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Advanced Łukasiewicz calculus and MV-algebras pp 227–238Cite as

Applications, Further Reading, Selected Problems

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Part of the book series: Trends in Logic ((TREN,volume 35))

Abstract

The borderline between Łukasiewicz logic and MV-algebras is vague. Proof-theoretic methods are fundamental in MV-algebra theory–and algebraic techniques pervade Łukasiewicz logic. Interpolation and amalgamation are symbiotic. Finite presentations of MV-algebras are finite axiomatizations of theories

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References

  1. Czelakowski, J. (2001). Protoalgebraic logics. Trends in logic, Vol. 10. Dordrecht: Kluwer.

    Google Scholar 

  2. Cignoli, R. L. O., D’Ottaviano, I. M. L., Mundici, D. (2000). Algebraic foundations of many-valued reasoning. Trends in logic, Vol. 7. Dordrecht: Kluwer.

    Google Scholar 

  3. Gispert, J., Mundici, D. (2005). MV-algebras: a variety for magnitudes with archimedean units. Algebra Universalis. Special issue in memoriam Wim Blok (Vol. 53.1, pp. 7–43).

    Google Scholar 

  4. Grigolia, R. S. Algebraic analysis of Łukasiewicz–Tarski’s n-valued logical systems. In [Wójcicki R, Malinowski G (Eds.). (1977). Selected papers on Łukasiewicz sentential calculi. Wroc\(\l\)aw: Ossolineum] (pp. 81–92).

    Google Scholar 

  5. Hay, L. S. (1963). Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic, 28, 77–86.

    Article  MathSciNet  Google Scholar 

  6. Wójcicki, R. (1973). On matrix representations of consequence operations of Łukasiewicz sentential calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 19: 239–247, reprinted. In [Wójcicki R, Malinowski G (Eds.). (1977). Selected papers on Łukasiewicz sentential calculi. Wroc\(\l\)aw: Ossolineum], pp. 101–111.

    Google Scholar 

  7. Czelakowski, J. (1979). “Large” matrices which induce finite consequence operations. Bulletin of the Section of Logic, 8, 79–82.

    MATH  MathSciNet  Google Scholar 

  8. Dziobiak, W., Kravchenko, A. V., Wojciechowski, P. J. (2009). Equivalents for a quasivariety to be generated by a single structure. Studia Logica, 91, 113–123.

    Article  MATH  MathSciNet  Google Scholar 

  9. Font, J. M., Rodriguez, A. J., Torrens, A. (1984). Wajsberg algebras. Stochastica, 8, 5–31.

    MATH  MathSciNet  Google Scholar 

  10. Czelakowski, J., Dziobiak, W. (1990). Congruence-distributive quasivarieties whose finitely subdirectly irreducible members form a universal class. Algebra Universalis, 27, 128–149.

    Article  MATH  MathSciNet  Google Scholar 

  11. Czelakowski, J. (1980). A remark on free products. Bulletin of the Section of Logic, 9.3, 125–130.

    MathSciNet  Google Scholar 

  12. Mundici, D. (1994). A constructive proof of McNaughton’s theorem in infinite-valued logics. Journal of Symbolic Logic, 59, 596–602.

    Article  MATH  MathSciNet  Google Scholar 

  13. Panti, G. (1995). A geometric proof of the completeness of the Łukasiewicz calculus. Journal of Symbolic Logic, 60, 563–578.

    Article  MATH  MathSciNet  Google Scholar 

  14. Mundici, D. (1988). Farey stellar subdivisions, ultrasimplicial groups and \(K_{0}\) of AF \(C^{\ast}\)-algebras. Advances in Mathematics, 68, 23–39.

    Article  MATH  MathSciNet  Google Scholar 

  15. Marra, V. (2000). Every abelian \(\ell\)-group is ultrasimplicial. Journal of Algebra, 225, 872–884.

    Article  MATH  MathSciNet  Google Scholar 

  16. Manara, C., Marra, V. Mundici, D. (2007). Lattice-ordered abelian groups and Schauder bases of unimodular fans. Transactions of the American Mathematical Society, 359, 1593–1604.

    Article  MATH  MathSciNet  Google Scholar 

  17. Baker, K. A. (1968). Free vector lattices. Canadian Journal of Mathematics, 20, 58–66.

    Article  MATH  Google Scholar 

  18. Beynon, W. M. (1974). Combinatorial aspects of piecewise linear maps. Journal of the London Mathematical Society, 31(2), 719–727.

    Article  MathSciNet  Google Scholar 

  19. Beynon, W. M. (1977). Applications of duality in the theory of finitely generated lattice-ordered abelian groups. Canadian Journal of Mathematics, 29, 243–254.

    Article  MATH  MathSciNet  Google Scholar 

  20. Mundici, D. (2008). The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete and Continuous Dynamical Systems, 21, 537–549.

    Article  MATH  MathSciNet  Google Scholar 

  21. Cabrer, L., Mundici D. Finitely presented lattice-ordered abelian groups with order-unit, arxiv-1006.4188

    Google Scholar 

  22. Effros, E. G., Handelman, D. E., Shen, C.-L. (1980). Dimension groups and their affine representations. American Journal of Mathematics, 102, 385–407.

    Article  MATH  MathSciNet  Google Scholar 

  23. Grillet, P. A. (1976). Directed colimits of free commutative semigroups. Journal of Pure and Applied Algebra, 9, 73–87.

    Article  MATH  MathSciNet  Google Scholar 

  24. Riečan, B., Mundici, D. Probability on MV-algebras, Chap. 21 in Pap, E. 2002, (Ed.), Handbook of Measure Theory (Vol. I, II), Amsterdam: North-Holland, pp. 869–909.

    Google Scholar 

  25. Pap, E. (2002). (Ed.), Handbook of measure theory (Vol.I, II). Amsterdam: North-Holland.

    Google Scholar 

  26. De Simone, A., Navara, M., Mundici, D. (2003). A Cantor–Bernstein theorem for \(\sigma\)-complete MV-algebras. Czechoslovak Mathematical Journal, 53(128), 437–447.

    Article  MATH  MathSciNet  Google Scholar 

  27. Di Nola, A., Navara, M. Cantor–Bernstein property for MV-algebras. In [Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara C., Marra, V. (Eds.). (2007). Algebraic and proof-theoretic aspects of non-classical logics. Lecture notes in artificial intelligence (Vol. 4460). Berlin: Springer] pp. 107–118.

    Google Scholar 

  28. Jakubík, J. (1999). Cantor–Bernstein theorem for \(MV\)-algebras. Czechoslovak Mathematical Journal, 49(124), 517–526.

    Article  MATH  MathSciNet  Google Scholar 

  29. Jakubík, J. (2001). A theorem of Cantor–Bernstein type for orthogonally \(\sigma\)-complete pseudo MV-algebras. Tatra Mountains Mathematical Publications, 22, 91–103.

    MATH  MathSciNet  Google Scholar 

  30. Barbieri, G., Weber, H. Measures on Clans and on MV-algebras, Chap. 22 in [Pap, E. (Ed.). (2002). Handbook of measure theory (Vol. I, II). Amsterdam: North-Holland], pp. 911–945.

    Google Scholar 

  31. Butnariu, D., Klement, E. P. (1995). Triangular norm-based measures and games with fuzzy coalitions. Dordrecht: Kluwer.

    Google Scholar 

  32. Butnariu, D., Klement, E. P. Triangular norm-based measures, Chap. 23 in [Pap, E. (Ed.). (2002). Handbook of measure theory (Vol. I, II). Amsterdam: North-Holland], pp. 947–1010.

    Google Scholar 

  33. Dvure˘enskij, A., Pulmannová, S. (2000). New Trends in Quantum Structures. Dordrecht: Kluwer.

    Google Scholar 

  34. Riečan, B., Neubrunn, T. (1997). Integral, measure, and ordering. Dordrecht: Kluwer.

    MATH  Google Scholar 

  35. Riečan, B. (2005). Kolmogorov–Sinaj entropy on MV-algebras. Internationtal Journal of Theoretical Physics, 44, 1041–1052.

    Article  MATH  Google Scholar 

  36. Kühr, J., Mundici, D. (2007). De Finetti theorem and Borel states in [0,1]-valued algebraic logic. International Journal of Approximate Reasoning, 46, 605–616.

    Article  MATH  MathSciNet  Google Scholar 

  37. Aguzzoli, S., Gerla, B. Marra, V. (2008). De Finetti’s no-Dutch-Book criterion for Gödel logic. Studia Logica, 90, 25–41.

    Article  MATH  MathSciNet  Google Scholar 

  38. Fedel, M., Keimel, K., Montagna, F., Roth, W. Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic, Forum Mathematicum. doi:10.1515/FORM.2011.123

  39. Panti, G. (2007). Bernoulli automorphisms of finitely generated free MV-algebras. Journal of Pure and Applied Algebra, 208, 941–950.

    Article  MATH  MathSciNet  Google Scholar 

  40. Panti, G. (2008). Invariant measures in free MV-algebras. Communications in Algebra, 36, 2849–2861.

    Article  MATH  MathSciNet  Google Scholar 

  41. Marra, V. (2009). The Lebesgue state of a unital abelian lattice-ordered group, II. Journal of Group Theory, 12, 911–922.

    Article  MATH  MathSciNet  Google Scholar 

  42. Bratteli, O. (1972). Inductive limits of finite-dimensional \(C^{\ast}\)-algebras. Transactions of the American Mathematical Society, 171, 195–234.

    MATH  MathSciNet  Google Scholar 

  43. Elliott, G. A. (1976). On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. Journal of Algebra, 38, 29–44.

    Article  MATH  MathSciNet  Google Scholar 

  44. Effros, E. G. (1980). Dimensions and \(C^{\ast}\)-algebras. Providence, RI: American Mathematical Society.

    Google Scholar 

  45. Chang, C. C. (1958). Algebraic analysis of many-valued logics. Transactions of the American Mathematical Society, 88, 467–490.

    Article  MATH  MathSciNet  Google Scholar 

  46. Behnke, H., Leptin, H. (1972). \(C^{\ast}\)-algebras with a two-point dual. Journal of Functional Analysis, 10, 330–335.

    Article  Google Scholar 

  47. Mundici, D. (1992). Turing complexity of the Behncke–Leptin \(C^{\ast}\)-algebras with a two-point dual. Annals of Mathematics and Artificial Intelligence, 26, 287–294.

    Article  MathSciNet  Google Scholar 

  48. Mundici, D. (1986). Interpretation of AF \(C^{\ast}\)-algebras in Łukasiewicz sentential calculus. Journal of Functional Analysis, 65, 15–63.

    Article  MATH  MathSciNet  Google Scholar 

  49. Boca, F. (2008). An AF algebra associated with the Farey tessellation. Canadian Journal of Mathematics, 60, 975–1000.

    Article  MATH  MathSciNet  Google Scholar 

  50. Bratteli, O., Joergensen, P., Kim, H. H., Roush, F. (2000). Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams. Ergodic Theory and Dynamical Systems, 20, 1639–1656.

    Article  MATH  MathSciNet  Google Scholar 

  51. Bratteli, O., Joergensen, P., Kim, H. H., Roush, F. (2001). Decidability of the isomorphism Problem for stationary AF algebras and the associated ordered simple dimension groups. Ergodic Theory and Dynamical Systems, 21, 1625–1655.

    Article  MathSciNet  Google Scholar 

  52. Mundici, D. (2004). Simple Bratteli diagrams with a Gödel-incomplete \(C^{\ast}\)-equivalence problem. Transactions of the American Mathematical Society, 356, 1937–1955.

    Article  MATH  MathSciNet  Google Scholar 

  53. Mundici, D., Panti, G. (2001). Decidable and undecidable prime theories in infinite-valued logic. Annals of Pure and Applied Logic, 108, 269–278.

    Article  MATH  MathSciNet  Google Scholar 

  54. Mundici, D., Tsinakis, C. (2008). Gödel incompleteness in AF C*-algebras. Forum Mathematicum, 20, 1071–1084.

    Article  MATH  MathSciNet  Google Scholar 

  55. Mundici, D. (2009). Recognizing the Farey–Stern–Brocot AF algebra. Rendiconti Lincei, Matematica e Applicazioni, 20, 327–338.

    Article  MATH  MathSciNet  Google Scholar 

  56. Cignoli, R., Elliott, G. A., Mundici, D. (1993). Reconstructing \(C^{\ast}\)-algebras from their Murray von Neumann orders. Advances in Mathematics, 101, 166–179.

    Article  MATH  MathSciNet  Google Scholar 

  57. Mundici, D., Panti, G. (1993). Extending addition in Elliott’s local semigroup. Journal of Functional Analysis, 171, 461–472.

    Article  MathSciNet  Google Scholar 

  58. Dvurečenskij, A. Measures on quantum structures. In [Pap, E. (Ed.). (2002). Handbook of measure theory (Vol. I, II). Amsterdam: North-Holland], pp. 827–868.

    Google Scholar 

  59. Foulis, D. J. (2000). MV and Heyting effect algebras. Invited papers dedicated to Maria Luisa Dalla Chiara, Part II. Foundations of Physics, 30, 1687–1706.

    Article  MathSciNet  Google Scholar 

  60. Galatos, N., Tsinakis, C. (2005). Generalized MV-algebras. Journal of Algebra, 283, 254–291.

    Article  MATH  MathSciNet  Google Scholar 

  61. Montagna, F., Tsinakis, C. (2010). Ordered groups with a conucleus. Journal of Pure and Applied Algebra, 214, 71–88.

    Article  MATH  MathSciNet  Google Scholar 

  62. Caicedo, X. Implicit Operations in MV-Algebras and the Connectives of Łukasiewicz Logic. In [Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara C., Marra, V. (Eds.). (2007). Algebraic and proof-theoretic aspects of non-classical logics. Lecture notes in artificial intelligence (Vol. 4460). Berlin: Springer], pp. 50-68.

    Google Scholar 

  63. Di Nola, A. Grigolia, R. (2004). On monadic MV-algebras. Annals of Pure and Applied Logic, 128, 125–139.

    Article  MATH  MathSciNet  Google Scholar 

  64. Jakubík, J. (2002). On product MV-algebras. Czechoslovak Mathematical Journal, 52, 797–810.

    Article  MATH  MathSciNet  Google Scholar 

  65. Montagna, F. (2000). An algebraic approach to propositional fuzzy logic. Journal of Logic, Language and Information, 9, 91–124.

    Article  MATH  MathSciNet  Google Scholar 

  66. Montagna, F. (2001). Functorial representation theorems for \(\hbox{MV}_\Updelta\) algebras with additional operators. Journal of Algebra, 238, 99–125.

    Article  MATH  MathSciNet  Google Scholar 

  67. Montagna, F., Panti, G. (2001). Adding structure to MV-algebras. Journal of Pure and Applied Algebra, 164, 365–387.

    Article  MATH  MathSciNet  Google Scholar 

  68. Di Nola, A., Flondor, P., Leustean, I. (2003). MV-modules. Journal of Algebra, 267, 21–40.

    Article  MATH  MathSciNet  Google Scholar 

  69. Flondor, P., Leustean, I. (2004). MV-algebras with operators (the commutative and the non-commutative case). Discrete Mathematics, 274, 41–76.

    Article  MATH  MathSciNet  Google Scholar 

  70. Di Nola, A., Gerla, B. (2005). Algebras of Łukasiewicz logic and their semiring reducts. Idempotent mathematics and mathematical physics. Contemporary Mathematics, 377, 131–144.

    MathSciNet  Google Scholar 

  71. Rump, W. (2008). L-algebras, self-similarity, and \(\ell\)-groups. Journal of Algebra, 320, 2328–2348.

    Article  MATH  MathSciNet  Google Scholar 

  72. Yang, Yi Chuan. (2006). \(\ell\) -Groups and Bézout domains, Thesis. University of Stuttgart. available at http://elib.uni-stuttgart.de/opus/volltexte/2006/2508/

    Google Scholar 

  73. Aguzzoli, S., Bova, S. (2010). The free \(n\)-generated BL-algebra. Annals of Pure and Applied Logic, 161, 1144–1170.

    Article  MATH  MathSciNet  Google Scholar 

  74. Dubuc, E. J., Poveda, Y. (2010). Representation theory of MV-algebras. Annals of Pure and Applied Logic, 161, 1024–1046.

    Article  MATH  MathSciNet  Google Scholar 

  75. Baaz, M., Hájek, P., Montagna, F., Veith, H. (2002). Complexity of t-tautologies. Annals of Pure and Applied Logic, 113, 3–11.

    Article  MATH  MathSciNet  Google Scholar 

  76. Galatos, N., Ono, H. (2006). Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL. Studia Logica, 83, 279–308.

    Article  MATH  MathSciNet  Google Scholar 

  77. Montagna, F. (2006). Interpolation and Beth’s property in propositional many-valued logics: A semantic investigation. Annals of Pure and Applied Logic, 141, 148–179.

    Article  MATH  MathSciNet  Google Scholar 

  78. Dvurečenskij, A. (2002). Pseudo MV-algebras are intervals in \(\ell\)-groups. Journal of the Australian Mathematical Society, 72, 427–445.

    Article  MATH  MathSciNet  Google Scholar 

  79. Esteva, F., Lluis Godo, Towards the Generalization of Mundici’s Gamma Functor to IMTL Algebras. The Linearly Ordered Case. In [Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara C., Marra, V. (Eds.). (2007). Algebraic and proof-theoretic aspects of non-classical logics. Lecture notes in artificial intelligence (Vol. 4460). Berlin: Springer]. pp. 127–137.

    Google Scholar 

  80. Dvurečenskij, A., Holland, C. W. (2007). Top varieties of generalized MV-algebras and unital lattice-ordered groups. Communications in Algebra, 35, 3370–3390.

    Article  MATH  MathSciNet  Google Scholar 

  81. Jakubík, J. (2007). Banaschewski’s theorem for generalized MV-algebras. Czechoslovak Mathematical Journal, 57, 1099–1105.

    Article  MATH  MathSciNet  Google Scholar 

  82. Leustean, I. (2008). A determination principle for algebras of \(n\)-valued Łukasiewicz logic. Journal of Algebra, 320, 3694–3719.

    Article  MATH  MathSciNet  Google Scholar 

  83. Ball, R., Georgescu, G., Leuştean, I. (2002). Cauchy completions of MV-algebras. Algebra Universalis, 47, 367–407.

    Article  MATH  MathSciNet  Google Scholar 

  84. Gehrke, M., Priestley, H. A. (2002). Non-canonicity of MV-algebras. Houston Journal of Mathematics, 28, 449–455.

    MATH  MathSciNet  Google Scholar 

  85. Georgescu, G., Leustean, I. (1998). Convergence in perfect MV-algebras. Journal of Mathematical Analysis and Applications, 228, 96–111.

    Article  MATH  MathSciNet  Google Scholar 

  86. Ciabattoni, A., Metcalfe, G. (2008). Density elimination. Theoretical Computer Science, 403, 328–346.

    Article  MATH  MathSciNet  Google Scholar 

  87. Metcalfe, G., Olivetti, N., Gabbay, D. M. (2005). Sequent and hypersequent calculi for abelian and Łukasiewicz logics. ACM Transactions on Computational Logic, 6(3), 578–613.

    Article  MathSciNet  Google Scholar 

  88. Metcalfe, G., Olivetti, N., Gabbay, D.M. (2009). Proof theory for fuzzy logics. Applied Logic Series (Vol. 36). New York: Springer.

    Google Scholar 

  89. Fermüller, C. (2008). Dialogue games for many-valued logics—an overview. Studia Logica, 90, 43–68.

    Article  MATH  MathSciNet  Google Scholar 

  90. Fermüller, C., Metcalfe, G. (2009). Giles’s game and the proof theory of Łukasiewicz logic. Studia Logica, 92, 27–61.

    Article  MATH  MathSciNet  Google Scholar 

  91. Jenei, S. Montagna, F. Rényi-Ulam game semantics for product logic and for the logic of cancellative hoops. In [Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara C., Marra, V. (Eds.). (2007). Algebraic and proof-theoretic aspects of non-classical logics. Lecture notes in artificial intelligence (Vol. 4460). Berlin: Springer]. pp. 231–246.

    Google Scholar 

  92. Glass, A. M. W., Madden, J. J. (1984). The word problem versus the isomorphism problem. Journal of the London Mathematical Society, 30(2), 53–61.

    Article  MATH  MathSciNet  Google Scholar 

  93. Chernavski, A.V., Leksine, V.P. (2006). Unrecognizability of manifolds. Annals of Pure and Applied Logic, 141, 325–335.

    Article  MathSciNet  Google Scholar 

  94. Massey, W. S. (1980). Singular homology theory. Graduate texts in mathematics (Vol. 70). New York: Springer.

    Google Scholar 

  95. Elliott, G. A., Mundici, D. (1993). A characterization of lattice-ordered abelian groups. Mathematische Zeitschrift, 213, 179–185.

    Article  MATH  MathSciNet  Google Scholar 

  96. Delzell, C. N., Madden, J. (1994). A completely normal spectral space that is not a real spectrum. Journal of Algebra, 169, 71–77.

    Article  MATH  MathSciNet  Google Scholar 

  97. Cignoli, R., Torrens, A. (1996). The poset of prime \(\ell\)-ideals of an abelian \(\ell\)-group with strong unit. Journal of Algebra, 184, 604–612.

    Article  MATH  MathSciNet  Google Scholar 

  98. Oda, T. (1978). Torus embeddings and applications. Tata Institute of Fundamental Research, Mumbay. Berlin: Springer.

    Google Scholar 

  99. Ewald, G. (1996). Combinatorial convexity and algebraic geometry. Graduate texts in mathematics (Vol. 168). Berlin: Springer.

    Book  Google Scholar 

  100. Gluschankof, D. (1992). Prime deductive systems and injective objects in the algebras of Łukasiewicz infinite-valued calculi. Algebra Universalis, 29, 354–377.

    Article  MATH  MathSciNet  Google Scholar 

  101. Aguzzoli, S., Marra, V. (2010). Finitely presented MV-algebras with finite automorphism group. Journal of Logic and Computation, 20(4), 811–822.

    Article  MATH  MathSciNet  Google Scholar 

  102. Busaniche, M., Mundici, D. (2007). Geometry of Robinson consistency in Łukasiewicz logic. Annals of Pure and Applied Logic, 147, 1–22.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics “Ulisse Dini”, University of Florence, Viale Morgagni 67 A, 50134, Florence, Italy

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Mundici, D. (2011). Applications, Further Reading, Selected Problems. In: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic, vol 35. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0840-2_20

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