# Ellis–Numakura lemma

In mathematics, the **Ellis–Numakura lemma** states that if *S* is a non-empty semigroup with a topology such that *S* is compact and the product is semi-continuous, then *S* has an idempotent element *p*, (that is, with *pp* = *p*). The lemma is named after Robert Ellis and Katsui Numakura.

## Applications

Applying this lemma to the Stone–Čech compactification *βN* of the natural numbers shows that there are idempotent elements in *βN*. The product on *βN* is not continuous, but is only semi-continuous (right or left, depending on the preferred construction, but never both).

## Proof

- By compactness and Zorn's Lemma, there is a minimal non-empty compact sub semigroup of
*S*, so replacing*S*by this sub semi group we can assume*S*is minimal. - Choose
*p*in*S*. The set*Sp*is a non-empty compact subsemigroup, so by minimality it is*S*and in particular contains*p*, so the set of elements*q*with*qp*=*p*is non-empty. - The set of all elements
*q*with*qp*=*p*is a compact semigroup, and is nonempty by the previous step, so by minimality it is the whole of*S*and therefore contains*p*. So*pp*=*p*.

## References

- Argyros, Spiros; Todorcevic, Stevo (2005),
*Ramsey methods in analysis*, Birkhauser, p. 212, ISBN 3-7643-7264-8 - Ellis, Robert (1958), "Distal transformation groups.",
*Pacific J. Math.*,**8**: 401–405, doi:10.2140/pjm.1958.8.401, MR 0101283 - Numakura, Katsui (1952), "On bicompact semigroups.",
*Math. J. Okayama University.*,**1**: 99–108, MR 0048467

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