Template:Short description Template:About Template:More references In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility.
An element a in a magma Template:Nowrap has the left cancellation property (or is left-cancellative) if for all b and c in M, Template:Nowrap always implies that Template:Nowrap.
An element a in a magma Template:Nowrap has the right cancellation property (or is right-cancellative) if for all b and c in M, Template:Nowrap always implies that Template:Nowrap.
An element a in a magma Template:Nowrap has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative.
A magma Template:Nowrap 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.
In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If a−1 is the left inverse of a, then Template:Nowrap implies Template:Nowrap, which implies Template:Nowrap by associativity.
For example, every quasigroup, and thus every group, is cancellative.
InterpretationEdit
To say that an element a in a magma Template:Math is left-cancellative, is to say that the function Template:Math is injective.<ref>Template:Cite book</ref> That the function g is injective implies that given some equality of the form Template:Math, 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 Template:Math. Put another way, for all x and y in M, if Template:Math, then Template:Math.<ref>Template:Cite book</ref>
Similarly, to say that the element a is right-cancellative, is to say that the function Template:Math is injective and that for all x and y in M, if Template:Math, then Template:Math.
Examples of cancellative monoids and semigroupsEdit
The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. Each of these is an example of a cancellative magma that is not a quasigroup.
Any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid that embeds 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. This remains valid even if the ring in question is noncommutative and/or nonunital.
Non-cancellative algebraic structuresEdit
Although the cancellation law holds for addition, subtraction, multiplication and division of real and complex numbers (with the single exception of multiplication by zero and division of zero by another number), 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 Template:Nowrap, then it does not follow that Template:Nowrap even if Template:Nowrap (take Template:Nowrap for example)
Matrix multiplication also does not necessarily obey the cancellation law. If Template:Nowrap and Template:Nowrap, then one must show that matrix A is invertible (i.e. has Template:Nowrap) before one can conclude that Template:Nowrap. If Template:Nowrap, then B might not equal C, because the matrix equation Template:Nowrap will not have a unique solution for a non-invertible matrix A.
Also note that if Template:Nowrap and Template:Nowrap and the matrix A is invertible (i.e. has Template:Nowrap), it is not necessarily true that Template:Nowrap. Cancellation works only for Template:Nowrap and Template:Nowrap (provided that matrix A is invertible) and not for Template:Nowrap and Template:Nowrap.