Cancellation law

In mathematics, the notion of cancellative is a generalization of the notion of invertible.

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 that 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 that 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.

A left-invertible element is left-cancellative, and analogously for right and two-sided.

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

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