locally modular lattices and locally distributive lattices

A locally modular (resp. locally distributive) lattice is a lattice with a congruence relation and each of whose equivalence class has sufficiently many elements and is a modular (resp. distributive) sublattice. Both the lattice of all closed subspaces of a locally convex space and the lattice of projections of a locally finite von Neumann algebra are locally modular. The lattice of all /^-topologies of an infinite set is locally distributive. Introduction. In this paper, a lattice L is called locally modular (resp. locally distributive) when L has a congruence relation 8 such that each equivalence class by 8 which contains sufficiently many elements is a modular (resp. distributive) sublattice. Any locally distributive lattice is locally modular evidently, and it is shown in §1 that any locally modular lattice is both upper and lower semimodular in the sense of Birkhoff [2]. Moreover in this section it is proved that both the lattice of all closed subspaces of a locally convex space and the lattice of all projections of a locally finite von Neumann algebra are locally modular. It was proved by Larson and Thron [5] that the lattice of all ^-topologies on an infinite set is both upper and lower semimodular. Generalizing this result, it is shown in §2 that the lattice of all 7\-topologies is locally distributive. Moreover, the final theorem of [5] is formulated as a theorem on locally distributive lattices. In the last section, we determine the form of standard elements in the dual of the lattice of T^-topologies. This result shows us that this lattice has infinitely many standard elements but has no neutral elements except Oand 1. 1. Locally modular lattices. An equivalence relation 8 in a lattice L is called a congruence relation when it satisfies the following condition: If ax = b¡ i8) and a2 = b2 (Ö) then ax v a2 = bx v b2 id) and ax A a2 = bx A b2 id). Received by the editors May 8, 1973. AMS (MOS) subject classifications (1970). Primary 06A30, 06A35; Secondary 46A05, 46L10, 54A10, 54D10.