declare_union yields wrong results

[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.

Depends on #31764

CC: @egourgoulhon @tscrim

Component: manifolds

Keywords: sets, union

Author: Matthias Koeppe

Branch/Commit: u/mkoeppe/declare_union_yields_wrong_results @ 743f114

Issue created by migration from https://trac.sagemath.org/ticket/30263

[mkoeppe]

comment:2

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

[mkoeppe]

comment:3

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]

Dependencies: #31680

[mkoeppe]

Description changed:

--- 
+++ 
@@ -9,11 +9,26 @@
 ----> 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
+
 ```
 
 I propose to allow more than two arguments for `declare_union`. Furthermore, something went wrong with the commutativity of the union. This should be investigated, too.

[mkoeppe]

Description changed:

--- 
+++ 
@@ -1,3 +1,4 @@
+(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')
@@ -29,6 +30,8 @@
 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
 ```
 
 I propose to allow more than two arguments for `declare_union`. Furthermore, something went wrong with the commutativity of the union. This should be investigated, too.

[mkoeppe]

comment:5

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]

Description changed:

--- 
+++ 
@@ -34,4 +34,14 @@
 False
 ```
 
-I propose to allow more than two arguments for `declare_union`. Furthermore, something went wrong with the commutativity of the union. This should be investigated, too.
+In this ticket, we extend `declare_union` as follows:
+- support more than two arguments 
+- support 0 arguments (alias: `declare_empty()`) - a subset is empty if it has an empty open cover
+- support 1 argument (alias: `declare_equal(other)`)
+
+We also introduce
+- `declare_disjoint(other)` - same as `self.intersection(other).declare_empty()`
+- `declare_disjoint_union(*sets)`
+
+Furthermore, something went wrong with the commutativity of the union. This should be investigated, too.
+

[mkoeppe]

Changed dependencies from #31680 to #31680, #31718

[mkoeppe]

Branch: u/mkoeppe/declare_union_yields_wrong_results

[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.

[mkoeppe]

Author: Matthias Koeppe