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

```