ACTA issues

On the algorithmic construction of the 1960 sectional complement

G. Grätzer, G. Klus, A. Nguyen

Acta Sci. Math. (Szeged) 77:1-2(2011), 35-45

Abstract. In 1960, G. Grätzer 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 {\it1960 sectional complement}. In 1999, G. Grätzer 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. Grätzer 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 Grätzer--Roddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper.

AMS Subject Classification (1991): 06C15, 06B10

Keyword(s): sectionally complemented lattice, sectional complement, finite

Received September 26, 2009, and in revised form August 6, 2010. (Registered under 318/2009.)