Aggregation Functions Constructions, Characterizations, and Functional Equations

Ordinal sums have been introduced in many different contexts, e.g., for posets, semigroups, t-norms, copulas, aggregation operators, or quite recently for hoops. In this contribution, we focus on ordinal sums of t-norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets. Necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t-norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t-norms. By such also the structure of the underlying bounded lattice is investigated. Further, it is shown that up to trivial cases there are no ordinal sum t-norms on product lattices in general. © 2006 Elsevier B.V. All rights reserved.