#
A functorial approach to Dedekind completions and the representation of
vector lattices and *l*-algebras by normal functions

##
G. Bezhanishvili, P. J. Morandi, B. Olberding

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.

Published 2016-12-19.

http://www.tac.mta.ca/tac/volumes/31/37/31-37.pdf

TAC Home