some care about algebra_generators

[fchapoton]

add type annotations to some of them

also change some of them to return Family objects or tuple

related to #41028 for Jordan algebras

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have created tests covering the changes.
• [x I have updated the documentation and checked the documentation preview.

[github-actions]

Documentation preview for this PR (built with commit dc3ce8c; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[cxzhong]

LGTM. Thank you. Now it fixes the infinite loop bug in python 3.14

[cxzhong]

┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.8.beta6, Release Date: 2025-10-06              │
│ Using Python 3.14.0. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
┏━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Warning: this is a prerelease version, and it may be unstable.     ┃
┗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┛
sage: F.<x,y,z> = FreeAlgebra(QQ)
sage: J = JordanAlgebra(F)
sage: J.gens()
Lazy family (Term map(i))_{i in Free monoid on 3 generators (x, y, z)}

It is the right output?

[fchapoton]

yes, this is the expected output.

[user202729]

Looking at the behavior, the reason for the change (test failure in Jordan algebra) is that in Python 3.14, tuple() no longer call __len__. So previously __len__ errors out, while now it just loop forever.

This change is caused by python/cpython#127758 which just modify the internal implementation of a function that converts from a sequence to a tuple.

[tscrim]

I am -1 on this change as the output of gens is expected to be a tuple (or at least behave like one).

[user202729]

I am -1 on this change as the output of gens is expected to be a tuple (or at least behave like one).

there are infinitely many generators.

the current behavior (Python ≤ 3.13) is to raise an error, because tuple() internally call __len__ which raise an error and short circuit the next step. In Python 3.14, tuple() no longer call __len__.

Maybe you'd prefer raising an error (and maybe tell the user to look at .algebra_generators()) instead?

( @vbraun note that this pull request is un-positive-reviewed for now. I don't think you have automatic notification on status change after first positive review. )

[tscrim]

I am -1 on this change as the output of gens is expected to be a tuple (or at least behave like one).

Maybe you'd prefer raising an error (and maybe tell the user to look at .algebra_generators()) instead?

Yes, I think that proposed behavior is best. (There is also an argument for moving away from the ambiguous gens() too.) It's possible we are not consistent about gens() (I might even have been guilty of not doing this myself...), and this might be something we should discuss setting a standard for. I haven't checked this at all. However, I feel like we should try to have some consistency...

[user202729]

in the case of J above, is J actually finitely generated as a Jordan algebra? It isn't obvious at all. (I suspect you can argue that it isn't, because suppose otherwise, J has a natural graded structure, there is a maximum degree of the generators, and you compare the dimension of the different levels or something)

On the other hand, we certainly are including some redundant entries in the list of algebra generators, for example

sage: import itertools
....: list(itertools.islice(J.algebra_generators(), 0, 10))
[1, x, y, z, x^2, x*y, x*z, y*x, y^2, y*z]
sage: J(x) * J(x)
x^2

so x^2 wouldn't be necessary to be included there.

[tscrim]

Yes, I agree that there are redundant generators, but this is allowed. It is not finitely generated for the $J(F_2)$ case. There is a natural (multi)grading on the Jordan algebra $J(A)$ inherited from the (multi)grading on the algebra $A$, which is clear from the definition of the product $x \circ y := (xy + yx)/2$. We can see that $j \circ j' = j' \circ j$ for any elements $j, j' \in J(A)$. Now suppose $F_2$ is the free algebra generated by $x,y$. I claim that we cannot find $x^k y$ for all $k > 1$ if we are generated by $M_k$ all monomials of total degree $\leq k$. To see this, we consider the multigrading $(k, 1)$, which is $k+1$ dimension spanned by $\{x^k y, x^{k-1}yx, x^{k-2}yx^2, \ldots, yx^k\}$. So the only possible products we can get are

$$2 \cdot x^j \circ x^{k-j} y = x^ky + x^{k-j} y x^j \qquad\qquad (0 < j \leq k).$$

In other words, this is a spanning set for the Jordan subalgebra of $F_2$ generated (as a Jordan algebra) by $M_k$, which is only $k$ dimensional. Hence we are missing a basis element in this multidegree that needs to be a generator, and so $J(F_2)$ is infinitely generated.

Moreover, determining an exact generating set for $J(F_2)$ seems nontrivial (much less for the general case).