Ellis–Nakamura lemma

In mathematics, the Ellis–Nakamura lemma states that if "S" is a non-empty semigroup with a topology such that"S" is compact and the product is left continuous, then "S" has an idempotent element "p", (that is, with "pp"="p").

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 left continuous.

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".

References

*citation|first=Spiros |last=Argyros
first2=Stevo|last2= Todorcevic
year= 2005
publisher=Birkhauser
ISBN =3764372648|page=212
*citation|id=MR|0101283
last=Ellis|first= Robert
title=Distal transformation groups.
journal=Pacific J. Math.|volume= 8|year= 1958 |pages=401--405
url=http://projecteuclid.org/euclid.pjm/1103039885

External links

*T. Tao [http://terrytao.wordpress.com/2008/01/21/254a-lecture-5-other-topological-recurrence-results/ lecture-5]