Ellis–Nakamura lemma
Encyclopedia
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Ellis–Numakura lemma states that if S is a non-empty semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
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).
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).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.
External links
- T. Tao lecture-5