Jürg Schmid 
A Boolean layer cake is an ordered set obtained from a Boolean lattice
by selecting any number of complete level sets from and
endowing their set union with the order inherited from . The first
Boolean layer cake considered in the literature seems to be the set of all
atoms and of all coatoms of in Dushnik and Miller (1941) - what is
known today as the standard example of an n-dimensional ordered set. Our
intention is (i) to survey the more recent literature on Boolean layer cakes
and (ii) to consider some properties of those among them which happen to be
lattices (a point of view which seems to be new in this context).
As for (i), the dominating part of the work done is about ordered sets
consisting of just two layers from , and within that part, the major
theme is their dimension. If n is odd, the two middle levels of have the same size, and an (open) conjecture now attributed to Havel
states that the covering graph of such an ordered sets always admits a
Hamiltonian path. Another question is which ordered sets allow an
embedding into or into a 2-layer cake. We also consider the
number of isotone self-maps and of order automorphisms of Boolean
layer cakes. Further, jump numbers of general layer cakes have been
determined.
As for (ii), a Boolean layer cake is a lattice iff it consists of
consecutive nontrivial levels of plus top and bottom. Concentrating
on the case of 2-layer cakes with levels 2 and 3, we show that these
lattices have a large number of large (in a sense to be specified) sublattices.
The enumeration and classification of such large sublattices sheds some light
on the sublattice spectrum question and shows that there is no polynomial
bound (in the size of the underlying lattice) for the number of maximal
sublattices of an arbitrary lattice. Sizes of generating sets and
computational aspects are also considered.