Internal direct product of integral ∨∨-distributive binary aggregation functions and the classification of all integral ∨∨-distributive binary aggregation functions of length 3

We present some relations between an external direct product and an internal direct product of a family of integral ∨∨-distributive binary aggregation functions. Relations between a direct decomposition of the unit and a family of pairwise comaximal elements in a lattice with an integral ∨∨-distributive binary aggregation function are investigated. Using these results, the classification of all integral ∨∨-distributive binary aggregation functions on a lattice of length 3 and a description of all lattices of length 3 with an integral ∨∨-distributive binary aggregation function are given.