Let L be any (finite) nontrivial distributive lattice. We write 
for
the number of proper maximal sublattices of L and 
for the number of
proper maximal (0,1)-sublattices of L. We are interested in the relations
between the numbers , and | L|. In [1], the
following results were proved:
(i) 
and (ii) 
iff L
is a chain.
The proof is based on Birkhoff duality (in the finite case), replacing finite distributive lattices by their dual finite ordered sets of join-irreducibles, and this is the point where critical pairs as considered in subsection 2.1again crop up - although in a slightly more general form. Here are the salient facts:
Let 
be any ordered set. Call an ordered pair (x,y) of elements
of P *critical iff 
and for all 
, u<x implies
and v>y implies 
. Every critical pair is *critical, but
not vice versa: A *critical pair may be comparable, and this happens exactly
iff y is the unique lower neighbor of x and x is the unique upper
neighbor of
y. Hashimoto [31] was the first to observe that there is a bijective
correpondence between the proper maximal (0,1)-sublattices of a finite
distributive lattice L on one side and the *critical pairs of the dual of L
on the other. See [1] for details; the results on resp.
cited above are obtained by a careful count of *critical pairs in
these dual orders.
Defining 
, we have 
and
iff L is a chain. 
may be made arbitrarily small:
Indeed, let 
, then the dual ordered set of L is the n-element
antichain 
. Now every pair of elements of is *critical, whence
for 
.
A natural question is whether the bound 
is characteristic for
finite distributive lattices. One of the main results of [2]
shows that this is not the case: Indeed, we have for any
finite bounded lattice L (where L is bounded iff it can be
obtained from the trivial lattice by a sequence of applications of Alan Day's
doubling construction for intervals). Every finite distributive lattice is
bounded, so this is a much better result whose proof, of course, has nothing to
do with counting *critical pairs since no duality theory is available.
Returning to the distributive case, we can give better estmates for
if we restrict the class of lattices considered. Denote by 
the
class of all n-generated distributive lattices ( 
), let 
be
the free distributive lattice on n generators and 
the n-element
chain. Then both and are in 
and
we claim that for any 
we have
. Indeed, if 
generates L, then the sets 
generate distinct proper
sublattices 
with 
which are contained in distinct
proper maximal sublattices 
. It follows that
. Turning to , note that its dual is 
.
Since no element of has a unique upper or lower neighbor,
*critical and critical pairs coincide for and by the count
given in 2.1.1 there are exactly n of them. So 
. Clearly,
, and thus 
. Of course, the exact value of
remains a mystery, but a reasonable estimate may be obtained from
Korshunov's asymptotic estimate for 
given in [36].
What is the range of outside bounded (and thus distributive)
lattices? Again, Boolean layer cakes provide a hint: Let 
denote, for the
moment, the layer cake P(0,1,2;n) realized on 
which is
obviously a lattice generated
by its atoms. It is not hard to check that the maximal sublattices of
missing, say, the atom 
are given by
for some 
. It follows that there are n-1 of them and thus 
,
giving 
. We see that 
although 
is neither a chain nor distributive, and that 
for
.
Of course, is not even modular. In [2], subspace lattices L
of projective planes are examined and it is shown that their count of maximal
sublattices is roughly 
, thus 
. In
fact, no modular lattices exceeding
this bound have been found so far. This fact and the examples in the preceding
paragraph motivated the investigation of Boolean layer cakes to test the
asymptotic behaviour of .