- Anticommutativity
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 inalgebra ,geometry ,mathematical analysis and, as a consequence inphysics : they are often called**antisymmetric operations**.**Definition**An $n$-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 $scriptstyle\; *:A^n\; longrightarrow\; mathfrak\{G\}$ from the set of all "n"-tuples of elements in a set "$A$" (where "$n$" is a general integer) to a group $scriptstylemathfrak\{G\}$ (whose operation is written in additive notation for the sake of simplicity),

**is anticommutative**if and only if:$x\_1*x\_2*dots*x\_n\; =\; sgn(sigma)\; x\_\{sigma(1)\}*x\_\{sigma(2)\}*dots*\; x\_\{sigma(n)\}\; qquad\; forall\backslash boldsymbol\{x\}\; =\; (x\_1,x\_2,dots,x\_n)\; in\; A^n$

where $scriptstylesigma:(n)longrightarrow(n)$ is an arbitrary

permutation of the set $(n)$ of first "$n$" non-zerointegers and $mathrm\{sgn\}(sigma)$ 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 thecodomain $scriptstylemathfrak\{G\}$ of the operation to be at least a group.Note that this is an

abuse of notation , since thecodomain of the operation needs only to be a group: "$-1$" has not a precise meaning since amultiplication is not necessarily defined on $scriptstylemathfrak\{G\}$.Particularly important is the case "$n\; =\; 2$".

**A**$scriptstyle\; *:A\; imes\; Alongrightarrow\; mathfrak\{G\}$binary operation **is anticommutative**if and only if:$x\_1\; *\; x\_2\; =\; -x\_2\; *\; x\_1\; qquadforall(x\_1,x\_2)in\; A\; imes\; A$

This means that $scriptstyle\; x\_1\; *\; x\_2$ is the inverse of the element $scriptstyle\; x\_2\; *\; x\_1$ in $scriptstylemathfrak\{G\}$.

**Properties**If the group $scriptstylemathfrak\{G\}$ is such that

:$mathfrak\{-a\}\; =\; mathfrak\{a\}\; iff\; mathfrak\{a\}\; =\; mathfrak\{0\}qquad\; forall\; mathfrak\{a\}\; in\; mathfrak\{G\}$

i.e. "the only element equal to its inverse is the

neutral element ", then for all the ordered tuples such that $x\_j\; =\; x\_i$ for at least two different index $i,j$:$x\_1*x\_2*dots*x\_n\; =\; mathfrak\{0\}$

In the case "$n\; =\; 2$" this means

:$x\_1*x\_1\; =\; x\_2*x\_2\; =\; mathfrak\{0\}$

**Examples**Anticommutative operators include:

*Subtraction

*Cross product

*Lie algebra

*Lie ring **ee also***

Commutativity

*Commutator

*exterior algebra

*Operation (mathematics)

*Symmetry in mathematics

*Particle statistics (for anticommutativity in physics).**References***Harvrefcol

Surname = Bourbaki

Given = Nicolas

Title = Algebra. Chapters 1-3

Publisher = Springer-Verlag

Place =Berlin ,Heidelberg ,New York

Year = 1989

Edition = paperback, ISBN 3-540-64243-9, chapter III, "Tensor algebra s,exterior algebra s,symmetric algebra s".**External links***

Weisstein , Eric W." [*http://mathworld.wolfram.com/Anticommutative.html Anticommutative*] ". FromMathWorld --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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### Look at other dictionaries:

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