chap2.txt

  
  2 Functions
  
  In  this chapter we describe the functions offered by NormalizInterface. All
  functions  supplied  by  this  package  start with Nmz. For examples see the
  chapter 'Examples'.
  
  
  2.1 Create a NmzCone
  
  To create a cone object use NmzCone.
  
  2.1-1 NmzCone
  
  NmzCone( list )  function
  Returns:  NmzCone
  
  Creates  a  NmzCone.  The  list  argument  should  contain an even number of
  elements,  alternating between a string and a integer matrix. The string has
  to  correspond to a Normaliz input type string and the following matrix will
  be interpreted as input of that type.
  
  Currently the following strings are recognized:
  
      integral_closure,
  
      polyhedron,
  
      normalization,
  
      polytope,
  
      rees_algebra,
  
      inequalities,
  
      strict_inequalities,
  
      signs,
  
      strict_signs,
  
      equations,
  
      congruences,
  
      inhom_inequalities,
  
      inhom_equations,
  
      inhom_congruences,
  
      dehomogenization,
  
      lattice_ideal,
  
      grading,
  
      excluded_faces,
  
      lattice,
  
      saturation,
  
      cone,
  
      offset,
  
      vertices,
  
      support_hyperplanes,
  
      cone_and_lattice,
  
      subspace.
  
  See the Normaliz manual for a detailed description.
  
    Example  
    gap> cone := NmzCone(["integral_closure",[[2,1],[1,3]]]);
    <a Normaliz cone>
  
  
  
  2.2 Use a NmzCone
  
  2.2-1 NmzHasConeProperty
  
  NmzHasConeProperty( cone, property )  function
  Returns:  whether the cone has already computed the given property
  
  See NmzConeProperty (2.2-6) for a list of recognized properties.
  
    Example  
    gap> NmzHasConeProperty(cone, "ExtremeRays");
    false
  
  
  2.2-2 NmzKnownConeProperties
  
  NmzKnownConeProperties( cone )  function
  Returns:  a   list   of  strings  representing  the  known  (computed)  cone
            properties
  
  Given  a  Normaliz  cone  object,  return  a  list of all properties already
  computed for the cone.
  
    Example  
    gap> NmzKnownConeProperties(cone);
    [ "Generators", "OriginalMonoidGenerators", "Sublattice" ]
  
  
  2.2-3 NmzSetVerboseDefault
  
  NmzSetVerboseDefault( verboseFlag )  function
  Returns:  the previous verbosity
  
  Set  the  global default verbosity state in libnormaliz. This will influence
  all NmzCone created afterwards, but not any existing ones.
  
  See also NmzSetVerbose (2.2-4)
  
  2.2-4 NmzSetVerbose
  
  NmzSetVerbose( cone, verboseFlag )  function
  Returns:  the previous verbosity
  
  Set the verbosity state for a cone.
  
  See also NmzSetVerboseDefault (2.2-3)
  
  2.2-5 NmzCompute
  
  NmzCompute( cone[, propnames] )  function
  Returns:  a boolean indicating success
  
  Start  computing properties of the given cone. The first parameter indicates
  a  cone object, the second parameter is either a single string, or a list of
  strings, which indicate what should be computed.
  
  The    single    parameter    version   is   equivalent   to   NmzCone(cone,
  ["DefaultMode"]).  See  NmzConeProperty  (2.2-6)  for  a  list of recognized
  properties.
  
    Example  
    gap> NmzKnownConeProperties(cone);
    [ "Generators", "OriginalMonoidGenerators", "Sublattice" ]
    gap> NmzCompute(cone, ["SupportHyperplanes", "IsPointed"]);
    true
    gap> NmzKnownConeProperties(cone);
    [ "Generators", "ExtremeRays", "SupportHyperplanes", "IsPointed",
      "IsDeg1ExtremeRays", "OriginalMonoidGenerators", "Sublattice",
      "MaximalSubspace" ]
    gap> NmzCompute(cone);
    true
    gap> NmzKnownConeProperties(cone);
    [ "Generators", "ExtremeRays", "SupportHyperplanes", "TriangulationSize",
      "TriangulationDetSum", "HilbertBasis", "IsPointed", "IsDeg1ExtremeRays",
      "IsIntegrallyClosed", "OriginalMonoidGenerators", "Sublattice",
      "ClassGroup", "MaximalSubspace"]
  
  
  2.2-6 NmzConeProperty
  
  NmzConeProperty( cone, property )  function
  Returns:  the result of the computation, type depends on the property
  
  Triggers the computation of the property of the cone and returns the result.
  If  the  property  was  already  known,  it is not recomputed. Currently the
  following strings are recognized as properties:
  
      Generators see NmzGenerators (2.3-10),
  
      ExtremeRays see NmzExtremeRays (2.3-9),
  
      VerticesOfPolyhedron see NmzVerticesOfPolyhedron (2.3-38),
  
      SupportHyperplanes see NmzSupportHyperplanes (2.3-33),
  
      TriangulationSize see NmzTriangulationSize (2.3-36),
  
      TriangulationDetSum see NmzTriangulationDetSum (2.3-35),
  
      Triangulation see NmzTriangulation (2.3-34),
  
      Multiplicity see NmzMultiplicity (2.3-28),
  
      RecessionRank see NmzRecessionRank (2.3-31),
  
      AffineDim see NmzAffineDim (2.3-1),
  
      ModuleRank see NmzModuleRank (2.3-27),
  
      HilbertBasis see NmzHilbertBasis (2.3-13),
  
      ModuleGenerators see NmzModuleGenerators (2.3-25),
  
      Deg1Elements see NmzDeg1Elements (2.3-4),
  
      HilbertSeries see NmzHilbertSeries (2.3-15),
  
      HilbertQuasiPolynomial see NmzHilbertQuasiPolynomial (2.3-14),
  
      Grading see NmzGrading (2.3-12),
  
      IsPointed see NmzIsPointed (2.3-22),
  
      IsDeg1ExtremeRays see NmzIsDeg1ExtremeRays (2.3-17),
  
      IsDeg1HilbertBasis see NmzIsDeg1HilbertBasis (2.3-18),
  
      IsIntegrallyClosed see NmzIsIntegrallyClosed (2.3-21),
  
      OriginalMonoidGenerators see NmzOriginalMonoidGenerators (2.3-29),
  
      IsReesPrimary see NmzIsReesPrimary (2.3-23),
  
      ReesPrimaryMultiplicity see NmzReesPrimaryMultiplicity (2.3-32),
  
      ExcludedFaces see NmzExcludedFaces (2.3-8),
  
      Dehomogenization see NmzDehomogenization (2.3-5),
  
      InclusionExclusionData see NmzInclusionExclusionData (2.3-16),
  
      ClassGroup see NmzClassGroup (2.3-2),
  
      ModuleGeneratorsOverOriginalMonoid                                 see
        NmzModuleGeneratorsOverOriginalMonoid (2.3-26),
  
      Sublattice  computes  the  efficient  sublattice  and  returns  a bool
        signaling   whether   the  computation  was  successful.  Actual  data
        connected  to  it  can  be  accessed by NmzRank (2.3-30), NmzEquations
        (2.3-7), NmzCongruences (2.3-3), and NmzBasisChange (2.3-51).
  
  Additionally  also  the  following compute options are accepted as property.
  They  modify  what  and  how  should  be  computed,  and return True after a
  successful computation.
  
      Approximate approximate the rational polytope by an integral polytope,
        currently only useful in combination with Deg1Elements.
  
      BottomDecomposition  use the best possible triangulation (with respect
        to the sum of determinants) using the given generators.
  
      DefaultMode  try  to  compute  what  is  possible  and do not throw an
        exception when something cannot be computed.
  
      DualModeactivates  the  dual  algorithm  for  the  computation  of the
        Hilbert  basis  and  degree  1 elements. Includes HilbertBasis, unless
        Deg1Elements   is   set.  Often  a  good  choice  if  you  start  from
        constraints.
  
      KeepOrder     forbids    to    reorder    the    generators.    Blocks
        BottomDecomposition.
  
  All  the  properties above can be given to NmzCompute (2.2-5). There you can
  combine  different properties, e.g. give some properties that you would like
  to know and add some compute options.
  
  See the Normaliz manual for a detailed description.
  
  2.2-7 NmzPrintConeProperties
  
  NmzPrintConeProperties( cone )  function
  
  Print  an  overview  of  all  known properties of the given cone, as well as
  their values.
  
  
  2.3 Cone properties
  
  2.3-1 NmzAffineDim
  
  NmzAffineDim( cone )  function
  Returns:  the affine dimension
  
  The  affine  dimension  of the polyhedron in inhomogeneous computations. Its
  computation is triggered if necessary.
  
  This   is   an   alias   for   NmzConeProperty(  cone,  "AffineDim"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-2 NmzClassGroup
  
  NmzClassGroup( cone )  function
  Returns:  the class group in a special format
  
  A  normal  affine  monoid  M  has  a well-defined divisor class group. It is
  naturally  isomorphic  to the divisor class group of K[M] where K is a field
  (or  any unique factorization domain). We represent it as a vector where the
  first entry is the rank. It is followed by sequence of pairs of entries n,m.
  Such  two  entries  represent a free cyclic summand (Z/nZ)^m. Not allowed in
  inhomogeneous computations.
  
  This   is   an   alias   for  NmzConeProperty(  cone,  "ClassGroup"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-3 NmzCongruences
  
  NmzCongruences( cone )  function
  Returns:  a matrix whose rows represent the congruences
  
  The  equations,  congruences  and  support hyperplanes together describe the
  lattice points of the cone.
  
  This is part of the cone property Sublattice.
  
  2.3-4 NmzDeg1Elements
  
  NmzDeg1Elements( cone )  function
  Returns:  a matrix whose rows are the degree 1 elements
  
  Requires   the   presence   of  a  grading.  Not  allowed  in  inhomogeneous
  computations.
  
  This   is   an  alias  for  NmzConeProperty(  cone,  "Deg1Elements"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-5 NmzDehomogenization
  
  NmzDehomogenization( cone )  function
  Returns:  the dehomgenization vector
  
  Only for inhomogeneous computations.
  
  This  is  an  alias  for  NmzConeProperty(  cone,  "Dehomogenization" ); see
  NmzConeProperty (2.2-6).
  
  2.3-6 NmzEmbeddingDimension
  
  NmzEmbeddingDimension( cone )  function
  Returns:  the embedding dimension of the cone
  
  The  embedding  dimension  is  the  dimension  of  the  space  in  which the
  computation  is  done. It is the number of components of the output vectors.
  This value is always known directly after the creation of the cone.
  
  2.3-7 NmzEquations
  
  NmzEquations( cone )  function
  Returns:  a matrix whose rows represent the equations
  
  The equations cut out the linear space generated by the cone. The equations,
  congruences  and support hyperplanes together describe the lattice points of
  the cone.
  
  2.3-8 NmzExcludedFaces
  
  NmzExcludedFaces( cone )  function
  Returns:  a matrix whose rows represent the excluded faces
  
  This   is  an  alias  for  NmzConeProperty(  cone,  "ExcludedFaces"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-9 NmzExtremeRays
  
  NmzExtremeRays( cone )  function
  Returns:  a matrix whose rows are the extreme rays
  
  This   is   an   alias  for  NmzConeProperty(  cone,  "ExtremeRays"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-10 NmzGenerators
  
  NmzGenerators( cone )  function
  Returns:  a matrix whose rows are the generators of cone
  
  This   is   an   alias   for  NmzConeProperty(  cone,  "Generators"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-11 NmzGeneratorOfInterior
  
  NmzGeneratorOfInterior( cone )  function
  Returns:  a vector representing the generator of the interior of cone
  
  If  cone  is Gorenstein, this function returns the generator of the interior
  of cone. If cone is not Gorenstein, an error is raised.
  
  2.3-12 NmzGrading
  
  NmzGrading( cone )  function
  Returns:  the grading vector
  
  This is an alias for NmzConeProperty( cone, "Grading" ); see NmzConeProperty
  (2.2-6).
  
  2.3-13 NmzHilbertBasis
  
  NmzHilbertBasis( cone )  function
  Returns:  a matrix whose rows are the Hilbert basis elements
  
  This   is   an  alias  for  NmzConeProperty(  cone,  "HilbertBasis"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-14 NmzHilbertQuasiPolynomial
  
  NmzHilbertQuasiPolynomial( cone )  function
  Returns:  the Hilbert function as a quasipolynomial
  
  The  Hilbert  function counts the lattice points degreewise. The result is a
  quasipolynomial  Q,  that is, a polynomial with periodic coefficients. It is
  given  as list of polynomials P_0, ..., P_(p-1) such that Q(i) = P_(i mod p)
  (i).
  
  This  is an alias for NmzConeProperty( cone, "HilbertQuasiPolynomial" ); see
  NmzConeProperty (2.2-6).
  
  2.3-15 NmzHilbertSeries
  
  NmzHilbertSeries( cone )  function
  Returns:  the Hilbert series as rational function
  
  The  result  consists of a list with two entries. The first is the numerator
  polynomial.  In  inhomogeneous  computations  this  can  also  be  a Laurent
  polynomial.  The  second list entry represents the denominator. It is a list
  of pairs [k_i, l_i]. Such a pair represents the factor (1-t^k_i)^l_i.
  
  This   is  an  alias  for  NmzConeProperty(  cone,  "HilbertSeries"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-16 NmzInclusionExclusionData
  
  NmzInclusionExclusionData( cone )  function
  Returns:  inclusion-exclusion data
  
  List of faces which are internally have been used in the inclusion-exclusion
  scheme.  Given  as a list pairs. The first pair entry is a key of generators
  contained  in  the  face  (compare  also  NmzTriangulation (2.3-34)) and the
  multiplicity  with  which  it  was  considered. Only available with excluded
  faces or strict constraints as input.
  
  This  is an alias for NmzConeProperty( cone, "InclusionExclusionData" ); see
  NmzConeProperty (2.2-6).
  
  2.3-17 NmzIsDeg1ExtremeRays
  
  NmzIsDeg1ExtremeRays( cone )  function
  Returns:  true if all extreme rays have degree 1; false otherwise
  
  This  is  an  alias  for  NmzConeProperty(  cone, "IsDeg1ExtremeRays" ); see
  NmzConeProperty (2.2-6).
  
  2.3-18 NmzIsDeg1HilbertBasis
  
  NmzIsDeg1HilbertBasis( cone )  function
  Returns:  true if all Hilbert basis elements have degree 1; false otherwise
  
  This  is  an  alias  for  NmzConeProperty( cone, "IsDeg1HilbertBasis" ); see
  NmzConeProperty (2.2-6).
  
  2.3-19 NmzIsGorenstein
  
  NmzIsGorenstein( cone )  function
  Returns:  whether the cone is Gorenstein
  
  Returns true if cone is Gorenstein, false otherwise.
  
  2.3-20 NmzIsInhomogeneous
  
  NmzIsInhomogeneous( cone )  function
  Returns:  whether the cone is inhomogeneous
  
  This value is always known directly after the creation of the cone.
  
  2.3-21 NmzIsIntegrallyClosed
  
  NmzIsIntegrallyClosed( cone )  function
  Returns:  true if the cone is integrally closed; false otherwise
  
  It  is  integrally closed when the Hilbert basis is a subset of the original
  monoid  generators.  So  it  is  only  computable if we have original monoid
  generators.
  
  This  is  an  alias  for  NmzConeProperty( cone, "IsIntegrallyClosed" ); see
  NmzConeProperty (2.2-6).
  
  2.3-22 NmzIsPointed
  
  NmzIsPointed( cone )  function
  Returns:  true if the cone is pointed; false otherwise
  
  This   is   an   alias   for   NmzConeProperty(  cone,  "IsPointed"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-23 NmzIsReesPrimary
  
  NmzIsReesPrimary( cone )  function
  Returns:  true if is the monomial ideal is primary to the irrelevant maximal
            ideal, false otherwise
  
  Only used with the input type rees_algebra.
  
  This   is  an  alias  for  NmzConeProperty(  cone,  "IsReesPrimary"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-24 NmzMaximalSubspace
  
  NmzMaximalSubspace( cone )  function
  Returns:  a matrix whose rows generate the maximale linear subspace
  
  This  is  an  alias  for  NmzConeProperty(  cone,  "MaximalSubspace"  ); see
  NmzConeProperty (2.2-6).
  
  2.3-25 NmzModuleGenerators
  
  NmzModuleGenerators( cone )  function
  Returns:  a matrix whose rows are the module generators
  
  This  is  an  alias  for  NmzConeProperty(  cone,  "ModuleGenerators" ); see
  NmzConeProperty (2.2-6).
  
  2.3-26 NmzModuleGeneratorsOverOriginalMonoid
  
  NmzModuleGeneratorsOverOriginalMonoid( cone )  function
  Returns:  a  matrix  whose  rows are the module generators over the original
            monoid
  
  A  minimal  system  of  generators of the integral closure over the original
  monoid. Requires the existence of original monoid generators. Not allowed in
  inhomogeneous computations.
  
  This       is       an       alias      for      NmzConeProperty(      cone,
  "ModuleGeneratorsOverOriginalMonoid" ); see NmzConeProperty (2.2-6).
  
  2.3-27 NmzModuleRank
  
  NmzModuleRank( cone )  function
  Returns:  the  rank  of  the module of lattice points in the polyhedron as a
            module over the recession monoid
  
  Only for inhomogeneous computations.
  
  This   is   an   alias   for  NmzConeProperty(  cone,  "ModuleRank"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-28 NmzMultiplicity
  
  NmzMultiplicity( cone )  function
  
  This   is   an  alias  for  NmzConeProperty(  cone,  "Multiplicity"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-29 NmzOriginalMonoidGenerators
  
  NmzOriginalMonoidGenerators( cone )  function
  Returns:  a matrix whose rows are the original monoid generators
  
  This  is  an  alias for NmzConeProperty( cone, "OriginalMonoidGenerators" );
  see NmzConeProperty (2.2-6).
  
  2.3-30 NmzRank
  
  NmzRank( cone )  function
  Returns:  the rank of the cone
  
  This value is the rank of the lattice generated by the lattice points of the
  cone.
  
  This is part of the cone property Sublattice.
  
  2.3-31 NmzRecessionRank
  
  NmzRecessionRank( cone )  function
  Returns:  the rank of the recession cone
  
  Only for inhomogeneous computations.
  
  This   is  an  alias  for  NmzConeProperty(  cone,  "RecessionRank"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-32 NmzReesPrimaryMultiplicity
  
  NmzReesPrimaryMultiplicity( cone )  function
  
  the  multiplicity of a monomial ideal, provided it is primary to the maximal
  ideal  generated  by  the  indeterminates.  Used  only  with  the input type
  rees_algebra.
  
  This is an alias for NmzConeProperty( cone, "ReesPrimaryMultiplicity" ); see
  NmzConeProperty (2.2-6).
  
  2.3-33 NmzSupportHyperplanes
  
  NmzSupportHyperplanes( cone )  function
  Returns:  a matrix whose rows represent the support hyperplanes
  
  The equations cut out the linear space generated by the cone. The equations,
  congruences  and support hyperplanes together describe the lattice points of
  the cone.
  
  This  is  an  alias  for  NmzConeProperty( cone, "SupportHyperplanes" ); see
  NmzConeProperty (2.2-6).
  
  2.3-34 NmzTriangulation
  
  NmzTriangulation( cone )  function
  Returns:  the triangulation
  
  This  returns  a  list  of  the maximal simplicial cones in a triangulation,
  i.e.,  a list of cones dividing the cone into simplicial cones. Each cone in
  the list is represented by a pair. The first entry of such a pair is the key
  of  the  simplex,  i.e.,  a  list of integers a_1,\dots,a_n referring to the
  NmzGenerators  (2.3-10)  (counting from 0) which are used in this simplicial
  cone. The second entry of each pair in the list is the absolute value of the
  determinant of the generator matrix of the simplicial cone.
  
  This   is  an  alias  for  NmzConeProperty(  cone,  "Triangulation"  );  see
  NmzConeProperty (2.2-6).
  
  2.3-35 NmzTriangulationDetSum
  
  NmzTriangulationDetSum( cone )  function
  Returns:  sum  of  the absolute values of the determinants of the simplicial
            cones in the used triangulation
  
  This  is  an  alias  for NmzConeProperty( cone, "TriangulationDetSum" ); see
  NmzConeProperty (2.2-6).
  
  2.3-36 NmzTriangulationSize
  
  NmzTriangulationSize( cone )  function
  Returns:  the number of simplicial cones in the used triangulation
  
  This  is  an  alias  for  NmzConeProperty(  cone, "TriangulationSize" ); see
  NmzConeProperty (2.2-6).
  
  2.3-37 NmzVerticesFloat
  
  NmzVerticesFloat( cone )  function
  Returns:  a  matrix  whose rows are the vertices of the polyhedron cone with
            float coordinates
  
  The  rows  of  this matrix represent the vertices of cone, printed as floats
  for  better  readability.  The result might be inexact, and should therefore
  not be used for computations.
  
  2.3-38 NmzVerticesOfPolyhedron
  
  NmzVerticesOfPolyhedron( cone )  function
  Returns:  a matrix whose rows are the vertices of the polyhedron
  
  This  is  an  alias for NmzConeProperty( cone, "VerticesOfPolyhedron" ); see
  NmzConeProperty (2.2-6).
  
  2.3-39 NmzConeDecomposition
  
  NmzConeDecomposition( cone )  function
  
  This is an alias for NmzConeProperty( cone, "ConeDecomposition" );
  
  2.3-40 NmzEmbeddingDim
  
  NmzEmbeddingDim( cone )  function
  
  This is an alias for NmzConeProperty( cone, "EmbeddingDim" );
  
  2.3-41 NmzExternalIndex
  
  NmzExternalIndex( cone )  function
  
  This is an alias for NmzConeProperty( cone, "ExternalIndex" );
  
  2.3-42 NmzGradingDenom
  
  NmzGradingDenom( cone )  function
  
  This is an alias for NmzConeProperty( cone, "GradingDenom" );
  
  2.3-43 NmzIntegerHull
  
  NmzIntegerHull( cone )  function
  
  This is an alias for NmzConeProperty( cone, "IntegerHull" );
  
  2.3-44 NmzInternalIndex
  
  NmzInternalIndex( cone )  function
  
  This is an alias for NmzConeProperty( cone, "InternalIndex" );
  
  2.3-45 NmzStanleyDec
  
  NmzStanleyDec( cone )  function
  
  This is an alias for NmzConeProperty( cone, "StanleyDec" );
  
  2.3-46 NmzSublattice
  
  NmzSublattice( cone )  function
  
  This is an alias for NmzConeProperty( cone, "Sublattice" );
  
  2.3-47 NmzUnitGroupIndex
  
  NmzUnitGroupIndex( cone )  function
  
  This is an alias for NmzConeProperty( cone, "UnitGroupIndex" );
  
  2.3-48 NmzWeightedEhrhartQuasiPolynomial
  
  NmzWeightedEhrhartQuasiPolynomial( cone )  function
  
  This       is       an       alias      for      NmzConeProperty(      cone,
  "NmzWeightedEhrhartQuasiPolynomial" );
  
  2.3-49 NmzWeightedEhrhartSeries
  
  NmzWeightedEhrhartSeries( cone )  function
  
  This is an alias for NmzConeProperty( cone, "NmzWeightedEhrhartSeries" );
  
  2.3-50 NmzWitnessNotIntegrallyClosed
  
  NmzWitnessNotIntegrallyClosed( cone )  function
  
  This is an alias for NmzConeProperty( cone, "WitnessNotIntegrallyClosed" );
  
  2.3-51 NmzBasisChange
  
  NmzBasisChange( cone )  function
  Returns:  a record describing the basis change
  
  The  result  record  r  has three components: r.Embedding, r.Projection, and
  r.Annihilator,  where the embedding A and the projection B are matrices, and
  the  annihilator  c  is  an  integer.  They  represent  the mapping into the
  effective  lattice  Z^n  -> Z^r, u ↦ (uB)/c and the inverse operation Z^r ->
  Z^n, v ↦ vA.
  
  This is part of the cone property Sublattice.