Automorphism groups of linearly ordered structures and endomorphisms of the ordered set (Q,≤) of rational numbers

We investigate the structure of the monoid of endomorphisms of the ordered set (Q,≤) of rational numbers. We show that for any countable linearly ordered set Ω, there are uncountably many maximal subgroups of End(Q,≤) isomorphic to the automorphism group of Ω. We characterize those subsets X of Q that arise as a retract in (Q,≤) in terms of topological information concerning X. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal subgroup of End(Q,≤), if and only if it is free abelian of finite rank.