Nearly Modular Orthocomplemented Lattices

Introduction. Let L be a complete, orthocomplemented lattice. We say that L is a dimension lattice if L is weakly modular and there is an equivalence relation on L satisfying the axioms A,B,C, and D' of Loomis [5]. We say that L is locally finite if every element of L is the join of finite elements. If L is a dimension lattice in which every element is finite, then L is modular. Conversely, Kaplansky [4] has shown that if L is a complete orthocomplemented modular lattice, then L is a continuous geometry. From this and the results of von Neumann [ 7] and Iwamura [ 3], it follows that L is a dimension lattice in which every element is finite. Thus we conclude that L is a finite dimension lattice if and only if L is a complete orthocomplemented modular lattice. The main purpose of this paper is to obtain a similar characterization of locally finite dimension lattices. To obtain such a characterization, we need to weaken the assumption that L is modular. It is natural to try the assumption that L is semimodular, but this is not enough. We need to know that in some sense enough modular pairs exist. For this reason, we define a modular element to be an element a such that [0,a] is a modular lattice and (x, a) is a modular pair for all x in LC). An atom is always a modular element, and the finite elements in a dimension lattice are modular. We say that an orthocomplemented lattice L is nearly modular, if L is weakly modular and semi-modular and every element of L is the join of modular elements. Our principal result is the following theoremi2).