A Pair of Four-Element Horizontal Generating Sets of a Partition Lattice

Let \( x \) and \( x \) 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 \(n≥ 7\) elements such that each of the four partitions has exactly \( n/2 \) 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 \( n/2-1\) instead of \( n/2\), too. To situate the paper within lattice theory, recall that the partition lattice \(Eq(A)\) of a set \(A\) consists of all partitions (equivalently, of all equivalence relations) of \(A\). For a natural number \(n\), \([n]\) and \(Eq(n)\) will stand for \(\{1,2,,n\}\) and \(Eq([n])\), respectively. In 1975, Heinrich Strietz proved that, for any natural number \(n≥ 3\), \(Eq (n)\) 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 \(Eq(n)\) horizontal if each member of \(X\) has the same height, denoted by \(h(X)\), in \(Eq (n)\); no such generating sets have been known previously. We prove that for each natural number \(n≥ 4\), \(Eq(n)\) has two four-element horizontal generating sets \(X\) and \(Y\) such that \(h(Y)=h(X) +1\); for \(n≥ 7\), \(h(X)= n/2 \).