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 {\displaystyle p} and q {\displaystyle q} are said to have the commutative property whenever
p q = q p {\displaystyle pq=qp}
The quasi-commutative property in matrices is defined as follows. Given two non-commutable matrices x {\displaystyle x} and y {\displaystyle y}
x y − y x = z {\displaystyle xy-yx=z}
satisfy the quasi-commutative property whenever z {\displaystyle z} satisfies the following properties:
x z = z x y z = z y {\displaystyle {\begin{aligned}xz&=zx\\yz&=zy\end{aligned}}}
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 × Y → X {\displaystyle f:X\times Y\to X} is said to be quasi-commutative if
f ( f ( x , y 1 ) , y 2 ) = f ( f ( x , y 2 ) , y 1 ) for all x ∈ X , y 1 , y 2 ∈ Y . {\displaystyle f\left(f\left(x,y_{1}\right),y_{2}\right)=f\left(f\left(x,y_{2}\right),y_{1}\right)\qquad {\text{ for all }}x\in X,\;y_{1},y_{2}\in Y.}
If f ( x , y ) {\displaystyle f(x,y)} is instead denoted by x ∗ y {\displaystyle x\ast y} then this can be rewritten as:
( x ∗ y ) ∗ y 2 = ( x ∗ y 2 ) ∗ y for all x ∈ X , y , y 2 ∈ Y . {\displaystyle (x\ast y)\ast y_{2}=\left(x\ast y_{2}\right)\ast y\qquad {\text{ for all }}x\in X,\;y,y_{2}\in Y.}
See also Commutative property – Property of some mathematical operations Accumulator (cryptography)
References