Definition 6

Let $E$ be a unital magma, $\mathsf{T}$ its law of composition, e its identity element and $x$ and $x'$ two elements of $E$. $x'$ is called a left inverse (resp. right inverse, resp. inverse) of $x$ if $x' \mathsf{T} x = e$ (resp. $x \mathsf{T} x' = e$, resp. $x' \mathsf{T} x = x \mathsf{T} x' = e$).

An element $x$ of $E$ is called left invertible (resp. right invertible, resp. invertible) if it has a left inverse (resp. right inverse, resp. inverse).

A monoid all of whose elements are invertible is called a group.

Symmetric and symmetrizable are sometimes used instead of inverse and invertible. When the law on E is written additively, we generally say negative instead of inverse.