Kronecker semigroups

Let S be a set with a distinguished element 0. Define a product \star on S by x \star y = x if x = y and 0 otherwise. Show that (S, \star) is a semigroup (called a Kronecker semigroup) and find all of its idempotents and ideals.

First, we note that \star is commutative and that 0 is a (two sided) zero, since if x \neq 0 we have x \star 0 = 0 and if x = 0 then x \star 0 = 0.

To see that \star 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.

Associativity of the Kronecker product on a set

So \star is associative, and (S, \star) is a semigroup.

Note that for all x \in S, we have x \star x = x by definition. So every element is idempotent.

Since S is commutative, the left, right, and two-sided ideals coincide.

Let A \subseteq S be a subset containing 0. If a \in A and s \in S, then either s \star a = a \in A or s \star a = 0 \in A. So A is an ideal.

Conversely, suppose A \subseteq S is an ideal. If A = S, then certainly 0 \in A. If A \neq S, then there exist elements a \in A and s \in S \setminus A, and we have 0 = s \star a \in A.

Hence the ideals of S are precisely the subsets which contain 0.

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