Commutative property

The commutative property is one of the three basic laws of regular algebras that greatly simplify equations and the derivation of their solutions. In essence, it assumes that one can commute, or interchange the positions of two variables $x$ and $y$ without changing the value of the result, (such as a product for multiplication, or a sum for addition), under an operation "$*$", that is: $x*y=y*x.$ Thus, commutativity can also be considered as a kind of symmetry caused by mirroring.

The associativity and distributivity laws are the other two simplifying properties of algebraic variables for operations such as multiplication and addition.

Notes

• 1.Through duality the three regular properties of an algebra are induced also in its corresponding geometric structure or 'space'. Thus, our graphical representations of two-dimensional and three-dimensional surfaces are commutative, and Geometry was implicitly thought as being commutative.

• 2. By extending through duality the concept of geometric structure to the dual 'space' of a noncommutative algebra, in 1979, the term "noncomutative geometry" has been established in mathematics, and an applied area of physical mathematics with the same name; the latter has been developed in a manner consistent with the Standard Model (SUSY) of modern physics.

• 3. Interestingly, certain operators of quantum mechanical observables, such as position and momentum do not commute, and therefore quantum mechanics is a noncommutative theory. In general, this noncommutation of certain pairs of quantum operators gives rise to the uncertainty relations or the Heisenberg Principle of Quantum Mechanics. On the other hand, in classical mechanics and Einstein's relativity theories the position and momentum operators commute, and position and momentum can be both determined simultaneously and precisely for any massive object.

• 4. In Category Theory a special form of commutativity occurs as a generalization of the commutativity property defining Abelian groups. More general categories that are dissimilar to the category of Abelian groups are called non-Abelian, or nonabelian.