Dedicated to Professor C. Tibiletti Marchionna on her 70th birthday In this paper a characterization of the regular co-semigroups whose congruence lattice is modular is given. The characterization obtained for such semigroups generalizes the one given by Munn for bisimple co-semigroups and completes a result of Baird dealing with the modularity of the sublattice of the congruence lattice of a simple regular co-semigroup consisting of congruences which are either idempotent separating or group congruences.

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In this paper a characterization of the regular to-semigroups whose congruence lattice is semimodular is given. The characterization obtained for such semigroups generalizes the one given by Scheiblich for bisimple ~o-semigroups. Notice that we use the definition of semimodularity which other authors call double covering property.~Xm, n ~ ~/m~m+l 9 9 9 ~n--I and, for all ne N, let ~n,n denote the identity automorphism Of Gn(modd).

In this paper conditions of M-symmetry, strong, semimodularity and θ-modularity for the congruence lattice L (S) of a regular ω-semigroup S are studied. They are proved to be equivalent to modularity. Moreover it is proved that the kernel relation is a congruence on L(S) if and only if L(S) is modular, generalizing an analogous result stated by Petrich for bisimple ω-semigroups.

Piochi in [10] gives a description of the least commutative congruence γ of an inverse semigroup in terms of congruence pairs and generalizes to inverse semigroups the notion of solvability. The object of this paper is to give an explicit construction of λ for simple regular ω-semigroups exploiting the work of Baird on congruences on such semigroups. Moreover the connection between the solvability classes of simple regular ω-semigroups and those of their subgroups is studied.