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

Abstract

We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},\le )$ of rational numbers. We show that for any countable linearly ordered set $\Omega$, there are uncountably many maximal subgroups of $\mathrm{End}(\mathbb{Q},\le )$ isomorphic to the automorphism group of $\Omega$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},\le )$ 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 $\mathrm{End}(\mathbb{Q},\le )$, if and only if it is free abelian of finite rank.

24 pages. Revised based on referee's comments