Lemma 47.4.15 (Idempotent Refinement).label Let $S$ be a semigroup and $f \in \beta S$ be idempotent, then there exists a minimal idempotent $e \in \beta S$ such that $ef = fe = e$.
Proof. By Proposition 5.16.3, $(\beta S)f$ is a closed left ideal in $S$. By Ellis’ Theorem, $(\beta S)f$ contains a minimal left ideal $L$, and a minimal idempotent $r \in L$. As $L \subset (\beta S)f$, there exists $t \in \beta S$ such that $r = tf$.
Let $e = fr \in L$, then $e^{2} = frfr = fr^{2} = fr = e$. Moreover,
\[ef = frf = ftf^{2} = ftf = fr = e = fr = ffr = fe\]
$\square$
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