EnumeratedFamily should use integers as keys

[gh-mwageringel]

This ticket solves the following problems involing EnumeratedFamily:

• The keys of an enumerated family should be integers. Currently, the elements of the family itself are returned.

  sage: f = Family(Permutations(4))
  sage: f.keys()
  Standard permutations of 4
  sage: list(f.keys()) == list(f)  # should not be True here
  True
  

  Because of this, looking up elements by key as in f[next(iter(f.keys()))] raises a type error. For example, this is used in DisjointUnionEnumeratedSets.

• Finite enumerated families should be in the category of FiniteEnumeratedSets(), while currently it is only in EnumeratedSets():

  sage: Family(Permutations(4)) in FiniteEnumeratedSets()  # should be True
  False
  

• Infinite enumerated families are currently indexed by NonNegativeIntegers() from sage.categories.example.infinite_enumerated_sets. This ticket replaces the usage of these example numbers by those in sage.sets.non_negative_integers. This also includes one instance in baxter_permutations.

[gh-mwageringel]

Note that this ticket is independent from, but compatible with #28273.

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New commits:

​4ba13d8

28274: fix keys and category of EnumeratedFamily

[chapoton]

Travis, any opinion on this proposal ?

[tscrim]

This is clearly a bug as it contradicts the documentation of EnumeratedFamily:

    :class:`EnumeratedFamily` turns an enumerated set ``c`` into a family
    indexed by the set `\{0,\dots, |c|-1\}`.

I don't understand this:

The category of finite enumerated families should be in FiniteEnumeratedSets(), while currently it is only in EnumeratedSets().

in both what is supposed to mean and what was changed.

[gh-mwageringel]

Replying to tscrim:

I don't understand this:

The category of finite enumerated families should be in FiniteEnumeratedSets(), while currently it is only in EnumeratedSets().

in both what is supposed to mean and what was changed.

What I meant is that the category should be a subcategory of FiniteEnumeratedSets() so that the following can be used to check whether the family is finite.

sage: Family(Permutations(4)) in FiniteEnumeratedSets()  # should be True
False

The category is set based on whether the type of the keys is known to be finite, such as for lists and tuples. I added range to this detection, which is the type of keys of EnumeratedFamily. This bug was probably introduced when six.moves.range was imported into the module, as ordinarily in Python 2 range returns a list.

[tscrim]

Okay, that helps quite a lot. Thanks.

[gh-mwageringel]

Thank you.