Invertible element
From Groupprops
This article defines a property of elements or tuples of elements with respect to a binary operation
Further information: inverse element
Definition
An element in a set with binary operation and neutral element is termed:
- left invertible if it has a left inverse, viz there exists an element such that
- right invertible it it has a right inverse, viz there exists an element such that
- 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