RFC: a good name the category of algebras that are not necessarily associative nor unital

[Nicolas M. Thiery]

Dear category fans,

One of the features introduced by the category patch #10963 is a new
category for algebras that are not necessarily associative nor unital.
This is a call for suggestions and votes for a good name for it.

- ``Algebras``: that's wikipedia's choice [1]. However using this name
would be backward incompatible, since ``Algebras'' in Sage currently
refers to associative unital algebras. At this point in time, I
don't want to open another can of worm on a ticket that is already
way too big. But we could think about it in a later ticket.

Note: many textbooks/papers use algebra as a short hand for
associative unital (and sometimes commutative) algebras; but they
usually specify this explicitly at the beginning, and they are each
in a smaller context than Sage's.

- ``NonAssociativeNonUnitalAlgebras``: that's what's currently
used in the patch. Of course this terminology is not great because
an associative algebra would then be a special case of a non
associative algebra ...

Note: I remember someone mentioning once that there was a tiny
difference between ``non-associative'' and ``not associative'' that
could possibly make this acceptable but I have no informed opinion
myself.

- ``MagmaticAlgebras``: this was suggested by Florent, referring to
the terminology used in the operad community; see e.g. 13.8 of
Loday&Valette [2]

- Something else?

Thanks for your feedback!

Cheers,
Nicolas

[1] http://en.wikipedia.org/wiki/Algebra_%28ring_theory%29
[2] http://math.unice.fr/~brunov/Operads.pdf

--
Nicolas M. Thi�ry "Isil" <nth...@users.sf.net>
http://Nicolas.Thiery.name/

[Nicolas M. Thiery]

On Wed, Jul 03, 2013 at 03:21:34PM +0200, Nicolas M. Thiery wrote:
> One of the features introduced by the category patch #10963 is a new
> category for algebras that are not necessarily associative nor unital.
> This is a call for suggestions and votes for a good name for it.

On a similar note: this ticket also introduces a category for sets
(E,+,*) where (E,+) is an additive magma, (E,*) is a magma, and *
distributes over +. In other words a ring with no axiom whatsoever but
distributivity. In the current patch, this category is dubbed
DistributiveMagmasAndAdditiveMagmas, by lack of creativity ...

Better suggestions welcome!

In the longer run, I'll also need a name for the same category,
without the distributivity axiom.

Cheers,
Nicolas

[David Kohel]

Dear Thierrry,

Definitely NonAssociativeNonUnitalAlgebras, or perhaps
NonassociateNonunitalAlgebras, since in English "non"
is a prefix not a negation (nonassociative, nonunital with
or without hyphen).

I don't find it a problem that AssociativeNonunitalAlgebras
should be a subcategory, and if Nonunital is dropped, then
it is not even a subcategory of (unital) Algebras since unital
algebras have more restrictive morphisms.

In an undergraduate course in Sydney, there was an
unfortunate choice of definition of (homo)morphism of
(unital) rings which didn't send 1 to 1. Consequently
I needed to redefine homomorphisms when I was teaching
commutative algebra.

I've no particular opinion on Magma terminology.

Cheers,

David

[Nicolas M. Thiery]

On Wed, Jul 03, 2013 at 06:47:12AM -0700, Travis Scrimshaw wrote:
> For the category of non-unital rings, how about Rngs? (I'm half joking.)

Actually that joke, for good or bad, is what's already been
implemented in successively Axiom, MuPAD, and Sage :-) They even had
Rigs. And Rgs.

But here we want to go further and remove all other axioms
(associativity, additive inverse, ...) but distributivity.

> Somewhat more serious, GeneralAlgebras/GeneralRings? I think
> overall we should be consistent between rings and algebras.

That would be a plus indeed.

> On the math side of things, doesn't a ring in general has to be
> distributive; if so, then I think (distributive) non-* rings
> should be called *Rings and non-distributive things should be
> MultiplicativeAndAdditiveMagmas (or maybe
> AdditiveAndMultiplicativeMagmas).

Thanks for your input.

> Also do we want/have a category for skew fields (a.k.a. division
> rings)?

sage: Rings().Division()
Category of division rings
sage: Rings().Division().Commutative()
Category of fields
sage: Rings().Division().Finite()
Category of finite fields

:-)