Maximality and Ultracompleteness in Normed Modules1

1. What follows is a drastically stripped down version of a paper completed several years ago. The original intention was to present a careful treatment of the developments centering around the maximally concept in classical valuation theory, using modern ideas and the natural setting of an additive group, and to apply the resulting theory to systematize and simplify some recent investigations into ordered groups. The various referees through whose hands the paper has since passed have finally succeeded in convincing the writer that there is insufficient interest in such a project to justify publication. Accordingly the present version confines itself strictly to an account of the new results obtained. Even before the first submission of this paper, there had appeared an article by Conrad [C] in which ordered groups are handled from the valuation standpoint, but which does not specifically make use of techniques from valuation theory. Subsequently, while this paper was bottled up in the refereeing process, Gravett [G] rederived Conrad's results, at least for the case of interest here, by generalizing the appropriate valuation theoretic arguments. What is still new in the present paper is the example of a nonunique maximal immediate extension, and the study of normed spaces over fields with a valuation, which can be used to treat ordered spaces over ordered fields.