Lattices of radicals of omega;-groups

Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández- Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps double-struck I sign, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps double-struck B sign in any universal class of ω-groups (of the same type). We show that double-struck I sign is a complete and modular lattice which contains atoms. In general, double-struck I sign is not atomic. double-struck I sign contains ℙ as a complete and atomic sublattice, whereas ℍ and double-struck B sign are not sublattices of double-struck I sign. In its own right, ℍ is a complete and atomic lattice and double-struck B sign is a complete lattice. We identify subclasses of double-struck I sign, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.