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.
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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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American Mathematical Society