Cartesian product of combinatorially rich sets -- algebraic, elementary and dynamical approaches

TL;DR

This paper proves that the Cartesian product of two $C$-sets or $CR$-sets retains their combinatorial richness using algebraic, elementary, and dynamical methods, extending previous results in topological dynamics and semigroup theory.

Contribution

It introduces a dynamical approach to show that the product of two $C$-sets or $CR$-sets remains within the same class, providing new proofs and extending known results.

Findings

Product of two $C$-sets is a $C$-set.

Product of two $CR$-sets is a $CR$-set.

Extension to essential $CR$-sets.

Abstract

Using the methods of topological dynamics, H. Furstenberg introduced the notion of a central set and proved the famous Central Sets Theorem. D. De, N. Hindman, and D. Strauss introduced $C$-set, satisfying the strong central set theorem. Using the algebraic structure of the Stone-\v{C}ech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two $C$-sets is a $C$-set. S. Goswami has proved the same result using the elementary characterization of $C$-sets. In this article, we will prove that the product of two $C$-sets is a $C$-set, using the dynamical characterization of $C$-sets. Recently, S. Goswami has proved that the Cartesian product of two $CR$-sets is a $CR$-set, which was a question posed by N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni in [Semigroup Forum 107 (2023)]. Here we also prove that the Cartesian product of two essential $CR$-sets is an essential $CR$-set.