Quasi-commutative property

In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions.

Applied to matrices

Two matrices $p$ and $q$ are said to have the commutative property whenever
$pq = qp$

The quasi-commutative property in matrices is definedNeal H. McCoy
On quasi-commutative matrices. ''Transactions of the American Mathematical Society, 36''(2), 327–340
as follows. Given two non-commutable matrices $x$ and $y$
$xy - yx = z$

satisfy the quasi-commutative property whenever $z$ satisfies the following properties:
$\begin xz &= zx \\ yz &= zy \end$

An example is found in the matrix mechanics introduced by Heisenberg as a version of quantum mechanics. In this mechanics, ''p'' and ''q'' are infinite matrices corresponding respectively to the momentum and position variables of a particle. These matrices are written out at Matrix mechanics#Harmonic oscillator, and z = iħ times the infinite unit matrix, where ħ is the reduced Planck constant.

Applied to functions

A function $f : X \times Y \to X$ is said to be Benaloh, J., & De Mare, M. (1994, January)
One-way accumulators: A decentralized alternative to digital signatures
In ''Advances in Cryptology – EUROCRYPT’93'' (pp. 274–285). Springer Berlin Heidelberg. if
$f\left(f\left(x, y_1\right), y_2\right) = f\left(f\left(x, y_2\right), y_1\right) \qquad \text x \in X, \; y_1, y_2 \in Y.$

If $f(x, y)$ is instead denoted by $x \ast y$ then this can be rewritten as:
$(x \ast y) \ast y_2 = \left(x \ast y_2\right) \ast y \qquad \text x \in X, \; y, y_2 \in Y.$

See also

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References

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Mathematical relations
Properties of binary operations