Re: Non-commutative algebra

• To: mathgroup at smc.vnet.net
• Subject: [mg13129] Re: [mg13053] Non-commutative algebra
• From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
• Date: Tue, 7 Jul 1998 03:44:32 -0400
• References: <199807042044.QAA02851@smc.vnet.net.>
• Sender: owner-wri-mathgroup at wolfram.com

```I replied to David Withoff's message before I had a chance to carefully
read the message from Daniel Lichtbau. Since this message essentially
deals with all the points I raised I am now quite satisfied.

Andrzej

At 4:29 PM -0500 98.7.6, Daniel Lichtblau wrote:
>Andrzej Kozlowski wrote:
>>
>> I have a question which (probably) can only be answered by a wri
>> insider. Since it concerns a "obsolete" version of Mathematica I
>> suppose it will not be seen as terribly important, but still I would be
>> very grateful if someone would satisfy my curiosity.
>>
>> Recently I was showing some computations in non-commutative algebra to
>> my students. This is very easy to do in Mathematica 3.0. Basically all
>> you need to do is to Clear the attribute Orderless in Times and use
>> Mathematica as usual. However, my university (for various reasons)
>> still has Mmma 2.2 installed on all the computers in the class where I
>> teach, so after making my notebook using 3.0 I tried the same
>> computations in 2.2. To my surprise the answers came out wrong! I soon
>> realized that Expand in Mathematica 2.2 has commutativity "built in"
>> quite independently from the Orderless attribute of Times. To see this
>> all you need to do is to evaluate:
>>
>> In[2]:=
>> Unprotect[Times]
>> Out[2]=
>> {Times}
>> In[3]:=
>> ClearAttributes[Times,Orderless]
>> In[4]:=
>> Protect[Times]
>> Out[4]=
>> {Times}
>> In[5]:=
>> Expand[(a+b)^2]
>>
>> In versions 3.0 and 2.2. In 3.0 you will correctly get:
>>
>> b*a + a^2 + a*b + b^2
>>
>> but in 2.2 you get
>>
>> a^2+2a*b+b^2
>>
>> I have looked through the documentation for Expand in both versions, and
>> through various accounts of the changes in Version 3.0 (e.g.
>> Mathematica Journal Vol. 6 Issue 4) but cannot find any mention of
>> commutativity. It seems to me very odd to deliberately "hard-wire"
>> commutativity in Expand. My question is: Why this behavious in version
>> 2? Was it ever recognized as a bug  and fixed v. 3 or was the fix just
>> a side-effect of some other changes?
>
>It was indeed recognized as a bug and fixed. Well, partly fixed. Power
>is still in need of some work, because it uses Times.
>
>In[9]:= ee = a*b;
>
>In[10]:= ClearAttributes[Times, Orderless]
>
>In[11]:= ee * ee
>Out[11]= a b a b
>
>In[12]:= ee^2
>          2  2
>Out[12]= a  b
>
>This matters e.g. if want to Expand the square of (x^2 + x*y + y^2).
>
>
>> (Fortunately by using Distribute I was able to define my own Expand in
>> 2.2 which does not assume commutativity, so this problem no longer has
>> a practical significance for me, it's just a matter of wanting to
>> understand what happened).
>>
>> Andrzej Kozlowski
>> ...
>
>
>We recommend that you use, say, NoncommutativeMultiply and attach rules
>for integer powers to it to do the expansion. It is virtually never safe
>to change the Orderless attributes of low-level operations such as
>Plus/Times/Power, and moreover (as you found) it is hard for us to
>obtain the correct functionality when you do make such changes.
>
>
>Daniel Lichtblau
>Wolfram Research

Andrzej Kozlowski
http://sigma.tuins.ac.jp/

```

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