The Construction of Fuzzy-valued t-norms and t-conorms

We are interested in generalizations of the concepts of t-norm and t-conorm. In a companion chapter in this volume [14] we have presented a family of hypert-norms ∧q and a family of hyper-t-conorms ∨p. The prefix hyper is used to indicate multi-valued operations, also known as hyperoperations (see [5, 14]), i.e. operations which map pairs of elements to sets of elements. See [14] for the construction of hyperoperations which generalize t-norms and t-conorms. These hyperoperations are crisp, i.e. their output is a crisp set. A natural generalization is to consider fuzzy hyperoperations, i.e. operations which map elements to fuzzy sets. Hence in this chapter we will present a procedure to construct fuzzy-valued t-norms and t-conorms. Relatively little work in this direction has appeared in the literature. A pioneering paper is [4] which introduces fuzzy hypoperations which induce fuzzy hypergroups. A fuzzy hypergroup, different from the one used by Corsini, is [13] and a version of fuzzy min and fuzzy max operations appears in [11, 12]. As explained also in [14], our motivation to study multi-valued and fuzzyvalued connectives is that, while fuzzy logic is a calculus of uncertain reasoning, not much attention has been paid to the case where uncertainty is