Algebraic semantics of refinement modal logic

We develop algebraic semantics of refinement modal logic using duality theory. Refinement modal logic has quantifiers that are interpreted using a refinement relation. A refinement relation is like a bisimulation, except that from the three relational requirements only ‘atoms’ and ‘back’ have to be satisfied. We study the dual notion of refinement on algebras and present algebraic semantics of refinement modal logic. To this end, we first present the algebraic semantics of action model logic quantifier, and we then introduce an algebraic model based on the semantics of the refinement quantifier in terms of the refinement relation. Then we show that refinement modal logic is sound and complete with respect to this algebraic semantics.