Ellis enveloping semigroups in real closed fields

TL;DR

This paper explores the structure of Ellis enveloping semigroups in real closed fields, introducing Boolean algebras of d-semialgebraic sets, and analyzing their properties and relationships with definable sets in o-minimal groups.

Contribution

It introduces the Boolean algebra of d-semialgebraic sets and establishes its connection with Ellis enveloping semigroups, highlighting differences and characterizations in real closed fields.

Findings

The Stone space of the Boolean algebra of d-semialgebraic sets is isomorphic to the Ellis enveloping semigroup.

In definably connected o-minimal groups, the family of sets coincides with externally definable sets in one dimension.

Differences between these families are shown to occur even in semialgebraic cases over the real algebraic numbers.

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.