Four generators of an equivalence lattice with consecutive block counts

TL;DR

This paper proves that for finite sets of size six or at least eight, the equivalence lattice can be generated by four elements with consecutive block counts, and discusses historical links to quasiorder lattices.

Contribution

It establishes the existence of four-element generating sets with consecutive block counts for certain finite sets and explores historical connections between equivalence and quasiorder lattices.

Findings

Equ$(A)$ has a four-element generating set with consecutive block counts for |A|=6 or |A|≥8.

The paper connects equivalence lattices to quasiorder lattices historically.

Provides conditions under which such generating sets exist.

Abstract

The block count of an equivalence $\mu\in$ Equ$(A)$ is the number blnum$(\mu)$ of blocks of (the partition corresponding to) $\mu$. We say that $X=\{\mu_1,\mu_2,\mu_3,\mu_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$(\mu_{i+1})$ = blnum$(\mu_{1})+i$ for $i\in\{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices.