Unoriented graphs of modular lattices
It is well-known that there exist infinite modular lattices possessing no non-trivial valuations. In this paper a class of modular lattices is defined and it is proved that each lattice belonging to has a nontrivial valuation. Next, a result of . Birkhoff concerning valuations on modular lattices of finite length is generalized.
Using a lattice-theoretical approach we find characterizations of modules with finite uniform dimension and of modules with finite hollow dimension.