Hyperboolean Algebras and Hyperboolean Modal Logic

To the memory of our colleague and friend George Gargov ABSTRACT. Hyperboolean algebras are Boolean algebras with operators, constructed as algebras of complexes (or, power structures) of Boolean algebras. They provide an algebraic semantics for a modal logic (called here a hyperboolean modal logic with a Kripke semantics is accordingly based on frames in which the worlds are elements of Boolean algebras and the relations correspond to the Boolean operations. We introduce and give a complete axiomatization of the hyperboolean modal logic, and show that is lacks the nite model property. The technique of axiomati-zation hinges upon the fact that a "diierence" operator is deenable in hyperboolean algebras, and makes use of additional non-Hilbert-style rules. Finally, we discuss a number of open questions and directions for further research.