On compact topological lattices of finite dimension

Abstract

Introduction. In [11], Kaplansky proved that a compact semisimple topological ring is isomorphic and homeomorphic with a cartesian direct sum of finite simple rings; this implies that any compact Boolean topological lattice is always totally disconnected. However, in the proof of his theorem, Kaplansky utilized duality theorem in the sense of a topological group. Professor A. D. Wallace had suggested the possibility of a proof of the latter theorem (Boolean ring case) which is independent of the duality theorem. In ?1, we show, without using the duality theorem, that any compact Boolean topological lattice of finite dimension is always totally disconnected. However, with the use of duality theorem, this theorem can be generalized as follows: a locally compact and locally convex Boolean topological lattice is totally disconnected. In the remainder of this section, we shall exhibit a sufficient condition for a compact complemented modular topological lattice to be totally disconnected. A. D. Wallace [14] conjectured that the center of a compact connected topological lattice L of dimension n contains at most 21 elements. L. W. Anderson [2] has shown that if L is distributive, then Wallace's conjecture is true. In ?2, we prove that the conjecture is always true. We shall show some necessary and sufficient conditions for a topological lattice to be isomorphic and homeomorphic with the Euclidean n-cell. In addition, several structure theorems of compact topological lattice are given. After introducing the definition of the ordinal sum of topological lattices, we shall find a necessary and sufficient condition for a compact connected distributive topological lattice to be decomposed into a finite ordinal sum of finite cartesian products of closed connected chains. The author wishes to express his gratitude to Professor A. D. Wallace for his helpful suggestions and kind criticisms. Preliminaries. The lattice operations of join and meet are designated by v and A , respectively. Set operations are indicated by rounded symbols: n, u and c stand for intersection, union and inclusion, respectively. The empty set is denoted by D. We define a topological lattice to be a pair (L, -r) where L is a lattice, and is a Hausdorff space under the topology z- in which the lattice operations A and v are continuous. For a pair of subsets A and B of L, we shall use A A B and A V B to denote the