Atoms and coatoms in three-generated lattices

Gábor Czédli 

Novi Sad Journal of Mathematics, Tome 52 (2022) no. 2.

Résumé

In addition to the unique cover $\var M^+$ of the variety of modular lattices, we also deal with those twenty-three \emph{known} covers of $\var M^+$ that can be extracted from the literature. For $\var M^+$ and for each of these twenty-three known varieties covering it, we determine what the pair formed by the number of atoms and that of coatoms of a three-generated lattice belonging to the variety in question can be. Furthermore, for each variety $\var W$ of lattices that is obtained by forming the join of some of the twenty-three varieties mentioned above, that is, for $2^{23}$ possible choices of $\var W$, we determine how many atoms a three-generated lattice belonging to $\var W$ can have. The greatest number of atoms occurring in this way is only six. In order to point out that this need not be so for larger varieties, we construct a $47\,092$-element three-generated lattice that has exactly eighteen atoms. In addition to purely lattice theoretical proofs, which constitute the majority of the paper, some computer-assisted arguments are also presented.

Détail

Publié le : 2022-08-30

DOI : 10.30755/NSJOM.12402

Classification : 06B99
Keywords: Three-generated lattice, number of atoms, many atoms, 18 atoms in a 3-generated lattice, atom spectrum of a lattice