Beyond Modalities: Sufficiency and Mixed Algebras

Abstract

In [24] a generalisation of relation algebras to Boolean algebras with normal and additive operators is introduced. These operators are the counterparts to the modal operators of possibility. In this paper we introduce a class of Boolean algebras with co-normal and co-additive operators referred to as sufficiency operators. They are the algebraic counterpart to the logical sufficiency operators introduced in [17] for an extension of modal logics. Next, we define a class of mixed algebras i.e., Boolean algebras with an additional modal operator and a sufficiency operator. We study representation and duality theory for these new classes of algebras. The motivation for those algebras comes from the problems of reasoning with incomplete information and spatial reasoning.