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

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Date
03/2019
Author
McPhee, Jillian Dawn
Mitchell, James David
Quick, Martyn
Keywords
QA Mathematics
T-NDAS
BDC
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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 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.
Citation
McPhee , J D , Mitchell , J D & Quick , M 2019 , ' Automorphism groups of linearly ordered structures and endomorphisms of the ordered set ( Q ,≤) of rational numbers ' , Quarterly Journal of Mathematics , vol. 70 , no. 1 , pp. 171-194 . https://doi.org/10.1093/qmath/hay043
Publication
Quarterly Journal of Mathematics
Status
Peer reviewed
DOI
https://doi.org/10.1093/qmath/hay043
ISSN
0033-5606
Type
Journal article
Rights
© 2018, the Author(s). Published by Oxford University Press. This work has been made available online in accordance with the publisher’s policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/https://doi.org/10.1093/qmath/hay043
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  • University of St Andrews Research
URI
http://hdl.handle.net/10023/15924

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