Theory of Symmetric Lattices

Of central importance in this book is the concept of modularity in lattices.

A lattice is said to be modular if every pair of its elements is a modular pair.

The properties of modular lattices have been carefully investigated by numerous mathematicians, including 1. von Neumann who introduced the important study of continuous geometry.

Continu- ous geometry is a generalization of projective geometry; the latter is atomistic and discrete dimensional while the former may include a continuous dimensional part.

Meanwhile there are many non-modular lattices. Among these there exist some lattices wherein modularity is symmetric, that is, if a pair (a,b) is modular then so is (b,a).

These lattices are said to be M-sym- metric, and their study forms an extension of the theory of modular lattices.

An important example of an M-symmetric lattice arises from affine geometry.

Here the lattice of affine sets is upper continuous, atomistic, and has the covering property.

Such a lattice, called a matroid lattice, can be shown to be M-symmetric.

We have a deep theory of parallelism in an affine matroid lattice, a special kind of matroid lattice. Further- more we can show that this lattice has a modular extension.