Antichain cutsets in real-ranked lattices

TL;DR

This paper demonstrates that in certain real-ranked lattices, antichain cutsets can be characterized as level sets under suitable gradings, extending to various continuous lattice structures.

Contribution

It establishes that antichain cutsets in rank supersolvable lattices and several continuous lattices are equivalent to level sets with appropriate gradings.

Findings

Antichain cutsets are level sets in rank supersolvable lattices.

Extension of the result to measurable Boolean, continuous partition, and projective geometry lattices.

Provides a method to find gradings where cutsets are level sets.

Abstract

We show that in a rank supersolvable lattice that is graded by a bounded real interval, any antichain cutset is a level set for some appropriately constructed grading. As a consequence, given an antichain cutset in any of the measurable Boolean lattice, a continuous partition lattice, or a continuous projective geometry, we may find a grading in which the cutset is a level set.