On the algorithmic construction of the 1960 sectional complement

TL;DR

This paper proves that a previously developed algorithm for constructing the 1960 sectional complement in finite lattices is deterministic, always producing the same unique complement, thus confirming its correctness and resolving open problems.

Contribution

It demonstrates that the algorithm's wide choices do not affect the outcome, ensuring a unique sectional complement consistent with the 1960 construction.

Findings

The algorithm always produces the same sectional complement regardless of choices.

The unique sectional complement obtained matches the 1960 sectional complement.

This confirms the algorithm's validity and solves related open problems.

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 $u \leq v$ 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.