Characterizations To The Archimedean And The Convexity Of Triangular Norms

In recent years,fuzzy mathematics has grown in importance as an advanced tool infuzzy optimization and control theory.The usual arithmetic operations on the reals canbe extended to rithmetic operations on the fuzzy intervals by means of Zadeh's exten-sion principal.This principal is based on a triangular norm T.One of the most importantproperties that can be satisfied by t-norms on the unit interval is the Archimedean prop-erty:continuous t-norms can be fully characterized by means of Archimedean t-norms,theArchimedean property is closely related to additive and multiplicative generators, etc.copulas have played an important role not only in probability theory and statistics, butalso in many other fields requiring the aggregation of incoming data, such multi-criteriadecision making, probabilistic metric spaces. Moreover, associative copulas are a well-known subclass of triangular norms.The open problem which is stated in this paper wouldinduce a new characterization of associative copulas.This paper we characterize the Archimedean property and answer the convexityOf the t-norms.In the former,we answer the convexity Of the t-norms and induce theconclusion:(1) For strict continuous Archimedean t-norms,we can completely prove T(max(x-α,0),min(x+α,1))≤T(x,x)(*) holds for all x∈[0,1] and for allα∈]0, 1/2[if and only ift is convex.(2) For nilpotent continuous Archimedean t-norms, we can disprove T(max(x-α, 0),min(x+α, 1))≤T(x,x)(*) for all x∈[0,1] and for allα∈]0,1/2[if and only if t isconvex.Moreover,reducing the condition, we also answer the question.In the latter,we char-acterize the Archimedean properties of t-norms and extend the property.(1) If t-norm T satisfies CCL and has no zero divisor,then T necessarily satisfies CL.(2) T is continuous Archimedean t-norm and has no zero-divisors then T is strict.