declare_union yields wrong results

[gh-mjungmath]

(for the visualization of the posets, use #31680 and install the dot2tex package - without that package, the layout is quite misleading)

sage: M = Manifold(3, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V'); W = M.open_subset('W')
sage: M.declare_union(U, V, W)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-3-c7f527dcc1ea> in <module>()
----> 1 M.declare_union(U, V, W)

TypeError: declare_union() takes 3 positional arguments but 4 were given

sage: def label(element): 
....:     try: 
....:         return element._name 
....:     except AttributeError: 
....:         return '[' + ', '.join(sorted(x._name for x in element)) + ']' 

sage: P = M.subset_poset(open_covers=True); P.plot(element_labels={element: label(element) for element in P})

sage: M.declare_union(U, V.union(W))
sage: P = M.subset_poset(open_covers=True); P.plot(element_labels={element: label(element) for element in P})

sage: U.union(V).union(W)
Open subset U_union_V_union_W of the 3-dimensional differentiable manifold M
sage: P = M.subset_poset(open_covers=True); P.plot(element_labels={element: label(element) for element in P})

sage: U.union(V.union(W))
3-dimensional differentiable manifold M
sage: P = M.subset_poset(open_covers=True); P.plot(element_labels={element: label(element) for element in P})  # same

sage: M.is_subset(U.union(V).union(W))                                                         
False

#31764 adds support for arbitrary unions. But the main issue here is the failing associativity of the union.

We fix it as follows.

When creating a union, we update the open covers of the supersets of the members, replacing the union members by the union.

[mkoeppe]

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.

[mkoeppe]

Unions of several subsets also appear in another form - as open covers (which are merely lists of subsets and not manifold subsets in their own right). One might ask whether this could be unified as well.

[mkoeppe]

An issue with the existing code is that through a sequence of declare_union calls, it is possible to obtain a situation in which two subsets must be the same, although declare_union makes a (not very effective) attempt to stop the user from doing so (by checking whether the two arguments are the same).

In terms of the poset of subsets that is defined by the _subsets (and _supersets) attributes (which is visualized by the code from #31680), this is quite problematic.

This could be solved by introducing equivalence classes of subsets - for example by keeping the _subsets relation acyclic and introducing an _equal_sets attribute; loops that update subsets will have to be modified. The elements of the poset introduced in #31680 then would be equivalence classes of subsets rather than subsets.

[mkoeppe]

Last 10 new commits:

​f095988	FiniteManifoldObjectFamily.__bool__: Remove, inherited from superclass after #31717
​5d87cee	Manifold{Object,Subset}FiniteFamily: Rename from FiniteManifold{Object,Subset}Family
​2f2ace2	src/doc/en/reference/manifolds/manifold.rst: Add sage.manifolds.family
​1aff58a	Fix up docstring markup
​b922066	ManifoldSubsetFiniteFamily: If all subsets are open, include 'open' in repr
​30271af	Fixup doctest
​adac07a	Merge #31680
​78cc27a	ManifoldSubset.open_covers: Change to generator, add optional arg 'trivial'; update uses
​e026e7a	ManifoldSubset.subset_digraph: Use open_covers method
​fca7e89	ManifoldSubset.declare_empty etc.

[git]

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

​215c378	ManifoldSubset.subset_{digraph,poset}: Add plots in documentation, remove # not tested
​84896c4	ManifoldSubsetFiniteFamily: Use full error message in doctest
​3364468	Merge #31680
​6c8374e	ManifoldSubset.open_covers: Add option supersets; use it to fix is_empty
​28650bc	Fix doctests
​26401dd	ManifoldSubset.declare_empty: Add plot

[git]

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

​fee8b99	ManifoldSubsetFiniteFamily.from_subsets_or_families: New constructor
​ba51202	ManifoldSubset.declare_union: Add option 'disjoint'

[git]

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

​9ca80d2	ManifoldSubset.{equal_subsets,supersets,superset_family}: New
​55d2aa8	ManifoldSubset.{subset,superset}_digraph: New option quotient; use it for {subset,superset}_poset
​4cf4a52	ManifoldSubset.declare_equal: New; use it for 1-arg ManifoldSubset.declare_union

[git]

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

​0f037ff	ManifoldSubset.intersection: Handle arbitrary intersections
​9c75d18	intersection: Refactor
​8d63c32	ManifoldSubset.intersection: Only pass name, latex_name to the final intersection
​0867b0c	ManifoldSubset.intersection: Go through ManifoldSubsetFiniteFamily to make the order of operations canonical
​de5f928	ManifoldSubset._union_subset: Factor out from ManifoldSubset.union
​4698da1	ManifoldSubset.union: Handle arbitrary unions
​51a89d0	ManifoldSubset.union: Add example and plots
​55a28ce	ManifoldSubset.declare_union: Handle arbitrary unions

[mkoeppe]

Last 10 new commits:

​0032bf9	ManifoldSubset.intersection: Go through ManifoldSubsetFiniteFamily to make the order of operations canonical
​282010b	ManifoldSubset._union_subset: Factor out from ManifoldSubset.union
​6772570	ManifoldSubset.union: Handle arbitrary unions
​edfea33	ManifoldSubset.union: Add example and plots
​fdd0aba	ManifoldSubset.declare_union: Handle arbitrary unions
​c2f484c	src/sage/manifolds/subset.py: import itertools
​a083569	ManifoldSubset._intersection_subset, _union_subset: Use declare_subset, declare_superset; fix doctest
​5d41b46	Merge #31764
​aceb0e8	ManifoldSubset._union_subset: Do cache store the union here
​743f114	ManifoldSubset._union_subset: Update open covers