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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 ofEnd  (  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 ofEnd  (  ℚ  ,  ≤  )    , if and only if it is free abelian of finite rank.

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