Four-element generating sets of partition lattices and their direct products

Abstract

Let n > 3 be a natural number. By a 1975 result of H. Strietz, the lattice Part(n) of all partitions of an n-element set has a four-element generating set. In 1983, L. Zadori gave a new proof of this fact with a particularly elegant construction. Based on his construction from 1983, the present paper gives a lower bound on the number ν(n) of four-element generating sets of Part(n). We also present a computer-assisted statistical approach to ν(n) for small values of n. In his 1983 paper, L. Zadori also proved that for n ≥ 7, the lattice Part(n) has a four-element generating set that is not an antichain. He left the problem whether such a generating set for n ∈ {5, 6} exists open. Here we solve this problem in negative for n = 5 and in affirmative for n = 6. Finally, the main theorem asserts that the direct product of some powers of partition lattices is four-generated. In particular, by the first part of this theorem, Part(n1) × Part(n2) is four-generated for any two distinct integers n1 and n2 that are at least 5. The second part of the theorem is technical but it has two corollaries that are easy to understand. Namely, the direct product Part(n) × Part(n + 1) × · · · × Part(3n − 14) is four-generated for each integer n ≥ 9. Also, for every positive integer u, the u-th the direct power of the direct product Part(n) × Part(n + 1) × · · · × Part(n + u − 1) is four-generated for all but finitely many n. If we do not insist on too many direct factors, then the exponent can be quite large. For example, our theorem implies that the 10127-th direct power of Part(1011) × Part(1012) × · · · × Part(2020) is four-generated.