Give an example of a semigroup of transformations having no idempotents.
We saw in this previous exercise that every finite semigroup contains idempotent elements. So any example we find will have to be transfinite.
Given , define by . Let be the subset . We claim that is a subsemigroup of the full semigroup of transformations on ; indeed, for all and , we have , so that is closed under composition. So is a semigroup of transformations.
If is idempotent, then for all , and so . This equation has no solutions in , so that does not contain an idempotent.