Abstract
It is well-known that every Archimedean Riesz space (vector lattice) can be embedded in a certain “minimal” Dedekind complete Riesz space (itsDedekind completion) and that this space is essentially unique. There are other nice properties that a Riesz space can enjoy besides Dedekind completeness; for example, the projection property, the principal projection property,σ-Dedekind completeness, and (relative) uniform completeness. It is shown that every Archimedean Riesz space has an essentially unique completion with respect to each of these properties. These completions can be viewed as universal objects in appropriate categories. As such, their uniqueness is obvious (universal objects are always unique), and their existence can be demonstrated very simply by working within the Dedekind completion. This approach is free of clutter since all it needs is theexistence of the Dedekind completion, and not its particular form (which can be quite complicated). By using the same methods within the universal completion, we can isolate further order completions; in a sense, every possible order completion can be obtained in this way, since the universal completion is the largest Riesz space in which the original space is order dense. As an added bonus, all of our results apply equally well to Archimedeanl-groups.
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Aliprantis, C.D., Langford, E. Order completions of Archimedean Riesz spaces andl-groups. Algebra Universalis 19, 151–159 (1984). https://doi.org/10.1007/BF01190426
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DOI: https://doi.org/10.1007/BF01190426