Cancellation property

In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility that does not rely on an inverse element.

An element ${\displaystyle a}$ in a magma ${\displaystyle (M,*)}$ has the left cancellation property (or is left-cancellative) if for all ${\displaystyle b}$ and ${\displaystyle c}$ in ${\displaystyle M}$ , ${\displaystyle a*b=a*c}$ always implies that ${\displaystyle b=c}$ .

An element ${\displaystyle a}$ in a magma ${\displaystyle (M,*)}$ has the right cancellation property (or is right-cancellative) if for all ${\displaystyle b}$ and ${\displaystyle c}$ in ${\displaystyle M}$ , ${\displaystyle b*a=c*a}$ always implies that ${\displaystyle b=c}$ .

An element ${\displaystyle a}$ in a magma ${\displaystyle (M,*)}$ has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative.

A magma ${\displaystyle (M,*)}$ is left-cancellative if all ${\displaystyle a}$ in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties.

In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If ${\displaystyle a^{-1}}$ is the left inverse of ${\displaystyle a}$ , then ${\displaystyle a*b=a*c}$ implies ${\displaystyle a^{-1}*(a*b)=a^{-1}*(a*c)}$, which implies ${\displaystyle b=c}$ by associativity.

For example, every quasigroup, and thus every group, is cancellative.