On the algorithmic construction of the 1960 sectional complement

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.