Complete congruences on lattices of varieties and of pseudovarieties

Three methods for the construction of all complete congruences on the lattice Lv(V) of subvarieties of a variety V are introduced. It is shown that there exists an order preserving embedding of the lattice of complete congruences on the lattice Lp(P) of all subpseudovarieties of a given pseudovariety P into the direct product of the lattices of complete congruences on lattices of subvarieties of varieties generated by members of P; thus there are methods for constructing all complete congruences on Lp(P). By way of application, 2אo complete congruences and complete endomorphisms are constructed on any lattice Lv(V), where V is a certain epigroup variety which includes all bands; there is an analogous application for the lattice of all pseudovarieties of semigroups.