Generalized central sets theorem for partial semigroups and vip systems
Anik Pramanick, MD Mursalim Saikh
TL;DR
This work addresses extending the Central Sets Theorem to the broader framework of adequate partial semigroups and VIP systems. It develops an algebraic–dynamical approach via the Stone–Čech compactification, employing $δS$, $J(S)$, and idempotents in $J(S)$ or $K(δS)$ to obtain structured finite-sum configurations inside central sets, and proves a Phulara-type theorem for adequate partial semigroups along with a VIP-system version. It then furnishes applications to combinatorial configurations, including first-entry matrix results and VIP-free formulations, demonstrating that central-set phenomena extend to new algebraic contexts. Overall, the results broaden the reach of central-set theory to partial semigroups and VIP systems, enabling multi-parameter constructions and richer combinatorial configurations in these settings.
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-$Č$ech compactification $β$$\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.