Irreducible divisor simplicial complexes

Abstract

For an integral domain $D$, the irreducible divisor graph $G_D\left(x\right)$ of a nonunit $x\in D$ gives a visual representation of the factorizations of $x$. Here we consider a higher-dimensional generalization of this notion, the irreducible divisor simplicial complex $S_D\left(x\right)$. We show how this new structure is a true generalization of $G_D\left(x\right)$, and show that it often carries more information about the element $x$ and the domain $D$ than its two-dimensional counterpart.