Sobriety and spatiality in categories of lattice-valued algebras. (English) Zbl 1261.18003
The paper provides an adjoint situation, with its corresponding maximal sobriety-spatiality equivalence, between the categories of lattice-valued topological spaces and localic lattice-valued algebras (in the sense of A. Di Nola and G. Gerla [Stochastica 11, No. 2–3, 137–150 (1987; Zbl 0681.08002)]). The author claims this to be an extension of the equivalence between the categories of sober topological spaces and spatial locales to the framework of \((L,M)\)-fuzzy topology of T. Kubiak and A. Šostak [“Foundations of the theory of \((L,M)\)-fuzzy topological spaces”, in: U. Bodenhofer (ed.) et al., Abstracts of the 30th Linz seminar on fuzzy set theory, Johannes Kepler Universität, Linz. 70–73 (2009)].
Reviewer: Jorge Picado (Coimbra)
MSC:
| 18B30 | Categories of topological spaces and continuous mappings (MSC2010) |
| 06D22 | Frames, locales |
| 08A72 | Fuzzy algebraic structures |
| 54A40 | Fuzzy topology |
Keywords:
categorical equivalence; categorically algebraic topology; \((L, M)\)-fuzzy topology; lattice-valued algebra; point-set lattice-theoretic topology; sober topological space; soft algebra; spatial locale; variety; adjoint situationCitations:
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