In mathematics, anticommutativity refers to the property of an operation being anticommutative, i.e. being non commutative in a precise way. Anticommutative operations are widely used in
algebra, geometry, mathematical analysisand, as a consequence in physics: they are often called antisymmetric operations.
An -ary operation is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation * is anticommutative if for all x and y, x*y = −y*x.
More formally, a map from the set of all "n"-tuples of elements in a set "" (where "" is a general integer) to a group (whose operation is written in additive notation for the sake of simplicity), is anticommutative if and only if
where is an arbitrary
permutationof the set of first "" non-zero integersand is its sign. This equality express the following concept
* the value of the operation is unchanged, when applied to all ordered tuples constructed by even permutation of the elements of a fixed one.
* the value of the operation is the inverse of its value on a fixed tuple, when applied to all ordered tuples constructed by odd permutation to the elements of the fixed one. The need for the existence of this inverse element is the main reason for requiring the
codomainof the operation to be at least a group.
Particularly important is the case "". A
binary operationis anticommutative if and only if
This means that is the inverse of the element in .
If the group is such that
i.e. "the only element equal to its inverse is the
neutral element", then for all the ordered tuples such that for at least two different index
In the case "" this means
Anticommutative operators include:
Symmetry in mathematics
Particle statistics(for anticommutativity in physics).
Surname = Bourbaki
Given = Nicolas
Title = Algebra. Chapters 1-3
Publisher = Springer-Verlag
Berlin, Heidelberg, New York
Year = 1989
Edition = paperback, ISBN 3-540-64243-9, chapter III, "
Tensor algebras, exterior algebras, symmetric algebras".
Weisstein, Eric W." [http://mathworld.wolfram.com/Anticommutative.html Anticommutative] ". From MathWorld--A Wolfram Web Resource.
* A.T. Gainov, " [http://eom.springer.de/A/a012580.htm Anti-commutative algebra] ", Springer-Verlag Online Encyclopaedia of Mathematics.
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