On Sat, 28 Oct 2006 16:39:10 -0700, David Harvey
<[EMAIL PROTECTED]> wrote:
> Another issue that slightly complicates this is base rings. When I do
> x * y, I don't always want to coerce into the same parent; sometimes
> I want to coerce x into the "base ring" of y (or vice versa), which
> will mean different things depending on what the rings/algebras are.
What are the rules for algebras going to be?
Suppose R and S are commutative rings (for simplicity) and M and N are
R modules and K is an S-module.
Choose x in M and y in N. Then coerce should work exactly as before.
All the _coerce_ maps are R-module homomorphisms, etc., and there
is nothing new going on here. Next assume that z in K. To do x + z,
we have to worry about base rings.
PROPOSED RULE: Suppose there is a coerce map (of rings!) R --> S.
Then to compute x+z we first compute Z = M.base_extend(S), which is an
S-module
such that _coerce_ on elements of M gives elements of Z in the natural way.
We are now do the add
Z._coerce_(x) + z
using the usual rules.
Next, consider multiplication. Suppose in addition that M, N, and K are
all
algebras. Suppose that x in M and z in K. Multiplication should work in
exactly the same way as addition above.
PROPOSED RULE: Suppose there is a coerce map (of rings!) R --> S.
Then to compute x*z we first compute Z = M.base_extend(S), which is an
S-module
equipped with the data that _coerce_ on elements of M gives elements of Z
in
the natural way. We are now do the add
Z._coerce_(x) + z
using the usual rules.
You know, when an object such as Z above decided that elements of some
other object (e.g., M) can be coerced to it, that should also be recorded
with M. And moreover, the process of recording this information with M
should propogate, e.g., each object from which M can coerce elements,
should be told that M can now also coerce elements to Z. This will
propogate
around until it stops, thus walking a component of some sort of reference
graph.
This will help very much to make the coercion graph transitive even
in the presence of automatically generated objects like base extensions.
-- William
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