Formal modules generated by polyhedra#
- class sage.geometry.polyhedron.modules.formal_polyhedra_module.FormalPolyhedraModule(base_ring, dimension, basis, category)#
Bases:
CombinatorialFreeModule
Class for formal modules generated by polyhedra.
It is formal because it is free – it does not know about linear relations of polyhedra.
A formal polyhedral module is graded by dimension.
INPUT:
base_ring
– base ring of the module; unrelated to the base ring of the polyhedradimension
– the ambient dimension of the polyhedrabasis
– the basis
EXAMPLES:
sage: from sage.geometry.polyhedron.modules.formal_polyhedra_module import FormalPolyhedraModule 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; bases of subspaces just use the indexing set \(0, \dots, d-1\), where \(d\) is the dimension:
sage: M_lower = M.submodule([M(I11)]); M_lower Free module generated by {0} over Rational Field sage: M_lower.print_options(prefix='S') 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 S[0] sage: M_lower.lift(y) [{[1]}]
Quotient space; bases of quotient space are families indexed by elements of the ambient 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.print_options(prefix='Q') sage: M_mod_lower.retract(x) Q[conv([0], [1])] + Q[conv([1], [2])] sage: M_mod_lower.retract(M(I01) - 2*M(I11) + M(I12)) == M_mod_lower.retract(M(I01) + M(I12)) True
- degree_on_basis(m)#
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 would only be 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