Anticommutativity

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.

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,

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

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