A family of monomial ideals with the persistence property

TL;DR

This paper introduces a family of monomial ideals derived from path graphs that possess the persistence property, explicitly describing their associated primes and stability indices, advancing understanding in combinatorial commutative algebra.

Contribution

The paper defines a new family of monomial ideals from path graphs and proves they have the persistence property, providing explicit descriptions of associated primes and stability indices.

Findings

All ideals in the family have the persistence property.

Explicit description of associated primes for all powers of the ideals.

Determination of the index of stability and stable associated primes.

Abstract

In this paper we introduce a family of monomial ideals with the persistence property. Given positive integers $n$ and $t$, we consider the monomial ideal $I=Ind_t(P_n)$ generated by all monomials $\textbf{x} ^F$, where $F$ is an independent set of vertices of the path graph $P_n$ of size $t$, which is indeed the facet ideal of the $t$-th skeleton of the independence complex of $P_n$. We describe the set of associated primes of all powers of $I$ explicitly. It turns out that any such ideal $I$ has the persistence property. Moreover the index of stability of $I$ and the stable set of associated prime ideals of $I$ are determined.