Quasi-commutative property

In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in certain 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 defined^[1] as follows. Given two non-commutable matrices x and y

$xy - yx = z$

satisfy the quasi-commutative property whenever z satisfies the following properties:

$xz = zx$

$yz = zy$

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.^[1] 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, defined as follows:

$f: X \times Y \rightarrow X$

is said to be quasi-commutative^[2] if for all $x \in X$ and for all $y_1, y_2 \in Y$,

$f(f(x,y_1),y_2) = f(f(x,y_2),y_1)$

See also

• Commutative property
• Accumulator (cryptography)

References

1. Neal H. McCoy. On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340.
2. 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.