New Classes Of Fuzzy Implication And Distributive Equations With Fuzzy Connectives

Fuzzy implications, as the generalization of the classical Boolean impli-cation, are one of important operations in fuzzy logic. It has been proved that fuzzy implications play an important role in approximate reasoning, fuzzy control and practical applications. Then many scholars show strong interest in them and their research include the construction of fuzzy impli-cations, the analysis of their properties and the characterization of fuzzy implications. Recently, distributive equations have become one of the focus-es because their solutions are important in theory and practical applications. Some authors study the distributivity between t-norms and t-conorms, some authors investigate the distributivity for more general operations.One of the aspects of the work is to construct fuzzy implications. In the literature, there are three ways to construct fuzzy implications:(â…°) From other fuzzy logic connectives, from whence we obtain, for in-stance, the families of (S,N)-, R-, QL-implications;(â…±) From monotone functions, from whence we obtain, for instance, the families of f-and g-implications proposed by Yager, and the h-implications introduced by Massanet and Torrens.(â…²) From given fuzzy implications such as the compositions of fuzzy implications and convex combinations of fuzzy implications.Based on the second way, this dissertation will introduce two new classes of fuzzy implications:(f,f’)-and (g,g’)-implications. The other main work in the paper is to discuss the distributivity of fuzzy connectives, including the distributivity of uninorms over nullnorms and the distributivity of fuzzy implications over continuous t-conorms.In order to unify and generate t-norms and t-conorms, Yager and Ry-balov introduced a new class of aggregation:uninorms. The usual classes of uninorms are the Umin and Umax class, idempotent uninorms, representable uninorms and uninorms continuous in (0,1)2. Nullnorms are also a general-ization of t-norms and t-conorms, which were presented by Calvo et al.As for the distributivity of uninorms and nullnorms, the results were first obtained by Mas and Qin. They studied the distributivity of the first two uninorms and nullnorms. Mas et al. proved that nullnorms are only distributive over idempotent uninorms and discussed the distributivity of nullnorms and uninorms belonging to Umin (Umax). While based on general idempotent uninorms, Qin and Zhao investigated the distributivity of idem-potent uninorms and nullnorms. As a result, Mas and Qin have completely resolved the distributivity of nullnorms over uninorms and the distributivity of the first two uninorms over nullnorms. So far, there is no work discussing the distributivity of the remaining two classes of uninorms over nullnorms. This dissertation will study the distributivity of representable uninorms and uninorms continuous in (0.1)2over nullnorms.The most important work is to study the distributivity of fuzzy impli-cations over continuous t-conorms, i.e., to research the distributive equation: I(x,Si(y,z))=S2(I(x,y),I(x,z)),r,y,z G [0,1], where Su S2are two ar-bitrarily given continuous t-conorms and/:[0,1]2â†'[0,1] is an unknown binary function which is increasing with respect to the second place. When S1and S2are both strict t-conorms or both nilpotent t-conorms, Baczynski and Balasubramaniam investigated the equation. They first presented the representation of I(x,·) for every fixed x∈[0,1] and then got its continuous solutions. They also showed that this distributive equation does not have con-tinuous solutions on [0,1]2and consequently gave its fuzzy implication solu-tions which are continuous on [0,1]2except at some special points. Baczynski and Balasubramaniam pointed out that their future work would focus on the this distributive equation with general continuous t-conorms. Later, based on one of Si and S2is strict and the other is nilpotent, Baczyriski used the similar way to discuss the equation and got its continuous solutions and non-continuous fuzzy implication solutions. With S2and Si being a contin-uous Archimedean t-conorm and an ordinal sum of continuous Archimedean t-conorms, respectively, Xie et al. continued to research the distributive e-quation and found out its continuous solutions and non-continuous fuzzy implication solutions. It is well known that if S is a continuous t-conorm if and only if S=max, or S is continuous Archimedean, or S is an ordinal sum of continuous Archimedean t-conorms. From the above, we know that all the work on this equation centers on continuous Archimedean t-conorms. This dissertation will investigate the distributive equation above with two continuous t-conorms given as ordinal sums.The paper contains the following four chapter. The details are as follows.Chapter1recalls briefly fuzzy connectives:t-norms, t-conorms, fuzzy negations and fuzzy implications. It gives the definitions of uninorms and nullnorms as well, which are generalizations of t-norms and t-conorms.Chapter2, based on unary functions on [0,1], introduces two new class-es of fuzzy implications:(f,f’)-and (g,g’)-implications. First we give the definitions of (f,f’)-implications and (g,g’)-implications, and explain that Yager’s implications are special cases of them. Then we study in detail some properties of:(f,f’)-and (g,g’)-implications, such as the properties of (NP),(EP),(OP) and (IP); prove that they are different from the usual known classes of fuzzy implications, namely, R-,(S,N)-, QL-and Yager’s impli-cations; research some classical logic tautologies (i.e., law of importation, contrapositive symmetry and distributivity over t-norms or t-conorms) for:(f,f’)-and (g,g’)-implications, and obtain a serial of satisfying results.Chapter3studies the distributivity of uninorms over nullnorms. The usual classes of uninorms are the Umin and Umax class, idempotent uninorms, representable uninorms and uninorms continuous in (0,1)2. The obtained results for the distributivity of uninorms over nullnorms are based on the first two uninorms, namely, Umin and Umax class, idempotent uninorms. This chapter researches the distributivity of uninorms continuous in (0,1)2and representable uninorms over nullnorms. First, with conjunctive uninorm continuous in (0,I)2, we prove that the absorbing element of the nullnor-m is an idempotent element the uninorm; we divide two cases to discuss the distributive equation and then get its solutions; moreover, when the null-norm is continuous, we obtain the sufficient and necessary conditions under which the distributive equation holds. Secondly, with disjunctive uninorms continuous in (0,1)2, we similarly obtain results. Finally, it is proved that any representable uninorm only has distributive property over t-norms and t-conorms.Chapter4discusses the distributivity of fuzzy implications over con-tinuous t-conorms, namely, the fuzzy functional equation:I(x. S1(y.z))=S2(I(x,y),I{x,z)), x.y,z∈[0,1], where S1, S2are two arbitrarily given continuous t-conorms and/:[0,1]22â†'[0.1] is an unknown binary function which is increasing with respect to the second place. There are papers which have studied the above equations with continuous Archimedean t-conorms. This chapter breaks through the limit of the Archimedean property of con-tinuous t-conorms to solve the distributive equation above. That is to say, we study the above equation with two continuous t-conorms given as ordinal sums. We first study its continuous solutions:if there is no summand of S2in the interval[/(1,0),I(1,1)], we get directly its sufficient and necessary conditions to the distributive equation; if there are summands of S2in the interval [I(1,0),I(1,1)], by defining a new concept called feasible correspon-dence and using this concept, we describe the solvability of the distributive equation above and characterize its general continuous solutions on [0,1]2. When I is restricted to fuzzy implications, it is showed that there is no con-tinuous solution to this equation. Then we characterize its fuzzy implication solutions which are continuous on (0,1] x [0,1]. The main innovations are summarized as follows.1. Based on unary functions, we introduced two new classes of fuzzy implications, which generalize the Yager’s implications. We proved that the two classes of fuzzy implications have good properties such as (NP) and (EP), and characterized some generalized tautologies with the two classes of fuzzy implications, including (CP),(LI) and distributive equations.2. We studied the distributivity of representable uninorms and uninorms continuous in (0,1)2over nullnorms. It is proved that representable uninorms are distributive with only t-norms and t-conorms. We characterized fully the solutions to the distributive equation with uninorms continuous in (0,1)2.3. The paper investigated the distributivity of fuzzy implications over continuous t-conorms. The work broke through the limit of continuous Archimedean t-conorms and resolved the distributive equations with two continuous t-conorms given as ordinal sums.