src/sage/tensor/modules/free_module_basis.py: Make Basis_abstract a s…

…ubclass of AbstractFamily

src/sage/tensor/modules/free_module_basis.py

@@ -31,10 +31,12 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
+from sage.rings.integer_ring import ZZ
+from sage.sets.family import AbstractFamily
 from sage.structure.unique_representation import UniqueRepresentation
-from sage.structure.sage_object import SageObject
 
-class Basis_abstract(UniqueRepresentation, SageObject):
+class Basis_abstract(UniqueRepresentation, AbstractFamily):
     """
     Abstract base class for (dual) bases of free modules.
     """
@@ -54,10 +56,59 @@ def __init__(self, fmodule, symbol, latex_symbol, indices, latex_indices):
         self._latex_symbol = latex_symbol
         self._indices = indices
         self._latex_indices = latex_indices
+        super().__init__(category=FiniteEnumeratedSets(), facade=fmodule)
+
+    def keys(self):
+        """
+        Return the keys (indices) of the family.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: e = M.basis('e')
+            sage: list(e.keys())
+            [0, 1, 2]
+        """
+        return self._fmodule.irange()
+
+    def values(self):
+        """
+        Return the basis elements of ``self``.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: e = M.basis('e')
+            sage: list(e.values())
+            [Element e_0 of the Rank-3 free module M over the Integer Ring,
+            Element e_1 of the Rank-3 free module M over the Integer Ring,
+            Element e_2 of the Rank-3 free module M over the Integer Ring]
+        """
+        return self._vec
+
+    def _element_constructor_(self, x):
+        """
+        Test whether ``x`` is an element of ``self``.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: e = M.basis('e')
+            sage: e(e[1])
+            Element e_1 of the Rank-3 free module M over the Integer Ring
+            sage: f = M.basis('f')
+            sage: e(f[1])
+            Traceback (most recent call last):
+            ...
+            ValueError: no common basis for the comparison
+        """
+        if x in self.values():
+            return x
+        raise ValueError(f'{x} is not in {self}')
 
     def __iter__(self):
         r"""
-        Return the list of basis elements of ``self``.
+        Return an iterator for the basis elements of ``self``.
 
         EXAMPLES::
 
@@ -83,8 +134,7 @@ def __iter__(self):
              Element e_2 of the Rank-3 free module M1 over the Integer Ring,
              Element e_3 of the Rank-3 free module M1 over the Integer Ring]
         """
-        for i in self._fmodule.irange():
-            yield self[i]
+        yield from self.values()
 
     def _test_iter_len(self, **options):
         r"""
@@ -125,6 +175,19 @@ def __len__(self):
         """
         return self._fmodule._rank
 
+    def cardinality(self):
+        r"""
+        Return the basis length, i.e. the rank of the free module.
+
+        EXAMPLES::
+
+            sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
+            sage: e = M.basis('e')
+            sage: e.cardinality()
+            3
+        """
+        return ZZ(self._fmodule._rank)
+
     def __getitem__(self, index):
         r"""
         Return the basis element corresponding to a given index.