Lattices of dominions in quasivarieties of Abelian groups

Let M be any quasivariety of Abelian groups, domGM (H) be the dominion of a subgroup H of a group G in M, and Lq(M) be the lattice of subquasivarieties of M. It is proved that domG M (H) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = {domGN(H) | N ∈ Lq(M)} forms a lattice under set-theoretic inclusion and the map φ : Lq(M) → L(G,H,M) such that φ(N) = dom GN (H) for any quasivariety N ∈ Lq(M)is an antihomomorphism of the lattice L q (M) onto the lattice L(G, H, M). © 2005 Springer Science+Business Media, Inc.