Let S be finite semigroup and let the set $K^{e}=${$a\in S: a^{p}=e ~{~\rm for~some} \quad p>0$}~ be a subsemigroup of $S$ corresponding to the idempotent $e\in S$. Moreover, let $S$ be the disjoint union of all $K^{e_i}$ for $e_i$ in $E(S)$.
Generally, the condition that $a\in K^{e_i}\setminus (K^{e_i})^{2}$ does not necessarily hold true for $a\in S\setminus S^{2}$? Are there any prerequisites that must be met to verify this implication?