Two embedding theorems for finite lattices
Juris Hartmanis · Proceedings of the American Mathematical Society · 1956
Introduction. One of the important unsolved problems of lattice theory is the problem of embedding every finite lattice in a finite partition lattice. The partition lattice can be considered as the lattice of subspaces of a suitable geometry. By slightly loosening the conditions on this geometry it will be shown that every finite lattice can be embedded in the lattice of subspaces of a finite geometry. Using this result it will be shown that every finite lattice can be embedded in the lattice of all geometries on a finite set. The lattice of all geometries will be shown to be again the lattice of subspaces of a geometry and it will be seen that its structure is similar to that of a partition lattice. This reduces the above mentioned problem to the problem of embedding every finite lattice of geometries in a finite partition lattice.