A binary operationMathworldPlanetmath” is said to be anticommutative if it satisfies the identityPlanetmathPlanetmathPlanetmath

yx=-(xy), (1)

where the minus denotes the element in the algebraMathworldPlanetmathPlanetmathPlanetmath in question.  This implies that  xx=-(xx),  i.e. xx must be the neutral element of the addition of the algebra:

xx=𝟎. (2)

Using the distributivity of “” over “+” we see that the indentity (2) also implies (1):


A well known example of anticommutative operations is the vector product in the algebra  (3,+,×),  satisfying


Also we know that the subtraction of numbers obeys identities

b-a=-(a-b),a-a= 0.

An important anticommutative operation is the Lie bracket.

Title anticommutative
Canonical name Anticommutative
Date of creation 2014-02-04 7:50:58
Last modified on 2014-02-04 7:50:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 17A01
Synonym anticommutative operation
Synonym anticommutativity
Related topic Supercommutative
Related topic AlternativeAlgebra
Related topic Subcommutative