Ellis enveloping semigroups in real closed fields (Q6542371)

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 externally definable subset of \(G(M) := G \cap M^n\) is a Boolean combination of \(d\)-definable sets.\N\NIn the final Section 4 the authors turn to the o-minimal context. If \(M\) is an o-minimal structure, then both semigroups are naturally isomorphic as Ellis semigroups if \(G\) is a definably connected one-dimensional definable group (Theorem 4.1). On the contrary, this is not the case (a) if \(M\) is an \(\aleph_0\)-saturated expansion of a real-closed field and \(G = (M^2, +)\) (Theorem 4.2); or (b) if \(M\) is an expansion of the field of real algebraic numbers \(\mathbb{R}_{\mathrm{alg}}\) and \(G = (\mathbb{R}_{\mathrm{alg}}^2, +)\) (Corollary 4.10). Also, Remark 4.11 provides an example of a definably compact group for which both semigroups are not naturally isomorphic as Ellis semigroups.