Generalized central sets theorem for partial semigroups and vip systems

TL;DR

This paper extends the Central Sets Theorem to a broader class of algebraic structures called partial semigroups and VIP systems, providing a more general framework for combinatorial and dynamical properties.

Contribution

It introduces a generalized version of the Central Sets Theorem applicable to partial semigroups and VIP systems, expanding its scope beyond traditional semigroups.

Findings

Established a generalized Central Sets Theorem for partial semigroups.

Extended the theorem to VIP systems, broadening its applicability.

Unified algebraic and dynamical characterizations of central sets.

Abstract

The Central sets theorem was first introduced by H. Furstenberg [F] in terms of Dynamical systems. Later Hindman and Bergelson extended the theorem using Stone-$\v{C}$ech compactification $\beta$$\mathbb{N}$ of $\mathbb{N}$. In [SY] algebraic characterization of Central sets was done for semigroup and equivalence of Dynamical and Algebraic characterizations were shown. D. De, N. Hindman, and D. Strauss proved a stronger version of the Central sets theorem for semigroup. D. Phulara generalized that theorem for commutative semigroup taking a sequence of Central sets. Recently J. Podder and S. Pal established the Phulara type generalization of Central sets theorem near zero [PP]. We did this for arbitrary adequate partial semigroup and VIP systems.