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

Given a lattice (X,≤,∧,∨) we define a multi-valued operation ∧ 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 ∨ . We show that, when P, Q are chosen appropriately, ∧Q,∨P (along with a standard negation) form a de Morgan pair. Furthermore ∧,∨ are order generating and (X,≤,∧,∨ ) is a superlattice, i.e. a multi-valued analog of a lattice.