Congruences on an inverse Bruck-Reilly monoid. I: Non-group congruences (Q1173961)

Let \(T\) be an inverse monoid with group of units \(G\) and let \(\alpha: T\to G\) be a homomorphism. The Bruck-Reilly extension \(S= \text{BR} (T, \alpha)\) is again an inverse monoid. Congruences \(\rho\) on \(S\) divide naturally into group congruences (for which \(S/\rho\) is a group) and nongroup congruences. The author gives a description of the nongroup congruences on \(\text{BR} (T, \alpha)\), the key elements in the description being ``crossover \(\alpha\)-congruences'' on \(T\). A congruence \(\rho\) is called an \(\alpha\)-congruence if (\(\forall x,y\in S\)) \(x\rho y\Rightarrow x\alpha \rho y\alpha\). It is a crossover \(\alpha\)-congruence if in addition there exists a proper ideal \(I\) of \(T\) such that \(\rho\) saturates \(I\) and (\(\forall x,y\in I\)) \(x\rho y\Leftrightarrow x\alpha\rho y\alpha\).