Diego’s theorem for nuclear implicative semilattices. (English) Zbl 1498.06007
Summary: We prove that the variety of nuclear implicative semilattices is locally finite, thus generalizing Diego’s Theorem. The key ingredients of our proof include the coloring technique and construction of universal models from modal logic. For this we develop duality theory for finite nuclear implicative semilattices, generalizing Köhler duality. We prove that our main result remains true for bounded nuclear implicative semilattices, give an alternative proof of Diego’s Theorem, and provide an explicit description of the free cyclic nuclear implicative semilattice.
MSC:
| 06A12 | Semilattices |
| 03B45 | Modal logic (including the logic of norms) |
| 03G10 | Logical aspects of lattices and related structures |
| 03G25 | Other algebras related to logic |
| 06D20 | Heyting algebras (lattice-theoretic aspects) |
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