non-commutative structureNonCommutativeStructureAndOperation2008-08-05 17:02:312009-02-02 06:56:27Definitionnon-commutative operationnon-commutative operationcommutative operatorsAbeliannon-Abelian % this is the default PlanetMath preamble. as your knowledge
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\newcommand{\hr}{{\hookrightarrow}}\begin{definition}
Let $(C,\circ)$ be a structure consisting of a \emph{class}, $C$, together with a \emph{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 \emph{noncommutative} if
\begin{align}
a \circ b \neq b \circ a
\end{align}
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 \emph{non-Abelian structure}, although the latter term is, in general, more often used to specify \PMlinkname{non-Abelian theories}{NonAbelianTheories}.\\
A binary operation that is not \PMlinkname{commutative}{Commutative} is said to be \emph{non-commutative} (or \emph{noncommutative}). Thus, a \emph{noncommutative structure} can be alternatively defined as any structure whose binary operation is not \PMlinkname{commutative}{Commutative} (that is, in the \PMlinkname{commutative}{Commutative} case one has
\begin{align}
a \circ b = b \circ a
\end{align}
for all $a,b$ pairs in $C$, and also that the operation $\circ$ is defined for all pairs in $C$).
\end{definition}
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.
\begin{remark}
A commutative group is also called \emph{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 \PMlinkname{alternative definition of an Abelian category}{AlternativeDefinitionOfAnAbelianCategory} .)
\end{remark}
An example of a non-commutative operation is the multiplication over $n \times n$ matrices.
Another example of a \emph{noncommutative algebra} is a general \PMlinkname{Clifford algebra}{CCliffordAlgebra}, which is of fundamental importance in the algebraic theory of observable quantum operators and also in quantum algebraic topology.