Please can sombody help me with the proof of this lemma, or even a construction of the proof? I will be glad for that.
Lemma: Show that if $\rho$ is an idempotent separating congruence of an inverse semigroup $S$, then $\operatorname{tr}(\rho)=\{ (e,e)\mid e\in E_S\},$ where $E_S$ is the set of idempotents of $S$ and $\operatorname{tr}(\rho)$ is the restriction of $\rho$ to $E_S$.
Note that: A congruence $\rho$ of a semigroup S is idempotent separating if for all $e,f\in E_{S}$, $$e\rho f \implies e=f$$
And S is an inverse semigroup iff for all $x\in S$ there exist a unique $x^{-1}\in S$ such that $$x=xx^{-1}x \text { and } x^{-1}=x^{-1}xx^{-1}.$$
Thanks.