Partial semigroup partial dynamical systems and Partial Central Sets

TL;DR

This paper extends the concept of central sets to partial semigroups using Stone-ech compactification, introducing partial dynamical systems and establishing their equivalence with algebraic characterizations.

Contribution

It introduces the notion of large sets in partial semigroups and defines Partial Semigroup Partial Dynamical Systems, linking topological and algebraic perspectives.

Findings

Characterization of central sets in partial semigroups.

Introduction of Partial Semigroup Partial Dynamical Systems.

Equivalence between topological and algebraic characterizations.

Abstract

H. Furstenberg defined Central sets in $\mathbb{N}$ by using the notions of topological dynamics, later Bergelson and Hindman characterized central sets in $\mathbb{N}$ and also in arbitrary semigroup in terms of algebra of Stone-\v{C}ech compactification of that set. We state the new notion of large sets in a partial semigroup setting and characterize the algebraic structure of the sets by using the algebra of Stone-\v{C}ech compactification. By using these notions, we introduce the \emph{Partial Semigroup Partial Dynamical System(PSPDS)} and show that topological dynamical characterization of central sets in a partial semigroup is equivalent to the usual algebraic characterization.