Hyper-Special Valued Lattice-Ordered Groups

Abstract

A lattice-ordered group G is hyper-special valued if it lies in the largest torsion class which is contained in the class of special-valued lattice-ordered groups. This is precisely the class of lattice-ordered g oups G such that for each g A G, every l-homomorphic image K of the principal convex l-subgroup generated by g has the feature that each 0 < x A K is the supremum of pairwise disjoint special elements. It is shown in this article that if G is hyper-special valued, then for each g ∈ G, the space of values Y(g) of g is a compact scattered space. This property naturally gives meaning to the notion of an α-special value of g: this is a value which corresponds to an isolated point of the α-th remainder in the Cantor-Bendixson sequence of Y(g). It is shown that, for each ordinal α, the set of a-special values of G forms a disjoint union of chains, which is at once an order ideal and a dual order ideal of the root system of all values of G. If G is projectable, then in addition the set of special values of G is also a disjoint union of chains which is an order ideal and a dual order ideal. An archimedean lattice-ordered group G with weak order unit u > 0, given its Yosida representation, such that u≡1 is hyper-special valued if and only if (a) G is projectable, (b) the Yosida space Y is scattered, and (c) for each g ∈ G the image of the function g has finitely many ∞'s as well as finitely many accumulations of 0.