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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
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.

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Additional Information

M. Droste
Affiliation: Institut für Algebra, Technische Universität Dresden, D-01062 Dresden, Germany

J. K. Truss
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England

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