Ellis enveloping semigroups in real closed fields
type: Item
instance of: scholarly article
title: Ellis enveloping semigroups in real closed fields
zbMATH Open document ID: 1542.14072
author: Elías Baro
author: D. Palacín
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2024-05-22T00:00:00Z
review text: An Ellis semigroup is a semigroup \((E, \cdot)\) which is a compact Hausdorff topological space such that for each \(y \in E\), the map \(x \mapsto x \cdot y\) is continuous. The present paper is devoted to the Ellis enveloping semigroup, which is an example of Ellis semigroup, constructed as follows. Let \(G\) be a group with the discrete topology acting on a compact Hausdorff space \(S\) by homeomorphisms. The action is given by a homomorphism \(G \rightarrow \mathrm{Homeo}(S)\) where \(g \mapsto \ell_g\). The Ellis enveloping semigroup \((E(S), \circ)\) is the closure of \(\{\ell_g\}_{g \in G}\) in \(S^S\), where \(S^S\) has the product topology, and \(\circ\) denotes the composition of functions.\N\NThen, consider \(G\) a definable group in a structure \(M\). The group \(G\) acts naturally on the compact Hausdorff space of types \(S_G(M)\), the space of ultrafilters of definable subsets of \(G\). Thus, this action has associated the Ellis enveloping semigroup \((E(S_G(M)), \circ)\). In order to study it, \textit{L. Newelski} in [Isr. J. Math. 190, 477--507 (2012; Zbl 1273.03118)] stated the convenience to work with externally definable sets. These are the counterpart in definable sets to externally semialgebraic sets in classical semialgebraic geometry. Then, he considered the action of \(G\) on the Stone space \(S^{\mathrm{ext}}_G(M)\) of ultrafilters of externally definable sets of \(M\). Then, \((E(S^{\mathrm{ext}}_G(M)), \circ)\) is \((S^{\mathrm{ext}}_G(M), \ast)\), where \(\ast\) is a model-theoretic operation called coheir product.\N\NIn this setting, the question is whether \(E(S^{\mathrm{ext}}_G(M)) \approx S^{\mathrm{ext}}_G(M)\) is naturally isomorphic to \(E(S_{G}(M))\) as Ellis semigroups. To answer this question is the goal of the paper under review. The main result is Corollary 3.4, which characterizes when both semigroups are canonically isomorphic in terms of Boolean combinations of \(d\)-definable sets, as follows: it holds if and only if every…
reviewed by: José Javier Etayo
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 37B05
zbMATH DE Number: 7851901
zbMATH Keywords: o-minimal
zbMATH Keywords: Ellis semigroup
zbMATH Keywords: semialgebraic
zbMATH Keywords: externally definable
MaRDI profile type: Publication
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DOI: 10.1007/S13398-024-01562-7
description: scientific article; zbMATH DE number 7851901