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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The pseudo MV-algebras were de®ned in [12] as non-commutative extensions of MV-algebras. In this paper we de®ne the local pseudo MV-algebras and we suggest a classi®cation for these structures. The subclass of perfect pseudo MV-algebras is deeply investigated. Extending the Di Nola±Lettieri result for MV-algebras [8], we prove that the category of l-groups is equivalent to a subcategory of perfect pseudo MV-algebras.

Our main issue was to understand the connection between Łukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of f MV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from [0, 1]. The proper quasi-variety generated by [0, 1], with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture.

We propose a general system that combines the powerful features of modal logic and many-sorted reasoning. Its algebraic semantics leads to a many-sorted generalization of boolean algebras with operators, for which we prove the analogue of the Jónsson-Tarski theorem. Our goal was to deepen the connections between modal logic and program verification, and we test the expressivity of our system by defining a small imperative language and its operational semantics.