Totally Ordered Finite Sets

AUTHORS:

  • Stepan Starosta (2012): Initial version
class sage.sets.totally_ordered_finite_set.TotallyOrderedFiniteSet(elements, facade=True)

Bases: sage.sets.finite_enumerated_set.FiniteEnumeratedSet

Totally ordered finite set.

This is a finite enumerated set assuming that the elements are ordered based upon their rank (i.e. their position in the set).

INPUT:

  • elements – A list of elements in the set
  • facade – (default: True) if True, a facade is used; it should be set to False if the elements do not inherit from Element or if you want a funny order. See examples for more details.

EXAMPLES:

sage: S = TotallyOrderedFiniteSet([1,2,3])
sage: S
{1, 2, 3}
sage: S.cardinality()
3

By default, totally ordered finite set behaves as a facade:

sage: S(1).parent()
Integer Ring

It makes comparison fails when it is not the standard order:

sage: T1 = TotallyOrderedFiniteSet([3,2,5,1])
sage: T1(3) < T1(1)
False
sage: T2 = TotallyOrderedFiniteSet([3,var('x')])
sage: T2(3) < T2(var('x'))
3 < x

To make the above example work, you should set the argument facade to False in the constructor. In that case, the elements of the set have a dedicated class:

sage: A = TotallyOrderedFiniteSet([3,2,0,'a',7,(0,0),1], facade=False)
sage: A
{3, 2, 0, 'a', 7, (0, 0), 1}
sage: x = A.an_element()
sage: x
3
sage: x.parent()
{3, 2, 0, 'a', 7, (0, 0), 1}
sage: A(3) < A(2)
True
sage: A('a') < A(7)
True
sage: A(3) > A(2)
False
sage: A(1) < A(3)
False
sage: A(3) == A(3)
True

But then, the equality comparison is always False with elements outside of the set:

sage: A(1) == 1
False
sage: 1 == A(1)
False
sage: 'a' == A('a')
False
sage: A('a') == 'a'
False

and comparisons are comparisons of types:

sage: for e in [1,'a',(0, 0)]:
...      f = A(e)
...      print e == f,
...      print cmp(e,f) == cmp(type(e),type(f)),
...      print cmp(f,e) == cmp(type(f),type(e))
False True True
False True True
False True True

This behavior of comparison is the same as the one of Element.

Since trac ticket #16280, totally ordered sets support elements that do not inherit from sage.structure.element.Element, whether they are facade or not:

sage: S = TotallyOrderedFiniteSet(['a','b'])
sage: S('a')
'a'
sage: S = TotallyOrderedFiniteSet(['a','b'], facade = False)
sage: S('a')
'a'

Multiple elements are automatically deleted:

sage: TotallyOrderedFiniteSet([1,1,2,1,2,2,5,4])
{1, 2, 5, 4}
Element

alias of TotallyOrderedFiniteSetElement

le(x, y)

Return True if \(x \le y\) for the order of self.

EXAMPLES:

sage: T = TotallyOrderedFiniteSet([1,3,2], facade=False)
sage: T1, T3, T2 = T.list()
sage: T.le(T1,T3)
True
sage: T.le(T3,T2)
True
class sage.sets.totally_ordered_finite_set.TotallyOrderedFiniteSetElement(parent, data)

Bases: sage.structure.element.Element

Element of a finite totally ordered set.

EXAMPLES:

sage: S = TotallyOrderedFiniteSet([2,7], facade=False)
sage: x = S(2)
sage: print x
2
sage: x.parent()
{2, 7}

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