Some new results about the smallest ideal of βS

摘要

We provide what we believe is the first nontrivial algebraic description of the smallest ideal of the Stone-ech compactification of a discrete semigroup. Specifically, given sets A and B, let S=R(A,B)=A x B with the discrete topology and the rectangular semigroup operation (a,b)(c,d)=(a,d). Then the smallest ideal of βS is isomorphic (but not homeomorphic) to R(βA,βB). We also determine exactly the topological center of βS. The minimal left ideals of βS all have isolated points. We derive several results about βS that must hold for any semigroup S if the minimal left ideals have isolated points.