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The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.
This article surveys a variety of approaches to integration theory that have arisen in the context of continuous domain theory. The presentation is set in a broader context of ‘positive’ analysis, analysis where order-theoretic notions and notions of positivity play a key role.
In this paper we sketch the earlier chapters in the history of the mathematical development of the theory of linearly ordered semigroups. The major contributions to the theory occurred in the 1950s and early 1960s, but, as we shall see, these developments were preceded by a scattering of earlier results, which are of broader historical interest with regard to the development of semigroup theory in general.
Introduction
This paper arose as a tribute to A. H. Clifford and significant portions of it were presented at a conference held in his honor on March 28–30, 1994 at Tulane University in New Orleans. Since the authors come from a background of topological semigroups, it seemed appropriate to focus on an aspect of semigroups, namely the theory of linearly ordered semigroups, in which Clifford made significant contributions and which has also impinged on the topological theory. Even though the work of Clifford in this area might not be recognized as the most important or foundational among the varied achievements of this great pioneer of semigroup theory, yet they form an interesting and significant contribution to the development of the theory of linearly ordered semigroups. Clifford wrote a total of five papers [14, 15, 16, 17, 18], all in the 1950s, on the topic of linearly ordered semigroups. One of these, which grew out of a 1957 hour address to the American Mathematical Society, gave a survey of the area as it existed at that time [17] and has proved a helpful reference in the preparation of this article.
A Hausdorff space X is said to be compactly generated (a k-space) if and only if the open subsets U of X are precisely those subsets for which K ∩ U is open in K for all compact subsets of K of X. We interpret this property as a duality property of the lattice O(X) of open sets of X. This view point allows the introduction of the concept of being quasicompactly generated for an arbitrary sober space X. The methods involve the duality theory of up-complete semilattices, and certain inverse limit constructions. In the process, we verify that the new concept agrees with the classical one on Hausdorff spaces.
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