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 $(S^{Lmc}-S)*S^{Lmc}$ is dense in $S^{Lmc}-S$.