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

TL;DR

This paper extends Sperner's theorem to various finite posets within powerset lattices, determines minimal generating sets for powers of distributive lattices, and explores applications in cryptography.

Contribution

It provides explicit calculations of Sperner numbers for specific posets and links these to minimal generating sets of lattice powers, advancing combinatorial and lattice theory.

Findings

Determined Sp(U,n) for all finite posets with 0 and 1.

Estimated Sp(U,n) for certain 3- and 4-element posets.

Calculated Gmin(D^k) for specific distributive lattices, with applications in cryptography.

Abstract

For a finite poset (partially ordered set) $U$ and a natural number $n$, let Sp$(U,n)$ 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 Sp$(U,n)$ 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 Nagy, J. Stahl, and W. T. Jr. Trotter. We determine Sp$(U,n)$ for all finite posets with 0 and 1, and we give reasonable estimates for the ``V-shaped'' 3-element poset and the 4-element poset with 0 and three maximal elements. For a lattice $L$, let Gmin($L$) 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 $k\mapsto$ Gmin($D^k)$ is the left adjoint of the function $n\mapsto$ Sp$(U,n)$. This allows us to determine 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 are appropriate for our 2021 cryptographic authentication protocol.