It is called cancellative if it is both left cancellative and right cancellative.
In infinity-groupoid theory
A monoid (-truncated-space) is called left cancellative if for all objects and the homotopy fiber of the functor , defined as , at is -truncated, and is called right cancellative if for all elements and the homotopy fiber of the functor , defined as at is -truncated. It is called cancellative if it is both left cancellative and right cancellative.