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

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, $I(\stackrel{˝}{)}$ its corresponding edge ideal in a polynomial ring in $n$ variables, and $\mathrm{I}\mathrm{c}\mathrm{m}(I(\stackrel{˝}{)})$ 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 $I(\stackrel{˝}{)}$ and its polarized ideal.