Locally convex topological lattices

Abstract

The main theorem of this paper is: Suppose that L is a topological lattice of finite breadth n. Then L can be embedded in a product of n compact chains if and only if L is locally convex and distributive. With this result it is then shown that the concepts of metrizability and separability are equivalent for locally convex, connected, distributive topological lattices of finite breadth. In [8] R. P. Dilworth proved that every distributive lattice of finite breadth n could be embedded (algebraically) in a product of n chains. Since finite breadth and distributivity are hereditary properties this result served to characterize distributive lattices of finite breadth. Dyer and Shields in [9] and Anderson in [4] (also see question 90 of [7]) asked if a result of a similar nature could be obtained for topological lattices, specifically: Can every compact, connected, metric, distributive topological lattice of breadth n be embedded in an n-cell? This question was answered affirmatively and more generally by Kirby A. Baker and the present author in [6]. For easier reference this result along with several consequences appears as Theorem 1.3 below. The major result of this paper is more nearly the topological analogue to Dilworth's theorem. We characterize the class of those topological lattices which can be embedded in a finite product of compact chains as the class of locally convex, distributive topological lattices of finite breadth. The class of locally convex topological lattices is rather large. For example, it contains all compact topological lattices [15], all locally compact and connected topological lattices [3], and all discrete lattices. This being the case our result contains those of [6] and [8]. We also show that the set of separating points of a locally convex, connected topological lattice is very well behaved. This fact together with our main theorem allows us to prove that separability and metrizability are equivalent for locally convex, connected, distributive topological lattices of finite breadth. The author wishes to express his gratitude to F. Burton Jones for his aid in the preparation of this paper. 1. Definitions and preliminary results. A topological lattice is a Hausdorff topological space with a pair of continuous maps A, V : L x L -* L such that Received by the editors January 10, 1970. AMS 1969 Subject Classifications. Primary 5456, 0665; Secondary 5453.