Cancellative element

This article defines a property of elements or tuples of elements with respect to a binary operation

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).

Relation with other properties

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. For full proof, refer: invertible implies cancellative in monoid