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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 valuations. For the natural valuation, these are the archimedean extensions ( a -extensions) first investigated by 12 and 30 . This leads to new results and new proofs of old results about a -extensions. Then we obtain new structure theorems for Δ-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. |
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