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# Finite posets

Here is some terminology used in this file:

- An order filter (or upper set) of a poset \(P\) is a subset \(S\) of \(P\) such that if \(x \leq y\) and \(x\in S\) then \(y\in S\).
- An order ideal (or lower set) of a poset \(P\) is a subset \(S\) of \(P\) such that if \(x \leq y\) and \(y\in S\) then \(x\in S\).

class `sage.categories.finite_posets.``FinitePosets`(base_category)

  Bases: `sage.categories.category_with_axiom.CategoryWithAxiom`

  The category of finite posets i.e. finite sets with a partial order structure.

  EXAMPLES:

      sage: FinitePosets()
      Category of finite posets
      sage: FinitePosets().super_categories()
      [Category of posets, Category of finite sets]
      sage: FinitePosets().example()
      NotImplemented

  See also

  `Posets`, `Poset()`

  TESTS:

      sage: C = FinitePosets()
      sage: C is Posets().Finite()
      True
      sage: TestSuite(C).run()

  class `ParentMethods`
    `antichains`()

      Return all antichains of `self`.

      EXAMPLES:

          sage: A = Posets.PentagonPoset().antichains(); A
          Set of antichains of Finite lattice containing 5 elements
          sage: list(A)
          [[], [0], [1], [1, 2], [1, 3], [2], [3], [4]]

    `birational_free_labelling`(linear_extension=None, prefix='x', base_field=None, reduced=False, addvars=None)

      Return the birational free labelling of `self`.

      Let us hold back defining this, and introduce birational toggles and birational rowmotion first. These notions have been introduced in [EP13] as generalizations of the notions of toggles (`order_ideal_toggle()`) and `rowmotion` on order ideals of a finite poset. They have been studied further in [GR13].

      Let \(\mathbf{K}\) be a field, and \(P\) be a finite poset. Let \(\widehat{P}\) denote the poset obtained from \(P\) by adding a new element \(1\) which is greater than all existing elements of \(P\), and a new element \(0\) which is smaller than all existing elements of \(P\) and \(1\). Now, a `mathbf{K}`-labelling of `P` will mean any function from \(\widehat{P}\) to \(\mathbf{K}\). The image of an element \(v\) of \(\widehat{P}\) under this labelling will be called the label of this labelling at \(v\). The set of all \(\mathbf{K}\)-labellings of \(P\) is clearly \(\mathbf{K}^{\widehat{P}}\).

      For any \(v \in P\), we now define a rational map \(T_v : \mathbf{K}^{\widehat{P}} \dashrightarrow \mathbf{K}^{\widehat{P}}\) as follows: For every \(f \in \mathbf{K}^{\widehat{P}}\), the image \(T_v f\) should send every element \(u \in \widehat{P}\) distinct from \(v\) to \(f(u)\) (so the labels at all \(u \neq v\) don’t change), while \(v\) is sent to

      \[\frac{1}{f(v)} \cdot \frac{\sum_{u \lessdot v} f(u)} {\sum_{u \gtrdot v} \frac{1}{f(u)}}\]

      (both sums are over all \(u \in \widehat{P}\) satisfying the respectively given conditions). Here, \(\lessdot\) and \(\gtrdot\) mean (respectively) “covered by” and “covers”, interpreted with respect to the poset \(\widehat{P}\). This rational map \(T_v\) is an involution and is called the (birational) `v`-toggle; see `birational_toggle()` for its implementation.

      Now, birational rowmotion is defined as the composition \(T_{v_1} \circ T_{v_2} \circ \cdots \circ T_{v_n}\), where \((v_1, v_2, \ldots, v_n)\) is a linear extension of \(P\) (written as a linear ordering of the elements of \(P\)). This is a rational map \(\mathbf{K}^{\widehat{P}} \dashrightarrow \mathbf{K}^{\widehat{P}}\) which does not depend on the choice of the linear extension; it is denoted by \(R\). See `birational_rowmotion()` for its implementation.

      The definitions of birational toggles and birational rowmotion extend to the case of \(\mathbf{K}\) being any semifield rather than necessarily a field (although it becomes less clear what constitutes a rational map in this generality). The most useful case is that of the `tropical semiring`, in which case birational rowmotion relates to classical constructions such as promotion of rectangular semistandard Young tableaux (page 5 of [EP13b] and future work, via the related notion of birational promotion) and rowmotion on order ideals of the poset ([EP13]).

      The birational free labelling is a special labelling defined for every finite poset \(P\) and every linear extension \((v_1, v_2, \ldots, v_n)\) of \(P\). It is given by sending every element \(v_i\) in \(P\) to \(x_i\), sending the element \(0\) of \(\widehat{P}\) to \(a\), and sending the element \(1\) of \(\widehat{P}\) to \(b\), where the ground field \(\mathbf{K}\) is the field of rational functions in \(n+2\) indeterminates \(a, x_1, x_2, \ldots, x_n, b\) over \(\mathbb Q\).

      In Sage, a labelling \(f\) of a poset \(P\) is encoded as a \(4\)-tuple \((\mathbf{K}, d, u, v)\), where \(\mathbf{K}\) is the ground field of the labelling (i. e., its target), \(d\) is the dictionary containing the values of \(f\) at the elements of \(P\) (the keys being the respective elements of \(P\)), \(u\) is the label of \(f\) at \(0\), and \(v\) is the label of \(f\) at \(1\).

      Warning

      The dictionary \(d\) is labelled by the elements of \(P\). If \(P\) is a poset with `facade` option set to `False`, these might not be what they seem to be! (For instance, if `P == Poset({1: [2, 3]}, facade=False)`, then the value of \(d\) at \(1\) has to be accessed by `d[P(1)]`, not by `d[1]`.)

      Warning

      Dictionaries are mutable. They do compare correctly, but are not hashable and need to be cloned to avoid spooky action at a distance. Be careful!

      INPUT:

      - `linear_extension` – (default: the default linear extension of `self`) a linear extension of `self` (as a linear extension or as a list), or more generally a list of all elements of all elements of `self` each occurring exactly once
      - `prefix` – (default: `'x'`) the prefix to name the indeterminates corresponding to the elements of `self` in the labelling (so, setting it to `'frog'` will result in these indeterminates being called `frog1, frog2, ..., frogn` rather than `x1, x2, ..., xn`).
      - `base_field` – (default: `QQ`) the base field to be used instead of \(\QQ\) to define the rational function field over; this is not going to be the base field of the labelling, because the latter will have indeterminates adjoined!
      - `reduced` – (default: `False`) if set to `True`, the result will be the reduced birational free labelling, which differs from the regular one by having \(0\) and \(1\) both sent to \(1\) instead of \(a\) and \(b\) (the indeterminates \(a\) and \(b\) then also won’t appear in the ground field)
      - `addvars` – (default: `''`) a string containing names of extra variables to be adjoined to the ground field (these don’t have an effect on the labels)

      OUTPUT:

      The birational free labelling of the poset `self` and the linear extension `linear_extension`. Or, if `reduced` is set to `True`, the reduced birational free labelling.

      REFERENCES:

      [EP13] | (1, 2) David Einstein, James Propp. Combinatorial, piecewise-linear, and birational homomesy for products of two chains. Arxiv 1310.5294v1.

      [EP13b] | David Einstein, James Propp. Piecewise-linear and birational toggling. Extended abstract for FPSAC 2014. http://faculty.uml.edu/jpropp/fpsac14.pdf

      [GR13] | (1, 2) Darij Grinberg, Tom Roby. Iterative properties of birational rowmotion I. http://web.mit.edu/~darij/www/algebra/skeletal.pdf

      EXAMPLES:

      We construct the birational free labelling on a simple poset:

          sage: P = Poset({1: [2, 3]})
          sage: l = P.birational_free_labelling(); l
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(l[1].items())
          [(1, x1), (2, x2), (3, x3)]

          sage: l = P.birational_free_labelling(linear_extension=[1, 3, 2]); l
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(l[1].items())
          [(1, x1), (2, x3), (3, x2)]

          sage: l = P.birational_free_labelling(linear_extension=[1, 3, 2], reduced=True, addvars="spam, eggs"); l
          (Fraction Field of Multivariate Polynomial Ring in x1, x2, x3, spam, eggs over Rational Field,
           {...},
           1,
           1)
          sage: sorted(l[1].items())
          [(1, x1), (2, x3), (3, x2)]

          sage: l = P.birational_free_labelling(linear_extension=[1, 3, 2], prefix="wut", reduced=True, addvars="spam, eggs"); l
          (Fraction Field of Multivariate Polynomial Ring in wut1, wut2, wut3, spam, eggs over Rational Field,
           {...},
           1,
           1)
          sage: sorted(l[1].items())
          [(1, wut1), (2, wut3), (3, wut2)]

          sage: l = P.birational_free_labelling(linear_extension=[1, 3, 2], reduced=False, addvars="spam, eggs"); l
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b, spam, eggs over Rational Field,
           {...},
           a,
           b)
          sage: sorted(l[1].items())
          [(1, x1), (2, x3), (3, x2)]
          sage: l[1][2]
          x3

      Illustrating the warning about facade:

          sage: P = Poset({1: [2, 3]}, facade=False)
          sage: l = P.birational_free_labelling(linear_extension=[1, 3, 2], reduced=False, addvars="spam, eggs"); l
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b, spam, eggs over Rational Field,
           {...},
           a,
           b)
          sage: l[1][2]
          Traceback (most recent call last):
          ...
          KeyError: 2
          sage: l[1][P(2)]
          x3

      Another poset:

          sage: P = Posets.SSTPoset([2,1])
          sage: lext = sorted(P)
          sage: l = P.birational_free_labelling(linear_extension=lext, addvars="ohai")
          sage: l
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, x4, x5, x6, x7, x8, b, ohai over Rational Field,
           {...},
           a,
           b)
          sage: sorted(l[1].items())
          [([[1, 1], [2]], x1), ([[1, 1], [3]], x2), ([[1, 2], [2]], x3), ([[1, 2], [3]], x4),
           ([[1, 3], [2]], x5), ([[1, 3], [3]], x6), ([[2, 2], [3]], x7), ([[2, 3], [3]], x8)]

      See `birational_rowmotion()`, `birational_toggle()` and `birational_toggles()` for more substantial examples of what one can do with the birational free labelling.

      TESTS:

      The `linear_extension` keyword does not have to be given an actual linear extension:

          sage: P = Posets.ChainPoset(2).product(Posets.ChainPoset(3))
          sage: P
          Finite poset containing 6 elements
          sage: lex = [(1,0),(0,0),(1,1),(0,1),(1,2),(0,2)]
          sage: l = P.birational_free_labelling(linear_extension=lex,
          ....:                                 prefix="u", reduced=True)
          sage: l
          (Fraction Field of Multivariate Polynomial Ring in u1, u2, u3, u4, u5, u6 over Rational Field,
           {...},
           1,
           1)
          sage: sorted(l[1].items())
          [((0, 0), u2),
           ((0, 1), u4),
           ((0, 2), u6),
           ((1, 0), u1),
           ((1, 1), u3),
           ((1, 2), u5)]

      For comparison, the standard linear extension:

          sage: l = P.birational_free_labelling(prefix="u", reduced=True); l
          (Fraction Field of Multivariate Polynomial Ring in u1, u2, u3, u4, u5, u6 over Rational Field,
           {...},
           1,
           1)
          sage: sorted(l[1].items())
          [((0, 0), u1),
           ((0, 1), u2),
           ((0, 2), u3),
           ((1, 0), u4),
           ((1, 1), u5),
           ((1, 2), u6)]

      If you want your linear extension to be tested for being a linear extension, just call the `linear_extension` method on the poset:

          sage: lex = [(0,0),(0,1),(1,0),(1,1),(0,2),(1,2)]
          sage: l = P.birational_free_labelling(linear_extension=P.linear_extension(lex),
          ....:                                 prefix="u", reduced=True)
          sage: l
          (Fraction Field of Multivariate Polynomial Ring in u1, u2, u3, u4, u5, u6 over Rational Field,
           {...},
           1,
           1)
          sage: sorted(l[1].items())
          [((0, 0), u1),
           ((0, 1), u2),
           ((0, 2), u5),
           ((1, 0), u3),
           ((1, 1), u4),
           ((1, 2), u6)]

      Nonstandard base field:

          sage: P = Poset({1: [3], 2: [3,4]})
          sage: lex = [1, 2, 4, 3]
          sage: l = P.birational_free_labelling(linear_extension=lex,
          ....:                                 prefix="aaa",
          ....:                                 base_field=Zmod(13))
          sage: l
          (Fraction Field of Multivariate Polynomial Ring in a, aaa1, aaa2, aaa3, aaa4, b over Ring of integers modulo 13,
           {...},
           a,
           b)
          sage: l[1][4]
          aaa3

      The empty poset:

          sage: P = Poset({})
          sage: P.birational_free_labelling(reduced=False, addvars="spam, eggs")
          (Fraction Field of Multivariate Polynomial Ring in a, b, spam, eggs over Rational Field,
           {},
           a,
           b)
          sage: P.birational_free_labelling(reduced=True, addvars="spam, eggs")
          (Fraction Field of Multivariate Polynomial Ring in spam, eggs over Rational Field,
           {},
           1,
           1)
          sage: P.birational_free_labelling(reduced=True)
          (Multivariate Polynomial Ring in no variables over Rational Field,
           {},
           1,
           1)
          sage: P.birational_free_labelling(prefix="zzz")
          (Fraction Field of Multivariate Polynomial Ring in a, b over Rational Field,
           {},
           a,
           b)

    `birational_rowmotion`(labelling)

      Return the result of applying birational rowmotion to the \(\mathbf{K}\)-labelling `labelling` of the poset `self`.

      See the documentation of `birational_free_labelling()` for a definition of birational rowmotion and \(\mathbf{K}\)-labellings and for an explanation of how \(\mathbf{K}\)-labellings are to be encoded to be understood by Sage. This implementation allows \(\mathbf{K}\) to be a semifield, not just a field. Birational rowmotion is only a rational map, so an exception (most likely, `ZeroDivisionError`) will be thrown if the denominator is zero.

      INPUT:

      - `labelling` – a \(\mathbf{K}\)-labelling of `self` in the sense as defined in the documentation of `birational_free_labelling()`

      OUTPUT:

      The image of the \(\mathbf{K}\)-labelling \(f\) under birational rowmotion.

      EXAMPLES:

          sage: P = Poset({1: [2, 3], 2: [4], 3: [4]})
          sage: lex = [1, 2, 3, 4]
          sage: t = P.birational_free_labelling(linear_extension=lex); t
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, x4, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(t[1].items())
          [(1, x1), (2, x2), (3, x3), (4, x4)]
          sage: t = P.birational_rowmotion(t); t
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, x4, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(t[1].items())
          [(1, a*b/x4), (2, (x1*x2*b + x1*x3*b)/(x2*x4)),
           (3, (x1*x2*b + x1*x3*b)/(x3*x4)), (4, (x2*b + x3*b)/x4)]

      A result of [GR13] states that applying birational rowmotion \(n+m\) times to a \(\mathbf{K}\)-labelling \(f\) of the poset \([n] \times [m]\) gives back \(f\). Let us check this:

          sage: def test_rectangle_periodicity(n, m, k):
          ....:     P = Posets.ChainPoset(n).product(Posets.ChainPoset(m))
          ....:     t0 = P.birational_free_labelling(P)
          ....:     t = t0
          ....:     for i in range(k):
          ....:         t = P.birational_rowmotion(t)
          ....:     return t == t0
          sage: test_rectangle_periodicity(2, 2, 4)
          True
          sage: test_rectangle_periodicity(2, 2, 2)
          False
          sage: test_rectangle_periodicity(2, 3, 5)  # long time
          True

      While computations with the birational free labelling quickly run out of memory due to the complexity of the rational functions involved, it is computationally cheap to check properties of birational rowmotion on examples in the tropical semiring:

          sage: def test_rectangle_periodicity_tropical(n, m, k):
          ....:     P = Posets.ChainPoset(n).product(Posets.ChainPoset(m))
          ....:     TT = TropicalSemiring(ZZ)
          ....:     t0 = (TT, {v: TT(floor(random()*100)) for v in P}, TT(0), TT(124))
          ....:     t = t0
          ....:     for i in range(k):
          ....:         t = P.birational_rowmotion(t)
          ....:     return t == t0
          sage: test_rectangle_periodicity_tropical(7, 6, 13)
          True

      Tropicalization is also what relates birational rowmotion to classical rowmotion on order ideals. In fact, if \(T\) denotes the `tropical semiring` of \(\ZZ\) and \(P\) is a finite poset, then we can define an embedding \(\phi\) from the set \(J(P)\) of all order ideals of \(P\) into the set \(T^{\widehat{P}}\) of all \(T\)-labellings of \(P\) by sending every \(I \in J(P)\) to the indicator function of \(I\) extended by the value \(1\) at the element \(0\) and the value \(0\) at the element \(1\). This map \(\phi\) has the property that \(R \circ \phi = \phi \circ r\), where \(R\) denotes birational rowmotion, and \(r\) denotes `classical rowmotion` on \(J(P)\). An example:

          sage: P = Posets.IntegerPartitions(5)
          sage: TT = TropicalSemiring(ZZ)
          sage: def indicator_labelling(I):
          ....:     # send order ideal `I` to a `T`-labelling of `P`.
          ....:     dct = {v: TT(v in I) for v in P}
          ....:     return (TT, dct, TT(1), TT(0))
          sage: all(indicator_labelling(P.rowmotion(I))
          ....:     == P.birational_rowmotion(indicator_labelling(I))
          ....:     for I in P.order_ideals_lattice(facade=True))
          True

      TESTS:

      Facade set to false works:

          sage: P = Poset({1: [2, 3], 2: [4], 3: [4]}, facade=False)
          sage: lex = [1, 2, 3, 4]
          sage: t = P.birational_free_labelling(linear_extension=lex); t
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, x4, b over Rational Field,
           {...},
           a,
           b)
          sage: t = P.birational_rowmotion(t); t
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, x4, b over Rational Field,
           {...},
           a,
           b)
          sage: t[1][P(2)]
          (x1*x2*b + x1*x3*b)/(x2*x4)
          sage: t = P.birational_rowmotion(t)
          sage: t[1][P(2)]
          a*b/x3

    `birational_toggle`(v, labelling)

      Return the result of applying the birational \(v\)-toggle \(T_v\) to the \(\mathbf{K}\)-labelling `labelling` of the poset `self`.

      See the documentation of `birational_free_labelling()` for a definition of this toggle and of \(\mathbf{K}\)-labellings as well as an explanation of how \(\mathbf{K}\)-labellings are to be encoded to be understood by Sage. This implementation allows \(\mathbf{K}\) to be a semifield, not just a field. The birational \(v\)-toggle is only a rational map, so an exception (most likely, `ZeroDivisionError`) will be thrown if the denominator is zero.

      INPUT:

      - `v` – an element of `self` (must have `self` as parent if `self` is a `facade=False` poset)
      - `labelling` – a \(\mathbf{K}\)-labelling of `self` in the sense as defined in the documentation of `birational_free_labelling()`

      OUTPUT:

      The \(\mathbf{K}\)-labelling \(T_v f\) of `self`, where \(f\) is `labelling`.

      EXAMPLES:

      Let us start with the birational free labelling of the “V”-poset (the three-element poset with Hasse diagram looking like a “V”):

          sage: V = Poset({1: [2, 3]})
          sage: s = V.birational_free_labelling(); s
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(s[1].items())
          [(1, x1), (2, x2), (3, x3)]

      The image of \(s\) under the \(1\)-toggle \(T_1\) is:

          sage: s1 = V.birational_toggle(1, s); s1
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(s1[1].items())
          [(1, a*x2*x3/(x1*x2 + x1*x3)), (2, x2), (3, x3)]

      Now let us apply the \(2\)-toggle \(T_2\) (to the old `s`):

          sage: s2 = V.birational_toggle(2, s); s2
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(s2[1].items())
          [(1, x1), (2, x1*b/x2), (3, x3)]

      On the other hand, we can also apply \(T_2\) to the image of \(s\) under \(T_1\):

          sage: s12 = V.birational_toggle(2, s1); s12
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, b over Rational Field,
           {...},
           a,
           b)
          sage: sorted(s12[1].items())
          [(1, a*x2*x3/(x1*x2 + x1*x3)), (2, a*x3*b/(x1*x2 + x1*x3)), (3, x3)]

      Each toggle is an involution:

          sage: all( V.birational_toggle(i, V.birational_toggle(i, s)) == s
          ....:      for i in V )
          True

      We can also start with a less generic labelling:

          sage: t = (QQ, {1: 3, 2: 6, 3: 7}, 2, 10)
          sage: t1 = V.birational_toggle(1, t); t1
          (Rational Field, {...}, 2, 10)
          sage: sorted(t1[1].items())
          [(1, 28/13), (2, 6), (3, 7)]
          sage: t13 = V.birational_toggle(3, t1); t13
          (Rational Field, {...}, 2, 10)
          sage: sorted(t13[1].items())
          [(1, 28/13), (2, 6), (3, 40/13)]

      However, labellings have to be sufficiently generic, lest denominators vanish:

          sage: t = (QQ, {1: 3, 2: 5, 3: -5}, 1, 15)
          sage: t1 = V.birational_toggle(1, t)
          Traceback (most recent call last):
          ...
          ZeroDivisionError: Rational division by zero

      We don’t get into zero-division issues in the tropical semiring (unless the zero of the tropical semiring appears in the labelling):

          sage: TT = TropicalSemiring(QQ)
          sage: t = (TT, {1: TT(2), 2: TT(4), 3: TT(1)}, TT(6), TT(0))
          sage: t1 = V.birational_toggle(1, t); t1
          (Tropical semiring over Rational Field, {...}, 6, 0)
          sage: sorted(t1[1].items())
          [(1, 8), (2, 4), (3, 1)]
          sage: t12 = V.birational_toggle(2, t1); t12
          (Tropical semiring over Rational Field, {...}, 6, 0)
          sage: sorted(t12[1].items())
          [(1, 8), (2, 4), (3, 1)]
          sage: t123 = V.birational_toggle(3, t12); t123
          (Tropical semiring over Rational Field, {...}, 6, 0)
          sage: sorted(t123[1].items())
          [(1, 8), (2, 4), (3, 7)]

      We turn to more interesting posets. Here is the \(6\)-element poset arising from the weak order on \(S_3\):

          sage: P = Posets.SymmetricGroupWeakOrderPoset(3)
          sage: sorted(list(P))
          ['123', '132', '213', '231', '312', '321']
          sage: t = (TT, {'123': TT(4), '132': TT(2), '213': TT(3), '231': TT(1), '321': TT(1), '312': TT(2)}, TT(7), TT(1))
          sage: t1 = P.birational_toggle('123', t); t1
          (Tropical semiring over Rational Field, {...}, 7, 1)
          sage: sorted(t1[1].items())
          [('123', 6), ('132', 2), ('213', 3), ('231', 1), ('312', 2), ('321', 1)]
          sage: t13 = P.birational_toggle('213', t1); t13
          (Tropical semiring over Rational Field, {...}, 7, 1)
          sage: sorted(t13[1].items())
          [('123', 6), ('132', 2), ('213', 4), ('231', 1), ('312', 2), ('321', 1)]

      Let us verify on this example some basic properties of toggles. First of all, again let us check that \(T_v\) is an involution for every \(v\):

          sage: all( P.birational_toggle(v, P.birational_toggle(v, t)) == t
          ....:      for v in P )
          True

      Furthermore, two toggles \(T_v\) and \(T_w\) commute unless one of \(v\) or \(w\) covers the other:

          sage: all( P.covers(v, w) or P.covers(w, v)
          ....:      or P.birational_toggle(v, P.birational_toggle(w, t))
          ....:         == P.birational_toggle(w, P.birational_toggle(v, t))
          ....:      for v in P for w in P )
          True

      TESTS:

      Setting `facade` to `False` does not break `birational_toggle`:

          sage: P = Poset({'x': ['y', 'w'], 'y': ['z'], 'w': ['z']}, facade=False)
          sage: lex = ['x', 'y', 'w', 'z']
          sage: t = P.birational_free_labelling(linear_extension=lex)
          sage: all( P.birational_toggle(v, P.birational_toggle(v, t)) == t
          ....:      for v in P )
          True
          sage: t4 = P.birational_toggle(P('z'), t); t4
          (Fraction Field of Multivariate Polynomial Ring in a, x1, x2, x3, x4, b over Rational Field,
           {...},
           a,
           b)
          sage: t4[1][P('x')]
          x1
          sage: t4[1][P('y')]
          x2
          sage: t4[1][P('w')]
          x3
          sage: t4[1][P('z')]
          (x2*b + x3*b)/x4

      The one-element poset:

          sage: P = Poset({8: []})
          sage: t = P.birational_free_labelling()
          sage: t8 = P.birational_toggle(8, t); t8
          (Fraction Field of Multivariate Polynomial Ring in a, x1, b over Rational Field,
           {...},
           a,
           b)
          sage: t8[1][8]
          a*b/x1

    `birational_toggles`(vs, labelling)

      Return the result of applying a sequence of birational toggles (specified by `vs`) to the \(\mathbf{K}\)-labelling `labelling` of the poset `self`.

      See the documentation of `birational_free_labelling()` for a definition of birational toggles and \(\mathbf{K}\)-labellings and for an explanation of how \(\mathbf{K}\)-labellings are to be encoded to be understood by Sage. This implementation allows \(\mathbf{K}\) to be a semifield, not just a field. The birational \(v\)-toggle is only a rational map, so an exception (most likely, `ZeroDivisionError`) will be thrown if the denominator is zero.

      INPUT:

      - `vs` – an iterable comprising elements of `self` (which must have `self` as parent if `self` is a `facade=False` poset)
      - `labelling` – a \(\mathbf{K}\)-labelling of `self` in the sense as defined in the documentation of `birational_free_labelling()`

      OUTPUT:

      The \(\mathbf{K}\)-labelling \(T_{v_n} T_{v_{n-1}} \cdots T_{v_1} f\) of `self`, where \(f\) is `labelling` and \((v_1, v_2, \ldots, v_n)\) is `vs` (written as list).

      EXAMPLES:

          sage: P = Posets.SymmetricGroupBruhatOrderPoset(3)
          sage: sorted(list(P))
          ['123', '132', '213', '231', '312', '321']
          sage: TT = TropicalSemiring(ZZ)
          sage: t = (TT, {'123': TT(4), '132': TT(2), '213': TT(3), '231': TT(1), '321': TT(1), '312': TT(2)}, TT(7), TT(1))
          sage: tA = P.birational_toggles(['123', '231', '312'], t); tA
          (Tropical semiring over Integer Ring, {...}, 7, 1)
          sage: sorted(tA[1].items())
          [('123', 6), ('132', 2), ('213', 3), ('231', 2), ('312', 1), ('321', 1)]
          sage: tAB = P.birational_toggles(['132', '213', '321'], tA); tAB
          (Tropical semiring over Integer Ring, {...}, 7, 1)
          sage: sorted(tAB[1].items())
          [('123', 6), ('132', 6), ('213', 5), ('231', 2), ('312', 1), ('321', 1)]

          sage: P = Poset({1: [2, 3], 2: [4], 3: [4]})
          sage: Qx = PolynomialRing(QQ, 'x').fraction_field()
          sage: x = Qx.gen()
          sage: t = (Qx, {1: 1, 2: x, 3: (x+1)/x, 4: x^2}, 1, 1)
          sage: t1 = P.birational_toggles((i for i in range(1, 5)), t); t1
          (Fraction Field of Univariate Polynomial Ring in x over Rational Field,
           {...},
           1,
           1)
          sage: sorted(t1[1].items())
          [(1, (x^2 + x)/(x^2 + x + 1)), (2, (x^3 + x^2)/(x^2 + x + 1)), (3, x^4/(x^2 + x + 1)), (4, 1)]
          sage: t2 = P.birational_toggles(reversed(range(1, 5)), t)
          sage: sorted(t2[1].items())
          [(1, 1/x^2), (2, (x^2 + x + 1)/x^4), (3, (x^2 + x + 1)/(x^3 + x^2)), (4, (x^2 + x + 1)/x^3)]

      Facade set to `False` works:

          sage: P = Poset({'x': ['y', 'w'], 'y': ['z'], 'w': ['z']}, facade=False)
          sage: lex = ['x', 'y', 'w', 'z']
          sage: t = P.birational_free_labelling(linear_extension=lex)
          sage: sorted(P.birational_toggles([P('x'), P('y')], t)[1].items())
          [(x, a*x2*x3/(x1*x2 + x1*x3)), (y, a*x3*x4/(x1*x2 + x1*x3)), (w, x3), (z, x4)]

    `directed_subsets`(direction)

      Return the order filters (resp. order ideals) of `self`, as lists.

      If `direction` is ‘up’, returns the order filters (upper sets).

      If `direction` is ‘down’, returns the order ideals (lower sets).

      INPUT:

      - `direction` – ‘up’ or ‘down’

      EXAMPLES:

          sage: P = Poset((divisors(12), attrcall("divides")), facade=True)
          sage: A = P.directed_subsets('up')
          sage: sorted(list(A))
          [[], [1, 2, 4, 3, 6, 12], [2, 4, 3, 6, 12], [2, 4, 6, 12], [3, 6, 12], [4, 3, 6, 12], [4, 6, 12], [4, 12], [6, 12], [12]]

      TESTS:

          sage: list(Poset().directed_subsets('up'))
          [[]]

    `is_lattice`()

      Returns whether this poset is both a meet and a join semilattice.

      EXAMPLES:

          sage: P = Poset([[1,3,2],[4],[4,5,6],[6],[7],[7],[7],[]])
          sage: P.is_lattice()
          True

          sage: P = Poset([[1,2],[3],[3],[]])
          sage: P.is_lattice()
          True

          sage: P = Poset({0:[2,3],1:[2,3]})
          sage: P.is_lattice()
          False

    `is_poset_isomorphism`(f, codomain)

      Return whether \(f\) is an isomorphism of posets from `self` to `codomain`.

      INPUT:

      - `f` – a function from `self` to `codomain`
      - `codomain` – a poset

      EXAMPLES:

      We build the poset \(D\) of divisors of 30, and check that it is isomorphic to the boolean lattice \(B\) of the subsets of \(\{2,3,5\}\) ordered by inclusion, via the reverse function \(f: B \to D, b \mapsto \prod_{x\in b} x\):

          sage: D = Poset((divisors(30), attrcall("divides")))
          sage: B = Poset(([frozenset(s) for s in Subsets([2,3,5])], attrcall("issubset")))
          sage: def f(b): return D(prod(b))
          sage: B.is_poset_isomorphism(f, D)
          True

      On the other hand, \(f\) is not an isomorphism to the chain of divisors of 30, ordered by usual comparison:

          sage: P = Poset((divisors(30), operator.le))
          sage: def f(b): return P(prod(b))
          sage: B.is_poset_isomorphism(f, P)
          False

      A non surjective case:

          sage: B = Poset(([frozenset(s) for s in Subsets([2,3])], attrcall("issubset")))
          sage: def f(b): return D(prod(b))
          sage: B.is_poset_isomorphism(f, D)
          False

      A non injective case:

          sage: B = Poset(([frozenset(s) for s in Subsets([2,3,5,6])], attrcall("issubset")))
          sage: def f(b): return D(gcd(prod(b), 30))
          sage: B.is_poset_isomorphism(f, D)
          False

      Note

      since `D` and `B` are not facade posets, `f` is responsible for the conversions between integers and subsets to elements of `D` and `B` and back.

      See also

      `FiniteLatticePosets.ParentMethods.is_lattice_morphism()`

    `is_poset_morphism`(f, codomain)

      Return whether \(f\) is a morphism of posets from `self` to `codomain`, that is

      \[x\leq y \Longrightarrow f(x) \leq f(y)\]

      for all \(x\) and \(y\) in `self`.

      INPUT:

      - `f` – a function from `self` to `codomain`
      - `codomain` – a poset

      EXAMPLES:

      We build the boolean lattice of the subsets of \(\{2,3,5,6\}\) and the lattice of divisors of \(30\), and check that the map \(b \mapsto \gcd(\prod_{x\in b} x, 30)\) is a morphism of posets:

          sage: D = Poset((divisors(30), attrcall("divides")))
          sage: B = Poset(([frozenset(s) for s in Subsets([2,3,5,6])], attrcall("issubset")))
          sage: def f(b): return D(gcd(prod(b), 30))
          sage: B.is_poset_morphism(f, D)
          True

      Note

      since `D` and `B` are not facade posets, `f` is responsible for the conversions between integers and subsets to elements of `D` and `B` and back.

      \(f\) is also a morphism of posets to the chain of divisors of 30, ordered by usual comparison:

          sage: P = Poset((divisors(30), operator.le))
          sage: def f(b): return P(gcd(prod(b), 30))
          sage: B.is_poset_morphism(f, P)
          True

      FIXME: should this be `is_order_preserving_morphism`?

      See also

      `is_poset_isomorphism()`

      TESTS:

      Base cases:

          sage: P = Posets.ChainPoset(2)
          sage: Q = Posets.AntichainPoset(2)
          sage: f = lambda x: 1-x
          sage: P.is_poset_morphism(f, P)
          False
          sage: P.is_poset_morphism(f, Q)
          False
          sage: Q.is_poset_morphism(f, Q)
          True
          sage: Q.is_poset_morphism(f, P)
          True

          sage: P = Poset(); P
          Finite poset containing 0 elements
          sage: P.is_poset_morphism(f, P)
          True

    `is_selfdual`()

      Returns whether this poset is self-dual, that is isomorphic to its dual poset.

      EXAMPLES:

          sage: P = Poset(([1,2,3],[[1,3],[2,3]]),cover_relations=True)
          sage: P.is_selfdual()
          False

          sage: P = Poset(([1,2,3,4],[[1,3],[1,4],[2,3],[2,4]]),cover_relations=True)
          sage: P.is_selfdual()
          True

          sage: P = Poset( {} )
          sage: P.is_selfdual()
          True

    `order_filter_generators`(filter)

      Generators for an order filter

      INPUT:

      - `filter` – an order filter of `self`, as a list (or iterable)

      EXAMPLES:

          sage: P = Poset((Subsets([1,2,3]), attrcall("issubset")))
          sage: I = P.order_filter([Set([1,2]), Set([2,3]), Set([1])]); I
          [{2, 3}, {1}, {1, 2}, {1, 3}, {1, 2, 3}]
          sage: P.order_filter_generators(I)
          {{2, 3}, {1}}

      See also

      `order_ideal_generators()`

    `order_ideal_complement_generators`(antichain, direction='up')

      Return the Panyushev complement of the antichain `antichain`.

      Given an antichain \(A\) of a poset \(P\), the Panyushev complement of \(A\) is defined to be the antichain consisting of the minimal elements of the order filter \(B\), where \(B\) is the (set-theoretic) complement of the order ideal of \(P\) generated by \(A\).

      Setting the optional keyword variable `direction` to `'down'` leads to the inverse Panyushev complement being computed instead of the Panyushev complement. The inverse Panyushev complement of an antichain \(A\) is the antichain whose Panyushev complement is \(A\). It can be found as the antichain consisting of the maximal elements of the order ideal \(C\), where \(C\) is the (set-theoretic) complement of the order filter of \(P\) generated by \(A\).

      `panyushev_complement()` is an alias for this method.

      Panyushev complementation is related (actually, isomorphic) to rowmotion (`rowmotion()`).

      INPUT:

      - `antichain` – an antichain of `self`, as a list (or iterable), or, more generally, generators of an order ideal (resp. order filter)
      - `direction` – ‘up’ or ‘down’ (default: ‘up’)

      OUTPUT:

      - the generating antichain of the complement order filter (resp. order ideal) of the order ideal (resp. order filter) generated by the antichain `antichain`

      EXAMPLES:

          sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
          sage: P.order_ideal_complement_generators([1])
          {2}
          sage: P.order_ideal_complement_generators([3])
          set()
          sage: P.order_ideal_complement_generators([1,2])
          {3}
          sage: P.order_ideal_complement_generators([1,2,3])
          set()

          sage: P.order_ideal_complement_generators([1], direction="down")
          {2}
          sage: P.order_ideal_complement_generators([3], direction="down")
          {1, 2}
          sage: P.order_ideal_complement_generators([1,2], direction="down")
          set()
          sage: P.order_ideal_complement_generators([1,2,3], direction="down")
          set()

      Warning

      This is a brute force implementation, building the order ideal generated by the antichain, and searching for order filter generators of its complement

    `order_ideal_generators`(ideal, direction='down')

      Return the antichain of (minimal) generators of the order ideal (resp. order filter) `ideal`.

      INPUT:

      - `ideal` – an order ideal \(I\) (resp. order filter) of `self`, as a list (or iterable); this should be an order ideal if `direction` is set to `'down'`, and an order filter if `direction` is set to `'up'`.
      - `direction` – `'up'` or `'down'` (default: `'down'`).

      The antichain of (minimal) generators of an order ideal \(I\) in a poset \(P\) is the set of all minimal elements of \(P\). In the case of an order filter, the definition is similar, but with “maximal” used instead of “minimal”.

      EXAMPLES:

      We build the boolean lattice of all subsets of \(\{1,2,3\}\) ordered by inclusion, and compute an order ideal there:

          sage: P = Poset((Subsets([1,2,3]), attrcall("issubset")))
          sage: I = P.order_ideal([Set([1,2]), Set([2,3]), Set([1])]); I
          [{}, {3}, {2}, {2, 3}, {1}, {1, 2}]

      Then, we retrieve the generators of this ideal:

          sage: P.order_ideal_generators(I)
          {{1, 2}, {2, 3}}

      If `direction` is ‘up’, then this instead computes the minimal generators for an order filter:

          sage: I = P.order_filter([Set([1,2]), Set([2,3]), Set([1])]); I
          [{2, 3}, {1}, {1, 2}, {1, 3}, {1, 2, 3}]
          sage: P.order_ideal_generators(I, direction='up')
          {{2, 3}, {1}}

      Complexity: \(O(n+m)\) where \(n\) is the cardinality of \(I\), and \(m\) the number of upper covers of elements of \(I\).

    `order_ideals_lattice`(as_ideals=True, facade=False)

      Return the lattice of order ideals of a poset `self`, ordered by inclusion.

      The lattice of order ideals of a poset \(P\) is usually denoted by \(J(P)\). Its underlying set is the set of order ideals of \(P\), and its partial order is given by inclusion.

      The order ideals of \(P\) are in a canonical bijection with the antichains of \(P\). The bijection maps every order ideal to the antichain formed by its maximal elements. By setting the `as_ideals` keyword variable to `False`, one can make this method apply this bijection before returning the lattice.

      INPUT:

      - `as_ideals` – Boolean, if `True` (default) returns a poset on the set of order ideals, otherwise on the set of antichains

      EXAMPLES:

          sage: P = Posets.PentagonPoset(facade = True)
          sage: P.cover_relations()
          [[0, 1], [0, 2], [1, 4], [2, 3], [3, 4]]
          sage: J = P.order_ideals_lattice(); J
          Finite lattice containing 8 elements
          sage: list(J)
          [{}, {0}, {0, 2}, {0, 2, 3}, {0, 1}, {0, 1, 2}, {0, 1, 2, 3}, {0, 1, 2, 3, 4}]

      As a lattice on antichains:

          sage: J2 = P.order_ideals_lattice(False); J2
          Finite lattice containing 8 elements
          sage: list(J2)
          [(0,), (1, 2), (1, 3), (1,), (2,), (3,), (4,), ()]

      TESTS:

          sage: J = Posets.DiamondPoset(4, facade = True).order_ideals_lattice(); J
          Finite lattice containing 6 elements
          sage: list(J)
          [{}, {0}, {0, 2}, {0, 1}, {0, 1, 2}, {0, 1, 2, 3}]
          sage: J.cover_relations()
          [[{}, {0}], [{0}, {0, 2}], [{0}, {0, 1}], [{0, 2}, {0, 1, 2}], [{0, 1}, {0, 1, 2}], [{0, 1, 2}, {0, 1, 2, 3}]]

      Note

      we use facade posets in the examples above just to ensure a nicer ordering in the output.

    `panyushev_complement`(antichain, direction='up')

      Return the Panyushev complement of the antichain `antichain`.

      Given an antichain \(A\) of a poset \(P\), the Panyushev complement of \(A\) is defined to be the antichain consisting of the minimal elements of the order filter \(B\), where \(B\) is the (set-theoretic) complement of the order ideal of \(P\) generated by \(A\).

      Setting the optional keyword variable `direction` to `'down'` leads to the inverse Panyushev complement being computed instead of the Panyushev complement. The inverse Panyushev complement of an antichain \(A\) is the antichain whose Panyushev complement is \(A\). It can be found as the antichain consisting of the maximal elements of the order ideal \(C\), where \(C\) is the (set-theoretic) complement of the order filter of \(P\) generated by \(A\).

      `panyushev_complement()` is an alias for this method.

      Panyushev complementation is related (actually, isomorphic) to rowmotion (`rowmotion()`).

      INPUT:

      - `antichain` – an antichain of `self`, as a list (or iterable), or, more generally, generators of an order ideal (resp. order filter)
      - `direction` – ‘up’ or ‘down’ (default: ‘up’)

      OUTPUT:

      - the generating antichain of the complement order filter (resp. order ideal) of the order ideal (resp. order filter) generated by the antichain `antichain`

      EXAMPLES:

          sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
          sage: P.order_ideal_complement_generators([1])
          {2}
          sage: P.order_ideal_complement_generators([3])
          set()
          sage: P.order_ideal_complement_generators([1,2])
          {3}
          sage: P.order_ideal_complement_generators([1,2,3])
          set()

          sage: P.order_ideal_complement_generators([1], direction="down")
          {2}
          sage: P.order_ideal_complement_generators([3], direction="down")
          {1, 2}
          sage: P.order_ideal_complement_generators([1,2], direction="down")
          set()
          sage: P.order_ideal_complement_generators([1,2,3], direction="down")
          set()

      Warning

      This is a brute force implementation, building the order ideal generated by the antichain, and searching for order filter generators of its complement

    `panyushev_orbit_iter`(antichain, element_constructor=<type 'set'>, stop=True, check=True)

      Iterate over the Panyushev orbit of an antichain `antichain` of `self`.

      The Panyushev orbit of an antichain is its orbit under Panyushev complementation (see `panyushev_complement()`).

      INPUT:

      - `antichain` – an antichain of `self`, given as an iterable.
      - `element_constructor` (defaults to `set`) – a type constructor (`set`, `tuple`, `list`, `frozenset`, `iter`, etc.) which is to be applied to the antichains before they are yielded.
      - `stop` – a Boolean (default: `True`) determining whether the iterator should stop once it completes its cycle (this happens when it is set to `True`) or go on forever (this happens when it is set to `False`).
      - `check` – a Boolean (default: `True`) determining whether `antichain` should be checked for being an antichain.

      OUTPUT:

      - an iterator over the orbit of the antichain `antichain` under Panyushev complementation. This iterator \(I\) has the property that `I[0] == antichain` and each \(i\) satisfies `self.order_ideal_complement_generators(I[i]) == I[i+1]`, where `I[i+1]` has to be understood as `I[0]` if it is undefined. The entries `I[i]` are sets by default, but depending on the optional keyword variable `element_constructors` they can also be tuples, lists etc.

      EXAMPLES:

          sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
          sage: list(P.panyushev_orbit_iter(set([1, 2])))
          [{1, 2}, {3}, set()]
          sage: list(P.panyushev_orbit_iter([1, 2]))
          [{1, 2}, {3}, set()]
          sage: list(P.panyushev_orbit_iter([2, 1]))
          [{1, 2}, {3}, set()]
          sage: list(P.panyushev_orbit_iter(set([1, 2]), element_constructor=list))
          [[1, 2], [3], []]
          sage: list(P.panyushev_orbit_iter(set([1, 2]), element_constructor=frozenset))
          [frozenset({1, 2}), frozenset({3}), frozenset()]
          sage: list(P.panyushev_orbit_iter(set([1, 2]), element_constructor=tuple))
          [(1, 2), (3,), ()]

          sage: P = Poset( {} )
          sage: list(P.panyushev_orbit_iter([]))
          [set()]

          sage: P = Poset({ 1: [2, 3], 2: [4], 3: [4], 4: [] })
          sage: Piter = P.panyushev_orbit_iter([2], stop=False)
          sage: Piter.next()
          {2}
          sage: Piter.next()
          {3}
          sage: Piter.next()
          {2}
          sage: Piter.next()
          {3}

    `panyushev_orbits`(element_constructor=<type 'set'>)

      Return the Panyushev orbits of antichains in `self`.

      The Panyushev orbit of an antichain is its orbit under Panyushev complementation (see `panyushev_complement()`).

      INPUT:

      - `element_constructor` (defaults to `set`) – a type constructor (`set`, `tuple`, `list`, `frozenset`, `iter`, etc.) which is to be applied to the antichains before they are returned.

      OUTPUT:

      - the partition of the set of all antichains of `self` into orbits under Panyushev complementation. This is returned as a list of lists `L` such that for each `L` and `i`, cyclically: `self.order_ideal_complement_generators(L[i]) == L[i+1]`. The entries `L[i]` are sets by default, but depending on the optional keyword variable `element_constructors` they can also be tuples, lists etc.

      EXAMPLES:

          sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
          sage: P.panyushev_orbits()
          [[{2}, {1}], [set(), {1, 2}, {3}]]
          sage: P.panyushev_orbits(element_constructor=list)
          [[[2], [1]], [[], [1, 2], [3]]]
          sage: P.panyushev_orbits(element_constructor=frozenset)
          [[frozenset({2}), frozenset({1})],
           [frozenset(), frozenset({1, 2}), frozenset({3})]]
          sage: P.panyushev_orbits(element_constructor=tuple)
          [[(2,), (1,)], [(), (1, 2), (3,)]]
          sage: P = Poset( {} )
          sage: P.panyushev_orbits()
          [[set()]]

    `rowmotion`(order_ideal)

      The image of the order ideal `order_ideal` under rowmotion in `self`.

      Rowmotion on a finite poset \(P\) is an automorphism of the set \(J(P)\) of all order ideals of \(P\). One way to define it is as follows: Given an order ideal \(I \in J(P)\), we let \(F\) be the set-theoretic complement of \(I\) in \(P\). Furthermore we let \(A\) be the antichain consisting of all minimal elements of \(F\). Then, the rowmotion of \(I\) is defined to be the order ideal of \(P\) generated by the antichain \(A\) (that is, the order ideal consisting of each element of \(P\) which has some element of \(A\) above it).

      Rowmotion is related (actually, isomorphic) to Panyushev complementation (`panyushev_complement()`).

      INPUT:

      - `order_ideal` – an order ideal of `self`, as a set

      OUTPUT:

      - the image of `order_ideal` under rowmotion, as a set again

      EXAMPLES:

          sage: P = Poset( {1: [2, 3], 2: [], 3: [], 4: [8], 5: [], 6: [5], 7: [1, 4], 8: []} )
          sage: I = Set({2, 6, 1, 7})
          sage: P.rowmotion(I)
          {1, 3, 4, 5, 6, 7}

          sage: P = Poset( {} )
          sage: I = Set({})
          sage: P.rowmotion(I)
          Set of elements of {}

    `rowmotion_orbit_iter`(oideal, element_constructor=<type 'set'>, stop=True, check=True)

      Iterate over the rowmotion orbit of an order ideal `oideal` of `self`.

      The rowmotion orbit of an order ideal is its orbit under rowmotion (see `rowmotion()`).

      INPUT:

      - `oideal` – an order ideal of `self`, given as an iterable.
      - `element_constructor` (defaults to `set`) – a type constructor (`set`, `tuple`, `list`, `frozenset`, `iter`, etc.) which is to be applied to the order ideals before they are yielded.
      - `stop` – a Boolean (default: `True`) determining whether the iterator should stop once it completes its cycle (this happens when it is set to `True`) or go on forever (this happens when it is set to `False`).
      - `check` – a Boolean (default: `True`) determining whether `oideal` should be checked for being an order ideal.

      OUTPUT:

      - an iterator over the orbit of the order ideal `oideal` under rowmotion. This iterator \(I\) has the property that `I[0] == oideal` and that every \(i\) satisfies `self.rowmotion(I[i]) == I[i+1]`, where `I[i+1]` has to be understood as `I[0]` if it is undefined. The entries `I[i]` are sets by default, but depending on the optional keyword variable `element_constructors` they can also be tuples, lists etc.

      EXAMPLES:

          sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
          sage: list(P.rowmotion_orbit_iter(set([1, 2])))
          [{1, 2}, {1, 2, 3}, set()]
          sage: list(P.rowmotion_orbit_iter([1, 2]))
          [{1, 2}, {1, 2, 3}, set()]
          sage: list(P.rowmotion_orbit_iter([2, 1]))
          [{1, 2}, {1, 2, 3}, set()]
          sage: list(P.rowmotion_orbit_iter(set([1, 2]), element_constructor=list))
          [[1, 2], [1, 2, 3], []]
          sage: list(P.rowmotion_orbit_iter(set([1, 2]), element_constructor=frozenset))
          [frozenset({1, 2}), frozenset({1, 2, 3}), frozenset()]
          sage: list(P.rowmotion_orbit_iter(set([1, 2]), element_constructor=tuple))
          [(1, 2), (1, 2, 3), ()]

          sage: P = Poset( {} )
          sage: list(P.rowmotion_orbit_iter([]))
          [set()]

          sage: P = Poset({ 1: [2, 3], 2: [4], 3: [4], 4: [] })
          sage: Piter = P.rowmotion_orbit_iter([1, 2, 3], stop=False)
          sage: Piter.next()
          {1, 2, 3}
          sage: Piter.next()
          {1, 2, 3, 4}
          sage: Piter.next()
          set()
          sage: Piter.next()
          {1}
          sage: Piter.next()
          {1, 2, 3}

          sage: P = Poset({ 1: [4], 2: [4, 5], 3: [5] })
          sage: list(P.rowmotion_orbit_iter([1, 2], element_constructor=list))
          [[1, 2], [1, 2, 3, 4], [2, 3, 5], [1], [2, 3], [1, 2, 3, 5], [1, 2, 4], [3]]

    `rowmotion_orbits`(element_constructor=<type 'set'>)

      Return the rowmotion orbits of order ideals in `self`.

      The rowmotion orbit of an order ideal is its orbit under rowmotion (see `rowmotion()`).

      INPUT:

      - `element_constructor` (defaults to `set`) – a type constructor (`set`, `tuple`, `list`, `frozenset`, `iter`, etc.) which is to be applied to the antichains before they are returned.

      OUTPUT:

      - the partition of the set of all order ideals of `self` into orbits under rowmotion. This is returned as a list of lists `L` such that for each `L` and `i`, cyclically: `self.rowmotion(L[i]) == L[i+1]`. The entries `L[i]` are sets by default, but depending on the optional keyword variable `element_constructors` they can also be tuples, lists etc.

      EXAMPLES:

          sage: P = Poset( {1: [2, 3], 2: [], 3: [], 4: [2]} )
          sage: sorted(len(o) for o in P.rowmotion_orbits())
          [3, 5]
          sage: sorted(P.rowmotion_orbits(element_constructor=list))
          [[[1, 3], [4], [1], [4, 1, 3], [4, 1, 2]], [[4, 1], [4, 1, 2, 3], []]]
          sage: sorted(P.rowmotion_orbits(element_constructor=tuple))
          [[(1, 3), (4,), (1,), (4, 1, 3), (4, 1, 2)], [(4, 1), (4, 1, 2, 3), ()]]
          sage: P = Poset({})
          sage: sorted(P.rowmotion_orbits(element_constructor=tuple))
          [[()]]

    `toggling_orbit_iter`(vs, oideal, element_constructor=<type 'set'>, stop=True, check=True)

      Iterate over the orbit of an order ideal `oideal` of `self` under the operation of toggling the vertices `vs[0], vs[1], ...` in this order.

      See `order_ideal_toggle()` for a definition of toggling.

      Warning

      The orbit is that under the composition of toggles, not under the single toggles themselves. Thus, for example, if `vs == [1,2]`, then the orbit has the form \((I, T_2 T_1 I, T_2 T_1 T_2 T_1 I, \ldots)\) (where \(I\) denotes `oideal` and \(T_i\) means toggling at \(i\)) rather than \((I, T_1 I, T_2 T_1 I, T_1 T_2 T_1 I, \ldots)\).

      INPUT:

      - `vs`: a list (or other iterable) of elements of `self` (but since the output depends on the order, sets should not be used as `vs`).
      - `oideal` – an order ideal of `self`, given as an iterable.
      - `element_constructor` (defaults to `set`) – a type constructor (`set`, `tuple`, `list`, `frozenset`, `iter`, etc.) which is to be applied to the order ideals before they are yielded.
      - `stop` – a Boolean (default: `True`) determining whether the iterator should stop once it completes its cycle (this happens when it is set to `True`) or go on forever (this happens when it is set to `False`).
      - `check` – a Boolean (default: `True`) determining whether `oideal` should be checked for being an order ideal.

      OUTPUT:

      - an iterator over the orbit of the order ideal `oideal` under toggling the vertices in the list `vs` in this order. This iterator \(I\) has the property that `I[0] == oideal` and that every \(i\) satisfies `self.order_ideal_toggles(I[i], vs) == I[i+1]`, where `I[i+1]` has to be understood as `I[0]` if it is undefined. The entries `I[i]` are sets by default, but depending on the optional keyword variable `element_constructors` they can also be tuples, lists etc.

      EXAMPLES:

          sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
          sage: list(P.toggling_orbit_iter([1, 3, 1], set([1, 2])))
          [{1, 2}]
          sage: list(P.toggling_orbit_iter([1, 2, 3], set([1, 2])))
          [{1, 2}, set(), {1, 2, 3}]
          sage: list(P.toggling_orbit_iter([3, 2, 1], set([1, 2])))
          [{1, 2}, {1, 2, 3}, set()]
          sage: list(P.toggling_orbit_iter([3, 2, 1], set([1, 2]), element_constructor=list))
          [[1, 2], [1, 2, 3], []]
          sage: list(P.toggling_orbit_iter([3, 2, 1], set([1, 2]), element_constructor=frozenset))
          [frozenset({1, 2}), frozenset({1, 2, 3}), frozenset()]
          sage: list(P.toggling_orbit_iter([3, 2, 1], set([1, 2]), element_constructor=tuple))
          [(1, 2), (1, 2, 3), ()]
          sage: list(P.toggling_orbit_iter([3, 2, 1], [2, 1], element_constructor=tuple))
          [(1, 2), (1, 2, 3), ()]

          sage: P = Poset( {} )
          sage: list(P.toggling_orbit_iter([], []))
          [set()]

          sage: P = Poset({ 1: [2, 3], 2: [4], 3: [4], 4: [] })
          sage: Piter = P.toggling_orbit_iter([1, 2, 4, 3], [1, 2, 3], stop=False)
          sage: Piter.next()
          {1, 2, 3}
          sage: Piter.next()
          {1}
          sage: Piter.next()
          set()
          sage: Piter.next()
          {1, 2, 3}
          sage: Piter.next()
          {1}

    `toggling_orbits`(vs, element_constructor=<type 'set'>)

      Return the orbits of order ideals in `self` under the operation of toggling the vertices `vs[0], vs[1], ...` in this order.

      See `order_ideal_toggle()` for a definition of toggling.

      Warning

      The orbits are those under the composition of toggles, not under the single toggles themselves. Thus, for example, if `vs == [1,2]`, then the orbits have the form \((I, T_2 T_1 I, T_2 T_1 T_2 T_1 I, \ldots)\) (where \(I\) denotes an order ideal and \(T_i\) means toggling at \(i\)) rather than \((I, T_1 I, T_2 T_1 I, T_1 T_2 T_1 I, \ldots)\).

      INPUT:

      - `vs`: a list (or other iterable) of elements of `self` (but since the output depends on the order, sets should not be used as `vs`).

      OUTPUT:

      - a partition of the order ideals of `self`, as a list of sets `L` such that for each `L` and `i`, cyclically: `self.order_ideal_toggles(L[i], vs) == L[i+1]`.

      EXAMPLES:

          sage: P = Poset( {1: [2, 4], 2: [], 3: [4], 4: []} )
          sage: sorted(len(o) for o in P.toggling_orbits([1, 2]))
          [2, 3, 3]
          sage: P = Poset( {1: [3], 2: [1, 4], 3: [], 4: [3]} )
          sage: sorted(len(o) for o in P.toggling_orbits((1, 2, 4, 3)))
          [3, 3]

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