Invertible element

Definition
An element $$a$$ in a set $$S$$ with binary operation $$*$$ and neutral element $$e$$ is termed:


 * left invertible if it has a left inverse, viz there exists an element $$b$$ such that $$b * a = e$$
 * right invertible it it has a right inverse, viz there exists an element $$b$$ such that $$a * b = e$$
 * invertible if it has a two-sided inverse

Note that we need to be particularly careful here. If an element possesses both a left inverse and a right inverse, that does not necessarily guarantee that the element possesses a two-sided inverse. The guarantee can, however, be given in the case of associativity.