In
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.
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, and thus every
group, 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 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 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 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 .
See also
*
Grothendieck group
*
Invertible element
*
Cancellative semigroup
*
Integral domain
References
{{DEFAULTSORT:Cancellation Property
Non-associative algebra
Properties of binary operations
Algebraic properties of elements
fr:Loi de composition interne#Réguliers et dérivés