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.
The main goal of this paper is to construct fuzzy connectives on algebraic completely distributive lattice(ACDL) by means of extending fuzzy connectives on the set of completely join-prime elements or on the set of completely meet-prime elements, and discuss some properties of the new fuzzy connectives. Firstly, we present the methods to construct t-norms, t-conorms, fuzzy negations valued on ACDL and discuss whether De Morgan triple will be kept. Then we put forward two ways to extend fuzzy implications and also make a study on the behaviors of R-implication and reciprocal implication. Finally, we construct two classes of infinitely ⋁-distributive uninorms and infinitely ⋀-distributive uninorms.
Distributivity of fuzzy implications over different fuzzy logic connectives have a very important role to play in efficient inferencing in approximate reasoning, especially in fuzzy control systems (see \cite{Combs_Andrews_1998,Jayaram2008} and \cite{Baczynski_Jayaram_2008}). Recently in some considerations connected with these distributivity laws, the following functional equation appeared (see \cite{Baczynski_Jayaram_2009})
f(min(x+y,a))=min(f(x)+f(y),b),
where a,b>0 and f:[0,a]→[0,b] is an unknown function. In this paper we consider in detail a generalized version of this equation, namely the equation
f(m1(x+y))=m2(f(x)+f(y)),
where m1,m2 are functions defined on some intervals of \R satisfying additional assumptions. We analyze the cases when m2 is injective and when m2 is not injective.
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.
Recently, Drygaś generalized nullnorms and t-operators and introduced semi-t-operators by eliminating commutativity from the axiom of t-operators. This paper is devoted to the study of the discrete counterpart of semi-t-operators on a finite totally ordered set. A characterization of semi-t-operators on a finite totally ordered set is given. Moreover, The relations among nullnorms, t-operators, semi-t-operators and pseudo-t-operators (i. e., commutative semi-t-operators) on a finite totally ordered set are shown.