Algebras and Relational Frames for Gödel Modal Logic and Some of its Extensions

Gödel modal logics can be seen as extensions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety \(\mathbb {GAO}\) of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper.