A Study Of Overlapping Functions And Their Associated Distributive Problems

Fuzzy logic,as a natural generalization of classical logic,has been one of the prin-ciple theories in the field of uncertainty theories and methods,and it has been widely used in artificial intelligence,computer science and technology,big data processing and many other fields.As a key part of fuzzy logic,fuzzy logic connectives have been constantly discussed and concerned by scholars at home and abroad because of their significant roles in applications of fuzzy logic.In recent years,as common fuzzy logic connectives,t-norms,t-conorms,uninorms and fuzzy implications defined on lattices or posets have been deeply studied and a series of achievements have been obtained.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.Overlap function,a new not necessarily associative aggregation function,was proposed by Bustince et al.in 2009 for its applications in image processing,clas-sification and decision making based on fuzzy preference relations.Once the function was put forward,it attracted the attention of many scholars,recently,not only in application but also in theory have overlap functions developed fast.Distributivity between two operators is a property that was already posed many years ago,and the study on the distributivity between aggregation functions and fuzzy logic connectives has been a hot spot in fuzzy logic for recent years,therefore,researches on the distribu-tivity between overlap functions and fuzzy logic connectives are not only a theoretical supplement to the theory of distributivity between operators,but also a promotion to the theoretical development of overlap functions themselves.This thesis explores the methods of constructing t-norms,t-conorms,uninorms and fuzzy implications on ACDL,and mainly studies the distributivity between overlap functions and any other operators.The structure of this thesis is organized as follows:Chapter One Preliminaries.This chapter reviews some basic concepts and results of lattice theory,and introduces some commonly used concepts and results of t-norms,t-conorms,uninorms,fuzzy implications,overlap and grouping functions.Chapter Two Constructions of uninorms and fuzzy implications on algebraiccompletely distributive lattice(ACDL).Firstly,the methods to construct t-norms,t-conorms and fuzzy negations on ACDL are presented by extending t-norms,t-conorms and fuzzy negations on the set of completely join-prime elements or the set of com-pletely meet-prime elements,and it is proved that the De Morgan law can be kept under specific conditions.Secondly,the ways to construct infinitely ∨-distributive uninorms and infinitely ∧-distributive uninorms are put forward.Finally,approaches to constructing fuzzy implications on ACDL by completely join-prime elements and completely meet-prime elements are given,meanwhile,combining with the behaviors of R-implication and Reciprocal implication,the relationships between fuzzy impli-cations obtained by different ways are investigated.Chapter Three Distributivity of fuzzy implications satisfying the contrapositive symmetry equation(CP property)over overlap functions.Firstly,the distributivity of fuzzy implications over multiplicatively generated overlap functions is studied,and it is proved that there is no continuous solution for this distributivity equation that is fuzzy implication,and a characterization for the case that fuzzy implications are continuous except for the point(0,0)is given.Secondly,the structures of fuzzy implications that simultaneously fulfill CP property and the distributivity equation when the overlap functions are multiplicatively generated are obtained under certain conditions.Fi-nally,considering the good algebraic properties of idempotent overlap function and Archimedean overlap function,the solutions of distributivity equations of fuzzy im-plications over overlap functions when the overlap functions are taken as idempotent overlap functions and Archimedean overlap functions are characterized,respectively.Chapter Four Distributivity of uninorms continuous in(0,1)2 over overlap func-tions.Firstly,the distributivity of t-conorms over overlap functions is studied,and its characterization is given.Secondly,based on the good properties of continuous t-norms,the distributivity of continuous t-norms over overlap functions are fully char-acterized.Finally,the distributivity of uninorms continuous in(0,1)2 over overlap functions is discussed:the necessary and sufficient conditions for the solutions of the distributivity equations when the uninorms belong in cumin are given;and a character-ization of the solutions of the distributivity equation when the uninorm is considered in cumax is given under certain conditions.Chapter Five Distributivity and conditional distributivity of overlap functions over uninorms.Firstly,the distributivity of overlap functions over uninorms is stud-ied,and the structures of overlap functions and uninorms that fulfill the distributivity equations are obtained.Secondly,the necessary and sufficient conditions are given for the conditional distributivity of overlap functions over continuous t-conorms.Final-ly,the conditional distributivity of overlap functions over uninorms with continuous underlying operators is fully characterized.