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Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structu
From: 
Gabriel Dos Reis 
Subject: 
Re: [Axiomdeveloper] 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 OOcentric 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 OOcentric 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: [Axiomdeveloper] Curiosities with Axiom mathematical structures, (continued)
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/04
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/07
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/07
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/08
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/09
 [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/09
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/09
 Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures,
Gabriel Dos Reis <=
 RE: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Bill Page, 2006/03/09
 Re: [Axiomdeveloper] BINGO,Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 Re: [Axiomdeveloper] BINGO, Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/10
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
 RE: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Bill Page, 2006/03/10
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 [Axiomdeveloper] Re: BINGO, Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/10