Let $M$ be a unital magma with neutral element $e$. Let $a$, $x \in M$.
$x$ is a left-inverse of $a$ if
\[xa = e.\]$x$ is a right-inverse of $a$ if
\[ax = e.\]$x$ is an inverse of $a$ if $x$ is both a left-inverse and a right-inverse of $a$.
$a$ is (left-/right-)invertible if it has a (left-/right-)inverse.
$a$ is uniquely (left-/right-)invertible if it has exactly one (left-/right-)inverse.
$M$ is (uniquely) (left-/right-)invertible if all elements of $M$ are (uniquely) (left-/right-)invertible.
If $M$ is uniquely invertible, we write $a^{-1}$ for the unique inverse of each $a\in M$.
The invertible subset of a magma $M$ is
\[\inv M = \{a \in M : \text{$a$ is invertible}\}.\]