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Archimedean Closures in Lattice-Ordered Groups

Published online by Cambridge University Press:  20 November 2018

Richard D. Byrd
Richard D. Byrd*
Affiliation:
University of Houston, Houston, Texas
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Conrad (10) and Wolfenstein (15; 16) have introduced the notion of an archimedean extension (a-extension) of a lattice-ordered group (l-group). In this note the class of l-groups that possess a plenary subset of regular subgroups which are normal in the convex l-subgroups that cover them are studied. It is shown in § 3 (Corollary 3.4) that the class is closed with respect to a-extensions and (Corollary 3.7) that each member of the class has an a-closure. This extends (6, p. 324, Corollary II; 10, Theorems 3.2 and 4.2; 15, Theorem 1) and gives a partial answer to (10, p. 159, Question 1). The key to proving both of these results is Theorem 3.3, which asserts that if a regular subgroup is normal in the convex l-subgroup that covers it, then this property is preserved by a-extensions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1004 - 1012
Copyright
Copyright © Canadian Mathematical Society 1969

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