Professor

Deputy Dean of the Doctoral School at the University of Silesia in Katowice

President of the Upper Silesian Branch of the Polish Mathematical Society

Secretary of the Polish Society for Fuzzy Sets

Faculty of Science and Technology

University of Silesia in Katowice

Bankowa 14

40-007 Katowice

POLAND

Abstract

Fuzzy implication functions are one of the main mathematical operations in fuzzy logic. They generalize the classical two-valued implication to fuzzy logic, where the truth values belong to the unit interval [0; 1]. This family of functions plays a significant role in the development of fuzzy systems, see [1]. Recently, the so-called family of power based implications was introduced in [2] as a new family in which most of its members satisfy the invariance with respect to powers of a continuous t-norm [2, 3], an important additional property in approximate reasoning. The class of power based implications was characterized through, among others, the multiplicative Sincov's equation I(x; y) I(y; z) = I(x; z) in a concrete sub-domain, see [4]. In my presentation I present the most recent results concerning the multiplicative Sincov's equation, as well as some its generalizations, cf.[5].

References

[1] Baczynski, M. and Jayaram, B. Fuzzy Implications, volume 231 of Studies in Fuzziness and Soft Computing. Springer, Berlin Heidelberg, 2008.

[2] Massanet, S., Recasens, J., and Torrens, J. Fuzzy implication functions based on powers of continuous t-norms. Internat. J. Approx. Reason., 83:265-279, 2017.

[3] Massanet, S., Recasens, J., and Torrens, J. Corrigendum to fuzzy implication functions based on powers of continuous t-norms [Int. J. Approx. Reason. 83 (2017) 265279]. Internat. J. Approx. Reason., 104:144-147, 2019.

[4] Massanet, S., Recasens, J., and Torrens, J. Some characterizations of T-power based implications. Fuzzy Sets and Systems, 359:42-62, 2019.

[5] Baczynski, M., Fechner, W., and Massanet, S. On a generalization of multiplicative Sincov's equation for fuzzy implication functions. Fuzzy Sets and Systems, 451:196-205, 2022.

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