Examples of posets

class sage.categories.examples.posets.FiniteSetsOrderedByInclusion

Bases: UniqueRepresentation, Parent

An example of a poset: finite sets ordered by inclusion.

This class provides a minimal implementation of a poset

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> P = Posets().example(); P
An example of a poset: sets ordered by inclusion

We conclude by running systematic tests on this poset:

Sage

sage: TestSuite(P).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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

Python

>>> from sage.all import *
>>> TestSuite(P).run(verbose = True)
running ._test_an_element() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_construction() . . . 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: ElementWrapper

wrapped_class

alias of Set_object_enumerated

an_element()

Return an element of this poset.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> B = Posets().example()
>>> B.an_element()
{1, 4, 6}

le(x, y)

Return whether \(x\) is a subset of \(y\).

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> P = Posets().example()
>>> P.le( P(Set([Integer(1),Integer(3)])), P(Set([Integer(1),Integer(2),Integer(3)])) )
True
>>> P.le( P(Set([Integer(1),Integer(3)])), P(Set([Integer(1),Integer(3)])) )
True
>>> P.le( P(Set([Integer(1),Integer(2)])), P(Set([Integer(1),Integer(3)])) )
False

class sage.categories.examples.posets.PositiveIntegersOrderedByDivisibilityFacade

Bases: UniqueRepresentation, 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

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

Python

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

>>> P(Integer(5))
5
>>> P(Integer(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

>>> Integer(3) in P
True
>>> Integer(0) in P
False

class element_class(X, category=None)

Bases: Set_object_enumerated, parent_class

A finite enumerated set.

le(x, y)

Return whether \(x\) is divisible by \(y\).

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> P = Posets().example("facade")
>>> P.le(Integer(3), Integer(6))
True
>>> P.le(Integer(3), Integer(3))
True
>>> P.le(Integer(3), Integer(7))
False