A representation of t-norms in interval-valued L-fuzzy set theory

In this paper we consider the lattice LI which has the closed subintervals of a complete lattice as elements. We give a representation theorem of t-norms on this lattice for which the partial mappings are join-morphisms in terms of t-norms on the underlying lattice. In fuzzy logic, t-norms which satisfy the residuation principle play an important role. Using our representation theorem, we represent t-norms on LI which satisfy the residuation principle and two border conditions in terms of t-norms on the underlying lattice. In the case that the underlying lattice of LI is the unit interval, we obtain characterizations of continuous t-norms which are natural extensions of t-norms on the unit interval and which have join-morphisms as partial mappings or alternatively satisfy the residuation principle.