# NonCommutativeMultiply

a**b**c

is a general associative, but noncommutative, form of multiplication.

# Examples

open allclose all

## Basic Examples(1)

Compare commutative multiplication with non-commutative multiplication:

Operations are associative:

## Applications(2)

Use NonCommutativeMultiply to represent composition in an algebra of differential operators.

The base case, where is a function, simply multiplies by :

The next two properties express linearity:

Here the operator is D. HoldPattern stops the derivative from acting on the double blank:

Composition of operators applied to an expression:

Power of an operator applied to an expression:

Apply these rules to derive the KdV equation for the Lax pair:

Build a function to expand non-commutative products. Distributivity with respect to Plus:

Handling the commutative product inside the non-commutative one:

Fall-back operation applied to everything else:

## Properties & Relations(2)

No automatic simplification rules exist for NonCommutativeMultiply:

Expand and Simplify do not operate on expressions with NonCommutativeMultiply:

## Possible Issues(1)

NonCommutativeMultiply of one argument, unlike Times, stays unevaluated:

Wolfram Research (1988), NonCommutativeMultiply, Wolfram Language function, https://reference.wolfram.com/language/ref/NonCommutativeMultiply.html.

#### Text

Wolfram Research (1988), NonCommutativeMultiply, Wolfram Language function, https://reference.wolfram.com/language/ref/NonCommutativeMultiply.html.

#### CMS

Wolfram Language. 1988. "NonCommutativeMultiply." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NonCommutativeMultiply.html.

#### APA

Wolfram Language. (1988). NonCommutativeMultiply. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NonCommutativeMultiply.html

#### BibTeX

@misc{reference.wolfram_2022_noncommutativemultiply, author="Wolfram Research", title="{NonCommutativeMultiply}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/NonCommutativeMultiply.html}", note=[Accessed: 14-August-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2022_noncommutativemultiply, organization={Wolfram Research}, title={NonCommutativeMultiply}, year={1988}, url={https://reference.wolfram.com/language/ref/NonCommutativeMultiply.html}, note=[Accessed: 14-August-2022 ]}