refuse to compare two ideals that we don't know how to compare

[yyyyx4]

See #37409: Most users seem to expect ideals to compare equal if and only if they are mathematically equal. The current behaviour alternates between mathematical comparison and "compare generators" comparison, depending on the base ring, which is (1) internally inconsistent and (2) unintuitive to most users.

In this patch I propose to fail by default in the ._richcmp_() method for generic ideals, except for trivial cases in which equality can easily be decided: If the parent ring does not implement a mathematically meaningful comparison operation for ideals, it should fail in a noticeable way, rather than silently falling back to a much weaker notion of equality.

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[user202729]

This is only a partial order... right? cf. how are ZZ[x,y] currently compared?

[yyyyx4]

Yes. As far as I can tell from a quick experiment, ideals of ℤ[x,y] are compared by set inclusion, while ideals of ℤ[x] are compared by generators. This patch renders the behaviour uniform in that regard. (It is possible that I missed some other rings that compare ideals in a non-mathematical way.)

[user202729]

wait, seriously.

sage: R.ideal(x) < R.ideal(y)
False
sage: R.ideal(x) > R.ideal(y)
False

which means…

sage: sorted([R.ideal(x), R.ideal(y)])
[Ideal (x) of Multivariate Polynomial Ring in x, y over Integer Ring,
 Ideal (y) of Multivariate Polynomial Ring in x, y over Integer Ring]
sage: sorted([R.ideal(y), R.ideal(x)])
[Ideal (y) of Multivariate Polynomial Ring in x, y over Integer Ring,
 Ideal (x) of Multivariate Polynomial Ring in x, y over Integer Ring]

at least with the previous behavior, sorting is order-invariant.

actually…

sage: sorted([frozenset({1}), frozenset({2})])
[frozenset({1}), frozenset({2})]
sage: sorted([frozenset({2}), frozenset({1})])
[frozenset({2}), frozenset({1})]

there is a precedent. Maybe it's actually fine.

sage: frozenset({frozenset({2}), frozenset({1})}) == frozenset({frozenset({1}), frozenset({2})})
True

most Python containers are hash-based instead of order-based so they're unaffected by non-total-ordering. (doctest output checker may suffer however)

[yyyyx4]

Yup. Even the comparison between standard Python sets is only partial. Related comments: #37409 (comment), #35546 (comment).

[user202729]

some other _richcmp_ returns NotImplemented (object) instead of raise. E.g. rings/ring_extension_morphism.pyx

[yyyyx4]

As far as I can tell, in this context, NotImplemented is just another way of saying False — which is not what we want.

[cxzhong]

Yes. Two ideals should be compared with respect to the set inclusion. I think it is naturally. You can not compare two ideals with like (x) and (y) in ZZ[x,y]. In that case, we should not return false. We should just throw an error said they are not comparable.

But I think we may need singular library to decide a>b false and b>a false. To raise a error.