If I have a monoid , and an element , then the following equivalence holds:

is a unit iff and are bijections onto .

Proof.Suppose is a unit. Let Then we have

1.

2. if then so

Conversely, if and are bijections onto then there exists such that and such that Thus has a left inverse and a right inverse, which is known to imply that is a unit.

Now let's say is a semigroup without an identity element. Can there be an element such that and are bijections? Is it possible when is finite? (Then it's equivalent to asking whether a semigroup without ideantity can have a cancellable element.) Is it possible when is infinite? Is it possible when we only demand thatat least oneof be a bijection onto