Support posets of some monomial ideals

TL;DR

This paper characterizes certain posets that can be realized as support posets of monomial ideals, providing explicit constructions and exploring their properties, with applications to forests and series-parallel ideals.

Contribution

It identifies classes of posets realizable as support posets of monomial ideals and constructs explicit examples, advancing understanding of the structure of these ideals.

Findings

Certain posets can be explicitly realized as support posets.

Support posets of specific monomial ideal families are characterized.

Application to forests and series-parallel ideals demonstrates practical relevance.

Abstract

The support poset of a monomial ideal $I\subseteq\mathbf{k}[x_1,\dots,x_n]$ encodes the relation between the variables $x_1,\dots,x_n$ and the minimal monomial generators of $I$. It is known that not every poset is realizable as the support poset of some monomial ideal. We describe some posets $P$ for which we can explicitly find at least one monomial ideal $I_P$ such that $P$ is the support poset of $I_P$. Also, for some families of monomial ideals we describe their support posets and study their properties. As an example of application we examine the relation between forests and series-parallel ideals.