Information Algebras

Abstract

In this chapter we present the algebraic systems which are the counterparts to the information frames defined in Sect. 3.10. In analogy to the classes of information frames, we present the major classes of algebras, namely the algebras of indistinguishability and distinguishability. All these algebras are Boolean algebras with some additional operators. The operators are intended to be the abstract counterparts to the information operators presented in Chap. 4. However, the standard theory of Boolean algebras with the normal and additive operators does not provide a sufficient framework for characterising the algebraic structures of all information operators. We extend the theory BAO of Boolean algebras with normal and additive operators to several other classes of operators in order to capture the features of the information operators. Similarly to the information relations, an important aspect of information operators is their dependence on varying subsets of attributes. So, in analogy to the relative frames, we introduce Boolean algebras with relative operators. Throughout this chapter the relative frames we refer to are of the form 〈W, (ℛ _P ) _{P⊆P A R} 〉.