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. Euclid, detail from The School of Athens by Raphael. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...

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. In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

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. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... A retraction is a public statement that confirms that a previously made statement was incorrect, invalid, or morally wrong. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

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. Zero can refer to several things. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...

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 x≠y. 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. The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

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. In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ...

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. In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

Matrix multiplication also does not necessarily obey the cancellation law. This article gives an overview of the various ways to multiply matrices. ...

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. In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Look up matrix in Wiktionary, the free dictionary. ...

See also

• invertible element