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.

Lukasiewicz Logic and Riesz Spaces

DI NOLA, Antonio;
2014

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4518457
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