Here is some terminology used in this file:
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
TESTS:
sage: C = FinitePosets()
sage: C is Posets().Finite()
True
sage: TestSuite(C).run()
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]]
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
(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:
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)
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:
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
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:
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
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:
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)]
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:
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'))
[[]]
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
Return whether \(f\) is an isomorphism of posets from self to codomain.
INPUT:
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.
Return whether \(f\) is a morphism of posets from self to codomain, that is
for all \(x\) and \(y\) in self.
INPUT:
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
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
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
Generators for an order filter
INPUT:
EXAMPLES:
sage: P = Poset((Subsets([1,2,3]), attrcall("issubset")))
sage: I = P.order_filter([Set([1,2]), Set([2,3]), Set([1])]); I
[{1}, {1, 3}, {1, 2}, {2, 3}, {1, 2, 3}]
sage: P.order_filter_generators(I)
{{2, 3}, {1}}
See also
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:
OUTPUT:
EXAMPLES:
sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
sage: P.order_ideal_complement_generators([1])
set([2])
sage: P.order_ideal_complement_generators([3])
set([])
sage: P.order_ideal_complement_generators([1,2])
set([3])
sage: P.order_ideal_complement_generators([1,2,3])
set([])
sage: P.order_ideal_complement_generators([1], direction="down")
set([2])
sage: P.order_ideal_complement_generators([3], direction="down")
set([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
Return the antichain of (minimal) generators of the order ideal (resp. order filter) ideal.
INPUT:
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
[{}, {1}, {3}, {2}, {1, 2}, {2, 3}]
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
[{1}, {1, 3}, {1, 2}, {2, 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 \(I\).
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:
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, 1}, {0, 1, 2}, {0, 2, 3}, {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.
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:
OUTPUT:
EXAMPLES:
sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
sage: P.order_ideal_complement_generators([1])
set([2])
sage: P.order_ideal_complement_generators([3])
set([])
sage: P.order_ideal_complement_generators([1,2])
set([3])
sage: P.order_ideal_complement_generators([1,2,3])
set([])
sage: P.order_ideal_complement_generators([1], direction="down")
set([2])
sage: P.order_ideal_complement_generators([3], direction="down")
set([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
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:
OUTPUT:
EXAMPLES:
sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
sage: list(P.panyushev_orbit_iter(set([1, 2])))
[set([1, 2]), set([3]), set([])]
sage: list(P.panyushev_orbit_iter([1, 2]))
[set([1, 2]), set([3]), set([])]
sage: list(P.panyushev_orbit_iter([2, 1]))
[set([1, 2]), set([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()
set([2])
sage: Piter.next()
set([3])
sage: Piter.next()
set([2])
sage: Piter.next()
set([3])
Return the Panyushev orbits of antichains in self.
The Panyushev orbit of an antichain is its orbit under Panyushev complementation (see panyushev_complement()).
INPUT:
OUTPUT:
EXAMPLES:
sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
sage: P.panyushev_orbits()
[[set([2]), set([1])], [set([]), set([1, 2]), set([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([])]]
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:
OUTPUT:
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 {}
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:
OUTPUT:
EXAMPLES:
sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
sage: list(P.rowmotion_orbit_iter(set([1, 2])))
[set([1, 2]), set([1, 2, 3]), set([])]
sage: list(P.rowmotion_orbit_iter([1, 2]))
[set([1, 2]), set([1, 2, 3]), set([])]
sage: list(P.rowmotion_orbit_iter([2, 1]))
[set([1, 2]), set([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()
set([1, 2, 3])
sage: Piter.next()
set([1, 2, 3, 4])
sage: Piter.next()
set([])
sage: Piter.next()
set([1])
sage: Piter.next()
set([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]]
Return the rowmotion orbits of order ideals in self.
The rowmotion orbit of an order ideal is its orbit under rowmotion (see rowmotion()).
INPUT:
OUTPUT:
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))
[[()]]
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:
OUTPUT:
EXAMPLES:
sage: P = Poset( ( [1,2,3], [ [1,3], [2,3] ] ) )
sage: list(P.toggling_orbit_iter([1, 3, 1], set([1, 2])))
[set([1, 2])]
sage: list(P.toggling_orbit_iter([1, 2, 3], set([1, 2])))
[set([1, 2]), set([]), set([1, 2, 3])]
sage: list(P.toggling_orbit_iter([3, 2, 1], set([1, 2])))
[set([1, 2]), set([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()
set([1, 2, 3])
sage: Piter.next()
set([1])
sage: Piter.next()
set([])
sage: Piter.next()
set([1, 2, 3])
sage: Piter.next()
set([1])
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:
OUTPUT:
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]