Distributivity of strong implications over conjunctive and disjunctive uninorms

summary:This paper deals with implications defined from disjunctive uninorms $U$ by the expression $I(x,y)=U(N(x),y)$ where $N$ is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a $t$-norm or a $t$-conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works when the implications are derived from $t$-conorms

On the distributivity equation for uni-nullnorms

summary:A uni-nullnorm is a special case of 2-uninorms obtained by letting a uninorm and a nullnorm share the same underlying t-conorm. This paper is mainly devoted to solving the distributivity equation between uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms and some binary operators, such as, continuous t-norms, continuous t-conorms, uninorms, and nullnorms. The new results differ from the previous ones about the distributivity in the class of 2-uninorms, which have not yet been fully characterized

Distributivity between extended nullnorms and uninorms on fuzzy truth values

This paper mainly investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. It presents the distributive laws between the extended nullnorm and t-conorm, and the left and right distributive laws between the extended generalization nullnorm and uninorm, where a generalization nullnorm is an operator from the class of aggregation operators with absorbing element that generalizes a nullnorm.Comment: 2

An extension of the ordering based on nullnorms

summary:In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the $F$-partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms

The quest for rings on bipolar scales

We consider the interval $]{-1},1[$ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t.~$0$ are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.

Intersections between some families of (U,N)- and RU-implications

(U,N)-implications and RU-implications are the generalizations of (S,N)- and R-implications to the framework of uninorms, where the t-norms and t-conorms are replaced by appropriate uninorms. In this work, we present the intersections that exist between (U,N)-implications and the different families of RU-implications obtainable from the well-established families of uninorms

Distributivity of ordinal sum implications over overlap and grouping functions

summary:In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given

Contribució a l'estudi de les uninormes en el marc de les equacions funcionals.

Les uninormes són uns operadors d'agregació que, per la seva definició, es poden considerar com a conjuncions o disjuncions, i que han estat aplicades a camps molt diversos. En aquest treball s'estudien algunes equacions funcionals que tenen com a incògnites les uninormes, o operadors definits a partir d'elles. Una d'elles és la distributivitat, que és resolta per les classes d'uninormes conegudes, solucionant, en particular, un problema obert en la teoria de l'anàlisi no-estàndard. També s'estudien les implicacions residuals i fortes definides a partir d'uninormes, trobant solució a la distributivitat d'aquestes implicacions sobre uninormes. Com a aplicació d'aquests estudis, es revisa i s'amplia la morfologia matemàtica borrosa basada en uninormes, que proporciona un marc inicial favorable per a un nou enfocament en l'anàlisi d'imatges, que haurà de ser estudiat en més profunditat.Las uninormas son unos operadores de agregación que, por su definición se pueden considerar como conjunciones o disjunciones y que han sido aplicados a campos muy diversos. En este trabajo se estudian algunas ecuaciones funcionales que tienen como incógnitas las uninormas, o operadores definidos a partir de ellas. Una de ellas es la distributividad, que se resuelve para las classes de uninormas conocidas, solucionando, en particular, un problema abierto en la teoría del análisis no estándar. También se estudian las implicaciones residuales y fuertes definidas a partir de uninormas, encontrando solución a la distributividad de estas implicaciones sobre uninormas. Como aplicación de estos estudios, se revisa y amplía la morfología matemática borrosa basada en uninormas, que proporciona un marco inicial favorable para un nuevo enfoque en el análisis de imágenes, que tendrá que ser estudiado en más profundidad.Uninorms are aggregation operators that, due to its definition, can be considered as conjunctions or disjunctions, and they have been applied to very different fields. In this work, some functional equations are studied, involving uninorms, or operators defined from them as unknowns. One of them is the distributivity equation, that is solved for all the known classes of uninorms, finding solution, in particular, to one open problem in the non-standard analysis theory. Residual implications, as well as strong ones defined from uninorms are studied, obtaining solution to the distributivity equation of this implications over uninorms. As an application of all these studies, the fuzzy mathematical morphology based on uninorms is revised and deeply studied, getting a new framework in image processing, that it will have to be studied in more detail