## Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics I (1998)

Venue: | Studia Logica |

Citations: | 12 - 6 self |

### BibTeX

@ARTICLE{Sofronie-stokkermans98dualityand,

author = {Viorica Sofronie-stokkermans},

title = {Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics I},

journal = {Studia Logica},

year = {1998},

volume = {64},

pages = {2000}

}

### OpenURL

### Abstract

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. Introduction In the study of non-classical propositional logics (and especially of modal logics) there are two main ways of defining interpretations or models. One possibility is to use algebras -- usually lattices with operators -- as models. Propositional variables are interpreted over elements of these algebraic models, an...

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Citation Context ...tions of satisfiability can be obtained in a natural way: Example 1 (E fde ) The fragment E fde of relevance logic has been proved sound and complete with respect to the variety of De Morgan algebras =-=[AB75]-=-. A De Morgan algebra is a distributive lattice endowed with a lattice antimorphism , which satisfies the De Morgan laws and the law of double negation. The Priestley duals of the De Morgan algebras a... |

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Citation Context ...ed in [SS98] in order to define a general notion of relational (not necessarily topological) models for certain logics, and a notion of satisfiability in such models. This type of results was used in =-=[SS97]-=- in the study of (first-order) finitely-valued logics. The Priestley dual of the algebra of truth values was used for obtaining efficient translations to clause forms and an automated theorem proving ... |

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Citation Context ...ul for instance in the study of the semantics of program languages, where the topology provides additional information about properties of states (e.g. verifiability, refutability, observability, cf. =-=[RB98]-=-). However, the use of topological models is not fully satisfactory in all situations, especially when one of the ultimate goals is automated theorem proving. Non-topological Kripke models are the typ... |

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Citation Context ...t fj 2 J j X j = Y i g 2 U . Proof : It follows from the fact that, since J is finite, the ultrafilter U is generated by some k 2 J , and therefore, by classical properties of ultraproducts (cf. e.g. =-=[BS71]-=-, p.124), Q j2J X j =U ' X k . 2 18 4.2.2 Elements of model theory The proof of Theorem 18, inspired by the idea used by Goldblatt in [Gol89], requires some elements of model theory, namely the use of... |

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