# Anticommutativity

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In mathematics, anticommutativity is the property of an operation with two or more arguments wherein swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric operations.

## Definition

An ${\displaystyle n}$-ary operation is anticommutative if swapping the order of any two arguments negates the result. For example, a binary operation ∗ is anti-commutative if for all x and y,

xy = −(yx).

More formally, a map ${\displaystyle \scriptstyle *:A^{n}\to {\mathfrak {G}}}$ from the set of all n-tuples of elements in a set A (where n is a non-negative integer) to a group ${\displaystyle \scriptstyle {\mathfrak {G}}}$ is anticommutative if and only if

${\displaystyle {x_{1}*x_{2}*\dots *x_{n}}=\operatorname {sgn}(\sigma )({x_{\sigma (1)}*x_{\sigma (2)}*\dots *x_{\sigma (n)}})\qquad \forall {\boldsymbol {x}}=(x_{1},x_{2},\dots ,x_{n})\in A^{n}}$

where ${\displaystyle \scriptstyle \sigma :(n)\to (n)}$ is an arbitrary permutation of the set (n) of the first n positive integers and ${\displaystyle \mathrm {sgn} (\sigma )}$ is its sign. This equality expresses the following concept:

Note that this is an abuse of notation, since the codomain of the operation needs only to be a group: "−1" does not have a precise meaning since a multiplication is not necessarily defined on ${\displaystyle \scriptstyle {\mathfrak {G}}}$.

Particularly important is the case n = 2. A binary operation ${\displaystyle \scriptstyle *:A\times A\to {\mathfrak {G}}}$ is anticommutative if and only if

${\displaystyle x_{1}*x_{2}=-(x_{2}*x_{1})\qquad \forall (x_{1},x_{2})\in A\times A}$

This means that x1x2 is the inverse of the element x2x1 in ${\displaystyle \scriptstyle {\mathfrak {G}}}$.

## Properties

If the group ${\displaystyle \scriptstyle {\mathfrak {G}}}$ is such that

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

i.e. the only element equal to its inverse is the neutral element, then for all the ordered tuples such that ${\displaystyle x_{j}=x_{i}}$ for at least two different index ${\displaystyle i,j}$

${\displaystyle x_{1}*x_{2}*\dots *x_{n}={\mathfrak {0}}}$

In the case ${\displaystyle n=2}$ this means

${\displaystyle x_{1}*x_{1}=x_{2}*x_{2}={\mathfrak {0}}}$

## Examples

Examples of anticommutative binary operations include: