Definition 5

An element $a$ of a magma $E$ is called left (resp. right) cancellable (or regular) if left (resp. right) translation by $a$ is injective. A left and right cancellable element is called a cancellable (or regular) element.

In other words, for $a$ to be cancellable under the law $\mathsf{T}$, it is necessary and sufficient that each of the relations $a \mathsf{T} x = a \mathsf{T} y, x \mathsf{T} a = y \mathsf{T} a$ imply $x = y$ (it is said that "a can be cancelled" from each of these equalities). If there exists an identity element $e$ under the law $\mathsf{T}$, it is cancellable under this law: the translations $\gamma_e$ and $\delta_e$ are then the identity mapping of $E$ onto itself.