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Antichain cutsets in real-ranked lattices

Stephan Foldes, Russ Woodroofe

TL;DR

The paper addresses how antichain cutsets in real-graded, rank-supersolvable lattices can always be realized as level sets under some grading. It introduces a constructive method to define a new grading sigma by analyzing a good chain and projecting to the cutset, proving A is a level set under sigma and sigma is a valid R-grading. The results extend to important continuous instances such as the measurable Boolean lattice, continuous partition lattices, and continuous projective geometries, under appropriate continuity assumptions. This yields a unifying approach to align combinatorial structure with graded decompositions in continuous lattice settings, with potential implications for analysis on these spaces.

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.

Antichain cutsets in real-ranked lattices

TL;DR

The paper addresses how antichain cutsets in real-graded, rank-supersolvable lattices can always be realized as level sets under some grading. It introduces a constructive method to define a new grading sigma by analyzing a good chain and projecting to the cutset, proving A is a level set under sigma and sigma is a valid R-grading. The results extend to important continuous instances such as the measurable Boolean lattice, continuous partition lattices, and continuous projective geometries, under appropriate continuity assumptions. This yields a unifying approach to align combinatorial structure with graded decompositions in continuous lattice settings, with potential implications for analysis on these spaces.

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.

Paper Structure

This paper contains 6 sections, 20 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and background
  3. Rank modularity and rank supersolvable lattices
  4. Continuity
  5. Real-graded vs discrete-graded supersolvable lattices
  6. Proof of Theorem \ref{['thm:Main']}

Key Result

Theorem 1.2

Let $R\subseteq\mathbb{R}$ be a bounded interval. If $L$ is a rank supersolvable lattice with respect to the $R$-grading $\rho$, and $A$ is an antichain cutset for $L$, then there is some grading $\sigma$ under which $A$ is a level set.

Figures (1)

  • Figure 4.1: Projection to $A$.

Theorems & Definitions (43)

  • Theorem 1.2
  • Corollary 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 33 more