# NonCommutativeMultiply a**b**c

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

# Details # Examples

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## 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: