Two congruence lattices of completely simple semigroups

Abstract

For a Rees matrix semigroupS with normalized sandwich matrix and ρ∈C(S), the congruence lattice ofS, we consider the lattice generated by {itpTl, pK, pTr, ptl, pk, ptr}. HerepT1 andptl are the upper and lower ends of the interval which makes up the ℐi-class of ρ, ℐi being the left trace relation onC(S). The remaining symbols have the analogous meaning relative to the kernel and the right trace relations. We also consider the lattice generated by {eTl, eK, eTr, ωtl, ωk, ωtr} where e and ω are the equality and the universal relations onS, respectively. In both cases, we find lattices “freest” relative to these lattices and represent them as distributive lattices with generators and relations.