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The aim of this short paper is to give a simple look to the historical development of logical connectives for fuzzy sets and fuzzy logic. Concepts that have been (and still are) in the core of extensive theoretical research, like conjunction, disjunction, complement, subsethood and fuzzy conditionals, are considered in an informal way. An extensive list of essential references helps the interested reader to find sources for deeper study of the main subjects. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new construction method of a fuzzy implication from two given ones, called threshold generation method, is introduced. It is a generalization of the way of construction of the recently introduced h-implications, which are fully characterized in this work. The threshold generation method allows to control, up to a certain level, the increasingness on the second variable of the fuzzy implication through an adequate scaling on that variable of the two given implications. The natural propagation of the most usual properties of fuzzy implications from the initial ones to the constructed implication is studied and the necessary and sufficient conditions in order to ensure this propagation are presented. In particular, the preservation of the contrapositive symmetry on threshold generated implications needs of another construction method of a fuzzy implication from a given one and a fuzzy negation. [Copyright &y& Elsevier]

Abstract: In this paper, the recently introduced construction method of a fuzzy implication from two given ones, called threshold generation method, is studied. This method preserves several of the most usual properties of fuzzy implications from the initial ones to the threshold generated implication. In particular, the preservation of the exchange principle, the law of importation and the distributivities with t-norms and t-conorms, stating the minimal necessary conditions to ensure this fact is studied. From these results, some properties of h-implications, as a particular case of threshold generated implications, are pointed out. [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]

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]