Ellis–Nakamura lemma
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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).
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[edit] Applications
Applying this lemma to the Stone-Cech compactification βN of the natural numbers shows that there are idempotent elements (other than 0) 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).
[edit] Proof
- By compactness, 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.
[edit] References
- Argyros, Spiros; Todorcevic, Stevo (2005), Ramsey methods in analysis, Birkhauser, p. 212, ISBN 3764372648
- Ellis, Robert (1958), "Distal transformation groups.", Pacific J. Math. 8: 401--405, MR0101283, http://projecteuclid.org/euclid.pjm/1103039885
- Numakura, Katsui (1952), "On bicompact semigroups.", Math. J. Okayama University. 1: 99--108, MR0048467
