A categorical analysis of compatibility frames

Abstract Negations as modal operators have been well investigated in the literature. There are various ways of providing semantics of negations in non-classical logic. Dunn has introduced semantics of negation in compatibility frames. In this work, we study the categories of compatibility frames and $K_{i}$ algebras—the algebraic semantics of minimal logic $K_{i}$. We establish categorical duality between $K_{i}$ algebras and descriptive general compatibility frames. Thereafter we introduce frames dual to De Morgan algebras and pseudocomplemented lattices in our framework. Correspondence of these frames with the existing frames in the literature, which are dual to these algebras, is also investigated. It is further extended to dualities between various subcategories of $K_{i}$ algebras and corresponding frames. Finally, it is shown that the duality generalizes to Stone’s duality in the case of ortho frames.