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Notes on divisible MV-algebras

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Abstract

In this notes, we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with \(\mathbb Q\)-vector lattices, we present the divisible hull as a categorical adjunction, and we prove a duality between finitely presented algebras and rational polyhedra.

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References

  • Baker KA (1968) Free vector lattices. Canad J Math 20:58–66

    Article  MathSciNet  MATH  Google Scholar 

  • Beynon WM (1975) Duality theorem for finitely generated vector lattices. Proc London Math Soc 31(3):114–128

    Article  MathSciNet  MATH  Google Scholar 

  • Birkoff G (1967) Lattice theory, 3rd edn. AMS Colloquium Publications, New york

  • Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467–490

    Article  MathSciNet  MATH  Google Scholar 

  • Chang CC (1959) A new proof of the completeness of the Łukasiewicz axioms. Trans Am Math Soc 93:74–80

    MATH  Google Scholar 

  • Cignoli R, D’Ottaviano IML, Mundici D (1999) Algebraic foundation of many valued reasoning. Kluwer Academic Publication, Dordrecht

    MATH  Google Scholar 

  • Diaconescu D, Leuştean I (2015) The Riesz hull of a semisimple MV-algebra. Mathematica Slovaka (special issue in honor of Antonio Di Nola) 65(4):801–816

    MathSciNet  MATH  Google Scholar 

  • Di Nola A, Dvurečenskij A (2001) Product MV-algebras. Mult-Valued Log 6:193–215

    MathSciNet  MATH  Google Scholar 

  • Di Nola A, Flondor P, Leustean I (2003) MV-modules. J Algebra 267(1):21–40

    Article  MathSciNet  MATH  Google Scholar 

  • Di Nola A, Leuştean I (2011) Łukasiewicz Logic and MV-algebras. In: Cintula P et al (eds) Handbook of Mathematical Fuzzy Logic. Studies in Logic. College Publications, London

    Google Scholar 

  • Di Nola A, Leuştean I (2014) Łukasiewicz logic and Riesz spaces. Soft Comput 18(12):2349–2363

    Article  MATH  Google Scholar 

  • Di Nola A, Lenzi G, Vitale G (2016) Riesz-McNaughton functions and Riesz MV-algebras of nonlinear functions. Fuzzy Sets and Systems. doi:10.1016/j.fss.2016.03.003

  • Gerla B (2001) Rational Łukasiewicz logic and divisible MV-algebras. Neural Netw World 10(11):579–584

    Google Scholar 

  • Lapenta S (2015) MV-algebras with products: connecting the Pierce-Birkhoff conjecture with Łukasiewicz logic, PhD Thesis

  • Lapenta S, Leuştean I (2015) Towards understanding the Pierce-Birkhoff conjecture via MV-algebras. Fuzzy Sets Syst 276:114–130

  • Lapenta S, Leuştean I (2016) Scalar extensions for the algebraic structures of Łukasiewicz logic. J Pure Appl Algebra 220:1538–1553

  • Madden J (1985) \(l\)-groups of piecewise linear functions, Ordered algebraic structures (Cincinnati, Ohio, 1982), Lecture Notes in Pure and Appl Math 99:117–124

  • Marra V, Spada L (2013) Duality, projectivity and unification in Łukasiewicz logic and MV-algebras. Ann Pure Appl Log 164(3):192–210

    Article  MATH  Google Scholar 

  • McNaughton R (1951) A theorem about infinite-valued sentential logic. J Symb Log 16:1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Montagna F (2000) An algebraic approach to propositional fuzzy logic. J Log Lang Inf 9:91–124

    Article  MathSciNet  MATH  Google Scholar 

  • Mundici D (1986) Interpretation of ACF*-algebras in Łukasiewicz sentential calculus. J Funct Anal 65:15–63

    Article  MathSciNet  MATH  Google Scholar 

  • Mundici D (1999) Tensor products and the Loomis-Sikorski theorem for MV-algebras. Adv Appl Math 22:227–248

    Article  MathSciNet  MATH  Google Scholar 

  • Mundici D (2011) Advances in Łukasiewicz calculus and MV-algebras, Trends in Logic, vol 35. Springer

  • Ovchinnikov S (2002) Max-min representation of piecewise linear functions. Beiträge zur Algebra und Geometrie - Contrib Algebra Geom 43(1):297–302

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

S. Lapenta acknowledges partial support from the Italian National Research Project (PRIN2010-11) entitled Metodi logici per il trattamento dellinformazione. I. Leuştean was supported by a Grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0730.

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Authors and Affiliations

  1. Department of Mathematics, University of Salerno, Via Giovanni Paolo II, 132, Fisciano, SA, Italy

    Serafina Lapenta

  2. Department of Computer Science, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei nr.14, sector 1 C.P., 010014, Bucharest, Romania

    Ioana Leuştean

Authors
  1. Serafina Lapenta
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  2. Ioana Leuştean
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Correspondence to Serafina Lapenta.

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Both authors declare that they had no conflict of interest in writing this paper.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by A. Di Nola.

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Lapenta, S., Leuştean, I. Notes on divisible MV-algebras. Soft Comput 21, 6213–6223 (2017). https://doi.org/10.1007/s00500-016-2339-z

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