Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting n-ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that the unary function is non-decreasing and distributive over special aggregation operators, for examples, continuous t-norms, continuous t-conorms and two classes of uninorms. Along this way of thinking, in this paper, we will investigate and fully characterize the following unary distributive functional equation f(U(x,y))=U(f(x),f(y)), where f:[0,1]→[0,1] is an unknown function but unnecessarily non-decreasing, a uninorm U∈Umin has a continuously underlying t-norm TU and a continuously underlying t-conorm SU. Our investigation shows that the key point is a transformation from this functional equation to the several known ones. Moreover, this equation has also non-monotone solutions completely different with already obtained ones. Finally, our results extend the previous ones about the Cauchy-like equation f(A(x,y))=B(f(x),f(y)), where A and B are some continuous t-norm or t-conorm.
In this paper, two construction methods have been proposed for uni-nullnorms on any bounded lattices. The difference between these two construction methods and the difference from the existing construction methods have been demonstrated and supported by an example. Moreover, the relationship between our construction methods and the existing construction methods for uninorms and nullnorms on bounded lattices are investigated. The charactertics of null-uninorms on bounded lattice L are given and a contruction method is presented.
In this paper, we analyze and characterize all solutions about α-migrativity properties of the five subclasses of 2-uninorms, i. e. Ck, C0k, C1k, C01, C10, over semi-t-operators. We give the sufficient and necessary conditions that make these α-migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for G∈Ck, the α-migrativity of G over a semi-t-operator Fμ,ν is closely related to the α-section of Fμ,ν or the ordinal sum representation of t-norm and t-conorm corresponding to Fμ,ν. But for the other four categories, the α-migrativity over a semi-t-operator Fμ,ν is fully determined by the α-section of Fμ,ν.
In this note, we point out that Theorem 3.1 as well as Theorem 3.5 in G. D. Çaylı and F. Karaçal (Kybernetika 53 (2017), 394-417) contains a superfluous condition. We have also generalized them by using closure (interior, resp.) operators.
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.
Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation U in the unit interval with the neutral element e∈[0,1]. If operation U is continuous, then e=0 or e=1. So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element e∈(0,1), which is continuous in the open unit square may be given in [0,1)2 or (0,1]2 as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7].