The two-sorted algebraic theory of states, and the universal states of MV-algebras

States of unital Abelian lattice-groups (normalised positive group homomorphisms to R) provide an abstraction expected-value operators. A well-known theorem due Mundici asserts that the category (with unit-preserving lattice-group as morphisms) is equivalent algebraic MV-algebras, and their homomorphisms. Through this equivalence, states correspond certain [0,1]-valued functionals on which are also known states. In paper we allow take values in any (or MV-algebra) rather than just R [0,1], respectively). We introduce a two-sorted theory whose models precisely MV-algebras. extend Mundici's equivalence one between MV-algebras with morphisms, with, again, morphisms. Thus, our may be regarded lattice-groups. As first main result, derive existence universal state MV-algebra from free algebras multi-sorted categories. The significance given algebra it provides most general operator algebra—a construct not available if insists real-valued. remaining part paper, seek concrete representations such begin by clarifying relationship affine representations: A→B coincides modification Choquet's representation (of corresponding A) if, only B semisimple. Locally finite semisimple, Boolean instances locally Our second result then has semisimple codomain, can thus described through adaptation representation.