Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory

In this paper we first give characterizations of the class of continuous t-norms on LI (where LI is the lattice of closed subintervals of the unit interval) which satisfy the residuation principle and which are a natural extension of a t-norm on the unit interval and which satisfy one of the following conditions: the negation generated by their residual implication is involutive; they are (weakly) Archimedean; they are (weakly) nilpotent. We fully characterize the class of continuous t-norms on LI which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are weakly Archimedean. We construct a separate representation for the t-norms in this class which are weakly nilpotent and for those which are not weakly nilpotent. Finally we give a characterization of the continuous t-norms on LI which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are strict.