Topological groups and semigroups form the basic building blocks of many different
areas of mathematics. The aim of this work is to determine if a general cancellative
semigroup can be given a left shift invariant topology. The theory behind a class
of topologies that can be created on a given group or semigroup is discussed.
The t-sequence proof of the Markov theorem is presented and this serves as a
catalyst for further inquiry. The algebra of the Stone-Cech compactification of a
discrete semigroup is utilized to prove the existence of certain ultrafilters, with
which topologies can be constructed