Finite atomic lattices and their monomial ideals

TL;DR

This paper explores the structure of finite atomic lattices and their monomial ideals through weak coordinatizations, super-atomic lattices, and specific labelings, providing new characterizations and algorithms.

Contribution

It introduces weaker notions of coordinatizations, characterizes all such weak coordinatizations, and develops an algorithm to enumerate super-atomic lattices in finite atomic lattices.

Findings

Characterization of all weak coordinatizations of finite atomic lattices.

Development of an algorithm to compute all super-atomic lattices in l(n).

Conditions for specific labelings to be (weak) coordinatizations.

Abstract

This paper primarily studies monomial ideals by their associated lcm-lattices. It first introduces notions of weak coordinatizations of finite atomic lattices which have weaker hypotheses than coordinatizations and shows the characterizations of all such weak coordinatizations. It then defines a finite super-atomic lattice in $\mathcal{L}(n)$, investigates the structures of $\mathcal{L}(n)$ by their super-atomic lattices and proposes an algorithm to calculate all the super-atomic lattices in $\mathcal{L}(n)$. It finally presents a specific labeling of finite atomic lattice and obtains the conditions that the specific labelings of finite atomic lattices are the weak coordinatizations or the coordinatizations by using the terminology of super-atomic lattices.