Denote the function on by . Let , and define a binary operator on by if and if . Show that is a semigroup and exhibit all of its idempotents and principal ideals. Does have a kernel?
First, we argue that is associative. To show this, we refer to the following tree diagram.
This diagram is to be read from left to right. Labels on an edge indicate an assumption that holds in all subsequent branches. Each path from the root to a leaf corresponds to a string of equalities, and together these imply that is associative.
So is a semigroup.
Suppose is idempotent. Then , and we have . Conversely, is clearly idempotent for all . So the idempotents in are precisely elements of the form with .
Next we claim that is commutative. Indeed, if , then and if then latex = (1, b \wedge a)$ .
We claim that the principal left ideal is . Indeed, if , then either or for some . But then . Conversely, consider with , and let be 0 if and 1 otherwise. Now if , then and if then . So .
In particular, for every element , there is an ideal of not containing (for instance, .) So has no kernel.