anticommutative

A binary operation “$\star$” is said to be anticommutative if it satisfies the identity

$y\star x=-(x\star y),$ (1)

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

$x\star x=\text{𝟎}.$ (2)

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

$$\text{𝟎}=(x+y)\star (x+y)=x\star x+x\star y+y\star x+y\star y=x\star y+y\star x$$

A well known example of anticommutative operations is the vector product in the algebra  $({ℝ}^3,+,\times )$,  satisfying

$$\overrightarrow{b}\times \overrightarrow{a}=-(\overrightarrow{a}\times \overrightarrow{b}),\overrightarrow{a}\times \overrightarrow{a}=\overrightarrow{0}.$$

Also we know that the subtraction of numbers obeys identities

$$b-a=-(a-b),a-a=\mathrm{\hspace{0.33em}0}.$$

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