
From:  Ralf Hemmecke 
Subject:  Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures 
Date:  Thu, 09 Mar 2006 23:30:46 +0100 
Useragent:  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 senseto 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 Monoidjust 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 monoidswhere 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 desirable?
A partial differential ring (0,1,+,*) with n derivations. ;) But maybe you prefer k automorphisms in order to get a difference algebra.
Ralf
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