On atoms and their density in the monoid of monomial ideals

@misc{bogdanovic2026atoms,
  title = {{On atoms and their density in the monoid of monomial ideals}},
  author = {Nikola Bogdanovic and Laura Cossu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17202},
  eprint = {2609.17202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17202},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On atoms and their density in the monoid of monomial ideals
AU  - Nikola Bogdanovic
AU  - Laura Cossu
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17202
UR  - https://arxiv.org/abs/2609.17202
AB  - We study the structure and distribution of atoms in the monoid ℳon(R) of nonzero monomial ideals of the polynomial ring R=K[X_1,…, X_N], with N≥ 2. We characterize the atoms with at most four minimal generators and introduce the density d(N,μ) of atoms among monomial ideals with μ minimal generators. For μ≤ 4, we prove the existence of this density and determine its asymptotic behavior as N→∞. In the bivariate case, we also obtain a uniform lower bound for the density of atoms in a divisor-closed submonoid of ℳon(R) generated by equigenerated ideals.
ER  -