Polar functions, III: On irreducible maps vs. essential extensions of Archimedean $ell$-groups with unit

Year, pages: 2003, 189 - 211

This paper further develops the notion of a polar function, extending the scope to sublattices of polars. The category of discourse is W, the category of archimedean $ell$-groups with designated unit and all $ell$-homomorphisms. The dual notion of a covering function of compact spaces is similarly extended.

The paper builds on the ideas of summand-inducing hulls, defining, for each invariant polar function $L$ on objects in W, an associated “$L$-essential extension”. With certain natural assumptions on the function $L$, there is, for each W-object $G$, a largest $L$-essential extension $G^<b>L$. Given $L$ and a W-object $G$, $widehatL(G)$ is the boolean algebra of all polars in $L(G)$ whose complements also lie in $L(G)$. This boolean center $widehatL$ of $L$ gives rise to the least summand-inducing hull $G[widehatL]$, which, under the same assumptions, is contained in $G^<b>L$.

In tandem with these ideas goes the development of $frak I$-irreducible maps of compact spaces, associated with an invariant covering function $frak I$.

Martinez, J. 2003. Polar functions, III: On irreducible maps vs. essential extensions of Archimedean $ell$-groups with unit. In Tatra Mountains Mathematical Publications, vol. 27, no.3, pp. 189-211. 1210-3195.

Martinez, J. (2003). Polar functions, III: On irreducible maps vs. essential extensions of Archimedean $ell$-groups with unit. Tatra Mountains Mathematical Publications, 27(3), 189-211. 1210-3195.