Quasi-uniformities on topological semigroups and bicompletion

Abstract.

We study the bicompletion of the quasi-uniformities that are induced in a natural way on a topological semigroup which has a neutral element. In particular, we show that if X is a topological semigroup, with neutral element, for which the left translations are open, then the bicompletion of the left quasi-uniformity of X can be considered a topological semigroup which contains the topological space X as a sup-dense subsemigroup. The bicompletion in the case that the left translations are not necessarily open is also discussed. In particular, both Abelian and left-cancellable topological semigroups are considered. For semigroups which are (left-)cancellable or which are locally totally bounded, theorems similar to those known from the classical theory of (para)topological groups are established.