Atoms and coatoms in three-generated lattices
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 223 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.