Cancellation property

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In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility.

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.

In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If a⁻¹ is the left inverse of a, then a ∗ b = a ∗ c implies a⁻¹ ∗ (a ∗ b) = a⁻¹ ∗ (a ∗ c) which implies b = c by associativity.

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