All ℓ-groups shall be abelian. An a-extension of an ℓ-group is an extension preserving the lattice of ideals; an ℓ-group with no proper a-extension is called a-closed. A hyperarchimedean ℓ-group is one for which each quotient is archimedean. This paper examines hyperarchimedean ℓ-groups with unit and their a-extensions by means of the Yosida representation, focussing on several previously open problems. Paul Conrad asked in 1965: If G is a-closed and M is an ideal, is G/M a-closed? And in 1972: If G is a hyperarchimedean sub-ℓ-group of a product of reals, is the f-ring which G generates also hyperarchimedean? Marlow Anderson and Conrad asked in 1978 (refining the first question above): If G is a-closed and M is a minimal prime, is G/M a-closed? If G is a-closed and hyperarchimedean and M is a prime, is G/M isomorphic to the reals? Here, we introduce some techniques of a-extension and construct a several parameter family of examples. Adjusting the parameters provides answers "No" to the questions above.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-2,
author = {Anthony W. Hager and Chawne M. Kimber},
title = {Some examples of hyperarchimedean lattice-ordered groups},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {107-122},
zbl = {1058.06020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-2}
}
Anthony W. Hager; Chawne M. Kimber. Some examples of hyperarchimedean lattice-ordered groups. Fundamenta Mathematicae, Tome 184 (2004) pp. 107-122. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm182-2-2/