Two congruence lattices of completely simple semigroups (Q1314867)

Let \(S\) be a completely simple semigroup and let \(T_ l\), \(T_ r\), \(K\) be the left trace, right trace, and kernel relations on the congruence lattice \({\mathcal C}(S)\). Let \(\rho\) be a congruence on \(S\) and let \(\rho T_ l\) \([\rho T_ r, \rho K]\) and \(\rho t_ l\) \([\rho t_ r, \rho k]\) be the greatest and least elements of the \(T_ l\) \([T_ r, K]\)-class of \(\rho\). The author describes the sublattice \(\Gamma_ \rho (S)\) of \({\mathcal C}(S)\) generated by the set \(\{\rho T_ l, \rho T_ r, \rho K, \rho t_ l, \rho t_ r, \rho k\}\). The lattice \(\Gamma_ \rho (S)\) depends on \(S\) and \(\rho\), of course. However, the ``freest'' possible case \(L\) of \(\Gamma_ \rho (S)\) is computed and each \(\Gamma_ \rho (S)\) is a homomorphic image of \(L\). The lattice \(L\) is finite and distributive, and a minimal finite presentation of \(L\) is given. The analogous problem is solved for the lattice generated by \(\{\varepsilon T_ l, \varepsilon T_ r, \varepsilon K, \omega t_ l, \omega t_ r, \omega k\}\) where \(\varepsilon\) and \(\omega\) denote the identical and universal relations, respectively.