Lukasiewicz Logic and Riesz Spaces

We initiate a deep study of Riesz MV-algebras which are MV-algebras endowed with a scalar multiplication with scalars from [0,1] . Extending Mundici’s equivalence between MV-algebras and ℓ -groups, we prove that Riesz MV-algebras are categorically equivalent to unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent to the class of commutative unital C ∗ -algebras. The propositional calculus RL that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz ∞ -valued propositional calculus and is complete with respect to evaluations in the standard model [0,1] . We prove a normal form theorem for this logic, extending McNaughton theorem for Ł ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in RL, and relate them with the analogue of de Finetti’s coherence criterion for RL