Editing Anticommutative property (section)

{{short description|Property of math operations which yield an inverse result when arguments' order reversed}}

In [[mathematics]], '''anticommutativity''' is a specific property of some non-[[commutative]] mathematical [[Operation (mathematics)|operations]]. Swapping the position of [[Binary operation|two arguments]] of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped arguments. The notion ''[[inverse element|inverse]]'' refers to a [[group (mathematics)|group structure]] on the operation's [[codomain]], possibly with another operation. [[Subtraction]] is an anticommutative operation because commuting the operands of {{nowrap|1=''a'' − ''b''}} gives {{nowrap|1=''b'' − ''a'' = −(''a'' − ''b'');}} for example, {{nowrap|1=2 − 10 = −(10 − 2) = −8.}} Another prominent example of an anticommutative operation is the [[Lie algebra|Lie bracket]].

In [[mathematical physics]], where [[symmetry (physics)|symmetry]] is of central importance, or even just in [[multilinear algebra]] these operations are mostly (multilinear with respect to some [[vector space|vector structures]] and then) called '''antisymmetric operations''', and when they are not already of [[arity]] greater than two, extended in an [[associative]] setting to cover more than two [[Argument of a function|arguments]].