Sobriety and spatiality in categories of lattice-valued algebras

Authors:
Sergey A. Solovyov
Affiliations:
Department of Mathematics, University of Latvia, Zellu iela 8, LV-1002 Riga, Latvia and Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris 29, LV-1459 Riga, Latvia
Venue:
Fuzzy Sets and Systems
Year:
2012

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Abstract

The paper provides an analogue of the famous equivalence between the categories of sober topological spaces and spatial locales for the framework of (L,M)-fuzzy topology of Kubiak and Sostak (and partly to that of Guido). To be more general, we replace locales with localic lattice-valued algebras in the sense of Di Nola and Gerla and use the respective generalized topological setting. As a result, it appears that the shift from crisp algebras to lattice-valued algebras weakens (resp. strengthens) considerably the classical (including the point-set lattice-theoretic setting of Rodabaugh) notion of sobriety (resp. spatiality).