Several combinatorial results generalized from one large subset of semigroups to infinitely many

TL;DR

This paper extends a combinatorial generalization from countably many to uncountably many C-sets in semigroups, broadening the scope of previous results and exploring new theoretical territory.

Contribution

It generalizes Phulara's idea to uncountably many C-sets, significantly expanding the applicability of combinatorial results in semigroup theory.

Findings

Generalization to uncountably many C-sets

Application of the idea to multiple combinatorial results

Further theoretical investigation into semigroup structures

Abstract

In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subset of the given semigroup to countably many. In this paper, we apply this idea to other combinatorial results to obtain corresponding generalizations, and do some further investigation. Moreover, we find that Phulara's generalization can be generalized further that can deal with uncountably many C-sets.