Algebraic Ordered Sets and Their Generalizations

We study order-theoretical, algebraic and topological aspects of compact generation in ordered sets. Today, algebraic ordered sets (a natural generalization of algebraic lattices) have their place not only in classical mathematical disciplines like algebra and topology, but also in theoretical computer sciences. Some of the main statements are formulated in the language of category theory, because the manifold facets of algebraic ordered sets become more transparent when expressed in terms of equivalences between suitable categories. In the second part, collections of directed subsets are replaced with arbitrary selections of subsets Z. Many results on compactness remain true for the notion of Z-compactness, and the theory is now general enough to provide a broad spectrum of seemingly unrelated applications. Among other representation theorems, we present a duality theorem encompassing diverse specializations such as the Stone duality, the Lawson duality, and the duality between sober spaces and spatial frames.

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