Convergence Completion of Partially Ordered Groups

As is well known, the (MacNeille, [7]) conditional completion by nonvoid cuts of a partially ordered group (which is the unique conditionally complete lattice each element of which is a join and a meet of the group’s elements) cannot in general (in fact whenever the group fails to be “Archimedean”) be made into a partially ordered group. There is a largest subset of the completion to which the group composition can be extended so as to achieve a partially ordered group (see Fuchs [6] or below). In the totally ordered case this subset may be attained intrinsically as the completion of the original group in its order-topology (Cohen-Goffman [4]). There is no suitable order-topology even for lattice-ordered groups; one can use certain down-directed subsets of positive elements with meet zero to induce order-theoretic topological group structures whose completions may then be shown to be canonically contained in the MacNeille completion (Banaschewski [1]); and more general such down-directed subsets to induce order-theoretic non-group convergence structures whose completions are also so contained - indeed, the totality of these suffice to attain the largest group subextension of the MacNeille completion of a commutative lattice-ordered group (Ibid.). This procedure has been identified as completion with respect to a form of order-convergence by Papangelou [8], who is then able to give a much more efficient proof of the same result. To extend this to partially ordered (possibly non-commutative) groups, it is necessary to isolate the appropriate notion of order-convergence and to devise a proof independent of the more special properties this notion has in l-groups. The result is to make every partially ordered group into a convergence group whose convergence completion is exactly the largest possible partially ordered group in the conditional (order) completion.