Cancellative element

Definition
An element $$a$$ in a magma $$(S,*)$$ (a set $$S$$ with binary operation $$*$$) is termed:


 * left-cancellative if whenever $$a * b = a * c$$, $$b = c$$
 * right-cancellative if whenever $$b * a = c * a$$, $$b = c$$
 * cancellative if it is both left and right cancellative

A magma where every element is left-cancellative (resp. right-cancellative, cancellative) is termed a left-cancellative magma (resp., right-cancellative magma, cancellative magma).

Stronger properties

 * Invertible element: In a monoid, any left invertible element is right cancellative, any right invertible element is left cancellative. Thus, any invertible element is cancellative.