| Abstract |
Associative aggregation operators on bounded lattices are special aggregation operators that have proven to be useful in many fields like fuzzy logics, expert systems, neural networks, data mining, and fuzzy system modeling. Nullnorms, uninorms, t-norms, t-conorms, and many other operations all belong to the class of associative aggregation operators. One of the typical constructions for associative aggregation operators on the unit interval [0,1] is the ordinal sum construction. As observed, in general, an ordinal sum construction may fail on a general bounded lattice. Motivated by the last observation, a new sum-type construction called lattice-based sum has been recently introduced by El-Zekey et al. [30]. In this thesis, based on the lattice-based sum of (bounded) lattices indexed by a (finite) lattice-ordered index set, new methods for constructing nullnorms and uninorms on bounded lattices, which are lattice-based sums of their summand sublattices, are developed. Subsequently, the obtained results are applied for building several new nullnorm and uninorm operations on bounded lattices. As a by-product, the lattice-based sum constructions of t-norms and t-conorms obtained by El-Zekey [31] are obtained in a more general setting where the lattice-ordered index set need not be finite and so-called t-subnorms (t-subconorms) can be used (with a little restriction) instead of t-norms (t-conorms) as summands. Furthermore, new idempotent nullnorms on bounded lattices, different from the ones given in [16], have been also obtained. We point out that, unlike [16], in our construction of the idempotent nullnorms, the underlying lattices need not be distributive. |
