Kernel and trace operators for extensions of Brandt semigroups. (Q2460062)

Let \(S\) be a regular semigroup. Let \(\rho\) be a congruence on \(S\), then let \(\rho_K\) and \(\rho^K\) denote the least and the greatest congruence on S having the same kernel as \(\rho\), respectively. The author puts the following question: what are the exact conditions on \(S\) in order that the map \(\rho\mapsto\rho_K\) (respectively \(\rho\mapsto\rho^K\) ) be a \(\vee\)- or \(\wedge\)-homomorphism of the congruence lattice \(C(S)\). The same question is put in the case when traces of congruences are considered instead of their kernels. With respect to this problem, semigroups \(S\) are investigated that are an ideal extension of a Brandt semigroup \(S_0\) by a Brandt semigroup \(S_1\). For this case, necessary and sufficient conditions on \(S\) are established in order that one or all of the mentioned maps be \(\vee\)- or \(\wedge\)-homomorphisms. The conditions are expressed directly in terms of the construction of an extension of \(S_0\) and \(S_1\).