Let be a set with a distinguished element . Define a product on by if and otherwise. Show that is a semigroup (called a Kronecker semigroup) and find all of its idempotents and ideals.
First, we note that is commutative and that is a (two sided) zero, since if we have and if then .
To see that is associative, we appeal to the following diagram. We have condensed four strings of equalities into a tree, to be read from top to bottom. A label on an equals sign indicates an assumption that holds in the arguments below it.
So is associative, and is a semigroup.
Note that for all , we have by definition. So every element is idempotent.
Since is commutative, the left, right, and two-sided ideals coincide.
Let be a subset containing 0. If and , then either or . So is an ideal.
Conversely, suppose is an ideal. If , then certainly . If , then there exist elements and , and we have .
Hence the ideals of are precisely the subsets which contain 0.