Unlike the uniform completion, the Dedekind completion of a vector lattice is not functorial. In order to repair the lack of functoriality of Dedekind completions, we enrich the signature of vector lattices with a proximity relation, thus arriving at the category pdv of proximity Dedekind vector lattices. We prove that the Dedekind completion induces a functor from the category bav of bounded archimedean vector lattices to pdv, which in fact is an equivalence. We utilize the results of Dilworth to show that every proximity Dedekind vector lattice D is represented as the normal real-valued functions on the compact Hausdorff space associated with D. This yields a contravariant adjunction between pdv and the category KHaus of compact Hausdorff spaces, which restricts to a dual equivalence between KHaus and the proper subcategory of pdv consisting of those proximity Dedekind vector lattices in which the proximity is uniformly closed. We show how to derive the classic Yosida Representation, Kakutani-Krein Duality, Stone-Gelfand-Naimark Duality, and Stone-Nakano Theorem from our approach.
Keywords: Vector lattice, $\ell$-algebra, uniform completion, Dedekind completion, compact Hausdorff space, extremally disconnected space, continuous real-valued function, normal real-valued function, proximity, representation
2010 MSC: 06F20; 46A40; 54E05; 54D30; 54G05
Theory and Applications of Categories, Vol. 31, 2016, No. 37, pp 1095-1133.