  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 $\Aut(\Omega,\leq)$.
