Convergence and Cauchy structures on lattice ordered groups

Ball, Richard N.

doi:10.1090/S0002-9947-1980-0567085-5

Convergence and Cauchy structures on lattice ordered groups
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Trans. Amer. Math. Soc. 259 (1980), 357-392 Request permission

Abstract:

This paper employs the machinery of convergence and Cauchy structures in the task of obtaining completion results for lattice ordered groups. §§1 and 2 concern l-convergence and l-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind-MacNeille completion. §5 concerns the polar convergence structure; the corresponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type $\mathcal {Y}$ hull, strictly larger than the lateral completion in general.

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