# 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

 $\displaystyle a\circ b\neq b\circ a$ (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

 $\displaystyle a\circ b=b\circ a$ (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.

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