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

arxiv（2023）

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Abstract

In , using the methods of topological dynamics, H. Furstenberg introduced the notion of central set and proved the famous Central Sets Theorem. D. De, N. Hindman and D. Strauss have introduced C-set in , satisfying the strong central set theorem. Using the algebraic structure of the Stone-Čech compactification of a discrete semigroup, N. Hindman and D. Strauss proved that the Cartesian product of two C-sets is a C-set. In , S. Goswami has proved the same result using the elementary characterization of C-sets. In this article, we prove that the product of two C-sets is a C-set, using the dynamical characterization of C-sets. Recently, in , S. Goswami has proved that 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 . Here we also prove that the Cartesian product of two essential CR-sets is an essential CR-set.

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