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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 with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C * -algebras. The propositional calculus RL that has Riesz MV-algebras as models is a conservative extension of Lukasiewicz ∞-valued propositional calculus and it 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 Lukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in RL and we relate them with the analogue of de Finetti's coherence criterion for RL.

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It is known that all subvarieties of MV-algebras are finitely axiomatizable. In the literature, one can find equational characterizations of certain subvarieties, such as MV -algebras. In this paper we write down equational bases for all MV-varieties n and prove a representation theorem for each subvariety.