Bases: sage.categories.category.Category
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.Facades), 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 #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
See also
TESTS:
sage: C = Posets()
sage: TestSuite(C).run()
Returns whether in this poset
INPUT:
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
Returns whether in this poset
INPUT:
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
Returns whether in this poset
INPUT:
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
Returns the lower covers of , that is the elements
such that
and there exists no
such that
.
EXAMPLES:
sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.lower_covers(15)
[3, 5]
Returns whether in this poset
INPUT:
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
Returns the order filter generated by a list of elements.
is an order filter if, for any
in
and
such that
, then
is in
.
EXAMPLES:
sage: B = Posets.BooleanLattice(4)
sage: B.order_filter([3,8])
[3, 7, 8, 9, 10, 11, 12, 13, 14, 15]
Returns the order ideal in self generated by gens.
EXAMPLES:
sage: B = Posets.BooleanLattice(4)
sage: B.order_ideal([7,10])
[0, 1, 2, 3, 4, 5, 6, 7, 8, 10]
Returns the order filter generated by an element x.
EXAMPLES:
sage: B = Posets.BooleanLattice(4)
sage: B.principal_order_filter(2)
[2, 3, 6, 7, 10, 11, 14, 15]
Returns the order ideal generated by an element x.
EXAMPLES:
sage: B = Posets.BooleanLattice(4)
sage: B.principal_order_ideal(6)
[0, 2, 4, 6]
Returns the upper covers of , that is the elements
such that
and there exists no
such that
.
EXAMPLES:
sage: D = Poset((divisors(30), attrcall("divides")))
sage: D.upper_covers(3)
[6, 15]
Returns examples of objects of Posets(), as per Category.example().
EXAMPLES:
sage: Posets().example()
An example of a poset: sets ordered by inclusion
sage: Posets().example("facade")
An example of a facade poset: the positive integers ordered by divisibility
Returns a list of the (immediate) super categories of self, as per Category.super_categories().
EXAMPLES:
sage: Posets().super_categories()
[Category of sets]