Constructions And Characterizations Of Triangular Norms And Their Extensions On Lattices

The theme of this thesis is the constructions and characterizations of triangular norms and their extensions on lattices.Menger first introduced the concept of triangular norms in the study of generalizing triangle inequality in statistical metric space.Now triangular norms and their dual operators,triangular conorms have became the perfect models of intersection and union in fuzzy set theory.Many classes of aggregation operators are derived from them such as uninorms,semi-uninorms,2-uninorms,nullnorms,.semi-t-operators and so on.These various kinds of operators can model the process of aggregating information,and play crucial role in the fields of decision making,pattern recognition,image processing,data mining,artificial intelligence.In recent years,the research of aggregation operators in lattice context draws more and more attention.Lattice is a special class of partially ordered set with fine properties.It is significant from both the perspectives of theoretic research and practical application.For example,the interval lattice is a perfect choice for handling inaccuracy data.Modular lattice and distributive lattice have smart representation theory.In the field of artificial intelligence,especially when we talk about the autonomic movement intelligence,the record and aggregation of environment information demand a higher level research on the non-Euclidean structure data context such as graph,lattice,manifold.These all reveal the significance of studying aggregation operators on lattices.In this thesis,we start from the triangular norm on bounded lattice and explore the constructions and characterizations of their extensions such as uninorms,nullnorms,semi-t-operators and 2-uninorms.Many innovative results are presented.In chapter 3,we propose the additive generator theorem on bounded lattices and discuss the characterizations of generators on finite cases.Additive generator theorem is an important construction method of triangular norms.It plays important role in the representation theory of continuous triangular norms on the unit interval,which is a powerful tool to solve many practical problems.We begin with the generalization of pseudo-inverse in partially ordered circumstances and figure out how the associativity of the constructed triangular norms is created with the help of the natural associativity of the operation+.As for the part of monotonic,we use the neutral element of triangular norm to obtain the necessary conditions of the function values when we let one of the two variables change in one fixed sub-chain.We finish the additive generator theorem with some properly selected sufficient conditions and discuss the relationship between the generators and the Archimedean property,nilpotent property,strict monotonic property of the yielded triangular norms.The theorem proposed in the article is a proper generalization of the classic additive generator theory on the real unit interval since many good properties are preserved.It also provides us with a perspective of connecting triangular norms and unary functions and makes it possible to build the representation theory for triangular norms on partially ordered circumstances.In chapter 4,we discuss the ordinal sum of triangular norms on bounded lattices.We analyse the necessity of the conditions used in h-ordinal sum and propose the necessary and sufficient conditions for this kind of the construction.Following the principle of neat and accurate,we modify the conditions for practical use.Based on the results above,we present a new type of ordinal sum of triangular norms on bounded lattices.It has neat form,handy conditions and can take many existing construction methods as its special cases.Comparing with h-ordinal sum,our method adapt to practical cases better and can construct a wilder range of triangular norms.In chapter 5,we research the representation of nullnorms on bounded lattices.We are inspired by the strong constrains of many existing construction methods and directly study the characterizations of nullnorms on bounded lattices.Since the value of nullnorm is comparable with the absorbing element as long as one of the two variables is comparable with the absorbing element,we choose quite a general subclass of nullnorms and achieve the complete characterization of its structure.In fact,the hardest point of the characterization locates on the case that at least one of the two variables is incomparable with the absorbing element.We fix the two underlying functions and introduce two order-preserving functions,which make it possible to represent the nullnorm as a compound.That provides us with a series of necessary conditions.The most important part is that these conditions are also sufficient for construction,which means we almost obtain a complete characterization of nullnorms on bounded lattices.The results in this section show a feasible way to build the representation theory for nullnorms on bounded lattices and have important theoretical value.In chapter 6,we study the constructions of uninorms on bounded lattices.The existing methods have many features in common,which inspire us to bring them into one same scope and present a generalized method.The construction method in this section is not only useful but also cover many existing constructions.We also try to characterize the local structure of uninorms and find that the structure of uninorms of two useful subclasses is closely related to the structure of lattice.It provides the basis for further study of the classification of uninorms.In chapter 7,we characterize the structure of semi-t-operators and 2-uninorms when the former is distributive over the latter.We first study the relationship of distributivity and idempotent property and simplify the structure of the evolved semi-t-operators.We choose five most useful subclasses of 2-uninorms and discuss the distributive equation.Finally we fully characterize the structure of them.The skills used in this section is also suitable for more general cases.