Re: [Axiom-developer] BINGO,Curiosities with Axiom mathematical structures

[William Sit]

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.

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, 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.

William