Cancellation Property - Interpretation

Interpretation

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. Moreover, we can be "constructive" with f taking the inverse in the range of g and sending the rest precisely to a.