On atoms and their density in the monoid of monomial ideals

We study the structure and distribution of atoms in the monoid $\mathcal{M}\mathrm{o}\mathrm{n}(R)$ of nonzero monomial ideals of the polynomial ring $R=K[X_1,\dots ,X_N]$, with $N\ge 2$. We characterize the atoms with at most four minimal generators and introduce the density $d(N,\mu )$ of atoms among monomial ideals with $\mu$ minimal generators. For $\mu \le 4$, we prove the existence of this density and determine its asymptotic behavior as $N\to \mathrm{\infty }$. In the bivariate case, we also obtain a uniform lower bound for the density of atoms in a divisor-closed submonoid of $\mathcal{M}\mathrm{o}\mathrm{n}(R)$ generated by equigenerated ideals.