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

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Abstract

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 characterise 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.
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Original languageEnglish
Pages (from-to)171-194
Number of pages24
JournalQuarterly Journal of Mathematics
Volume70
Issue number1
Early online date28 Aug 2018
DOIs
Publication statusPublished - Mar 2019

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