ACTA issues

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

Delbrin Ahmed, Gábor Czédli

Acta Sci. Math. (Szeged) 87:3-4(2021), 415-427

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 Zádori's method is used.

DOI: 10.14232/actasm-021-303-1

AMS Subject Classification (1991): 06B99

Keyword(s): quasiorder lattice, lattice of preorders, minimum-sized generating set, four-generated lattice, $(1+1+2)$-generated lattice, Zádori's method

received 3.5.2021, revised 19.5.2021, accepted 19.5.2021. (Registered under 53/2021.)