Study On Semi-uninorms And Other Related Operators

Fuzzy logic, as a generalization of classical logic, has been one of the principle theories in the field of uncertain theories and methods. It has been successfully applied in several fields on artificial intelligence. In recent years, fuzzy logic operators and their distributive functions attract a lot of atten-tion from researchers. Many interesting achievements are put forward on the theories and their applications. For example, the study on conjunctive and disjunctive fuzzy operators is gradually extended from triangular norms (t-norms) and triangular conorms (t-conorms) to uninorms, nullnorms, copulas and so on; several classical implications are characterized completely or par-tially; some new classes of implications and construction methods continue to appear; distributive functions of some classical operators are studied widely. All of these researches provides a solid foundation for practical applications.Commutativity and associativity have always drawn much attention in studies. But some researchers pointed out that non-commutative fuzzy oper-ators play important role in certain case and some researchers said that:"if one works with binary conjunctions and there is no need to extend them for three or more arguments, as happens, for example, in the inference pattern called generalized modus ponens, associativity of the conjunction is an un-necessarily restrictive condition." Inspired by these facts, non-commutative t-norms, non-commutative triangular conorms, semi-copulas are put forward and investigated. With the further study, some researcher introduced semi-uninorms, which is non-commutative and non-associative, and discussed their properties and residual implications derived from semi-uninorms. Many re-searchers were interested in this topic and studied the properties, applications and construction on semi-uninorms and related operators, respectively. The study on semi-uninorms becomes a focus in fuzzy logic field gradually. The author will go into details of this class of connective in the dissertation.This dissertation is organized as follows.Chapter 1 introduces the research background and status about fuzzy operators and their distributive functions and migrativities. The main con-tents of this dissertation is presented.For self-contained Chapter 2 proposes the concepts and conclusions about some classical conjunctive and disjunctive operators, fuzzy negations, fuzzy implications, distributive functions and migrativities. At last, seni-uninorms defined on finite chain, their structure and basic properties is dis-cussed. The equivalence among smooth, Lipschitz condition and "intermedi-ate value theorem" and properties of diagonal section are pointed out.In Chapter 3 the author introduces the (Us, N)-implications, where Us is a disjunctive semi-uninorm, a commutative semi-uninorm or pseudo-uninorm, respectively, and N is a fuzzy negation. Some properties and char-acterizations of these three classes of (Us,N)-implications are given out. The author also investigates the intersections between residual implications and (Us,N)-implications derived from some semi-uninorms.The distributivity of (Us,N)-implications over semi-uninorms deduces to the distributivity between two semi-uninorms. Since, Chapter 4 discusses the distributivity between two semi-uninorms Us1 and Us2-The sufficient and necessary conditions such that Us1 is left or right distributive over Us2 are given out, where US1 ∈{Nemin, Nemax, Nerep}.Based on the sufficient and necessary conditions in Chapter 4, the dis- tributivity of (Us,N)-implications is investigated in Chapter 5. One section examines four classes of distributive functions of (Us,N)-implications over conjunctive and disjunctive semi-uninorms and the other section examines four classes of distributive functions of (Us, N)-implications over t-norms and t-conorms, respectively.