Quasiidentities implying modularity and related conditions in lattices of overcommutative semigroup varieties

A semigroup variety is called overcommutative if it contains the variety of all commutative semigroups. For an arbitrary quasivariety L of modular lattices, we give a complete description of overcommutative semigroup varieties whose lattice of overcommutative subvarieties belongs to L. Analogous results are obtained for (weak) upper semimodularity and (weak) lower semimodularity. We clarify an interdependence between some of these properties of an overcommutative variety and permutability, 1,5-permutability or weak permutability of subcommutative fully invariant congruences on its free objects. No proofs are included.