Groups of complexes of a representable lattice-ordered group

Abstract

In 1954 N. Kimura proved that each idempotent in a semigroup is contained in a unique maximal subgroup of the semigroup and that distinct maximal subgroups are disjoint [13] (or see [6, pp. 21–23]). This generalized earlier results of Schwarz [14] and Wallace [15]. These maximal subgroups are important in the study of semigroups. If G is a group, then the collectionS(G) of nonempty complexes ofGis a semigroup and it is natural to inquire what properties ofGare inherited by the maximal subgroups ofS(G). There seems to be very little literature devoted to this subject. In [5, Theorem 2], with certain hypotheses placed on an idempotent, it was shown that ifGis a lattice-ordered group (“1-group”) then a maximal subgroup ofS(G) containing an idempotent satisfying these conditions admits a natural lattice-order. The main result of this note (Theorem 1) is that ifGis a representable1-group andEis a normal idempotent ofS(G) and a dual ideal of the latticeG, then the maximal subgroup ofS(G) containingEadmits a representable lattice-order.