Ordered Algebraic Structures pp 275-299 | Cite as

# 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.

## Keywords

Covering Function Boolean Algebra Compact Space Polar Function Compact Hausdorff Space## Preview

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