John F. Berglund

Compact right topological semigroups and generalizations of almost periodicity

Preliminaries.- The structure of compact semigroups.- Subspaces of C(S) and compactifications of S.- Fixed points and left invariant means on subspaces of C(S).- Examples.

Filters and the weak almost periodic compactification of a discrete semigroup

The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.

Compact semitopological inverse Clifford Semigroups

An inverse Clifford Semigroup is a semilattice of groups. Conditions are given for constructing a compact semitopological (separately continuous multiplication) inverse Clifford semigroup on a compact Hausdorff semilattice. The conditions are necessary and sufficient for decomposing a compact inverse Clifford semigroup containing a dense subgroup and locally compact maximal groups into its semilattice of groups. A catalogue of examples is given to demonstrate the construction while exhibiting various pathologies.

A class of semigroups having almost trivial multiplications.

In this note we completely describe the structure of algebraic semigroups S with Sx=S or Sx degenerate and xS=S or xS degenerate for each x∈S. We then apply our results in characterizing the separately continuous multiplications on a topological space whose only self-maps are either surjective or have degenerate image. In particular, we find that any locally compact Hausdorff space with this property can admit only trivial separately continuous multiplications. Examples of spaces satisfying this property are certain continua discovered by H. Cook [1]. The authors are indebted to Karl H. Hofmann and James T. Rogers for conversations elucidating the topological implications of Theorem 1.