Could you, please, help me understand the reason why we are interested in trace of a congruence and the kernel's congruence when we're talking about the congruences on inverse semigroup. Also, I have read that congruences of an inverse semigroup are classified according to their traces? could you explain that?
Note that trace $(p)= \{ (e,f): e,f \in E(S)\}$, where $E(S)$ is the set of idempotents.