(1+1+2)-generated lattices of quasiorders

TL;DR

This paper proves that the lattice of all quasiorders on an n-element set is (1+1+2)-generated for specific values of n, extending previous results and identifying twenty-four new such cases.

Contribution

The paper extends the class of n for which Quo(n) is (1+1+2)-generated, using an extended Zádori's method to include new values of n.

Findings

Quo(3) is (1+1+2)-generated trivially

Quo(6) is (1+1+2)-generated with 209,527 elements

Quo(n) is (1+1+2)-generated for n=11 and all n≥13

Abstract

A lattice is $(1+1+2)$-generated if it has a four-element generating set such that exactly two of the four generators are comparable. We prove that the lattice Quo$(n)$ of all quasiorders (also known as preorders) of an $n$-element set is $(1+1+2)$-generated for $n=3$ (trivially), $n=6$ (when Quo(6) consists of $209\,527$ elements), n=11, and for every natural number $n\geq 13$. In 2017, the second author and J. Kulin proved that Quo$(n)$ is $(1+1+2)$-generated if either $n$ is odd and at least $13$ or $n$ is even and at least $56$. Compared to the 2017 result, this paper presents twenty-four new numbers $n$ such that Quo$(n)$ is $(1+1+2)$-generated. Except for Quo(6), an extension of Z\'adori's method is used.