Rocky Mountain Journal of Mathematics

Finite atomic lattices and their monomial ideals

Peng He and Xue-ping Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 30 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • D. Bayer, I. Peeva and B. Sturmfels, Monomial resolutions, Math. Res. Lett. 5 (1998), 31–46.
  • G. Birkhoff and O. Frink, Representations of lattices by sets, Trans. Amer. Math. Soc. 64 (1948), 299–316.
  • J.R. Büchi, Representation of complete lattices by sets, Portugal. Math. 11 (1952), 151–167.
  • Timothy B.P. Clark, Poset resolutions and lattice-linear monomial ideals, J. Algebra 323 (2010), 899–919.
  • P. Crawley and R.P. Dilworth, Algebraic theory of lattices, Prentice Hall, Englewood Cliffs, NJ, 1973.
  • S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1–25.
  • V. Gasharov, I. Peeva and W. Volkmar, The ${\lcm}$-lattice in monomial resolutions, Math. Res. Lett. 6 (1999), 521–532.
  • L. Katthän, Stanley depth and simplicial spanning trees, J. Alg. Combin. 42 (2015), 507–536.
  • S. Mapes, Finite atomic lattices and their relationship to resolutions of monomial ideals, Ph.D. dissertation, Columbia University, New York, 2009.
  • S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra 379 (2013), 259–276.
  • S. Mapes and L. Piechnik, Constructing monomial ideals with a given minimal resolution, Rocky Mountain J. Math. 47 (2017), 1963–1985.
  • J. Phan, Order properties of monomial ideals and their free resolutions, Ph.D. dissertation, Columbia University, New York, 2006.
  • A.B. Tchernev, Representations of matroids and free resolutions for multigraded modules, Adv. Math. 208 (2007), 75–134.
  • M. Velasco, Minimal free resolutions that are not supported by a CW-complex, J. Algebra 319 (2008), 102–114.