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 defined as follows. Given two non-commutable matrices $$x$$ and $$y$$ $$xy - yx = z$$

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

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, 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  if $$f\left(f\left(x, y_1\right), y_2\right) = f\left(f\left(x, y_2\right), y_1\right) \qquad \text{ for all } x \in X, \; y_1, y_2 \in Y.$$

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