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

Let $(Ω, \leq)$ be a totally ordered set. We prove that if $\Aut(Ω,\leq)$ is transitive and satisfies the same first-order sentences as $\Aut(\RR,\leq)$ (in the language of lattice-ordered groups) then $Ω$ 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 $\Aut(Ω,\leq)$.

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