# 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:

$x\star x=\text{\U0001d7ce}.$ | (2) |

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

$$\text{\U0001d7ce}=(x+y)\star (x+y)=x\star x+x\star y+y\star x+y\star y=x\star y+y\star x$$ |

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