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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, and thus every
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
under addition. The non-negative integers form a cancellative
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
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 In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, 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: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
and
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s (with the single exception of multiplication by
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

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 In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

cross product
of two vectors does not obey the cancellation law. If , then it does not follow that even if .
Matrix multiplication In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
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 groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Invertible element In the branch of abstract algebra known as ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations def ...
*
Cancellative semigroupIn mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality (mathematics), equality of the form ''a''·'' ...
*
Integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...


References

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