Fixing up doctests and some other compatibility issues.

src/sage/algebras/clifford_algebra.py

@@ -385,27 +385,11 @@ def supercommutator(self, x):
         return ret
 
 
-class CliffordAlgebraIndices(Parent):
+class CliffordAlgebraIndices(UniqueRepresentation, Parent):
     r"""
     A facade parent for the indices of Clifford algebra.
     Users should not create instances of this class directly.
     """
-    def __call__(self, el):
-        r"""
-        EXAMPLES::
-
-            sage: from sage.algebras.clifford_algebra import CliffordAlgebraIndices
-            sage: idx = CliffordAlgebraIndices(7)
-            sage: idx([1,3,6])
-            0101001
-            sage: E = ExteriorAlgebra(QQ, 7)
-            sage: B = E.basis()
-        """
-        if not isinstance(el, Element):
-            return self._element_constructor_(el)
-        else:
-            return Parent.__call__(self, el)
-
     def __init__(self, Qdim):
         r"""
         Initialize ``self``.
@@ -419,12 +403,12 @@ def __init__(self, Qdim):
             sage: idx._cardinality
             128
             sage: i = idx.an_element(); i
-            0
+            1111
             sage: type(i)
             <class 'sage.data_structures.bitset.FrozenBitset'>
         """
         self._nbits = Qdim
-        self._cardinality = 2**Qdim
+        self._cardinality = 2 ** Qdim
         # the if statement here is in case Qdim is 0.
         category = FiniteEnumeratedSets().Facade()
         Parent.__init__(self, category=category, facade=True)
@@ -456,6 +440,22 @@ def _element_constructor_(self, x):
         if isinstance(x, int):
             return FrozenBitset((x,))
 
+    def __call__(self, el):
+        r"""
+        EXAMPLES::
+
+            sage: from sage.algebras.clifford_algebra import CliffordAlgebraIndices
+            sage: idx = CliffordAlgebraIndices(7)
+            sage: idx([1,3,6])
+            0101001
+            sage: E = ExteriorAlgebra(QQ, 7)
+            sage: B = E.basis()
+        """
+        if not isinstance(el, Element):
+            return self._element_constructor_(el)
+        else:
+            return Parent.__call__(self, el)
+
     def cardinality(self):
         r"""
         Return the cardinality of ``self``.
@@ -556,7 +556,7 @@ def __contains__(self, elt):
         EXAMPLES::
 
             sage: from sage.algebras.clifford_algebra import CliffordAlgebraIndices
-            sage: idx = CliffordAlgebraIndices(3);
+            sage: idx = CliffordAlgebraIndices(3)
             sage: int(8) in idx  # representing the set {4}
             False
             sage: int(5) in idx  # representing the set {1,3}
@@ -572,6 +572,32 @@ def __contains__(self, elt):
             return False
         return elt.capacity() <= self._nbits
 
+    def _an_element_(self):
+        """
+        Returns an element of ``self``.
+
+        EXAMPLES::
+
+            sage: from sage.algebras.clifford_algebra import CliffordAlgebraIndices
+            sage: idx = CliffordAlgebraIndices(0)
+            sage: idx._an_element_()
+            0
+            sage: idx = CliffordAlgebraIndices(1)
+            sage: idx._an_element_()
+            1
+            sage: idx = CliffordAlgebraIndices(2)
+            sage: idx._an_element_()
+            01
+            sage: idx = CliffordAlgebraIndices(3)
+            sage: idx._an_element_()
+            11
+        """
+        if not self._nbits:
+            return FrozenBitset()
+
+        from sage.combinat.subset import SubsetsSorted
+        X = SubsetsSorted(range(self._nbits))
+        return FrozenBitset(X.an_element())
 
 class CliffordAlgebra(CombinatorialFreeModule):
     r"""

src/sage/categories/filtered_modules_with_basis.py

@@ -178,9 +178,9 @@ def basis(self, d=None):
 
                 sage: E.<x,y> = ExteriorAlgebra(QQ)
                 sage: E.basis()
-                Lazy family (Term map from Subsets of {0, 1} to
+                Lazy family (Term map from Subsets of {0,1} to
                  The exterior algebra of rank 2 over Rational Field(i))_{i in
-                 Subsets of {0, 1}}
+                 Subsets of {0,1}}
             """
             if d is None:
                 from sage.sets.family import Family

src/sage/categories/graded_algebras_with_basis.py

@@ -168,10 +168,10 @@ def one_basis(self):
 
                     sage: A.<x,y> = ExteriorAlgebra(QQ)
                     sage: A.one_basis()
-                    ()
+                    0
                     sage: B = tensor((A, A, A))
                     sage: B.one_basis()
-                    ((), (), ())
+                    (0, 0, 0)
                     sage: B.one()
                     1 # 1 # 1
                 """

src/sage/modules/with_basis/invariant.py

@@ -109,7 +109,7 @@ class FiniteDimensionalInvariantModule(SubmoduleWithBasis):
         sage: M = algebras.Exterior(QQ, 'x', 3)
         sage: def cyclic_ext_action(g, m):
         ....:     # cyclically permute generators
-        ....:     return M.prod([M.monomial((g(j+1)-1,)) for j in m])
+        ....:     return M.prod([M.monomial(FrozenBitset([g(j+1)-1])) for j in m])
 
     If you care about being able to exploit the algebra structure of the
     exterior algebra (i.e. if you want to multiply elements together), you
@@ -398,7 +398,7 @@ def _mul_(self, other):
 
                 sage: G = CyclicPermutationGroup(3); G.rename('G')
                 sage: M = algebras.Exterior(QQ, 'x', 3)
-                sage: def on_basis(g,m): return M.prod([M.monomial((g(j+1)-1,)) for j in m])  # cyclically permute generators
+                sage: def on_basis(g,m): return M.prod([M.monomial(FrozenBitset([g(j+1)-1])) for j in m])  # cyclically permute generators
                 sage: R = Representation(G, M, on_basis, category=Algebras(QQ).WithBasis().FiniteDimensional(), side='right')
                 sage: I = R.invariant_module(); I.rename('I')
                 sage: B = I.basis()
@@ -466,7 +466,6 @@ def _acted_upon_(self, scalar, self_on_left=False):
                 sage: g = G.an_element(); g
                 (1,2,3)
                 sage: M = CombinatorialFreeModule(QQ, [1,2,3])
-                sage: E = algebras.Exterior(QQ, 'x', 3)
                 sage: from sage.modules.with_basis.representation import Representation
                 sage: R = Representation(G, M, lambda g,x: M.monomial(g(x)))
                 sage: I = R.invariant_module()
@@ -480,7 +479,8 @@ def _acted_upon_(self, scalar, self_on_left=False):
                 [2*B[0], 2*B[0], 2*B[0]]
 
 
-                sage: def on_basis(g,m): return E.prod([E.monomial((g(j+1)-1,)) for j in m])  # cyclically permute generators
+                sage: E = algebras.Exterior(QQ, 'x', 3)
+                sage: def on_basis(g,m): return E.prod([E.monomial(FrozenBitset([g(j+1)-1])) for j in m])  # cyclically permute generators
                 sage: R = Representation(G, E, on_basis, category=Algebras(QQ).WithBasis().FiniteDimensional())
                 sage: I = R.invariant_module()
                 sage: B = I.basis()
@@ -528,7 +528,7 @@ def _acted_upon_(self, scalar, self_on_left=False):
                 sage: [b._acted_upon_(G((1,3,2)), self_on_left=True) for b in I.basis()]
                 [B[0]]
 
-                sage: def on_basis(g,m): return E.prod([E.monomial((g(j+1)-1,)) for j in m])  # cyclically permute generators
+                sage: def on_basis(g,m): return E.prod([E.monomial(FrozenBitset([g(j+1)-1])) for j in m])  # cyclically permute generators
                 sage: R = Representation(G, E, on_basis, category=Algebras(QQ).WithBasis().FiniteDimensional(), side='right')
                 sage: I = R.invariant_module()
                 sage: B = I.basis()
@@ -687,7 +687,7 @@ class FiniteDimensionalTwistedInvariantModule(SubmoduleWithBasis):
 
         sage: G = SymmetricGroup(3); G.rename('S3')
         sage: E = algebras.Exterior(QQ, 'x', 3); E.rename('E')
-        sage: def action(g,m): return E.prod([E.monomial((g(j+1)-1,)) for j in m])
+        sage: def action(g,m): return E.prod([E.monomial(FrozenBitset([g(j+1)-1])) for j in m])
         sage: from sage.modules.with_basis.representation import Representation
         sage: EA = Representation(G, E, action, category=Algebras(QQ).WithBasis().FiniteDimensional())
         sage: T = EA.twisted_invariant_module([2,0,-1])

src/sage/modules/with_basis/representation.py

@@ -276,7 +276,7 @@ def __init__(self, semigroup, module, on_basis, side="left", **kwargs):
             sage: G = CyclicPermutationGroup(3)
             sage: M = algebras.Exterior(QQ, 'x', 3)
             sage: from sage.modules.with_basis.representation import Representation
-            sage: on_basis = lambda g,m: M.prod([M.monomial((g(j+1)-1,)) for j in m]) #cyclically permute generators
+            sage: on_basis = lambda g,m: M.prod([M.monomial(FrozenBitset([g(j+1)-1])) for j in m]) #cyclically permute generators
             sage: from sage.categories.algebras import Algebras
             sage: R = Representation(G, M, on_basis, category=Algebras(QQ).WithBasis().FiniteDimensional())
             sage: r = R.an_element(); r
@@ -451,9 +451,9 @@ def product_by_coercion(self, left, right):
             ...
             TypeError: unsupported operand parent(s) for *:
              'Representation of The Klein 4 group of order 4, as a permutation
-             group indexed by Subsets of {0, 1, 2, 3} over Rational Field' and
+             group indexed by Subsets of {0,1,...,3} over Rational Field' and
              'Representation of The Klein 4 group of order 4, as a permutation
-             group indexed by Subsets of {0, 1, 2, 3} over Rational Field'
+             group indexed by Subsets of {0,1,...,3} over Rational Field'
 
             sage: from sage.categories.algebras import Algebras
             sage: category = Algebras(QQ).FiniteDimensional().WithBasis()