In contrast to the preceding two subsections, we will briefly consider a
property of arbitrary lattice layer cakes here.
According to Fact 3.2.1 a Boolean layer cake P is a lattice iff
or 
with 
. This
lattice is nonboolean iff either 1<r or s<n-1 (or both). Taking up the
theme of subsection 2.4, we set out to characterize the
lattice automorphisms resp. endomorphisms of such nonboolean lattice Boolean
layer cakes. Endomorphisms will not be required to preserve top and
bottom.
Automorphisms are easy: A map 
is a lattice automorphism iff it is
an order automorphism. If r<s, P has at least two adjacent nontrivial
layers and so the only order automorphism of P are those induced by
permutations of the base set realizing P, by Proposition 2.4.1.
If r=s, then obviously 
with 
, where 
denotes the k-diamond (an antichain
of length k with top and bottom added). So the order automorphisms of P in
this case are those given by permutations of the k atoms of .
Proof:
Assume, without loss of generality, that with s<n-1
realized on a set 
(the case 1<r is handled dually). Consider a
nontrivial congruence relation 
on P. So there exist 
such that 
, 
and 
.
Assume 
. Hence 
. Pick any points 
and 
. It is not hard to see that we may find
such that 
, 
,
and 
, 
. Since is
a congruence, 
, that is,
by the arithmetic of P. But now the set intersection
of all such sets 
- that is, their infimum in P - is obviously
A, so we conclude that 
whenever A belongs to some
nontrivial -class.
If also 1<r, the dual argument will prove 
and
thus collapses top and bottom of P. If r=1, observe that
in this case. Now 
or 
. But
then 
by the first half of this proof and we are
done again.
So Proposition 2.4.4 takes a different form in the lattice setting:
