Our tour d'horizon of Boolean layer cakes has probably raised more open questions than it has provided answers. These questions have mostly been specified at their proper place in section 2 and are accessible through the literature cited there. In fact, only the jump number problem (see 2.5) has been solved completely for arbitrary layer cakes, whereas the other questions considered have been answered only for restricted classes of such orders. This is even more the case for the results reported on in section 3: It would be interesting to know what can be carried over to lattices of type P(0,k,k+1,n;n) for 0<k<k+1<n or even the general layer cake lattices given by 3.2.1.
Boolean layer cakes are defined by picking certain subsets - the level sets -
from a Boolean algebra. One might feel tempted to see what happens if we
replace the level sets by similarly defined subsets of a Boolean algebra.
Possible candidates include (disjoint) cutsets which always contain a
maximal antichain as proved by Duffus, Sands and Winkler in [16]. Also,
the Boolean algebra in question might be cut up into antichains in a different
way: Lonc [41] has proved that for any 0<m<n, the truncated Boolean
lattice 
may be partitioned into antichains of size
m except for at most m-1 elements which also form an antichain.
It is also quite natural to consider nonboolean settings: Level sets of distributive lattices have been considered by Gierz and Hergert in [25] and [26] in the case of distributive lattices of breadth 3. These sets have interesting geometric properties and arise also in the context of the so-called bandwidth problem. It would be interesting to see what the properties of layer cakes cut from distributive lattices are.
Finally, we mention another concept of ``layered'' ordered sets: Call an
ordered set 
a k-layer order ( 
) iff P can be
partitioned into k antichains 
such that (1) every element of
is below every element of 
for 
and 
,
and (2) no element of is above an element of 
for
. Kleitman and Rothschild prove in [35] that almost all
n-element orders are 3-layer orders, and that almost all 3-layer orders on
n elements have about n/2 elements in the middle layer and about n/4
elements in the outer two layers. Such orders habe been used recently by
Brightwell, Prömel and Steger in [6] to estimate the average number
of linear extensions of a partial order.
Our last topic is the real thing; accompanied by a nice cup of coffee it might help to digest the contents of this survey. The following construction for a layer cake is fairly standard; see, e.g., Crocker [8].
The most famous consequence of Construction 4.1.1 is given by
