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

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