A lower bound for congruence representations (Q1368669)

For every distributive lattice \(D\) with \(n\) join-irreducible elements there exists a finite lattice \(L\) such that the congruence lattice of \(L\) is isomorphic to \(D\) and its size is minimal. The maximum of these numbers is denoted by cr\((n)\). In Proc. Am. Math. Soc. 123, 2619-2623 (1995; Zbl 0842.06007), \textit{G. Grätzer}, \textit{H. Lakser} and \textit{E. T. Schmidt} proved that \(\text{cr} (n)<3 (n+1)^2\). Later, \textit{G. Grätzer}, \textit{I. Rival} and \textit{N. Zaguia} proved that the exponent is sharp [ibid. 123, 1959-1961 (1995; Zbl 0822.06009)]. In an unpublished paper, Y. Zhang gave a better lower bound for \(\text{cr}(n)\). Using a result of \textit{R. Freese} [ibid. 125, 3457-3463 (1997; Zbl 0885.06003)] the authors give the lower bound \((1/16) \cdot (n^2/ \log_2(n))\), which is much better that Zhang's one.