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 $\mathbb {R}$, 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

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

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