Operators on the lattice of pseudovarieties of finite semigroups

As the first step in a study of the lattice L(F) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, Auinger, Hall and the present authors recently introduced fourteen complete congruences on L(F). Such congruences provide a framework from which to study L(F) both locally and globally. For each such congruence rho and each U is an element of L(F) the rho-class of U is an interval [U-rho, U-rho]. This provides a family of operators of the form U --> U-rho on L(F) that reveal important relationships between elements of L(F). Various aspects of these operators are considered including characterizations of U-rho, bases of pseudoidentities for U-rho instances of commutativity (U-rho)(sigma) = (U sigma)(rho), well as the semigroups generated trp certain pairs of such operators.