Four-element generating sets of partition lattices and their direct products

TL;DR

This paper explores the minimal generating sets of partition lattices and their products, providing bounds, solving open problems, and demonstrating that certain large direct products are four-generated, with implications for algebraic structure understanding.

Contribution

It establishes new bounds on the number of four-element generating sets, solves open problems about generating sets for specific n, and proves that certain large direct products of partition lattices are four-generated.

Findings

Lower bounds on the number of four-element generating sets for Part(n)

Existence of non-antichain four-element generating sets for n=6

Certain large direct products of partition lattices are four-generated

Abstract

Let $n>3$ be a natural number. By a 1975 result of H. Strietz, the lattice Part$(n)$ of all partitions of an $n$-element set has a four-element generating set. In 1983, L. Z\'adori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number $\nu(n)$ of four-element generating sets of Part$(n)$. We also present a computer assisted statistical approach to $\nu(n)$ for small values of $n$. In his 1983 paper, L. Z\'adori also proved that for $n\geq 7$, the lattice Part$(n)$ has a four element generating set that is not an antichain. He left the problem whether such a generating set for $n\in\{5,6\}$ exists open. Here we solve this problem in negative for $n=5$ and in affirmative for $n=6$. Finally, the main theorem asserts that the direct product of some powers of partition lattices is four-generated. In particular, by the first part of this theorem, Part$(n_1)\times$ Part$(n_2)$ is four-generated for any two distinct integers $n_1$ and $n_2$ that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(3n-14)$ is four-generated for each integer $n\geq 9$. Also, for every positive integer $u$, the $u$-th the direct power of the direct product Part$(n)$ $\times$ Part$(n+1)$ $\times\dots\times$ Part$(n+u-1)$ is four-generated for all but finitely many $n$. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the $10^{127}$-th direct power of Part$(1011)$ $\times$ Part$(1012)$ $\times \dots \times$ Part$(2020)$ is four-generated.