Abstract
Associated with any [0,1]-valued propositional logic with a complete algebraic semantics, one can consider algebras of families of fuzzy sets over a classical universe, endowed with the appropriate operations. For the three most important schematic extensions of Hájek’s Basic (Fuzzy) Logic, we investigate the existence and the structure of such algebras of fuzzy sets in the corresponding algebraic varieties. In the general case of Basic Logic itself, and in sharp contrast to the three aforementioned extensions, we show that there actually exist different, incomparable notions of algebras of fuzzy sets.
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References
Aglianò, P., Ferreirim, I.M.A., Montagna, F.: Basic hoops: an algebraic study of continuous t-norms. Studia Logica 87(1), 73–98 (2007)
Aglianò, P., Montagna, F.: Varieties of BL-algebras I: general properties. Journal of Pure and Applied Algebra 181, 105–129 (2003)
Amer, K.: Equationally complete classes of commutative monoids with monus. Algebra Universalis 18(1), 129–131 (1984)
Belluce, L.P.: Semisimple algebras of infinite valued logic and bold fuzzy set theory. Canad. J. Math. 38(6), 1356–1379 (1986)
Burris, S., Sankappanavar, H.P.: A course in universal algebra. Graduate Texts in Mathematics, vol. 78. Springer, New York (1981)
Butnariu, D., Klement, E.P.: Triangular norm-based measures and games with fuzzy coalitions. Kluwer Academic Publishers, Dordrecht (1993)
Chang, C.C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)
Cignoli, R.L.O., D’Ottaviano, I.M.L., Mundici, D.: Algebraic foundations of many-valued reasoning. Trends in Logic—Studia Logica Library, vol. 7. Kluwer Academic Publishers, Dordrecht (2000)
Cignoli, R., Torrens, A.: An algebraic analysis of product logic. Mult.-Valued Log. 5(1), 45–65 (2000)
Cignoli, R., Torrens, A.: Free cancellative hoops. Algebra Universalis 43(2-3), 213–216 (2000)
Esteva, F., Godo, L.: Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets and Systems 124(3), 271–288 (2001)
Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Logic 39(2), 103–124 (2000)
Esteva, F., Godo, L., Montagna, F.: Equational characterization of the subvarieties of BL generated by t-norm algebras. Studia Logica 76(2), 161–200 (2004)
Flaminio, T., Marchioni, E.: T-norm-based logics with an independent involutive negation. Fuzzy Sets and Systems 157(24), 3125–3144 (2006)
Glass, A.M.W.: Partially ordered groups. Series in Algebra, vol. 7. World Scientific Publishing Co. Inc., Singapore (1999)
Hájek, P.: Metamathematics of fuzzy logic. Trends in Logic—Studia Logica Library, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)
Montagna, F.: Generating the variety of BL-algebras. Soft Computing 9, 869–874 (2005)
Johnstone, P.T.: Stone spaces. Cambridge Studies in Advanced Mathematics, vol. 3. Cambridge University Press, Cambridge (1982)
Kalman, J.: Lattices with involution. Trans. Amer. Math. Soc. 87, 485–491 (1958)
Klement, E.P., Mesiar, R., Pap, E.: Triangular norms. Trends in Logic—Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)
Trillas, E.: Negation functions in the theory of fuzzy sets. Stochastica 3(1), 47–60 (1979)
Rosenstein, J.G.: Linear orderings. Pure and Applied Mathematics, vol. 98. Academic Press Inc.[Harcourt Brace Jovanovich Publishers], London (1982)
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
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Aguzzoli, S., Gerla, B., Marra, V. (2009). Algebras of Fuzzy Sets in Logics Based on Continuous Triangular Norms. In: Sossai, C., Chemello, G. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2009. Lecture Notes in Computer Science(), vol 5590. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02906-6_75
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DOI: https://doi.org/10.1007/978-3-642-02906-6_75
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02905-9
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