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Universal sequences for the order-automorphisms of the rationals

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J. Hyde, J. Jonusas, J. D. Mitchell, Y. H. Peresse

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Abstract

In this paper, we consider the group Aut(Q,≤) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q,≤) is contained in an N-generated subgroup of Aut(Q,≤) for some fixed N ∈ N. We show that the least such N is 2. Moreover, for every countable subset of Aut(Q,≤), we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that a and b freely generate the free semigroup {a,b}+ consisting of the non-empty words over a and b. Then we show that there exists a sequence of words w1, w2,... over {a,b} such that for every sequence f1, f2, ... ∈ Aut(Q,≤) there is a homomorphism φ : {a,b}+ → Aut(Q,≤) where (wi)φ=fi for every i.

As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of Aut(Q,≤) is uncountable, or equivalently that Aut(Q,≤) has uncountable cofinality and Bergman's property.
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Original languageEnglish
Pages (from-to)21-37
Number of pages17
JournalJournal of the London Mathematical Society
Volume94
Issue number1
Early online date13 May 2016
DOIs
Publication statusPublished - Aug 2016

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