The first-order theory of $\ell$-permutation groups

TL;DR

This paper characterizes when the automorphism group of a totally ordered set shares the same first-order properties as that of the real line, leading to the conclusion that the underlying sets are isomorphic.

Contribution

It extends previous results by linking first-order properties of automorphism groups to the isomorphism of the underlying ordered sets.

Findings

Automorphism groups of certain ordered sets can be characterized by their first-order theories.

If the automorphism group of an ordered set is elementarily equivalent to that of the real line, the sets are isomorphic.

The study introduces new methods involving centralizers and coloured chains in automorphism groups.

Abstract

Let $(\Omega, \leq)$ be a totally ordered set. We prove that if $\Aut(\Omega,\leq)$ is transitive and satisfies the same first-order sentences as $\Aut(\RR,\leq)$ (in the language of lattice-ordered groups) then $\Omega$ and $\RR$ are isomorphic ordered sets. This improvement of a theorem of Gurevich and Holland is obtained as one of many consequences of a study of centralizers and coloured chains associated with certain transitive subgroups of $\Aut(\Omega,\leq)$.