recognizable series: properly deal with zero

Author: dkrenn

src/sage/combinat/recognizable_series.py

@@ -1105,6 +1105,11 @@ def minimized(self):
             sage: all(c == d and v == w
             ....:     for (c, v), (d, w) in islice(zip(iter(S), iter(M)), 20))
             True
+
+        TESTS::
+
+            sage: Rec((Matrix([[0]]), Matrix([[0]])), vector([1]), vector([0])).minimized().linear_representation()
+            ((), Finite family {0: [], 1: []}, ())
         """
         return self._minimized_right_()._minimized_left_()
 
@@ -1341,6 +1346,11 @@ def _rmul_(self, other):
             sage: 1 * E is E
             True
 
+        ::
+
+            sage: 0 * E is Seq2.zero()
+            True
+
         We test that ``_rmul_`` and ``_lmul_`` are actually called::
 
             sage: def print_name(f):
@@ -1359,9 +1369,11 @@ def _rmul_(self, other):
             _lmul_
             2-regular sequence 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...
         """
+        P = self.parent()
+        if other.is_zero():
+            return P._zero_()
         if other.is_one():
             return self
-        P = self.parent()
         return P.element_class(P, self.mu, other*self.left, self.right)
 
     def _lmul_(self, other):
@@ -1398,6 +1410,11 @@ def _lmul_(self, other):
             sage: E * 1 is E
             True
 
+        ::
+
+            sage: E * 0 is Seq2.zero()
+            True
+
         The following is not tested, as `MS^i` for integers `i` does
         not work, thus ``vector([m])`` fails. (See :trac:`21317` for
         details.)
@@ -1414,9 +1431,11 @@ def _lmul_(self, other):
             sage: M  # not tested
             sage: M.linear_representation()  # not tested
         """
+        P = self.parent()
+        if other.is_zero():
+            return P._zero_()
         if other.is_one():
             return self
-        P = self.parent()
         return P.element_class(P, self.mu, self.left, self.right*other)
 
     @minimize_result
@@ -1913,6 +1932,28 @@ def some_elements(self):
                     break
                 yield self(mu, *LR)
 
+    @cached_method
+    def _zero_(self):
+        r"""
+        Return the zero element of this :class:`RecognizableSeriesSpace`,
+        i.e. the unique neutral element for `+`.
+
+        TESTS::
+
+            sage: Rec = RecognizableSeriesSpace(ZZ, [0, 1])
+            sage: Z = Rec._zero_(); Z
+            0
+            sage: Z.linear_representation()
+            ((), Finite family {0: [], 1: []}, ())
+        """
+        from sage.matrix.constructor import Matrix
+        from sage.modules.free_module_element import vector
+        from sage.sets.family import Family
+
+        return self.element_class(
+            self, Family(self.alphabet(), lambda a: Matrix()),
+            vector([]), vector([]))
+
     @cached_method
     def one_hadamard(self):
         r"""
@@ -1985,13 +2026,7 @@ def _element_constructor_(self, data,
             ValueError: left or right vector is None
         """
         if isinstance(data, int) and data == 0:
-            from sage.matrix.constructor import Matrix
-            from sage.modules.free_module_element import vector
-            from sage.sets.family import Family
-
-            return self.element_class(
-                self, Family(self.alphabet(), lambda a: Matrix()),
-                vector([]), vector([]))
+            return self._zero_()
 
         if type(data) == self.element_class and data.parent() == self:
             element = data