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# Toric lattices

This module was designed as a part of the framework for toric varieties (`variety`, `fano_variety`).

All toric lattices are isomorphic to \(\ZZ^n\) for some \(n\), but will prevent you from doing “wrong” operations with objects from different lattices.

AUTHORS:

- Andrey Novoseltsev (2010-05-27): initial version.
- Andrey Novoseltsev (2010-07-30): sublattices and quotients.

EXAMPLES:

The simplest way to create a toric lattice is to specify its dimension only:

    sage: N = ToricLattice(3)
    sage: N
    3-d lattice N

While our lattice `N` is called exactly “N” it is a coincidence: all lattices are called “N” by default:

    sage: another_name = ToricLattice(3)
    sage: another_name
    3-d lattice N

If fact, the above lattice is exactly the same as before as an object in memory:

    sage: N is another_name
    True

There are actually four names associated to a toric lattice and they all must be the same for two lattices to coincide:

    sage: N, N.dual(), latex(N), latex(N.dual())
    (3-d lattice N, 3-d lattice M, N, M)

Notice that the lattice dual to `N` is called “M” which is standard in toric geometry. This happens only if you allow completely automatic handling of names:

    sage: another_N = ToricLattice(3, "N")
    sage: another_N.dual()
    3-d lattice N*
    sage: N is another_N
    False

What can you do with toric lattices? Well, their main purpose is to allow creation of elements of toric lattices:

    sage: n = N([1,2,3])
    sage: n
    N(1, 2, 3)
    sage: M = N.dual()
    sage: m = M(1,2,3)
    sage: m
    M(1, 2, 3)

Dual lattices can act on each other:

    sage: n * m
    14
    sage: m * n
    14

You can also add elements of the same lattice or scale them:

    sage: 2 * n
    N(2, 4, 6)
    sage: n * 2
    N(2, 4, 6)
    sage: n + n
    N(2, 4, 6)

However, you cannot “mix wrong lattices” in your expressions:

    sage: n + m
    Traceback (most recent call last):
    ...
    TypeError: unsupported operand parent(s) for '+':
    '3-d lattice N' and '3-d lattice M'
    sage: n * n
    Traceback (most recent call last):
    ...
    TypeError: elements of the same toric lattice cannot be multiplied!
    sage: n == m
    False

Note that `n` and `m` are not equal to each other even though they are both “just (1,2,3).” Moreover, you cannot easily convert elements between toric lattices:

    sage: M(n)
    Traceback (most recent call last):
    ...
    TypeError: N(1, 2, 3) cannot be converted to 3-d lattice M!

If you really need to consider elements of one lattice as elements of another, you can either use intermediate conversion to “just a vector”:

    sage: ZZ3 = ZZ^3
    sage: n_in_M = M(ZZ3(n))
    sage: n_in_M
    M(1, 2, 3)
    sage: n == n_in_M
    False
    sage: n_in_M == m
    True

Or you can create a homomorphism from one lattice to any other:

    sage: h = N.hom(identity_matrix(3), M)
    sage: h(n)
    M(1, 2, 3)

Warning

While integer vectors (elements of \(\ZZ^n\)) are printed as `(1,2,3)`, in the code `(1,2,3)` is a `tuple`, which has nothing to do neither with vectors, nor with toric lattices, so the following is probably not what you want while working with toric geometry objects:

    sage: (1,2,3) + (1,2,3)
    (1, 2, 3, 1, 2, 3)

Instead, use syntax like

    sage: N(1,2,3) + N(1,2,3)
    N(2, 4, 6)

class `sage.geometry.toric_lattice.``ToricLatticeFactory`

  Bases: `sage.structure.factory.UniqueFactory`

  Create a lattice for toric geometry objects.

  INPUT:

  - `rank` – nonnegative integer, the only mandatory parameter;
  - `name` – string;
  - `dual_name` – string;
  - `latex_name` – string;
  - `latex_dual_name` – string.

  OUTPUT:

  - lattice.

  A toric lattice is uniquely determined by its rank and associated names. There are four such “associated names” whose meaning should be clear from the names of the corresponding parameters, but the choice of default values is a little bit involved. So here is the full description of the “naming algorithm”:

  1. If no names were given at all, then this lattice will be called “N” and the dual one “M”. These are the standard choices in toric geometry.
  2. If `name` was given and `dual_name` was not, then `dual_name` will be `name` followed by “*”.
  3. If LaTeX names were not given, they will coincide with the “usual” names, but if `dual_name` was constructed automatically, the trailing star will be typeset as a superscript.

  EXAMPLES:

  Let’s start with no names at all and see how automatic names are given:

      sage: L1 = ToricLattice(3)
      sage: L1
      3-d lattice N
      sage: L1.dual()
      3-d lattice M

  If we give the name “N” explicitly, the dual lattice will be called “N*”:

      sage: L2 = ToricLattice(3, "N")
      sage: L2
      3-d lattice N
      sage: L2.dual()
      3-d lattice N*

  However, we can give an explicit name for it too:

      sage: L3 = ToricLattice(3, "N", "M")
      sage: L3
      3-d lattice N
      sage: L3.dual()
      3-d lattice M

  If you want, you may also give explicit LaTeX names:

      sage: L4 = ToricLattice(3, "N", "M", r"\mathbb{N}", r"\mathbb{M}")
      sage: latex(L4)
      \mathbb{N}
      sage: latex(L4.dual())
      \mathbb{M}

  While all four lattices above are called “N”, only two of them are equal (and are actually the same):

      sage: L1 == L2
      False
      sage: L1 == L3
      True
      sage: L1 is L3
      True
      sage: L1 == L4
      False

  The reason for this is that `L2` and `L4` have different names either for dual lattices or for LaTeX typesetting.

  `create_key`(rank, name=None, dual_name=None, latex_name=None, latex_dual_name=None)

    Create a key that uniquely identifies this toric lattice.

    See `ToricLattice` for documentation.

    Warning

    You probably should not use this function directly.

    TESTS:

        sage: ToricLattice.create_key(3)
        (3, 'N', 'M', 'N', 'M')
        sage: N = ToricLattice(3)
        sage: loads(dumps(N)) is N
        True
        sage: TestSuite(N).run()

  `create_object`(version, key)

    Create the toric lattice described by `key`.

    See `ToricLattice` for documentation.

    Warning

    You probably should not use this function directly.

    TESTS:

        sage: key = ToricLattice.create_key(3)
        sage: ToricLattice.create_object(1, key)
        3-d lattice N

class `sage.geometry.toric_lattice.``ToricLattice_ambient`(rank, name, dual_name, latex_name, latex_dual_name)

  Bases: `sage.geometry.toric_lattice.ToricLattice_generic`, `sage.modules.free_module.FreeModule_ambient_pid`

  Create a toric lattice.

  See `ToricLattice` for documentation.

  Warning

  There should be only one toric lattice with the given rank and associated names. Using this class directly to create toric lattices may lead to unexpected results. Please, use `ToricLattice` to create toric lattices.

  TESTS:

      sage: N = ToricLattice(3, "N", "M", "N", "M")
      sage: N
      3-d lattice N
      sage: TestSuite(N).run()

  `ambient_module`()

    Return the ambient module of `self`.

    OUTPUT:

    - `toric lattice`.

    Note

    For any ambient toric lattice its ambient module is the lattice itself.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: N.ambient_module()
        3-d lattice N
        sage: N.ambient_module() is N
        True

  `dual`()

    Return the lattice dual to `self`.

    OUTPUT:

    - `toric lattice`.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: N
        3-d lattice N
        sage: M = N.dual()
        sage: M
        3-d lattice M
        sage: M.dual() is N
        True

    Elements of dual lattices can act on each other:

        sage: n = N(1,2,3)
        sage: m = M(4,5,6)
        sage: n * m
        32
        sage: m * n
        32

  `plot`(**options)

    Plot `self`.

    INPUT:

    - any options for toric plots (see `toric_plotter.options`), none are mandatory.

    OUTPUT:

    - a plot.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: N.plot()
        Graphics3d Object

class `sage.geometry.toric_lattice.``ToricLattice_generic`(base_ring, rank, degree, sparse=False)

  Bases: `sage.modules.free_module.FreeModule_generic_pid`

  Abstract base class for toric lattices.

  `construction`()

    Return the functorial construction of `self`.

    OUTPUT:

    - `None`, we do not think of toric lattices as constructed from simpler objects since we do not want to perform arithmetic involving different lattices.

    TESTS:

        sage: print ToricLattice(3).construction()
        None

  `direct_sum`(other)

    Return the direct sum with `other`.

    INPUT:

    - `other` – a toric lattice or more general module.

    OUTPUT:

    The direct sum of `self` and `other` as \(\ZZ\)-modules. If `other` is a `ToricLattice`, another toric lattice will be returned.

    EXAMPLES:

        sage: K = ToricLattice(3, 'K')
        sage: L = ToricLattice(3, 'L')
        sage: N = K.direct_sum(L); N
        6-d lattice K+L
        sage: N, N.dual(), latex(N), latex(N.dual())
        (6-d lattice K+L, 6-d lattice K*+L*, K \oplus L, K^* \oplus L^*)

    With default names:

        sage: N = ToricLattice(3).direct_sum(ToricLattice(2))
        sage: N, N.dual(), latex(N), latex(N.dual())
        (5-d lattice N+N, 5-d lattice M+M, N \oplus N, M \oplus M)

    If `other` is not a `ToricLattice`, fall back to sum of modules:

        sage: ToricLattice(3).direct_sum(ZZ^2)
        Free module of degree 5 and rank 5 over Integer Ring
        Echelon basis matrix:
        [1 0 0 0 0]
        [0 1 0 0 0]
        [0 0 1 0 0]
        [0 0 0 1 0]
        [0 0 0 0 1]

  `intersection`(other)

    Return the intersection of `self` and `other`.

    INPUT:

    - `other` - a toric (sub)lattice.dual

    OUTPUT:

    - a toric (sub)lattice.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns1 = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Ns2 = N.submodule([N(1,4,9), N(9,2,0)])
        sage: Ns1.intersection(Ns2)
        Sublattice <N(54, 12, 0)>

    Note that if one of the intersecting sublattices is a sublattice of another, no new lattices will be constructed:

        sage: N.intersection(N) is N
        True
        sage: Ns1.intersection(N) is Ns1
        True
        sage: N.intersection(Ns1) is Ns1
        True

  `quotient`(sub, check=True, positive_point=None, positive_dual_point=None)

    Return the quotient of `self` by the given sublattice `sub`.

    INPUT:

    - `sub` – sublattice of self;
    - `check` – (default: True) whether or not to check that `sub` is a valid sublattice.

    If the quotient is one-dimensional and torsion free, the following two mutually exclusive keyword arguments are also allowed. They decide the sign choice for the (single) generator of the quotient lattice:

    - `positive_point` – a lattice point of `self` not in the sublattice `sub` (that is, not zero in the quotient lattice). The quotient generator will be in the same direction as `positive_point`.
    - `positive_dual_point` – a dual lattice point. The quotient generator will be chosen such that its lift has a positive product with `positive_dual_point`. Note: if `positive_dual_point` is not zero on the sublattice `sub`, then the notion of positivity will depend on the choice of lift!

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Q = N/Ns
        sage: Q
        Quotient with torsion of 3-d lattice N
        by Sublattice <N(1, 8, 0), N(0, 12, 0)>

    See `ToricLattice_quotient` for more examples.

  `saturation`()

    Return the saturation of `self`.

    OUTPUT:

    - a `toric lattice`.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([(1,2,3), (4,5,6)])
        sage: Ns
        Sublattice <N(1, 2, 3), N(0, 3, 6)>
        sage: Ns_sat = Ns.saturation()
        sage: Ns_sat
        Sublattice <N(1, 0, -1), N(0, 1, 2)>
        sage: Ns_sat is Ns_sat.saturation()
        True

  `span`(*args, **kwds)

    Return the span of the given generators.

    INPUT:

    - `gens` – list of elements of the ambient vector space of `self`.

    OUTPUT:

    - submodule spanned by `gens`.

    Note

    The output need not be a submodule of `self`, nor even of the ambient space. It must, however, be contained in the ambient vector space.

    See also `span_of_basis()`, `submodule()`, and `submodule_with_basis()`,

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N.gen(0)])
        sage: Ns.span([N.gen(1)])
        Sublattice <N(0, 1, 0)>
        sage: Ns.submodule([N.gen(1)])
        Traceback (most recent call last):
        ...
        ArithmeticError: Argument gens (= [N(0, 1, 0)])
        does not generate a submodule of self.

  `span_of_basis`(*args, **kwds)

    Return the submodule with the given `basis`.

    INPUT:

    - `basis` – list of elements of the ambient vector space of `self`.

    OUTPUT:

    - submodule spanned by `basis`.

    Note

    The output need not be a submodule of `self`, nor even of the ambient space. It must, however, be contained in the ambient vector space.

    See also `span()`, `submodule()`, and `submodule_with_basis()`,

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.span_of_basis([(1,2,3)])
        sage: Ns.span_of_basis([(2,4,0)])
        Sublattice <N(2, 4, 0)>
        sage: Ns.span_of_basis([(1/5,2/5,0), (1/7,1/7,0)])
        Sublattice <(1/5, 2/5, 0), (1/7, 1/7, 0)>

    Of course the input basis vectors must be linearly independent:

        sage: Ns.span_of_basis([(1,2,0), (2,4,0)])
        Traceback (most recent call last):
        ...
        ValueError: The given basis vectors must be linearly independent.

class `sage.geometry.toric_lattice.``ToricLattice_quotient`(V, W, check=True, positive_point=None, positive_dual_point=None)

  Bases: `sage.modules.fg_pid.fgp_module.FGP_Module_class`

  Construct the quotient of a toric lattice `V` by its sublattice `W`.

  INPUT:

  - `V` – ambient toric lattice;
  - `W` – sublattice of `V`;
  - `check` – (default: `True`) whether to check correctness of input or not.

  If the quotient is one-dimensional and torsion free, the following two mutually exclusive keyword arguments are also allowed. They decide the sign choice for the (single) generator of the quotient lattice:

  - `positive_point` – a lattice point of `self` not in the sublattice `sub` (that is, not zero in the quotient lattice). The quotient generator will be in the same direction as `positive_point`.
  - `positive_dual_point` – a dual lattice point. The quotient generator will be chosen such that its lift has a positive product with `positive_dual_point`. Note: if `positive_dual_point` is not zero on the sublattice `sub`, then the notion of positivity will depend on the choice of lift!

  OUTPUT:

  - quotient of `V` by `W`.

  EXAMPLES:

  The intended way to get objects of this class is to use `quotient()` method of toric lattices:

      sage: N = ToricLattice(3)
      sage: sublattice = N.submodule([(1,1,0), (3,2,1)])
      sage: Q = N/sublattice
      sage: Q
      1-d lattice, quotient of 3-d lattice N by Sublattice <N(1, 0, 1), N(0, 1, -1)>
      sage: Q.gens()
      (N[0, 0, 1],)

  Here, `sublattice` happens to be of codimension one in `N`. If you want to prescribe the sign of the quotient generator, you can do either:

      sage: Q = N.quotient(sublattice, positive_point=N(0,0,-1)); Q
      1-d lattice, quotient of 3-d lattice N by Sublattice <N(1, 0, 1), N(0, 1, -1)>
      sage: Q.gens()
      (N[0, 0, -1],)

  or:

      sage: M = N.dual()
      sage: Q = N.quotient(sublattice, positive_dual_point=M(0,0,-1)); Q
      1-d lattice, quotient of 3-d lattice N by Sublattice <N(1, 0, 1), N(0, 1, -1)>
      sage: Q.gens()
      (N[0, 0, -1],)

  TESTS:

      sage: loads(dumps(Q)) == Q
      True
      sage: loads(dumps(Q)).gens() == Q.gens()
      True

  `Element`

    alias of `ToricLattice_quotient_element`

  `base_extend`(R)

    Return the base change of `self` to the ring `R`.

    INPUT:

    - `R` – either \(\ZZ\) or \(\QQ\).

    OUTPUT:

    - `self` if \(R=\ZZ\), quotient of the base extension of the ambient lattice by the base extension of the sublattice if \(R=\QQ\).

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Q = N/Ns
        sage: Q.base_extend(ZZ) is Q
        True
        sage: Q.base_extend(QQ)
        Vector space quotient V/W of dimension 1 over Rational Field where
        V: Vector space of dimension 3 over Rational Field
        W: Vector space of degree 3 and dimension 2 over Rational Field
        Basis matrix:
        [1 0 0]
        [0 1 0]

  `coordinate_vector`(x, reduce=False)

    Return coordinates of x with respect to the optimized representation of self.

    INPUT:

    - `x` – element of `self` or convertable to `self`.
    - `reduce` – (default: False); if True, reduce coefficients modulo invariants.

    OUTPUT:

    The coordinates as a vector.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Q = N.quotient(N.span([N(1,2,3), N(0,2,1)]), positive_point=N(0,-1,0))
        sage: q = Q.gen(0); q
        N[0, -1, 0]
        sage: q.vector()  # indirect test
        (1)
        sage: Q.coordinate_vector(q)
        (1)

  `dimension`()

    Return the rank of `self`.

    OUTPUT:

    Integer. The dimension of the free part of the quotient.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Q = N/Ns
        sage: Q.ngens()
        2
        sage: Q.rank()
        1
        sage: Ns = N.submodule([N(1,4,0)])
        sage: Q = N/Ns
        sage: Q.ngens()
        2
        sage: Q.rank()
        2

  `dual`()

    Return the lattice dual to `self`.

    OUTPUT:

    - a `toric lattice quotient`.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([(1, -1, -1)])
        sage: Q = N / Ns
        sage: Q.dual()
        Sublattice <M(1, 0, 1), M(0, 1, -1)>

  `gens`()

    Return the generators of the quotient.

    OUTPUT:

    A tuple of `ToricLattice_quotient_element` generating the quotient.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Q = N.quotient(N.span([N(1,2,3), N(0,2,1)]), positive_point=N(0,-1,0))
        sage: Q.gens()
        (N[0, -1, 0],)

  `is_torsion_free`()

    Check if `self` is torsion-free.

    OUTPUT:

    - `True` is `self` has no torsion and `False` otherwise.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Q = N/Ns
        sage: Q.is_torsion_free()
        False
        sage: Ns = N.submodule([N(1,4,0)])
        sage: Q = N/Ns
        sage: Q.is_torsion_free()
        True

  `rank`()

    Return the rank of `self`.

    OUTPUT:

    Integer. The dimension of the free part of the quotient.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Q = N/Ns
        sage: Q.ngens()
        2
        sage: Q.rank()
        1
        sage: Ns = N.submodule([N(1,4,0)])
        sage: Q = N/Ns
        sage: Q.ngens()
        2
        sage: Q.rank()
        2

class `sage.geometry.toric_lattice.``ToricLattice_quotient_element`(parent, x, check=True)

  Bases: `sage.modules.fg_pid.fgp_element.FGP_Element`

  Create an element of a toric lattice quotient.

  Warning

  You probably should not construct such elements explicitly.

  INPUT:

  - same as for `FGP_Element`.

  OUTPUT:

  - element of a toric lattice quotient.

  TESTS:

      sage: N = ToricLattice(3)
      sage: sublattice = N.submodule([(1,1,0), (3,2,1)])
      sage: Q = N/sublattice
      sage: e = Q(1,2,3)
      sage: e
      N[1, 2, 3]
      sage: e2 = Q(N(2,3,3))
      sage: e2
      N[2, 3, 3]
      sage: e == e2
      True
      sage: e.vector()
      (4)
      sage: e2.vector()
      (4)

  `set_immutable`()

    Make `self` immutable.

    OUTPUT:

    - none.

    Note

    Elements of toric lattice quotients are always immutable, so this method does nothing, it is introduced for compatibility purposes only.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([N(2,4,0), N(9,12,0)])
        sage: Q = N/Ns
        sage: Q.0.set_immutable()

class `sage.geometry.toric_lattice.``ToricLattice_sublattice`(ambient, gens, check=True, already_echelonized=False)

  Bases: `sage.geometry.toric_lattice.ToricLattice_sublattice_with_basis`, `sage.modules.free_module.FreeModule_submodule_pid`

  Construct the sublattice of `ambient` toric lattice generated by `gens`.

  INPUT (same as for `FreeModule_submodule_pid`):

  - `ambient` – ambient `toric lattice` for this sublattice;
  - `gens` – list of elements of `ambient` generating the constructed sublattice;
  - see the base class for other available options.

  OUTPUT:

  - sublattice of a toric lattice with an automatically chosen basis.

  See also `ToricLattice_sublattice_with_basis` if you want to specify an explicit basis.

  EXAMPLES:

  The intended way to get objects of this class is to use `submodule()` method of toric lattices:

      sage: N = ToricLattice(3)
      sage: sublattice = N.submodule([(1,1,0), (3,2,1)])
      sage: sublattice.has_user_basis()
      False
      sage: sublattice.basis()
      [
      N(1, 0, 1),
      N(0, 1, -1)
      ]

  For sublattices without user-specified basis, the basis obtained above is the same as the “standard” one:

      sage: sublattice.echelonized_basis()
      [
      N(1, 0, 1),
      N(0, 1, -1)
      ]

class `sage.geometry.toric_lattice.``ToricLattice_sublattice_with_basis`(ambient, basis, check=True, echelonize=False, echelonized_basis=None, already_echelonized=False)

  Bases: `sage.geometry.toric_lattice.ToricLattice_generic`, `sage.modules.free_module.FreeModule_submodule_with_basis_pid`

  Construct the sublattice of `ambient` toric lattice with given `basis`.

  INPUT (same as for `FreeModule_submodule_with_basis_pid`):

  - `ambient` – ambient `toric lattice` for this sublattice;
  - `basis` – list of linearly independent elements of `ambient`, these elements will be used as the default basis of the constructed sublattice;
  - see the base class for other available options.

  OUTPUT:

  - sublattice of a toric lattice with a user-specified basis.

  See also `ToricLattice_sublattice` if you do not want to specify an explicit basis.

  EXAMPLES:

  The intended way to get objects of this class is to use `submodule_with_basis()` method of toric lattices:

      sage: N = ToricLattice(3)
      sage: sublattice = N.submodule_with_basis([(1,1,0), (3,2,1)])
      sage: sublattice.has_user_basis()
      True
      sage: sublattice.basis()
      [
      N(1, 1, 0),
      N(3, 2, 1)
      ]

  Even if you have provided your own basis, you still can access the “standard” one:

      sage: sublattice.echelonized_basis()
      [
      N(1, 0, 1),
      N(0, 1, -1)
      ]

  `dual`()

    Return the lattice dual to `self`.

    OUTPUT:

    - a `toric lattice quotient`.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: Ns = N.submodule([(1,1,0), (3,2,1)])
        sage: Ns.dual()
        2-d lattice, quotient of 3-d lattice M by Sublattice <M(1, -1, -1)>

  `plot`(**options)

    Plot `self`.

    INPUT:

    - any options for toric plots (see `toric_plotter.options`), none are mandatory.

    OUTPUT:

    - a plot.

    EXAMPLES:

        sage: N = ToricLattice(3)
        sage: sublattice = N.submodule_with_basis([(1,1,0), (3,2,1)])
        sage: sublattice.plot()
        Graphics3d Object

    Now we plot both the ambient lattice and its sublattice:

        sage: N.plot() + sublattice.plot(point_color="red")
        Graphics3d Object

`sage.geometry.toric_lattice.``is_ToricLattice`(x)

  Check if `x` is a toric lattice.

  INPUT:

  - `x` – anything.

  OUTPUT:

  - `True` if `x` is a toric lattice and `False` otherwise.

  EXAMPLES:

      sage: from sage.geometry.toric_lattice import (
      ...     is_ToricLattice)
      sage: is_ToricLattice(1)
      False
      sage: N = ToricLattice(3)
      sage: N
      3-d lattice N
      sage: is_ToricLattice(N)
      True

`sage.geometry.toric_lattice.``is_ToricLatticeQuotient`(x)

  Check if `x` is a toric lattice quotient.

  INPUT:

  - `x` – anything.

  OUTPUT:

  - `True` if `x` is a toric lattice quotient and `False` otherwise.

  EXAMPLES:

      sage: from sage.geometry.toric_lattice import (
      ...     is_ToricLatticeQuotient)
      sage: is_ToricLatticeQuotient(1)
      False
      sage: N = ToricLattice(3)
      sage: N
      3-d lattice N
      sage: is_ToricLatticeQuotient(N)
      False
      sage: Q = N / N.submodule([(1,2,3), (3,2,1)])
      sage: Q
      Quotient with torsion of 3-d lattice N
      by Sublattice <N(1, 2, 3), N(0, 4, 8)>
      sage: is_ToricLatticeQuotient(Q)
      True

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