Several inequalities regarding sdepth

TL;DR

This paper establishes bounds for the Stanley depth of various operations on monomial ideals and explores equivalent forms of the Stanley Conjecture related to these ideals.

Contribution

It provides new bounds for Stanley depth of sums, intersections, colon ideals, and quotients of monomial ideals, and offers equivalent formulations of the Stanley Conjecture.

Findings

Derived bounds for sdepth of I+J, I∩J, and related modules.

Presented equivalent forms of Stanley Conjecture for monomial ideals.

Enhanced understanding of Stanley depth behavior in monomial ideal operations.

Abstract

We give several bounds for $sdepth_S(I+J)$, $sdepth_S(I\cap J)$, $sdepth_S(S/(I+J))$, $sdepth_S(S/(I\cap J))$, $sdepth_S(I:J)$ and $sdepth_S(S/(I:J))$ where $I,J\subset S=K[x_1,...,x_n]$ are monomial ideals. Also, we give several equivalent forms of Stanley Conjecture for $I$ and $S/I$, where $I\subset S$ is a monomial ideal.