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r""" 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`. """ #***************************************************************************** # Copyright (C) 2011 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from sage.misc.abstract_method import abstract_method from sage.categories.category_with_axiom import CategoryWithAxiom
class FinitePosets(CategoryWithAxiom): r""" 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
.. SEEALSO:: :class:`~sage.categories.posets.Posets`, :func:`Poset`
TESTS::
sage: C = FinitePosets() sage: C is Posets().Finite() True sage: TestSuite(C).run()
"""
class ParentMethods:
########################################################################## # Properties of this poset
def is_lattice(self): r""" Return whether the poset is a lattice.
A poset is a lattice if all pairs of elements have both a least upper bound ("join") and a greatest lower bound ("meet") in the poset.
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
sage: P = Poset({1: [2, 3, 4], 2: [5, 6], 3: [5, 7], 4: [6, 7], 5: [8, 9], ....: 6: [8, 10], 7: [9, 10], 8: [11], 9: [11], 10: [11]}) sage: P.is_lattice() False
TESTS::
sage: P = Poset() sage: P.is_lattice() True
.. SEEALSO::
- Weaker properties: :meth:`~sage.combinat.posets.posets.FinitePoset.is_join_semilattice`, :meth:`~sage.combinat.posets.posets.FinitePoset.is_meet_semilattice` """ (self.has_bottom() and self.is_join_semilattice()))
def is_self_dual(self): r""" Return whether the poset is *self-dual*.
A poset is self-dual if it is isomorphic to its dual poset.
EXAMPLES::
sage: P = Poset({1: [3, 4], 2: [3, 4]}) sage: P.is_self_dual() True
sage: P = Poset({1: [2, 3]}) sage: P.is_self_dual() False
TESTS::
sage: P = Poset() sage: P.is_self_dual() True
.. SEEALSO::
- Stronger properties: :meth:`~sage.combinat.posets.lattices.FiniteLatticePoset.is_orthocomplemented` (for lattices) - Other: :meth:`~sage.combinat.posets.posets.FinitePoset.dual` """ # Two quick checks before full isomorphic test. return False
from sage.misc.superseded import deprecated_function_alias is_selfdual = deprecated_function_alias(24048, is_self_dual)
########################################################################## # Properties of morphisms
def is_poset_isomorphism(self, f, codomain): r""" 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.
.. SEEALSO:: :meth:`FiniteLatticePosets.ParentMethods.is_lattice_morphism` """ # Not injective # Not surjective
def is_poset_morphism(self, f, codomain): r""" Return whether `f` is a morphism of posets from ``self`` to ``codomain``, that is
.. MATH::
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``?
.. SEEALSO:: :meth:`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
"""
########################################################################## # About order ideals, order filters and the like
def order_ideal_generators(self, ideal, direction='down'): r""" 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`. """ else: for y in covers(x)))
def order_filter_generators(self, filter): r""" 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}}
.. SEEALSO:: :meth:`order_ideal_generators` """
def order_ideal_complement_generators(self, antichain, direction='up'): r""" 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`.
:meth:`panyushev_complement` is an alias for this method.
Panyushev complementation is related (actually, isomorphic) to rowmotion (:meth:`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 """ else:
panyushev_complement = order_ideal_complement_generators
def rowmotion(self, order_ideal): r""" 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 (:meth:`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) {} """
def birational_free_labelling(self, linear_extension=None, prefix='x', base_field=None, reduced=False, addvars=None): r""" 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 [EP2013]_ as generalizations of the notions of toggles (:meth:`order_ideal_toggle`) and :meth:`rowmotion <rowmotion>` on order ideals of a finite poset. They have been studied further in [GR2013]_.
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
.. MATH::
\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 :meth:`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 :meth:`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 :class:`tropical semiring <sage.rings.semirings.tropical_semiring.TropicalSemiring>`, in which case birational rowmotion relates to classical constructions such as promotion of rectangular semistandard Young tableaux (page 5 of [EP2013b]_ and future work, via the related notion of birational *promotion*) and rowmotion on order ideals of the poset ([EP2013]_).
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.
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 :meth:`birational_rowmotion`, :meth:`birational_toggle` and :meth:`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 lattice 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) """ else: else: # Now, ``PR`` is the polynomial ring in `n + 2` indeterminates # (or more, if ``addvars`` was set; or less, if ``reduced`` is # ``True``) over ``base_field``. # The first `n + 2` of these indeterminates are named # ``a, x1, x2, ..., xn, b`` (if ``reduced`` is ``False``). # These will label the vertices of `\widehat{P}`. else: # So ``xs`` is the list ``[x1, x2, ..., xn]``. else: # So ``a`` and ``b`` are the labels at `0` and `1`. # ``FF`` is the field of rational functions.
def birational_toggle(self, v, labelling): r""" 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 :meth:`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 :meth:`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 """ # Construct the harmonic sum ``x`` of the labels at the # elements covering ``v``: else: # ``FF.sum``, not ``sum``, see trac #15591. # Construct the sum ``y`` of the labels at the elements # covered by ``v``: else: # Now, transform the label at v:
def birational_toggles(self, vs, labelling): r""" 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 :meth:`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 :meth:`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)] """
def birational_rowmotion(self, labelling): r""" Return the result of applying birational rowmotion to the `\mathbf{K}`-labelling ``labelling`` of the poset ``self``.
See the documentation of :meth:`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 :meth:`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 [GR2013]_ 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 :class:`tropical semiring <sage.rings.semirings.tropical_semiring.TropicalSemiring>` 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 :meth:`classical rowmotion <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 """
def panyushev_orbits(self, element_constructor = set): r""" Return the Panyushev orbits of antichains in ``self``.
The Panyushev orbit of an antichain is its orbit under Panyushev complementation (see :meth:`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()]] """ # TODO: implement a generic function taking a set and # bijections on this set, and returning the orbits.
def rowmotion_orbits(self, element_constructor = set): r""" Return the rowmotion orbits of order ideals in ``self``.
The rowmotion orbit of an order ideal is its orbit under rowmotion (see :meth:`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)) [[()]] """
def toggling_orbits(self, vs, element_constructor = set): r""" Return the orbits of order ideals in ``self`` under the operation of toggling the vertices ``vs[0], vs[1], ...`` in this order.
See :meth:`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] """ # TODO: implement a generic function taking a set and # bijections on this set, and returning the orbits.
def panyushev_orbit_iter(self, antichain, element_constructor=set, stop=True, check=True): r""" Iterate over the Panyushev orbit of an antichain ``antichain`` of ``self``.
The Panyushev orbit of an antichain is its orbit under Panyushev complementation (see :meth:`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: next(Piter) {2} sage: next(Piter) {3} sage: next(Piter) {2} sage: next(Piter) {3} """ # TODO: implement a generic function taking a set and # bijections on this set, and returning an orbit of a given # element. raise ValueError("the given antichain is not an antichain") else:
def rowmotion_orbit_iter(self, oideal, element_constructor=set, stop=True, check=True): r""" 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 :meth:`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: next(Piter) {1, 2, 3} sage: next(Piter) {1, 2, 3, 4} sage: next(Piter) set() sage: next(Piter) {1} sage: next(Piter) {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]] """ # TODO: implement a generic function taking a set and # bijections on this set, and returning an orbit of a given # element. raise ValueError("the given order ideal is not an order ideal") else:
def toggling_orbit_iter(self, vs, oideal, element_constructor=set, stop=True, check=True): r""" 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 :meth:`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: next(Piter) {1, 2, 3} sage: next(Piter) {1} sage: next(Piter) set() sage: next(Piter) {1, 2, 3} sage: next(Piter) {1} """ # TODO: implement a generic function taking a set and # bijections on this set, and returning an orbit of a given # element. raise ValueError("the given order ideal is not an order ideal") else:
def order_ideals_lattice(self, as_ideals=True, facade=None): r""" 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 - ``facade`` -- Boolean or ``None`` (default). Whether to return a facade lattice or not. By default return facade lattice if the poset is a facade poset.
EXAMPLES::
sage: P = posets.PentagonPoset() 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}]]
sage: P = Poset({1:[2]}) sage: J_facade = P.order_ideals_lattice() sage: J_nonfacade = P.order_ideals_lattice(facade=False) sage: type(J_facade[0]) == type(J_nonfacade[0]) False """ for antichain in self.antichains()] facade=facade) else: def is_above(a, xb):
@abstract_method(optional = True) def antichains(self): r""" 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]] """
def directed_subsets(self, direction): r""" 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')) [[]] """ raise ValueError("Direction must be either 'up' or 'down'.") |