On the distributivity of uninorms over nullnorms

This paper investigates the functional equation of distributivity of uninorms over nullnorms. We consider the case where the uninorm is continuous in (0,1)^2 or is representable. It has been proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributive equation holds. And then combining the structures of the uninorm and the nullnorm, we give the characterization of the pair of the functions of the uninorm and the nullnorm. Moreover, when the nullnorm is continuous, we obtain the sufficient and necessary conditions under which the distributive equation holds.

[1]  Fengxia Zhang,et al.  On the distributivity of fuzzy implications over continuous Archimedean t-conorms and continuous t-conorms given as ordinal sums , 2012, Fuzzy Sets Syst..

[2]  Daniel Ruiz,et al.  Distributivity and conditional distributivity of a uninorm and a continuous t-conorm , 2006, IEEE Transactions on Fuzzy Systems.

[3]  Joan Torrens,et al.  On locally internal monotonic operations , 2003, Fuzzy Sets Syst..

[4]  Joan Torrens,et al.  On Distributivity and Modularity in de Morgan Triplets , 1996, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[5]  Shi-kai Hu,et al.  The structure of continuous uni-norms , 2001, Fuzzy Sets Syst..

[6]  Joan Torrens,et al.  t-Operators , 1999, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[7]  J. Balasubramaniam,et al.  On the distributivity of implication operators over T and S norms , 2004, IEEE Transactions on Fuzzy Systems.

[8]  Enric Trillas,et al.  On almost distributive Łukasiewicz triplets , 1992 .

[9]  E. Trillas,et al.  On the law [p/spl and/q/spl rarr/r]=[(p/spl rarr/r)V(q/spl rarr/r)] in fuzzy logic , 2002 .

[10]  Ronald R. Yager,et al.  Uninorm aggregation operators , 1996, Fuzzy Sets Syst..

[11]  Joan Torrens,et al.  The distributivity condition for uninorms and t-operators , 2002, Fuzzy Sets Syst..

[12]  Ronald R. Yager,et al.  Structure of Uninorms , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[13]  Bernard De Baets,et al.  The functional equations of Frank and Alsina for uninorms and nullnorms , 2001, Fuzzy Sets Syst..

[14]  Qin Feng,et al.  The distributive equations for idempotent uninorms and nullnorms , 2005, Fuzzy Sets Syst..