Homogeneity Of Overlap(Grouping) Functions And Triangular Norms

This thesis mainly discusses the homogeneity,quasi-homogeneity and pseudo-homogeneity of overlap functions and the related aggregation functions.First,it gives a positive answer to an open problem raised by Dimuro et al.in[Fuzzy Sets and Systems,2016,287:76-96]by constructing an example,and establishes three necessary and sufficient conditions under which an overlap function can be obtained by a(F,T)-distortion for a positive and continuous triangular norm T and a(pseudo)automorphism T.Secondly,it points out that some results on homogeneity of functions in[Fuzzy Sets and Systems,2016,287:76-96]are invalid and studies the relevant problems by the multiplicative generator pair.Moreover,it investigates the homogeneity of overlap functions by using multiplicative generator pairs and additive generator pairs,respectively.Thirdly,it characterizes pseudo-homogeneous overlap and grouping functions,respectively,constructs representations of a given pseudo-homogeneous overlap and grouping functions in terms of the single-argument functions induced by the given overlap and grouping functions,respectively,and proves that the pseudo-homogeneity and quasi-homogeneity are equivalent for overlap and grouping functions,respectively.It then introduces the concept of a pseudo-homogeneous generator pair of overlap function which is applied to construct an overlap function.Finally,this thesis suggests the concept of pseudo-homogeneous triangular norm without any restriction on F,obtains the explicit expressions of all continuous pseudo-homogeneous triangular norms,and verifies that the existence of discontinuous pseudo-homogeneous triangular norm by an example.It also presents that the quasi-homogeneous triangular norms found by Ebanks et al.in[International Journal of Mathematics and Mathematical Sciences,1998,21:351-358]are exactly continuous,which leads us to explore the noncontinuous quasi-homogeneous triangular norms,and shows that only TD is noncontinuous quasi-homogeneous.Furthermore,it investigates the explicit expressions of pseudo-homogeneous overlap functions through(F,T)-distortions.