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Łukasiewicz 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 \(\ell \)-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 \({\mathbb R}{\mathcal L}\) that has Riesz MV-algebras as models is a conservative extension of Łukasiewicz \(\infty \)-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 \({\mathbb R}{\mathcal L},\) and relate them with the analogue of de Finetti’s coherence criterion for \({\mathbb R}{\mathcal L}\).

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I. Leuştean was supported by the strategic grant POSDRU/89/1.5/S/58852, cofinanced by ESF within SOP HRD 2007-2013. Part of the research has been carried out while visiting the University of Salerno. The investigations from Sect. 5 were initiated in 2009, when V. Marra (University of Milan) pointed out Kakutani’s theorem and its relevance to the theory of Riesz MV-algebras.

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Correspondence to Ioana Leuştean.

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Communicated by L. Spada.

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Di Nola, A., Leuştean, I. Łukasiewicz logic and Riesz spaces. Soft Comput 18, 2349–2363 (2014).

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  • Riesz MV-algebra
  • Łukasiewicz logic
  • Piecewise linear function
  • Quasi-linear combination

Mathematics Subject Classification (2010)

  • 06D35
  • 03B50