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\(\mu \)MV-algebras: An approach to fixed points in Łukasiewicz logic. (English) Zbl 1183.06006
Chang introduced MV-algebras to give a purely algebraic proof of the completeness of the Łukasiewicz axioms. In his paper [J. Funct. Anal. 65, 15–63 (1986; Zbl 0597.46059)], the present reviewer established a categorical equivalence between MV-algebras and unital lattice-ordered abelian groups. Divisible MV-algebras are the correspondents of divisible unital lattice-ordered abelian groups. By further equipping a divisible MV-algebra with the well-known \(\Delta\)-operator, one has divisible MV\(_\Delta\)-algebras. The author proves that by expanding MV-algebras to structures allowing minimal and maximal fixed points, one obtains a term-equivalent variant of divisible MV\(_\Delta\)-algebras. He then derives various kinds of results on these algebras, such as subdirect representation, completeness, amalgamation and a representation of free algebras. For background on MV-algebras see the monograph [R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)].

MSC:
06D35 MV-algebras
03B50 Many-valued logic
03G25 Other algebras related to logic
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[1] Aglianó, P.; Montagna, F., Varieties of BL algebras I: general properties, J. pure appl. algebra, 181, 105-129, (2003) · Zbl 1034.06009
[2] Baaz, M.; Veith, H., Interpolation in fuzzy logic, Arch. math. logic, 38, 461-489, (1999), URL \(\langle\)http://www.springerlink.com/openurl.asp?genre=article&eissn=1432-0665&volume=38&issue=7&spage=461/⟩ · Zbl 0936.03026
[3] Beckmann, A.; Preining, N., Linear Kripke frames and Gödel logics, J. symbolic logic, 72, 1, 26-44, (2007) · Zbl 1118.03016
[4] Chang, C., Algebraic analysis of many valued logic, Trans. amer. math. soc., 88, 467-490, (1958), \(\langle\)http://links.jstor.org/sici?sici=0002-9947%28195807%2988%3A2%3C467%3AAAOMVL%3E2.0.CO%3B2-A/⟩ · Zbl 0084.00704
[5] C. Chang, A new proof of the completeness of the Łukasiewicz axioms, Trans. Amer. Math. Soc. 93(74-80). · Zbl 0093.01104
[6] R. Cignoli, I. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-valued Reasoning, Trends in Logic, Studia Logica Library, Vol. 7, Kluwer Academic, Dordrecht, 2000.
[7] Esteva, F.; Godo, L.; Montagna, F., The \(\operatorname{L} \Pi\) and \(\operatorname{L} \Pi \frac{1}{2}\) logics: two complete fuzzy systems joining łukasiewicz and product logics, Arch. math. logic, 40, 39-67, (2001) · Zbl 0966.03022
[8] Flaminio, T., NP-containment for the coherence test of assessments of conditional probability: a fuzzy logical approach, Arch. math. logic, 46, 3-4, 301-319, (2007) · Zbl 1110.03012
[9] B. Gerla, Many-valued logics of continuous t-norms and their functional representation, Ph.D. thesis, Universià di Milano, 2001.
[10] Gerla, B., Rational łukasiewicz logic and divisible MV algebras, Neural network world, 11, 159-194, (2001)
[11] Gurevich, Y.S.; Kokorin, A.I., Universal equivalence of order abelian groups, Algebra i logika, 2, 37-39, (1963)
[12] P. Hájek, Metamathematics of fuzzy logic, Trends in Logic, Studia Logica Library, Vol. 4, Kluwer Academic, Berlin, 1998.
[13] Höhle, U., Commutative residuate l-monoids, (), 53-106 · Zbl 0838.06012
[14] Kozen, D., Results on the propositional mu-calculus, Theoret. comput. sci., 27, 333-354, (1983) · Zbl 0553.03007
[15] J. Łukasiewicz, A. Tarski, Untersuchungen über den aussagenkalkül, Comptes Rendus de la Societe de Sciences et des Letters de Varsovie iii (23) (1930) 1-21.
[16] McKenzie, R., On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model, J. symbolic logic, 40, 186-196, (1975) · Zbl 0316.02052
[17] McKenzie, R.; McNulty, G.; Taylor, W., Algebras lattices varieties, (1987), Wadsworth and Brooks/Cole Monterey CA
[18] Montagna, F., Functorial representation theorems for \(\textit{MV}_\Delta\) algebras, J. algebra, 238, 99-125, (2001) · Zbl 0987.06012
[19] Montagna, F., Interpolation and Beth’s property in propositional many-valued logics: a semantic investigation, Ann. pure appl. logic., 141, 1-2, 148-179, (2006) · Zbl 1094.03011
[20] Montagna, F.; Sacchetti, L., Kripke-style semantics for many-valued logics, Math. logic quater., 49, 6, 629-641, (2003) · Zbl 1035.03010
[21] Mundici, D., Interpretation of \(\textit{AFC}^*\) algebras in łukasiewicz, sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059
[22] Mundici, D., A constructive proof of mcnaughton theorem in infinite-valued logic, J. symbolic logic, 59, 2, 596-602, (1994), URL \(\langle\)http://links.jstor.org/sici?sici=0022-4812%28199406%2959%3A2%3C596%3AACPOMT%3E2.0.CO%3B2-K/⟩ · Zbl 0807.03012
[23] Rose, A.; Rosser, J., Fragments of many-valued statement calculi, Trans. amer. math. soc., 87, 1-53, (1958) · Zbl 0085.24303
[24] L. Spada, \(\operatorname{L} \Pi\) logic with fixed points, preprint (2006).
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