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

TL;DR

This paper explores the structure of endomorphisms of the rational numbers' order, revealing uncountably many maximal subgroups isomorphic to automorphism groups of countable linear orders, and characterizing which groups appear as automorphism groups.

Contribution

It characterizes the maximal subgroups of endomorphisms of $(Q, \, ext{≤})$ and classifies automorphism groups of countable linear orders as free abelian groups of finite rank.

Findings

Uncountably many maximal subgroups are isomorphic to automorphism groups of countable linear orders.

Characterization of subsets of $Q$ that are retracts in terms of topology.

Automorphism groups of countable linear orders are exactly free abelian groups of finite rank.

Abstract

We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly 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 a retract in $(\mathbb{Q},{\leq})$ 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 $\operatorname{End}(\mathbb{Q},{\leq})$, if and only if it is free abelian of finite rank.