In this article we apply the technique of the lateral analysis on vector lattices developed in [24, 25] to the study of orthogonally bi-additive operators defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W. First, we resolve the open problem stated in [11], showing that there exists a lateral preideal \(\mathcal {I}\) of \(E\times F\) such that \(\mathcal {I}\) cannot be equal to the kernel of a positive orthogonally bi-additive operator from \(E\times F\) to any separable Banach lattice W. We also prove that every lateral preideal \(\mathcal {I}\) of \(E\times F\) is the kernel of a positive orthogonally bi-additive operator \(T:E\times F\rightarrow W\) taking values in some Dedekind complete vector lattice W. Finally, we obtain the criterion of the disjointness of two positive orthogonally bi-additive operators \(T_1,T_2:E\times F\rightarrow W\) .
