Examples of posets

class sage.categories.examples.posets.FiniteSetsOrderedByInclusion

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

An example of a poset: finite sets ordered by inclusion

This class provides a minimal implementation of a poset

EXAMPLES:

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

We conclude by running systematic tests on this poset:

sage: TestSuite(P).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_new() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass
running ._test_some_elements() . . . pass
class Element

Bases: sage.structure.element_wrapper.ElementWrapper

wrapped_class

alias of sage.sets.set.Set_object_enumerated

an_element()

Returns an element of this poset

EXAMPLES:

sage: B = Posets().example()
sage: B.an_element()
{1, 4, 6}
le(x, y)

Returns whether \(x\) is a subset of \(y\)

EXAMPLES:

sage: P = Posets().example()
sage: P.le( P(Set([1,3])), P(Set([1,2,3])) )
True
sage: P.le( P(Set([1,3])), P(Set([1,3])) )
True
sage: P.le( P(Set([1,2])), P(Set([1,3])) )
False
class sage.categories.examples.posets.PositiveIntegersOrderedByDivisibilityFacade

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

An example of a facade poset: the positive integers ordered by divisibility

This class provides a minimal implementation of a facade poset

EXAMPLES:

sage: P = Posets().example("facade"); P
An example of a facade poset: the positive integers ordered by divisibility

sage: P(5)
5
sage: P(0)
Traceback (most recent call last):
...
ValueError: Can't coerce `0` in any parent `An example of a facade poset: the positive integers ordered by divisibility` is a facade for

sage: 3 in P
True
sage: 0 in P
False
class element_class(X)

Bases: sage.sets.set.Set_object_enumerated, sage.categories.finite_sets.FiniteSets.parent_class

A finite enumerated set.

le(x, y)

Returns whether \(x\) is divisible by \(y\)

EXAMPLES:

sage: P = Posets().example("facade")
sage: P.le(3, 6)
True
sage: P.le(3, 3)
True
sage: P.le(3, 7)
False