Let , and define a binary operator on by if and otherwise. Show that is a semigroup and exhibit all of its principal ideals. Does have a kernel? Compute the orders of the elements of .
Note first that is commutative.
Given a triple of elements in , each is either less than 5 or not. This leads us to consider 8 cases. Let be at least 5, and let be less than 5.
- (Without loss of generality, .)
- .
So is associative, and is a semigroup. Since is commutative, the left, right, and two-sided ideals of coincide. Recall that the principal left ideal generated by is .
If , then .
If , then
In particular, for every element in , there is a principal ideal which does not contain . (For example, .) So the kernel of is empty.
If , then , so that has order 1. If , then has infinite order.