Economic production systems may break up into subeconomies of goods and processes that can function independently of each other. This article first explores the various kinds of decompositions that may exist in production schemes, nonlinear as well as linear. Some recent techniques developed in graph theory are adopted to ascertain the decompositions possible for a given production system and the precedence relations between the subeconomies of the decomposition. Finally, I show how the concept of "tearing" from simultaneous equations theory might be used to analyze square input-output models for potential decompositions.