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 (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the right cancellation property (or is right-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma 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. A left-invertible element is left-cancellative, and analogously for right and two-sided. For example, every quasigroup, and thus every

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

, 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 (equally non-negative) integers form a cancellative

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'' ...

under addition. The non-negative integers form a cancellative

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...

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. In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a

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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. Note that this remains valid even if the ring in question is noncommutative and/or nonunital.

Non-cancellative algebraic structures

Although the cancellation law holds for addition, subtraction, multiplication and division of

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and complex numbers (with the single exception of multiplication by

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...

and division of zero by another number), there are a number of algebraic structures where the cancellation law is not valid. The

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...

of two vectors does not obey the cancellation law. If , then it does not follow that even if .

Matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...

also does not necessarily obey the cancellation law. If and , then one must show that matrix A is ''invertible'' (i.e. has ) before one can conclude that . If , then B might not equal C, because the

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equation will not have a unique solution for a non-invertible matrix A. Also note that if and and the matrix A is ''invertible'' (i.e. has ), it is not necessarily true that . Cancellation works only for and (provided that matrix A is ''invertible'') and not for and .

References

{{DEFAULTSORT:Cancellation Property Non-associative algebra Properties of binary operations Algebraic properties of elements fr:Loi de composition interne#Réguliers et dérivés

Interpretation

Examples of cancellative monoids and semigroups

Non-cancellative algebraic structures

See also

References