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

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