# Posets¶

class sage.categories.posets.Posets(s=None)

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

EXAMPLES:

sage: Posets()
Category of posets
sage: Posets().super_categories()
[Category of sets]
sage: P = Posets().example(); P
An example of a poset: sets ordered by inclusion


The partial order is implemented by the mandatory method le():

sage: x = P(Set([1,3])); y = P(Set([1,2,3]))
sage: x, y
({1, 3}, {1, 2, 3})
sage: P.le(x, y)
True
sage: P.le(x, x)
True
sage: P.le(y, x)
False


The other comparison methods are called lt(), ge(), gt(), following Python’s naming convention in operator. Default implementations are provided:

sage: P.lt(x, x)
False
sage: P.ge(y, x)
True


Unless the poset is a facade (see Sets.Facade), one can compare directly its elements using the usual Python operators:

sage: D = Poset((divisors(30), attrcall("divides")), facade = False)
sage: D(3) <= D(6)
True
sage: D(3) <= D(3)
True
sage: D(3) <= D(5)
False
sage: D(3) < D(3)
False
sage: D(10) >= D(5)
True


At this point, this has to be implemented by hand. Once trac ticket #10130 will be resolved, this will be automatically provided by this category:

sage: x < y      # todo: not implemented
True
sage: x < x      # todo: not implemented
False
sage: x <= x     # todo: not implemented
True
sage: y >= x     # todo: not implemented
True

class ElementMethods
Finite
class ParentMethods
CartesianProduct
directed_subset(elements, direction)

Return the order filter or the order ideal generated by a list of elements.

If direction is ‘up’, the order filter (upper set) is being returned.

If direction is ‘down’, the order ideal (lower set) is being returned.

INPUT:

• elements – a list of elements.
• direction – ‘up’ or ‘down’.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.directed_subset([3, 8], 'up')
[3, 7, 8, 9, 10, 11, 12, 13, 14, 15]
sage: B.directed_subset([7, 10], 'down')
[0, 1, 2, 3, 4, 5, 6, 7, 8, 10]

ge(x, y)

Return whether $$x \ge y$$ in the poset self.

INPUT:

• x, y – elements of self.

This default implementation delegates the work to le().

EXAMPLES:

sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.ge( 6, 3 )
True
sage: D.ge( 3, 3 )
True
sage: D.ge( 3, 5 )
False

gt(x, y)

Return whether $$x > y$$ in the poset self.

INPUT:

• x, y – elements of self.

This default implementation delegates the work to lt().

EXAMPLES:

sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.gt( 3, 6 )
False
sage: D.gt( 3, 3 )
False
sage: D.gt( 3, 5 )
False

is_antichain_of_poset(o)

Return whether an iterable o is an antichain of self.

INPUT:

• o – an iterable (e. g., list, set, or tuple) containing some elements of self

OUTPUT:

True if the subset of self consisting of the entries of o is an antichain of self, and False otherwise.

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: sorted(P.list())
[1, 2, 3, 4, 6, 12]
sage: P.is_antichain_of_poset([1, 3])
False
sage: P.is_antichain_of_poset([3, 1])
False
sage: P.is_antichain_of_poset([1, 1, 3])
False
sage: P.is_antichain_of_poset([])
True
sage: P.is_antichain_of_poset([1])
True
sage: P.is_antichain_of_poset([1, 1])
True
sage: P.is_antichain_of_poset([3, 4])
True
sage: P.is_antichain_of_poset([3, 4, 12])
False
sage: P.is_antichain_of_poset([6, 4])
True
sage: P.is_antichain_of_poset(i for i in divisors(12) if (2 < i and i < 6))
True
sage: P.is_antichain_of_poset(i for i in divisors(12) if (2 <= i and i < 6))
False

sage: Q = Poset({2: [3, 1], 3: [4], 1: [4]})
sage: Q.is_antichain_of_poset((1, 2))
False
sage: Q.is_antichain_of_poset((2, 4))
False
sage: Q.is_antichain_of_poset((4, 2))
False
sage: Q.is_antichain_of_poset((2, 2))
True
sage: Q.is_antichain_of_poset((3, 4))
False
sage: Q.is_antichain_of_poset((3, 1))
True
sage: Q.is_antichain_of_poset((1, ))
True
sage: Q.is_antichain_of_poset(())
True


An infinite poset:

sage: from sage.categories.examples.posets import FiniteSetsOrderedByInclusion
sage: R = FiniteSetsOrderedByInclusion()
sage: R.is_antichain_of_poset([R(set([3, 1, 2])), R(set([1, 4])), R(set([4, 5]))])
True
sage: R.is_antichain_of_poset([R(set([3, 1, 2, 4])), R(set([1, 4])), R(set([4, 5]))])
False

is_chain_of_poset(o, ordered=False)

Return whether an iterable o is a chain of self, including a check for o being ordered from smallest to largest element if the keyword ordered is set to True.

INPUT:

• o – an iterable (e. g., list, set, or tuple) containing some elements of self
• ordered – a Boolean (default: False) which decides whether the notion of a chain includes being ordered

OUTPUT:

If ordered is set to False, the truth value of the following assertion is returned: The subset of self formed by the elements of o is a chain in self.

If ordered is set to True, the truth value of the following assertion is returned: Every element of the list o is (strictly!) smaller than its successor in self. (This makes no sense if ordered is a set.)

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: sorted(P.list())
[1, 2, 3, 4, 6, 12]
sage: P.is_chain_of_poset([1, 3])
True
sage: P.is_chain_of_poset([3, 1])
True
sage: P.is_chain_of_poset([1, 3], ordered=True)
True
sage: P.is_chain_of_poset([3, 1], ordered=True)
False
sage: P.is_chain_of_poset([])
True
sage: P.is_chain_of_poset([], ordered=True)
True
sage: P.is_chain_of_poset((2, 12, 6))
True
sage: P.is_chain_of_poset((2, 6, 12), ordered=True)
True
sage: P.is_chain_of_poset((2, 12, 6), ordered=True)
False
sage: P.is_chain_of_poset((2, 12, 6, 3))
False
sage: P.is_chain_of_poset((2, 3))
False

sage: Q = Poset({2: [3, 1], 3: [4], 1: [4]})
sage: Q.is_chain_of_poset([1, 2], ordered=True)
False
sage: Q.is_chain_of_poset([1, 2])
True
sage: Q.is_chain_of_poset([2, 1], ordered=True)
True
sage: Q.is_chain_of_poset([2, 1, 1], ordered=True)
False
sage: Q.is_chain_of_poset([3])
True
sage: Q.is_chain_of_poset([4, 2, 3])
True
sage: Q.is_chain_of_poset([4, 2, 3], ordered=True)
False
sage: Q.is_chain_of_poset([2, 3, 4], ordered=True)
True


Examples with infinite posets:

sage: from sage.categories.examples.posets import FiniteSetsOrderedByInclusion
sage: R = FiniteSetsOrderedByInclusion()
sage: R.is_chain_of_poset([R(set([3, 1, 2])), R(set([1, 4])), R(set([4, 5]))])
False
sage: R.is_chain_of_poset([R(set([3, 1, 2])), R(set([1, 2])), R(set([1]))], ordered=True)
False
sage: R.is_chain_of_poset([R(set([3, 1, 2])), R(set([1, 2])), R(set([1]))])
True

sage: T.is_chain_of_poset((T(3), T(4), T(7)))
False
sage: T.is_chain_of_poset((T(3), T(6), T(3)))
True
sage: T.is_chain_of_poset((T(3), T(6), T(3)), ordered=True)
False
sage: T.is_chain_of_poset((T(3), T(3), T(6)))
True
sage: T.is_chain_of_poset((T(3), T(3), T(6)), ordered=True)
False
sage: T.is_chain_of_poset((T(3), T(6)), ordered=True)
True
sage: T.is_chain_of_poset((), ordered=True)
True
sage: T.is_chain_of_poset((T(3),), ordered=True)
True
sage: T.is_chain_of_poset((T(q) for q in divisors(27)))
True
sage: T.is_chain_of_poset((T(q) for q in divisors(18)))
False

is_order_filter(o)

Return whether o is an order filter of self, assuming self has no infinite ascending path.

INPUT:

• o – a list (or set, or tuple) containing some elements of self

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: sorted(P.list())
[1, 2, 3, 4, 6, 12]
sage: P.is_order_filter([4, 12])
True
sage: P.is_order_filter([])
True
sage: P.is_order_filter({3, 4, 12})
False
sage: P.is_order_filter({3, 6, 12})
True

is_order_ideal(o)

Return whether o is an order ideal of self, assuming self has no infinite descending path.

INPUT:

• o – a list (or set, or tuple) containing some elements of self

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: sorted(P.list())
[1, 2, 3, 4, 6, 12]
sage: P.is_order_ideal([1, 3])
True
sage: P.is_order_ideal([])
True
sage: P.is_order_ideal({1, 3})
True
sage: P.is_order_ideal([1, 3, 4])
False

le(x, y)

Return whether $$x \le y$$ in the poset self.

INPUT:

• x, y – elements of self.

EXAMPLES:

sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.le( 3, 6 )
True
sage: D.le( 3, 3 )
True
sage: D.le( 3, 5 )
False

lower_covers(x)

Return the lower covers of $$x$$, that is, the elements $$y$$ such that $$y<x$$ and there exists no $$z$$ such that $$y<z<x$$.

EXAMPLES:

sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.lower_covers(15)
[3, 5]

lt(x, y)

Return whether $$x < y$$ in the poset self.

INPUT:

• x, y – elements of self.

This default implementation delegates the work to le().

EXAMPLES:

sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.lt( 3, 6 )
True
sage: D.lt( 3, 3 )
False
sage: D.lt( 3, 5 )
False

order_filter(elements)

Return the order filter generated by a list of elements.

A subset $$I$$ of a poset is said to be an order filter if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \ge x$$, then $$y$$ is in $$I$$.

This is also called the upper set generated by these elements.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.order_filter([3,8])
[3, 7, 8, 9, 10, 11, 12, 13, 14, 15]

order_ideal(elements)

Return the order ideal in self generated by the elements of an iterable elements.

A subset $$I$$ of a poset is said to be an order ideal if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \le x$$, then $$y$$ is in $$I$$.

This is also called the lower set generated by these elements.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.order_ideal([7,10])
[0, 1, 2, 3, 4, 5, 6, 7, 8, 10]

order_ideal_toggle(I, v)

Return the result of toggling the element v in the order ideal I.

If $$v$$ is an element of a poset $$P$$, then toggling the element $$v$$ is an automorphism of the set $$J(P)$$ of all order ideals of $$P$$. It is defined as follows: If $$I$$ is an order ideal of $$P$$, then the image of $$I$$ under toggling the element $$v$$ is

• the set $$I \cup \{ v \}$$, if $$v \not\in I$$ but every element of $$P$$ smaller than $$v$$ is in $$I$$;
• the set $$I \setminus \{ v \}$$, if $$v \in I$$ but no element of $$P$$ greater than $$v$$ is in $$I$$;
• $$I$$ otherwise.

This image always is an order ideal of $$P$$.

EXAMPLES:

sage: P = Poset({1: [2,3], 2: [4], 3: []})
sage: I = Set({1, 2})
sage: I in P.order_ideals_lattice()
True
sage: P.order_ideal_toggle(I, 1)
{1, 2}
sage: P.order_ideal_toggle(I, 2)
{1}
sage: P.order_ideal_toggle(I, 3)
{1, 2, 3}
sage: P.order_ideal_toggle(I, 4)
{1, 2, 4}
sage: P4 = Posets(4)
sage: all(all(all(P.order_ideal_toggle(P.order_ideal_toggle(I, i), i) == I
....:               for i in range(4))
....:     for P in P4)
True

order_ideal_toggles(I, vs)

Return the result of toggling the elements of the list (or iterable) vs (one by one, from left to right) in the order ideal I.

See order_ideal_toggle() for a definition of toggling.

EXAMPLES:

sage: P = Poset({1: [2,3], 2: [4], 3: []})
sage: I = Set({1, 2})
sage: P.order_ideal_toggles(I, [1,2,3,4])
{1, 3}
sage: P.order_ideal_toggles(I, (1,2,3,4))
{1, 3}

principal_lower_set(x)

Return the order ideal generated by an element x.

This is also called the lower set generated by this element.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.principal_order_ideal(6)
[0, 2, 4, 6]

principal_order_filter(x)

Return the order filter generated by an element x.

This is also called the upper set generated by this element.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.principal_order_filter(2)
[2, 3, 6, 7, 10, 11, 14, 15]

principal_order_ideal(x)

Return the order ideal generated by an element x.

This is also called the lower set generated by this element.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.principal_order_ideal(6)
[0, 2, 4, 6]

principal_upper_set(x)

Return the order filter generated by an element x.

This is also called the upper set generated by this element.

EXAMPLES:

sage: B = posets.BooleanLattice(4)
sage: B.principal_order_filter(2)
[2, 3, 6, 7, 10, 11, 14, 15]

upper_covers(x)

Return the upper covers of $$x$$, that is, the elements $$y$$ such that $$x<y$$ and there exists no $$z$$ such that $$x<z<y$$.

EXAMPLES:

sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.upper_covers(3)
[6, 15]

example(choice=None)

Return examples of objects of Posets(), as per Category.example().

EXAMPLES:

sage: Posets().example()
An example of a poset: sets ordered by inclusion


super_categories()
Return a list of the (immediate) super categories of self, as per Category.super_categories().
sage: Posets().super_categories()