Abstract
In their paper [29], Rose and Rosser gave a proof of the completeness of the infinite-valued sentential calculus of Lukasiewicz. Their proof is syntactic in nature. In two subsequent papers [7], [8], Chang introduced MV algebras (many-valued algebras), and gave an algebraic proof of the completeness theorem using these structures. Thus, MV algebras were originally introduced as algebraic counterparts of many-valued logic, just as Boolean algebras are the algebraic counterpart of classical, two-valued, logic. Recently, however, MV algebras have found novel and surprising applications, and today they are studied per se. As proved in [19], MV algebras are categorically equivalent to abelian lattice-groups with strong unit. They are also equivalent to bounded commutative BCK algebras [20], and to several other mathematical structures. Composition with the Grothendieck functor K O yields a one-one correspondence between countable MV algebras and approximately finite-dimensional (AF) C*-algebras whose Murray von Neumann ordering of projections is a lattice order. This correspondence has many applications [10], [19], [21], [22], [23]. AF C*-algebras are the mathematical counterpart of quantum spin systems.
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References
L.P. Belluce, Semisimple algebras of infinite valued logic and bold fuzzy set theory, Canadian J. Math., 38 (1986) 1356–1379.
L.P. Belluce, A. Di Nola, A. Lettieri, Local MV-algebras, preprint.
L.P.A Belluce, A. Di Nola, S. Sessa, On the prime spectrum of MV-algebras, preprint.
E.R. Berlekamp, Block coding for the binary symmetric channel with noiseless, delayless feedback, In: “Error Correcting Codes”, Wiley, New York, 1968, pp.61–68.
A. Bigard, K. Keimel, S. Wolfenstein, “Groupes et anneaux reticules”, Lecture Notes in Mathematics, 608, 1977.
G. Birkhoff, “Lattice Theory”, Amer. Math. Soc. Colloquium Publ., Vol. 25, 1967.
C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc, 88 (1958) 467–490.
C.C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc, 93 (1959) 74–80.
C.C. Chang and H. J. Keisler, “Model Theory”, North-Holland, Amsterdam, 1973.
R. Cignoli, G. A. Elliott, D. Mundici, Reconstructing C*-algebras from their Murray von Neumann orders, Advances in Mathematics, to appear.
J. Czyzowicz, D. Mundici, A. Pelc, Ulam’s searching game with lies, J. Combinatorial Theory, Series A, 52 (1989) 62–76.
A. Di Nola, Representation and reticulation by quotients of MV-algebras, Ricerche di Matematica (Naples), to appear.
A. Di Nola, G. Gerla, Non-standard fuzzy sets, Fuzzy Sets and Systems, 18 (1986) 173–181.
A. Di Nola, A. Lettieri, Perfect MV-algebras are categorically equivalent to abelian l-groups, preprint.
G. Grätzer, “Universal Algebra”, 2nd ed., Springer Verlag, New York, 1979.
R. Grigolia, Algebraic analysis of Lukasiewicz-Tar ski n-valued logical systems, In: “Selected Papers on Lukasiewicz Sentential Calculi” (R. Wòjcicki and G. Malinowski, Editors), Polish Acad. of Sciences, Ossolineum, Wroclaw, 1977, pp. 81–92.
P. Mangani, On certain algebras related to many-valued logics (Ital.), Bollettino Unione Matematica Italiana, (4)8, (1973) 68–78.
R. McNaughton, A theorem about infinite-valued sentential logic, Journal of Symbolic Logic, 16 (1951) 1–13.
D. Mundici, Interpretation of AF C*-algebras in Lukasiewicz sentential calculus, J. Functional Analysis, 65 (1986) 15–63.
D. Mundici, MV-algebras are categorically equivalent to bounded commutative BCK-algebras, Math. Japonica, 31 (1986) 889–894.
D. Mundici, Farey stellar subdivisions, ultrasimplicial groups, and K 0 of AF C*-algebras, Advances in Mathematics, 68 (1988) 23–39.
D. Mundici, Free products in the category of abelian l-groups with strong unit, Journal of Algebra, 113 (1988) 89–109.
D. Mundici, The C*-algebras of three-valued logic, In: “Proceedings Logic Colloquium’ 88, Studies in Logic and the Foundations of Mathematics”, North-Holland, Amsterdam (1989) pp. 61–77.
D. Mundici, The complexity of adaptive error-correcting codes, Lecture Notes in Computer Science, 533 (1991) 300–307.
D. Mundici, Ulam games, Lukasiewicz logic, and AF C*-algebras, Fundamenta Informaticae, to appear.
D. Mundici, The logic of Ulam’s game with lies, “Cambridge Studies in Probability, Induction and Decision Theory”, to appear.
D. Mundici, A constructive proof of McNaughton’s theorem, to appear.
A. Pelc, Solution of Ulam’s problem on searching with a lie, J. Combinatorial Theory, Series A, 44 (1987)129–140.
A. Rose, J. B. Rosser, Fragments of many-valued statement calculi, Trans. Atner. Math. Soc, 87 (1958) 1–53.
S.M. Ulam, “Adventures of a Mathematician”, Scribner’s, New York, 1976, p.281.
A. Tarski, J. Lukasiewicz, Investigations into the sentential calculus, In: “Logic, Semantics, Metamathematics”, Oxford University Press 1956, pp.38–59. Reprinted by Hackett Publishing Company, 1981.
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Di Nola, A. (1993). MV Algebras in the Treatment of Uncertainty. In: Lowen, R., Roubens, M. (eds) Fuzzy Logic. Theory and Decision Library, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2014-2_12
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