2 CONGRUENCE LATTICES OF COMPLETELY SIMPLE SEMIGROUPS

For a Rees matrix semigroup S with normalized sandwich matrix and rho is an element of C(S), the congruence lattice of S, we consider the la ttice generated by {rho T-l, rho K, rho T-r, rho t(t), rho k, rho t(r) ). Here rho T-t and rho t(t) are the upper and lower ends of the inter val which makes up the F-l-class of rho, F-l, being the left trace rel ation on C(S). The remaining symbols have the analogous meaning relati ve to the kernel and the right trace relations. We also consider the l attice generated by {epsilon T-l epsilon K, epsilon T-r, omega t(l), o mega(k), omega t(r)} where epsilon and omega are the equality and the universal relations on S, respectively. In both cases, we find lattice s ''freest'' relative to these lattices and represent them as distribu tive lattices with generators and relations.