Axioms for semigroups: L,R,J,H-trivial, aperiodic

[nthiery]

This ticket defines the usual Green axioms for semigroups, and provides stubs for the corresponding categories:

sage: Semigroups().LTrivial()
Category of ltrivial semigroups
sage: Semigroups().LTrivial().RTrivial()
Category of jtrivial semigroups
sage: Semigroups().HTrivial().Finite()
Category of finite aperiodic semigroups
sage: Semigroups().Commutative().RTrivial()
Category of commutative jtrivial semigroups

This is a first step toward #18001.

CC: @hivert @saliola @avirmaux @sagetrac-sage-combinat

Component: algebra

Author: Nicolas M. Thiéry

Branch/Commit: 6158f0e

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/18265

[nthiery]

Branch: u/nthiery/semigroups/axioms-18265

[nthiery]

Work Issues: Create a ticket and reference it in the doc

[nthiery]

New commits:

2e678e6

18265: setup axioms for L,R,J,H trivial and aperiodic semigroups

[nthiery]

Commit: 2e678e6

[nthiery]

comment:4

I am setting it to needs review to get the patchbot to pick it up.

[tscrim]

comment:5

It would be good for the new files to have the standard copyright template: http://doc.sagemath.org/html/en/developer/coding_basics.html#headings-of-sage-library-code-files.

Patchbot report has also come in and states errors in

sage -t --long src/sage/categories/semigroups.py  # 5 doctests failed

along with docbuilding issues.

[sagetrac-git]

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

f5185bc	Merge branch 'develop' into semigroups/axioms-18265
ba12c41	18265: minor doc improvement, non verbose TestSuite, marked category example as not implemented (in this ticket)

[sagetrac-git]

Changed commit from 2e678e6 to ba12c41

[sagetrac-git]

Changed commit from ba12c41 to 2dc3f8a

[sagetrac-git]

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

282be1f	18265: added reference to #20515 + some explanations
01a9efe	18265: fixed the copyright headers of the new files
2dc3f8a	18265: added the new categories to the reference manual

[sagetrac-git]

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

77776ed	18265: fixed monoid to semigroup in several spots of the documentation
18baeef	18265: uncapitalize module title

[sagetrac-git]

Changed commit from 2dc3f8a to 18baeef

[sagetrac-git]

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

61cf0f4

18265: fixed references to wikipedia

[sagetrac-git]

Changed commit from 18baeef to 61cf0f4

[sagetrac-git]

Changed commit from 61cf0f4 to 7943a39

[sagetrac-git]

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

7943a39

18265: proofread the html documentation

[nthiery]

Changed work issues from Create a ticket and reference it in the doc to none

[nthiery]

comment:12

Thanks Travis for having a look. The patchbot should go green now, and I have finished cleaning things up.

Up for review for real now!

[tscrim]

comment:13

Some comments:

• Do we want these to be axioms since these properties for subcategories do not make as much sense (e.g., for groups, L-trivial = R-trivial = J-trivial = category consisting only of the trivial group, correct?)
• Typo "preoder".
• Do we want to spend some time trying to beautiful axiom printing (i.e., "J-trivial" instead of "jtrivial")?
• I would like to see the defining properties to the category level documentation, not only in the subcategory methods.

[sagetrac-git]

Changed commit from 7943a39 to d09fe3b

[sagetrac-git]

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

d09fe3b

18265: typo: preoder -> preorder

[nthiery]

comment:15

Hi Travis,

Thanks for the review.

Replying to @tscrim:

• Do we want these to be axioms since these properties for subcategories do not make as much sense (e.g., for groups, L-trivial = R-trivial = J-trivial = category consisting only of the trivial group, correct?)

We want them to be axioms to interact nicely with each other: there
will be many interesting combinations between the Finite / X-trivial /
Unital axioms (and more will come, like Regular, ...). In fact
implementing those was one of my original motivation for implementing
axioms in Sage :-)

Indeed Groups().XTrivial() is, well, trivial, so that's boring. But
this is only polluting the namespace of the Groups() category, not the
groups themselves, so that's a minimal annoyance.

• Typo "preoder".

Thanks for catching. Fixed!

• Do we want to spend some time trying to beautiful axiom printing
  (i.e., "J-trivial" instead of "jtrivial")?

This would be cute, indeed. I'll have a quick look if it's trivial to
do. Otherwise, we can leave it for later; that will be easy to change
anyway.

• I would like to see the defining properties to the category level
  documentation, not only in the subcategory methods.

I agree that documentation at the category level would be highly
desirable. But I don't want to duplicate manually the documentation.
That's a general problem that requires a general solution by having a
nice automatically generated default documentation for categories with
axioms. See also the related:

#16363

In the mean time the subcategory methods are the most appropriate
location for the doc, since this makes it accessible for all
subcategories by XXX.Jtrivial?

---

New commits:

d09fe3b

18265: typo: preoder -> preorder

[tscrim]

comment:16

Replying to @nthiery:

Replying to @tscrim:

• Do we want these to be axioms since these properties for subcategories do not make as much sense (e.g., for groups, L-trivial = R-trivial = J-trivial = category consisting only of the trivial group, correct?)

We want them to be axioms to interact nicely with each other: there
will be many interesting combinations between the Finite / X-trivial /
Unital axioms (and more will come, like Regular, ...). In fact
implementing those was one of my original motivation for implementing
axioms in Sage :-)

I think we should also add that an inverse H-trivial monoids implies J-trivial.

Indeed Groups().XTrivial() is, well, trivial, so that's boring. But
this is only polluting the namespace of the Groups() category, not the
groups themselves, so that's a minimal annoyance.

Considering there is exactly 1 X-trivial group (up to isomorphism), there won't be any additional pollution from Parent/ElementMethods either. So I agree, minimal annoyance.

• Do we want to spend some time trying to beautiful axiom printing
  (i.e., "J-trivial" instead of "jtrivial")?

This would be cute, indeed. I'll have a quick look if it's trivial to
do. Otherwise, we can leave it for later; that will be easy to change
anyway.

Let me know.

[sagetrac-git]

Changed commit from d09fe3b to 06d9bb4

[sagetrac-git]

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

06d9bb4

18265: fixed category_with_axiom.uncamelcase to also split on single capitals; e.g. JTrivial is displayed as j trivial, not jtrivial

[nthiery]

comment:18

Replying to @tscrim:

I think we should also add that an inverse R-trivial monoids implies J/H-trivial.

Yes, but only when we will have the "inverse" axiom in the sense of
local inverses in semigroups. Our current Inverse axiom is about
having a global inverse (it's indeed annoying that we have this naming
conflict, but there is nothing much we can do about it).

Of course an R-trivial monoid with a global inverse is J/H trivial,
but that's not super interesting :-)

This would be cute, indeed. I'll have a quick look if it's trivial to
do. Otherwise, we can leave it for later; that will be easy to change
anyway.

Let me know.

I made a quick fix, so that JTrivial is displayed as j trivial and not
jtrivial; it only slightly better, but requires no special casing and
makes for a more consistent implementation of uncamelcasing.

It's been discussed several times that axioms like AdditiveAssociative
would be better displayed as additive-associative, to raise
ambiguities in expressions such as:

    Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas

This would make the j trivial into j-trivial as a by-product
without any special casing. However this is a relatively invasive
change (it will require updating a bunch of doctests"), so I vote for
doing this in a separate ticket.

Do you agree that this is good enough for now?

[tscrim]

Reviewer: Travis Scrimshaw

[tscrim]

comment:19

Replying to @nthiery:

Replying to @tscrim:

I think we should also add that an inverse R-trivial monoids implies J/H-trivial.

Yes, but only when we will have the "inverse" axiom in the sense of
local inverses in semigroups. Our current Inverse axiom is about
having a global inverse (it's indeed annoying that we have this naming
conflict, but there is nothing much we can do about it).

Of course an R-trivial monoid with a global inverse is J/H trivial,
but that's not super interesting :-)

I'm somewhat leaning towards having the implication for the global inverse, but considering the category of X-trivial groups is, well, trivial, I don't feel strongly about this. So if you don't feel like doing it, then you can set this to a positive review.

This would make the j trivial into j-trivial as a by-product
without any special casing. However this is a relatively invasive
change (it will require updating a bunch of doctests"), so I vote for
doing this in a separate ticket.

Yes, I believe this should be on a separate ticket.

[sagetrac-git]

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

6158f0e

18265: added the (dummy) implication H-trivial -> J-trivial for groups

[sagetrac-git]

Changed commit from 06d9bb4 to 6158f0e

[nthiery]

comment:21

Replying to @tscrim:

I'm somewhat leaning towards having the implication for the global inverse, but considering the category of X-trivial groups is, well, trivial, I don't feel strongly about this. So if you don't feel like doing it, then you can set this to a positive review.

Done and pushed!

[tscrim]

comment:22

Thank you.

[vbraun]

Changed branch from u/nthiery/semigroups/axioms-18265 to 6158f0e