Cancellation property

In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a*b = a*c always implies b = c.

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

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

A magma (M,*) has the left cancellation property (or is left-cancellative) if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties.

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

To say that an element a in a magma (M,*) is left-cancellative, is to say that the function g: x |-> a*x is injective, so a set monomorphism but as it is a set endomorphism it is a set section, i.e. there is a set epimorphism f such f(g(x))=f(a*x)=x for all x, so f is a retraction. (The only injective function which has not inverse goes from the empty set to a non empty set, so it can't be undone). Moreover, we can be "constructive" with f taking the inverse in the range of g and sending the rest precisely to a.