In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element a in a magma has the left cancellation property if for all b and c in M, always implies that. An element a in a magma has the right cancellation property if for all b and c in M, always implies that. An element a in a magma has the two-sided cancellation property if it is both left- and right-cancellative. A magma has the left cancellation property 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.
Interpretation
To say that an element a in a magma is left-cancellative, is to say that the function 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. Put another way, for all x and y in M, if a * x = a * y, then x = y.
Examples of cancellative monoids and semigroups
The positive 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 will obey the cancellative law. In a different vein, the multiplicative semigroup of elements of a ring that are not zero divisors has the cancellation property. Note that this remains valid even if the ring in question is noncommutative and/or nonunital.
Although the cancellation law holds for addition, subtraction, multiplication and division of real and complex numbers, there are a number of algebraic structures where the cancellation law is not valid. The cross product of two vectors does not obey the cancellation law. If, then it does not follow that even if. Matrix multiplication also does not necessarily obey the cancellation law. If and, then one must show that matrixA is invertible before one can conclude that. If, then B might not equalC, because the matrix equation will not have a unique solution for a non-invertible matrix A. Also note that if and and the matrixA is invertible, it is not necessarily true that. Cancellation works only for and and not for and.
