Induced Triangular Norms and Negations on Bounded Lattices

Some relevant notions in fuzzy set theory are those of triangular-(t)- norm, t-conorm, and negation, which provide a systematic way of defining set-theoretic operations, or, from other point of view, logical connectives. For instance, the majority of fuzzy implications are directly derived from these operators, so they play a prominent role in fuzzy control theory or in approximate reasoning. This incites the search of suitable t-norms, t-conorms, and negations for solving each specific problem. In this article, we propose a procedure, that we call induction, for designing them on spaces of lattice-valued maps. Concretely, for each family of operators (t-norms, t-conorms, or negations) indexed in the domain set, we may induce an operator of the same kind, so that our method offers a great flexibility in the design task. It may be applied to well-known fuzzy objects as interval-valued or type-2 fuzzy sets. Nevertheless, the theory is formally developed for arbitrary bounded lattices.