A note-question on partitions of semigroups

TL;DR

This paper investigates a specific partition problem in semigroup theory, proving that certain conditions guarantee the existence of particular subsets and finite sets with intersection properties.

Contribution

It provides an affirmative answer to the partition problem for semigroups when the semigroup is finite or when the partition has two parts.

Findings

The problem has an affirmative solution for finite semigroups.

The problem also has an affirmative solution when the partition has two parts.

The results extend understanding of partitions in semigroup structures.

Abstract

Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in \mathbb{N}$, do there exist $A\in \mathcal{P}$ and a subset $F$ of $S$ such that $S=F ^{-1} \{x \in S: x A \bigcap A\neq\emptyset\}$ and $|F |\leq n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.