mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the notion of cancellative is a generalization of the notion of

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

. An element ''a'' in a

magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

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 In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...

, 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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

. 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. 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. 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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...

, 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

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

and

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s (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 Multiplication, multiplying digits to the left of 0 by th ...

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 ...

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 s ...

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

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

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