P. F. Conrad has obtained some
properties of archimedean extensions (a-extensions) of lattice ordered groups
(l-groups). In particular, Conrad proved that every abelian l-group has an 𝒜-closure
(an abelian a-extension which has no proper abelian a-extension). D. Khuon proved
that every l-group has an a-closure (an a-extension which has no proper
a-extension). Using a slightly different definition, Conrad and Bleier defined an
a∗-extension of an l-group and proved that every abelian l-group has an
a∗-closure and every archimedean l-group has a unique a∗-closure. These
results have been extended to another class of l-groups by Glass and Holland
(unpublished).
The purpose of this paper is to extend the l-group results to the class of directed
interpolation groups. The obvious definitions give rise to some negative results; the
situation for abelian 𝒫-groups is more propitious and it is proved that any such
group has an 𝒜-closure in this class. However, taking less direct definitions of
a-extensions and a∗-extensions gives 𝒜-closures and 𝒜∗-closures in restricted classes
of abelian directed interpolation groups.
