Atoms and coatoms in three-generated lattices

TL;DR

This paper classifies possible atom and coatom counts in three-generated lattices within certain varieties, identifies the maximum number of atoms in these lattices, and constructs a large example with 18 atoms, using theoretical and computational methods.

Contribution

It determines atom and coatom counts for three-generated lattices in specific varieties and constructs a large lattice with many atoms, extending understanding of lattice structures.

Findings

Maximum of six atoms in varieties formed by joins of 23 known covers.

Existence of a 47,092-element lattice with 18 atoms.

Combination of theoretical and computer-assisted proofs.

Abstract

In addition to the unique cover $M^+$ of the variety of modular lattices, we also deal with those twenty-three known covers of $M^+$ that can be extracted from the literature. For $M^+$ 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 $W$ 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 $W$, we determine how many atoms a three-generated lattice belonging to $W$ 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.