Invertible element

This article defines a property of elements or tuples of elements with respect to a binary operation

Further information: inverse 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. For full proof, refer: Inverse element#Equality of left and right inverses