Let be a semigroup. Show that is left simple if and only if for all , and is simple if and only if for all .
Recall that a semigroup is called left simple if the only left ideal is itself.
Suppose is left simple. Now , so that is a left ideal. Hence . Conversely, suppose for every . Let be a left ideal. Now , so we have for some . Then , and thus . So is left simple.
Similarly, we can show that is right simple if and only if for all .
Now suppose is simple. (That is, it has no nontrivial ideals.) Let . Now and , so that is an ideal. Hence . Now suppose for all . If is a two-sided ideal, then we have for some . Now , so that . Hence is simple.