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.