Filters and semigroup compactification properties

Stone-Čech compactifications derived from a discrete semigroup S can be considered as the spectrum of the algebra ℬ(S) or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Čech compactifications derived from a discrete semigroup. It seems that filters can play a role in the study of general semigroup compactifications too. In the present paper, first we review the characterizations of semigroup compactifications in terms of filters and then extend some of the results in Papazyan (Semigroup Forum 41:329–338, 1990) concerning the Stone-Čech compactification to a semigroup compactification associated with a Hausdorff semitopological semigroup.