The Distributivity Of Extension Operators On Type-2 Fuzzy Sets And Related Studies

This thesis deals with the distributivity between extension operators of fuzzy truth values and the algebraic structures of lattice functions.It first concentrates on the distributive laws between extended t-norms and t-conorms on fuzzy truth values under the condition that the t-norm is conditionally distributive over the t-conorm or the t-conorm is conditionally distributive over the t-norm.It then presents the distributive laws between extended t-norms and t-conorms,which are then used to discuss the distributive laws between extended uninorms and t-norms,and the distributive laws between extended uninorms and t-conorms,respectively,and it investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm.It also discusses the distributive laws between the extended nullnorm and t-conorm,and the left and right distributive laws between the extended generalized nullnorm and uninorm,and gives necessary and sufficient conditions for the distributivity of extended operations on fuzzy truth values.Secondly,it investigates Z-extended overlap functions and grouping functions on fuzzy truth values.It presents some properties of the Z-extended overlap functions and grouping functions,and uses these properties to show the representations of the Z-extended overlap functions and grouping functions.Then it obtains the conditions under which the distributive laws between the Z-extended overlap functions and grouping functions are true.Thirdly,it induces a partial order by the extended t-norm on membership functions of fuzzy sets of type-2.It shows the relationship between the partial order induced by the extended t-norm and the partial order induced by the extended minimum and maximum,and investigates the related algebraic structures of fuzzy truth values.Finally,It investigates the algebraic structures of functions between bounded lattices.It constructs a subset of the set of functions that is both a bisemilattice and a Birkhoff system under the convolution operations and then shows another subset of the set of functions that is a bounded distributive lattice under the convolution operations while the domain lattice of the functions does not need to be distributive.