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In mathematics, 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 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 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.
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. (The only injective function which has no inverse goes from the empty set to a non empty set, so it can't be undone). Moreover, we can be "constructive" with f taking the inverse in the range of g and sending the rest precisely to a.
Non-cancellative algebras
Although, with the single exception of multiplication by zero and division of zero by another number, the cancellation law holds for addition, subtraction, multiplication and division of real and complex numbers, there are a number of algebras where the cancellation law is not valid.
The vector dot product is perhaps the simplest example. In this case, for an arbitrary nonzero vector a, the product a·b can equal another dot product a·c even if b≠c. This occurs because the dot product relates to the angle between two vectors as well as their magnitude, and a change in one can, in effect, counterbalance the other to produce equal products for unequal vectors.
For the same reason, the cross product of two vectors also does not obey the cancellation law. If axb = axc, then it does not follow that b=c even if a≠0.
However, if both a·b=a·c and axb=axc, then one can conclude that b=c. This is because for dot and cross products to be simultaneously equal, then both a·(b-c) and ax(b-c) must be zero by the distributive law. This means that both the sine and cosing of the angle between a and (b-c) must be zero, which is not possible because sin2x+cos2x is identically 1.
Matrix multiplication also does not necessarily obey the cancellation law.
If AB=AC and A≠O, then one must show that matrix A is invertible (ie. has det(A)≠0) before one can conclude that B=C. If det(A)=0, then B may not equal C, because the matrix equation AX=B will not have a unique solution for a non-invertible matrix.
See also
"Cancellation property"