Trac #18223: cartesian products with orders

Create cartesian products which have a particular order
(lexicographically or component-wise) on its elements.

This also incorporates the proposed changes of #18586 (won't be fixed;
see discussion there), namely passes keyword arguments on in
`cartesian_product` and adds `extra_category`.

URL: http://trac.sagemath.org/18223
Reported by: dkrenn
Ticket author(s): Daniel Krenn
Reviewer(s): Benjamin Hackl, Vincent Delecroix

src/doc/en/reference/combinat/module_list.rst

@@ -162,6 +162,7 @@ Comprehensive Module list
     sage/combinat/permutation_nk
     sage/combinat/posets/__init__
     sage/combinat/posets/all
+    sage/combinat/posets/cartesian_product
     sage/combinat/posets/elements
     sage/combinat/posets/hasse_diagram
     sage/combinat/posets/incidence_algebras

src/sage/categories/covariant_functorial_construction.py

@@ -201,7 +201,7 @@ def _repr_(self):
         """
         return "The %s functorial construction"%self._functor_name
 
-    def __call__(self, args):
+    def __call__(self, args, **kwargs):
         """
         Functorial construction application
 
@@ -220,7 +220,7 @@ def __call__(self, args):
         args = tuple(args) # a bit brute force; let's see if this becomes a bottleneck later
         assert(all( hasattr(arg, self._functor_name) for arg in args))
         assert(len(args) > 0)
-        return getattr(args[0], self._functor_name)(*args[1:])
+        return getattr(args[0], self._functor_name)(*args[1:], **kwargs)
 
 class FunctorialConstructionCategory(Category): # Should this be CategoryWithBase?
     """

src/sage/categories/posets.py

@@ -683,6 +683,9 @@ def is_antichain_of_poset(self, o):
             """
             return all(not self.lt(x,y) for x in o for y in o)
 
+        CartesianProduct = LazyImport(
+            'sage.combinat.posets.cartesian_product', 'CartesianProductPoset')
+
     class ElementMethods:
         pass
         # TODO: implement x<y, x<=y, x>y, x>=y appropriately once #10130 is resolved

src/sage/categories/sets_cat.py

@@ -1400,10 +1400,32 @@ def _test_cardinality(self, **options):
         # Functorial constructions
 
         CartesianProduct = CartesianProduct
-        def cartesian_product(*parents):
+        def cartesian_product(*parents, **kwargs):
             """
             Return the cartesian product of the parents.
 
+            INPUT:
+
+            - ``parents`` -- a list (or other iterable) of parents.
+
+            - ``category`` -- (default: ``None``) the category the
+              cartesian product belongs to. If ``None`` is passed,
+              then
+              :meth:`~sage.categories.covariant_functorial_construction.CovariantFactorialConstruction.category_from_parents`
+              is used to determine the category.
+
+            - ``extra_category`` -- (default: ``None``) a category
+              that is added to the cartesian product in addition
+              to the categories obtained from the parents.
+
+            - other keyword arguments will passed on to the class used
+              for this cartesian product (see also
+              :class:`~sage.sets.cartesian_product.CartesianProduct`).
+
+            OUTPUT:
+
+            The cartesian product.
+
             EXAMPLES::
 
                 sage: C = AlgebrasWithBasis(QQ)
@@ -1423,10 +1445,31 @@ def cartesian_product(*parents):
                 sage: C.category()
                 Join of Category of rings and ...
                     and Category of Cartesian products of commutative additive groups
+
+            ::
+
+                sage: cartesian_product([ZZ, ZZ], category=Sets()).category()
+                Category of sets
+                sage: cartesian_product([ZZ, ZZ]).category()
+                Join of
+                Category of Cartesian products of commutative rings and
+                Category of Cartesian products of enumerated sets
+                sage: cartesian_product([ZZ, ZZ], extra_category=Posets()).category()
+                Join of
+                Category of Cartesian products of commutative rings and
+                Category of posets and
+                Category of Cartesian products of enumerated sets
             """
-            return parents[0].CartesianProduct(
-                parents,
-                category = cartesian_product.category_from_parents(parents))
+            category = kwargs.pop('category', None)
+            extra_category = kwargs.pop('extra_category', None)
+
+            category = category or cartesian_product.category_from_parents(parents)
+            if extra_category:
+                if isinstance(category, (list, tuple)):
+                    category = tuple(category) + (extra_category,)
+                else:
+                    category = category & extra_category
+            return parents[0].CartesianProduct(parents, category=category, **kwargs)
 
         def algebra(self, base_ring, category=None):
             """

src/sage/combinat/posets/__init__.py

@@ -15,6 +15,8 @@
 
 - :ref:`sage.combinat.posets.incidence_algebras`
 
+- :ref:`sage.combinat.posets.cartesian_product`
+
 - :ref:`sage.combinat.tamari_lattices`
 - :ref:`sage.combinat.interval_posets`
 - :ref:`sage.combinat.shard_order`