Sperner Theorems for Unrelated Copies of Posets and Generating Distributive Lattices

For a finite poset (partially ordered set) \(U\) and a natural number \(n\), let \(S(U,n)\) denote the largest number of pairwise unrelated copies of \(U\) in the powerset lattice (AKA subset lattice) of an \(n\)-element set. If \(U\) is the singleton poset, then \(S(U,n)\) was determined by E. Sperner in 1928; this result is well known in extremal combinatorics. Later, exactly or asymptotically, Sperner's theorem was extended to other posets by A.P. Dove, J.R. Griggs, G.O.H. Katona, D.J. Stahl, and W.T.Jr. Trotter. We determine \(S(U,n)\) for all finite posets with 0 and 1, and we give reasonable estimates for the "\(V\)-shaped" 3-element poset and, mainly, for the 4-element poset with 0 and three maximal elements. For a lattice \(L\), let \(GₘᵢₙL\) denote the minimum size of generating sets of \(L\). We prove that if \(U\) is the poset of the join-irreducible elements of a finite distributive lattice \(D\), then the function \(k↦ Gₘᵢₙ{D^k}\) is the left adjoint of the function \(n↦ S(U,n)\). This allows us to determine \(Gₘᵢₙ{D^k}\) in many cases. E.g., for a 5-element distributive lattice \(D\), \(Gₘᵢₙ{D²⁰²³}=18\) if \(D\) is a chain and \(Gₘᵢₙ{D²⁰²³}=15\) otherwise. The present paper, another recent paper, and a 2021 one indicate that large direct powers of small distributive lattices could be of interest in cryptography.