The Distributivity For Uninorms

The aggregation is a process of merging/combining a collection of data into a representative value and a function that preforms this process is called an aggregation function.An aggregation process occurs in many situations like in decision making or in statistical and economic measurement by aggre-gating expert's opinions or by synthesizing judgements.The common feature of all these problems is that an aggregation function that carries out this ag-gregation process satisfies a system of functional equations,which usually depends on the context where they are going to be applied,For example,in multi-criteria decision making,if all criteria are satisfied in the same degree,then also the global score should be also this degree.In fact,the aggregation function with the above property satisfies idempotency function,that is,the aggregation value of the same elements is still this element.There are many different classes of aggregation functions and their elec-tion usually depends on the context where they are going to be applied.Based on their behaviors and functions,there are four classes of aggregation opera-tors:conjunctive aggregation operators,in which triangular norms,short for t-norms,are standard examples;disjunctive aggregation operators,in which triangular conorms,short for t-conorms,are standard examples;averaging aggregation operators,in which OWA operators and type-integral aggrega-tion operator are standard examples;and mixed aggregation operators,in which uninorms and nullnorms(also called t-operators)are standard exam-ples.Mixed aggregation operators are a very special class of aggregation operators,which simultaneously preserve the characterizations of conjunc-tive aggregation operators,disjunctive aggregation operators and averaging aggregation operators.Uninorms as mixed aggregation operators have good behaviors,and their model interpretation is as follows:there exists a thresh-old value in aggregation process.From the point of view of mathematics,this threshold value is called neutral or identity element,which means that"let the other inputs decide on this one,I neither support nor detract." By providing all inputs below the identity element,the aggregation process is conjunctive;the aggregation process is disjunctive when all inputs are above the identity element;and for the another cases,the aggregation process is averaging.From the algebraic point of view,t-norms and t-conorms are,re-spectively,Abelian ordered semigroups defined on the unit interval with neu-tral elements 0 and 1.However,uninorms are Abelian ordered semigroups defined on the unit interval with any given element as the neutral element.From the point of view of applications,uninorms have proved to be useful in a wide range of fields like aggregation of information,expert systems,neural networks,fuzzy system modeling,pseudo-analysis and measure theory,fuzzy mathematical morphology,fuzzy sets and fuzzy logic,approximate reason-ing,and so on.Since uninorms are always conjunctive or disjunctive they have been extensively studied in the framework of logical connectives and they have been used in fuzzy sets theory and fuzzy logic.Therefore,in the framework of logical connectives,uninorms as logical connectives have been extensively studied.Due to this great number of applications,uninorms have also been studied from the purely theoretical point of view.In this way,one of the main topics is directed towards the characterization of those uninorms that verify certain properties that may be useful in each context.The study of these properties for certain aggregation functions usually involves the res-olution of functional equations.Examples of this research line are the study of the properties of idempotency,modularity,Frank and Alsina equations,migrativity,and distributivity.Distributivity between two operators is a property that was already posed many years ago,for example:the multiplication is distributive over the addition in a ring and the multiplication is distributive over the union in a lattice.The models that need to satisfy the distributivity equation are summarized as follows:there are two different kinds of behavior that can be interpreted as a pair of binary aggregation functions and the relevance assumption of these two binary aggregation functions are the distributivity equation,for example,the logical "and" and "or" in fuzzy logic;the pseudo-addition and pseudo-multiplication functions in pseudo-analysis;and gamble and joint receipt in utility theory.This work is devoted to the study of the distributivity for uninorms,that is,looking for the uninorm solution of the distributivity equation.Uninorms only need to satisfy four axioms,i.e.the commutativity,the associativity,the monotonicity and the neutral element.Thus,there is no good character-ization for general uninorms.In fact,the general uninroms cannot be char-acterized.However,with some additional conditions,the five most studied classes of uninorms,i.e.uninorms in Umin and Umax,idempotent uninorms,representable uninorms,uninorms continuous in the open unit square and locally internal uninorms,are obtained.And the above classes of uninorms have been completely characterized.In literature,concerning the distributiv-ity,only these five most studied classes of uninorms are considered because of their good characterizations.The main ideas of these research are that by means of their good characterizations the necessary and sufficient conditions such that the distributivity equation holds are obtained,in other words,the further characterizations of these two uninorms are obtained by means of their own characterizations.Therefore,these results strictly depend on the characterizations of these two uninorms.For general uninorms without good characterizations,the methods and technique to deal with the distributivity for the five most studied classes of uninorms are invalid,that is,the distribu-tivity for uninorms that do not lie in five most studied classes of uninorms is still missing.The above classes of uninorms have been completely charac-terized,however,the corresponding results concerning the distributivity for these uninroms are scatted,non-system and very restrictive.The five most studied classes of uninorms are a small part of uninorms.Indeed,all the five most studied classes of uninorms are locally interval on the boundary.In lit-erature,the uninroms,which are not locally interval on the boundary,have appeared,however,their characterizations are still missing.In this work,several new methods and technique are employed to deal with the distribu-tivity for general uninorms.It means that the condition that both uninorms belong to the five most studied classes of uninorms is not demanded.Having a neutral element is an important axiom of uninorms.When two uninorms have the same neutral element,the distributivity equation becomes very spe-cial,which can be completely solved(see Chapter 2 for details).That is the first problem that is solved by this paper.The cases are more complex and it is difficult to completely solve the distributivity equation when the neutral elements of uninorms are different.In this work,we solve the distributivity equation in a much weaker condition,that is,only one uninorm is required to belong to the five most studied classes of uninorms.Therefore,the idea can be divided into two parts,namely,the distributivity for the most studied classes of uninorms over the general uninorms(see Chapter 3 for details)and the distributivity for the general uninorms over the most studied classes of uninorms(see Chapter 4 for details).We will see that many new solutions appear from this new point of view that were not included in the previous approaches.Of course,they also appear many other matching solutions with those already known.Compiling all of these results is a goal of this work.Therefore,the goal of this paper is twofold,i.e.showing new solutions and compiling all solutions of the distributivity equation.Moreover,these ap-proaches can be used to completely solve the migrativity and modularity of uninorms.