  We introduce topological invariants of semi-decompositions (e.g. filtrations,
semi-group actions, multi-valued dynamical systems, combinatorial dynamical
systems) on a topological space to analyze semi-decompositions from a dynamical
systems point of view. In fact, we construct Morse hyper-graphs and abstract
weak elements of semi-decompositions. Moreover, the Morse hyper-graphs of the
set of sublevel sets of a Morse function of a compact manifold is the Reeb
graph of such function as abstract multi-graphs. The abstract weak element
space for a simplicial complex is the face poset. The abstract weak element
spaces of positive orbits of acyclic directed graphs are their abstract
directed multi-graphs.
