Hi,
> > Given algebras A and B, I would return C and the two maps m1: A -> C
> > and m2: B -> C:
>
> > (C, m1, m2) = A.tensor_product(B,ring=R)
>
> I might prefer to have something like
>
> C = A.tensor_product(B, ring=R)
>
> and then one can do
>
> C.coerce_map_from(A)
> C.coerce_map_from(B)
This would require a new tensor product class in order to represent C.
I think it might be desirable when taking the tensor product of two
matrix algebras to return a matrix algebra, or when taking a tensor
product of two polynomial rings to return a polynomial ring.
The other option is to have a genuine class which wraps these
rings as you indicate.
The suggested syntax above, and here:
> One would rarely need the maps themselves, as one
> could do C(a) and C(b), or even c + a, and the coercion model would
> handle it all well (here a, b, and c live in A, B, and C resp. of
> course).
assumes that A is distinct from B (or that there are no canonical
coercions from A to B or vice versa). However a common
construction is precisely to form a self tensor product or power
C = \otimes_{i=1}^n A, in which case the choice of the 2 or n
maps A -> C can not be decided by without specification of
the map.
> C would probably be implemented one of David Roe's wrapper
> classes, so Q tensor Z[x] would be Q[x], but would still know its
> factors.
A good design is very important.
In fact this is a vey generic categorical construction of (a sum or
coproduct in the category of rings). We should first consider how
general products and coproducts should be constructed, and set
up a common infrastructure and syntax. It needs to be (1) easy
and natural to use, (2) mathematically correct and complete.
The homomorphisms for products and coproducts are essential
parts of the construction so they should be an accessible part
of the design. The homomorphisms themselves should be fast
to apply, and their design should take into account canonical
homomorphisms.
--David
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