Lower Bound for the Number of 4-Element Generating Sets of Direct Products of Two Neighboring Partition Lattices

Abstract

Abstract H. Strietz proved in 1975 that the minimum size of a generating set of the partition lattice Part(n) on the n-element set (n ≥ 4) equals 4. This classical result forms the foundation for this study. Strietz's results have been echoed by L. Zádori (1983), who gave a new elegant proof confirming the outcome. Several studies have indeed emerged henceforth concerning four-element generating sets of partition lattices. More recently more studies have presented the approach for the lower bounds on the number λ(n) of the four-element generating sets of Part(n) and statistical approach to λ(n) for small values of n. Also, G. Czédli and the present author have recently proved that certain direct products of partition lattices are also 4-generated. In a recent paper, G. Czédli has shown that this result has connection with information theory. On this basis, here we give a lower bound on the number ν(n) of 4-element generating sets of the direct product Part(n) × Part(n + 1) for n ≥ 7 using the results from previous studies. For n = 1, . . . , 5, we use a computer-aided approach; it gives exact values for n = 1, 2, 3, 4 but we need a statistical method for n = 5.