# non-commutative structure

###### Definition 0.1.

Let $(C,\circ )$ be a structure^{} consisting of a *class*, $C$, together with a *binary operation ^{}* $\circ $ defined for pairs of objects in $C$ (or elements of $C$ when the latter is a small class, i.e., a set). The structure– and the operation

^{}$\circ $– are said to be

*noncommutative*if

$a\circ b\ne b\circ a$ | (0.1) |

for either at least some or all of the $a,b$ pairs in $C$ for which the operation is defined.

A structure that is noncommutative is also called sometimes a *non-Abelian structure ^{}*, although the latter term is, in general, more often used to specify non-Abelian theories (http://planetmath.org/NonAbelianTheories).

A binary operation that is not commutative

^{}(http://planetmath.org/Commutative) is said to be

*non-commutative*(or

*noncommutative*). Thus, a

*noncommutative structure*can be alternatively defined as any structure whose binary operation is not commutative (http://planetmath.org/Commutative) (that is, in the commutative (http://planetmath.org/Commutative) case one has

$a\circ b=b\circ a$ | (0.2) |

for all $a,b$ pairs in $C$, and also that the operation $\circ $ is defined for all pairs in $C$).

An example of a commutative structure is the field of real numbers– with two commutative operations in this case–
which are the addition^{} and multiplication over the reals.

###### Remark 0.1.

A commutative group is also called *Abelian*, whereas a category^{} with structure that has commutative diagrams^{} is not necessarily Abelian –unless it does satisfy the Ab1 to Ab6 axioms that define an Abelian category^{} (or equivalently, if it has the properties specified in Mitchell’s alternative definition of an Abelian category (http://planetmath.org/AlternativeDefinitionOfAnAbelianCategory) .)

An example of a non-commutative operation is the multiplication over $n\times n$ matrices.
Another example of a *noncommutative algebra* is a general Clifford algebra^{} (http://planetmath.org/CCliffordAlgebra), which is of fundamental importance in the algebraic theory of observable quantum operators and also in quantum algebraic topology.

Title | non-commutative structure |

Canonical name | NoncommutativeStructure |

Date of creation | 2013-03-22 18:18:06 |

Last modified on | 2013-03-22 18:18:06 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 24 |

Author | bci1 (20947) |

Entry type | Definition |

Classification | msc 55-00 |

Classification | msc 18-00 |

Synonym | noncommutative |

Synonym | nonabelian |

Synonym | non-Abelian |

Related topic | Commutative |

Related topic | QuantumTopos |

Defines | non-commutative operation |