Cartesian product of rings

[nthiery]

This ticket lifts up cartesian product features to the categories recently created in #10963, where they belong. Thanks to this and to the definition of CartesianProducts for the Distributive axiom, we have now:

sage: Fields().CartesianProducts()
Join of Category of rings
    and Category of Cartesian products of distributive magmas and additive magmas
    and Category of Cartesian products of semigroups 
    and Category of Cartesian products of unital magmas
    and Category of Cartesian products of additive inverse additive unital additive magmas 
    and Category of Cartesian products of additive commutative additive magmas 
    and Category of Cartesian products of additive semigroups

And this works smoothly for all variants of rings (semirings, ...).

This fixes a piece of : #15425.

CC: @sagetrac-sage-combinat @nathanncohen @videlec @dimpase

Component: categories

Keywords: cartesian product

Author: Nicolas M. Thiéry

Branch/Commit: eced39e

Reviewer: Nathann Cohen

Issue created by migration from https://trac.sagemath.org/ticket/16405

[nthiery]

Branch: u/nthiery/cartesian_product_of_rings

[nthiery]

New commits:

8b547db

16405: cartesian product of rings and generalization of many cartesian product features

[nthiery]

Commit: 8b547db

[nathanncohen]

comment:3

Is it normal that we see

    and Category of Cartesian products of additive inverse additive unital additive magmas 
    and Category of Cartesian products of additive commutative additive magmas 
    and Category of Cartesian products of additive semigroups

instead of "Additive Group" ?

Nathann

[nathanncohen]

comment:4

Helloooooooooooo !!

Some comments/questions:

• What are sub/neg doing in AdditiveMagma().AdditiveUnital() ?

• consistancy -> consistency

• class nesting is cool and everything but you have to scroll non-stop with
  stuff like emacs.... Do you have a trick to know in which sub/sub/sub/sub
  class you are writing your code ?

• You implement the neg function for cartesian product of AdditiveInverse
  Magma, but don't you also need to say somewhere what exactly is the "1"
  element of a cartesian product of UnitalMagma ?

• You do things like :

  Magmas().Unital().Inverse()
  

  It feels like the syntax should be Magmas().Unital().Inverse(), which would imply Unital. What do you think ?

• maybe I misread but you seem to implement __invert__ in
  Magmas().Unital().CartesianProduct(). Shouldn't it be several lines above, in
  Magmas().Unital().Inverse().CartesianProduct() ?

Nathann

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

27f6c9a

16405: better handling and documentation of partially defined `__neg__` and __invert__

[sagetrac-git]

Changed commit from 8b547db to 27f6c9a

[nthiery]

comment:7

Salut Nathann,

Thanks for the quick review!

Replying to @nathanncohen:

Some comments/questions:

• What are sub/neg doing in AdditiveMagma().AdditiveUnital() ?
• maybe I misread but you seem to implement __invert__ in
  Magmas().Unital().CartesianProduct(). Shouldn't it be several lines above, in
  Magmas().Unital().Inverse().CartesianProduct() ?

The notion of inverse (resp. negation) can be defined as soon as there
is a unit, thought some elements might not have inverses, or have non
unique inverses. So the method __invert__ (resp. __neg__) makes
sense, even if it's only partially defined.

I just pushed a fix that better handles those partially defined
inverse / negation for cartesian products, and clarifies the
documentation.

• consistancy -> consistency

Fixed!

• class nesting is cool and everything but you have to scroll non-stop with
  stuff like emacs.... Do you have a trick to know in which sub/sub/sub/sub
  class you are writing your code ?

Alas no. Code folding is really a life saver here, but I still need to
dig through the emacs packages to find one that implements such
folding with a reasonable interface.

• You implement the neg function for cartesian product of AdditiveInverse
  Magma, but don't you also need to say somewhere what exactly is the "1"
  element of a cartesian product of UnitalMagma ?

I think this was there. In any cases, right now we have:

Magmas.Unital.CartesianProducts.ParentMethods.one
Magmas.Unital.CartesianProducts.ElementMethods.__invert__

AdditiveMagmas.AdditiveUnital.CartesianProducts.ParentMethods.zero
AdditiveMagmas.AdditiveUnital.CartesianProducts.ElementMethods._neg_

• You do things like :

  Magmas().Unital().Inverse()
  

  It feels like the syntax should be Magmas().Unital().Inverse(), which would imply Unital. What do you think ?

You mean Magmas().Inverse() as a shorthand for
Magmas().Unital().Inverse()? I wondered about this, but so far
preferred to only define the Inverse axiom if there exists a unit.

One thing is that there exist a notion of inverse semigroups where
an element is only requested to have a "local" inverse (technically it
has to be invertible in some submonoid whose unit need not be a unit
for the whole semigroup). So I'd rather leave the room free for future
extensions.

Cheers,
Nicolas

---

New commits:

27f6c9a

16405: better handling and documentation of partially defined `__neg__` and __invert__

---

New commits:

27f6c9a

16405: better handling and documentation of partially defined `__neg__` and __invert__