Ordered sets obtained from a finite Boolean lattice B by selecting a number of complete level sets of B and imposing the induced order on the subset of B so defined seem to be a common source of problems as different as difficult - even in the case of such an order consisting of just two layers. We call such orders Boolean layer cakes for the obvious graphical reason.
As ordered sets, they have been investigated from quite different points of view. It is the purpose of section 2 to survey the more recent literature for results on ordered sets of this type, to the extent we are aware of them. The major part of the work done is devoted to orders consisting of just two layers of a Boolean lattice, even with further constraints (adjacent layers, middle layers). Main themes are the order dimension, the existence of Hamiltonian cycles, the types of suborders, isotone selfmaps and jump numbers of such orders.
Some layer cakes are even lattices, see Fact 3.2.1. As far as we know, they have never been looked at from this point of view. A striking feature of lattices consisting of the 2-element and the 3-element subsets of some finite set (plus top and bottom) is their large number of large (in a sense to be specified) sublattices. We devote section 3 to the study of some properties of such lattices.
Consequently, this paper is of a somewhat hybrid nature. Surveyed results are cited with full bibliographical references but no hints to proof. ``Fact'' is used to label statements which we feel are known but for which we don't have a direct reference; here, sketches of proofs are included. The traditional ``Proposition-Proof''scheme is reserved to statements which we believe to be new.
Our notation is fairly standard. All ordered sets considered are finite unless
ecplicitly stated. We write 
for the Boolean lattice with n atoms
and think of it as being explicitly realized as the collection of all
subsets of an n-element set X, with set inclusion as order relation. An
i-set is any set with i elements. So the elements of are
arranged in layers, the i-th layer consisting of all i-subsets of X
( 
). Given any two natural numbers 
, we write [r,s]
for 
.
Ordered sets alias orders are written as 
or shortly P if there
is no need to emphasize the order relation 
. For 
, 
means that x and y are not comparable, 
that x is a lower
neighbor of y. Given r natural numbers 
, we
write 
for the Boolean layer cake obtained from
by selecting layers 
(and the induced order), and
call it a r-layer cake. If
we think of a specific n-set X used to represent , we say that
is realized on X.