Some New Results On Overlap And Grouping Functions

Recently,overlap and grouping functions,as two new cases of nonassociative fuzzy-logic connectives,have been investigated by many researchers for applications in im-age processing,classification problems and decision making based on fuzzy preference relations.This paper mainly introduces some new results on these two new cases of nonassociative fuzzy logic connectives.More precisely,this paper contains the following contents:(1)We introduce the concept of multiplicative generator pair for overlap functions and investigate the migrativity,homogeneity,idempotency,Archimedean and cancella-tion properties for the overlap functions obtained by such multiplicative generator pairs.In addition,we give the definition of multiplicative generator pair for group-ing functions and show some related results by the duality of overlap and grouping functions.(2)We generalize the concepts of overlap and grouping functions to the notions of in-terval overlap functions and interval grouping functions and investigate some vital related properties.In particular,we give the construction methods to obtain inter-val overlap functions and interval grouping functions,which are the best interval representation of the overlap and grouping functions,respectively.We also intro-duce the concepts of interval additive generators of interval overlap functions and interval grouping functions,which offer convenience for the selection of proper in-terval overlap functions or interval grouping functions in concrete problems.In the meantime,we prove that the best interval representation of the additive generator pair of an additively generated overlap function is an interval additive generator pair of the best interval representation of this overlap function and the case for grouping function is analogous.(3)We generalize the α-migrativity of any overlap function O from the usual formu-la O(αx,y)=O(x,αy)to the so-called(α,O*,O(?0)-migrativity O(O*(α,x),y)=O(x,O(?)(α,y)),where O*and O(?)are two fixed overlap functions.And then,we in-vestigate the(α,O*,O(?))-migrativity of an overlap function by taking O*and O(?)as the minimum overlap function and give an equivalent characterization of it by the ordinal sum of overlap functions.In addition,we propose the(α,O*,O(?))-migrativity of an overlap function by taking O*and O(?)as the p-product overlap function and show an equivalent characterization of it by its additive generator pair.In particu-lar,we obtain an equivalent characterization of the usual a-migrativity of an overlap function by its additive generator pair.We also discuss the(α,O*,O(?))-migrativity of an overlap function by taking O*and O(?)as the 1-product overlap function and p-product overlap function,respectively,and give two characterizations of it by its additive generator pair.(4)We introduce the concept of(α,O)-migrative uninorms over any fixed overlap func-tion 0.In addition,we show equivalent characterizations of the(α,O)-migrativity equation when the uninorm U belongs to one of the certain classes(e.g.,μmin μmax,the family of idempotent uninorms,representable uninorms or uninorms continu-ous on]0,1[2).We also give the notion of(a,G)-migrative uninorms over any fixed grouping function G and propose the(a,G)-migrativity equation by an analogous way.In addition,we discuss the(α,O)-migrative and(α,G)-migrative nullnorms and obtain equivalent characterizations of the related(α,O)-migrativity and(α,G)-migrativity equations,respectively.(5)We investigate the four basic distributive laws of fuzzy implications over overlap and grouping functions when the fuzzy implications become residual implications derived from overlap functions,(G,N)-implications for grouping functions G and fuzzy negations N or QL-operations derived from overlap functions,grouping func-tions and fuzzy negations,respectively.In addition,we extend the related results when the fuzzy implications are considered as the residual implications,(G,N)-implications or QL-operations in the four basic distributive laws of fuzzy impli-cations over overlap and grouping functions to the arbitrary fuzzy implications satisfying some desirable algebraic properties.(6)We show some new results for additively generated overlap and grouping functions.And then,we propose the two basic distributive laws of fuzzy implication functions over additively generated overlap and grouping functions.(7)We introduce the notions of pseudo-homogeneous overlap and grouping functions,which can be regarded as the generalizations of the concepts of homogeneous and quasi-homogenous overlap and grouping functions,respectively.And then,we in-vestigate the homogeneity,quasi-homogeneity and pseudo-homogeneity of an over-lap function when it belongs to one of the classes of idempotent overlap functions,multiplicatively generated overlap functions or the ordinal sum of overlap functions.In addition,we also show an analogous discussion for grouping functions.