A Study On Constructions Of T-norms And Properties Of T-norms-related Operators

Triangular norms is a commutative ordered monoid in unit interval [0,1] with neutral element 1, and it is applicable broadly in the theory of statistic metric space, fuzzy logic, the theory of fuzzy sets, information aggregation and many fields of artificial intelligence. The theory and the application of triangular norms has been one of the hottest topics in research for a long time. As a special kind of algebraic structure, the constructions of triangular norms are crucial in the theory of the structure of triangular norms. Additionally, many are interested in the triangular norms based function equations for better applications of triangular norms. From the two mentioned aspects, the paper deals with the constructions of t-norms and properties of t-norms related operators.Chapter 1 introduces the basic notions and properties of triangular norms and related operators,and some constructing methods of triangular norms .Chapter 2 mainly investigates the rotation and the rotation-annihilation constructions of triangular norms, basing on the weak negations. Firstly the basic properties of weak negations are analyzed; then the rotation construction based on weak negations are given, where the involved weak negations have only one discontinuity point t = sup{s∈[0,1]|N(s)>s}; As a partial generalization of Jenei's rotation construction, not all of the weak negations are suitable for the rotation construction, which is shown with counterexamples in Section2.3. Then the rotation-annihilation construction based on weak negations are given, where the involutive set of the involved weak negations are required to have a subset [a,1](a<1). At the end, a question in [1] is answered and several open questions on the construction and decomposition of triangular norms are proposed.Chapter 3 mainly discusses the properties of triangular norms and its related operators. Firstly, the iterative boolean-like law I(x,y) = I(x,I(x,y)) of S_D{T_D) - QL -implications is discussed and the sufficient and necessary conditions for S_D(T_D) - QL -implications to satisfy iterative boolean-like law are given; then the iterative boolean-like law S(a,b) = S(T(a,b),T(S{a,b), N(T(a,b)))) is investigated and two sufficient conditions to satisfy the iterative boolean-like law are gotten.