General Boolean layer cakes, that is, cakes obtained by picking an arbitrary collection of level sets from a Boolean algebra, seem to occur in the literature at just one point: In [24], Füredi and Reuter calculate their jump number and determine an optimal linear extension. The author is indebted to Klaus Reuter for bringing [24] to his attention.
Given an ordered set 
and a linear extension L of 
, a
jump (of L) is a pair (x,y) of elements of P such that 
in but
x is covered by y in L. Let s(P;L) be the number of jumps of L; then
the jump number of P, denoted by s(P), is the minimal value of s(P;L)
where L runs over all linear extensions L of . Call a linear extension
L of optimal iff s(P;L)=s(P). It is a nontrivial fact that
optimal linear extensions exist at all. Chapter 9 of [48] is a good
general reference.
Let 
be any subset of 
and suppose
. Write Q for the Boolean layer cake
. The main result of [24] states that
Suppose Q is realized as collection of subsets of 
. For
define 
iff max 
. Then
is a linear order on Q extending 
(a reverse lexicographic
ordering). It is shown in [24] that L is even optimal. The proof rests
on a result from extremal set theory which is itself is only accessible - up
to now - by methods of multilinear algebra.