Edgewise Cohen-Macaulay connectivity of partially ordered sets

TL;DR

This paper demonstrates that face lattices of convex polytopes exhibit a strong Cohen-Macaulay property, remaining Cohen-Macaulay after removing edges in any closed interval, and explores the concept of edgewise Cohen-Macaulay connectivity in posets.

Contribution

It introduces and investigates the concept of edgewise Cohen-Macaulay connectivity for posets, extending known properties of face lattices of convex polytopes.

Findings

Face lattices of convex polytopes are strongly Cohen-Macaulay.

Removing edges in any closed interval preserves Cohen-Macaulayness.

Discussion of examples and open questions in the area.

Abstract

The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen--Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen--Macaulay poset of the same rank. A corresponding notion of edgewise Cohen--Macaulay connectivity for partially ordered sets is investigated. Examples and open questions are discussed.