Lattice properties of the lattice of lattice uniformities
by
Hans Weber
Dept. of Mathematics and Computer Science, University of Udine

Since FN-topologies are used as a powerful tool in measure theory, it becomes clear that several important theorems in measure theory have a purely topological core. In this talk I'm interested in decomposition theorems of modular functions on lattices. ( m is called modular if m(x) + m(y) = m(x \cap y) + m(x \cup y).) For that we study lattice properties of the lattice LU(L) of all lattice uniformities on a lattice L. It turns out: Whenever a certain sublattice of LU(L) is a Boolean algebra, we obtain a decomposition theorem for modular functions on L. This yields in particular a decomposition theorem in non commutative measure theory (for measures on orthomodular lattices) and in fuzzy measure theory (for measures on MV-algebras). In this topological approach also the concept of independent topologies is important. Two topologies s and t on X are called independent if the diagonal {(x, x): x in X } is dense in the product (X, s) ×(X, t).

Date received: August 1, 2001