Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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 $\mathbb{R}$, 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?)

  • 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

Similar Articles

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

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

American Mathematical Society