Let be a set. A transformation of is a function from to .

If and are transformations on , then their product is defined (writing functions on the right) by .

With this definition, the set of all transformations on becomes a semigroup, the full semigroupf of transformations on , denoted .

More generally, a semigroup of transformations is any subsemigroup of a full set of transformations.

When is finite, say , then the transformation which maps to (with , of course) is often written:

With this notation it is quite easy to calculate products. For example, if , then

When is infinite, say , then this notation is still useful for illustration in cases where the transformation pattern is apparent. For example, if is given by , we can write