| Fuzzy implications are one of basic connectives in both theoretical and applied communities of fuzzy set theory.The well known fuzzy implications are usually constructed in a specific way from t-norms,t-conorms and fuzzy negations,and according to construction methods,they can be roughly classified into five classes,namely(S,N)-implication,R-implication,QL-implication,Yager’s implication and ordinal sum implication.So far,the above-mentioned fuzzy implications all have been gotten rapid developments.For example,according to the structure of(S,N)－implications,(U,N)-implications,(D,N)-implications,(G,N)-implications and so on have been put forward;according to the method of Yager’s implications,(f,g)－implications,(g,min)-implications,(g,u)-implications and so on have also been put forward.But on the other hand,fuzzy coimplications,as the dual operators of fuzzy implications,have received,however,little attention in the literature.At the same time,many studies show that not all properties of fuzzy implications can be dualized in the standard way to fuzzy coimplications while fuzzy coimplications play key roles in quantum B-algebras,BCK-algebras,Heyting-Brouwer logic,fuzzy expert systems,etc.The main content of this article is divided into two parts.Firstly,some properties of(G,N)-implications are further studied,and the main results are some characterizations of(G,N)-implications.Secondly,based on overlap function and fuzzy negation,the notion of(O,N)-coimplication is introduced,and basic properties and characterizations of some subclasses of(O,N)-coimplications are obtained.The concrete content is as follows:Chapter One:Preliminaries.In this chapter,some basic concepts and relevant conclusions of fuzzy implication,fuzzy coimplication,grouping function and overlap function are given.Chapter Two:(G,N)-implication and its characterization.Based on the known results of(G,N)-implications,some properties of(G,N)-implications are further studied,and the main results are some characterizations of(G,N)-implica-tions.At last,mutual relations between(G,N)-implications and their(pseudo-)φ-conjugates are discussed,characterizations of them are given.Chapter Three:(O,N)-coimplication and its characterization.Firstly,the notion of(O,N)-coimplication is proposed.Some properties of(O,N)-coimplications are discussed.The sufficient and necessary conditions that(O,N)-coimplications meet OP(ordering property)and IP(identity principle),respectively,are given.Sec-ondly,some characterizations of some particular subclasses of(O,N)-coimplications are studied,including the characterizations of(O,N)-coimplication which is based on WLI(the weak law of importation).Thirdly,mutual relations between(O,N)-coimplications and their(pseudo-)φ-conjugates are discussed,characterizations of them are given.At last,mutually determinative relations between(G,N)-implications and(O,N)-coimplications are discussed. |
