Antichain cutsets of strongly connected posets

TL;DR

This paper generalizes the characterization of antichain cutsets from finite Boolean lattices to a broad class of strongly connected posets and related structures, revealing new structural insights.

Contribution

It extends the known characterization of antichain cutsets to strongly connected posets of locally finite height and related structures, broadening the scope of previous results.

Findings

Antichain cutsets in strongly connected posets are characterized by level sets.

The results apply to semimodular, supersolvable lattices, and Bruhat orders.

A generalization to strongly connected hypergraphs is also provided.

Abstract

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we get such a characterization for semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We also consider a generalization to strongly connected hypergraphs having finite edges.