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Epicompletetion of archimedean l–groups and vector lattices with weak unit

Part of: Ordered structures Topological linear spaces and related structures Peculiar spaces Maps and general types of spaces defined by maps General theory of categories and functors

Published online by Cambridge University Press:  09 April 2009

Richard N. Ball  and
Anthony W. Hager
Richard N. Ball
Affiliation:
Department of Mathematics, Boise State UniversityBoise, Idaho 83725, U.S.A.
Anthony W. Hager
Affiliation:
Department of Mathematics Wesleyan UniversityMiddletown, Connecticut 06457, U.S.A.
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Abstract

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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 2c 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 essentialBcompletion. 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.

MSC classification

Secondary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces 46A40: Ordered topological linear spaces, vector lattices 54C40: Algebraic properties of function spaces 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 54G05: Extremally disconnected spaces, $F$-spaces, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 1 , February 1990 , pp. 25 - 56
Copyright
Copyright © Australian Mathematical Society 1990

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