The Role of Gluing Constructions in Modular Lattice Theory
Alan Day, Ralph S. Freese · Birkhäuser Boston eBooks · 1990
In the 1930’s and 1940’s lattice theory was often broken into three subdivisions: distributive lattice theory, modular lattice theory, and the theory of all lattices. A question about lattices could usually be formulated for each of these subdivisions. Of the three resulting questions, the one about modular lattices almost always proved to be the most difficult. The problem of embedding a lattice into a complemented lattice was an example of such a problem. It is trivial to see that every lattice can be embedded into a complemented lattice, and Birkhoff’s representation theorem [1] shows that every distributive lattice can be embedded in a complemented distributive lattice. However the problem of embedding modular lattices into complemented modular lattices remained open for some time. R. P. Dilworth and Marshall Hall addressed this problem in their 1944 paper [23], showing, in fact, that there are finite modular lattices which cannot be embedded into a complemented modular lattice.