A bitopological duality for some subordination Boolean algebras

S4-subordination algebras are a generalization of the closure algebras. In this paper, we give a topological representation for S4-subordination algebras by means of bitopological spaces $\left<X,\tau,\tau_{S}\right>$, where $\left<X,\tau\right>$ is a Stone space and $\tau_{S}$ is a topology that enables the characterization of the subordination relation. We apply this bitopological representation to give a characterization of S5-subordination algebras and lattice subordinations. We also show that there exists a bijective correspondence between congruence compatible with the subordination and certain closed subsets of the Stone space $\left<X,\tau\right>$ that are also saturated sets of the space $\left<X,\tau_{S}\right>$. Additionally, we explore two types of morphisms between S4-subordination algebras: one based on Boolean homomorphisms and another based on meet-homomorphisms. Finally, we provide a topological representation for each type of morphism.