 Cancellation property

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 leftcancellative) 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 rightcancellative) 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 twosided cancellation property (or is cancellative) if it is both left and rightcancellative.
A magma (M,*) has the left cancellation property (or is leftcancellative) if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or twosided cancellative properties.
A leftinvertible element is leftcancellative, and analogously for right and twosided.
For example, every quasigroup, and thus every group, is cancellative.
Contents
Interpretation
To say that an element a in a magma (M,*) is leftcancellative, 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.
Examples of cancellative monoids and semigroups
The positive (equally nonnegative) integers form a cancellative semigroup under addition. The nonnegative 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.
Noncancellative 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 cross product of two vectors does not obey the cancellation law. If a×b = a×c, then it does not follow that b=c even if a≠0.
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 (i.e. has det(A)≠0) before one can conclude that B=C. If det(A)=0, then B might not equal C, because the matrix equation AX=B will not have a unique solution for a noninvertible matrix A.
See also
 Grothendieck group
 Invertible element
 Cancellative semigroup
Categories: Nonassociative algebra
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