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    <p>OK, writing rules that play nice with the attributes of <code>NonCommutativeMultiply</code> is sometimes a hassle. Here's an alternate method which introduces a helper <code>NCM</code> that does not have the rules and attributes of <code>NonCommutativeMultiply</code> associated with it.</p> <p>The following code also incorporates the last couple of your questions. <a href="https://stackoverflow.com/q/4988323/421225">(1)</a> <a href="https://stackoverflow.com/q/5001368/421225">(2)</a></p> <pre><code>Unprotect[NonCommutativeMultiply]; Clear[NonCommutativeMultiply] (* Factor out numerics -- could generalize to some ScalarQ *) nc:NonCommutativeMultiply[a__]/;MemberQ[{a},_?NumericQ]:=NCMFactorNumericQ[NCM[a]]/.NCM-&gt;NonCommutativeMultiply (* Simplify Powers *) b___**a_^n_.**a_^m_.**c___:=NCM[b,a^(n+m),c]/.NCM-&gt;NonCommutativeMultiply (* Expand Brackets *) nc:NonCommutativeMultiply[a___,b_Plus,c___]:=Distribute[NCM[a,b,c]]/.NCM-&gt;NonCommutativeMultiply (* Sort Subscripts *) c___**Subscript[a_, i_]**Subscript[b_, j_]**d___/;i&gt;j:=c**Subscript[b, j]**Subscript[a, i]**d Protect[NonCommutativeMultiply]; Unprotect[NCM]; Clear[NCM] NCMFactorNumericQ[nc_NCM]:=With[{pos=Position[nc,_?NumericQ,1]},Times@@Extract[nc,pos] Delete[nc,pos]] NCM[a_]:=a NCM[]:=1 Protect[NCM]; </code></pre> <p>Note that <code>NCMFactorNumericQ</code> is fast because it works in a single pass, but the rule associated with it <code>nc:NonCommutativeMultiply[a__]/;MemberQ[{a},_?NumericQ]</code> is slow, because the Flat attribute means that it does a stupid number of checks using <code>NumericQ</code>. If you really want more speed and have large expressions, then you should just manually apply the <code>Sort</code> and <code>Factor</code> routines, so that Mathematica does less pattern checks.</p>
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    1. PORedefine Noncommutative Multiplication in Mathematica
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    1. USCommunity
    1. USSimon
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    1. CO@Simon, So I think I have a definition for multiplication that I like. I did not use (* Simplify Powers *) and (* Expand Brackets *), I will apply them later. Something I notice about (* Simplify Powers *), if you say multiply identical expressions that are the sum of many terms together using **, it squares (^2) the expression instead of multiplying it out with noncommutative multiplication. Again this is not a problem since I am not using it. An issue I am running into is negative variables at the beginning of an expression. I am getting `c+(-q)**a` instead of `c-q**a`.
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      1. USCantormath
    2. CO@Simon, So I think I have a definition for multiplication that I like. I did not use `(Simplify Powers )` and `(Expand Brackets )`, I will apply them later. Something I notice about `(Simplify Powers )`, if you say multiply identical expressions that are the sum of many terms together using `**`, it squares `(^2)` the expression instead of multiplying it out with `**`. Again this is not a problem since I am not using it. An issue I am running into is negative variables at the beginning of an expression. I am getting `b**c+(-q)**a` instead of say `b**c-q**a`.
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    3. CO@Cantormath: The power problem comes from the `Flat` attribute of `**`. I didn't use such a rule in my work, and an not sure how to get around it. Maybe just manually apply that type of simplification at the end of the evaluation.
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