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

2016/07/13 by Jillian Dawn McPhee, McPhee, Jillian D., James D. Mitchell +3
Computer Science · Mathematics · #03C50 #06A05 #20B22 #20B27 #20M20 (Primary) #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1607.03655

openalex publication_date 2016/07/13 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28

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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