Definition 1. Let $M$ be a unital magma with neutral element $e$. Let $a$, $x \in M$.
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$x$ 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 $a$.
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$a$ is (uniquely) (left-/right-)invertible if it has a (unique) (left-/right-)inverse.
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$M$ 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 $a\in M$.