Find all of the left- and right ideals of a zero semigroup and of a left zero semigroup.
Let be a zero semigroup. That is, there exists such that for all . In particular, is commutative, so that left and right ideals coincide.
Let be a subset containing 0. (In particular, latex a \in A$ and , we have . So , and thus is an ideal.
Now let be a left zero semigroup. That is, for all , we have .
Suppose is a left ideal. Since , there exists an element . Then for all , we have . So in fact , and is the only left ideal. (I.e., S is left simple.)
Now let be a nonempty subset with . For all , we have . So , and thus is a right ideal. That is, every nonempty subset of is a right ideal.