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Compatible Tight Riesz Orders II

Published online by Cambridge University Press:  20 November 2018

A. M. W. Glass
A. M. W. Glass*
Affiliation:
Bowling Green State University, Bowling Green, Ohio
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R. N. Ball (unpublished) and G. E. Davis and C. D. Fox [1] established that if Ω is a doubly homogeneous totally ordered set, the l-group A (Ω) of all orderpreserving permutations of Ω endures a compatible tight Riesz order. Specifically T = {g ∈ A(Ω)+ : supp (g) is dense in Ω} is a compatible tight Riesz order for A(Ω). Using this fact, I inserted Theorem 3.7 into [2; MR 53 (1977), #13070] at the galley proof stage. (It was also included in MR 54 (1977), #7350 and [3; p. 472].) Theorem 3.7 stated: Let Ω be homogeneous. Then A(Ω) endures a compatible tight Reisz order if and only if Ω is dense. I stated that it was obvious that if Ω were homogeneous and discrete, A(Ω) could not endure a compatible tight Riesz order. This “obvious” is neither obvious nor true.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 304 - 307
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Davis, G. E. and Fox, C. D., Compatible tight Riesz orders on the group of automorphisms of an 0-2 homogeneous set, Canadian J. Math.. 28 (1976), 1076–1081.Google Scholar
2. Glass, A. M. W., Compatible tight Riesz orders, Canadian J. Math.. 28 (1976), 186–200.Google Scholar
3. Glass, A. M. W., Ordered Permutation Groups, Bowling Green State University, 1976.Google Scholar
4. Reilly, N. R., Compatible tight Riesz orders and prime subgroups, Glasgow Math. J. 14 (1973), 146–160.Google Scholar