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arXiv · 2104.14653

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

Delbrin Ahmed·Gábor Czédli
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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.

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Delbrin Ahmed, Gábor Czédli. 2021-04-29. (1+1+2)-generated lattices of quasiorders. https://arxiv.org/abs/2104.14653

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