Definition 0.1. Let (C,∘) be a structure consisting of a class, C, together with a binary
operation ∘ 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 ∘– are said to be noncommutative if
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
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 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 .)
An example of a non-commutative operation is the multiplication over n × 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.
