Epicompletetion of archimedean <i>l</i>–groups and vector lattices with weak unit

Abstract

Abstract In the category W of archimedean l –groups with distinguished weak order unit, with unitpreserving l –homorphism, 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 M/N , where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier 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 ∈ B . This describes the epicompletetions of a give G (that is, epicomplete objects epically 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 2 c non-eqivalent 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 σ-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 homorphism 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 σ -ideal, called Z(Y(G) ), and the embedding G ≦ B(y(G))/Z(Y(G) ) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.