Almost finite-valued lattice-ordered groups
by
Michael Darnel
Indiana University, South Bend
Coauthors: Paul Conrad (University of Kansas)

Almost finite-valued l-groups were introduced by Chen and Conrad [CC] in studying special-valued l-subgroups of lattice-ordered groups. Among their results were that maximal special-valued l-subgroups of abelian l-groups must be weakly saturated, the class of almost finite-valued l-groups forms a quasitorsion class, and several characterizations of almost finite-valued l-groups.

In this article, we show that maximal special-valued l-subgroups of representable l-groups must be saturated, show that maximal almost finite-valued l-subgroups of abelian l-groups must be saturated, and show the finite-valued radical of an l-group must be contained in all maximal almost finite-valued l-subgroups. A condition is given to ensure that each maximal almost finite-valued l-subgroup of an abelian l-group contains a maximal finite-valued l-subgroup. An s-extension of a special-valued l-group is defined and an explicit description is given of the s-closure of a special-valued l-group. These results are then applied to the quasitorsion class of almost finite-valued lattic-ordered groups.

Date received: February 16, 2000