Abstract
Triangular norms or t-norms, in short, and automorphisms are very useful to fuzzy logics in the narrow sense. However, these notions are usually limited to the set [0,1].
In this paper we will consider a generalization of the t-norm notion for arbitrary bounded lattices as a category where these generalized t-norms are the objects and generalizations of automorphisms are the morphisms of the category. We will prove that, this category is an interval category, which roughly means that it is a Cartesian category with an interval covariant functor.
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References
Alsina, C., Trillas, E., Valverde, L.: On non-distributive logical connectives for fuzzy set theory. Busefal 3, 18–29 (1980)
Asperty, A., Longo, G.: Categories, Types and Structures: An introductionn to category theory for the working computer scientist. MIT Press, Cambridge (1991)
Barr, M., Well, C.: Category Theory for Computing Scientist. Prentice Hall International (UK) Ltda, Englewood Cliffs (1990)
Bedregal, B.C., Santos, H.S., Callejas-Bedregal, R.: T-Norms on Bounded Lattices: t-norm morphisms and operators. In: IEEE Int. Conference on Fuzzy Systems, Vancouver, July 16-21, 2006, pp. 22–28. IEEE Computer Society Press, Los Alamitos (2006)
Bedregal, B.C., Takahashi, A.: The best interval representation of t-norms and automorphisms. Fuzzy Sets and Systems 157, 3220–3230 (2006)
Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Management Science 17, B141–B164 (1970)
Bustince, H., Burillo, P., Soria, F.: Automorphisms, negations and implication operators. Fuzzy Sets and Systems 134, 209–229 (2003)
Callejas-Bedregal, R., Bedregal, B.C.: Interval categories.In: de Janeiro, R. (ed.) Proceedings of IV Workshop on Formal Methods, pp. 139–150 (2001), (available in http://www.dimap.ufrn.br/~bedregal/main-publications.html )
Cornelis, G., Deschrijver, G., Kerre, E.E.: Advances and Challenges in Interval-Valued Fuzzy Logic. Fuzzy Sets and Systems 157, 622–627 (2006)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)
De Baets, B., Mesiar, R.: Triangular norms on product lattices. Fuzzy Sets and Systems 104, 61–75 (1999)
De Cooman, G., Kerre, E.E.: Order norms on bounded partially ordered sets. Journal Fuzzy Mathematics 2, 281–310 (1994)
Eytan, M.: Fuzzy Sets: A topos-logical point of view. Fuzzy Sets and Systems 5, 47–67 (1981)
Gehrke, M., Walker, C., Walker, E.: De Morgan systems on the unit interval. International Journal of Intelligent Systems 11, 733–750 (1996)
Goguen, J.: L-fuzzy sets. Journal of Mathematics Analisys and Applications 18, 145–174 (1967)
Gottwald, S.: Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches. Studia Logica 84, 23–50 (2006)
Grätzer, G.: General Lattice Theory. Academic Press, New York (1978)
Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publisher, Dordrecht (1998)
Höhle, U.: GL-quantales: Q-valued sets and their singletons. Studia Logica 61, 123–148 (1998)
Kulisch, U., Miranker, W.: Computer Arithmetic in Theory and Practice. Academic Press, London (1981)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic publisher, Dordrecht (2000)
Klement, E.P., Mesiar, R.: Semigroups and Triangular Norms. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, and Probalistic Aspects of Triangular Norms, Elsevier, Amsterdam (2005)
Menger, K.: Statical metrics. Proc. Nat. Acad. Sci 37, 535–537 (1942)
Ray, S.: Representation of a Boolean algebra by its triangular norms. Matheware & Soft Computing 4, 63–68 (1997)
Schweizer, B., Sklar, A.: Associative functions and statistical triangle inequalities. Publ. Math. Debrecen 8, 169–186 (1961)
Van Gasse, B., Cornelis, G., Deschrijver, G., Kerre, E.E.: On the Properties of a Generalized Class of t-Norms in Interval-Valued Fuzzy Logics. New Mathematics and Natural Computation 2, 29–42 (2006)
Walters, R.F.C.: Categories and Computer Sciences. Cambridge University Press, Cambridge (1991)
Wang, Z.D., Yu, Y.D.: Pseudo t-norms an implication operators on a complete Brouwerian lattice. Fuzzy Sets and Systems 132, 113–124 (2002)
Wyler, O.: Lecture Notes on Topoi and Quasitopoi. World Scientific, Singapore (1991)
Yager, R.R.: An approach to inference in approximate reasoning. International Journal on Man-Machine Studies 13, 323–338 (1980)
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
Zuo, Q.: Description of strictly monotonic interval and/or operations. In: APIC’S Proceedings: International Workshop on Applications of Interval Computations, pp. 232–235 (1995)
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Bedregal, B.C., Callejas-Bedregal, R., Santos, H.S. (2007). Bounded Lattice T-Norms as an Interval Category. In: Leivant, D., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2007. Lecture Notes in Computer Science, vol 4576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73445-1_3
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DOI: https://doi.org/10.1007/978-3-540-73445-1_3
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