Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c 3 A, the ``closed under countable composition hull of A,'' as C(Y l A), where Y l A is the Yosida locale of A, and as the largest essential reflection of A.
Positivity – Springer Journals
Published: Jan 1, 2005
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