ACTA issues

Notes on planar semimodular lattices. IV. The size of a minimal congruence lattice representation with rectangular lattices

G. Gr├Ątzer, E. Knapp

Acta Sci. Math. (Szeged) 76:1-2(2010), 3-26

Abstract. Let $D$ be a finite distributive lattice with $n$ join-irreducible elements. In Part III, we proved that $D$ can be represented as the congruence lattice of a special type of planar semimodular lattices of $O(n^3)$ elements, we called {\it rectangular}. In this paper, we show that this result is best possible. Let $D$ be a finite distributive lattice whose order of join-irreducible elements is a balanced bipartite order on $n$ elements. Then any rectangular lattice $L$ whose congruence lattice is isomorphic to $D$ has at least $k n^3$ elements, for some constant $k > 0$.

AMS Subject Classification (1991): 06C10, 06B10

Keyword(s): semimodular lattice, planar, congruence, rectangular

Received February 4, 2008, and in revised form February 11, 2009. (Registered under 53/2010.)