Some generalizations of the singular and hyperarchimedean conditions in W
by
Chawne M. Kimber
Wesleyan University and Lafayette College
Coauthors: Anthony W. Hager (Wesleyan University)

W is the category whose objects are archimedean l-groups (G, u) with designated weak unit, u > 0, and whose morphisms preserve the unit. Let (G, u) be in W and let YG be the compact Hausdorff space of values of the unit. Then G < D(YG); that is, G is represented as a group of continuous functions on YG that take values in the two-point compactification of the reals and are real-valued on a dense set.

In 1998, Hager and Martinez showed that a W-object is singular precisely when all its elements are integer-valued. When pursuing a-extensions of a singular object in the same year, Kimber and McGovern quickly came to consider the W-subclass of bounded away l-groups. This class, BA, is defined by: for every element g, there exists a positive real number r such that if g(p) is a positive real number then g(p) > r. Clearly, singular l-groups are in BA; since the hyperarchimedean l-groups are the bounded away l-groups with strong unit, they are also in BA.

We describe twelve new subclasses of BA and attempt to say something interesting about each one. For instance, also contained in BA are the W-objects that are greatest integer closed. That is, we define [g](p) to be the greatest integer in g(p) and say that G is greatest integer closed if each [g] is in G. (Note that G is bounded away if and only if each [g] is continuous on YG.) If G is greatest integer closed and K is any hyperarchimedean extension of the convex l-subgroup generated by the unit, then the l-group generated by G and K in D(YG) is an a-extension of G.

Date received: February 7, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caed-09.