Polar Functions, I: The Summand-Inducing Hull of an Archimedean l-Group with Unit

Abstract

This is an introduction to the concept of a polar function on W, the category of archimedean l-groups with designated unit, and to that of its dual, a covering function on compact spaces.

Let X be a subalgebra of the boolean algebra of polars P(G) of the W-object G. An essential extension H of G is said to be an X-splitting extension of G if the extension of each K ∈ P(G) to H is a cardinal summand. The least X-splitting extension G[X] of G is studied here. Dually, one considers a compact Hausdorff space, and a subalgebra k of R(X), the boolean algebra of all regular closed sets. A k-cover Y of X is represented by an irreducible map g : Y→X subject to the condition that clyg^-1(int ^X A) is clopen, for each A ∈ k. There is a minimum k-cover. To each subalgebra X of polars of G there corresponds canonically a subalgebra k of regular closed sets of the Yosida space YG of G, in such a way that the Yosida space of G[X] is the minimum k-cover of YG. This general setup is applied to some well known situations, to recapture constructions such as the projectable hull. On the topological side one may recover the cloz cover of a compact space.

A function which assigns to each G a subalgebra X(G) of polars of G is called a polar function. The dual notion for compact spaces is the covering function: assigning to each space X a subalgebra k(X) of regular closed sets. By transfinitely iterating the basic constructions of least X-splitting extensions and minimum k-covers, one obtains their idempotent closures, X ^b and k ^b, respectively. These closures give rise to, respectively, hull classes of archimedean l-groups and covering classes of compact spaces.