On The Construction Of Four Types Of Associative Aggregation Operators On Bounded Lattices

Due to the close connection between fuzzy set theory and order theory,the study of associative aggregation operators(such as t-norms,t-conorms,uninorms,nullnorms and uni-nullnorms)on more general structures(such as bounded lattices and bounded posets)has become a contemporary and hot research topic in the broader setting of aggregation theory,and has resulted in numerous research papers.This thesis mainly focuses on the construction of t-norms,t-conorms,uninorms and nullnorms on bounded lattices.Analogous to the study of ordinal sums of t-norms and t-conorms on the real unit interval [0,1],ordinal sums of t-norms and t-conorms on bounded lattices have been extensively investigated in the literature.Generally,ordinal sums of t-norms and tconorms constructed on bounded lattices are not necessarily t-norms and t-conorms,respectively,since they do not satisfy the associativity.One task of this thesis is to consider ordinal sums of t-norms and t-conorms on bounded lattices with one summand such that they are t-norms and t-conorms,respectively.More specifically:(1)by amending the formula of an ordinal sum of t-norms on a bounded lattice,we extend a t-norm from a subinterval of a bounded lattice to that bounded lattice;we can also extend a t-conorm from a subinterval of a bounded lattice to that bounded lattice in an analogous way;(2)based on the modified method of ordinal sums of t-norms and t-conorms in(1),we provide a method of finite recursion for constructing t-norms and t-conorms;(3)a sufficient condition is provided for extending a t-norm from a bounded sublattice of a bounded lattice to that bounded lattice.In particular,if a bounded lattice and its a bounded sublattice have the same bottom element,then we present a necessary and sufficient condition to characterize the extension of a t-norm from a bounded sublattice of a bounded lattice to that bounded lattice.Compared with t-norms and t-conorms,the unique feature of uninorms is that they allow the neutral element to lie anywhere in the real unit interval [0,1].In this way,they are a natural generalization of the notions of t-norms and t-conorms.We would like to discuss the class of uninorms on bounded lattices analogous to their research on the real unit interval [0,1].Thus,the second task of this thesis is to construct uninorms on bounded lattices.It will be helpful for exploring the class of uninorms on bounded lattices.More specifically:(1)we construct a uninorm on a bounded lattice with a given underlying t-norm on a subinterval of that bounded lattice;(2)we construct a uninorm on a bounded lattice with a given underlying t-norm and a given underlying t-conorm on the corresponding subintervals of that bounded lattice.Note that,for the above two construction methods,this thesis also respectively presents a dual method for constructing uninorms.Also nullnorms are a natural generalization of the notions of t-conorms and tnorms.Not surprisingly,we would like to discuss the class of nullnorms on bounded lattices.Therefore,the last task of this thesis is to construct nullnorms on bounded lattices.It will be helpful for investigating the class of nullnorms on bounded lattices.More specifically: based on the simultaneous use of a closure operator and an interior operator on a bounded lattice,we propose a more general method for constructing a nullnorm on a bounded lattice with a given underlying t-conorm on a join sub-semilattice(depending on the closure operator)of that bounded lattice and a given underlying tnorm on a meet sub-semilattice(depending on the interior operator)of that bounded lattice.We also present a dual construction method.