# Rank Bounded Hibi Subrings for Planar Distributive Lattices

## TL;DR

This paper investigates the algebraic properties of rank bounded Hibi subrings derived from planar distributive lattices, showing conditions for quadratic Gr"obner bases and linear resolutions, thus advancing understanding of their algebraic structure.

## Contribution

It characterizes when rank bounded Hibi subrings have quadratic Gr"obner bases and linear resolutions in planar distributive lattices, extending previous algebraic results.

## Key findings

- Bounded Hibi subrings of planar lattices have quadratic Gr"obner bases.
- Characterization of planar lattices where all rank bounded subrings have linear resolutions.
- Identification of linearly related Hibi subrings within the lattice framework.

## Abstract

Let $L$ be a distributive lattice and $R[L]$ the associated Hibi ring. We show that if $L$ is planar, then any bounded Hibi subring of $R[L]$ has a quadratic Gr\"obner basis. We characterize all planar distributive lattices $L$ for which any proper rank bounded Hibi subring of $R[L]$ has a linear resolution. Moreover, if $R[L]$ is linearly related for a lattice $L$, we find all the rank bounded Hibi subrings of $R[L]$ which are linearly related too.