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

Abstract

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