Reconstructing Structures from Topological Symmetry Monoids

Definitions and mathematical statement

Let $A$ be an infinite first-order structure with underlying set $|A|$. Define \[ \operatorname{Aut}(A)=\{g:|A|\to |A|:g\text{ is an automorphism of }A\} \] and \[ \operatorname{Emb}_{\mathrm{el}}(A)=\{e:|A|\to |A|:e\text{ is an elementary embedding}\}. \] Here an elementary embedding is an injective map such that, for every first-order formula $\varphi(x_1,\ldots,x_n)$ in the language of $A$ and every tuple $\bar a\in |A|^n$, \[ A\models\varphi(\bar a)\quad\Longleftrightarrow\quad A\models\varphi(e(\bar a)). \] Composition makes these objects a group and a monoid, respectively. For a conventional precise interpretation of their topology, give $|A|$ the discrete topology and give both objects the subspace topology inherited from $|A|^{|A|}$. Thus basic neighborhoods of a map $f$ specify its values on a finite subset $F\subseteq |A|$: \[ U(f,F)=\{h:h\!\upharpoonright_F=f\!\upharpoonright_F\}, \] where $h$ ranges over the relevant group or monoid. The supplied question does not explicitly specify this topology, so its use here is a convention requiring confirmation.

Reconstruction question. Determine what information about $A$ is determined by the topological group $\operatorname{Aut}(A)$, and separately by the topological monoid $\operatorname{Emb}_{\mathrm{el}}(A)$. More precisely, for a specified class $\mathcal K$ of infinite structures, determine the consequences for $A,B\in\mathcal K$ of each of \[ \operatorname{Aut}(A)\cong_{\mathrm{top.grp}}\operatorname{Aut}(B), \qquad \operatorname{Emb}_{\mathrm{el}}(A)\cong_{\mathrm{top.mon}}\operatorname{Emb}_{\mathrm{el}}(B). \] These denote algebraic isomorphisms that are homeomorphisms. Equivalently, determine which properties $P$ satisfy, for all $A,B\in\mathcal K$, \[ \operatorname{Aut}(A)\cong_{\mathrm{top.grp}}\operatorname{Aut}(B) \ \Longrightarrow\ (P(A)\Longleftrightarrow P(B)), \] and which satisfy the analogous implication for elementary-embedding monoids.

The accompanying talk title specifically concerns $\omega$-categorical structures: a structure is $\omega$-categorical when its complete first-order theory has exactly one countably infinite model up to isomorphism. The broader supplied statement concerns infinite structures without this restriction. The intended class $\mathcal K$, language restrictions, cardinalities, and precise reconstruction target are unresolved. In particular, the record does not specify a single conjectured implication to isomorphism or to bi-interpretability. Bi-interpretability means mutual first-order interpretability with the composite interpretations definably isomorphic to the respective original structures; the permitted use of parameters is also unspecified.

The supplied dated curation reports reconstruction up to bi-interpretability in the $\omega$-categorical setting and says that stronger exact reconstruction is not general. This is recorded as a qualified source assessment, not asserted here as a verified theorem or as a fully specified conjecture.

Short English statement

Ask which features of an infinite structure are determined by its automorphism group or its monoid of elementary self-embeddings, including their topology. The talk context singles out $\omega$-categorical structures, but the supplied question leaves the comparison class and the desired notion of reconstruction unspecified. This draft therefore states a reconstruction programme requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30006515$, source problem code catalogue entry, title: Reconstructing Structures from Topological Symmetry Monoids. The registry labels the problem partially_solved and the statement corrected_verified; these are supplied curation labels. Its statement-verification field reports that the official report was checked and the question isolated and repaired; no independent source verification is claimed here.
• Supplied report citation: Homogeneous Structures: Model Theory meets Universal Algebra (2026), Oberwolfach Reports, pp. 3063--3124, DOI 10.4171/owr/2025/58. https://doi.org/10.4171/owr/2025/58. The registry separately gives proposed year 2025; the relationship between these dates is not resolved in this draft.
• Supplied original-statement attribution: Paolo Marimon, On the topological reconstruction of monoids of elementary embeddings of $\omega$-categorical structures, joint work with J. de la Nuez Gonzales, Zaniar Ghadernezhad, and Michael Pinsker. The registry has no separate named proposer.
• Supplied literature reference: Ahlbrandt--Ziegler, Quasi finitely axiomatizable totally categorical theories. https://doi.org/10.1016/0168-0072(86)90037-0.
• Supplied literature reference: de la Nuez Gonzalez et al., Minimal and intrinsic topologies on monoids. https://arxiv.org/abs/2603.28419.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that topological automorphism groups and elementary-embedding monoids recover $\omega$-categorical structures up to bi-interpretability, while stronger exact reconstruction is not general. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its time-sensitive status and the hypotheses and scope of the cited claims require independent checking.