Cartesian product of some combinatorially rich sets

TL;DR

This paper provides a combinatorial proof that certain large, combinatorially rich sets in semigroups retain their properties when taking finite Cartesian products, extending prior algebraic results.

Contribution

It offers a combinatorial proof for the preservation of J-sets and C-sets under finite Cartesian products, complementing algebraic approaches.

Findings

Finite Cartesian products of J-sets and C-sets are also J-sets and C-sets.

Provides a combinatorial alternative to algebraic proofs.

Extends known results to a broader combinatorial context.

Abstract

N. Hindman and D. Strauss had shown that, for discrete semigroups, the cartesian product of two central sets are central. They also proved that the product of J- sets and C-sets are also J-set and C-set and characterized when the infnite product of these sets are preserved. To prove these results they used the algebraic structure of Stone-Cech compactification of discrete semigroups. In this work we will give a combinatorial proof of the preserveness of those large sets under fnite cartesian product.