Priestley Duality, a Sahlqvist Theorem and a Goldblatt-Thomason Theorem for Positive Modal Logic

In [12] the study of Positive Modal Logic (PML) is initiated using standard Kripke semantics and the positive modal algebras (a class of bounded distributive lattices with modal operators) are introduced. The minimum system of Positive Modal Logic is the (∧,∨, 2, 3,⊥,>)-fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calculus and extensions of it can be obtained by adding sequents as new axioms. In [6] a new semantics for PML is proposed to overcome some frame incompleteness problems discussed in [12]. The frames of this semantics consists of a set of indexes, a quasi-order on them and an accessibility relation. The models are obtained by using increasing valuations relatively to the quasiorder of the frame. This semantics is coherent with the dual structures obtained by developing the Priestley duality for positive modal algebras, one of the topics or the present paper, and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic. The present paper is devoted to the study of the mentioned duality as well as to proving some d-persistency results as well as a Sahlqvist Theorem for sequents and the semantics proposed in [6]. Also a GoldblattThomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved.