non-commutative structure

Definition 0.1.

Let (C,) be a structureMathworldPlanetmath consisting of a class, C, together with a binary operationMathworldPlanetmath 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 operationMathworldPlanetmath – are said to be noncommutative if

abba (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 structurePlanetmathPlanetmath, although the latter term is, in general, more often used to specify non-Abelian theories (
A binary operation that is not commutativePlanetmathPlanetmathPlanetmath ( 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 ( (that is, in the commutative ( case one has

ab=ba (0.2)

for all a,b pairs in C, and also that the operation 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 additionPlanetmathPlanetmath and multiplication over the reals.

Remark 0.1.

A commutative group is also called Abelian, whereas a categoryMathworldPlanetmath with structure that has commutative diagramsMathworldPlanetmath is not necessarily Abelian –unless it does satisfy the Ab1 to Ab6 axioms that define an Abelian categoryMathworldPlanetmathPlanetmathPlanetmath (or equivalently, if it has the properties specified in Mitchell’s alternative definition of an Abelian category ( .)

An example of a non-commutative operation is the multiplication over n×n matrices. Another example of a noncommutative algebra is a general Clifford algebraMathworldPlanetmath (, 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