Abstract
From the birth of fuzzy sets theory, several extensions have been proposed changing the possible membership values. Since fuzzy connectives such as t-norms and negations have an important role in theoretical as well as applied fuzzy logics, these connectives have been adapted for these generalized frameworks. Perhaps, an extension of fuzzy logic which generalizes the remaining extensions, proposed by Joseph Goguen in 1967, is to consider arbitrary bounded lattices for the values of the membership degrees. In this paper we extend the usual way of constructing fuzzy negations from t-norms for the bounded lattice t-norms and prove some properties of this construction.
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References
Alsina, C., Frank, M.J., Schweizer, B.: Associative Functions: Triangular Norms And Copulas. World Scientific Publishing Company (2006)
Alsina, C., Trillas, E., Valverde, L.: On non-distributive logical connectives for fuzzy set theory. Busefal 3, 18–29 (1980)
Baczyński, M., Jayaram, B.: Fuzzy Implications. Springer, Heidelberg (2008)
De Baets, B., Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104(1), 61–75 (1999)
Bedregal, B.C.: On interval fuzzy negations. Fuzzy Sets and Systems 161, 2290–2313 (2010)
Bedregal, B.C., Santos, H.S., Callejas-Bedregal, R.: T-norms on bounded lattices: t-norm morphisms and operators. In: Proceedings of 2006 IEEE International Conference on Fuzzy Systems, Vancouver, Canada, pp. 22–28 (2006)
Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1973)
Chang, C.C.: Algebraic analisys of many valued logics. Trans. of the American Mathematical Society 88, 467–490 (1958)
De Cooman, G., Kerre, E.E.: Order norms on bounded partially ordered sets. Fuzzy Methematics 2, 281–310 (1994)
Da Costa, C.G., Bedregal, B.C., Doria Neto, A.D.: Relating De Morgan triples with Atanassov’s intuitionistic De Morgan triples via automorphisms. Int. J. Approximate Reasoning 52(4), 473–487 (2010)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Goguen, J.: L-fuzzy sets. Journal of Mathematical Analysis and Applications 18(1), 145–174 (1967)
Grätzer, G.: Lattice Theory: First Concepts and Distributive Lattices. Dover Publications (2009)
Grätzer, G.: Lattice Theory: Foundation. Birkhäuser (2011)
Hájek, P.: Metamathematics of Fuzzy Logic. Springer, Heildelberg (2001)
Hungerford, T.W.: Algebra. Springer, New York (1974)
Karaçal, F.: On the directed decomposability of strong negations and S-implications operators on product lattices. Information Sciences 176(20) (2006)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht (2000)
Menger, K.: Statistical metrics. Proc. Nat. Academic Science 28(12), 535–537 (1942)
Mesiar, R., Komorníková, M.: Aggregation Functions on Bounded Posets. In: 35 Years of Fuzzy Sets Theory: Celebratory volume Dedicated to the Retirement of Etiene E. Kerre, pp. 3–17. Springer (2010)
Nguyen, H.T., Walker, E.A.: A First Course in Fuzzy Logics, 2nd edn. CRC Press, Boca Raton (2000)
Palmeira, E.S., Bedregal, B.: Extension of fuzzy logic operators defined on bounded lattices via retractions. Computers and Mathematics with Applications 63, 1026–1038 (2012)
Reiser, R.H.S., Dimuro, G.P., Bedregal, B., Santos, H.S., Callejas-Bedregal, R.: S-implications on complete lattice and the interval constructor. Tendências em Matemática Aplicada e Computacional – TEMA 9(1), 143–154 (2008)
Schweizer, B., Sklar, A.: Associative functions and abstract semigroups. Publ. Math. Dedrecen 10, 69–81 (1963)
Trillas, E.: Sobre funciones de negación en la teoria de los conjuntos difusos. Stochastica 3(1), 47–59 (1979)
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
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Bedregal, B., Beliakov, G., Bustince, H., Fernandez, J., Pradera, A., Reiser, R.H.S. (2012). Negations Generated by Bounded Lattices t-Norms. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 299. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31718-7_34
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DOI: https://doi.org/10.1007/978-3-642-31718-7_34
Publisher Name: Springer, Berlin, Heidelberg
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