dc.contributor.author | Hyde, James | |
dc.contributor.author | Jonusas, Julius | |
dc.contributor.author | Mitchell, J. D. | |
dc.contributor.author | Peresse, Y. H. | |
dc.date.accessioned | 2016-12-07T16:51:40Z | |
dc.date.available | 2016-12-07T16:51:40Z | |
dc.date.issued | 2016-05-13 | |
dc.identifier.citation | Hyde , J , Jonusas , J , Mitchell , J D & Peresse , Y H 2016 , ' Universal sequences for the order-automorphisms of the rationals ' , Journal of the London Mathematical Society . https://doi.org/10.1112/jlms/jdw015 | |
dc.identifier.issn | 1469-7750 | |
dc.identifier.other | ArXiv: http://arxiv.org/abs/1401.7823v4 | |
dc.identifier.uri | http://hdl.handle.net/2299/17397 | |
dc.description | This is the accepted version of the following article: J. Hyde, J. Jonušas, J. D. Mitchell, and Y. Péresse, Universal sequences for the order-automorphisms of the rationals, J. London Math. Soc., first published online May 13, 2016 which has been published in final form at doi:10.1112/jlms/jdw015 | |
dc.description.abstract | In this paper, we consider the group Aut$(\mathbb{Q}, \leq)$ of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Kh\'elif states that every countable subset of Aut$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q}, \leq)$ for some fixed $N\in\mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q}, \leq)$, 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 $w_1, w_2,\ldots$ over $\{a,b\}$ such that for every sequence $f_1, f_2, \ldots\in\,$Aut$(\mathbb{Q}, \leq)$ there is a homomorphism $\phi:\{a,b\}^{+}\to$ Aut$(\mathbb{Q},\leq)$ where $(w_i)\phi=f_i$ 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$(\mathbb{Q}, \leq)$ is uncountable, or equivalently that Aut$(\mathbb{Q}, \leq)$ has uncountable cofinality and Bergman's property. | en |
dc.format.extent | 210161 | |
dc.language.iso | eng | |
dc.relation.ispartof | Journal of the London Mathematical Society | |
dc.subject | Group Theory | |
dc.subject | Infinite Combinatorics | |
dc.title | Universal sequences for the order-automorphisms of the rationals | en |
dc.contributor.institution | School of Physics, Astronomy and Mathematics | |
dc.description.status | Peer reviewed | |
dc.identifier.url | http://arxiv.org/pdf/1401.7823v4.pdf | |
rioxxterms.versionofrecord | 10.1112/jlms/jdw015 | |
rioxxterms.type | Journal Article/Review | |
herts.preservation.rarelyaccessed | true | |