Question 1:Is there a classification of finite lattices which admit a multiplication making them into a finite multiplicative lattices? (see https://encyclopediaofmath.org/wiki/Multiplicative_lattice for a definition)? Is there a good way to find all multiplications on a finite lattices that make it into a multiplicative lattice via a computer?

If it helps, we can assume that they are distributive.

Question 2: Is there such a classification if we drop the commutativity assumption in the definition?

Question 1:Is there a classification of finite lattices which admit a multiplication making them into a finite multiplicative lattices? (see https://encyclopediaofmath.org/wiki/Multiplicative_lattice for a definition)? Is there a good way to find all multiplications on a finite lattices that make it into a multiplicative lattice via a computer?

If it helps, we can assume that the lattices are distributive in the classical sense.

Question 2: Is there such a classification if we drop the commutativity assumption in the definition?