Ideal Towards Exploring Piecewise Syndetic Sets.

The Stone-Čech compactification is the largest/maximal compactification of a Tychonoff space. For a discrete space, the Stone-Čech compactification may be revealed as the set of ultrafilters on that space identifying elements of the discrete space with principal ultrafilters. Further, any semigroup operation on a discrete space can be elegantly extended to the space of ultrafilters which makes it a compact right topological semigroup possessing the discrete semigroup in its topological center. This phenomenon provides an excellent synthesis of algebra and topology that has many fascinating applications in Ramsey theory as well as in topological dynamics involving various notions of sets such as IP sets, piecewise syndetic sets, central sets etc. of a discrete semigroup.

This talk will cover an array of visualizations of ultrafilters belonging to the smallest two-sided ideal of the Stone-Čech compactification of a discrete semigroup in terms of variants of central sets. Indeed, each arrangement of the array will bloom a representation of piecewise syndetic sets. Moreover, a characterization of piecewise syndetic sets is exhibited in the framework of topological dynamical system, highlighting the utility of ultrafilter attributes. As there is a one-to-one corresponding between non-empty closed subsets of Stone-Čech compactification of a discrete semigroup and filters on that semigroup, the discussion is concluded by exploring the aforesaid results in the ambiance of filters.