Stanley depth and the lcm-lattice

TL;DR

This paper demonstrates that the Stanley depth and usual depth of monomial ideal quotients are fundamentally linked to the structure of their lcm-lattice, enabling simplified proofs and new insights.

Contribution

It establishes that both depths are determined by the lcm-lattice and introduces a generalization of polarization results, along with a classification of monomial ideals based on their lcm-lattice.

Findings

Depths are monotonic with respect to lcm-lattice maps

Provided a uniform proof framework for Stanley depth results

Characterized all monomial ideals with a given lcm-lattice

Abstract

In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients $I/J$ of monomial ideals $J\subset I$, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in the reference [IKMF14]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.