Remove category, merge method degree_on_basis into the class

src/sage/geometry/polyhedron/modules/formal_polyhedra_module.py

@@ -5,7 +5,6 @@
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.modules.with_basis.subquotient import SubmoduleWithBasis, QuotientModuleWithBasis
 from sage.categories.graded_modules_with_basis import GradedModulesWithBasis
-from sage.categories.polyhedra_modules import PolyhedraModules
 
 class FormalPolyhedraModule(CombinatorialFreeModule):
     r"""
@@ -18,12 +17,12 @@ class FormalPolyhedraModule(CombinatorialFreeModule):
 
     INPUT:
 
-     - ``base_ring`` - base ring of the module; unrelated to the
-       base ring of the polyhedra
+    - ``base_ring`` -- base ring of the module; unrelated to the
+      base ring of the polyhedra
 
-     - ``dimension`` - the ambient dimension of the polyhedra
+    - ``dimension`` -- the ambient dimension of the polyhedra
 
-     - ``basis`` - the basis
+    - ``basis`` -- the basis
 
     EXAMPLES::
 
@@ -86,7 +85,7 @@ def __classcall__(cls, base_ring, dimension, basis, category=None):
         if isinstance(basis, list):
             basis = tuple(basis)
         if category is None:
-            category = PolyhedraModules(base_ring) & GradedModulesWithBasis(base_ring)
+            category = GradedModulesWithBasis(base_ring)
         return super(FormalPolyhedraModule, cls).__classcall__(cls,
                                                                base_ring=base_ring,
                                                                dimension=dimension,
@@ -98,3 +97,42 @@ def __init__(self, base_ring, dimension, basis, category):
         Construct a free module generated by the polyhedra in ``basis``.
         """
         super(FormalPolyhedraModule, self).__init__(base_ring, basis, prefix="", category=category)
+
+    def degree_on_basis(self, m):
+        r"""
+        The degree of an element of the basis is defined as the dimension of the polyhedron.
+
+        INPUT:
+
+        - ``m`` -- an element of the basis (a polyhedron)
+
+        EXAMPLES::
+
+            sage: from sage.geometry.polyhedron.modules.formal_polyhedra_module import FormalPolyhedraModule
+            sage: def closed_interval(a,b): return Polyhedron(vertices=[[a], [b]])
+            sage: I01 = closed_interval(0, 1); I01.rename("conv([0], [1])")
+            sage: I11 = closed_interval(1, 1); I11.rename("{[1]}")
+            sage: I12 = closed_interval(1, 2); I12.rename("conv([1], [2])")
+            sage: I02 = closed_interval(0, 2); I02.rename("conv([0], [2])")
+            sage: M = FormalPolyhedraModule(QQ, 1, basis=[I01, I11, I12, I02])
+
+        We can extract homogeneous components::
+
+            sage: O = M(I01) + M(I11) + M(I12)
+            sage: O.homogeneous_component(0)
+            [{[1]}]
+            sage: O.homogeneous_component(1)
+            [conv([0], [1])] + [conv([1], [2])]
+
+        We note that modulo the linear relations of polyhedra, this is only a filtration,
+        not a grading, as the following example shows.
+
+            sage: X = M(I01) + M(I12) - M(I02)
+            sage: X.degree()
+            1
+
+            sage: Y = M(I11)
+            sage: Y.degree()
+            0
+        """
+        return m.dimension()