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

(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.
A binary operation that is not 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 (that is, in the commutative case one has

(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, which is of fundamental importance in the algebraic theory of observable quantum operators and also in quantum algebraic topology.