Triangular norms which are meet-morphisms in interval-valued fuzzy set theory

In this paper we study t-norms on the lattice of closed subintervals of the unit interval. Unlike for t-norms on a product lattice for which there exists a straightforward characterization of t-norms which are join-morphisms, respectively meet-morphisms, the situation is more complicated for t-norms in interval-valued fuzzy set theory. In previous papers several characterizations were given of t-norms in interval-valued fuzzy set theory which are join-morphisms and which satisfy additional properties, but little attention has been paid to meet-morphisms. Therefore, in this paper, we focus on t-norms which are meet-morphisms. We consider a general class of t-norms and investigate under which conditions t-norms belonging to this class are meet-morphisms. We also characterize the t-norms which are both a join- and a meet-morphism and which satisfy an additional border condition.

► Triangular norms in interval-valued fuzzy set theory which are meet-morphisms. ► For continuous t-norms, the notions of sup-(inf-) and join-(meet-)morphism coincide. ► The meet-morphisms in a large class of t-norms are characterized. ► The t-norms which are both a join- and meet-morphism are partially characterized.