Generalised states: a multi-sorted algebraic approach to probability

Abstract

We introduce a generalised notion of state as an additive map from a Boolean algebra of events to an arbitrary MV-algebra. Generalised states become unary operations in two-sorted algebraic structures that we call state algebras. Since these, as we show, form an equationally defined class of algebras, universal-algebraic techniques apply. We discuss free state algebras, their geometric representation, and their connection with the theory of affine representations of lattice groups.