A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE

Abstract

Let $\lfloor x \rfloor$ and $\lceil x\rceil$ denote the lower integer part and the upper integer part of a real number $x$, respectively. Our main goal is to construct four partitions of a finite set $A$ with $n\geq 7$ elements such that each of the four partitions has exactly $\lceil n/2\rceil$  blocks and any other partition of $A$ can be obtained from the given four by forming joins and meets in a finite number of steps. We do the same with $\lceil n/2\rceil-1$ instead of $\lceil n/2\rceil$, too. To situate the paper within lattice theory, recall that the partition lattice $\mathrm{Eq}(A)$ of a set $A$ consists of all partitions (equivalently, of all equivalence relations) of $A$. For a natural number $n$, $[n]$ and $\mathrm{Eq}(n)$ will stand for $\{1,2,\dots,n\}$ and $\mathrm{Eq}([n])$, respectively. In 1975, Heinrich Strietz proved that, for any natural number $n\geq 3$, $\mathrm{Eq} (n)$ has a four-element generating set; half a dozen papers have been devoted to four-element generating sets of partition lattices since then. We give a simple proof of his just-mentioned result. We call a generating set $X$ of $\mathrm{Eq}(n)$ horizontal if each member of $X$ has the same height, denoted by $h(X)$, in $\mathrm{Eq} (n)$; no such generating sets have been known previously. We prove that for each natural number $n\ge 4$, $\mathrm{Eq}(n)$ has two four-element horizontal generating sets $X$ and $Y$ such that $h(Y)=h(X) +1$; for $n\geq 7$, $h(X)= \lfloor n/2 \rfloor$