sage.geometry.polyhedron.modules.formal_polyhedra_module: New

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

@@ -0,0 +1,90 @@
+"""
+Formal modules generated by polyhedra.
+"""
+
+from sage.combinat.free_module import CombinatorialFreeModule
+from sage.modules.with_basis.subquotient import SubmoduleWithBasis, QuotientModuleWithBasis
+
+class FormalPolyhedraModule(CombinatorialFreeModule):
+    """
+    Class for formal modules generated by polyhedra.
+
+    It is formal because it is free -- it does not know
+    about linear relations of polyhedra.
+
+    INPUT:
+
+     - ``base_ring`` - base ring of the module; unrelated to the
+       base ring of the polyhedra
+
+     - ``dimension`` - the ambient dimension of the polyhedra
+
+     - ``basis`` - the basis
+
+    EXAMPLES::
+
+        sage: def closed_interval(a,b): return Polyhedron(vertices=[[a], [b]])
+
+    A three-dimensional vector space of polyhedra::
+
+        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: basis = [I01, I11, I12]
+        sage: M = FormalPolyhedraModule(QQ, 1, basis=basis); M
+        Free module generated by {conv([0], [1]), {[1]}, conv([1], [2])} over Rational Field
+        sage: M.get_order()
+        [conv([0], [1]), {[1]}, conv([1], [2])]
+
+    A one-dimensional subspace::
+
+        sage: M_lower = M.submodule([M(I11)]); M_lower
+        Free module generated by {0} over Rational Field
+        sage: M_lower.is_submodule(M)
+        True
+        sage: x = M(I01) - 2*M(I11) + M(I12)
+        sage: M_lower.reduce(x)
+        [conv([0], [1])] + [conv([1], [2])]
+        sage: M_lower.retract.domain() is M
+        True
+        sage: y = M_lower.retract(M(I11)); y
+        B[0]
+        sage: M_lower.lift(y)
+        [{[1]}]
+
+    Quotient space::
+
+        sage: M_mod_lower = M.quotient_module(M_lower); M_mod_lower
+        Free module generated by {conv([0], [1]), conv([1], [2])} over Rational Field
+        sage: M_mod_lower.retract(x)
+        B[conv([0], [1])] + B[conv([1], [2])]
+        sage: M_mod_lower.retract(M(I01) - 2*M(I11) + M(I12)) ==  M_mod_lower.retract(M(I01) + M(I12))
+        True
+
+    """
+
+    @staticmethod
+    def __classcall__(cls, base_ring, dimension, basis, category=None):
+        r"""
+        Normalize the arguments for caching.
+
+        TESTS::
+
+            sage: FormalPolyhedraModule(QQ, 1, ()) is FormalPolyhedraModule(QQ, 1, [])
+            True
+        """
+        if isinstance(basis, list):
+            basis = tuple(basis)
+        if category is None:
+            category = GradedModulesWithBasis(base_ring)
+        return super(FormalPolyhedraModule, cls).__classcall__(cls,
+                                                               base_ring=base_ring,
+                                                               dimension=dimension,
+                                                               basis=basis,
+                                                               category=category)
+
+    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)