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We investigate the structure of the monoid of endomorphisms of the orderedset $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countablelinearly ordered set $\Omega$, there are uncountably many maximal subgroups of$\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of$\Omega$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as aretract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning$X$. Finally, we establish that a countable group arises as the automorphismgroup of a countable linearly ordered set, and hence as a maximal subgroup of$\operatorname{End}(\mathbb{Q},{\leq})$, if and only if it is free abelian offinite rank.

Automorphism groups of linearly ordered structures and endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers