Decomposition of topologies on lattices and hyperspaces

The notion of decomposable topology is introduced in a partially ordered set, and in particular in the lattice C(X) of all closed subsets (ordered by reverse inclusion) of a topological space X, which is also called the hyperspace of X. This notion is closely related to the concepts, defined in the same framework, of lower, upper and strong upper topology. In this paper we investigate decomposability and unique decomposability of the main hyperspace topologies, and of topologies which are defined on some quite natural lattices or semilattices.