Albert Stralka

Dimension Raising Maps in Topological Algebra

In 1883 Cantor introduced perfect sets on the real line and described, in a footnote, as his first example of a nowhere dense perfect set, C, the set which now bears his name [2, p. 165 ft., notably p. 207]. It was only a year later, in 1884 that he discussed in a letter to the editor of Acta Mathematica [3, notably pp. 255, 256 of [2]] a class of functions from an interval to the real line which were monotone, continuous and were constant on each of the components of the complement of a perfect set. In particular, as a special example he produced a function f : I -+ I from the interval to itself which is most quickly recalled by depicting its graph:

Locally convex topological lattices

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 Dilworths 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.

ESSENTIAL COMPLETIONS OF DISTRIBUTIVE LATTICES

The salient feature of the essential completion process is that for most common distributive lattices it will yield a completely distributive lattice. In this note it is shown that for those distributive lattices which have at least one completely distributive essential extension the essential completion is minimal among the completions by infinitely distributive lattices. Thus in its setting the essential completion of a distributive lattice behaves in much the same way as the one-point compactification of a locally compact topological space does in it s setting.

The lattice of ideals of a compact semilattice

It is shown that, if L is a compact distributive topological lattice with enough continuous join-preserving maps into I to separate points, then there is a continuous lattice homomorphism from JI(L), the lattice of M-closed subsets of L, onto L. If J(L), the set of join-irreducible elements of L, is a compact semilattice then L is iseomorphic with .1(J(L)).

A characterization of full sublattices of finite dimensional Euclidean space

Abstract Full sublattices of finite dimensional Euclidean space are defined to be closures of bounded, open, connected sublattices of R n . They are the basic building blocks in a theory for infinite distributive lattices. In this paper, using a variant of the Bergman double projection theorem, we show that a sublattice of R n is full if and only if each of its triple projections into R 3 is full. This implies that for many problems concerning sublattices of Euclidean n -space it suffices to study low dimensional examples.

The Zariski topology and essential extensions of semilattices

In this paper we wish to study the relationship between the concepts of essential extension and Zariski topology in the theory of semilattices. This work was prompted by comments made by A. Day on some earlier work by the authors, comments to the effect that papers by Burns and Lakser [l] and by Horn and Kimura [6] might contribute to a useful investigation of the Zariski topology on semilattices. The particular questions which motivate this paper are: (1) For which semilattices is the Zariski topology Hausdorff? and (2) What significance does the possession of the Hausdorff separation axiom for its Zariski topology have for a semilattice? In seeking to answer these questions we make much use of the essential completion of a semilattice. The essential completion of a semilattice S may be realized as the smallest subsemilattice of the injective hull of S which contains S and has the appropriate completeness properties.

Distributive lattices with sufficiently many chain intervals

The idea of reductivity is introduced and basic properties are established. The major result is that a distributive lattice is reductive if and only if it has an essential extension which is a product of chains.