Research On Several Types Of Related Function Equations With Fuzzy Implication

As a natural extension of classical Boolean implication,fuzzy implications are one class of the main logical connectives of fuzzy logic.It plays an important role in theoretical research and many practical applications,especially in mathematical fuzzy logic,approximate reasoning,fuzzy control system,fuzzy relational equation and image processing.The wide application platform and the large demand for fuzzy implications have promoted the rapid development of fuzzy implications in the past few decades.Therefore,more and more scholars are committed to the study of various related theories of fuzzy implication,and have achieved a series of fruitful results.Since fuzzy implication requires only the monotonicity axiom and takes the same values with Boolean implication at the vertices of the unit square,many researchers have put forward a lot of methods for the construction of fuzzy implication.Among the common methods are:(1)Fuzzy implications derived from other fuzzy logical connectives by following various but equivalent representations of Boolean implication in terms of fuzzy negations,conjunctions and disjunctions;(2)Fuzzy implications generated by generator functions and various generalizations;(3)Various of ordinal sum implications.At the same time,the study of functional equations related to fuzzy implication is also a hot topic at present.This kind of problem is usually carried out in two directions:one is to find the aggregation operator solutions of the corresponding equations for given fuzzy implications,and the other is to find the fuzzy implication solutions of the corresponding equations for given aggregation operators.Based on the existing results,this dissertation mainly introduces the research results of several kinds of fuzzy implications related functional equations such as distributive equation,migrative equation and the law of importation.The main research contents are as follows:1.The(s,ξ)-generated implications,by means of a pair of additive and multiplicative generators of triangular conorms is constructed.It is proved that this kind of fuzzy implication is different from Yager’s f-and g-implications,as well as h-,k-generated implications and(S,N)-,R-implications,and the relationships between these fuzzy implications are discussed.In addition,some important algebraic properties satisfied by this kind of fuzzy implication are also studied,such as(NP),(EP),(CP),(OP)and so on.Finally,generalizations of three classical logic tautologies with this kind of implications,viz.,law of importation,T-conditionality and distributivity over t-norms and t-conorms are investigated.2.We investigate the four basic distributive law equations of ordinal sum implications proposed by Su et al.,over two new cases of non-associative fuzzy logic connectives,overlap and grouping functions,respectively.The main results are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications.And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are obtained respectively.3.The migrativity equation of continuous t-conorms over fuzzy implications is introduced,which also provides an answer to Fodor’s question on how to define the migrativity of t-conorms.We first describe completely the α-migrativity of continuous t-conorms defined by Baczynski et al.in their latest work,which proves to be the standard dual to the migrativity of t-norms over the product t-norm,but provides a new perspective to study the migrativity of t-conorms.And then,we turn to characterize the migrativity of t-conorms over several well-known specific fuzzy implications,of which most are no longer dual to the migrativity of t-norms,and show some interesting results.Further,we define α-migrativity of continuous t-conorms over general fuzzy implications and discuss migrativity equation when a is the fixed point of natural negation of a fuzzy implication I,satisfying I(α,α)≥α.Also obtain the characterizations of solutions to migrativity equations by the ordinal sum of t-conorms.Finally,We discuss the migrativity of several kinds of common uninorms over a special class of fuzzy implication.4.We first give the definition of fuzzy implication I satisfying the law of importation equation(LIO)with any fixed overlap function O.And then,some results about general fuzzy implications satisfying the(LIO)with O are given.We also study the characterizations of fuzzy implication IN,O generated by a fuzzy negation N and taking O=OM,Op respectively which satisfying(LIO).In addition,we characterize the(G,N)-implication obtained by a grouping function G and a strong negation N satisfying(LIO)with a fixed overlap function O,RO-implication obtained from an overlap function O∈Ox by satisfying(LIO*)with another overlap function O*,and last a QL-operator constructed from the tuple(O,G,N)by satisfying(LIO*)with another overlap function O*.