Weakly uniform completely O-simple semigroups of matrix type (Q1057371)

A weakly uniform semigroup is a locally compact semigroup which satisfies a certain translation condition. The class of uniform semigroups forms a large subclass of the class of locally compact semigroups and includes discrete and compact semigroups. Let \(H^ 0\) be a topological group with isolated zero, C and D be topological Hausdorff spaces, \(f: C\times D\to H^ 0\) be a continuous function such that for every \(c\in C\) and for every \(d\in D\), the sets f(c,D) and f(C,d) are non-zero. Let \((H^ 0,C,D,f)\) denote a topological completely O-simple semigroup of matrix type with isolated zero. The author proves the following: A locally compact semigroup \((H^ 0,C,D,f)\) is weakly uniform if and only if its sandwich-function f is equicontinuous. This result together with Rees theorem for weakly uniform semigroups yields the following structure theorem: A weakly uniform completely O-simple semigroup is topologically isomorphic to some locally compact completely O-simple semigroup of matrix type with isolated zero and the equicontinuous sandwich-function. Conversely, any locally compact completely O-simple semigroup of matrix type with isolated zero and the equicontinuous sandwich-function is weakly uniform.