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Jul 3, 2013, 9:21:34 AM7/3/13

to sage-...@googlegroups.com, sage-a...@googlegroups.com, sage-comb...@googlegroups.com

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/

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/

Jul 3, 2013, 9:38:00 AM7/3/13

to sage-...@googlegroups.com, sage-a...@googlegroups.com, sage-comb...@googlegroups.com

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
> 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.

(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

Jul 3, 2013, 9:40:51 AM7/3/13

to sage-a...@googlegroups.com

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

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

Jul 3, 2013, 10:03:09 AM7/3/13

to Travis Scrimshaw, sage-comb...@googlegroups.com, sage-...@googlegroups.com, sage-a...@googlegroups.com

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

:-)

> 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

:-)

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