Rigid chains admitting many embeddings
M. Droste and J. K. Truss
Proc. Amer. Math. Soc. 129 (2001), 1601-1608
October 31, 2000
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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 , 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.
Droste, The existence of rigid measurable spaces, Topology
Appl. 31 (1989), no. 2, 187–195. MR 994410
Droste, Super-rigid families of strongly
Blackwell spaces, Proc. Amer. Math. Soc.
103 (1988), no. 3,
947662 (89e:28006), http://dx.doi.org/10.1090/S0002-9939-1988-0947662-2
M. Droste, D. Kuske, R. McKenzie, R. Pöschel, Complementary closed relational clones are not always Krasner clones, Algebra Universalis, to appear.
Dugas and Rüdiger
Göbel, Applications of abelian groups and model theory to
algebraic structures, Infinite groups 1994 (Ravello), de Gruyter,
Berlin, 1996, pp. 41–62. MR 1477163
Dushnik and E.
W. Miller, Concerning similarity transformations
of linearly ordered sets, Bull. Amer. Math.
Soc. 46 (1940),
0001919 (1,318g), http://dx.doi.org/10.1090/S0002-9904-1940-07213-1
M. W. Glass, Ordered permutation groups, London Mathematical
Society Lecture Note Series, vol. 55, Cambridge University Press,
Cambridge, 1981. MR 645351
G. Rosenstein, Linear orderings, Pure and Applied Mathematics,
vol. 98, Academic Press Inc. [Harcourt Brace Jovanovich Publishers],
New York, 1982. MR 662564
M. Solovay, Real-valued measurable cardinals, Axiomatic set
theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los
Angeles, Calif., 1967), Amer. Math. Soc., Providence, R.I., 1971,
pp. 397–428. MR 0290961
- M. Droste, The existence of rigid measurable spaces, Topology and its Applications 31 (1989), 187-195. MR 90b:28001
- M. Droste, Super-rigid families of strongly Blackwell spaces, Proc. Amer. Math. Soc. 103 (1988), 803-808. MR 89e:28006
- M. Droste, D. Kuske, R. McKenzie, R. Pöschel, Complementary closed relational clones are not always Krasner clones, Algebra Universalis, to appear.
- 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
- B. Dushnik and E. W. Miller, Concerning similarity transformations of linearly ordered sets, Bull. Amer. Math. Soc. 46 (1940), 322-326. MR 1:318g
- A. M. W. Glass, Ordered Permutation Groups, London Mathematical Society, Lecture Notes, 55, Cambridge University Press, 1981. MR 83j:06004
- J. G. Rosenstein, Linear Orderings, Academic Press, 1982. MR 84m:06001
- 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
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Institut für Algebra, Technische Universität Dresden, D-01062 Dresden, Germany
J. K. Truss
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
linearly ordered set,
Received by editor(s):
July 7, 1999
Received by editor(s) in revised form:
September 15, 1999
October 31, 2000
Research supported by a grant from the British-German Academic Collaboration Programme.
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