The Imbedding Problem for Modular Lattices

Abstract

It is trivially true that an arbitrary lattice may be imbedded in a complemented lattice. We need only adjoin a unit and null elements if they do not already exist and a single element which is a complement of each of the elements not the unit of null element. For distributive lattices, the imbedding problem is not so trivial, but is contained in the representation theorem which asserts that any distributive lattice is isomorphic with a ring of sets (Birkhoff (1), Mac Neille (1)). The corresponding problem of imbedding a modular lattice in a complemented modular lattice is an outstanding problem in lattice theory. We exhibit here an example of a modular lattice which cannot be imbedded in any complemented modular lattice. However we will be concerned primarily with the isometric problem for finite dimensional modular lattices; that is, the problem of imbedding a finite dimensional modular lattice in a complemented modular lattice of the same dimensionality.