My question is probably a little bit silly, but still.. The definition of $0$-simple semigroup states, that a semigroup $S$ with zero is called $0$-simple, if $\{0\}$ and $S$ are it's only ideals and $S^2\neq\{0\}$.
The second requirement excludes the two-element null semigroup from the class of $0$-simple semigroups and I am wondering what is the particual reason to do that? Could it be because there are some properties and statements that hold for every $0$-simple semigroup, but do not hold for the two element null semigroup?