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For a finite poset (partially ordered set) $U$ and a natural number $n$, letSp$(U,n)$ denote the largest number of pairwise unrelated copies of $U$ in thepowerset lattice (AKA subset lattice) of an $n$-element set. If $U$ is thesingleton poset, then Sp$(U,n)$ was determined by E. Sperner in 1928; thisresult is well known in extremal combinatorics. Later, exactly orasymptotically, Sperner's theorem was extended to other posets by A. P. Dove,J. R. Griggs, G. O. H. Katona, D Nagy, J. Stahl, and W. T. Jr. Trotter. Wedetermine Sp$(U,n)$ for all finite posets with 0 and 1, and we give reasonableestimates for the ``V-shaped'' 3-element poset and the 4-element poset with 0and three maximal elements.  For a lattice $L$, let Gmin($L$) denote the minimum size of generating setsof $L$. We prove that if $U$ is the poset of the join-irreducible elements of afinite distributive lattice $D$, then the function $k\mapsto$ Gmin($D^k)$ isthe left adjoint of the function $n\mapsto$ Sp$(U,n)$. This allows us todetermine Gmin($D^k)$ in many cases. E.g., for a 5-element distributive lattice$D$, Gmin($D^{2023})=18$ if $D$ is a chain and Gmin($D^{2023})=15$ otherwise.  It follows that large direct powers of small distributive lattices areappropriate for our 2021 cryptographic authentication protocol.

Sperner theorems for unrelated copies of some partially ordered sets in  a powerset lattice and minimum generating sets of powers of distributive  lattices