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(0.1) |
for either at least some or all of the
pairs in
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
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for all
pairs in
, and also that the operation
is defined for all pairs in
).
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
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.
As of this snapshot date, this entry was owned by bci1.