Filters and the Weakly Almost Periodic Compactification of a Semitopological Semigroup

Let S be a semitopological semigroup. The wap? compactification of semigroup S, is a compact semitopological semigroup with certain universal properties relative to the original semigroup, and the Lmc? compactification of semigroup S is a universal semigroup compactification of S, which are denoted by Swap and SLmc respectively. In this paper, an internal construction of the wap?compactification of a semitopological semigroup is constructed as a space of z?filters. Also we obtain the cardinality of Swap and show that if Swap is the one point compactification then (SLmc ? S) ? SLmc is dense in SLmc ? S.