://pith.science/paper/EM3PLAOH

@misc{pith2026251201086,
  author       = {Pith},
  title        = {Pith review of: A guide to topological reconstruction on endomorphism monoids and polymorphism clones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EM3PLAOH}},
  note         = {Machine review of arXiv:2512.01086}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2512.01086 by the authors.

Figure 1
Figure 1. Implications amongst automatic continuity (AC) and automatic homeomorphicity (AH) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Pith tools

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