A note-question on partitions of semigroups

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 $S=F^{-1}\{x\in S:xA\bigcap A\ne \varnothing \}$ and $|F|\le n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.