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MV-algebras and Partially Cyclically Ordered Groups

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Abstract

We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are the wound-rounds of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We generalize these results to some non-linearly ordered MV-algebras, for example hyperarchimedean MV-algebras.

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References

  1. Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Lecture Notes, p 608. Springer, New York (1977)

    Book  Google Scholar 

  2. Chang, C.C.: A new proof of the completeness of Łukasiewicz axioms. Trans. Amer. Math. Soc. 93, 74–80 (1959)

    MathSciNet  MATH  Google Scholar 

  3. Cignoli, R., D’Ottavio, I.M.L., Mundici, D.: Algebraic Fundations of Many-Valued Reasoning. Kluwer, Boston (2000)

    Book  Google Scholar 

  4. Feferman, S., Vaught, R.: The first-order properties of algebraic systems. Fund. Math. 47, 57–103 (1959)

    Article  MathSciNet  Google Scholar 

  5. Fuchs, L.: Partially Ordered Algebraic Systems, International Series of Monographs in Pure and Applied Mathematics, vol. 28. Pergamon Press, New York (1963)

    Google Scholar 

  6. Giraudet, M., Leloup, G., Lucas, F.: First order theory of cyclically ordered groups. Ann. Pure. Appl. Logic 169(9), 896–927 (2018)

    Article  MathSciNet  Google Scholar 

  7. Glass, A.M.W.: Partially Ordered Groups, Series in Algebra, vol. 7. World Scientific, Singapore (1999)

    Book  Google Scholar 

  8. Glusschankov, D.: Cyclic ordered groups and MV-algebras, Czechoslovak. Math. J. 43 (n∘2), 249–263 (1993)

    MathSciNet  Google Scholar 

  9. Hodges, W.: Model theory, Encyclopedia of Mathematics. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  10. Jaffard, P.: Contribution à l’étude des groupes ordonnés. J. Math. Pures Appl. 32, 203–280 (1953)

    MathSciNet  MATH  Google Scholar 

  11. Jakubík, J., Pringerová, C.: Representation of cyclically ordered groups. Časop. Pěstov. Matem. 113, 184–196 (1988)

    Article  MathSciNet  Google Scholar 

  12. Leloup, G.: Pseudo-c-archimedean and pseudo-finite cyclically ordered groups. Math. Log. Quart. 65(4), 412–443 (2019)

    Article  MathSciNet  Google Scholar 

  13. Munidici, D.: Interpretation of AF C*-Algebras in Łukasiewicz Calculus and MV-algebras. J. Funct. Anal. 65, 15–31 (1986)

    Article  MathSciNet  Google Scholar 

  14. Zabarina, I., Pestov, G.: Śverkovskii’s theorem. Sibrisk. Math. Zh. 25(4), 46–53 (1984)

    MATH  Google Scholar 

  15. Zakon, E.: n∘1. Trans. Am. Math. Soc. 99, 21–40 (1961)

    Article  MathSciNet  Google Scholar 

Download references

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  1. Laboratoire Manceau de Mathématiques, Faculté des Sciences, Avenue Olivier Messiaen, 72085, LE MANS, CEDEX, France

    Gérard Leloup

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  1. Gérard Leloup
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Correspondence to Gérard Leloup.

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Leloup, G. MV-algebras and Partially Cyclically Ordered Groups. Order 39, 323–359 (2022). https://doi.org/10.1007/s11083-021-09578-z

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  • DOI: https://doi.org/10.1007/s11083-021-09578-z

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