Trac #16125: Implement method _an_element_ for Words

{{{
sage: W = Words(4)
sage: w = W([3,1,2,1])
sage: w in ZZ
NotImplementedError                       Traceback (most recent call
last)
[...]
NotImplementedError: please implement _an_element_ for Words over {1, 2,
3, 4}
}}}

URL: http://trac.sagemath.org/16125
Reported by: virmaux
Ticket author(s): Aladin Virmaux
Reviewer(s): Frédéric Chapoton

src/sage/algebras/shuffle_algebra.py

@@ -301,9 +301,9 @@ def coproduct(self, S):
 
             sage: F = ShuffleAlgebra(QQ,'ab')
             sage: S = F.an_element(); S
-            B[word: ] + 2*B[word: a] + 3*B[word: b]
+            2*B[word: ] + 2*B[word: a] + 3*B[word: b]
             sage: F.coproduct(S)
-            B[word: ] # B[word: ] + 2*B[word: ] # B[word: a] + 3*B[word: ] # B[word: b]
+            2*B[word: ] # B[word: ] + 2*B[word: ] # B[word: a] + 3*B[word: ] # B[word: b]
              + 2*B[word: a] # B[word: ] + 3*B[word: b] # B[word: ]
             sage: F.coproduct(F.one())
             B[word: ] # B[word: ]
@@ -319,9 +319,9 @@ def counit(self,S):
 
             sage: F = ShuffleAlgebra(QQ,'ab')
             sage: S = F.an_element(); S
-            B[word: ] + 2*B[word: a] + 3*B[word: b]
+            2*B[word: ] + 2*B[word: a] + 3*B[word: b]
             sage: F.counit(S)
-            1
+            2
         """
         return S.monomial_coefficients().get(Word(), 0)
 

src/sage/categories/modules_with_basis.py

@@ -1055,7 +1055,7 @@ def _an_element_(self):
                     sage: B = HopfAlgebrasWithBasis(QQ).example(); B
                     An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
                     sage: A.an_element(), B.an_element()
-                    (B[word: ] + 2*B[word: a] + 3*B[word: b], B[()] + 2*B[(2,3)] + 3*B[(1,2)] + B[(1,2,3)])
+                    (2*B[word: ] + 2*B[word: a] + 3*B[word: b], B[()] + 2*B[(2,3)] + 3*B[(1,2)] + B[(1,2,3)])
                     sage: cartesian_product((A, B, A)).an_element()           # indirect doctest
                     2*B[(0, word: )] + 2*B[(0, word: a)] + 3*B[(0, word: b)]
                 """

src/sage/combinat/words/words.py

@@ -181,6 +181,27 @@ def _element_classes(self):
             'Word_iter': word.Word_iter
             }
 
+    def _an_element_(self):
+        r"""
+        Return an element of self.
+
+        EXAMPLES::
+
+            sage: W = Words(4)
+            sage: W.an_element()
+            word:
+
+        TESTS:
+
+        Check that :trac:`16125` is fixed::
+
+            sage: W = Words(4)
+            sage: w = W([3,1,2,1])
+            sage: w in ZZ
+            False
+        """
+        return self([])
+
     def __call__(self, data=None, length=None, datatype=None, caching=True, **kwds):
         r"""
         Construct a new word object with parent self.