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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.

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    @HassanMuhammad: An answer like what? I wasn't trying to ask you to accept any of my own answers in particular. Actually it was more that I had noticed a couple of good answers by ymar which you hadn't accepted yet. But now that you have mentioned that I confused you, I would like to know where and how. It would be best if you comment on the relevant question and/or answer. (I won't be replying until tomorrow though, because it's late here.)2012-03-26

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