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 s(wap) and S-Lmc 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 s(wap) and show that if s(wap) is the one point compactification then (S-Lme 5) * (S-Lmc - S) * S-Lmc is dense in S-Lmc - S.