On the algorithmic construction of the 1960 sectional complement

arXiv2208.03745MaRDI QIDQ6407205

Athena Nguyen, G. Grätzer, G. Klus

Publication date: 7 August 2022

Abstract: In 1960, G. Gr"atzer and E.,T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice $L$. For $uleqv$ in $L$, they constructed a sectional complement, which is now called the emph{1960 sectional complement}. In 1999, G. Gr"atzer and E.,T. Schmidt discovered a very simple way of constructing a sectional complement in the ideal lattice of a chopped lattice made up of two sectionally complemented finite lattices overlapping in only two elements -- the Atom Lemma. The question was raised whether this simple process can be generalized to an algorithm that finds the 1960 sectional complement. In 2006, G.~Gr"atzer and M. Roddy discovered such an algorithm -- allowing a wide latitude how it is carried out. In this paper we prove that the wide latitude apparent in the algorithm is deceptive: whichever way the algorithm is carried out, it~produces the same sectional complement. This solves, in fact, Problems 2 and 3 of the Gr"atzer-Roddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper.

Mathematics Subject Classification ID

Complemented lattices, orthocomplemented lattices and posets (06C15) Lattice ideals, congruence relations (06B10)

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