Cancellation property

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. A left-invertible element is left-cancellative, and analogously for right and two-sided. If a−1 is the inverse of a, then a ∗ b = a ∗ c implies a−1 ∗ a ∗ b = a−1 ∗ a ∗ c which implies b = c. 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. That the function g is injective implies that given some equality of the form a ∗ x = b, where the only unknown is x, there is only one possible value of x satisfying the equality. More precisely, we are able to define some function f, the inverse of g, such that for all x f(g(x)) = f(a ∗ x) = x. Put another way, for all x and y in M, if a * x = a * y, then x = y. The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law. In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property.

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