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
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 that at least one of be a bijection onto