Abstract
Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Aczél J. (1996) Lectures on functional equations and their applications. New York, Academic Press
Alsina, C. (1997). On connectives in fuzzy logic satisfying the condition S (T 1 (x, y), T 2 (x, N(y))) = x. Proceedings of the FUZZ’IEEE-97, Barcelona, pp.149–153.
Alsina, C. (1996). As you like them: connectives in fuzzy logic, ISMVL 96, Santiago de Compostela. In Proceedings of the ISMVL, Univ. Santiago de Compostela, 1–7.
Alsina C., Frank M.J., Schweizer B. (2006). Associative functions: triangular norms and copulas, Hackensack, London, Singapore: World Scientific
Alsina, C., & Trillas, E. (1999). On (S,N)-implications in fuzzy logic consistent with T-conjunctions. In Proceedings of the 1999 EUSFLAT-ESTYLF joint conference, Univ. Illes Balears, Palma, pp. 425–428.
Alsina C., Trillas E. (2003) When (S, N)-implications are (T, T 1)-conditional functions?. Fuzzy Sets and Systems 134: 305–310
Alsina C., Trillas E., Valverde L. (1983). On some logical connectives for fuzzy set theory. Journal of Mathematical Analysis and Applications, 93(1): 15–26
Alsina C., Trillas E. (2002). On the functional equation S 1 (x, y) = S 2 (x, T(N(x), y)). In: Darczy Z., Ples Z. (eds). Functional equations results and advances. Kluwer, Dordrecht, pp. 323–334
Frank M.J. (1979). On the simultaneous associativity of F(x, y) and x + y − F(x, y). Aequationes Mathamatics 19: 194–226
Frank, M. J. (1981), An equation which links associative functions. Abstracts American Mathematical Society, 1, p. 128.
Klement E.P., Mesiar R., Pap E. (2000). Triangular norms. Dordrecht, Kluwer
Schweizer B., Sklar A. (1983). Probabilistic metric spaces. New York, Elsevier North-Holland
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alsina, C., Trillas, E. On the law S(S(x, y), T(x, y)) = S(x, y) of fuzzy logic. Fuzzy Optim Decis Making 6, 99–107 (2007). https://doi.org/10.1007/s10700-007-9006-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-007-9006-x