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

${\displaystyle pq=qp}$

The quasi-commutative property in matrices is defined[1] as follows. Given two non-commutable matrices x and y

${\displaystyle xy-yx=z}$

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

${\displaystyle xz=zx}$

${\displaystyle yz=zy}$

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:

${\displaystyle f:X\times Y\rightarrow X}$

is said to be quasi-commutative[2] if for all ${\displaystyle x\in X}$ and for all ${\displaystyle y_{1},y_{2}\in Y}$,

${\displaystyle f(f(x,y_{1}),y_{2})=f(f(x,y_{2}),y_{1})}$

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.