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Ellis enveloping semigroups in real closed fields

Elías Baro, Daniel Palacín

Abstract

We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgrebraic functions representing Boolean combinations of d-semialgebraic sets.

Ellis enveloping semigroups in real closed fields

Abstract

We introduce the Boolean algebra of d-semialgebraic (more generally, d-definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgrebraic functions representing Boolean combinations of d-semialgebraic sets.
Paper Structure (7 sections, 13 theorems, 85 equations)

This paper contains 7 sections, 13 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Set-up on Ellis semigroups of Stone spaces
  3. Translations by external elements
  4. Ellis envelopes in o-minimal structures
  5. One-dimensional groups
  6. Arbitrary dimension in an $\aleph_0$-saturated model
  7. Arbitrary dimension in the real algebraic numbers

Key Result

Theorem A

Let $M$ be an arbitrary structure. The Ellis semigroups $S_G^{\rm ext}(M)$ and $E(S_G(M))$ are canonically isomorphic if and only if every externally definable subset of $G(M)$ is a (positive) Boolean combination of $d$-definable sets.

Theorems & Definitions (35)

  • Theorem A: Corollary \ref{['C:Equiv-Ellis-ext']}
  • Theorem B: Theorem \ref{['T:1dim']} and \ref{['main']}, Corollary \ref{['C:Alg']}
  • Definition 2.1
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • proof
  • Lemma 2.9
  • proof
  • ...and 25 more