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.

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

The quasi-commutative property in matrices is defined^[1] as follows. Given two non-commutable matrices ${\displaystyle x}${\displaystyle x} and ${\displaystyle y}${\displaystyle y} $${\displaystyle xy-yx=z}$${\displaystyle xy-yx=z}

satisfy the quasi-commutative property whenever ${\displaystyle z}${\displaystyle z} satisfies the following properties: $${\displaystyle {\begin{aligned}xz&=zx\\yz&=zy\end{aligned}}}$${\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.^[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.

A function ${\displaystyle f:X\times Y\to X}${\displaystyle f:X\times Y\to X} is said to be quasi-commutative^[2] if $${\displaystyle f\left(f\left(x,y_{1}\right),y_{2}\right)=f\left(f\left(x,y_{2}\right),y_{1}\right)\qquad {\text{ for all }}x\in X,\;y_{1},y_{2}\in Y.}$${\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 ${\displaystyle f(x,y)}${\displaystyle f(x,y)} is instead denoted by ${\displaystyle x\ast y}${\displaystyle x\ast y} then this can be rewritten as: $${\displaystyle (x\ast y)\ast y_{2}=\left(x\ast y_{2}\right)\ast y\qquad {\text{ for all }}x\in X,\;y,y_{2}\in Y.}$${\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.}

• Commutative property – Property of some mathematical operations
• Accumulator (cryptography)

• Mathematical relations
• Properties of binary operations