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Disiblizing normal valued lattice-ordered groups
by
W. Charles Holland
Bowling Green State University
It is a classical result that every group can be embedded in a divisible group, that is, a group G such that every equation xn = g has a solution x in G for every n > 0 and every g in G. It has long been known that the same is true for lattice-ordered groups, that is, every lattice-ordered group can be embedded in a divisible lattice-ordered group. An open question is whether every totally ordered group can be embedded in a divisible totally ordered group. Every abelian lattice-ordered group can be embedded in a divisible abelian lattice-ordered group, and a similar result holds for many other varieties (equationally defined classes). In this paper we extend this result to the important variety of normal valued lattice-ordered groups, which is the largest proper variety.
Theorem: Every normal valued lattice-ordered group can be embedded in a divisible normal valued lattice-ordered group.
The proof uses the generalized wreath product and various results of Holland, S. H. McCleary, and J. A. Read.
Date received: January 31, 2002
Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cain-09.