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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."
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