On Some Properties of LCM-Lattices of Edge Ideals of k-Uniform Hypergraphs

TL;DR

This paper explores the combinatorial and algebraic properties of lcm-lattices associated with edge ideals of hypergraphs, focusing on conditions for Boolean, modular, or complemented structures and effects of polarization.

Contribution

It establishes new conditions for the lcm-lattice properties of hypergraph edge ideals and extends results to product lattices and polarization effects.

Findings

Identifies conditions for Boolean, modular, and complemented lcm-lattices.

Extends results to product of lcm-lattices in the complemented case.

Studies the impact of polarization on lcm-lattices.

Abstract

In this article, we investigate the combinatorial and algebraic properties of the lcm-lattice associated with the edge ideal of a hypergraph. Let $\H$ be a hypergraph, $I(\H)$ its corresponding edge ideal in a polynomial ring in $n$ variables, and $\mathrm{Icm}(I(\H))$ the associated lcm-lattice. We establish conditions under which the lcm-lattice of an edge ideal is Boolean, modular, or complemented. Furthermore, we extend these results to the case of the product of lcm-lattices in the complemented case. Additionally, we study the effects of polarization on the lcm-lattices of $I(\H)$ and its polarized ideal.