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Re: [Axiom-developer] Re: BINGO,Curiosities with Axiom mathematical stru
From: |
Ralf Hemmecke |
Subject: |
Re: [Axiom-developer] Re: BINGO,Curiosities with Axiom mathematical structures |
Date: |
Tue, 14 Mar 2006 12:24:48 +0100 |
User-agent: |
Thunderbird 1.5 (X11/20051201) |
Hi Bill,
Hmmm... surely this must have been considered in the design of
Aldor. Are there no Aldor categories like DIRPCAT that take a
member of a domain as a parameter?
All categories in libaldor either take no parameter or just domain
parameters.
In libalgebra, I could only find
define ResidueClassRing(R: CommutativeRing, p: R): Category
define DirectProductCategory(dim: MachineInteger, T: ExpressionType):
So it the same problem occurs.
> This seems like a natural mathematical construction to me.
I somehow agree, but unfortunately, we have no clear statement from the
original designers about how Aldor's "Category" relates to mathematical
categories or order-sorted algebras. There is only the PhD thesis of
Doye. Maybe we should try to contact and ask people that appear in the
original Axiom book personally about it. A bit of history would
certainly be of value for Axiom.
If you look at DirProdCat as a function then clearly
DirProdCat(2,Z) = DirProd(3,Z) = with{first:%->Z}
so why would one want another output?
----------- dirprod.as
#include "aldor"
#include "aldorio"
define DirProdCat(n: Integer, R: Type): Category == with {
first: % -> R;
}
Well, clearly the application of the category 'DIRPCAT(2,INT'
makes good sense in Axiom, don't you think?
Well, since (I think) the category of pairs of rings makes sense, it
should also make sense in Axiom.
How else would you propose to define this category?
I have no idea, how to define this category in Axiom in another way.
But obviously, there is a problem if one wants to check whether a domain
satisfies that category.
How could we do without it in defining the domain 'DIRPROD(2,INT)'?
Hmm, I guess you would not like
define DirectProductCategory(T: Ring): Category == ...
since that bears less information.
To me this is the
reason why we want another output from your example Aldor
program. I would conclude that in fact internally (as you
already said above):
DirProdCat(2,Z) ~= with{first:%->Z}
There must be something additional on the right-hand side
that is not reflected in the syntax of the category constant
'with' clause.
Well, you are in some sense right. As defined in the Aldor User Guide
Section 7.5 if you define
define CatA: Category == with {foo: % -> %}
DomA: with {foo: % -> %} == add {foo(x:%): % == x}
then "DomA has CatA" returns false, because DomA is not explicityly
declared to be of type CatA. There is this invisible symbol %% for each
category from which a domain inherits. So in that sense
CatA ~= with{foo: %->%}
Ralf
PS: Enjoy the output
aldor -grun -laldor cattest.as
A has CatA : F
A has CatA 3 : F
A has CatA 3 : F
A has CatA Integer: T
A has CatA Boolean: F
-----------------
B has CatA : F
B has CatA 2 : T
B has CatA 3 : T
B has CatA Integer: F
B has CatA Boolean: F
of the following program... ;-)
It seems that domains as parameters are treated differently than
elements. Look especially at
define CatA(R: Type): Category == with {bar: () -> ()}
the "with" does not involve R in any way.
---- cattest.as ------------------
#include "aldor"
#include "aldorio"
define CatA: Category == with {foo: %->%}
define CatA(R: Type): Category == with {bar: () -> ()}
define CatA(n: Integer): Category == with {baz: () -> ()}
DomA(R: Type): CatA(R) == add {
bar(): () == {}
}
DomA(n: Integer): CatA(n) == add {
baz(): () == {}
}
main(): () == {
macro A == DomA Integer;
macro B == DomA 2;
import from Integer;
stdout << "A has CatA : " << (A has CatA) << newline;
stdout << "A has CatA 3 : " << (A has CatA 2) << newline;
stdout << "A has CatA 3 : " << (A has CatA 3) << newline;
stdout << "A has CatA Integer: " << (A has CatA(Integer)) << newline;
stdout << "A has CatA Boolean: " << (A has CatA(Boolean)) << newline;
stdout << "-----------------" << newline;
stdout << "B has CatA : " << (B has CatA) << newline;
stdout << "B has CatA 2 : " << (B has CatA 2) << newline;
stdout << "B has CatA 3 : " << (B has CatA 3) << newline;
stdout << "B has CatA Integer: " << (B has CatA(Integer)) << newline;
stdout << "B has CatA Boolean: " << (B has CatA(Boolean)) << newline;
}
main();
- RE: [Axiom-developer] Curiosities with Axiom mathematical structures, (continued)
- RE: [Axiom-developer] Curiosities with Axiom mathematical structures, Bill Page, 2006/03/10
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
- [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10
- RE: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/11
- Re: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/11
- Re: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Re: BINGO,Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
- RE: [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/13
- Re: [Axiom-developer] Re: BINGO,Curiosities with Axiom mathematical structures,
Ralf Hemmecke <=
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, William Sit, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
- RE: [Axiom-developer] Curiosities with Axiom mathematical structures, Bill Page, 2006/03/13