Daniel Ruiz-Aguilera

Some remarks on the characterization of idempotent uninorms

In this paper the characterization of idempotent uninorms given in [21] is revisited and some technical aspects are corrected. Examples clarifying the situation are given and the same characterization is translated in terms of symmetrical functions. The particular cases of left-continuity and right-continuity are studied retrieving the results in [7].

S- and R-implications from uninorms continuous in ]0,1[2 and their distributivity over uninorms

This work deals with two kinds of implications derived from uninorms continuous in ]0,1[^2: 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.

IDEMPOTENT UNINORMS ON FINITE ORDINAL SCALES

In this paper we characterize all idempotent uninorms defined on a finite ordinal scale. It is proved that any such discrete idempotent uninorm is uniquely determined by a decreasing function from the set of scale elements not greater than the neutral element to the set of scale elements not smaller than the neutral element, and vice versa. Based on this one-to-one correspondence, the total number of discrete idempotent uninorms on a finite ordinal scale of n + 1 elements is equal to 2n.

A survey on the existing classes of uninorms

This paper wants to be a compilation of the different existing classes of uninorms. From their introduction, uninorms have been extensively studied not only as aggregation operators but also as logical connectives. The study of both aspects has produced many results on this kind of operators and many different classes have appeared. This work does not aim to be exhaustive since it is not possible to recall all known results about all classes of uninorms in a reduced space. Thus, we only want to state the general research lines of these classes in the two main frameworks where uninorms have been studied: the unit interval (0, 1) and the discrete setting. However, we will also compile the references where more details about all the existing classes of uninorms can be found, for the convenience of the interested reader. Uninorms in other frameworks are also recalled and finally, a section devoted to applications of uninorms is included.

Migrative uninorms and nullnorms over t-norms and t-conorms

In this paper the notions of α-migrative uninorms and nullnorms over a fixed t-norm T and over a fixed t-conorm S are introduced and studied. All cases when the uninorm U lies in any one of the most usual classes of uninorms are analyzed, characterizing all solutions of the migrativity equation for all possible combinations of U and T and for all possible combinations of U and S. A similar study is done for nullnorms.

On the Choice of the Pair Conjunction–Implication Into the Fuzzy Morphological Edge Detector

In this paper, the fuzzy morphological gradients from the fuzzy mathematical morphologies based on t-norms and conjunctive uninorms are deeply analyzed in order to establish which pair of conjunction and fuzzy implications are optimal, in accordance with their performance in edge detection applications. A novel three-step algorithm based on the fuzzy morphology is proposed. The comparison is performed by means of the so-called Pratts figure of merit. In addition, a statistical analysis is carried out to study the relationship between the different configurations and to establish a classification of the conjunctions and implications considered. Both the objective measure and the statistical analysis conclude that the pairs nilpotent minimum t-norm and the Kleene-Dienes implication, and the idempotent uninorm obtained with the classical negation as a generator and its residual implication, are the best configurations in this approach, because they also obtain competitive results with respect to other approaches.

A characterization of a class of uninorms with continuous underlying operators

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.

On Migrative t-Conorms and Uninorms

In this paper the notions of α-migrative t-conorms over a fixed t-conorm S 0, and α-migrative uninorms over another fixed uninorm U 0 with the same neutral element are introduced. All continuous t-conorms that are α-migrative over the maximum, the probabilistic sum and the Łukasiewicz t-conorm are characterized. Uninorms belonging to one of the classes \({\cal U}_{\min}\), \({\cal U}_{\max}\), idempotent or representable that are α-migrative over a uninorm U 0 in \( {\cal U}_{\min}\) or \({\cal U}_{\max}\) are also characterized.

Edge-Images Using a Uninorm-Based Fuzzy Mathematical Morphology: Opening and Closing

In this paper a fuzzy mathematical morphology based on fuzzy logical operators is proposed and the Generalized Idempotence (GI) property for fuzzy opening and fuzzy closing operators is studied. It is proved that GI holds in fuzzy mathematical morphology when the selected fuzzy logical operators are left-continuous uninorms (including left-continuous t-norms) and their corresponding residual implications, generalizing known results on continuous t-norms. Two classes of left-continuous uninorms are emphasized as the only ones for which duality between fuzzy opening and fuzzy closing holds. Implementation results for these two kinds of left-continuous uninorms are included. They are compared with the classical umbra approach and the fuzzy approach using t-norms, proving that they are specially adequate for edge detection.

On the distributivity property for uninorms

Abstract 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.