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Ellis Lemma on idempotent elements asserts that:

Lemma (Ellis). Every compact semigroup has an idempotent.

The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.

Let $S$ be a compact semigroup. Pick by Zorn’s Lemma a minimal compact subsemigroup $R \subseteq S$ and an arbitrary $s \in R$. Then $Rs$ is also a compact subsemigroup and $Rs \subseteq R$. Hence $Rs = R$. Let $P = \{x \in R : xs = s\}$. Then $P \neq \emptyset$, since $s \in Rs$. Note that $P$ is also a compact subsemigroup of $S$. Hence $P = R$ and therefore $s^2 = s$.

It appears that Zorn's Lemma is required to obtain a minimal compact subsemigroup $R \subseteq S$. I do not understand why, as it appears to me that we can simply let: $$ R := \bigcap \{Q \subseteq S : Q \text{ is a compact subsemigroup of $S$}\} $$ What is the error in reasoning here?

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    $\begingroup$ What if $R=\emptyset$? Note that in Zorn Lemma approach, the nested intersection of non-empty compact sets is non-empty, but without the nestedness you may get the empty set. $\endgroup$
    – Nick S
    Feb 28, 2022 at 5:42
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    $\begingroup$ A simple counterexample would be the discrete semigroup $\{a,b,ab\}$ where $a,b$ are commuting idempotents. Then $\{a\}$ and $\{b\}$ are the minimal subsemigroups but the intersection of the two is empty. $\endgroup$
    – François G. Dorais
    Feb 28, 2022 at 6:35
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    $\begingroup$ Is it known whether the axiom of choice is necessary for this result? $\endgroup$
    – Yair Hayut
    Feb 28, 2022 at 9:08
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    $\begingroup$ @YairHayut according to a comment of this answer, the Ultrafilter Lemma is sufficient for Ellis' Lemma. $\endgroup$
    – Clement Yung
    Feb 28, 2022 at 9:44
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    $\begingroup$ So should Todorcevic have written “Pick by Zorn’s lemma a minimal non-empty compact subsemigroup”, or is he working with the convention that all semigroups are non-empty? $\endgroup$
    – Peter LeFanu Lumsdaine
    Feb 28, 2022 at 14:06

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The issue is that $R$ could be empty. When you apply Zorn's lemma, a nested intersection of non-empty compact sets is non-empty, guaranteeing minimal non-empty elements.

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