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[edit]

An 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  \scriptstyle *:A^n \to \mathfrak{G} from the set of all n-tuples of elements in a set A (where n is a general integer) to a group  \scriptstyle\mathfrak{G} 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\boldsymbol{x} = (x_1,x_2,\dots,x_n) \in A^n

where  \scriptstyle\sigma:(n)\to(n) is an arbitrary permutation of the set (n) of the first n positive integers and \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  \scriptstyle\mathfrak{G} .

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

 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  \scriptstyle\mathfrak{G} .

Properties[edit]

If the group  \scriptstyle\mathfrak{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[edit]

Examples of anticommutative binary operations include:

See also[edit]

References[edit]

External links[edit]