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The law of importation is an important property of fuzzy implication functions because of its interesting applications in approximate reasoning and image processing. In this paper, as a continuation of the paper [S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with respect to a fixed uninorm—Part I,” IEEE Trans. Fuzz Syst. , to be published.], the characterization of all fuzzy implication functions that satisfy the law of importation with respect to a given uninorm $U$ , and having an $\alpha$ -section that is a continuous fuzzy negation is given for the cases when the uninorm $U$ lies in one of the most used classes of uninorms. As the case when the uninorm $U$ is in ${\mathcal U}_{\min }$ was already solved in [S. Massanet, D. Ruiz-Aguilera, and J. Torrens, “Characterization of a class of fuzzy implication functions satisfying the law of importation with respect to a fixed uninorm—Part I,” IEEE Trans. Fuzz Syst., to be published.], this paper focuses in the cases when $U$ is an idempotent uninorm, a representable uninorm, or a uninorm continuous in the open unit square. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose a method to construct aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers from a couple of discrete aggregation functions. The interest on these discrete fuzzy numbers lies on the fact that they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms. Finally, a linguistic decision making model based on a pair of aggregation functions defined on discrete fuzzy numbers is given. [Copyright &y& Elsevier]

Abstract: In this paper aggregation functions defined on the set of all discrete fuzzy numbers whose support is a subset of consecutive natural numbers are introduced and the particular cases of uninorms and nullnorms are studied in detail. These aggregation functions are constructed from discrete aggregation functions (defined on a finite chain) and they are applied to the aggregation of subjective evaluations. [Copyright &y& Elsevier]

Abstract: Some open problems on fuzzy implications dealing with the so-called importation law are studied and totally or partially solved in this work. A weaker version of the law of importation, called the weak law of importation, is introduced. The relationships of these two properties and the exchange principle are studied. In particular, it is proved that the law of importation is stronger than the exchange principle. On the other hand, the three properties are equivalent for some kind of fuzzy implications, those that satisfy a boundary property. Along this study, new characterizations of (S,N)-implications, R-implications and their counterparts for uninorms based on the weak law of importation are showed. [Copyright &y& Elsevier]

Although the concept of reverse is given in (Publ. Math. Debrecen 12 (1973) 21–39) not only for a t-norm but also for any associative operator, the definition does not work in so general a case. In this paper, the concept of reverse is extended in an appropriate way to some kinds of associative operators like t-conorms, uninorms and t-operators. All continuous and reversible t-conorms are characterized and, after some results, an analogous characterization is also given for all semi-continuous reversible uninorms and all continuous reversible t-operators. In the last case, reversibility becomes a kind of transformation that converts a uninorm into a t-operator and viceversa. Finally, related to reversibility, uninorms and t-operators of moderate growth are studied. [Copyright &y& Elsevier]

In this paper all uninorms locally internal in the region A ( e ) (given by the complement in [ 0 , 1 ] 2 of [ 0 , e ] 2 ∪ [ e , 1 ] 2 , where e is the neutral element of the uninorm) having continuous underlying operators are studied and characterized, by distinguishing some cases. When the underlying t-norm and t-conorm are not given by ordinal sums, it is proved that uninorms locally internal in A ( e ) are in fact all possible uninorms with these underlying operators (except when both the t-norm and the t-conorm are strict in which case there is also the class of representable uninorms), leading to a finite number of possibilities. When at least one of the continuous underlying operators is given by an ordinal sum, again there are other possible uninorms than those that are locally internal in A ( e ) , but all uninorms with this property are also characterized. In this case, infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm. [ABSTRACT FROM AUTHOR]

Distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives. For this reason, the distributivity property has been extensively studied for several families of operations like triangular norms and conorms, some kinds of uninorms and nullnorms (also called t-operators) and even for some generalizations of them. In this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms, leading to many new solutions. [ABSTRACT FROM AUTHOR]