A note-question on partitions of semigroups

Given a semigroup $S$ and an $n$-partition $\mathcal{P}$ of $S$, $n\in $\mathbb{N}$$,dothereexist$A\in $\mathcal{P}$$andasubset$F$of$S$ such that $S=F^{-1}\{x\in S:xA\bigcap A\mathrm{\ne }\mathrm{\varnothing }\}$ and $\mathrm{\mid }F\mathrm{\mid }\le n$? We give an affirmative answer provided that either $S$ is finite or $n=2$.