Combinatorially rich sets in partial semigroups

TL;DR

This paper explores the concept of combinatorially rich (CR) sets within partial semigroups, examining their properties, relationships to other size notions, and highlighting unexpected differences among these concepts.

Contribution

It introduces and analyzes CR sets in adequate partial semigroups, clarifying their relation to J sets and other size notions, revealing surprising distinctions.

Findings

CR sets are distinct from J sets in certain contexts.

CR sets exhibit unique combinatorial properties in partial semigroups.

The study uncovers unexpected differences among various size notions in partial semigroups.

Abstract

There are several notions of size for semigroups that have natural analogues for partial semigroups. Among these are thick, syndetic, central, piecewise syndetic, IP, J, and the more recently introduced notion of combinatorially rich, abbreviated CR. We investigate the notion of CR set for adequate partial semigroups, its relation to other notions, especially J sets, and some surprising differences among them.