Nearly Modular Orthocomplemented Lattices

摘要

Let L be a complete, weakly modular, orthocomplemented lattice. A modular element in L is an element a such that (O, a) is a modular lattice and (x,a) is a modular pair for all x. The lattice L is nearly modular if it is semi-modular and every element in L is the join of modular elements. The following is a key result. Theorem. If L is nearly modular a is a modular element, and b is perspective to a, then b is modular. Using this theorem and some continuity theorems involving modular elements the following theorems are proved. Theorem. L is nearly modular if an only if L is a locally finite dimension lattice. Theorem. L is semi-modular and contains a minimal element a with central cover 1 if and only if L is a type I dimension lattice. A simple construction, the horizontal sum, is defined and used to show that neither of the conditions in the definition of nearly modular can be dropped. (Author)