Antichain cutsets in real-ranked lattices

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.

💡 Research Summary

The paper investigates antichain cutsets in lattices that are graded by real intervals, extending a well‑known discrete result to a continuous setting. In finite supersolvable lattices every antichain cutset coincides with a level set of the natural rank function. The authors ask whether an analogous statement holds for continuous lattices where many different rank functions (gradings) may exist.

Key definitions: an R‑grading of a poset L is a map ρ : L → R (R a subset of the real line or the extended reals) that restricts to an order‑isomorphism on every maximal chain. An element m is rank‑modular if for all x, ρ(x ∨ m)+ρ(x ∧ m)=ρ(x)+ρ(m). A maximal chain consisting entirely of rank‑modular elements is called a chief chain; a lattice possessing such a chain is termed rank‑supersolvable.

The paper’s main theorem (Theorem 1.2) states: if R is a bounded real interval, L is a rank‑supersolvable lattice equipped with an R‑grading ρ, and A⊂L is an antichain cutset (every maximal chain meets A), then there exists another grading σ of L such that A becomes a σ‑level set. The construction of σ proceeds by exploiting the continuity properties of the maps λ↦ρ(m_λ∧z) and λ↦ρ(m_λ∨z) for elements of the chief chain m_λ and arbitrary z. Lemma 3.3 and Lemma 3.4 show these maps are Lipschitz, hence continuous. Using this, the authors define a monotone, continuous transformation φ : R→R that collapses the interval of ρ‑values occupied by A to a single point while preserving order elsewhere. Setting σ(x)=φ(ρ(x)) yields a new grading under which A is exactly a level set.

When R is unbounded, Lemma 3.7 and Lemma 3.8 allow the addition of artificial top or bottom elements to L, extending the grading to the extended reals and preserving the Lipschitz property, so the same construction works.

The paper applies the theorem to three important families of continuous lattices:

1. Measurable Boolean lattice B(X, μ) on a finite measure space (X, μ). The measure μ itself provides a