Some Properties of the Lattice $\mathbb{L}_k$

Abstract

It is the aim of this first section above $\mathbb{L}_k$ to basically elaborate on the differences between $\mathbb{L}_2$, which we already completely determined in Chapter 3, and $\mathbb{L}_k$ for k ≥ 3. In particular, we deal with cardinality statements (both about $\mathbb{L}_k$, than also about chains and antichains in $\mathbb{L}_k$) and with the embedding of $\mathbb{L}_k$ into $\mathbb{L}_{k'}$. Furthermore, the question, which cardinalities have the lattices $\mathbb{L}_k^ \downarrow (A)$ for maximal classes A of P_k, is clarified. In the last section of chapter, there are remarks about “strategies” during the determination of “manageable” sublattices of $\mathbb{L}_k$. In addition, two examples of theorems are proven as one can determine certain subsets of $\mathbb{L}_k$.

Notice that one can find further properties of $\mathbb{L}_k$ , for instance, in [Bul-K-S-S 95], [Bul-K-S-S-S 2001], [Bul 99a,b], and [Bul 2001].