On the quasi-depth of squarefree monomial ideals and the sdepth of the monomial ideal of independent sets of a graph

TL;DR

This paper introduces the concept of quasi-depth to bound the Stanley depth of monomial ideal quotients and computes it for specific classes, also analyzing the Stanley depth of ideals from graph independent sets.

Contribution

It proposes the quasi-depth as a practical upper bound for Stanley depth and applies it to various squarefree monomial ideals and graph-related ideals.

Findings

Quasi-depth provides an effective upper bound for Stanley depth.

Computed quasi-depth for several classes of squarefree monomial ideals.

Analyzed Stanley depth of monomial ideals from graph independent sets.

Abstract

If $J\subset I$ are two monomials ideals, we give a practical upper bound for the Stanley depth of $J/I$, which we call it the \emph{quasi-depth} of $J/I$. Also, we compute the quasi-depth of several classes of square free monomial ideals. We also study the Stanley depth of the monomial ideal associated of the independent sets of a graph.