Given a finite semigroup $S$, let $I$ be a $\mathcal{J}$-minimal ideal, where $\mathcal{J}$ is equivalence relation brought about by the Green relation $\leq_{\mathcal{J}}$, where given $a,b \in S$, $a \mathcal{J} b$ if and only if $a \leq_{\mathcal{J}} b$ and $b \leq_{\mathcal{J}} a$.
I would like to ask, does $I$ have an idempotent ? ... and since $I \subseteq I'$ for any other ideal $I' \subseteq S$, does each ideal have an idempotent ?
Every finite semigroup contains an idempotent, and ideals are subsemigroups.
To see the first statement note that it's enough to find $a\in S$ with $a^n=a$ for some $n\in\Bbb N$: if $n=2$ we are done, otherwise $a'=a^{n-1}$ is idempotent.
Now consider any $a\in S$ and look at the sequence $a,a^2,a^4,\ldots$. Since $S$ is finite there must be a repetition, so for some $m,l$ we have $a^{2^m}=a^{2^l}$. Assume wlog $m>l$ and consider $a'=a^{2^{l}}$ and $n=2^{m-l}$, so that $a'^n=a'$ and we are done.
The "using a nuke for a mosquito" approach to the first statement is to note that every finite semigroup is a compact topological semigroup in the discrete topology all of whose ideals are closed, and then use Ellis theorem that a compact semitopological semigroup has an idempotent.
