Splitting properties in Archimedeanl-groups

Abstract

Throughout this paper anl-group will always mean an archimedean lattice-ordered group and we shall confine our attention to such groups. Anl-groupsplitsif it is a cardinal summand of eachl-group that contains it as anl-ideal. Suppose thatGis anl-subgroup of anl-groupH. ThenGislargeinHorHis anessential extensionofGif for eachl-ideal L≠0 ofH, L∩G≠0.Gisessentiallyclosed if it does not admit any proper essential extension. Conrad (1971) proved that each essentially closedl-group splits, but not conversely.