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

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)...

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Main Authors: Article 2021 Acta scientiarum mathematicarum 87 No. 3-4 Matematika, Algebra 10.14232/actasm-021-303-1 http://acta.bibl.u-szeged.hu/75848
Summary: 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 ≥ 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. 415-427 2064-8316