Vol. 44, No. 2, 1973

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Archimedean extensions of directed interpolation groups

Andrew M. W. Glass

Vol. 44 (1973), No. 2, 515–521

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 These results have been extended to another class of l-groups by Glass and Holland (unpublished).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.

Mathematical Subject Classification
Primary: 06A55
Received: 1 October 1971
Revised: 29 August 1972
Published: 1 February 1973
Andrew M. W. Glass