Characteristic subgroups of lattice-ordered groups

Abstract

Characteristic subgroups of an l l -group are those convex l l -subgroups that are fixed by each l l -automorphism. Certain sublattices of the lattice of all convex l l -subgroups determine characteristic subgroups which we call socles. Various socles of an l l -group are constructed and this construction leads to some structure theorems. The concept of a shifting subgroup is introduced and yields results relating the structure of an l l -group to that of the lattice of characteristic subgroups. Interesting results are obtained when the l l -group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of real-valued functions on a root system and determine those vector lattices in which every l l -ideal is characteristic. The automorphism group of the vector lattice of all continuous real-valued functions (almost finite real-valued functions) on a topological space (a Stone space) is shown to be a splitting extension of the polar preserving automorphisms by the ring automorphisms. This result allows us to construct characteristically simple vector lattices. We show that self-injective vector lattices exist and that an archimedean self-injective vector lattice is characteristically simple. It is proven that each l l -group can be embedded as an l l -subgroup of an algebraically simple l l -group. In addition, we prove that each representable (abelian) l l -group can be embedded as an l l -subgroup of a characteristically simple representable (abelian) l l -group.