Dynamical characterization of central sets in adequate partial semigroups

Using the methods from topological dynamics, H. Furstenberg introduced the notions of Central sets and proved the famous Central Sets Theorem which is the simultaneous extension of the van der Waerden and Hindman Theorem. Later N. Hindman and V. Bergelson found an equivalent formulation of Central sets in the set of natural numbers in terms of the algebra of the Stone-\v{C}ech compactification of discrete semigroups. The general case was proved by H. Shi and H. Yang. Using the notions of ultrafilters, J. McLeod introduced the notions of Central sets for commutative adequate partial semigroups, however for noncommutative cases, Central sets can be defined similarly. In this article, introducing the notions of topological dynamics for partial semigroup actions, we find an equivalent dynamical characterization of central sets in partial semigroups\footnote{Recently in \cite{GTG}, authors attempted to do the same but in a different approach.}. Throughout our article, we follow the approach of N. Hindman and D. Strauss [N. Hindman, and D. Strauss: Algebra in the Stone-\v Cech compactification: theory and applications, second edition, de Gruyter, Berlin, 2012.].