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 \(\ell \)-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 \({\mathbb R}{\mathcal L}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Amato P, Di Nola A, Gerla B (2005) Neural networks and rational McNaughton functions. J Multi Val Logic Soft Comput 11:95–110
Banaschewski B (1976) The duality between \(M\)-spaces and compact Hausdorf spaces. Math Nachr 75:41–45
Bigard A, Keimel K, Wolfenstein S (1977) Groupes et anneaux réticulés, Lectures Notes in Mathematics, vol 608, Springer-Verlag, Berlin
Burris S, Ankappanavar HP (1982) A course in universal algebra. Springer, Berlin
Castro JL, Trillas E (1998) The logic of neural networks, mathware and softcomputing, 5(1): 2327
Chang CC (1959) A new proof of the completeness of the Łukasiewicz axioms. Trans Amer Math Soc 93:74–80
Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic Foundations of many-valued reasoning. Kluwer, Dordrecht
de Finetti B (1931) Sul significato soggettivo della probabilitá. Fundamenta Mathematicae 17:289–329
Di Nola A, Dvurečenskij A (2001) Product MV-algebras. Mult Val Log 6:193–215
Di Nola A, Leuştean I, Flondor P (2003) MV-modules. J Algebra 267:21–40
Di Nola A, Leuştean I (2011) Riesz MV-algebras and their logic, In: Proceedings of EUSFLAT-LFA 2011, pp 140–145
Flondor P, Leuştean I (2004) MV-algebras with operators: the commutative and the non-commutative case. Discr Math 274(1–3):41–76
Fremlin DH (1974) Topological Riesz spaces and measure theory. Cambridge University Press, Cambridge
Gelfand I, Neimark M (1943) On the embedding of normed rings into the ring of operators in Hilbert space. Mat Sbornik 12(54):197–213
Gerla B (2001) Rational Łukasiewicz logic and DMV-algebras. Neural Netw World 11:579–584
Johnstone PT (1982) Stone spaces. Cambridge University Press, Cambridge
Kakutani S (1941) Concrete representations of abstract (M)-spaces (a characterization of the space of continuous functions). Ann Math 42:994–1024
Khalkhali M (2009) Basic noncommutative geometry. European Mathematical Society Publishing House, Europe
Kühr J, Mundici D (2007) De Finetti theorem and Borel states in [0, 1]-valued algebraic logic. Int J Approx Reasn 46(3):605–616
Labuschagne CCA, van Alten CJ (2007) On the variety of Riesz spaces. Indagationes Mathematicae 18(1):61–68
Leuştean I (2004) Contributions to the theory of MV-algebras: MV-modules, Editura Universitara, 2009. PhD Thesis, University of Bucharest, Bucharest
Luxemburg WAJ, Zaanen AC (1971) Riesz spaces I. North-Holland
Marra V (2009) private communication
McNaughton R (1951) A theorem about infinite-valued sentential logic. J Symbol Log 16:1–13
Meyer-Nieberg P (1991) Banach lattices. Universitext. Springer Verlag, Berlin
Montagna F (2005) Subreducts of MV-algebras with product and product residuation. Algebra Universalis 53:109–137
Mundici D (1986) Interpretation of AF C*-algebras in Łukasiewicz sentential calculus. J Funct Anal 65:15–63
Mundici D (1995) Averaging the truth value Łukasiewicz logic. Studia Logica 55:113–127
Mundici D (2006) Bookmaking over infinite-valued events. Int J Approx Reasn 43(3):223–240
Mundici D (2011) Advanced Łukasiewicz calculus and MV-algebras, trends in logic, vol 35. Springer, Berlin
Ovchinnikov S (2002) Max–Min representation of piecewise linear functions. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry 43:297–302
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Spada.
Rights and permissions
About this article
Cite this article
Di Nola, A., Leuştean, I. Łukasiewicz logic and Riesz spaces. Soft Comput 18, 2349–2363 (2014). https://doi.org/10.1007/s00500-014-1348-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-014-1348-z