Kernel and Trace Operators for Extensions of Brandt Semigroups

Abstract

Let S be an (ideal) extension of a Brandt semigroup S0 by a Brandt semigroup S1 and let $\mathcal{C}(S)$ denote the congruence lattice of S. For $\rho \in \mathcal{C}(S),$ denote by $\rho _K$ and $\rho ^K$ the least and the greatest congruences on S with the same kernel as $\rho ,$ respectively, and let $\rho _T$ and $\rho ^T$ have the analogous meaning relative to trace. We establish necessary and sufficient conditions on S in order that one or more of the operators $\rho \longrightarrow \rho _K, \quad \rho \longrightarrow \rho ^K, \quad \rho \longrightarrow \rho _T, \quad \rho \longrightarrow \rho^T$ be $\vee$ - or $\wedge$ -homomorphisms on $\mathcal{C}(S).$ The conditions are expressed directly in terms of a construction of an extension of S0 and S1 and the proofs make use of a construction of congruences on S expressed by means of congruences on S0 and S1.