Finite lattice posets¶

class sage.categories.finite_lattice_posets.FiniteLatticePosets(base_category)[source]¶

Bases: CategoryWithAxiom

The category of finite lattices, i.e. finite partially ordered sets which are also lattices.

EXAMPLES:

Sage

sage: FiniteLatticePosets()
Category of finite lattice posets
sage: FiniteLatticePosets().super_categories()
[Category of lattice posets, Category of finite posets]
sage: FiniteLatticePosets().example()
NotImplemented

Python

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

class ParentMethods[source]¶

Bases: object

irreducibles_poset()[source]¶

Return the poset of meet- or join-irreducibles of the lattice.

A join-irreducible element of a lattice is an element with exactly one lower cover. Dually a meet-irreducible element has exactly one upper cover.

This is the smallest poset with completion by cuts being isomorphic to the lattice. As a special case this returns one-element poset from one-element lattice.

See also

completion_by_cuts().

EXAMPLES:

Sage

sage: # needs sage.combinat sage.graphs sage.modules
sage: L = LatticePoset({1: [2, 3, 4], 2: [5, 6], 3: [5],
....:                   4: [6], 5: [9, 7], 6: [9, 8], 7: [10],
....:                   8: [10], 9: [10], 10: [11]})
sage: L_ = L.irreducibles_poset()
sage: sorted(L_)
[2, 3, 4, 7, 8, 9, 10, 11]
sage: L_.completion_by_cuts().is_isomorphic(L)
True

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.graphs sage.modules
>>> L = LatticePoset({Integer(1): [Integer(2), Integer(3), Integer(4)], Integer(2): [Integer(5), Integer(6)], Integer(3): [Integer(5)],
...                   Integer(4): [Integer(6)], Integer(5): [Integer(9), Integer(7)], Integer(6): [Integer(9), Integer(8)], Integer(7): [Integer(10)],
...                   Integer(8): [Integer(10)], Integer(9): [Integer(10)], Integer(10): [Integer(11)]})
>>> L_ = L.irreducibles_poset()
>>> sorted(L_)
[2, 3, 4, 7, 8, 9, 10, 11]
>>> L_.completion_by_cuts().is_isomorphic(L)
True

is_lattice_morphism(f, codomain)[source]¶

Return whether f is a morphism of posets from self to codomain.

A map \(f : P \to Q\) is a poset morphism if

\[x \leq y \Rightarrow f(x) \leq f(y)\]

for all \(x,y \in P\).

INPUT:

f – a function from self to codomain
codomain – a lattice

EXAMPLES:

We build the boolean lattice of \(\{2,2,3\}\) and the lattice of divisors of \(60\), and check that the map \(b \mapsto 5 \prod_{x\in b} x\) is a morphism of lattices:

Sage

sage: D = LatticePoset((divisors(60), attrcall("divides")))             # needs sage.graphs sage.modules
sage: B = LatticePoset((Subsets([2,2,3]), attrcall("issubset")))        # needs sage.graphs sage.modules
sage: def f(b): return D(5*prod(b))
sage: B.is_lattice_morphism(f, D)                                       # needs sage.graphs sage.modules
True

Python

>>> from sage.all import *
>>> D = LatticePoset((divisors(Integer(60)), attrcall("divides")))             # needs sage.graphs sage.modules
>>> B = LatticePoset((Subsets([Integer(2),Integer(2),Integer(3)]), attrcall("issubset")))        # needs sage.graphs sage.modules
>>> def f(b): return D(Integer(5)*prod(b))
>>> B.is_lattice_morphism(f, D)                                       # needs sage.graphs sage.modules
True

We construct the boolean lattice \(B_2\):

Sage

sage: B = posets.BooleanLattice(2)                                      # needs sage.graphs
sage: B.cover_relations()                                               # needs sage.graphs
[[0, 1], [0, 2], [1, 3], [2, 3]]

Python

>>> from sage.all import *
>>> B = posets.BooleanLattice(Integer(2))                                      # needs sage.graphs
>>> B.cover_relations()                                               # needs sage.graphs
[[0, 1], [0, 2], [1, 3], [2, 3]]

And the same lattice with new top and bottom elements numbered respectively \(-1\) and \(3\):

Sage

sage: G = DiGraph({-1:[0], 0:[1,2], 1:[3], 2:[3], 3:[4]})               # needs sage.graphs
sage: L = LatticePoset(G)                                               # needs sage.graphs sage.modules
sage: L.cover_relations()                                               # needs sage.graphs sage.modules
[[-1, 0], [0, 1], [0, 2], [1, 3], [2, 3], [3, 4]]

sage: f = {B(0): L(0), B(1): L(1), B(2): L(2), B(3): L(3)}.__getitem__  # needs sage.graphs sage.modules
sage: B.is_lattice_morphism(f, L)                                       # needs sage.graphs sage.modules
True

sage: f = {B(0): L(-1),B(1): L(1), B(2): L(2), B(3): L(3)}.__getitem__  # needs sage.graphs sage.modules
sage: B.is_lattice_morphism(f, L)                                       # needs sage.graphs sage.modules
False

sage: f = {B(0): L(0), B(1): L(1), B(2): L(2), B(3): L(4)}.__getitem__  # needs sage.graphs sage.modules
sage: B.is_lattice_morphism(f, L)                                       # needs sage.graphs sage.modules
False

Python

>>> from sage.all import *
>>> G = DiGraph({-Integer(1):[Integer(0)], Integer(0):[Integer(1),Integer(2)], Integer(1):[Integer(3)], Integer(2):[Integer(3)], Integer(3):[Integer(4)]})               # needs sage.graphs
>>> L = LatticePoset(G)                                               # needs sage.graphs sage.modules
>>> L.cover_relations()                                               # needs sage.graphs sage.modules
[[-1, 0], [0, 1], [0, 2], [1, 3], [2, 3], [3, 4]]

>>> f = {B(Integer(0)): L(Integer(0)), B(Integer(1)): L(Integer(1)), B(Integer(2)): L(Integer(2)), B(Integer(3)): L(Integer(3))}.__getitem__  # needs sage.graphs sage.modules
>>> B.is_lattice_morphism(f, L)                                       # needs sage.graphs sage.modules
True

>>> f = {B(Integer(0)): L(-Integer(1)),B(Integer(1)): L(Integer(1)), B(Integer(2)): L(Integer(2)), B(Integer(3)): L(Integer(3))}.__getitem__  # needs sage.graphs sage.modules
>>> B.is_lattice_morphism(f, L)                                       # needs sage.graphs sage.modules
False

>>> f = {B(Integer(0)): L(Integer(0)), B(Integer(1)): L(Integer(1)), B(Integer(2)): L(Integer(2)), B(Integer(3)): L(Integer(4))}.__getitem__  # needs sage.graphs sage.modules
>>> B.is_lattice_morphism(f, L)                                       # needs sage.graphs sage.modules
False

See also

is_poset_morphism()

join_irreducibles()[source]¶

Return the join-irreducible elements of this finite lattice.

A join-irreducible element of self is an element \(x\) that is not minimal and that can not be written as the join of two elements different from \(x\).

EXAMPLES:

Sage

sage: L = LatticePoset({0:[1,2],1:[3],2:[3,4],3:[5],4:[5]})             # needs sage.graphs sage.modules
sage: L.join_irreducibles()                                             # needs sage.graphs sage.modules
[1, 2, 4]

Python

>>> from sage.all import *
>>> L = LatticePoset({Integer(0):[Integer(1),Integer(2)],Integer(1):[Integer(3)],Integer(2):[Integer(3),Integer(4)],Integer(3):[Integer(5)],Integer(4):[Integer(5)]})             # needs sage.graphs sage.modules
>>> L.join_irreducibles()                                             # needs sage.graphs sage.modules
[1, 2, 4]

See also

Dual function: meet_irreducibles()
Other: double_irreducibles(), join_irreducibles_poset()

join_irreducibles_poset()[source]¶

Return the poset of join-irreducible elements of this finite lattice.

A join-irreducible element of self is an element \(x\) that is not minimal and can not be written as the join of two elements different from \(x\).

EXAMPLES:

Sage

sage: L = LatticePoset({0:[1,2,3],1:[4],2:[4],3:[4]})                   # needs sage.graphs sage.modules
sage: L.join_irreducibles_poset()                                       # needs sage.graphs sage.modules
Finite poset containing 3 elements

Python

>>> from sage.all import *
>>> L = LatticePoset({Integer(0):[Integer(1),Integer(2),Integer(3)],Integer(1):[Integer(4)],Integer(2):[Integer(4)],Integer(3):[Integer(4)]})                   # needs sage.graphs sage.modules
>>> L.join_irreducibles_poset()                                       # needs sage.graphs sage.modules
Finite poset containing 3 elements

See also

Dual function: meet_irreducibles_poset()
Other: join_irreducibles()

meet_irreducibles()[source]¶

Return the meet-irreducible elements of this finite lattice.

A meet-irreducible element of self is an element \(x\) that is not maximal and that can not be written as the meet of two elements different from \(x\).

EXAMPLES:

Sage

sage: L = LatticePoset({0:[1,2],1:[3],2:[3,4],3:[5],4:[5]})             # needs sage.graphs sage.modules
sage: L.meet_irreducibles()                                             # needs sage.graphs sage.modules
[1, 3, 4]

Python

>>> from sage.all import *
>>> L = LatticePoset({Integer(0):[Integer(1),Integer(2)],Integer(1):[Integer(3)],Integer(2):[Integer(3),Integer(4)],Integer(3):[Integer(5)],Integer(4):[Integer(5)]})             # needs sage.graphs sage.modules
>>> L.meet_irreducibles()                                             # needs sage.graphs sage.modules
[1, 3, 4]

See also

Dual function: join_irreducibles()
Other: double_irreducibles(), meet_irreducibles_poset()

meet_irreducibles_poset()[source]¶

Return the poset of join-irreducible elements of this finite lattice.

A meet-irreducible element of self is an element \(x\) that is not maximal and can not be written as the meet of two elements different from \(x\).

EXAMPLES:

Sage

sage: L = LatticePoset({0:[1,2,3],1:[4],2:[4],3:[4]})                   # needs sage.graphs sage.modules
sage: L.join_irreducibles_poset()                                       # needs sage.graphs sage.modules
Finite poset containing 3 elements

Python

>>> from sage.all import *
>>> L = LatticePoset({Integer(0):[Integer(1),Integer(2),Integer(3)],Integer(1):[Integer(4)],Integer(2):[Integer(4)],Integer(3):[Integer(4)]})                   # needs sage.graphs sage.modules
>>> L.join_irreducibles_poset()                                       # needs sage.graphs sage.modules
Finite poset containing 3 elements

See also

Dual function: join_irreducibles_poset()
Other: meet_irreducibles()