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

Abstract

We introduce a two-sorted algebraic theory whose models are states of MV-algebras and, to within a categorical equivalence that extends Mundici's well-known one, states of Abelian lattice-groups with (strong order) unit. We discuss free states, and their relation to the universal state of an~MV-algebra. We clarify the relationship of such universal states with the theory of affine representations of lattice-groups. Main result: The universal state of any locally finite MV-algebra---in particular, of any Boolean algebra---has semisimple codomain.