Lattice posets

class sage.categories.lattice_posets.LatticePosets[source]

Bases: Category

The category of lattices, i.e. partially ordered sets in which any two elements have a unique supremum (the elements’ least upper bound; called their join) and a unique infimum (greatest lower bound; called their meet).

EXAMPLES:

sage: LatticePosets()
Category of lattice posets
sage: LatticePosets().super_categories()
[Category of posets]
sage: LatticePosets().example()
NotImplemented

>>> from sage.all import *
>>> LatticePosets()
Category of lattice posets
>>> LatticePosets().super_categories()
[Category of posets]
>>> LatticePosets().example()
NotImplemented

See also

Posets, FiniteLatticePosets, LatticePoset()

Finite[source]

alias of FiniteLatticePosets

class ParentMethods[source]

Bases: object

join(x, y)[source]

Return the join of \(x\) and \(y\) in this lattice.

INPUT:

  • x, y – elements of self

EXAMPLES:

sage: D = LatticePoset((divisors(60), attrcall("divides")))             # needs sage.graphs sage.modules
sage: D.join( D(6), D(10) )                                             # needs sage.graphs sage.modules
30

>>> from sage.all import *
>>> D = LatticePoset((divisors(Integer(60)), attrcall("divides")))             # needs sage.graphs sage.modules
>>> D.join( D(Integer(6)), D(Integer(10)) )                                             # needs sage.graphs sage.modules
30
meet(x, y)[source]

Return the meet of \(x\) and \(y\) in this lattice.

INPUT:

  • x, y – elements of self

EXAMPLES:

sage: D = LatticePoset((divisors(30), attrcall("divides")))             # needs sage.graphs sage.modules
sage: D.meet( D(6), D(15) )                                             # needs sage.graphs sage.modules
3

>>> from sage.all import *
>>> D = LatticePoset((divisors(Integer(30)), attrcall("divides")))             # needs sage.graphs sage.modules
>>> D.meet( D(Integer(6)), D(Integer(15)) )                                             # needs sage.graphs sage.modules
3
super_categories()[source]

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

EXAMPLES:

sage: LatticePosets().super_categories()
[Category of posets]

>>> from sage.all import *
>>> LatticePosets().super_categories()
[Category of posets]