Über Morphismen halbmodularer Verbände.
In this paper we generalize a result of V. N. Salij concerning direct product decompositions of lattices which are complete and uniquely complemented.
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.
Here we consider the weak congruence lattice of an algebra with the congruence extension property (the CEP for short) and the weak congruence intersection property (briefly the WCIP). In the first section we give necessary and sufficient conditions for the semimodularity of that lattice. In the second part we characterize algebras whose weak congruences form complemented lattices.
Using a lattice-theoretical approach we find characterizations of modules with finite uniform dimension and of modules with finite hollow dimension.