t-Operators and uninorms on a finite totally ordered set

This paper presents a detailed study of two classes of operators on a finite totally ordered set of labels L: t-operators and uninorms. Both kinds of operators (on [0, 1]) are introduced as generalizations of t-norms and t-conorms. We characterize these operators on L as special combinations of operators of directed algebras in a similar way as they are characterized in the case of [0, 1] as special combinations of t-norms and t-conorms. We also study duality of these operators with respect to the only negation N on L, and we give the number of different t-operators and uninorms that exist on L, related to the number of elements in L.