Interpolation properties in the extensions of the logic of inequality

We consider the modal logics $wK4$ and $DL$ as well as the corresponding weakly transitive modal algebras and $DL$-algebras. We prove that there exist precisely 16 amalgamable varieties of $DL$-algebras. We find a criterion for the weak amalgamation property of varieties of weakly transitive modal algebras, solve the deductive interpolation problem for extensions of the logic of inequality $DL$, and obtain a weak interpolation criterion over $wK4$.