Orderable groups and semigroup compactifications

TL;DR

This paper explores new connections between orderability of groups and topological dynamics, introducing analogs of algebraic orderability for topological groups and examining their implications for compact spaces and semigroups.

Contribution

It proposes novel concepts linking group orderability with topological dynamics, expanding the theoretical framework and suggesting potential applications to discrete groups.

Findings

Introduces order-preserving actions on compact spaces.

Defines analogs of algebraic orderability for topological groups.

Raises questions for further research in group orderability and dynamics.

Abstract

Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups.