Boolean layer cakes. Proceedings ORDAL '96

Abstract

A Boolean layer cake is an ordered set obtained from a Boolean lattice 2 n by selecting any number of complete level sets from 2 n and endowing their set union with the order inherited from 2 n . The first Boolean layer cake considered in the literature seems to be the set of all atoms and of all coatoms of 2 n in Dushnik and Miller (1941, Amer. J. Math. 63, 600–610) — what is known today as the standard example of an n-dimensional ordered set. Our intention is 1. (i) to survey the more recent literature on Boolean layer cakes and 2. (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 2 n , and within that part, the major theme is their dimension. If n is odd, the two middle levels of 2 n 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 2 n 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 2 n 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.