Valuations of Lattice-Ordered Groups

✍ Scribed by Paul F. Conrad; Michael R. Darnel; David G. Nelson

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 379 KB
• Volume: 192
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6891

✦ Synopsis

In this paper, we introduce the concept of a valuation mapping of an l-group G onto a distributive lattice and use such valuations to investigate the structure of G. Then we examine the maximal immediate extensions of G with respect to these Ž valuations. For the natural valuation, these are the archimedean extensions a-. Ž extensions first investigated by P. Conrad 1966, J. Indiana Math. Soc. 30, . Ž . 131᎐160 and S. Wolfenstein 1970, dissertation, University of Paris . This leads to new results and new proofs of old results about a-extensions. Then we obtain new structure theorems for ⌬-extensions of l-groups. In another paper, it will be shown that this valuation theory determines all the torsion classes of l-groups that have invariant torsion radicals. This includes most of the well known and interesting torsion classes that are not l-varieties.