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Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structu
From: |
Gabriel Dos Reis |
Subject: |
Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures |
Date: |
14 Mar 2006 01:30:45 +0100 |
William Sit <address@hidden> writes:
| Martin:
|
| Good work (I'm impressed :-). I think your solution is in fact the way
| mathematicians build up algebraic structures: first define the underlying set
| and the basic operations, then prove that the axioms of certain algebraic
| structure are satisfied. Here you declare (rather than prove) these using
| Aldor's 'extend' mechanism.
Yes, and remarkable they don't use OO-centric view that hinder
evolution and improvements.
|
| The beauty of your solution is that the notation (naming of the operations) is
| defined in the domain itself and then the algebraic properties declared by
| extension.
|
| The slight disadvantage is that these structural operations must be listed
with
| each domain,
no, only the independent parameters necessary to define the algebraic
structure need to be passed as parameter. The rest is function of
those and default defined (or just listed) in the interface.
| and there is no default way (yet). So perhaps in MyMonoid, with
| parameters, there can be a default syntax, such as:
|
| MyMonoid(T:Type, default{*:(T,T)->T)}):Category
| MyDualMonoid(T:Type, default{*:(T,T)->T, o:(T,T)->T}):Category ==
| with{MyMonoid(T); MyMonoid(T, o)};
| MyAbelianMonoid(T:Type, default{+:(T,T)->T}):Category ==
| with{commutative(+); MyMonoid(T, +), ...};
| MyRing(T:Type, default{*:(T,T)->T, +:(T,T)->T}:Category ==
| with{MyAbelianMonoid(T), MyMonoid(T), ...};
| MyCommutativeRing(T:Type, default{*:(T,T)->T, +:(T,T)->T}:Category ==
| with{MyAbelianMonoid(T), MyAbelianMonoid(T,*), ...};
|
| (the above omits the units for the operations, which could be added easily) so
| that
|
| extend MyInteger: MyRing(MyInteger) == add;
|
| would make sense.
It suffers from the old OO-centric view, which is causing the problem
I mentioned in the start of this thread. The problem here is that
you're focusing on the carrier set (MyInteger), intead of focusing on the
operations that give structure to the set.
-- Gaby
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, (continued)
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, William Sit, 2006/03/04
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/07
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, William Sit, 2006/03/07
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/08
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/09
- [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/09
- Re: [Axiom-developer] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/09
- Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10
- Re: [Axiom-developer] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
- Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures,
Gabriel Dos Reis <=
- RE: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/09
- Re: [Axiom-developer] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
- Re: [Axiom-developer] BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
- [Axiom-developer] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10
- Re: [Axiom-developer] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
- 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