Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-05T13:00:56.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Wreath Products of Nonoverlapping Lattice Ordered Groups

Published online by Cambridge University Press:  20 November 2018

John A. Read
John A. Read*
Affiliation:
Canadian Coast Guard College, Sydney, N.S.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

One of the fundamental tools in the theory of totally ordered groups is Hahn’s Theorem (a detailed discussion may be found in Fuchs [3]), which asserts, roughly, that every abelian totally ordered group can be embedded in a lexicographically ordered (unrestricted) direct sum of copies of the ordered group of real numbers. Almost any general question regarding the structure of abelian totally ordered groups can be answered by reference to Hahn’s theorem. For the class of nonabelian totally ordered groups, a theorem which parallels Hahn’s Theorem is given in [5], and states that each totally ordered group can be o-embedded in an ordered wreath product of subgroups of the real numbers. In order to extend this theorem to include an “if and only if” statement, one must consider lattice ordered groups, as an ordered wreath product of subgroups of the real numbers is, in general, not totally-ordered, but lattice ordered.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 5 , February 1975 , pp. 713 - 722
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Byrd, R. D., Archimidean Closures in Lattice-Ordered Groups, Canadian J. Math. 21 (1969). 1004–1012.Google Scholar
2. Byrd, R. D. and Lloyd, J. T., Closed Subgroups and Complete Distributivity in Lattice-Ordered Groups, Math. Zeitschr, 101 (1967), 123–130.Google Scholar
3. Fuchs, L., Partially Ordered Algebraic Systems, Pergamon Press, New York, 1963.Google Scholar
4. Holland, W. Charles, The Lattice-Ordered Groups of Auotmorphisms of an Ordered Set, Michigan Math. J. 10 (1963), 399–408.Google Scholar
5. Holland, W. Charles and McCleary, S. H., Wreath Products of Ordered Permutation Groups, Pacific J. Math. 31. (1969), 703–716.Google Scholar
6. Read, J. A., Nonoverlapping Lattice-Ordered Groups, Ph.D. thesis, The University of Wisconsin, Madison, Wisconsin, 1971.Google Scholar
7. Wolfenstein, S., Valeurs normales dans un groupe réticulé, Accademia Nazionale dei Lincei, 44, (1968), 337–342.Google Scholar