## Rocky Mountain Journal of Mathematics

- Rocky Mountain J. Math.
- Volume 48, Number 8 (2018), 2503-2542.

### Finite atomic lattices and their monomial ideals

Peng He and Xue-ping Wang

#### 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 of 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.

#### Article information

**Source**

Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2503-2542.

**Dates**

First available in Project Euclid: 30 December 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.rmjm/1546138819

**Digital Object Identifier**

doi:10.1216/RMJ-2018-48-8-2503

**Mathematical Reviews number (MathSciNet)**

MR3894991

**Zentralblatt MATH identifier**

06999272

**Subjects**

Primary: 06D05: Structure and representation theory 13D02: Syzygies, resolutions, complexes

**Keywords**

Monomial ideal finite atomic lattice coordinatization weak coordinatization super-atomic lattice labeling

#### Citation

He, Peng; Wang, Xue-ping. Finite atomic lattices and their monomial ideals. Rocky Mountain J. Math. 48 (2018), no. 8, 2503--2542. doi:10.1216/RMJ-2018-48-8-2503. https://projecteuclid.org/euclid.rmjm/1546138819