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

Full-text PDF

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.

**1.**M. Droste, The existence of rigid measurable spaces, Topology and its Applications 31 (1989), 187-195. MR**90b:28001****2.**M. Droste, Super-rigid families of strongly Blackwell spaces, Proc. Amer. Math. Soc. 103 (1988), 803-808. MR**89e:28006****3.**M. Droste, D. Kuske, R. McKenzie, R. Pöschel, Complementary closed relational clones are not always Krasner clones, Algebra Universalis, to appear.**4.**M. Dugas, R. Göbel: Applications of abelian groups and model theory in algebraic structures, in: `Infinite Groups' (de Giovanni, Newell, eds.), de Gruyter and Co., Berlin, New York, 1996, 41-62. MR**98m:20030****5.**B. Dushnik and E. W. Miller, Concerning similarity transformations of linearly ordered sets, Bull. Amer. Math. Soc. 46 (1940), 322-326. MR**1:318g****6.**A. M. W. Glass, Ordered Permutation Groups, London Mathematical Society, Lecture Notes, 55, Cambridge University Press, 1981. MR**83j:06004****7.**J. G. Rosenstein, Linear Orderings, Academic Press, 1982. MR**84m:06001****8.**R. M. Solovay, Real-valued measurable cardinals, in D. S. Scott (ed.) Axiomatic set theory, Proc. Symp. Pure Math. XIII Part 1, Amer. Math. Soc. 1971, 397-428. MR**45:55**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
06A05

Retrieve articles in all journals with MSC (2000): 06A05

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