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 | References | Similar Articles | Additional Information

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 , and there is a rather different construction of dense rigid chains of cardinality , 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 , 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

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