Let $S$ be a semigroup and $X$ be a non-empty subset of it. Note that for a right zero semigroup $S$ and every $x\in S$ we have $xS=S$. So for a proper non-empty subset $X\subset S$ we have $S=\cup_{x\in X}xS=\cup_{y\in S\setminus X}yS$. On the other hand, for every $s\in S$, $sS$ is not contained in $X$ or $S\setminus X$.
Is it known that when it does not happen? I mean under which condition on $S$, every principal right ideal of $S$ does not intersect both $X$ and $S\setminus X$?
Thanks in advance