Atoms in four-element generating sets of partition lattices

Abstract

Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove that each element of Part($n$) with height one or two (in particular, each atom) belongs to a four-element generating set. Furthermore, our construction leads to a concise and easy proof of a 1996 result of the author stating that the lattice of partitions of a countably infinite set is four-generated as a complete lattice. In a recent paper "Generating Boolean lattices by few elements and exchanging session keys", see https://doi.org/10.30755/NSJOM.16637, the author establishes a connection between cryptography and small generating sets of some lattices, including Part($n$). Hence, it is worth pointing out that by combining a construction given here with a recent paper by the author, "Four-element generating sets with block count width at most two in partition lattices", available at https://tinyurl.com/czg-4gw2, we obtain many four-element generating sets of Part($n$).