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Rigid chains admitting many embeddings


Authors: M. Droste and J. K. Truss
Journal: Proc. Amer. Math. Soc. 129 (2001), 1601-1608
MSC (2000): Primary 06A05
DOI: https://doi.org/10.1090/S0002-9939-00-05702-6
Published electronically: October 31, 2000
MathSciNet review: 1814086
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Abstract: A chain (linearly ordered set) is rigid if it has no non-trivial automorphisms. The construction of dense rigid chains was carried out by Dushnik and Miller for subsets of $\mathbb{R}$, and there is a rather different construction of dense rigid chains of cardinality $\kappa$, an uncountable regular cardinal, using stationary sets as `codes', which was adapted by Droste to show the existence of rigid measurable spaces. Here we examine the possibility that, nevertheless, there could be many order-embeddings of the chain, in the sense that the whole chain can be embedded into any interval. In the case of subsets of $\mathbb{R}$, an argument involving Baire category is used to modify the original one. For uncountable regular cardinals, a more complicated version of the corresponding argument is used, in which the stationary sets are replaced by sequences of stationary sets, and the chain is built up using a tree. The construction is also adapted to the case of singular cardinals.


References [Enhancements On Off] (What's this?)

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Additional Information

M. Droste
Affiliation: Institut für Algebra, Technische Universität Dresden, D-01062 Dresden, Germany
Email: droste@math.tu-dresden.de

J. K. Truss
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
Email: pmtjkt@leeds.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-00-05702-6
Keywords: Chain, linearly ordered set, rigid, embedding, meagre, stationary
Received by editor(s): July 7, 1999
Received by editor(s) in revised form: September 15, 1999
Published electronically: October 31, 2000
Additional Notes: Research supported by a grant from the British-German Academic Collaboration Programme.
Communicated by: Alan Dow
Article copyright: © Copyright 2000 American Mathematical Society

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