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In this paper discrete uninorms U such that their underlying t-norm T and t-conorm S are smooth are characterized. The different cases combining when T is the minimum or the Łukasiewicz t-norm and S is the maximum or the Łukasiewicz t-conorm, are given separately and the number of discrete uninorms with these underlying operators is given. The same is done for the general case when T and/or S are smooth, but ordinal sums. [ABSTRACT FROM AUTHOR]

Abstract: This work deals with two kinds of implications derived from uninorms continuous in : S-implications and R-implications. In both cases, the general expression of such implications is found and several properties are studied. In particular, the distributivity of the S- and R-implications over conjunctive and disjunctive uninorms is investigated. [Copyright &y& Elsevier]

This article deals with the construction of relevancy transformation (RET) operators for fuzzy systems. The notion of pseudo-duality is introduced to obtain new RET operators, and t-norms, t-conorms, nullnorms, and uninorms are used in different ways for the same purpose. Finally, several other methods to construct new RET operators from old ones are pointed out. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 155–171, 2006. [ABSTRACT FROM AUTHOR]

Abstract: In this paper a method of defining commutative semicopulas from fuzzy negations is introduced. Some properties are investigated that lead to understand these semicopulas as non-associative generalizations of the Łukasiewicz t-norm. In particular, it is proved that some well known examples of copulas and t-norms can be obtained by this method. Moreover, any commutative semicopula constructed by this method can be always obtained from a negation N which is symmetric with respect to the diagonal. Then, those symmetric fuzzy negations N for which the corresponding semicopula is a copula are characterized. Also, several examples of symmetric negations N are given such that the corresponding semicopula is a t-norm. [Copyright &y& Elsevier]

Abstract: The law of importation is exhaustively studied for fuzzy implications derived from uninorms. All widely used classes of implications, that is (U,N), RU, QL and D-implications derived from most usual classes of uninorms are investigated. Along this study many new solutions of the law of importation appear different from those already known for implications derived from t-norms and t-conorms. [Copyright &y& Elsevier]

Abstract The translation of the classical Non-Contradiction (NC) principle, as well as its dual law: the Excluded-Middle (EM) principle, to the framework of fuzzy logic leads to a well-known functional equation involving appropriate aggregation and negation functions. When the involved negation is the natural negation of a fuzzy implication function, the NC principle becomes specially important because then it is a necessary condition for the fulfillment of the Modus Ponens inequality, as well as for the residuation property for residuated implications. In this paper, the functional equation corresponding to the NC principle is deeply studied in the case when both, the aggregation function and the fuzzy negation, are as general as possible. Moreover, the results are applied to the case when the fuzzy negation is the natural negation of a fuzzy implication function, extending this study to the most usual classes of fuzzy implication functions. [ABSTRACT FROM AUTHOR]

Abstract Functional equations involving aggregation functions play an important role in fuzzy sets and fuzzy logic theory. The modularity equation is a kind of restricted general associative equations. In the literature, the already known results concerning the modularity equation for two uninorms are based on the assumption that both uninorms lie in one of the most studied classes of uninorms. In this study we explore the modularity equation involving uninorms in a most general setting. Specifically, we investigate the modularity between two uninorms in the cases when one uninorm belongs to any of the most studied classes of uninorms, but the other one is any uninorm with no further assumptions. Further, many new solutions appear from this new point of view that were not included in previous approaches. [ABSTRACT FROM AUTHOR]

Since their introduction in 2012 and 2013 by P. Grzegorzewski, probabilistic S -implications and survival S -implications have attracted the efforts of some researchers due to the connection that they represent between probability theory and fuzzy logic. In this paper, the characterizations of these two families of fuzzy implication functions are presented jointly with the characterization of the material implications derived from co-copulas and the classical negation. Even more, it is proved that the three families are actually the same. Thus every result concerning one of these families can be straightforwardly rewritten in terms of the other two families. [ABSTRACT FROM AUTHOR]