Anticommutativity

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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 analysis and, as a consequence in physics: they are often called antisymmetric operations.

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[edit] 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 \scriptstyle\mathfrak{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\boldsymbol{x} = (x_1,x_2,\dots,x_n) \in A^n

where \scriptstyle\sigma:(n)\longrightarrow(n) is an arbitrary permutation of the set (n) of first n non-zero integers and sgn(σ) is its sign. This equality express the following concept

Note that this is an abuse of notation, since the codomain of the operation needs only to be a group: " − 1" has not 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\longrightarrow \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 \scriptstyle x_1 * x_2 is the inverse of the element \scriptstyle x_2 * x_1 in \scriptstyle\mathfrak{G} .

[edit] Properties

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 xj = xi 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}

[edit] Examples

Anticommutative operators include:

[edit] See also

[edit] References

[edit] External links

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