Quasi-commutative property#Applied to functions

Two matrices $p$ and $q$ are said to have the commutative property whenever

$$pq\; =\; qp$$

The quasi-commutative property in matrices is definedNeal H. McCoy. [https://www.ams.org/journals/tran/1934-036-02/S0002-9947-1934-1501746-8/S0002-9947-1934-1501746-8.pdf On quasi-commutative matrices. Transactions of the American Mathematical Society, 36(2), 327–340]. as follows. Given two non-commutable matrices $x$ and $y$

$$xy\; -\; yx\; =\; z$$

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

$$\backslash begin\{align\}$$

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#Harmonic oscillator, and z = iħ times the infinite unit matrix, where ħ is the reduced Planck constant.

A function $f\; :\; X\; \backslash times\; Y\; \backslash to\; X$ is said to be {{visible anchor|quasi-commutative}}Benaloh, J., & De Mare, M. (1994, January). [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.13.8287&rep=rep1&type=pdf One-way accumulators: A decentralized alternative to digital signatures]. In Advances in Cryptology – EUROCRYPT’93 (pp. 274–285). Springer Berlin Heidelberg. if

$$f\backslash left(f\backslash left(x,\; y\_1\backslash right),\; y\_2\backslash right)\; =\; f\backslash left(f\backslash left(x,\; y\_2\backslash right),\; y\_1\backslash right)\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X,\; \backslash ;\; y\_1,\; y\_2\; \backslash in\; Y.$$

If $f(x,\; y)$ is instead denoted by $x\; \backslash ast\; y$ then this can be rewritten as:

$$(x\; \backslash ast\; y)\; \backslash ast\; y\_2\; =\; \backslash left(x\; \backslash ast\; y\_2\backslash right)\; \backslash ast\; y\; \backslash qquad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X,\; \backslash ;\; y,\; y\_2\; \backslash in\; Y.$$

• {{annotated link|Commutative property}}
• {{annotated link|Accumulator (cryptography)}}

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Category:Mathematical relations

Category:Properties of binary operations