Some properties of lattices of radical classes of lattice-ordered groups

Abstract

Aradical class ℛ of lattice-ordered groups (l-groups) is a class closed under taking convexl-subgroups, joins of convexl-subgroups, andl-isomorphic images. Imposing various other closure conditions leads to many specific types of radical classes (e.g., torsion classes). For several of these types, the complete latticeT of radical classes of that type has been studied, and such latticesT are our object of study here. We give the characteristic properties of closed-kernel radical mappings and polar kernel radical mappings. We prove in many instances thatT isrelatively polarized, that is, for any\(U\)],ℛ eT with\(U\)] ⫅ℛ there exists a unique largest ℛ′e T such thatℛ ∩ ℛ′=\(U\)], and often we are able to explicitly identifyℛ′. By using these properties we characterize meet irreducibility in the latticeT of polar kernel radical classes.