Toko, Wilson Bombe2013-04-042013-04-042013-04-04http://hdl.handle.net/10539/12624A thesis submitted in ful llment of the requirements for the degree of Doctor of Philosophy in Mathematics School of Mathematics University of the Witwatersrand Johannesburg October, 2012Let S be an in nite discrete semigroup and S the Stone- Cech compacti cation of S. The operation of S naturally extends to S and makes S a compact right topological semigroup with S contained in the topological center of S. The aim of this thesis is to present the following new results. 1. If S embeddable in a group, then S contains 22jSj pairwise incomparable semiprincipal closed two-sided ideals. 2. Let S be an in nite cancellative semigroup of cardinality and U(S) the set of uniform ultra lters on S. If > !, then there is a closed left ideal decomposition of U(S) such that the corresponding quotient space is homeomorphic to U( ). If = !, then for any connected compact metric space X, there is a closed left ideal decomposition of U(S) with the quotient space homeomorphic to X.enStone-Cech compactification.Thesis