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Identifier

St001617: Lattices ⟶ ℤ

Values

([],1) => 0

([(0,1)],2) => 1

([(0,2),(2,1)],3) => 2

([(0,1),(0,2),(1,3),(2,3)],4) => 2

([(0,3),(2,1),(3,2)],4) => 3

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 1

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 3

([(0,4),(2,3),(3,1),(4,2)],5) => 4

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 1

([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 1

([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 1

([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 2

([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 3

([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 4

([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 2

([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 4

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1

([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 4

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 2

([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 2

([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3

([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5

([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 3

([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1

([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 1

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 0

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 1

([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 2

([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 2

([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 2

([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 3

([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 1

([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 2

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 2

([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 5

([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 3

([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 4

([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 1

([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 1

([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 3

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 0

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 1

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 4

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 1

([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 3

([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 2

([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 1

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 2

([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 5

([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 2

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 2

([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 2

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 1

([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 0

([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 3

([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 4

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 3

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 1

([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 2

([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 3

([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => 2

([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 4

([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 2

([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 2

([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 3

([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 2

([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 5

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => 1

([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 4

([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 2

([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 4

([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 3

([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 6

([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 3

([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 3

([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 5

([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1

([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8) => 1

([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8) => 0

([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8) => 0

([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8) => 1

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8) => 2

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,4),(6,1),(6,2),(6,3),(6,5)],8) => 2

([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8) => 1

([(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,4),(7,6)],8) => 2

([(0,6),(1,7),(2,7),(3,7),(4,7),(6,1),(6,2),(6,3),(6,4),(7,5)],8) => 3

([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2),(6,5)],8) => 1

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8) => 1

([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8) => 1

([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(7,6)],8) => 2

([(0,6),(2,7),(3,7),(4,7),(5,1),(6,2),(6,3),(6,4),(7,5)],8) => 4

([(0,5),(1,7),(2,6),(3,6),(4,1),(5,2),(5,3),(5,4),(6,7)],8) => 2

([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3),(5,7),(6,7)],8) => 3

([(0,5),(1,7),(2,7),(3,6),(4,3),(4,7),(5,1),(5,2),(5,4),(7,6)],8) => 3

([(0,5),(2,7),(3,6),(4,6),(5,2),(5,3),(5,4),(6,7),(7,1)],8) => 3

([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,4),(7,3)],8) => 2

([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(6,1),(6,4),(6,5)],8) => 2

([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8) => 1

([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8) => 0

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Description

The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient Mp00196The modular quotient of a lattice..
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

References

[1] Birkhoff, G. Lattice theory MathSciNet:0227053
[2] Fleischer, I., Traynor, T. Group-valued modular functions MathSciNet:0654397

Code

def projective_prime_intervals(L):
    assert L.is_modular()
    I = [tuple(r) for r in L.cover_relations()]
    D = DisjointSet(I)
    for a in L:
        for b, c in Subsets(L.upper_covers(a), 2):
            assert a == L.meet(b, c), "The meet of %s and %s is not %s" % (b, c, a)
            d = L.join(b, c)
            D.union((a,b), (c,d))
            D.union((a,c), (b,d))
    return D

def modular_quotient(L):
    L = L.relabel()
    C = L.congruences_lattice()
    c = C.subposet([c for c in C if (L.quotient(c)).is_modular()]).bottom()
    return L.quotient(c)

def statistic(L):
    L_modular = modular_quotient(L)
    return projective_prime_intervals(L_modular).number_of_subsets()

Created

Sep 29, 2020 at 17:15 by Martin Rubey

Updated

Feb 08, 2021 at 23:11 by Martin Rubey