Antichain cutsets of real-ranked supersolvable lattices

A (nontrivial) antichain cutset in a poset is an antichain that cuts every maximal chain into two parts.  In a finite, ranked, nice-enough poset such as the Boolean lattice or partition lattice, every antichain cutset is a level set for the rank.  In a continuously ranked lattice, such as the measurable Boolean lattice on [0,1], there are generally many choices of rank function, having different families of level sets.

In recent work with Stephan Foldes, we show that for any antichain cutset in a real-ranked supersolvable lattice, there is a rank function (generally different from the initially-presented one) under which the cutset is a level set.