On Transformation Semigroups
by
Hideomi Ikeshoji
Coauthors: Ta Sun Wu

Let (X, T) be a transformation semigroup with compact Hausdorff phase space X, and let E(X) be the enveloping semigroup, i.e., the closure of T (identified as a subset of X^X) in the product space X^X. (X, T) is called point-transitive if

D(X) = { a in X | aT   = X } =/= \emptyset,

where [`aT] is the orbit-closure of a. If (X, T) is point-transitive and a in D(X), there exists a closed right ideal M in E(X) with the following property: (i) aM=X, and (ii) M is minimal with respect to the condition (i). We call such an M an a-minimal ideal and we extend Ellis's results on the structure of minimal right ideals to those of a-mimimal ideals.

As a byproduct we also show the following proposition.

Proposition Let I be a compact Hausdorff space with a semigroup structure. Assume that left multiplication Lp : I --> I is a continuous surjection for each p in I. Then each Lp : I --> I is bijective and so a homeomorphism.

Date received: July 2, 1997

Copyright © 1997 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 # caao-76.