I'm studying Howie's Fundamentals of Semigroup Theory and during the characterization of congruences on inverse semigroups the following property is used a few times:
Let $S$ be an inverse semigroup. Let $\rho$ be a congruence on $S$. Then $(a,b) \in \rho$ $\Rightarrow$ $(a^{-1},b^{-1})\in \rho$.
I tried a few things, but I can't seem to prove this. Any help would be greatly appreciated.