Relative size of subsets of a semigroup

TL;DR

This paper introduces and characterizes relative large, thick, and prethick subsets within semigroups using ultrafilters, exploring their properties and partition behaviors.

Contribution

It develops a framework for relative subset sizes in semigroups with ultrafilter characterizations and analyzes partition properties of these subsets.

Findings

Ultrafilter characterizations of relative large, thick, and prethick subsets

Partition properties of subsets in semigroups

Introduction of relative subset concepts with respect to filters

Abstract

Given a semigroup $S$, we introduce relative (with respect to a filter $\tau$ on $S$) versions of large, thick and prethick subsets of $S$, give the ultrafilter characterizations of these subsets and explain how large could be some cell in a finite partition of a subset $A\in\tau$.