Combinatorially rich sets in partial semigroups
Arpita Ghosh, Neil Hindman
TL;DR
The paper extends the theory of combinatorially rich sets from semigroups to adequate partial semigroups by defining $J$-sets and $CR$-sets within the compact right topological extension $\delta S$, and relating them to classical largeness notions via ultrafilter and finite-pattern characterizations. It establishes that $CR$-sets imply $J$-sets, but $PS$-sets need not be $CR$, and it provides concrete sufficient conditions (property $(\dagger)$ and $(\ddagger)$) that guarantee $CR$-ness, including countable examples that illuminate limitations of the theory. The paper further shows that CR is preserved under Cartesian products and develops the necessary combinatorial machinery (e.g., $\Theta$-sets and IP$_r^*$-arguments) to transfer CR-ness to product structures, while also presenting simplifications of the CR/J descriptions in the countable and commutative cases. Together, these results deepen the structural understanding of combinatorial richness in partial semigroups and lay groundwork for applications to product constructions and countable analogues.
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.