## Rocky Mountain Journal of Mathematics

### Finite atomic lattices and their monomial ideals

#### 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

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

#### 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

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