Combinatorial Problems in Lattice Theory

In a finite lattice with n elements is there always a
join irreducible element whose filter has size at most n/2?
This is equivalent to the old problem: if a finite collection of
finite sets is closed under union is there an element in at least half
of the sets?
This would certainly be true if the average size of the filter above a join
irreducible element is at least n/2. But J. B. and I used our
lattice program to find a counter example to this average
conjecture.
The weaker conjecture:
{(a,x) : a in J(L), x in L, a
<= x} <= J(L) L/2 + (J(L)  1)/2
which would also imply the problem has a positive answer, is also
false; see the second image.
A counter example to the average conjecture
A counter example to the weak average conjecture
18 June 1996
Ralph Freese
ralph@math.hawaii.edu