# Abstract:

Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see \cite{Combs_Andrews_1998,Jayaram2008} and \cite{Baczynski_Jayaram_2008}). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see \cite{Baczynski_Jayaram_2009}) $$f(\min(x+y,a))=\min(f(x)+f(y),b),$$ where $a,b>0$ and $f\colon[0,a]\to[0,b]$ is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation $$f(m_1(x+y))=m_2(f(x)+f(y)),$$ where $m_1,m_2$ are functions defined on some intervals of $\R$ satisfying additional assumptions. We analyze the cases when $m_2$ is injective and when $m_2$ is not injective.

# Keywords:

fuzzy implication, distributivity, fuzzy connectives, functional equations

# Classification:

03B52, 03E72, 39B99

# References:

1. M. Baczyński: On a class of distributive fuzzy implications. Int. J. Uncertain. Fuzziness Knowledge-Based Systems 9 (2001), 229-238.   CrossRef
2. M. Baczyński: On the distributivity of fuzzy implications over continuous and Archi\-medean triangular conorms. Fuzzy Sets and Systems 161 (2010), 1406-1419.   CrossRef
3. M. Baczyński: On the distributivity of fuzzy implications over representable uninorms. Fuzzy Sets and Systems 161 (2010), 2256-2275.   CrossRef
4. M. Baczyński and B. Jayaram: Fuzzy Implications. Studies in Fuzziness and Soft Computing 231, Springer, Berlin Heidelberg 2008.   CrossRef
5. M. Baczyński and B. Jayaram: On the distributivity of fuzzy implications over nilpotent or strict triangular conorms. IEEE Trans. Fuzzy Syst. 17 (2009), 590-603.   CrossRef
6. M. Baczyński and F. Qin: Some remarks on the distributive equation of fuzzy implication and the contrapositive symmetry for continuous, Archimedean t-norms. Int. J. Approx. Reason. 54 (2013), 290-296.   CrossRef
7. M. Baczyński, T. Szostok and W. Niemyska: On a functional equation related to distributivity of fuzzy implications. In: 2013 IEEE International Conference on Fuzzy Systems (FUZZ IEEE 2013) Hyderabad 2013, pp. 1-5.   CrossRef
8. J. Balasubramaniam and C. J. M. Rao: On the distributivity of implication operators over {T} and {S} norms. IEEE Trans. Fuzzy Syst. 12 (2004), 194-198.   CrossRef
9. W. E. Combs and J. E. Andrews: Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans. Fuzzy Syst. 6 (1998), 1-11.   CrossRef
10. W. E. Combs: Author's reply. IEEE Trans. Fuzzy Syst. 7 (1999), 371-373.   CrossRef
11. W. E. Combs: Author's reply. IEEE Trans. Fuzzy Syst. 7 (1999), 477-478.   CrossRef
12. B. De Baets: Fuzzy morphology: A logical approach. In: Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, and Neural Network Approach (B. M. Ayyub and M. M. Gupta, eds.), Kluwer Academic Publishers, Norwell 1997, pp. 53-68.   CrossRef
13. S. Dick and A. Kandel: Comments on Combinatorial rule explosion eliminated by a fuzzy rule configuration". IEEE Trans. Fuzzy Syst. 7 (1999), 475-477.   CrossRef
14. M. González-Hidalgo, S. Massanet, A. Mir and D. Ruiz-Aguilera: Fuzzy hit-or-miss transform using the fuzzy mathematical morphology based on T-norms. In: Aggregation Functions in Theory and in Practise (H. Bustince et al., eds.), Advances in Intelligent Systems and Computing 228, Springer, Berlin - Heidelberg 2013, pp. 391-403.   CrossRef
15. B. Jayaram: Rule reduction for efficient inferencing in similarity based reasoning. Int. J. Approx. Reason. 48 (2008), 156-173.   CrossRef
16. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
17. M. Kuczma: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality. Państwowe Wydawnictwo Naukowe (Polish Scientific Publishers) and Uniwersytet Śl\c aski, Warszawa-Kraków-Katowice 1985.   CrossRef
18. C. H. Ling: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189-212.   CrossRef
19. J. M. Mendel and Q. Liang: Comments on Combinatorial rule explosion eliminated by a fuzzy rule configuration". IEEE Trans. Fuzzy Syst. 7 (1999), 369-371.   CrossRef
20. F. Qin, M. Baczyński and A. Xie: Distributive equations of implications based on continuous triangular norms (I). IEEE Trans. Fuzzy Syst. 20 (2012), 153-167.   CrossRef
21. F. Qin and L. Yang: Distributive equations of implications based on nilpotent triangular norms. Int. J. Approx. Reason. 51 (2010), 984-992.   CrossRef
22. D. Ruiz-Aguilera and J. Torrens: Distributivity of strong implications over conjunctive and disjunctive uninorms. Kybernetika 42 (2006), 319-336.   CrossRef
23. D. Ruiz-Aguilera and J. Torrens: Distributivity of residual implications over conjunctive and disjunctive uninorms. Fuzzy Sets and Systems 158 (2007), 23-37.   CrossRef
24. E. Trillas and C. Alsina: On the law $[(p\wedge q)\to r]=[(p\to r)\vee(q\to r)]$ in fuzzy logic. IEEE Trans. Fuzzy Syst. 10 (2002), 84-88.   CrossRef