Sorry for the late replies, was on a conference trip.
On Aug 24, 1:33 pm, "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr>
wrote:
> - The problem with the map a -> (a,0) is only that 1_A is mapped to
> (1_A,0) which is not 1_{A\oplusB} = (1_A,1_B), right?
This is an "easy to spot" problem. The deep underlying problem is the
universal property.
> Otherwise said, the category of NonUnitalAlgebras (which is not yet
> implemented in Sage) indeed has adirectsum?
No, as John points the problem is with the universal property. The
"filler map" required by such property always exists (and it is
unique) at the level of the underlying vector spaces, but there is no
way to guarantee that it is a morphism of algebras (in general, it
isn't).
Back to terminology, what it is implemented is a categorical product,
which in this category (algebras) coincides with the vector spaces
direct product. If we want to implement a categorical coproduct, then
we are aiming for the free product, not any "direct sum".
> - I am in the train right now, without appropriate references. For,
> say, monoids (additive, or multiplicative), how should the
> operation of taking cartesian products be called?
It is also called direct product, same as in groups (where we have
direct and semi-direct products).
Cheers,
Javier
PS: I am cross-posting this to sage-combinat-devel. Maybe we should
move the discussion there?
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