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

TL;DR

This paper determines the minimum number of generators needed for many direct powers of free distributive lattices, providing exact values or bounds, with implications for lattice theory and cryptology.

Contribution

It offers new exact and estimated values for the minimal generating sets of direct powers of free distributive lattices, extending previous results and connecting to cryptology.

Findings

Gm(FD(r)^k) values determined for many (r,k) pairs

Provides bounds and estimates for the number of unrelated poset copies

Establishes connections between lattice generation and cryptology

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 $300$ and $6000$, 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.