Minimum-sized generating sets of the direct powers of free distributive lattices

Abstract

For a finite lattice \(L\), let Gm(\(L\)) denote the least \(n\) such that \(L\) can be generated by \(n\) elements. For integers \(r>2\) and \(k>1\), denote by FD\((r)^k\) the \(k\)-th direct power of the free distributive lattice FD(\(r\)) on \(r\) generators. We determine Gm(FD\((r)^k\)) for many pairs \((r,k)\) either exactly or with good accuracy by giving a lower estimate that becomes an upper estimate if we increase it by 1. For example, for \((r,k)=(5,25\,000)\) and \((r,k)=(20,\ 1.489\cdot 10^{1789})\), Gm(FD\((r)^k\)) is \(22\) and \(6\,000\), respectively. To reach our goal, we give estimates for the maximum number of pairwise unrelated copies of some specific posets (called full segment posets) in the subset lattice of an \(n\)-element set. In addition to analogous earlier results in lattice theory, a connection with cryptology is also mentioned among the motivations.