The existence of disjoint smallest ideals in the left continuous and right continuous structures on \Beta S.
by
Shea D. Burns
Howard University

Given a discrete semigroup (S, ·), the operation on S can be extended to the Stone-Cech compactification \betaS so that (\betaS, ·) is a compact right topological semigroup with S contained in its topological center. (That is, for each p in \betaS, the function \rho_p:\betaS --> \betaS, defined by \rho_p(x)=x·p, is continuous, and for each s in S, the function \lambda_s:\betaS --> \betaS, defined by \lambda_s(x)=s·x, is continuous.) The operation can also be extended making \betaS a left topological semigroup. (We shall denote this extension by \diamond.) If S is commutative, there is no essential difference. That is, for any p and q in \betaS, p·q=q\diamondp. In particular, if S is commutative, the smallest ideals, K(\betaS, ·) and K(\betaS, \diamond), are identical.

Previous research has shown that in any event these ideals must be close. That is, K(\betaS, ·) \cap cl K(\betaS, \diamond) =/= \emptyset and K(\betaS, \diamond) \cap cl K(\betaS, ·) =/= \emptyset. We show that they can be disjoint. Specifically, if S is the free semigroup on two generators, then K(\betaS, ·) \cap K(\betaS, \diamond)=\emptyset.

Date received: June 2, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacl-38.