Why is two-element null semigroup excluded from the $0$-simple semigroup definition? 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?

We can consider $S$ as $0$-simple, but then we must add in the Sushkevich-Rees theorem (and other theorems) the words: "except such $S$ that $S^2=0$". It is the same reason because of which the axiom "$0\ne 1$" is used in the theory of fields.
