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Re: [Axiom-developer] Curiosities with Axiom mathematical structures

From: Ralf Hemmecke
Subject: Re: [Axiom-developer] Curiosities with Axiom mathematical structures
Date: Thu, 09 Mar 2006 23:30:46 +0100
User-agent: Thunderbird 1.5 (X11/20051201)

On 03/09/2006 03:46 PM, Martin Rubey wrote:
I wouldn't want to ask "Integer has Monoid", since this doesn't make any sense
to me. I'd like to ask "Integer has Monoid(Integer, *)" or "Integer has Monoid(*)"

Well, if one interprets Monoid as the category of monoids then

  Integer has Monoid

just say that the integers (now the question is whether you mean the integers with the additive or the multiplicative structure) are an object in the category of monoids.

Integer is a name for a structure with carrier set

{0, 1, -1, 2, -2, ...}

and operations {+, *, 0, 1, ...}.

Integer is certainly not the carrier set alone.
How would you mathematically express that the integers belong to the category of monoids? You would probably say that

F(Integer) is an object in the category of monoids

where F is a functor from the category of rings (or rather the category in which Integer really lives) that forgets every extra structure of a ring an just selects a monoid structure. Yes, the functor F decides whether you mean the additive or the multiplicative structure.

I hope, some category experts correct me, if I am wrong. I'm not so fluent in that language.

Anyway there is clearly something missing in the "has" construction if that would have to be written mathematically.

Simply think of a category Foo with hundreds of exported function, would you
like to write

   Dom has Foo(f1, f2, ..., f100)

no, but wait a moment: It is obvious to me that I don't want to have all
exported functions as parameters. Only certain "defining" functions, like:

Integer has Monoid(*, 1);
Integer has Ring(+, *, 1);

Can you think of an example where more than, say 5, parameters would be 

A partial differential ring (0,1,+,*) with n derivations. ;-) But maybe you prefer k automorphisms in order to get a difference algebra.


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