A new and simpler noncommutative central sets theorem

TL;DR

This paper presents a simpler yet equally powerful version of the noncommutative central sets theorem, improving the clarity and accessibility of the existing complex formulation.

Contribution

It introduces a more straightforward formulation of the noncommutative central sets theorem, enhancing understanding and potential applications.

Findings

Simplified the noncommutative central sets theorem

Maintained the theorem's strength and applicability

Provided clearer proof and formulation

Abstract

Using dynamics, Furstenberg defined the concept of a central subset of positive integers and proved several powerful combinatorial properties of central sets. Later using the algebraic structure of the Stone-\v{C}ech compactification, Bergelson and Hindman, with the assistance of B. Weiss, generalized the notion of a central set to any semigroup and extended the most important combinatorial property of central sets to the central sets theorem. Currently the most powerful formulation of the central sets theorem is due to De, Hindman, and Strauss in [3, Corollary 3.10]. However their formulation of the central sets theorem for noncommutative semigroups is, compared to their formulation for commutative semigroups, complicated. In this paper I prove a simpler (but still equally strong) version of the noncommutative central sets theorem in Corollary 3.3.