Given a matroid over E={1,2,...,n}, consider for each basis B the sets that contain B*, the set of elements of B that are (relatively to B) not internally active, and that are contained in B', the union of B with the set of the elements of E that externally active. These sets form the classes of a partition of the power set of E into intervals. In a series of articles around 1980, Dawson studies a generalization of this decomposition, where general subsets of the power set replace the set of bases of the matroid. We present some new results, including the characterization in simple terms of the partitions into intervals that arise from this construction and a proof of the essential uniqueness of the same construction.