Coinciding the ideals in a semigroup

Question

Kindly, I am asking to light my mind by some leading hints:

Can minimal and maximal ideals in a finite non-commutative semigroup $S$ coincide?

Thanks for the time!

Answer 1

If you just consider ideals, every finite non-commutative group is a solution to your question. If you consider proper ideals, the minimal examples are the monoid $\{1, a, b\}$ with $aa = ba = a$ and $ab = bb = b$ and its dual version, the monoid $\{1, a, b\}$ with $aa = ab = a$ and $ba = bb = b$.