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Sobriety and spatiality in varieties of algebras. (English) Zbl 1177.54004

This paper extends the fundamental adjunction between the categories of topological spaces and locales (that paved the ground for the development of locale theory and point-free topology), and its analogue for fuzzy topology, to an arbitrary variety of algebras A. For this, the author introduces the category \(Q\)-Top of \(Q\)-topological spaces and \(Q\)-continuous maps, for an algebra \(Q\) of A. The corresponding notions of \(Q\)-sobriety and \(Q\)-spatiality are studied in detail and the obtained results are illustrated in the category of algebras over a given unital commutative quantale.

MSC:

54A40 Fuzzy topology
06F07 Quantales
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18B99 Special categories
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