Bases: sage.categories.category_with_axiom.CategoryWithAxiom
The category of finite lattices, i.e. finite partially ordered sets which are also lattices.
EXAMPLES:
sage: FiniteLatticePosets()
Category of finite lattice posets
sage: FiniteLatticePosets().super_categories()
[Category of lattice posets, Category of finite posets]
sage: FiniteLatticePosets().example()
NotImplemented
See also
FinitePosets, LatticePosets, LatticePoset
TESTS:
sage: C = FiniteLatticePosets()
sage: C is FiniteLatticePosets().Finite()
True
sage: TestSuite(C).run()
INPUT:
Returns whether \(f\) is a morphism of posets form self to codomain, that is
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: D = LatticePoset((divisors(60), attrcall("divides")))
sage: B = LatticePoset((Subsets([2,2,3]), attrcall("issubset")))
sage: def f(b): return D(5*prod(b))
sage: B.is_lattice_morphism(f, D)
True
We construct the boolean lattice \(B_2\):
sage: B = Posets.BooleanLattice(2)
sage: B.cover_relations()
[[0, 1], [0, 2], [1, 3], [2, 3]]
And the same lattice with new top and bottom elements numbered respectively \(-1\) and \(3\):
sage: L = LatticePoset(DiGraph({-1:[0], 0:[1,2], 1:[3], 2:[3],3:[4]}))
sage: L.cover_relations()
[[-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__
sage: B.is_lattice_morphism(f, L)
True
sage: f = { B(0): L(-1),B(1): L(1), B(2): L(2), B(3): L(3) }.__getitem__
sage: B.is_lattice_morphism(f, L)
False
sage: f = { B(0): L(0), B(1): L(1), B(2): L(2), B(3): L(4) }.__getitem__
sage: B.is_lattice_morphism(f, L)
False
See also
Returns 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: L = LatticePoset({0:[1,2],1:[3],2:[3,4],3:[5],4:[5]})
sage: L.join_irreducibles()
[1, 2, 4]
See also
Returns 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: L = LatticePoset({0:[1,2,3],1:[4],2:[4],3:[4]})
sage: L.join_irreducibles_poset()
Finite poset containing 3 elements
See also
Returns 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: L = LatticePoset({0:[1,2],1:[3],2:[3,4],3:[5],4:[5]})
sage: L.meet_irreducibles()
[1, 3, 4]
See also
Returns 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: L = LatticePoset({0:[1,2,3],1:[4],2:[4],3:[4]})
sage: L.join_irreducibles_poset()
Finite poset containing 3 elements
See also