J Austral Math Soc Ser A 48 pp25--56, 1990.
(Received 12 November 1987; revised 19 July 1988)
In the category W of archimedean l-groups with distinguished weak order unit, with unit-preserving l-homomorphisms, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the form M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earler work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G Î B. This describes the epicompletions of a given G (that is, epicomplete objects especially containing G). First, we note that an epicompletion of G is just a "B-completion", that is, a minimal extension of G by a B-object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-equivalent such extensions.) Then (we show) the B-completions, or epicompletions, of G are exactly the quotients of the l-group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by s-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G £ B(Y(G))/N(G) lifts any homomorphism from G to a B-object. (The existence, though not the nature, of such a "reflective" epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essential B-completion. There is an intermediate s-ideal, called Z(Y(G)), and the embedding G £ B(Y(G))/Z(Y(G)) is a s-embedding, and the functorial for s-homomorphisms. The situationstands in strong analogy to the theory in Boolean algebras of free s-algebras and s-extensions, though there are crucial differences.
1980 AMS Subject Classification (1985 Revision): 06F20, 46A40, 54C40, 18A40, 54G05
Last Modified: Wed Feb 19 10:27:51 2003