A Family of Multi-valued t-norms and t-conorms

Summary. Given a lattice (X, ≤, ∧, ∨) we de.ne a multi-valued operation ∧Q which is analogous to a t-norm (i.e. it is commutative, associative, has one as a neutral element and is monotone). The operation is parametrized by the set Q, hence we actually obtain an entire family of such multi-valued t-norms. Similarly we define a family of multi-valued t-conorms ∨P. We show that, when P, Q are chosen appropriately, ∧Q, ∨P (along with a standard negation) form a de Morgan pair. Furthermore ∧Q, ∨P are order generating and (X, ≤, ∧Q, ∨P) is a superlattice, i.e. a multi-valued analog of a lattice.