Bases of closure systems over residuated lattices. (English) Zbl 1327.03047
Summary: We present results on bases of closure systems over residuated lattices, which appear in applications of fuzzy logic. Unlike the Boolean case, the situation is not straightforward as there are two non-commuting generating operations involved. We present a decomposition theorem for a general closure operator and utilize it for computing generators and bases of the closure system. We show that bases are not unique and may in general have different sizes, and obtain a constructive description of the size of a largest base. We prove that if the underlying residuated lattice is a chain, all bases have the same size.
MSC:
| 03G25 | Other algebras related to logic |
| 03B52 | Fuzzy logic; logic of vagueness |
| 06A15 | Galois correspondences, closure operators (in relation to ordered sets) |
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