The Order Relation Of Uninorms And A Class Of Conditionally Distributive Equations

The process of aggregating multiple input data into a single output is called data fusion(also known as information fusion).In the whole process of data fusion,aggregation operator plays a key role.As typical examples of conjunctive and disjunctive aggregation operators,triangular norms and triangular conorms have been widely used in many fields.As a unified and generalized form of them,uninorms(nullnorms)have been widely used in the fields of decision support system,sentiment analysis,logical reasoning and pattern recognition in recent years because of its excellent algebraic properties.Most of the existing literature studies are confined to the structural characterization of the usual uninorms(nullnorms)and the related functional equations.In this paper,we focus on the study of uninorms and discuss the order relation of uninorms and the conditionally distributive equation of nullnorms over uninorms.This paper is divided into two parts: the first part includes the order relations of four classes of uninorms,including: representable uninorms,minimum(maximum)uninorms,idempotent uninorms and uninorms with continuous Archimedean underlying operators.The relation between the order of the uninorms and that of the corresponding underlying operators is analyzed,and the formula to additive generators of the representable uninorms is given.In the second part,the complete solution to the conditionally distributive equation with continuous nullnorms is given.Moreover,the existence of the solution to the conditionally distributive equation for the uninorms with continuous Archimedean underlying operators is analyzed preliminarily.