Interpolation properties in the extensions of the logic of inequality

Abstract

We consider the modal logics wK4 and DL as well as the corresponding weakly transitive modal algebras and DL-algebras. We prove that there exist precisely 16 amalgamable varieties of DL-algebras. We find a criterion for the weak amalgamation property of varieties of weakly transitive modal algebras, solve the deductive interpolation problem for extensions of the logic of inequality DL, and obtain a weak interpolation criterion over wK4.The interpolation theorem, proved by Craig in 1957 for the classical predicate logic, initiated the study of interpolation in various formal theories. In the classical predicate logic the interpolation theorem of Craig has several equivalent statements, all becoming inequivalent for modal logics. In this regard a few cases of interpolation properties were formulated: the Craig interpolation property (CIP), the deductive interpolation property (IPD), the restricted interpolation property (IPR), and finally the weak interpolation property (WIP) [1].It is known that CIP and IPD are decidable [2] over the logic S4 but undecidable [3, 4] over K4, while WIP is decidable [5] over K4. The proofs of these facts base on the existence of a duality isomorphism between the class of extensions of a logic and the class of subvarieties of the corresponding variety of algebras (see [6] for instance). Therefore, the study of interpolation reduces to studying the amalgamation property of varieties of modal algebras.Maksimova studied various cases of the amalgamation property of varieties of modal algebras. In particular, she showed [2] that precisely 50 varieties of topoboolean algebras are amalgamable, among which 38 enjoy the strong amalgamation property. Maksimova found [7] the necessary and sufficient conditions for the amalgamation property, which enables us to reduce the question of the presence of the properties for this variety of modal algebras to the consideration of a subclass of finitely generated and finitely indecomposable algebras.The joint article of the author and Maksimova [8] deals mainly with weakly transitive modal algebras and DL-algebras. These classes of modal algebras became known in connection with the study of the modal logics wK4 and DL [9-11] presenting their natural algebraic semantics. In [8] we showed that the class of weakly transitive modal algebras coincides with the class of simple DL-algebras and found a full description for finitely generated simple DL-algebras and their embeddings. We proved that the variety of DL-algebras is not weakly amalgamable and the corresponding modal logics wK4 and DL lack all interpolation properties. In this article we study the amalgamation property of varieties of DL-algebras in more detail.In Section 2 we prove that the family of varieties of DL-algebras has the cardinality of the continuum. We establish the existence of a bijective correspondence between the varieties of DL-algebras and the downward closed subclasses of the class K (DL) = {V m n | n + m > 0}. In Section 3 we obtain necessary conditions for the amalgamation propert...