Algebraic ordered sets and their generalizations. (English) Zbl 0791.06007

Erné, Marcel

Rosenberg, Ivo (ed.) et al., Algebras and orders. Proceedings of the NATO Advanced Study Institute and Séminaire de mathématiques supérieures, Montréal, Canada, July 29 - August 9, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 389, 113-192 (1993).

Summary: 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 science. 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 \(\mathcal Z\). Many results on compactness remain true for the notion of \({\mathcal Z}\)-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.
For the entire collection see [Zbl 0778.00036].

MSC:

06A06	Partial orders, general
06A15	Galois correspondences, closure operators (in relation to ordered sets)
06-02	Research exposition (monographs, survey articles) pertaining to ordered structures
06B23	Complete lattices, completions
54F05	Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
18B99	Special categories

Keywords:

chains; directed sets; categories of ordered sets; compact generation in ordered sets; algebraic ordered sets; collections of directed subsets; representation theorems; duality theorem; sober spaces; spatial frames