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

Question

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?

Answer 1

Your question makes sense. Here are two elementary properties of $0$-simple semigroups that are not satisfied by the two-element null semigroup.

1. $S^2 = S$
2. $SsS = S$ for every $s \not= 0$

It turns out that, for a semigroup with $0$ in which $\{0\}$ and $S$ are the unique ideals, the condition $S^2 = S$ is equivalent to $S^2 \not= \{0\}$. Therefore one could replace the condition $S^2 \not= \{0\}$ by $S^2 = S$ in the definition of a $0$-simple semigroup. This might look less artificial than the usual definition.

Answer 2

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.